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Springer Series in

MATERIALS SCIENCE

Springer-Verlag Berlin Heidelberg GmbH

34

Springer Series in

MATERIALS SCIENCE

Editors: R. Hull . R. M. Osgood, Jr .. H. Sakaki . A. Zunger

26 Gas Source Molecular Beam Epitaxy Growth and Properties ofPhosphorus Containing III-V Heterostructures By M. B. Panish and H. Temkin

27 Physics ofNew Materials Editor: F. E. Fujita 2nd Edition

28 Laser Ablation Principles and Applications Editor: J. C. Miller

29 Elements of Rapid Solidification Fundaments and Applications Editor: M. A. Otooni

30 Process Technology for Semiconductor Lasers Crystal Growth and Microprocesses By K. Iga and S. Kinoshita

31 Nanostructures and Quantum Effects By H. Sakaki and H. Noge

32 Nitride Semiconductors and Devices ByH. Morko<;:

33 Supercarbon Synthesis, Properties and Applications Editors: S. Yoshimura and R. P. H. Chang

34 Computational Materials Design Editor: T. Saito

35 Macromolecular Science and Engineering New Aspects Editor: Y. Tanabe

36 Ceramics Mechanical Properties, Failure Behaviour, Materials Selection By D. Munz and T. Fett

37 Technology and Applications of Amorphous Silicon Editor: R. A. Street

Volumes 1-25 are listed at the end of the book.

T. Saito (Ed.)

Computational Materials Design

With 212 Figures and 43 Tables

Springer

Dr. Tetsuya Saito National Research Institute for Metals 1-2-1 Sengen, Tsukuba-shi 305 Ibaraki-ken, Japan

Series Editors:

Prof. Alex Zunger NREL National Renewable Energy Laboratory 1617 Cole Boulevard Golden Colorado 80401-3393, USA

Prof. R. M. Osgood, Jr. Microelectronics Science Laboratory Department of Electrical Engineering Columbia University Seeley W. Mudd Building New York, NY 10027, USA

ISSN 0933-033X

Prof. Robert Hull University of Virginia Dept. of Materials Science and Engineering Thornton Hall Charlottesville, VA 22903-2442, USA

Prof. H. Sakaki Institute ofIndustrial Science University of Tokyo 7-22-1 Roppongi, Minato-ku Tokyo 106, Japan

ISBN 978-3-642-08404-1 ISBN 978-3-662-03923-6 (eBook) DOI 10.1007/978-3-662-03923-6

Library ofCongress Cataloging-in-Publication Data

Computational materials design / T. Saito, (ed.). p. cm. - (Springer series in materials sciences ; vol. 34)

Includes bibliographical references and index.

1. Materials-Mathematical models. 2. Continuum mechanics. I. Saito, T. (Tetsuya), 1941- . II. Series: Springer series in materials science; v. 34. TA404·8.C66 1999 620.1'1'015118-dc21 98-53845

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Preface

Materials science is the basic research subject for all the engineering fields, and the development of advanced materials contributes to dramatic evolution of technologies. In design and development of materials, however, painstaking empirical methods are unavoidable unless there are appropriate guidelines. On the other hand, there has been enormous advancement in the speed and capability of computers. These improvements have offered exciting prospects for the new research field 'computational materials science'. Computation has now become an essential tool in predicting and explaining various phenomena in terms of materials science, such as microstructural evolution and mechanical strength, and it is approaching its goal, namely that it will give us complete guidelines for developing materials suitable for certain applications.

This book has arisen from the above circumstances. It intro duces the current research activities within 'computational materials design' in Japan, describing (1) the significance of computational materials design, (2) its position today and (3) intended or expected breakthroughs in materials research. The chapters are largely independent of one another and readers interested in only one topic will find that reading only one or more chapters is sufficient.

In the first chapter, we review the recent applications of first-principles electronic-structure methods to magnetic and semiconductor materials. The following two chapters deal with the design of high-temperature alloys using various simulation techniques. Not just microstructural information but also the mechanical properties of Ni-base superalloys (Chap. 2), and Ti alloys and intermetallic compounds (Chap. 3) are predicted using regression analysis, the cluster variation method, Monte Carlo simulation, central atoms model, etc. Chapters 4 and 5 describe the calculation method and utilization of phase diagrams, which provide very important information in materials design. As typical examples, Chap. 4 intro duces the adoption of CALPHAD (CALculation of PHAse Diagram) to design compound semiconductors and solder materials, and Chap. 5 presents the utilization of phase diagrams in microstructural control in Fe-base alloys.

Chapters 6 and 7 concern changes in microstructure and mechanical properties in time under so me environment al conditions. Chapter 6 introduces the design of fusion reactor materials wh ich are severely irradiated anel change very differently to other high-temperature structural materials. Chapter 7 introduces more fundamental and theoretical approaches. It deals with theories of first-order phase transitions and their application to the simulation of microstructural evolution. The last two chapters deal with the finite element method, which is very effective in modeling localized stress distribution and deformation behavior of materials. Chapter 8 presents the analysis of deformation behavior in void-

VI Preface

containing materials, and Chapter 9 analyzes the interface problems in continuous fiber ceramic composites.

This is a timely textbook for university students, young researchers, and engineers of general engineering fields. We hope that the reader stimulated by this book will pioneer a new field of 'computational materials design'.

Tsukuba, March 1999 Tetsuya Saito Deputy Director-General National Research Institute for Metals

Table of Contents

Electronic Structure Theory for Condensed Matter Systems

Takahisa Ohno and Tamio Oguchi ................................... 1 1.1 Introduction.................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Overview of Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Structural and Magnetic Properties of Fe16N2 .................... 6 1.4 Magnetism of Platinum 3d-Transition-Metal Intermetallics . . . . . . . .. 10 1.5 Semiconductor Nanostructures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14 1.6 Semiconductor Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20

Design ofNi-Base Superalloys

Hiroshi Harada and Hideyuki Murakami ............... . . . . . . . . . . . . . .. 39 2.1 Introduction... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39 2.2 Mathematical Models Based on Database . . . . . . . . . . . . . . . . . . . . . . .. 44 2.3 Theoretical Models Based

on Statistical Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 54 2.4 Towards an Open Laboratory for Materials Design . . . . . . . . . . . . . . .. 66 2.5 Conclusions......................... . . . . . . . . . . . . . . . . . . . . . . . .. 67

Design of Titanium Alloys, Intermetallic Compounds and Heat Resistant Ferritic Steels

Hidehiro Onodera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 71 3.1 Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 71 3.2 Design of Superplastic Titanium Alloys . . . . . . . . . . . . . . . . . . . . . . . . .. 72 3.3 Design of Heat Resistant Oe + 0e2 Titanium Alloys . . . . . . . . . . . . . . . .. 86 3.4 Alloy Design Based on the Prediction

of Atomic Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 91 3.5 Summary .................................................... 102

CALPHAD Approach to Materials Design

Hiroshi Ohtani .................................................... 105 4.1 Outline of the CALPHAD Method .............................. 105 4.2 Procedures of the CALPHAD Method .......................... 106 4.3 Phase Diagrams of the Compound Semiconductors ................ 112 4.4 Phase Diagrams of the Sold er Alloys ............................ 122

VIII

Phase Equilibria and Microstructural Control in Iron-base Alloys

Table of Contents

Kiyohito Ishida ................................................... 135 5.1 Introduction .................................................. 135 5.2 Solidification Process .......................................... 136 5.3 Low Alloy Steels .............................................. 143 5.4 High Alloy Steels ............................................. 151 5.5 Functional Iron Alloys ......................................... 155 5.6 Concluding Remarks .......................................... 159

Computational Approach to the Fusion Reactor Materials Tetsuji Noda and Johsei Nagakawa .................................. 163 6.1 Introduction .................................................. 163 6.2 Radiation Damage in Fusion Reactor Materials ................... 164 6.3 Simulation of Transmutation and Induced Activity

of Structural Materials ........................................ 178

Modeling of Microstructural Evolution in Alloys Yoshiyuki Saito ................................................... 195 7.1 Introduction .................................................. 195 7.2 Theoretical Background ........................................ 196 7.3 Macroscopic Modeling of Microstructure ......................... 209 7.4 Mesoscopic Modeling of Microstructure .......................... 214 7.5 Industrial Applications ........................................ 222

Finite Element Analysis of the Deformation in Materials Containing Voids H. Shiraishi ...................................................... 227 8.1 Introduction .................................................. 227 8.2 Formulation of the Large Plastic Deformation

Finite Element Method ........................................ 228 8.3 Analysis of the Effect of Void Lattice

on Stress-Strain Relationship ................................... 235 8.4 Application to Helium Bubble Grain Boundary Embrittlement ...... 246 8.5 Discussion ................................................... 251 8.6 Conclusion ................................................... 254

N umerical Analysis of the Interface Problem in Continuous Fiber Ceramic Composites Y. Kagawa and C. Masuda . ......................................... 257 9.1 Introduction .................................................. 257 9.2 Crack-Fiber Interaction Problem ............................... 258 9.3 Evaluation of Interface Shear Stress: FEA Analysis ................ 265 9.4 Interface Debonding Criterion .................................. 276 9.5 Conclusion ................................................... 287

Index ............................................................ 291


Takahisa Ohno1 and Tamio Oguehi2

1 National Research Institute for Metals, Tsukuba 305, Japan 2 Department of Materials Science, Hiroshima University,

Higashihiroshima 739, Japan

Abstract. Recent application of the first-principles electronic-structure method to magnetic and semiconductor materials is reviewed. As an application to magnetic materials, the crystal structure of a ferromagnet, Fe16N 2, is optimized by minimizing total energy. An interesting cooperative structural relaxation behavior is found. Magnetic properties of Fe16N2 are discussed by analyzing the local magnetic moments and hyperfine fields at the iron and nitrogen sites. Magnetism of TPt3 (T= V, Cr, Mn, Fe, Co and Ni) intermetallics is also studied from first principles. By including the spin-orbit interaction as the second variation in a one-electron equation, the orbital magnetic moments are evaluated in addition to the spin counterparts. A microscopic mechanism for the systematic variation of the Pt-site orbital moments in the se ries is given and the importance of mixing effect between Pt and 3d transitionmetal d states is emphasized. As an application to semiconductor materials, exotic electronic properties of novel semiconductor nanostructures, GaAs/Ge superlattices and Si quantum wires, are predicted from firstprinciples. The detailed atomic structures and stabilities of both As-rich and Ga-rich GaAs(OOl) surfaces are determined from the standpoint of equilibrium energetics and theoretical simulations of scanning tunneling microscopy images. Microscopic diffusion processes of Si adatoms on a hydrogen-terminated Si(OOl) surface are studied and the effect of hydrogen termination on Si homoepitaxial growth is discussed. Diffusion constants, that is, pre-exponential factors and activation energies, of Ga and Al adatoms on a As-rich GaAs(OOl) surface are evaluated from first-principles calculations of migration potentials. On the basis of these calculated results, stochastic Monte Carlo simulations for a GaAs-AIAs binary system are performed.

1.1 Introduction

The density functional theory proposed by Hohenberg and Kühn [1] provides a püwerful tool to prediet properties of eondensed matter systems from first prineiples without using any empirie al parameters. The field of first-prineiples eleetronie-strueture theory has been developed sinee the proposal of the density functional theory and has matured within the last deeade. A large variety of theoretieal teehniques has been invented and sueeessfully applied tü many systems including metals, semieonduetors, insulators, organie materials, and so on.

Springer Series in Materials Science Volume 34, Ed. by T. Saito © Springer-Verlag Beflin Heidelberg 1999

2 Electronic Structure Theory for Condensed Matter Systems

We are now able to calculate electronic structures of materials and predict their electronic, optical, and magnetic properties with great precision. The static and dynamical structural properties such as lattice constants, bulk moduli, phonon frequencies, and various reaction processes also can be evaluated.

To design a material which exhibits a desired property is one of the major goals of materials science. Recently, novel semiconductor structures such as superlattices and quantum wires have been fabricated by using state-of-the-art growth techniques, in expectation of exhibiting novel and exotic properties. For materials design it is essential both to predict properties of the designed material and to find methods to fabricate it. The first-principles electronic-structure theory contributes not only to prediction of materials properties, but also to development of methods to fabricate materials by revealing the underlying growth mechanisms.

In this paper we will review our recent applications of first- principles electronic-structure theory. We first describe the density functional theory in Sect. l.2. Among several ab initio computational techniques, the full-potential linearized augmented plane wave method and the pseudopotential method, which are widely used for magnetic and semiconductor materials, respectively, are briefiy reviewed.

We review our applications to magnetic materials in Sects. l.3 and 1.4. Giant magnetic moments have been recently found in several magnetic multilayer and film systems. In this context, the electronic structure and magnetism of Fe16N2 is studied from first principles by optimizing structural parameters in Sect. l.3. We found that the stable structure could be reasonably understood by considering a cooperative relaxation mechanism. However, no giant spin magnetic moment as great as that observed experimentally has been obtained, although there exists some enhancement in the magnetic moments. Another important new feature in recent research on magnetism is a direct observation of orbital magnetic moments in ferromagnets by the magnetic circular dichroism (MCD) technique. In consultation with the so-called orbital-moment and spin-moment sum rules, the size and relative direction of the orbital and spin magnetic moments can be separately extracted. In Sect. 1.4, the orbital and spin magnetic moments in a series of platinum 3d-transition-metal intermetallics have been investigated by including the spin-orbit interaction as the second variation. The magnetic moments calculated for the intermetallics are compared with MCD results and a trend found through the series is discussed in relation to the electronic structure.

Applications to semiconductor materials are reviewed in Sects. l.5 and l.6. Novel semiconductor nanostructures, for example, quantum wells, wires, and dots, have been recently fabricated by using state-of the-art growth techniques. In Sects. l.5.1 and l.5.2, exotic electronic properties of semiconductor nanostructures, GaAs/Ge hetero-valent superlattices and Si quantum wires, respectively, are predicted, by first-principles electronic-structure methods. The migration process of adatoms on a semiconductor surface and their incorporation into growth sites plays an important role in the growth of epitaxial films. Understanding these fundamental processes at the atomic level. will develop methods

Takahisa Ohno and Tamio Oguchi 3

to manipulate the surface dynamics and offer the exciting possibility to tailor thin-film growth for specific materials applications. In Sect. 1.6, we will describe theoretical investigations of atomic structures and dynamics on semiconductor surfaces. The atomic structures of GaAs(OOl) surfaces under both As-rich and Ga-rich growth conditions are determined from first-principles in Sect. 1.6.1. The determination of surface structures is aprerequisite to investigations of surface dynamics. Surface diffusion processes both of Si adatoms on the hydrogenterminated Si(001) surface and of Ga adatoms on the As-rich GaAs(OOl) surface are studied as applications to surface dynamics in Sect. 1.6.2.

1.2 Overview of Computational Methods

A condensed matter system of interest consists of a number of nuclei and electrons, which interact with each other. In order to study properties of the system from first principles, we usuaHy assume two major approximations, the adiabatic approximation and the one-electron approximation. The former is an adiabatic separation of the motions of nuclei and electrons and the problem is reduced to finding the many-body electronic state, mostly its ground state, for a given nucleus geometry. The latter is an approximation adopted in solving the many-body Schrödinger equation and the resulting one-electron equation can be generally written as

(1)

where 1i is the Planck constant divided by 27r, m is the mass of an electron, Ci

is the eigenvalue of the i-th one-electron state and 'l/Ji (r) is the corresponding wavefunction. An effective potential Veff (r) derived by the one-electron approximation, is assumed. Therefore, for a particular material, problems are two-fold, how to derive the one-electron potential Veff (r) and how to solve the one-electron equation (1).

A local spin density approximation or its gradient corrected version to the density functional theory [1,2] is such a one-electron approximation, which provides an efficient and reliable prescription for practical calculations [3], as described brieflT in Sect. 1.2.1. Among several ab initio (non-empirical) methods to solve the one-electron equation for condensed matter, mostly crystalline systems, fuH-potential linear plane wave (FLAPW) and pseudopotential (PP) methods are briefly reviewed in Sects. 1.2.2 and 1.2.3.

1.2.1 Density F\lllctional Theory

According to the density functional theory of Hohenberg and Kohn [1], the nondegenerate ground state of an inhomogeneous electron system is a functional of the electron density n( r). For example, the total energy of the ground state E can be given as

E = E[n(r)]. (2)

4 Electronic Structure Theory for Condensed Matter Sy~telll~

A virtue of the density functional theory is that the variational principle hold::; at the correct ground-state density, namely

6E[n(r)] 6n(r) = 0,

und er a conservation of the total number of electrons

(:3 )

(4)

The density functional theory is a rigorous theory concerning the ground state of an inhomogeneous electron system but its correct functional form is unknown. However, the functional form is well studied in the case of a homogeneous electron gas system, for which the local density approximation gives the proper description.

Following Kohn and Sham's [2] formulation, the total energy can be formally written as a sum of the kinetic energy of a non-interacting electron system with the same electron density n( r), Ts [n( r)], the classical electrostatic energy, U[n(r)], the exchange-correlation energy, Exc[n(r)], which includes all cffects of many-body origins, and the interaction energy with the external potential Vext (r)

E[n(r)] = Ts[n(r)] + U[n(r)] + Exc[n(r)] + J Vext(r)n(r)d:'r (5)

U[n(r)] = e2 J ~(r)n(~? d:l rd3 r ' , (6) 2 r-r

where e is the unit charge and the external potential Vext (r) is usually given by nucleus charges.

The variational principle with a local density approximation for the exchangecorrelation energy

(7)

where Exc (n) is the exchange-correlation energy density of the homogeneous elcctron gas ofthe density n, leads to the one-electron equation (1) with an effective potential represented as

( ) _ () 6U[n(r)] 6Exc [n(r)] Veff r - Vext r + 6n(r) + 6n(r)

- () 2 J n(r/) d3 I [dnExc(n)] - Vext r + e I _ r'l r + d .

r n n=n(r) (8)

Then, the electron density is given by summing the occupied states to givc the correct number of electrons

ace.

(9)


The one-electron equation (1) with (9) is a nonlinear problem with respect to the electron density and may be solved in an iterative procedure (self-consistent field calculation).

1.2.2 FLAPW Method

There are two difficulties in solving (1) self-consistently with (9) for a crystalline system. These are general ways to represent the wavefunction 'l/Ji (r) and to solve the Poisson equation given as the second term in the right-hand side of (8). Formally, the plane wave (PW) basis functions

~k + K (r) = n-1 / 2 exp{i(k + K) . r} (10)

can be used to expand the wavefunction in any crystalline systems and the Poisson equation can be easily solved by Fourier representations for the charge density and potential functions. However, the tremendously large numbers of PWs required make practical computations impossible unless pseudopotentials are used instead of the true ion core as described in Sect. 1.2.3.

An augmented plane wave (APW) method [4] removes such a problem by augmenting PW with spherical wave functions defined around each atom Rn as

k + K _ -k + K '" [ k + K -k + K ] cf> (r)-cf> (r)+ ~ cf>aRm (r-Ra)-cf>aRm (r-Rrx) , nRm

(11)

(12)

where ~~R;;: K is a shperical-wave expansion of ~k + K, RaR is the radial function and Y Rm is the spherical harmonics. In the APW method, the so-called muffin-tin approximation is usually assumed for the form of the potential functions. The APW method is very efficient and reliable for closely-packed systems with a small unit cell but be comes untrustworthy in other complicated systems because of nonlinearity of the secular equation and inaccuracy of the potential form.

The full-potential linearized augmented plane wave (FLAPW) method is a highly precise and robust technique for electronic-structure calculations in the sense of being element independent and structure independent. The radial function RaR is represented in an energy-independent form by a linear method [5,6] and no shape approximation (full-potential), such as the muffin-tin approximation, is assumed [7,8]. The FLAPW method has been applied for a variety of systems, including complex compounds, surfaces and interfaces. Recently, formulation of atomic forces within the FLAPW scheme has been proposed and its capability opens up structural optimizations and dynamical simulations [9- 13].

Applications of the FLAPW method to magnetic materials have been very actively carried out with use of a local spin-density approximation (LSDA) to the density functional theory. By a self-consistent treatment of core-electron states, which can be rigorously done in an all-electron scheme such as the FLAPW

6 Electrünic Structure Theüry für Cündensed Matter Systems

method, very accurate calculations of core polarization of the co re states and hyperfine fields are possible. An example will be given in Sect. 1.3.

It is widely known that the spin magnetic moment may be well reproduced within LSDA. By inclusion of the spin-orbit coupling as the second variation, the orbital magnetic moment can be estimated qualitatively in some magnetic materials, especially transition-metal elements, although an orbital polarization effect is missing in LSDA and Hund's second rule may not be obtained in an atomic limit. Magnetism of transition-met al intermetallics including trends in the orbital and spin magnetic moments will be presented in conjunction with recent magnetic-circular-dichroism measurements in Sect. 1.4.

1.2.3 Ab Initio Pseudopotentials

Most physical properties of materials depend on the valence electrons to a much greater extent than on the core electrons. The pseudopotential approximation takes advantage of this by removing the core electrons from the system and by including their effects in a pseudopotential which acts on the valence electrons. The ab initio pseudopotential is constructed such that its scattering properties for the valence electrons are identical to those for the valence electrons in the trlle system including all electrons. The valence wavefunctions of the pseudopotential have no radial no des in the core region because of the lack of orthogonality between the core wave functions. This means that the pseudopotential is much weaker than the true ionic potential. The weakness of the pseudopotential enables us to expand the wavefunctions using a reasonable number of plane wave basis functions.

The most general form of an ab initio pseudopotential is

Vpseudo = L Ilm)Vi(lml, (13) Im

where 11m) are the spherical harmonics and Vi is the pseudopotential for angular moment um I. When the pseudopotential Vpseudo acts on the electronic wavefunction, it decomposes the wavefunction into spherical harmonics, each of which is then multiplied by the relevant pseudopotential Vi. A variety of methods for the construction of ab initio pseudopotentials have been proposed [14-17]. These ab initio pseudopotentials are capable of describing the scattering due to the trlle ion cores in a variety of atomic environments. The ab initio pseudopotential techniques are widely applied to study electronic structures and dynamical properties of a variety of materials, especially semiconductors, as will be described in Sects. 1.5 and 1.6.

1.3 Structural and Magnetic Properties of Fe16N2

Fe16N2 has been reported as a magnet with a large magnetization up to 3.5 fJ'B [18-23]. Several band-theoretical calculations [24-28] have been carried out but


a

Fig.1. Crystal structure of Fe16N2. Open eircles with 1, 2 and 3 denote Fe[4(e)], Fe[8(h)] and Fe[4(d)], respectively, and filled eircles N[2(a)].

have given the average magnetic moment of only 2.4-2.5 JLB for an experimentally proposed crystal structure. However, the iron atomic positions have not been precisely settled experimentally because of its thermodynamically metastable phase. We have studied electronic structure and magnetic properties of Fe16N2' optimizing the structural parameters by using the atomic-force FLAPW method [29]. Results of the study are shown below.

The proposed crystal structure of Fe16N2 [30] is body-centred tetragonal (bct) with ordered nitrogen atoms at the octahedral interstitial sites, as illustrated in Fig. 1, and the atomic positions are listed in Table 1. In this study, we first optimize the internal structural parameters of the iron sites, x and z, by computing atomic forces, and then calculate local spin magnetic moments and hyperfine fields at the iron and nitrogen sites.

Table 1. The positions of iron and nitrogen sites of Fe16N2.

Site Positions

Fe[4(e)] O,O,z; ~,~,~ + z; O,O,-z; ~,~,~ - z

Fe[8(h)] x,x,O; ~ + x,~ + x,~; x,-x,O; ~ + x,~ - x,~

Fe[4(d)] O,~,~; ~,O,~; ~,O,~; O,~,~ N[2(a)] 0,0,0; ~,~,~


1.3.1 Structural Properties

Information on atomic forces for given coordinates is very helpful in performing structural optimization for a complicated system, as well as the total energy of the system. The formulation of the atomic forces has been proposed within the all-electron FLAPW scheme [9-12]. As already shown in our previous applications [13], the atomic forces are sufficiently accurate and reliable with a moderate number of plane waves. Since the only two parameters are degrees of freedom to be determined for fixed lattice constants (a = 5.72 A and c = 6.29 A) in the present case, we simply compute atomic forces for several different parameters and fit the results near the equilibrium to a form of

Fai = - L kaißj L1Rßj + O(L1R2 ),

ßj (14)

where L1Rßj is the displacement of the atom ß in the direction j from its equilibrium position.

The presently optimized structural parameters, x and z, and two sets of experimentally proposed ones (called hereinafter Jack-l and Jack-2) [30] are listed in Table 2. Table 3 summarizes the distance of neighboring atomic pairs. The nearest neighbor (NN) Fe[4(e)]-Fe[8(h)] distance of Jack-l, 2.341 A, is much shorter than the NN distance of bcc iron, 2.477 A. Therefore, the crystal structure of the Jack-l structure is expected to relax so as to increase the distance. After the optimization, the Fe[4(e)]-Fe[8(h)] distance becomes closer to the NN distance of bcc Fe. Similar expansion might happen for another neighboring distance Fe[4(e)]-Fe[4(e)] of Jack-1. Concerning the distances between iron and nitrogen, Fe[4(e)]-N[2(a)] becomes shorter by the optimization, being remarkably close to the NN iron-nitrogen distance of Fe4N, 1.898 A.

Table 2. Structural parameters of Fe16 N2 .

Jack-l Jack-2 Present

x 0.25 0.222 0.243

z 0.3125 0.306 0.293

These pro ces ses of structural relaxation can be understood as a cooperative relaxation mechanism starting from the ideal bct structure, as depicted in Fig. 2. Looking at the ideal structure with x = 0.25 and Z = 0.25, neighboring Fe[4(e)] and nitrogen atoms seem to be too close to each other. The Fe[4(e)] atoms move first farther from the nitrogen. Due to this movement, the nearest two Fe[(e)] atoms get closer and squeeze the nearest Fe[8(h)] atoms toward a nitrogen site at the opposite side. As a result, the Fe[8(h)]-N[2(a)] distance becomes shorter than the ideal one.

The optimized structure is found to be more stable than Jack-l by 62.4 mRy per Fe16N2. It is known experimentally that Fe16N2 is a metastable phase and


Table 3. Atom-pair distance of Fe16N2 in A.

Jack-1 Jack-2 Present

Fe[4( e )]-Fe[8( h)] 2.820 2.632 2.695

2.341 2.559 2.453

Fe[8(h)]-Fe[4(d)] 2.562 2.572 2.562

Fe[4( e) ]-Fe[4( e)] 2.359 2.441 2.604

Fe[8(h)]-Fe[8(h)] 2.860 2.540 2.780

2.860 3.180 2.940

Fe[4(e)]-N[2(a)] 1.966 1.925 1.843

Fe[8( h )]-N[2( a)] 2.022 1.796 1.966

Fig. 2. Schematic drawing of the cooperative relaxation mechanism.

should decompose into ferrite iron and Fe4N. We have shown that the total energy of the optimized Fe16N 2 is slightly higher than that of bcc iron and Fe4N by 0.8 mRy per Fe16N2.

1.3.2 Magnetic Properties

The calculated local spin magnetic moments at the iron and nitrogen sites and the average moment per iron atom are summarized together with experimental data [21,22] in Table 4. It is clear in Table 4 that the magnitude of the loc:al moments changes only slightly by the optimization but the average moment does not change at all. The average moment is in good agreement with the previous FLAPW result (2.37 /LB) [28], showing not such a large moment as the 3.5 /LB experimentally observed [21].

The calculated Fermi-contact terms in the hyperfine fields are listed with experimental data [22,31] in Table 5. The Fermi-contact term turns out to he more sensitive to the structural change than the local moments. Especially, the


Table 4. The magnetic moments of Fe16 N2 in jLB.

Experiments Present

Fe[4(e)]

Fe[8(h)] Fe[4(d)] N[2(a)] Interstitial

1.3

2.5

3.8

Average per iron 2.5, 3.5

Jack-1 Optimized

2.043 2.002

2.295 2.295

2.759 2.757

-0.058 -0.043

0.500 0.572

2.403 2.403

Table 5. The hyperfine fields of Fe16N2 in kG. The experimental value for N[2(a)] is taken from that in Fe4 N.

Experiments Present

Jack-1 Optimized

Fe[4(e)] -296 -246.4 -245.6

Fe[8(h)] -316 -245.3 -254.2

Fe[4(d)] -399 -345.7 -331.9

N[2(a)] -9.32 5.10 -7.58

value at the nitrogen site changes from positive to negative by the optimization. It is quite interesting that the hyperfine field at nitrogen sites calculated far the optimized Fe16N2 coincides weIl with that observed for Fe4N [31]. The calculated Fermi-contact terms at the iron sites seem to underestimate the observed ones systematically by ab out 20%, as seen in bcc iron (the experimental value is -339 kG while the calculated one is -265 kG). This underestimation may be attributed to a poor description of LSDA for the core polarization [32].

1.4 Magnetism of Platinum 3d-'fransition-Metal Intermetallics

Platinum 3d-transition-metal intermetallic compounds, TPt3 (T = V, Cr, Mn, Fe and Co), all have common CU3Au-type crystal structure but show different kinds of magnetism [33-38]. MnPt3 and CoPt3 are ferromagnets while FePt3 is an antiferromagnet. VPt3 and CrPt3 are ferrimagnets, in wh ich the Pt local moments align antiparallel to the moments on the 3d-element sites. Recent magnetic circular x-ray dichroism (MCD) measurements of these compounds display an interesting variation in the orbital and spin magnetic moments on the Pt site [39,40]. Among the data, it is quite remarkable to note that, in CrPt3, the Pt magnetic moment is dominated by its orbital component and coupled with the Cr spin moment in an antiparallel way while, in MnPt3, the Pt orbital mo-


ment vanishes completely and sm all but parallel spin moment to the Mn spin moment takes place.

Here, we investigate such systematic trends seen in the orbital and spin magnetic moment of the TPt3 compounds [41]. The spin moments are selfconsistently calculated by an all-electron scalar-relativistic version of FLAPW method within LSDA while the orbital moments are evaluated in a perturbative way by an additional inclusion of the spin-orbit interaction. The perturbative technique has been tested for ferromagnetic 3d transition met als [42-44]. However, it has been well known that the orbital moments are often underestimated by up to 50%. Nevertheless, the present technique may give a qualitative description for the systematic trends seen in the orbital moments of TPt3. In the calculation, we assume a ferromagnetic ordering for all TPt3 including FePt3, which is actually an antiferromagnet. This assumption enables us to study qualitative systematic trends in the series of TPt3 .

•

• o f----.ß<.=.'" "c..." "-"-" "_..t;._" "_" _" "..;..0" "C=;" ~"="=" "=" "c-" ".=..t;..,-"...c" "ce:" """" "=" "---l

V Cr Mn Fe Co

Transitionmetal

Fig.3. Calculated total (open pentagon), orbital (open triangle) and spin (open dia" mond) magnetic moments on the 3d-met al site of TPt3. Neutron data are indicated by filled pentagons.

1.4.1 Orbital and Spin Moments at the 3d-Metal Sites

Calculated orbital and spin magnetic moments of TPt3 are shown in Fig. 3. It can be seen that the magnetic moments at the 3d-met al sites are dominated by its spin components in all TPt3. The calculated spin moments are in fair agreement with neutron experiments [38] except for so me discrepancy in CrPt3 . Linear


and symmetrie behaviors centered at MnPh can be seen in the spin magnetic moments at the 3d-metal sites. This may be easily understood by the occupancy of the 3d bands. The 3d states are situated near the top of the Pt 5d bands, forming relatively narrow bands. As one moves from the lighter to the heavier 3d elements, the up-spin 3d band is first filled in MnPt3 and then electrons will start to occupy the down-spin 3d band, resulting in linear and symmetrie behaviors of the spin moments at the 3d-metal sites mentioned above. As a typieal example of the density of states (DOS), the partial 3d-metal-site and Pt-site projected DaSs of MnPt3 are shown in Fig. 4.

-E 6 ~ 4 c:::: t 'Ci 2 UJ

>- 0 ~ CD 2 1ii

~ - 4 UJ -CI) 0 6 C

-0.6 -0.4 -0.2 0 0.2

Energy(Ry)

Fig. 4. Partial density ofstates ofMnPt3. Thick solid (broken) lines denote the m = +2 (m = -2) components on Pt while thin solid (broken) lines the m = +2 (m = -2) components on Mn.

1.4.2 Orbital and Spin Moments at the Pt Sites

Figure 5 illustrates orbital and spin moments on the Pt sites calculated for TPt3 together with experimental moments extracted from Maruyama's [39,40] MCD measurements. The Pt spin moments arise from hybridization of the Pt d states with the 3d bands. Since the up-spin 3d bands in MnPt3, FePt3 and COPt3 are almost filled, the down-spin holes in the Pt d states hybridized with the empty 3d bands give rise to the spin moments on the Pt sites parallel to those of the 3d-metal sites. The reason why the Pt spin moment is not the largest for MnPt3, where the 3d spin moment has a peak, is because in FePt3 and COPt3, the hybridization of the Pt d states with the empty 3d bands becomes stronger than in MnPh due to smaller 3d exchange splitting and introduces larger holes in the Pt down-spin states. In VPt3 and CrPt3, on the other hand, the number of d up-spin holes on the Pt sites is greater than that of down-spin ones because

Takahisa Ohno and Tamio Oguchi

-:f 0.4 -"E

o

Q)

E o 0.2 E (.) ...

-:.;:::::; 'b.-------- a

~ 0 t-------'~_____=v_----__J CD ca :!: 0"

V Cr Mn Fe Co

Transition metal

13

Fig.5. Calculated total (open pentagon), orbital (open tri angle) and spin (open diamond) magnetic moments the Pt si te of TPt3. MCD data are indicated by filled symbols.

of larger hybridization of the Pt and 3d-metal d bands. (See the partial DOS of CrPt3 in Fig. 6.)

-E 6 .9 tU 4 c t 'Ci 2 Cf)

>. 0 ([ -Cf)

2 (J)

E 4 t Cf) -CI)

0 6 C

-0.6 -0.4 -0.2 0 0.2

Energy(Ry)

Fig. 6. Partial density of states of CrPt3. Thick solid (broken) lines denote the m = +2 (m = -2) components on Pt while thin solid (broken) lines the m = +2 (m = -2) components on Cr.


The magnitude of the calculated orbital moment is about half of the observed one. This underestimation is comparable with that seen in ferromagnetic 3d transition met als [42-44]. The mechanism of the systematic trends in the Pt orbital moments can be understood from the calculated DOS. Due to the large spinorbit interaction of Pt, the m = ±2 and m = ±1 components of the Pt d bands split significantly, as shown for MnPh and CrPt3 in Figs. 4 and 6, respectively. Roughly speaking, the 5d holes on Pt are dominated by the m = 2 and m = - 2 components in the up-spin and down-spin bands, respectively. Therefore, the difference between the up-spin and down-spin holes on Pt, which corresponds to the Pt spin moment, determines the major feature of the systematic variation in the orbital moments of the series. In CrPt3, particularly, the Pt up-spin holes amplified by the hybridization result in negative orbital moments on Pt despite the relatively small spin moment, as demonstrated in Fig. 6.

1.5 Semiconductor Nanostructures

Recent progress of epitaxial growth techniques with atomic scale controllability such as molecular beam epitaxy (MBE) and metal organic chemical vapor deposition (MOCVD) of semiconductors has led to the fabrication of high-quality optoelectronic devices or novel nanostructures, for example, quantum wells, wires, and dots. First-principles pseudopotential methods have predicted novel and exotic properties of GaAs/Ge superlattices and Si quantum wires, as described in Sects. 1.5.1 and 1.5.2.

1.5.1 GaAs/Ge Superlattices

Metastable (GaAsh-x(Ge2)x alloy is known to have a specific feature in which the direct band gap, as a function of atomic concentration x, exhibits much larger bowing than conventional III-V alloy systems [45]. Hetero-valent systems such as a combination of Ga, Ge, and As atoms have a possibility of exhibiting attractive characteristics, not only of fundamental interest, but also for device applications. We have systematically investigated the electronic structure and thermodynamic stability of hetero-valent GaAs/Ge superlattices [46-48].

We consider various GaAs/Ge superlattices with different orientations and atomic configurations: (OOl)--oriented GaAsGeGe, GaGeAsGe, (1l1)-oriented GaAsGeGe, GaGeGeAs, and GaGeAsGe, which are shown in Fig. 7. These superlattices have the same atomic concentration, whereas they differ in their concentration of interatomic bonds. There are two types of interatomic bonds: homo bonds (Ga-As and Ge-Ge) and hetero bonds (Ga-Ge and Ge-As) , in which the total number of valence electrons is equal to eight or differs from eight, respectively.

Figure 8 presents the electronic structures calculated using the ab initio pseudopotential method for the GaAs/Ge superlattices shown in Fig. 7. These electronic structures differ greatly from each other in spite of having the same atomic


• (001 )-orienled

(0) GaAsGeGe (b) GaGeAsGe

• (t11)-oriented

• As

Ge o Ga

(e) GaAsGeGe Cd) GaGeGeAs (e) GaGeAsGe

15

Fig. 7. Atomic structures of (a) (OOl)-oriented GaAsGeGe, (b) GaGeAsGe, (c) (111)oriented GaAsGeGe, (d) GaGeGeAs, and (e) GaGeAsGe superlattices. Open, solid, and dotted circles represent Ga, As, and Ge atoms, respectively.

2

- 0 ~ --2 >- L......-_~

Cl .... Q.I

Jj 2

o -2

Zx r Z Wave Vector

R

(e

2 t::::::::::::=:::~

o

210----..: ~~P'l

o l:====---:::::=~r==---~ -2

M r Wave Vector

z

Fig.8. Calculated electronic structures for (a) (OOl)--oriented GaAsGeGe, (b) GaGeAsGe, (c) (111)-oriented GaAsGeGe, (d) GaGeGeAs, and (e) GaGeAsGe superlattices. The relevant superlattice structures far (a) and (b) correspond ta structures with n/N=0.5 and 0.25, respectively.


1lJ'0 > 1J' 0·5 -'" W

o

~\. ~

~ • \ . I · · · \ i

I . \

0·5 n/N

Oeale.

6. Expt.

1·0

Fig.9. Calculated band-gap energy ratio E g / E:v (open circle) for GaAs/Ge superlattices as a function of n/N, where E:v is the atomic concentration averaged band-gap energy. Experimental results (open triangle) for (GaAsh-x(Ge2)x are also plotted in this figure.

concentration. The (111)-oriented GaAsGeGe superlattice exhibits a semiconducting character with a band gap. For the other GaAsjGe superlattices, however, the band gaps disappear in spite of a combination of constituent semiconductor elements. In this way, these electronic structures of GaAsjGe superlattices differ significantly from the electronic structures of GaAs and Ge.

Figure 9 presents the calculated band-gap energy ratio Egj E~v, where E:v is the atomic-concentration averaged band-gap energy. The results are summarized as a function of the concentration of interatomic bonds n/N, where n is the number of hetero interatomic bonds (Ga-Ge and Ge-As bonds), and N is the total number of bonds in the superlattice. It is noted that thc hetero Ga-Ge and Ge-As bonds do not exist in ordinary III-V alloy systems. The calculated results compare favorably with the experimental results for band-gap energies of (GaAsh-x(Ge2)x alloys [45], where the value of n/Nis derived from the atomic concentration x based on the analytical relationship njN = 6x(l-x )/(x+3) [49]. Figure 9 implies that superlattices in the range of n/N::; 0.5 show semiconducting characteristics and that the band-gap energy decreases with increasing n/N. It is also predicted that metallic phases appear in the range of n/N::::: 0.5, including (111 )-oriented GaGeGeAs with n/N = 0.75. We note that the range of n/N::::: 0.5 has not been achieved experimentally in (GaAsh-x(Ge2)x alloys [45]. In this way, the GaAsjGe superlattices exhibit various electronic properties, frOln semiconducting to metallic, depending on the concentration of interatomic bonds.


In particular, Ge-As bonds are found to be responsible for the shrinkage of the band-gap energy in the GaAs/Ge system [48].

The calculated electronic and structural properties for hetero-valent GaAs/Ge superlattices with different orientations and atomic configurations are summarized in Table 6. The calculated excess energies are positive, which means that these superlattices are thermodynamically unstable and may exist in a metastable phase. This table indicates that the band-gap energy and the thermodynamic stability can be controlled by changing the concentration of hetero interatomic bonds. The GaAs/Ge superlattice has a smaller band-gap energy and becomes less stable, as the hetero-bond concentration increases. The concentration of interatomic bonds is crucial in designing the properties of hetero-valent systems, instead of the atomic concentration used in homo-valent systems. Table 6 provides the guiding principles for controlling the material properties of heterovalent systems.

Table 6. Electronic and structural properties of GaAs/Ge superlattices with different orientations and atomic configurations. The number of interatomic bonds, concentra-tion n/N of the Ga-Ge and Ge-As bonds, lattice constant a (Ä), excess energy i1E (meV /atom), and energy-band gap E g (eV) are shown. S means semiconducting char-acteristics and M me ans metallic characteristics.

Ga-As Ge-Ge Ga-Ge Ge-As n/N a i1E E g S/M

GaAs 8 0 0 0 0 5.653 0 1.44 S Ge 0 8 0 0 0 5.658 0 0.99 S

(111) GaAsGeGe 3 3 1 1 0.25 5.643 75.9 1.11 S (001) GaAsGeGe 2 2 2 2 0.50 5.643 78.7 0 M (111) GaGeGeAs 1 1 3 3 0.75 5.644 82.0 0 M (001) GaGeAsGe 0 0 4 4 1.00 5.632 153.4 0 M (111) GaGeAsGe 0 0 4 4 1.00 5.633 153.8 0 M

1.5.2 Si Quantum Wires

Silicon-based light-emitting devices are expected to offer many new possibilities, but first the major issue of the indirect band-gap character of bulk silicon must be tackled. One promising approach of overcoming this is to fabricate low dimensional fine Si structures. Visible light emission has been observed from short-period Si/Ge superlattices [50], ultrafine Si particles [51], polysilanes [52], and highly porous Si [53]. Among these structures, porous Si has attracted a great amount of interest because of its stable and efficient visible light emission, which is reported for a porous Si containing Si wires with a width less than 30 A [53].

We have investigated the electronic and optical properties of Si wires focusing on the quantum-confinement effects, and examined the possibility of light ernis-

18

5' Q) ->-E> Q) c:

W


3.2

3.1

0.2 0.1

...---kz

0.0 r

C1,C2

81,82

A2 A1

Fig. 10. Electronic structure near the band gap region for a 3 x 3 Si wire. The atomic structure of the 3 x 3 Si wire is also plot ted in the inset. The open (solid) circles describe the hydrogen (silicon) atoms. The electronic structures for the 4 x 4, and 5 x 5 Si wires have almost the same characteristics as those of the 3 x 3 wire.

sion from Si wires [54]. We employ a model of a crystalline Si wire along the (001) direction (hereafter denoted the z-axis), and a plane view of the atomic structure is illustrated in the inset of Fig. 10. The Si wires with 3 x 3,4 x 4, and 5 x 5 structures are considered, which have the characteristic diameter L,l of 7.7, 11.5, and 15.3 A, respectively. Surface Si atoms are assumed to be terminated by hydrogen atoms as indicated by earlier experiments [53]. Our interest is in two physical quantities: (a) the energy of optical interband transitions and (b) the oscillator

strength of the transitions. The former is given by nWQß = E;;!r, - E~!citon where E;;!r, is the difference in energy between a conduction band (whose index is a)

and a valence band ß, and E~!citon is the binding energy of the geometricallyrestricted exciton associated with these two bands. Here, we consider the onedimensional (lD) exciton whose relative wavefunction is squeezed by the 2D confinement in the Si wire. In this case, the oscillator strength per unit volume of the wire for an excitonic transition is proportional to

(15)

Here dQß(c) == ('I/'ßlg· p I'I/'Q) is the interband-transition dipole matrix element with polarization unit-vector g (g = x, y, z) and moment um operator p, and 'I/' is the band wave function. A wave function of relative motion of the 1D exciton is described as <p~~(() with ( being the relative coordinate of an electron and a hole.

Figure 10 shows the calculated electronic structure of a 3 x 3 Si wire. One of the most important features is that the band gap is direct at the zone center. The main six conduction bands originate from the six valleys in bulk Si, and


these structures are weIl explained in terms of the mixing and the zone folding ofthe bulk Si bands. The four lowest conduction band states, Al, A2 , BI, and B2

originate from the four transverse vaIleys of bulk Si. The conduction band state Cl reaches the minimum at point Cm , which corresponds to the longitudinal vaIley of bulk Si. The four transverse vaIleys of the Si wire are lower in energy than the two longitudinal vaIleys, making the band gap direct. The direct bandgap characteristic can be explained in terms of the anisotropie effective mass of bulk Si [54].

Another important feature of the band structure is that the direct band gap is opticaIly allowed. This is because the quantum confinement and surface scattering in the Si wire result in a finite matrix element daß (c) between the zonecenter hole state and thc electron state folded from near the zone edge of bulk Si. When the Si wire decreases in width, these effects are significantly enhanced. Thus the enlargement in daß(c) depends strongly on the wire width. The bandgap energy is also enlargcd with decreasing Si wire width. The calculated results of band energy, effective mass, and transition matrix element for Si wires with 3 x 3, 4 x 4, and 5 x 5 structures are tabulated in Table 7.

Table 7. The band energy E~e, (eV) measured from the valence-band-maximum, the effective mass m* in the unit of free electron mass, and the interband matrix element J"'ß(c) == Id"'ß(c)/dGaAsl (where c = x, y, z) relative to the direct transition in bulk GaAs are listed for the 3 x 3, 4 x 4, and 5 x 5 Si wires. The wavefunction symmetry is also presented: the 0"0 symmetry is the identity, O"x (O"y) is the mirror symmetry with respect to the y-z (x-z) plane, and O"xy equals O"xO"y. In the Table, v stands for x or y.

Si wire 3x3 4x4 5x5

L-L 7.7 A 11.5 A 15.3 A

a,ß ß - -E~ap m* dcxß(z) dcxß(v) E~e, m* J"'ß(z) J"'ß(v) E~e, m* J"'ß(z) J"'ß(v)

Al 0"0 3.11 0.3 0 0.01 2.47 0.3 0 0.003 2.06 0.3 0 0.001

A2 O"xy 3.16 0.3 0 0.01 2.52 0.3 0 0.003 2.11 0.3 0 0.001

BI o"x 3.23 1.5 0.3 0 2.54 0.6 0.09 0 2.11 0.4 0.02 0

B2 O"y 3.23 1.5 0.3 0 2.54 0.6 0.09 0 2.11 0.4 0.02 0

Cm 3.31 2.62 2.25

VI o"x 0 1.1 0 1.0 0 1.2

V2 O"y 0 1.1 0 1.0 0 1.2

On the basis of the electronic band properties, we can discuss Wannier excitonic effects in the Si wires within the effective-mass approximation [55]. The ID exciton effects are found to enhance the oscillator strength of the Si wire hy two or three orders of magnitude, compared with the 3D exciton in bulk GaAs. The values of E~!iton are typicaIly as large as a few hundred meV.

20 Electronic Structure Theory far Condensed Matter Systems

Table 8. The optical transition energies nWaß (e V) and the relative oscillator strength 1]:'~ (z) and 1]:'~ (l/) with l/ = x, y for Si wires.

Si wire 3x3 4x4 5x5

Li. 7.7 A 11.5 A 15.3 A

a-ß nWaß 1]:'~ (z) 1]:'~ (l/) nWaß 1]:'~(z) 1]:'~(l/) nWaß 1]:'~ (z) 1]:'~ (l/)

Al-VI 2.48 0 0.2 2.08 0 0.003 1.76 0 0.0002

A2-VI 2.53 0 0.2 2.13 0 0.003 1.81 0 0.0002

BI-VI 2.49 140 0 2.11 2.6 0 1.78 0.06 0

Table 8 shows nWaß = E~e, - E:!citon and r];~(E) for several interband transitions in the Si wires. The relative oscillator strength 7];~ (E) is defined as

7];~(E) == f~~(E)/ f6~As' where f6~As is the oscillator strength of an optical transition in the bulk GaAs. The values of nWaß are in the visible range, changing from red to green as the width decreases. It is noted that both the band-mixing effect and the ID exciton effect remarkably enhance the oscillator strength of the Si wire, compared with bulk GaAs. The Si wires are expected to exhibit visible and strong light emission.

1.6 Semiconductor Surfaces

Semiconductor surfaces are of great interest both from a fundamental and an applied point of view. Surfaces are unique systems in that they have many more degrees of structural frecdom compared with bulk crystals and in that transport of materials, for example, adsorption and desorption of molecules, occurs frequentlyon surfaces. Most optoelectronic devices take advantage of the properties of surfaces and interfaces of semiconductors. The recent reduction in device size has been enhancing the importance of surface/interface properties. In these circumstances, there has been an increasing stimulus to understand thc structural and dynamical properties of semiconductor surfaces on a microscopic level. A large variety of surface sensitive experimental techniques such as scanning tunneling microscopy (STM) have been developed. The theory of semiconductor surfaces has also matured concomitantly with the experimental developments. In this section, we describe theoretical investigations of atomic structurcs and dynamics on semiconductor surfaces.

1.6.1 Surface Structures

The surface of asolid no longer has the three-dimensional periodicity of the crystal and the surface atoms relax from what would be their ideal bulk positions. Because the bonding in semiconductors is both strong and highly directional, relaxation can occur, such as the surface atoms move to eliminate their dangling bonds. One prominent example is the (001) surface of silicon, where the surface


Si atoms relax to form Si-Si dimers. At the surfaee of a eompound semieonduetor sueh as GaAs, there exist dangling bonds of both eation and anion atoms. The eh arge transfer from eation to anion atoms ean induee surfaee relaxation of eompound semieonductors. We have investigated the atomie struetures of GaAs(OOl) surfaees und er both As-rieh and Ga-rieh growth eonditions by using first-prineiples eleetronie structure theory.

GaAs(OOl) As-rich Surfaces. The GaAs(OOl) surfaee is very interesting, not only in terms of surfaee physies but also in semieonduetor teehnology of thinfilm growth. The GaAs(OOl) surfaee manifests a sequenee of reeonstruetions dependent on surfaee stoiehiometry ranging from the As-rieh c( 4 x 4) strueture to the Ga-rieh (4 x 2) reeonstruetion. Among them, the As-rieh (2 x 4) surfaee is of most teehnologieal interest sinee moleeular beam epitaxy (MBE) growth usually begins and ends with this surfaee und er As-rieh growth eonditions. The atomie structure of the (2 x 4) surfaee has been intensively investigated [56-59]. It was reported that this symmetry exists over a range of surfaee stoiehiometry. Eleetron diffraction measurements aetually showed three different (2 x 4) phases (0:, ß, and I)' although it eould not determine the struetures definitely sinee the unit eell was relatively large [57].

The prineipal model for the (2 x 4) surfaee is the vaeaney model, whieh assumes that some As dimers are missing in a (2 x 4)unit eell. A tight-binding total energy ealculation has shown that the (2 x 4) unit eell with three As dimers and one missing dimer is energetieally favorable [56]. This structure obeys the electran counting model, whieh is a guiding prineiple popularly used to determine low energy surfaee struetures of eompound semieonductors. The eleetron eounting model requires that a surfaee strueture is stable where the number of eleetrons per unit eell will exaetly fill all As dangling bonds and empty all Ga dangling bonds. The existenee of missing As dimers has been eonfirmed by reeent STM images [58,59].

We have determined the detailed atomie struetures and stabilities of As-rieh surfaees from the standpoint of equilibrium energeties by using first-prineiples total-energyealculations [60]. Some reeonstruction models for As-rieh GaAs(OOl) surfaees having (2 xl), (2 x 4), c(2 x 8), c(4 x 4), and (3 x 1) symmetries are eonsidered, where the As eoverage ranges from 8 = 0.5, through 2/3, 0.75, l.0, and finally up to l. 75. The atomie structures of some of the (2 x 4) surfaees are sehematieally shown in Fig. 1l. The strueture Ha eontains three As dimers and one dimer vaeaney per (2 x 4) unit eell (8 = 0.75), whieh is denoted as the .8 phase. When basie (2 x 4) units are arranged in anti-phase along the missingdimer rows, the strueture has the c(2 x 8) symmetry. The Hd strueture is denoted as the ß2 phase, sinee there are aetually three As dimers per (2 x 4) unit eell (that is, two in the top layer and one in the third layer) like in the ß phase. The c( 4 x 4) strueture eonsists of three As ad-dimers on top of a eomplete As monolayer (8 = l.75). Furthermore, two Ga-rieh (4 x 2) phases are eonsidered; the (4 x 2)ß surfaee eonsists of three Ga dimers in the top layer and the (4 x 2)(J2 surfaee eontains two Ga dimers in the top layer and one in the third layer.

22 Electronic Structure Theory für Cündensed Matter Systems

(a) S~~~~~: ~:,Q.7,5 , , , ,

, , ------------

(d) (2x4 )-ß2: 8=0.5

(b) (2x4)-al: 8=0.5 r - - - - - - - - - --I , , LHJ' , , , , , , , , ,

LHJ, , , , , , , , , ------------

(e) (2x4)-y. 8=1.0 r - - - - - - - - ---I , ,

(c) (2x4)-a2: 8=0.5 r-----------I , , , ,

, ------------

Fig. 11. Schematic atomic structures of As-rich GaAs(00l)-(2 x 4) surfaces whose As coverages e are 0.5, 0.75, and 1.0. Filled and open circles denote Ga and As atoms, respectively.

The formation energies of the GaAs(OOl) surfaees are plotted in Fig. 12 as a function of the As ehemieal potential. In this figure, the ehemieal potential is measured relative to that of bulk As, J.LAs(bulk)' It is noted that the energy differenee between the (2 x 4)ß and ß2 surfaees is very small; the latter is lower in energy by only 0.01 eV per (1 x 1) unit eell than the former. Both surfaees have three As dimers per (2 x 4) unit eell and the loeal bonding environment of eaeh atom in the two surfaees is essentially identieal, as shown in Fig. 11. The faetor whieh determines the relative stability of the (2 x 4)ß and ß2 surfaees is of great interest. Sinee the two surfaees satisfy the eleetron eounting rule, eaeh As dimer-atom is negatively eharged and eaeh threefold-eoordinated Ga atom is positively eharged in the (2 x 4) unit eell. The eleetrostatie interactions between these eharged atoms favors the (2 x 4)ß2 phase [61]. On the other hand, the surfaee stress indueed by the As-dimer formation is more diffieult to be relaxed in the (2 x 4)ß2 phase beeause one As-dimer is formed but no As atom is missing in the third layer of the (2 x 4)ß2 surfaee. Thus, the surfaee stress favors the (2 x 4)ß strueture. The eaneellation of the effects of eleetrostatie interaetion and surfaee stress leads to the small energy differenee between the As-rieh (2 x 4)ß and ß2 surfaees. The same holds for the relative stability of the Ga-rieh (4 x 2)ß and ß2 phases; they have almost the same energy.

Consequently, the (2 x 4)ß2 surfaee having two As dimers in the top layer and one in the third layer is found to be the most stable As-rieh strueture in


0.2 ,----~'""'O"""T""-..--.::__--------r--_____,

~ ,...... X ,...... ;> -0.1

~ -0.2

~-0.3 $-<

~ -0.4 Q) C -0.5

o '.g -0.6

§ -0.7

o -0.8 ~

-0.9

-1. 0 '---'------1.---'-'----'-----'-_-'--....I----'-_.1.--U.---'----I

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2

Chemie al potential f.l(As) (e V)

23

Fig.12. Surface formation energy per (1 x 1) unit cell for GaAs(OOl) surfaces. The vertical dashed lines indicate the range of the As chemical potential. The horizontal line at 0.0 eV is the surface energy of the GaAs(001)-Ga(1 x 2) surface. The origin of the As chemical potential is taken to be that of bulk As.

the range -0.54 < /.LAs< -0.28 eV, and may eorrespond to the well ordered two-As-dimer structure in the reeent STM images [59]. There is a possibility that the (2 x 4)ß surfaee with three As dimers in the top layer may appear in some MBE growth eonditions [57,58], sinee the energy differenee between the (2 X 4)ß and ß2 surfaees is very small. The c(2 X 8)ß surfaee has almost the same energy as the (2 x 4)ß surfaee, whieh indieates that the basie (2 x 4) units ean be arranged both in phase and in anti-phase. When /.LAs exeeeds -0.28 eV, the more As-rieh c(4 x 4) phase beeomes stable with respeet to the (2 x 4)ß2 surfaee. For -0.74 < /.LAs< -0.54 eV the (2 x 4)ß2 surfaee beeomes unstable with respeet to the more Ga-rieh (4 x 2)ß or ß2 surfaees.

GaAs(OOl) Ga-rieh Surfaees. Little attention has been directed toward the Ga-rieh surfaee reeonstruetions such as the (4 x 2) and (4 x 6) phases, whieh is mainly due to the diffieulty in preparing these Ga-rieh surfaees using eonventional MBE. The study on the struetures of the Ga-rieh surfaees is of inereasing importanee in order to eomprehensively understand the growth meehanism of the GaAs(OOl) surfaee, sinee the growing front must be terminated alternately with the As-rieh (2 x 4) and the Ga-rieh (4 x 2) surfaees.

There are two distinetly different models for the Ga-rieh (4 x 2) surfaee as shown in Fig. 13. One is the Ga model [62] eonsisting of two Ga dimers on the top layer and one Ga dimer at the third layer, whieh eorresponds to the (4 x 2)/32 surfaee. The other is the As model [63], which has two As dimers in the top layer


(a) Ga-model (b) As-model

Fig.13. Schematic atomic structures of the Ga-rich GaAs(OOl)-(4 x 2) surface: (a) the Ga model and (b) the As model. Filled and open circles denote Ga and As atoms, respectively.

and two Ga dimers in the second layer in a unit cell. The STM image of the Ga-rich (4 x 2) surface shows that the faint features are positioned in between the brighter features along the [110] direction [63,64]. If the faint and brighter features are identified as Ga and As atoms, respectively, the STM image cannot be explained readily by the Ga model since the Ga and As atoms should line up in the Ga model. If STM simply maps the surface atomic geometry, this would be true. However, STM does not map merely the atomic geometry of the surface but the local density of states near the Fermi level. The correct interpretation of the STM image requires knowledge of surface local density of states near the Fermi level. In order to resolve the discrepancy in the Ga and As models, we have performed a first-principles total-energy calculation [65].

It is found that the As model is energetically much more unstable with respect to the Ga model for the allowed range of the As chemical potential. Thus, the As model can be safely ruled out. The Ga model, that is the Ga-rich (4 x 2){32 surface, is stable in the range -0.74 < /-lAs< -0.54 eV. The calculated level charge densities for the Ga model structure are shown in Fig. 14. The 91st band is the HOMO (highest occupied molecular orbital) band and has acharge localized at the As atoms in the second layer (peak B in the figure). The contribution from the Ga dimer on the top layer becomes noticeable only at the deep 87th band, whose charge distribution peaks at the middle of the Ga dimer (peak C in the figure). As the energy level increases from the 87th band to the 91st band, the contribution from the Ga dimer gradually disappears and the charge derived from the As atoms becomes more dominant. The charge distributions of the 90th, 89th, and 88th bands basically overlap with that of the 91st HOMO band. The 92nd band is the LUMO (lowest unoccupied molecular orbital) band derived from the Ga dangling bonds and should be basically empty for the ideal semiconducting (4 x 2) surface.

By comparing the calculated surface energy levels and the filled-state STM imaging condition at about -2 V [63,64], it is determined that alliocal density


92nd (LUMO) 89th

91st(HOMO) 88th

90th

Fig. 14. Calculated charge density from the (4 x 2) Ga model structure at 0.9 A above the top layer Ga dimer position for the 92nd (LUMO), 91st (HOMO), 90th, 89th, 88th, and 87th bands.

of states between the 87th and 92nd bands contribute to the tunneling current to form the STM image of the (4 x 2) surface. The 91st HOMO band makes the most significant contribution to the tunneling current because of the lower potential barrier for tunneling, compared with the 90th, 89th, 88th, and 87th bands. Thus, the As atoms in the second layer are imaged as individual brighter protrusions (peak B), whereas the Ga dimer on the top layer is observed as a single faint hump (peak C) instead of a pair-like feature.

Since there are significant amounts of surface defects, such as adsorbates and vacancies, on the MBE grown (4 x 2) surface [64], the 92nd LUMO band is partially filled due to the charge transfer from these surface defects and contributes to the tunneling current. The charge distribution of the 92nd band is localized at one of the Ga dangling bond positions (peak A in Fig. 14), which is due to the existence of the Ga dimer at the third layer. The Ga dimer atoms on the top layer asymmetrically contribute to the STM image.

The integrated local density of states from the 87th band to the 92nd band, which is proportional to the tunneling current density, is presented in Fig. 15. It is clear that the local density of states (the STM image) does not simply correspond to the atomic geometry of the Ga model. The extent of the LUMO band contribution to the filled-state STM image depends on the amount of the charge transfer to the Ga dimers. Since the calculated density of states shown in Fig. 14 is the mapping of the charge distribution at only 0.9 A away from the surface while the STM image reflects the charge distribution approximately 10 A away, peaks A and B may shift away from the atomic position to so me

26 Electronic Structure Theüry für Cündensed Matter Systems

Fig. 15. Integrated lücal density üf states from the 87th band tü the 92nd band üf the (4 x 2) Ga model.

extent. Realizing these effects, the STM image reported [63,64J can be explained weIl in terms of the (4 x 2) Ga model.

1.6.2 Surface Dynamics

The migration process of individual adatoms on a semiconductor surface and their incorporation into growth sites of the surface play important roles in controlling atomic arrangements during epitaxial growth. Understanding these fundamental pro ces ses at the atomic level will develop methods to manipulate the surface dynamics and offer the exciting possibility to tailor thin-film growth for specific materials applications. The dynamical properties of surfaces are, however, much more difficult to investigate experimentally, compared with the static ones such as surface atomic structures. The first-principles electronicstructure theory is one of the most powerful methods to investigate these problems. We will describe our recent studies on surface diffusion both of Si adatoms on the hydrogen-terminated Si(OOl) surface and of Ga adatoms on the As-rich GaAs(OOl) surface.

Diffusion of Si on H-terminated Si(OOl). Hydrogen termination has attracted considerable interest because of the possibility of changing the growth mode of epitaxial semiconductor films. Adsorption of hydrogen can alter the surface properties such as the surface energy, the adatom diffusion, and the nucleation processes, and result in the modification of the morphology of the epitaxial films [66-69J. Recently, the breakdown of Si homoepitaxial growth has been reported in low temperature Si molecular beam epitaxy (MBE) on the hydrogenated Si(OOl) surfaces [66, 67J. Three mechanisms have been proposed to explain the disruptive effects of the H termination. The first one suggests that the suppression in surface Si mobility caused by the H termination results in the


defective epitaxial growth [66J. The second one claims that the breakdown in epitaxy correlates with the formation of Si dihydrides on the surface [67J. The third one, which is based on ab initio calculations [70J, suggests that surface H atoms hardly segregate from the surface and hinder the incorporation of Si adatoms into growth sites. Experimental results, however, indicate that H atoms segregate from the Si(OOl) surface [67J. The role of surface H atoms in Si homoepitaxy has not been understood satisfactorily at the microseopie level.. We have theoretically investigated Si adsorption and diffusion on the H-terminated Si(001)-(2 X 1) surface by using first-principles total-energy calculation techniques [71J. Figure 16 shows the structure of the H-terminated Si(00l)-(2 x 1) surface, which corresponds to the H coverage of one monolayer (ML) [72J.

[1101

[110]

Fig.16. Top view of the H-terminated Si(OOl)-(2 x 1) surface. The large and small open circles denote the top- and second-layer Si atoms, respectively. The solid circles denote the terminating H atoms. The (2 x 1) unit cell is indicated by the solid line and the c( 4 x 4) supereell by the dashed line. The lines a and bare the boundaries of the (2 x 1) unit cell and the line c halves the unit cell perpendicular to the Si dimer bond.

Figure 17 presents the total energy curve as a function of the height of the adsorbing Si atom, when the Si atom is deposited over one Si dimer atom. As the Si atom moves close to the surface, the total energy decreases gradually and then decreases drastically by about 1.2 eV. This discontinuous drop of the total energy corresponds to the transfer of the H atom from the Si dimer atom to the adsorbing Si atom. The bond between the H atom and the Si dimer atom is broken and a new bond is formed between the H atom and the adsorbing Si atom. The adsorbing Si atom moves further downward and finally forms a bond to the Si dimer atom. In this way, the Si atom adsorbs on the monohydride Si(OOl) surface by spontaneously capturing one nearest neighbor H atom, which

28

A

B

Electronic Structure Theory far Condensed Matter Systems

o

I ~~~~~~~~~

2.4 2.8 3.2 3.6 Height of Si adatom(Ä)

Fig. 17. Total energy curve when the Si atom is deposited above one Si dimer atom of the H-terminated Si(OOl)-(2 xl) surface, as a function of the height of the deposited Si atom. Three geometries appearing during the Si adsorption are also presented in the side view. The open, solid, and gray circles denote the substrate Si, H atoms, and the deposited Si atoms, respectively.

is consistent with experiments [67]. The geometry of the Si adatom capturing one H atom is named the H-capture geometry hereafter.

Figure 18 shows the total energy surface of the Si adatom in the H-capture geometry. The contour map is plotted in the surface region near the bared Si dimer atom. The bared Si atom me ans the substrate Si atom from which the terminating H atom is detached by the Si adsorption. It is noted that the contour map loses the original (2 x 1) periodicity of the H-terminated Si surface due to the H detachment from the surface. There are three almost degenerated stable sites; the S, M, and E sites. The E site is very elose to the dimer center D site. In these geometries, given in Fig. 19, the Si adatom binds to one H atom and one Si dimer atom. The Si adatom migrates among these stable sites with an energy barrier of 0.5 eV.

The capture of H atoms gives another degree of freedom to the Si adsorption on the H-covered Si surface, compared with that on the bare Si surface. Besides the H-capture geometry, there are two other adsorption geometries where the Si adatom captures no and two H atoms, which are denoted by the no- and H2 -

capture geometry, respectively. Although the H-capture geometry is the lowest in energy in most of the surface region, the other geometries also play important roles at so me positions. The total energy surfaces for these kinds of adsorption geometries are connected by the transfer of H atoms between the Si adatom and the substrate Si atoms.

The H mobility not only realizes the most stable adsorption geometry, but assists the surface migration of the Si adatom. On the dimer center D site, the H2-capture geometry (Fig. 19d) is more stable by 0.5 eV than the H-capture oue


1.85 eV

1.65eV

1.45eV

1.25 eV

1.05eV

0.85eV

0.65eV

0.45 eV

0.25eV

0.05 eV

OeV

Fig.18. Total energy surface of the Si adatom in the H-capture geometryon the Hterminated Si(001)-(2 X 1) surface. The contour map is plot ted near the (1 x 1) surface region containing the Si dimer atom from which the H atom is detached. The (1 x 1) cell is indicated by the white line. The open and solid circles denote the substrate Si and H atoms, respectively. The S, M, and E sites are three almost degenerated stable adsorption sites. The D is the dimer center site. The Y is the minimum-energy site along the line b. The first contour is plotted at 0.05 eV with respect to the energy of the S site and the contour spacing is 0.2 eV.

(Fig. 19a). The initial Si adsorption, however, results in the H-capture geometry instead of the H2-capture one. The migration of one H atom after the adsorption can transform the former geometry to the latter, as shown in Fig. 20. The barrier for the transformation from the H- to the H2-capture geometry by the H displacement is 0.2 eV and that for the reverse transformation is 0.7 eV. It is noted that the diffusion barrier of H adatoms is measured to be more than 2.0 e V on the Si(OOl) surface [73]. The H mobility is considerably enhanced by the Si adsorption on the H-terminated Si(OOl) surface, since the Si adatom comes so elose to the substrate Si atom that one H atom can be transferred between them.

There are two possible migration processes in which the Si adatom in the H2-

capture geometry diffuses from the most stable dimer center D site. One is the process in which the Si adatom migrates from the dimer center while remaining in the H2-capture geometry. The other is the process assisted by the H mobility, in which the Si adatom is transformed to the H-capture geometry by releasing one H atom as shown in Fig. 20 and then migrates on the energy surface of the H-capture geometry (Fig. 18). The energy barrier for the Si diffusion in the H2-capture geometry is more than 2.0 eV, while the barrier for the H release


Fig.19. Top views of the stable geometries for the Si adatom on the H-terminated Si(00l)-(2 x 1) surface; (a), (b), and (c) are the H-capture geometries at the D (E), M, and S sites, respectively, (d) is the H2-capture geometry at the D site.

is 0.7 e V and that for the Si diffusion in the H-capture geometry is 0.5 e V. Consequently, the Si adatom prefers to migrate from the dimer center in the H-capture geometry by releasing one H atom.

When the Si adatom in the H-capture geometry diffuses into the neighbor (1 x 1) cell in which the Si dimer atom is H-terminated, the energy abruptly increases as shown in Fig. 18. This means that the Si adatom hardly migrates into the next cell as long as it remains in the H-capture geometry. The alternative process of the Si migration into the next cell is the following one, which is assisted by the H mobility. When the Si adatom in the H-capture geometry arrives at the boundary a or b line of the (1 x 1) cell, the transformation to the no-capture geometry occurs by releasing the H atom back to the bared Si dimer atom, as shown in Fig. 21. Then the Si adatom migrates into the next cell and captures one H atom from the Si dimer atom in the cell to form the H-capture geometry again. The H release has an energy barrier of 1.0 eV (Fig. 21) and 1.2 eV at the boundary M and Y sites, respectively. The Si adatom migrates across the boundary c line at the D site, via the H2-capture geometry by capture and release of one H atom; the Si adatom captures another H from the front Si atom of the dimer and then releases one H to the back Si atom. The barrier for this H displacement is 0.7 eV (Fig. 20).

Consequently, the Si adatom migrates on the monohydride Si(OOI) surface by repeated capture and release of H atoms. From the barrier heights of the H movement and the energy difference of 0.5 eV between the H- and H2-capture stable geometries, the Si diffusion barriers are estimated to be 1.5 e V and 1. 7 e V along the direction parallel and perpendicular to the dimer rows, respectively. These barriers are much larger compared with that on the bare Si(OOl) surface


A

B

> 0.4 (l)

~ O b.O t> ~ -° . 6 u..L.L.L ............................ LU..U-.............. -LU..LJ ° 0.8 1.6 2.4

H migration length(Ä)

C

31

Fig. 20. Total energy curve of the H migration during the transformation between the H- and H2 - capture geometries at the dimer center D site, as a function of the displacement of the H atom. The geometries appearing during the H migration are also presented in the top view.

A i l tti c (l) i: c:: .:: U.l ·····1···········:-··········1···········

o 1 i i

o 1 2 3 4 H migration length(Ä)

c

Fig. 21. Total energy curve of the H migration during the transformation between the H- and no-capture geometries at the M site, as a function of the displacement of the H atom. The geometries appearing during the H migration are also presented in the top view.


[74, 75]. The Si adatom hardly migrates, apart from the Si dimer atom from which the H atom is detached. The reduction of the surface Si mobility would increase the surface roughness of the epitaxial Si film. The increased surface roughness seems to intrinsieally lead to the crystalline to amorphous transition of the Si film on the H-terminated Si(OOl) surface [66], as well as on the bare surface. As a result, the Si homoepitaxy on the monohydride surface may be disrupted due to the reduction of the Si migration.

Diffusion of Ga on GaAs(OOl). The surface diffusion of Ga adatoms on the As-rieh GaAs(001)-(2 x 4) surface is one ofthe most important rate limiting processes in MBE growth of GaAs under As-rich conditions, and has been intensively investigated by using the RHEED intensity oscillation technique [76-79]. There is, however, a large discrepancy between the reported values of diffusion constants: the activation energy of Ga-adatom diffusion on the As-rich GaAs(OOl) surface is estimated to be from 1.3 eV [76] to 4.0 eV [77]. Neither the anisotropy in the surface diffusion nor the effects of surface reconstruction on the surface diffusion have been clarified. The RHEED intensity oscillations indicate twodimensional nucleation on terraces of a vicinal GaAs surface and provide macroscopic information on the surface diffusion including the influence of surface defects such as steps and interactions between adatoms themselves. It is difficult experimentally to obtain information on the pure migration of individual adatoms on the GaAs surface. First-principles theoretical studies, therefore, are necessary to investigate the microseopie processes of adatom diffusion. We have performed parameter-free calculations of the surface diffusion of cation adatoms on the As-rich GaAs(00l)-(2 x 4) surface [80].

The diffusion constant is usually expressed in the following Arrhenius form:

(16)

where Do is the pre-exponential and i1E is the diffusion activation energy. The activation energy i1E can be evaluated by using static first-principles totalenergy calculations. These calculations, however, cannot give a value for the pre-exponential Do which contains entropy terms. A more recent approach, firstprinciples molecular-dynamics simulation, is a promising tool to obtain diffusion constants directly, but is limited to systems with small activation energies at high temperatures. We have calculated the diffusion constant by combining classical transition-state theory with first-principles total-energy calculations [81].

The surface diffusion constant D of an adatom is rigorously given by

(17)

where ni is the probability for the adatom to be at a stable site Xi and rij 18

the jump rate from the Xi site to the X j site. The jump rate can be decomposed into the directional flux rg through a saddle surface separating the Xi and X j sites and a dynamical correction factor ~ which accounts for immediate


return diffusion jumps and eorrelated multiple jumps. The directional flux l~~ is expressed by using the transition-state theory [82] as follows:

r O = (kBT) 1/2 I5ij exp[-E(x)/kBT]d2 x tJ 27rJ-l lVi exp[-E(x)/kBT]d3 x .

(18)

Here J-l is the redueed mass of the diffusing adatom and E(x) is the total energy obtained with the adatom loeated at x and all other surfaee atoms relaxed. The three- and two-dimensional integrations are performed, respeetively, over the volume Vi eentered at the stable Xi site and the saddle surfaee Sij separating the Xi and X j sites. In deriving (18), it is assumed that the vibrational frequeneies of the surfaee atoms depend only weakly on the position of the diffusing adatom. We also set the dynamieal eorreetion factor ~ equal to unity in the present ealeulations.

The ealculated migration potential of a Ga adatom on the As-rieh GaAs (001)-(2 x 4)ß surfaee is presented in Fig. 22. A (4 x 4) surfaee supereell, whieh eontains two (2 x 4)ß structures, is used in the ealculation as shown in this figure, where the adsorption sites in the surfaee unit eell are indieated by letters. The most stable adsorption site of aGa adatom is the long-bridge site (F site) whieh is loeated between two As-dimer rows. The other long-bridge site in the dimer region (FB site) and the bridge site in the missing dimer region (B site) eorrespond to loeal minima on the potential surfaee, whieh are 0.43 eV and 0.64 eV higher than the absolute minimum F site. The migration potential surfaee of an Al adatom has eharaeteristies similar to that of a Ga adatom, while the former has larger undulation than the latter. For the ealculation of the diffusion eonstant, we define the volumes Vi eentered at the loeal minimum sites Xi and the saddle surfaees Sij separating the Xi and X j sites using a Wigner-Seitz-like eonstruetion.

ellOJ

: ........ " ... B· · ·C .. · B·· ·" · · · ··

~

Fig. 22. Calculated migration potential of a Ga adatom on the reconstructed GaAs(OOl)-(2x4)ß surface. A (4 x 4) surface supereell used in the calculation is also shown, in wh ich several adsorption sites are indicated by letters of the alphabet.

34

10

4

10

10"' (jJ

---Q 10

~

Q

10 -10

10-12

10


0.8 1.0 12

---.- Ga(x) ____ Ga(y)

--0-- Al(x)

- -0- - Al(y)

.. Neave

Ohta

o

1.4 1.6

10 3 /T(K)

1.8

Fig. 23. Calculated diffusion constants of Ga and Al adatoms on the GaAs(OOl)-(2x4) surface, for the two principal directions along (x) and perpendicular (y) to the Asmissing dimer rows. Experimental data by Neave et al. (ref. 76) and Ohta et al. (ref. 79) are also plotted for Ga adatoms.

Generally, the diffusion constant will be different for the two principal directions on the surface. These principal directions on the GaAs(00l)-(2x4) surface are along and perpendicular to the As-missing dimer rows, respectively. The calculated diffusion constants of Ga and Al adatoms for the two principal direet ions are shown in Fig. 23. All of the diffusion constants follow an Arrhenius behavior. The activation energy and pre-exponential for diffusion of Ga adatoms are found to be 0.85 eV and 2.1 x 10-2 cm2 s-1 along the As-missing dimer rows, and 1.05 eV and 2.5 x 10-2 cm2 S-1 perpendicular to the missing dimer rows. The diffusion of Ga adatoms is anisotropie because of the reconstruction of the GaAs(OOl) surface, and the direction of fast diffusion is parallel to the As-missing dimer rows. The Al-adatom diffusion exhibits anisotropy similar to that of the Ga-adatom diffusion. The activation energies and pre-exponentials for Al-adatom diffusion are found to be 0.96 eV and 3.3 x 10-2 cm2 S-1, and 1.20 eV and 5.4 x 10-2 cm2 s-1, respectively, for the directions along and perpendicular to the As-missing dimer rows. The Al adatoms diffuse several times more slowly than the Ga adatoms in the same directions because of the larger activation energies.

The diffusion constants obtained by the RHEED experiments are several orders of magnitude sm aller than the calculated values. This large discrepancy may arise from the macroscopic effects included in the RHEED measurements such as the infiuence of surface defects and roughness. It is desirable to determine the diffusion constants on the GaAs(OOl) surface by more microseopie methods that


can measure the pure migration of single adatoms, such as scanning tunneling microscopy and field ion microscopy.

To investigate the macroscopic dynamical behavior of cation adatoms and the difference between Ga and Al adatoms on the As-stabilized GaAs(001)-(2x4) surface, we performed stochastic Monte Carlo (MC) simulations for a GaAsAIAs binary system on the basis of the calculated results for diffusion constants and migration potentials [83]. In doing so, we took account of the dependence of the calculated migration potential on the adatom coverage, which means that the favorable sites for adatoms change from sites on the dimer region to those on the missing dimer region as the coverage increases [84].

Figure 24 shows the ratio of the number of codeposited Ga and Al adatoms n for Alo.5Gao.5As in the dimer region and missing dimer region to the total number of surface lattice sites N as a function of adatom coverage e. It is seen from Fig. 24 that cation adatoms favorably exist in the dimer region at very low coverages (e < 0.1) and then the number of cation adatoms in the missing dimer region linearly increases with an increase of coverage. The number of Ga adatoms in the missing dimer region increases more rapidly than that of Al adatoms. This implies that the Ga adatoms migrate faster to the missing dimer region than the Al adatoms with the increase in coverage, which reflects the larger diffusivity of Ga adatoms than Al adatoms on the GaAs(OOl) surface.

(a)

~ Ga adatam

0.08 '- dimer region missing dimer region

Z 0.06 ~ :~ ~

0.04 r- c 0 o

••• "l," 0 0

o ~ ....... ~ o 0.05 0.1

w D c

o co

. . •

0.15

Caverage

•• ~.

::1 0-' o c •• ..

) !

0.2 0.25

(b)

" t AI adatom

0.08 dimer region I ~ missing dimer region

~ 2J

ce 1

~~.06 f ::J :J 1

i ~ :::t::J::J • • c: r 0 0

~ 0.04 t ~ •• ~~

.. 0

~ 0 • • < 0.02 r- ;J • -i

I- ~ I

0 , r D ••• • •

. •• , I

o ". ' 0 0.05 0.1 0.15 0.2 0.25

Caverage

Fig. 24. Ratio of the number of (a) Ga and (h) Al adatoms in the dimer region (open square) and missing dimer region (closed square) to the total number of surface lattice sites as a function of surface coverage. These are obtained by Me simulation at 873 K and growth rate 2 ML S-l on the As-stabilized GaAs(001)~(2x4) surface.

Figure 25 shows the respective atomic arrangements of Ga and Al adatoms at the coverage e = 0.10 and 0.25 on the As-stabilized GaAs(001)-(2x4) surface. It is clearly seen in Fig. 25a that randomly impinging cations predominantly


(a) 8=0.10 (b) 8=0.25

Fig.25. Snapshots of atomic arrangements for Alo.5Gao.5As relative to adatom coverage (a) () = 0.10 and (h) () = 0.25. The white, black, and gray circles denote Ga adatoms, Al adatoms, and surface As atoms, respectively.

occupy the lattice sites on As dimers (dimer region) at B = 0.10. As the coverage increases, the favorable lattice sites for adatoms change from sites in the dimer region to those along the missing dimer row as shown in Fig. 25b. This refiects the coverage dependence of the migration potential.

Acknowledgments

One of the authors (T. Oguchi) would like to thank H. Sawada, A. Nogami, T. Matsumiya, K. Iwashita and T. Jo for collaboration on the present issues. The other author (T. Ohno) would like to give his thanks to K. Shiraishi, T. Ito, A. Taguchi, T. Ogawa, J. Nara, and T. Sasaki for collaborating on the present studies.

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(1991). 23. M. Takahashi, H. Shoji, H. Takahashi, T. Wakiyama, M. Kinoshita, W. Ohta: IEEE

Trans. Magn. 29, 3040 (1993). 24. A. Sakuma: J. Magn. Mater. 102, 127 (1991). 25. S. Matter: Z. Phys. B 87, 91 (1992). 26. B.I. Min: Phys. Rev. B 46, 8232 (1992). 27. S. Ishida, K Kitawase, S. Fujii, S. Asano: J. Phys., Condens. Matter 4,765 (1992). 28. R Coehoorn, G.H.O. Daalderrop, H.J.F. Jansen: Phys. Rev. B 48, 3830 (1993). 29. H. Sawada, A. Nogami, T. Matsumiya, T. Oguchi: Phys. Rev. B 50, 10004 (1994). 30. KH. Jack: Proc. R Soc. Lond. A 208, 216 (1951). 31. T. Minamisono, Y. Nojiri, K. Matusta, T. Iwayama, M. Fujinaga: Hyperfine Inter

act. 34, 299 (1987). 32. H. Akai, M. Akai, S. Blügel, B. Drittler, H. Ebert, K Terakura, R. Zeller, P.H.

Dederichs: Prog. Theor. Phys. Suppl. No. 101, 11 (1990). 33. G.E. Bacon, J. Crangle: Proc. Phys. Soc. A 272, 387 (1963). 34. F. Menzinger, A. Paoletti: Phys. Rev. 143, 365 (1966). 35. B. Antonini, F. Lucari, F. Menzinger, A. Paoletti: Phys. Rev. 187, 611 (1969). 36. M. Kawakami, T. Goto: J. Phys. Soc. Jpn. 46, 1492 (1979). 37. S.K Burke, B.D. Rainford, D.E.G. Williams, P.J. Brown, D.A. Hukin: J. Magn.

Mater. 15-18,505 (1980). 38. J.G. Booth, in Ferromagnetic Materials, Vol. 4, edited by E.P. Wohlfarth and

KH.J. Buschow (North-Holland, Amsterdam, 1988). 39. H. Maruyama, F. Matsuoka, K Kobayashi, H. Yamazaki: Physica B 208&209,

787 (1995). 40. H. Maruyama, F. Matsuoka, K Kobayashi, H. Yamazaki: J. Magn. Mater. 140-

144, 43 (1995). 41. K Iwashita, T. Oguchi, T. Jo: Phys. Rev. B 54, 1159 (1996). 42. H. Ebert, P. Strange, B.L. Gyorffy: J. Phys. F 18, L135 (1988). 43. O. Eriksson, B. Johansson, RC. Albers, A.M. Boring, M.S.S. Brooks: Phys. Rev.

B 42, 2707 (1990). 44. G.Y. Guo, H. Ebert, W.M. Temmerman, P.J. Durharn: Phys. Rev. B 50, 3861

(1994). 45. J.E. Greene: J. Vac. Sci. Technol. B 1, 229 (1983). 46. T. Ohno: Solid State Commun. 74, 7 (1990). 47. T. Ito, T. Ohno: Surf. Sci. 267, 87 (1992). 48. T. Ohno, T. Ito: Phys. Rev. B 47, 16336 (1993). 49. H. Holloway, L.C. Davis: Phys. Rev. Lett. 53,830 (1984); L.C. Davis, H. Holloway:

Phys. Rev. B 35, 2767 (1987).

38 Elcctronic Structure Theory for Condensed Matter Systems

50. T. P. Pearsall et a!.: Phys. Rev. Lett. 58, 729 (1987). 51. H. Takagi et a!.: App!. Phys. Lett. 56, 2379 (1990). 52. R D. Miller, J. Michl: Chem. Rev. 89, 1359 (1989); H. Tachibana et a!.: Solid

State Commun. 75, 5 (1990). 53. A. G. Cullis, L. T. Canham: Nature 353, 335 (1991); L. T. Canham: App!. Phys.

Lett. 57, 1046 (1990); V. Lehmann, U. Gosele, ibid. 58, 856 (1991); A. Halimaoui et a!., ibid. 59, 304 (1991); S. Gardelis et a!., ibid. 59, 2118 (1991); Y. Kato, T. Ito, A. Hiraki: Jpn. J. App!. Phys. 27, L1406 (1988).

54. T. Ohno, K Shiraishi, T. Ogawa: Phys. Rev. Lett. 69, 2400 (1992). 55. T. Ogawa, T. Takagahara: Phys. Rev. B 44, 8138 (1991); Surf. Sci. 263, 506 (1992). 56. D.J. Chadi: J. Vac. Sci. Techno!. A 5, 834 (1987). 57. H.H. Farrell, C.J. Palmstrom: J. Vac. Sci. Techno!. B 8, 903 (1990). 58. M.D. Pashley, KW. Haberern, W. Friday, J.M. Woodall, P.D. Kirchner: Phys.

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Lett. 74, 3177 (1995). 65. T. Ohno: Surf. Sci. 357-358, 265 (1996). 66. D.P. Adams, S.M. Yalisove, D.J. Eaglesharn: App!. Phys. Lett. 63, 3571 (1993). 67. M. CopeI, RM. Tromp: Phys. Rev. Lett. 72, 1236 (1994). 68. A. Sakai, T. Tatsumi: App!. Phys. Lett. 64, 52 (1994). 69. J.E. Vasek, Z. Zhang, C.T. Salling, M.G. Lagally: Phys. Rev. B 51, 17207 (1995). 70. T. Ogitsu, T. Miyazaki, M. Fujita, M. Okazaki: Phys. Rev. Lett. 75, 4226 (1995). 71. J. Nara, T. Sasaki, T. Ohno: Phys. Rev. Lett. 79, 4421 (1997). 72. J.J. Boland: Phys. Rev. Lett. 65, 3325 (1990). 73. K Sinniah, M.G. Sherman, L.B. Lewis, W.H. Weinberg, J.T. Yates, Jr.,

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Design of Ni-Base Superalloys

Hiroshi Harada and Hideyuki Murakami

3rd Research Group (Advanced High Temperature Materials), National Research Institute for Metals

1-2-1 Sengen, Tsukuba-shi, Ibaraki 305, Japan.

Abstract. Computer models for designing Ni-base superalloys have been developed. A mathematical model using regression equations based on microstructure and property databases has been established and successfully applied to alloy developments. Several types of Ni-base superalloys including new generation single crystal (SC) superalloys have been designed using the model. Evaluation tests have shown their superior high temperature properties.

A theoretical modeling of equilibrium states in multi-component Nibase superalloys has also become possible by employing statistical thermodynamics with using interatomic potentials, namely, cluster variation method (CVM). The "( and "(' phase compositions, site occupations of alloying elements in the phases, lattice parameters, and so on were calculated self-consistently and successfully verified using atom-probe field ion microscopy (APFIM). The same set of interatomic potentials are used in Monte Carlo simulation (MCS) of microstructural evolution in multi-component Ni-base superalloys.

An open laboratory for materials design (OLMD), which enables one to use our computer models and databases within NRIM for superalloys, is now open to the public on our world wide web (WWW) site.

2.1 Introduction

2.1.1 Ni-base Superalloy

The temperature capability of Ni-base superalloys has been improved year by year, as a means to improving thermal efficiencies in aeroengines and power stations, as shown in Fig. 1. As the new process is introduced, i.e. casting processes for conventionally cast (CC), directionally solidified (DS), and then single crystal (SC) blades, the chemical compositions were optimized for each process to realize the superior properties than before. Thus the temperature capability has been improved by 350°C in the last 50 years.

Figure 2 shows a model SC hollow blade for a hypersonic jet engine. Turbine blades are normally air cooled inside so as to be used in hot ('" 1500°C) gas. Even so the met al temperature increases up to 1000°C or higher. Also, the rotation speed is as high as the speed of sound at the blade tip, causing a large centrifugal stress in the blade. Thus the creep strength is the most important property required for the turbine blade materials, as well as other mechanical

Springer Series in Materials Science Volume 34. Ed. by T. Saito © Springer-Ver1ag Berlin Heidelberg J 999

40 Design of Ni-Base Superalloys

.U 1200 r-------r-------~------~------~------~----~ -"' Cl. S.C

~ r- 1100 ........... ~ ... " ............. ~ . ..... " ......... \ .. , , , · . C'?

, ,

-"' GI ~ C.C 1000 ..... ... .. ~ ..... " ...... " ~ · . C. GI

· . : ~: ___ ...",.,-;:;;;;;:-iR;;~

GI ... U .I: CI

900 CI CI ... ... 0 -GI

0 Wrought A OS(NRIM)

800 0 CC <> sc _ .. -- ..... :::J -"' ... • CC(NRIM) • SC(NRIM)

Ll. OS GI C.

E GI ~

700 1940 1950 1960 1970 1980 1990 2000

Year

Fig. 1. Improvement in temperature capability of Ni-base superalloys.

Fig. 2. A model SC hollow blade made of alloy TMS-75 designed at NRIM.

Hiroshi Harada and Hideyuki Murakami 41

Fig. 3. Typical ideal microstructure in Ni-base superalloys.

properties, and the environment al properties including oxidation and hot corrosion resistance.

Figure 3 shows the typical ideal microstructure of an SC superalloy observed by transmission electron microscopy (TEM). The matrix is the "( phase which is an fcc solid solution based on Ni. The cuboidal precipitates are the "(' phase which is an L1 2 ordered structure based on Ni3Al. As both phases have fcc based structures and have an intentional smalliattice misfit (-0.4 % at 1050°C, a .. y' < a,), a beautiful alignment of coherent "(' precipitates can be achieved through solution and ageing heat treatments.

Ni-base superalloys, however, contain many substitutional alloying elements apart from Al, including Co, Cr, Mo, W, Ti, Nb, Ta, Hf, Re and so on. Depending on the combination and balance of these major alloying additions, the microstructural parameters such as combinations and fractions of "( and "(' phases, the absolute value and sign of lattice misfit, and so on, can change widely, resulting in drastic changes in creep strength.

Figure 4 depicts experimental data showing the effect of "(' phase fraction on the creep rupture life [1,2]. This shows the peak creep rupture strength at around 62.5 - 75% in "(' phase fraction. Having optimum microstructures similar to that in Fig. 3, these alloys are far stronger than the component "( and "(' single phase alloys, due to the interphase interface preventing dislocation motion at high temperatures. This is the nature of so-called precipitation hardening in Ni-base superalloys.

2.1.2 Historical Background of Alloy Design

As Ni-base superalloys contain high amounts of alloying elements, serious problems often happened with phase stability. The so-called topologically-close-

42

1000

500

'L

Il

'I. 100

50 u

'I.

l

l

/ , ,

• o


-0- I 8000C d45MPa e--tr- 1000 oe 117MPa -.. - 1000 oe 98MPa

..n-

/.,.

"'" v/ '",,\, ,.---", '

/ ~ r\. ~ ,- '\.\ ü , , (")

, ;:::. \' ~ Q) , c , , 0

u c

f 25 50 75 100

Designed amount of i (mol%)

Fig. 4. Experimental data showing the effect of "(' phase fraction on creep rupture strength. Aseries of alloys on a "( / "(' tie-line was designed and tested.

packed phases (TCP phases: (T, p, Laves, etc.) precipitated with plate or rodlike shape, resulting in a significant degradation in mechanical properties during high temperature services. The Phacomp (phase computation) was developed as a technique to predict this TCP phase formation. The concept of "electron vacancy number", which was defined by Pauling [3] for transition metals, was extended to all the alloying elements used in superalloys. The me an electron vacancy number N v is then calculated by (1), where h is a mole fraction of the i-th element in the "( phase and Nv(i) is the electron vacancy number assigned to the i-th alloying element,

(1)

It was found that for a"( phase with an N v value exceeding a criterion, e.g. 2.5, the TCP phase tended to precipitate [4]. Although there were exceptions, this empirical approach was used as Phacomp when designing new alloys.

To apply Phacomp to practical alloys which normally contain the "(' phase and also some minor phases, including carbides and barides, it was necessary to predict the "( phase composition from the alloy composition. A simple assumption that all the so-called "(' phase formers, Ti, Nb, Ta, etc., are solutioned in the "(' phase and so on, far instance, was employed to calculate the fraction and composition of the "(' phase, and then the "( phase. This whole procedure was called Phacomp. After some modifications including a sophisticated one by Barrows and Newkirk [5] had been conducted, a major revision was proposed by


Morinaga et al. [6] using the Md-electron level instead of the Nu. They reported that the mean Md value for the alloy composition can be used to predict TCP phase formation more accurately than Nu. Using Md together with a bond order, Bo, as guidelines, SC superalloys are being developed by Murata et al. [7].

The prediction of TCP formation is an important aspect to reject alloys with undesirable microstructures, which have inferior high temperature properties. However, in order to design superalloys with superior high temperature properties, alloy designers should control the ,,/ phase fraction, lattice misfit and so on by selecting the optimum combinations and balances of alloying additions. Thus, it is rather more important to predict such fundamental microstructural parameters from the alloy composition.

A mathematical approach was first tried by Dreshfield [8] for predicting the "! /"!' equilibrium. Using "! and "!' phase compositions in experimental alloys, "! / h+"!') and h+"!') h' phase boundaries in the multi-component phase diagram and the tie-lines connecting the boundaries were expressed as a set of regression equations. It was demonstrated with the equations that equilibrium "! and "!' compositions were successfully calculated from the alloy composition. Mongeau and Wall ace [9] proposed another approach by solving the simultaneous equations of the partitioning coefficients of all the alloying elements to obtain "! and "!' fractions and compositions. Motivated by these series of mathematical modeling studies, a model was established on our microstructure and property database, so that high temperature properties as well as the equilibrium phase compositions can be predicted from the alloy composition [10-12]. This model has been used successfully for the design of several types of Ni-base superalloys for advanced gas turbines and jet engines. The details are described in Sect. 2.2.

A theoretical approach employing thermodynamics was initiated by Kaufmann and Nesor [13]. A prediction of the phase diagram was successfully made although the "!' phase was treated as a line compound in this work. It was only recently that the thermodynamics database for the Thermo-Calc program was established by Saunders [14]. Using this database, more sophisticated phase equilibrium calculation among ,,!, "!', liquidus, carbides, borides and even TCP phases including 0', fl, and Laves phases, has become possible for Ni-Co-Cr-MoW-AI-Ti-Nb-Ta-Hf-C-B-Zr multi-component system. An example is shown in Fig.5.

A theoretical calculation based on statistical thermodynamics using interatomic potentials, namely, the cluster variation method (CVM), has become possible. The principle methodology of CVM was established by Kikuchi [15] and successfully applied to the Ni-Al binary system by Sanchez et al. [16]. This method was extended to Ni-AI-X ternary systems by Enomoto and Harada [17] and then to the multi-component system by Enomoto et al. [18]. The CVM calculation describes not only the equilibrium compositions of the "! and "!' phases, but also equilibrium atomic arrangements, lattice parameters and lattice misfit, all self-consistently. As the lattice misfit is one of the key parameters when designing superalloys, the CVM has a great advantage in this aspect. The same interatomic potentials are also used for the MCS of microstructural evolution

44

100

80

~ 60 ~ ..c c.. ~

~ 40 ~

20

o 600

Borides Carbides


1200 Temperature (OC)

Liq

1500

Fig. 5. Thermodynamical calculation for multi-component equilibrium for a commercial superalloy U720.

in multi-component Ni-base superalloys by Saito [19]. The details of CVM and MCS are explained in Seet. 2.3 with the results of experimental verifications.

2.2 Mathematical Models Based on Database

2.2.1 Phase Calculation

An empirical approach using the database and regression analysis has been established [12] and used praetically for alloy development [20,21].

The l'+y' region in multi-component nickel-base superalloys is presented schematically in Fig. 6. The 1" hypersurface in this figure, on which the composition of 1" in equilibrium with I' should lie, is expressed by a regression equation (2), where XI is a concentration of the v-th element in the 1" phase and ai is the coefficient,

Const. = Lai· X;. (2)


Ni

Co, Cr, Mo, W, Re Ti, Nb, Ta, Hf

Ni AI 3

45

Fig. 6. Schematically presented /,+/,' region in a multi-component phase diagram.

MCC=0.96

20

O~--------~--------~----~ o 10 20

AI (at%) calc.

Fig. 7. Agreement between calculated and analyzed Al concentrations in the /" phase.


For this regression analysis, " compositions analyzed in 30 different alloys heated at 900°C for 1500 h were used together with a,' composition from Ni-Al binary phase diagram at 900°C. The multiple correlation coefficient (MCC) of this equation was 0.96. Figure 7 shows an excellent agreement between calculated and analyzed Al concentrations in the " phase. The agreement holds in a wide range, from 6.5 to 23.4 at.%, covering from the Ni-Al binary alloys to highly solid solutioned multi-component commercial alloys. In order to calculate the composition of the , phase which is equilibrated to the " phase on the " hypersurface, a partitioning coefficient is defined as,

(3)

where K i is the partitioning coefficient of the i-th element and Xi is the concentration of the i-th element in the , phase. The partitioning coefficient for each element is derived by regression analysis. The 31 pairs of the , and " compositions were again used for regression analysis and the K i was expressed as a function of the " composition, XI , as

K i = Const. + L bij . Xj. (4)

4 MCC=O.98

3

ctS c ctS

0 2 '::.(:.ü

2 3 4

K calc. Co

Fig.8. Agreement between analyzed and calculated partitioning coefficients, K;, for Co.


Table 1. Calculated "(' (top) and "( (bottom) compositions in comparison with experimentally determined compositions.

Composition (at.%) F Alloy Phase

Co Cr Mo W Al Ti Nb Ta Hf (at.%)

Calc. "(' 3.8 3.1 0.4 0.5 13.0 7.7 0.9 0.9 -

Calc."( 11.4 26.8 1.4 1.0 3.4 1.3 0.3 0.2 41

-IN738LC

Anal. "(' 3.8 3.3 0.3 0.5 12.9 7.4 0.9 0.9 -42

Calc."( 11.7 27.2 1.5 0.9 3.2 1.4 0.3 0.2 -

Calc. "(' 5.5 2.7 1.4 - 18.1 1.3 - 1.8 0.6

Calc."( 14.9 15.9 5.6 7.0 0.2 0.4 0.1 54

- -B1900Hf

Anal. "(' 5.4 3.4 2.2 - 19.7 1.2 - 1.8 0.6

Anal. "( 14.8 14.8 4.7 5.5 0.3 0.4 0.1 52

- -Calc. "(' 6.8 3.2 0.2 2.9 16.6 1.6 - 1.1 0.6

Calc."( 16.8 19.7 0.6 3.2 5.2 0.3 0.3 0.1 64

-MM247DS

Anal. "(' 6.4 3.3 0.2 2.8 17.3 1.7 - 1.2 0.6

Anal. "( 16.2 17.5 0.5 3.2 5.5 O. 0.3 0.1 59

-

Calc. "(' 3.1 3.2 - 0.9 15.8 2.7 - 5.6 -

Calc. "( 8.8 26.5 2.1 3.3 0.5 1.3 64

- - -Alloy454

Anal. "(' 2.9 3.1 0.8 16.8 2.7 5.3 - - -

Anal. "( 8.2 23.6 2.0 3.3 0.7 1.4 59

- - -

Calc. "(' 5.1 2.9 - 2.0 13.0 7.7 - 1.1 -

Calc."( 14.5 24.3 3.7 3.3 1.4 0.3 67

- - -TM-53

Anal. "(' 5.6 3.5 - 1.6 11.5 7.6 - 1.2 -

Anal. "( 13.6 26.5 4.2 3.1 1.5 0.2 66

- - -

Calc. "(' 5.2 2.3 - 4.7 17.3 - - 2.5 -

Calc."( 13.2 14.6 7.7 3.6 0.6 64

- - - -TMS-l I

Anal. "( 5.5 2.6 - 5.0 16.2 - - 2.4 -

Anal. "( 13.6 15.5 7.4 3.9 0.6 68

- - - -

Figure 8 shows an excellent agreement between analyzed and calculated partitioning coefficients, K i , für Co. For Cr, W, and Mo, good regression equations were obtained. Für Ti, Nb, Ta and Re, there was no clear composition dependence of the partitioning coefficient. For these elements, constant values were used as partitioning coefficients.

By using the equations above, the "( and "(' equilibrium fractions and compositions are calculated by a successive iteration. The result of the calculation is shown in Table 1 with the results of electron probe microanalysis (EPMA). There is an excellent agreement between the calculated and measured values.


100r-----~-----,------~----~----~

~80 <ft j (;

<ft Ti "C ~ ::J 1{l4 ~ U500(760V. U700(982)

InconeI700(870) . /

WASPALi( Rene41 (760) 20 (760VO AF1753(900)

X75(704j" TMS-1 (1240) Rene95(1120)

20 60 Calculated (at%)

80 100

Fig. 9. Agreement between calculated and measured "(' fractions. Temperatures are given in parentheses.

The partitioning coefficients are dependent on temperature. Also, the " surface in the phase diagram moves as the temperature changes. These temperature dependence terms were taken into account in the equations. Figure 9 shows a good agreement between calculated and measured " fractions for a wide range of chemical compositions at different temperatures. The temperature dependence of the " fraction is estimated for several commercial superalloys and results are shown in Fig. 10. The change of equilibrium " fraction at up to the solvus trmperature is weil presented for several typical superalloys.

When C and/or B atoms are added in the alloy, it is necessary to calculate fractions and compositions of carbides (MC, M6 C, M23 C6 ) and borides (M3B2 ,

M5B3 ) under assumptions similar to those in the Phacomp procedure. After deducting the elements forming these minor phases, the , and " equilibrium phase calculation described above is performed.

2.2.2 Property Calculation

On the basis of the phase calculation, structural parameters and properties are calculated. Table 2 shows such parameters and properties that can be predicted. Most of them are predicted by regression equations using the database.

For the creep strength of SC superalloys, as an example, creep rupture data of 34 different SC alloys were used for the regression analysis. The creep rupture


80

==----70 ~-</l.l.Oy' Sf:lll "s" 041. '9B

60 ~ S-\'2

;:? ~f:lf:l<. ~50 ~ VOo ~ f:lsll c ~u>; •• g40 (.)

q,'>,j! cu ......

~~~ -- 30 0'0.-?-

20 (/6:

va 10

0 700 800 900 1000 1100 1200 1300 1400

Temperature (0C)

Fig. 10. The temperature dependence of "(' fraction estimated by the equations.

49

Table 2. Structural parameters and high temperature properties predicted in the program on the basis of the phase calculation.

Structural parameters

Fractions and compositions of the phases

· "( phase, "(' phase

· Carbides (MC, M23 C6, M6C)

· Borides (M3B2, M5 B3)

Phase stability

Properties

Conventionally cast alloy

· Creep rupture life (at 1000°C, 117 MPa)

· Tensile property (at 900°C)

Single crystal alloy

Lattice parameter of"( phase and "(' phase· Creep rupture life (at 1040°C,

(at room temperature) 137 MPa and at 850°C, 400 MPa)

Alloy density Powder metallurgy or forged alloy

Liquidus temperature . Tensile property (at 760°C)

Solidus temperature

Incipient melting temperature

Perfect solution treatment window

Common to all the structures

· Hot corrosion resistance


Table 3. Regression coefficients of the creep rupture life at 1040 °C and at 137 MPa, expressed as a function of, composition, " volume fraction and ,/,' lattice misfit.

- 0.059Xbo - 0.161Xbr + 0.027X~o + 0.254X~ log(T) = 3.00f----------------------

+ 0.087 X~i + 0.298Xf.rb + 0.420X~b - 0.012 F - 3.098 15

0.1 10 t-test*

10 0.1 1

F-test* 0.5

MCC 0.92

" composition X; (at.%), " fraction F (at.%), and ,h' lattice misfit 15 (%).

*Probabilities (%) of error are sm aller than the shown values.

life at 1040°C and at 137 MPa is expressed by the analysis as a function of "I' composition, "I' volume fraction and "1/"1' lattice misfit. The coefficients and the constant value of the equation are shown in Table 3. It is remarked in the table that the effect of lattice parameter, as weH as other parameters, is very large. The coefficient of the lattice misfit suggests that the rupture li fe becomes longer by a factor of 2 for each 0.1% negative lattice misfit change (a,' < a,). This suggests the effect of the large negative lattice misfit on the formation of a rafted structure which increased the creep strength.

Table 4. Regression coefficients for hot corrosion resistance, for a crucible test and a burner rig test, as functions of alloy composition (at %).

= - 1.13 - 0.41 Hf - 0.13 Cr +0.01 Ti +0.01 Nb log[tML]

+ 0.02 Co + 0.11 Al + 0.34 W + 0.44 Mo + 0.67 Ta

10 1 - -t-test*

- - 0.1 1 0.1

F-test 0.5

MCC 0.91

= 1.38 - 1.42 Hf - 0.16 Ti - 0.03 Cr - 0.01 Co log[tp ]

- 0.01 Al + 0.04 W + 0.09 Ta + 0.14 Mo + 0.24Nb

- 10 - -t-test*

- - - - -

F-test 1

MCC 0.95

tML: Metalloss (mm) by 20 h crucible test.

t p : Penetration depth [ratio to IN 738 (=1)] in a burner rig test.


Composions and fractions

of yand y , phases

by successive iteration using eqs, of y , surface

and partiti ing ratios

AI/oy composition with results of

phase and property calculation

Input

Phase

"'IO" Property calculation

Properties

Selection and

sorting

Output

YES Try again?

NO

Fig. 11. The flow chart of our alloy design computer program.

51

For predicting hot corrosion resistance, as another example, a so-called crucible test was conducted at NRIM and regression analysis was carried out using data from 42 alloys. The result is shown in Table 4 with another regression equation derived from burner rig data conducted elsewhere. The equations indicate that Hf and Cr (and Ti) improve the hot corrosion resistance, while the addition of W, Ta, Mo, which are effective in increasing creep strength, are extremely detrimental for hot corrosion.

All the prediction equations for the phase equilibrium and properties were combined each other to establish our alloy design program (ADP).

2.2.3 Alloy Design and Development

Aseries of Ni-base superalloys were designed using our mathematical model established on a database using regression analysis [20-23]. Figure 11 shows the flow chart of the alloy design computer program. The program is composed of two sub-programs, namely, the analyzing program and the searching program. The analyzing program is used to calculate microstructural parameters and high temperature properties from given alloy chemical compositions. The searching program is used to find chemical compositions of alloys with the most desirable


Table 5. The design condition far alloys with very high creep rupture strengtlls as weIl as comparatively low alloy densities.

Condition 1 2 3 4

Cr content 2: 7 at.% 2: 7 at.% 2: 7 at% 2: 7at%

"(' content (900°C) 60 at.% 60 at.% 60 at.% 60 at.%

H. T. Window 2: 40°C 2: 40°C 2: 40°C 2: 40°C

SI (900°C) ::;1.1 ::; 0.9 ::;1.1 ::;1.1

Lattice misfit (RT) 2: -0.2% 2: -0.2% 2: 0% 2: -0.2%

Alloy density (RT) ::; 8.1 ::; 8.1 ::; 8.6 ::; 8.6

properties among aIl the possible combinations and balances of the aIloying additions in the multi-component system.

An example of the aIloy design using ADP is as foIlows [20]. Four aIloys having very high creep rupture strengths, as weIl as comparatively low aIloy densities, were designed by the searching program. The design condition for each aIloy is set as Table 5. Here, SI (solution index) is a criterion for the phase stability, which we use instead of N v . The lattice misfit is the value at room temperature. At high temperatures, such as 1040°C, the misfit normaIly shifts negatively by 0.2~0.3%. Thus, the design condition aIlows a smaIl negative lattice misfit.

The strongest aIloy was then searched for in the Ni-Co-Cr-Mo-W-Al-Ti-NbTa-Hf system by the calculation equation for the creep rupture life at 1040°C and 137 MPa.

Table 6. The chemical composition of the best alloys selected by the program, with some structural parameters and creep rupture life predicted by the program.

Composition (wt %), baI. Ni. Condition Alloy

Co Cr Mo W Al Ti Nb Ta Hf

1 TMS-61 0 11.4 5.9 0 5.02.1 1.90.9 0

2 TMS-62 0 7.9 6.9 0 5.8 0.9 2.4 0 0

3 TMS-63 0 6.9 7.5 0 5.8 0 0 8.4 0

4 TMS-64 0 6.5 8.4 1.05.8 0 o 6.7 0

(continued)

"(' content H.T. Lattice Density Creep rupture

window SI misfit life at 1040°C

(at.%) (OC) (%) (gcm- 3 ) -137 MPa (h)

60 53 1.10 -0.20 8.09 1755

60 78 0.90 -0.18 8.10 1646

60 42 1.10 0.01 8.48 3587

60 57 1.10 -0.17 8.48 7078

Hiroshi Harada and Hideyuki M urakami

:2 -co c

10000

~ 1000 'C Q) a. x

UJ

100 100

1040°C 137Mpa

.. Aj2

Ä

1000

Calculated (h)

53

• PWA .. CMSX A TUT 0 Hitachi

• TMS

10000

Fig. 12. The relationship between calculated and experimental creep rupture lives, showing a fairly good agreement.

The chemical composition of the best alloys selected are shown in Table 6 with some structural parameters and the creep rupture life also predicted by the program. Experimental evaluation showed that the alloys thus designed, cast and heat treated had ideal microstructures with very weIl aligned cuboidal " phase of 0.5 mm in size.

Figure 12 shows the agreement between calculated and experimental creep rupture lives, showing a fairly good agreement. The designed alloys are thus found to be light er and stronger than the existing SC superaIloys, as shown in Fig. 13.

Using the program, the strongest alloy was searched for in the whole , and ,'two phase region in the Ni-Co-Cr-Mo-W-Al-Ti-Nb-Ta-Hf-Re multi-component system. The result of the calculation is summarized in Fig. 14. It is predicted here that the temperature capability is improved by changing the lattice misfit towards greater negativity, although there is a limitation in practice. If the lattice misfit is too large, the coherency is easily broken down during heat treatment. Also, a discontinuous coarsening occurs to reduce the strength. Taking these experimental results into consideration, the maximum negative lattice misfit should be set at around -0.2% (at room temperature ). Thus the temperature capability of the strongest alloy with 5% Cr is 1100°C which is higher than that of PWA1484 by 50°C. Reducing Cr below 5% might also be worth considering, since the calculation predicts the temperature capability as high as 1130°C with no Cr containing alloy, which is 80°C higher than the PWA1484.

54

6' 20 E (.)

El tU a.. 6

a. 10 -.t= 0 0 0

p 0 ... 0

b

0


SC83 _---TMS-63. __ - - _.0 - -tMS-6

TMS-62. _----·0 COPWA1484 __ <:J - - - - TUT92 ° CMSX-4

.0 _ - - - - - - - - - AM3 °ALLOY454 - RR2000 °SRR99

8.0

° IN738CC

°MM247DS

° MM247CC

8.5 9.0

P (gcm 3)

Fig. 13. Specific creep strengths and densities of designed alloys, with those of existing alloys. The designed alloys are lighter and stronger than existing SC superalloys.

The same alloy design methodology has been applied to design conventionally cast (CC), directionally solidified (DS), oxide dispersion hardening (ODS) and powder metallurgy (PM) processed superalloys [22]. The typical alloy compositions and experimental creep rupture temperature capabilities are shown in Table 7, compared with those of reference commercial alloys. The alloys designed show higher temperature capabilities than commercial ones.

The alloy design program described above is now used not only in NRIM but also in other institut ions including a major jet engine manufacturer for their alloy development.

2.3 Theoretical Models Based on Statistical Thermodynamics

2.3.1 Cluster Variation Method

The mathematical approach using regression equations and databases has been very effective in practical alloy development work. In order to improve the accuracy of our alloy design program and thus to develop alloys with furt her superior properties, however, a more theoretical approach was thought to be needed. For this purpose, modeling of Ni-base superalloys on an atomic scale using the cluster variation method (CVM) was conducted with some experimental work for verification of the calculation.

Hiroshi Harada and Hideyuki M urakami 55

Table 7. The typical alloy compositions and experimental creep rupture temperature capabilities, with those of reference commercial alloys. CC, DS and ODS alloys contain small amounts of C, Band Zr for grain boundary strengthening. ODS alloys contain Y203 dispersoid.

Alloy Co Cr Mo W Al Ti Nb Ta Hf Re Temperature

capability (0C)

CC MarM247 10 8 0.6 10 5.5 1 - 3 1.5 - 957

TM-321 8.2 8.1 - 12.6 5 0.8 - 4.7 0.9 - 973

DS CM186LC 9 6 0.5 8 5.7 0.7 - 3 1.4 3 1008

TMD-103 12 3 2 6 6 - - 6 0.1 5 1038

SC CMSX-4 9 6.5 0.6 6 5.6 1 - 6.5 0.1 3 1035

CMSX-10 3 2 0.4 5 5.70.20.1 8 0.03 6 1040

TMS-75 12 3 2 6 6 - - 6 0.1 5 1065

TMS-82 7.8 5 3.4 8.7 5.2 0.5 - 4.4 0.1 2.4 1075

ODS MA6000 - 15.2 2 4 4.6 2.5 - 2 - - 1068

TMO-20 8.7 4.3 1.5 11.6 5.6 1.1 - 6 - - >1170

Lattice m islit at 1050 °C

1200 - +

137 MPa, 1000 h

Creep rupture

1150 '" O%Cr

Ü ~

~ 15 <tI a. <tI 1100 0

~ ~ ~ Q) a. E Q) 1050 t-

~ ~ Predi lion

"" ~ ?... '"' ~ TMS-64 TMS-

~S-71 ~ TMS-75 TMS-26

TMS-63 ~ 0 0

n Hen ,NB

IpWA148 0 0 CMSX- CMSX- 0

CMSX-4

Rene N5 0 PWA14~0

0 ~ 1000

-0.6 -OA -0.2 o 0.2 OA 0.6

Lattice misfil at room temperature (%)

Fig. 14. Effect of lattice misfit on the temperature capability of SC superalloys, showing that negative lattice misfit improves the creep strength.

56 Design of Ni-Base 811peralloys

Computer Modeling. The equilibrium between "( and "(' phases, including site occupancy of alloying elements in the phases, lattice parameters and so on, was calculated using CVM [17,18, 24J. The principle of this method is as follows. The configurational entropy is calculated by using the tetrahedron nearest neighbor atom cluster shown in Fig. 15. The entropy per lattice point of the "(' phase is written as

(5)

where z is the probability of finding atoms, i, j, k and l, in a tetrahedron nearest neighbor atom cluster. The summation is taken with all component atom species. The Al and Ni sublattices are denoted by superseripts A and B, respectively. Then, y is the probability of finding atoms i and j in a nearest neighbor atom pair between the two sublattices (A-B) or in the Ni sublattices (B-B); x is the probability of finding an atom i at a lattice point belonging to the Al sublattice (A) or Ni sublattice (B); k is the Boltzmann constant, and L(() = (In ( - (.

Q:AI

o :Ni

Fig. 15. Tetrahedron nearest neighbor atom cluster for calculating the configurational entropy.

The enthalpy is calculated by using the Lennard-Jones pair potential, e(r), as schematically shown in Fig. 16 and the potential parameters listed in Table 8;


>-~ Q) c: Q)

cti :;:J c: Q) -o c..

Interatomic distance

Fig. 16. The Lennard-Jones pair potential, e(r).

only the nearest neighbor pair interaction is taken into aeeount, namely:

H"I' = ~ 2:: eij (r) (yDB + y~B) , "J

57

(6)

where z = 12 is the nearest neighbor co ordination number in the fee lattice. The atom configuration whieh minimizes the grand potential:

[2"1' = H"I' - TB"I' + PV"I' + ~ L /-Li (xt + 3xn

is sought under the restrictive condition:

LzD~IBB = l. ijkl

(7)

(8)

Here, T, P, V"I' and /-Li are temperature, pressure, atomic volume and chemieal potential of the i-th eomponent, respectively. All the eorresponding equations for the , phase are obtained by removing the superseripts A and B from the above equations. The equilibrium state between the , and " phases ean be obtained by equalizing the grand potentials thus calculated for the phases, under the same value of P, T, and /-L. The lattiee parameter of the phase can be calculated by finding the mean interatomie distanee whieh gives the minimum enthalpy of the phase.


Table 8. Normalized Lennard-Jones potential parameters far CVM and MCS calculations.

Ni Al Ti Cr Co Mo Ta W Re e~j / el]yi-Ni

1 1 1.384 0.982 1.002 1.275 1.543 1.56 1.42 Ni

0.7660.909 0.76 0.91 1.035 1.262 1.238 1.238 Al

1.095 1.089 1.248 1.336 1.095 1.549 1.549 Ti

Ni 1 0.902 0.968 1.15 1.353 1.659 1.359 Cr

Al 1.0531.149 0.9981.365 1.6 1.509 1.409 Co

Ti 1.0351.1221.174 1.3561.7871.791 1.791 Mo

Cr 1.018 1.073 1.08 1.032 1. 797 1.901 1.901 Ta

Co 1.007 1.009 1.062 1.021 1.181 1.96 1.96 W

Mo 1.028 1.147 1.156 1.07 1.157 1.095 1.804 Re

Ta 1.049 1.165 1.186 1.073 1.187 1.137 1.18

W 1.029 1.15 1.151 1.041 1.155 1.126 1.155 1.132

Re 1.027 1.133 1.151 1.038 1.025 1.05 1.104 1.05 1.104

Tij jTNi-Ni Ni Al Ti Cr Co Mo Ta W Re

An example of the calculation output for a multi-component single crystal nickel-base superalloy TMS-63 is shown in Fig. 17. In this figure, the alloy name, temperature, pressure and bulk chemical composition were input, and the rest is the output of the calculation; CP and C are the "t' and "( chemical compositions, respectively. CP / C is the partitioning ratio of the alloying elements. The chemical compositions of the Al site and Ni site are shown with partitioning ratios to the Ni site; for instance, 53% of Cr atoms in "(' is predicted to be located at the Ni site, whereas only 7% of Mo atoms is located at the Ni site. The "(' mole fraction is ab out 67%. The lattice misfit is predicted to be slightly negative at high temperature, e.g. 1079°C.

ALlDY:'IMS-63 (PR:GRAMME=m:xi.=t DATA=xxmef. test) TEMPERA'lURE: 1079 . ODar PRESSURE: 1. OA'lM ELEMENr BULK GP G GP/G AL-SI'IE NI-SlTE 3 N/(A+3N) FRAGP

NI AT% 72.00 73.19 69.82******* 4.65 96.04 98.41 64.7 AL AT% 12.80 15.70 6.71 2.34 61.75 0.35 1.66 67.7 CR AT% 7.80 4.64 14.38 0.32 8.65 3.30 53.36 67.5 MO AT% 4.60 2.97 7.90 0.38 10.97 0.31 7.73 67.0 TA AT% 2.80 3.50 1.18 2.96 13.98 0.01 0.25 69.7

IATTlCE PARAMETER IKJI': 3.6264 3.6393 MISFIT( %): -0.353 IATTlCE PARAMETER Rr : 3.5768 3.5779 MISFIT(%): -0.032

Fig.17. Example of the output from CVM calculation for multi-component single crystal nickel-base superalloy, TMS-63.


Fig. 18. FIM image showing atomic layers on low index planes as coaxial rings. The dark line across the center was confirmed as the interphase interface.

Experimental Verification. The equilibrium state of single crystal superalloys was analyzed using APFIM and compared with the calculation [25,26]. Very sharp needle-like sampies were prepared by electropolishing of thin rods cut from the alloy sampies heated at 1040°C for 2650 hand quenched in water. The sampie tip temperature was kept below 40 K during the analysis.

Figure 18 shows the FIM image. Each spot corresponds to an atom. The atomic layers on low index planes are c1early visible as coaxial rings. The dark line across the center of this figure was confirmed, using chemical analysis, to be the "( lY' interface. The observed continuity of {200} planes across the phase interface is consistent with the coherency between the "( and "(' phases.

The results of chemical analysis of the phases are compared with the estimation by CVM in Fig. 19. There is a very good agreement among the calculations (CVM and ADP) and the experimental results (APFIM).

The site occupancy of the solute elements is determined from the chemical analysis of the Ni+AI mixed layers and Ni-rich layers obtained by progressively stripping {200} planes of the "(' phase. The sequence of this analysis is shown in Fig. 20 for the alloy CMSX-4. The result is compared with the CVM calculation in Fig. 21. This figure demonstrates a very good agreement between the theoretical calculation and the experimental determination, except for the difference in Ni content in the Al site, which is suggesting the need for a slight modification

60

TMS-63 T hase ~~~~~~~==~

ADP

APFIM

o

o

Ni

20 40 60 80 Composilion (al%)

20 40 60 80 Composition (al%)

100 0

100 0


20 40 60 80 Composilion (al%)

100

20 40 60 80 100 Composilion(at%)

Fig.19. Agreement among the calculated (CVM and ADP) and the experimentally determined (APFIM) phase compositions.

of interatomic potential parameters in the Lennard- Jones pair potentials used in the calculation.

Figure 22 shows a comparison between the estimated and experimentally determined lattice misfit values; an excellent agreement is obtained in a wide range of misfit values.

CVM was thus found to be a very powerful tool for modeling the equilibrium atomic configuration in the superalloys. The CVM program is now being applied for designing next generation superalloys, e.g. Ir [27]. On the basis of the CVM calculation, estimations of microstructures and mechanical properties are expected to be made possible on the next stage.

2.3.2 Monte Carlo Simulation

By using the same interatomic potentials esta:blished in the CVM work, an atomic level Monte Carlo simulation (MCS) of the microstructural evolution in Ni-base superalloys was initiated for ternary systems by Saito [19], and for multi-component systems by Saito and Harada [28] and Murakami et a1. [30].

For MCS the initial structure may be generated by randomly assigning atoms to lattice sites. The kinetics of ordering may be controlled by the direct exchange of a randomly selected atom with one of its neighboring atoms (Kawasaki dy-


12

(/) 8 0 E o ..... ro 40

«

o

"'-:JM,-~""" AI Ni

" , -IT-- I 0 : CJ

: c:x:O PDl ~ c(JD: ,

0: I A I

1:.', : I . : -

I • ~ ~ • : . I I

50 1 00

Mixed Ni layer ' , , <:>

: ' CPoJ~~ .n-=~,,-;(!!) , oODl _0 Q ' , :

~ ~D :00 : :~ :b, ~ "aD(J: :61 1::

I:.~ , , ~ : . : .. ,.: .~ : .. g8~

150 200

<> Mo

'" W • Ti • Ta • Re

2 50

Ni atoms

61

Fig. 20. The sequence of the APFIM site occupation analysis of the solute elements in the " phase, obtained by progressively stripping {200} planes of the " phase.

O : AI

Q:Ni

AP~FM Cr.H ,R

Mo T Ni CO '

AI site Ti

AI

/

Ni site

Fig. 21. Agreement between the theoretically predicted and the experimentally determined site occupation in CMSX-4.


0.6

1050°C

0.4

01+AI

0.2 0 ~ / !?... ä1 CMSX-2' - 0.0 g20Y c Q)

E 0 211 .L: Q) / a. - 0.2 MM200/0221 ><

UJ o OCMSX-2 210,0 °210-Nb - 0.4 0

0.0 0.2 0.4 0.6

Calculated (%)

Fig.22. Comparison between the estimated and experimentally determined lattice misfit values, showing an excellent agreement in a wide range of misfit values.

namics [30]). The prob ability W that an exchange trial is accepted is given by:

W = exp (-i1H/kT) / [1 + exp (-i1H/kT)] , (9)

where i1H is the change in energy associated with the exchange of the atoms, k is the Boltzmann constant and T is the absolute temperature. i1H is calculated by taking account of the contributions from only first nearest neighbor atoms.

The simulations were performed on a 163 unit cell mesh (16 384 atoms) mainly for visualization, and on a 323 unit cell mesh (131072 atoms) mainly for determination of the "( and "(' phase compositions and for the investigation of the site occupancy behavior of alloying elements in the "(' phase. Periodic boundary conditions were employed. It should be noted that all the lattice positions in the system were fixed; i.e. no relaxation effects were considered in the calculation. The simulations were carried out until the atomic configurations were regarded as equilibrium (typically 20000 Monte Carlo steps); the total enthalpy change of the system may be negligibly small after 20000 Monte Carlo steps.

Figure 23 shows the arrangement of atoms in TMS-71 on the 22nd to the 25th layers, normal to the <100> direction. The atom species in the figure are identified by their size and brightness. Since pure Ni layers and Ni-Al mixed layers must appear in turn along <100> for pure fully-ordered Ni3AI, from the result of MCS it is thus possible to identify the ordered "(' phase, as indicated by arrows in this figure.


22

24

• Ni - Al .er Co

. . . . . .. .. • • • . . . . .. . .. . . . . .. . . . . ... ....... . . .. . .. . . . .. . .. '

23

25

• Ho • Re

63

Fig. 23. The result of MCS showing the arrangement of atoms on the selected 22nd to the 25th layers, normal to the <100> direction. The area of ordered ""/ phase is indicated by arrows.

Although the , and " phases may be visualized by MCS, the determination of the , and " phase composition requires careful analysis based on the identification of neighboring atoms. For example, if the first neighbor atoms are mainly Ni and the second neighbor atoms are mainly Al, the selected atom is deemed to be in the Al site in the " lattice. Regions which are not assigned as


Re Ta Oamma pha •• Re Ta Gamma p<",me phase

MCS Nt NJ

AR'IM

CVM

ADP

40 60 60 40 60 80 100 CoqlOSIIJOn (al '!!t) Compositlon (at %)

Fig.24. The , and " compositions estimated by MCS for TMS-71, compared with CVM calculation and APFIM analysis. A good agreement is obtained, particularly with the " phase composition.

" regions are regarded as the , phase. The , and " compositions estimated by MCS for alloy TMS-71 is compared with CVM calculation and APFIM analysis in Fig. 24. An excellent agreement is obtained, particularly with the "t' phase composition.

The partitioning behavior of alloying elements, 'k', defined in Sect. 2.2.1 was examined. Figure 25 summarizes the partitioning behavior of alloying elements for alloys CMSX-4, MC2, TMS-63 and TMS-71 showing a good agreement, except for the partitioning behavior of Mo, obtained by MCS. It is also found that the partitioning tendencies predicted by MCS are not as pronounced as those obtained by other methods. Further composition analysis showed that Ti and Ta preferentially partition into the,' phase while Co, Cr, Wand Re have preference to partition into the , phase. CVM, ADP and APFIM analysis showed that Mo tends to partition into the , phase.

In order to determine the site occupation behavior, a parameter oS' was defined as:

(10)

where PiAl and Ch , are the occupancy probabilities for the Al site and concentration in the " phase of an alloying element 'i', respectively. Since the number of Ni sites is three times that of Al sites for Ni3AI, Si = 1, 2 and 4 correspond to the occupancy behavior of an alloying element 'i' for the cases in which atoms are randomly distributed in the two sites, an equal number of atoms enter the two lattices and all atoms substitute for the Al sites, respectively.

Figure 26 summarizes the site occupancy of alloying elements in CMSX-4, MC2, TMS-63 and TMS-71. Generally, good agreement is obtained except for some discrepancies between estimated and experimental data in the case of Co and Ta, and estimated site substitution behavior of W obtained by MCS and CVM. Site occupation determination thus revealed that, generally, Mo, Ta, W,


(I) 04SX~

"- r-=I---==-~~~ W

TI

I: .... Cr

T,

AI

ra

Cr

AI

GImma.,...,..pllue (c) T1oIS-63

~ ~------~~~====~ 0.1 1 10 Glrrmo ptime k Gamma ",,"oe

I

W

r. Mo

Co er Ti

AI

Ni

0.1

Re

r.

AI

Ni

65

(b) MeZ

10 GImma."..,. phi .. GammapIIeM

(d) TMS-71

.AIlP Cl CVM . ~

O MCS

10

Fig. 25. The partitioning behavior of alloying elements in alloys CMSX-4, MC2, TMS-63 and TMS-71, showing a good agreement, except for the partitioning behavior of Mo by MCS.

Re

W

T. ..., j ~

j

li

AI

N

,

.

ra

Mo

er

Al

N.

0,

.. .. _ ... + . ... ~ - I~~I + •.•.

t ,

ttllte AI"'l. S

(c) So l' occ._ for TNS-63

.. t. AI .. t.

(b) SilO CICCUIW'CY Ior MCZ

w .;. Ta ;

'.'

"'" i

j Co . ......

I ~~I Cr ...,. ~ TI

AI

NJ

NI .. te AI SIt. S

(d) Sit. occuopncy for TNS·71

R. • TI

No

jCo .1.. ... · .. ·· .... ·1 ~I · e, .!

AI

Ni

IJ. ~ S

NI SIlfl: Alsne

Fig. 26. The site occupancy of alloying elements in CMSX-4, MC2, TMS-63 and TMS-71, showing reasonable agreement.


Re and Ti tend to occupy the Al sites whilst Co has a preference for the Ni site. The site occupancy of Cr is composition dependent.

From the investigation of phase composition and site occupancy of alloying elements in the "(' phase, there are still some discrepancies (1) between experimental analyses and numerical simulations, such as the site occupancy of Cr in the "(' phase, and (2) between CVM and MCS, such as partitioning behavior of alloying elements (typically Mo) and the site occupancy of Re in the "(' phase for TMS-71 and the site occupancy of W in the "(' phase for all the other alloys.

The discrepancy (1) may be attributed to the determination of potential parameters for Cr. Further investigation of the experimental conditions should also be conducted and (2) is caused by the difficulty in distinguishing "( and "(' phases from each other in MCS. As mentioned before, the MCS were carried out using fixed lattice positions and identical pair potentials for both "( and "(' phases. In addition, the number of atoms used for the calculation may not be large enough to describe the atomic configurations of the two phases for multicomponent alloys.

Figure 27 shows an example of the two-dimensional atomic configurations on a (100) layer for TMS-71 obtained by MCS for a 323 unit cell system. Here, some Mo-Re cluster with 3-8 atom numbers are indicated by the arrows. Since the pair potential between Mo and Re is very binding, Mo-Re atom pairs easily form in TMS-71 during MCS. W atoms behave similarly to Re for all the other alloys; e.g. Mo-W clusters are formed during calculations. These clusters frequently appear in the vicinity of "( / "(' interfaces or even embedded in the "(' precipitates. Superalloys containing large amounts of solute elements tend to form TCP phases such as the JL phase which has very high concentration of W and Mo. For example, it has been reported for MC2 that the JL phase precipitates from the "( matrix, preferentially in the region where the W level is maximum and that the JL phase particles are embedded in the "(' phase, due to the depletion of the matrix of elements Cr, Co and Mo leads to the transformation "( -+ "(' + JL. The Mo-W clusters in MCS predictions may suggest the JL phase formation in this alloy.

2.4 Towards an Open Laboratory for Materials Design

Computer networking is now one of the most effective ways to help computational materials science. An open laboratory for materials design (0 LMD), which enables materials scientists and engineers to use the computer models and databases within NRIM for superalloys, is now partly open to the public on the world wide web (WWW) site (http://www.nrim.go.jp:8080/open/usr/harada/3gindex.html) [31]. Figure 28 shows the outline ofthe OLMD and Fig. 29 shows an example of the result obtained by the mathematical model, namely, alloy design program, which is open in the OLMD. This result shows that creep rupture life of CMSX-2 at 1040°C and 137 MPa is improved by increasing Co and/or Ta contents, although other properties such as phase stability should be examined at the same time. We expect that OLMD will enhance the international collab-


• NI - Al Co • HO eTa • Re

Fig.27. Example of the two-dimensional atomic configurations on a (200) layer for TMS-71 obtained by MCS for a 323 unit cell system, predicting Mo-Re cluster formations.

orations on superalloys and other high temperature materials on the common use of the materials design computer programs and related databases.

2.5 Conc1usions

A mathematical model using regression equations based on microstructure and property databases was established and applied to alloy developments. Several types of Ni-base superalloys including new generation single crystal (SC) superalloys have been designed using the model. Evaluation tests have shown their superior high temperature properties.

A theoretical model using the cluster variation method (CVM) with LennardJones pair potentials was established for modeling the equilibrium atomic configuration in multi-component superalloys. The same potentials were used for

68

RemOIC c::JC ::

SimuLato

Remote Multi-media database n:ferring

~ • I t EKperimental data

t- -


other research Institute

Remote computing and experiment

Fig. 28. The outline of our open laboratory for materials design on the WWW site.

dispersed alloy ••••••••••••••••••••••••••••••••••••••••••••••••••••••••• ••••••••••• ••••••••••••••••••••••• •••••••••••.•....•••••••••••••• J

1 ;;~;·~1;;·!'~l .. ~~.:~.:~~.:~.I~~ .. ~_~~~~::~~.~.~: ............... .1

;.~.::.~;::~.~.~.=~.~= .. ~~~~:~.~.~~~~.~:~~_ . .Jy. ... ~!~f!.~.~JJL.~'L[_~~Hlj

Fig. 29. An example of calculation by the mathematical model, showing the effects of Co and Ta contents on the creep rupture life of CMSX-2 at 1040 °C and at 137 MPa.


Monte Carlo simulations (MCS) of microstructural evolution in Ni-base superalloys. These calculations were successfully verified by atom-probe field ion microscopy (APFIM). The Ni-base superalloys design work at NRIM is partly open in the open laboratory for materials design (OLMD) on the world wide web (WWW) site.

Acknowledgement

The work introduced in this chapter has been carried out in a long term collaboration. Professor M. Yamazaki of Teikyo University of Science, who initiated the alloy design work at NRIM in 1975 and directed the research until 1992, should firstly be mentioned for his foresight and distinguished services. Prof. M. Enomoto of Ibaraki University (CVM), Prof. Y. Saito of Waseda University (MCS) and Dr. H.K.D.H. Bhadeshia of University of Cambridge (APFIM) are to be mentioned for their major contributions. We would also like to thank Ms. Y. Hiradate for re-drawing most of the figures. Apart of this work was performed in a collaborative research proj ect , "Atomic Arrangement Design and Control Project", between the University of Cambridge, the Research Development Corporation of Japan (JRDC, nOw JST) and NRIM.

References

1. H. Harada, M. Yamazaki, Y. Koizumi: Tetsu to Hagane (in Japanese), 65, 1049 (1979).

2. H. Harada, M. Yamazaki, Y. Koizumi, N. Sakuma, N. Furuya, H. Kamiya: Proeeedings of a Conferenee on High Temperature Alloys for Gas Turbines 1982, Liege, Belgium, 4-6 Oetober 1982 (D. Reidel Publishing Company), p.721.

3. L. Pauling: Phys. Rev., 54, 899 (1938). 4. W. J. Boesh, J. S. Slaney: Met. Prog., 86, 109 (1964). 5. R. G. Barrows, J. B. Newkirk: Met. Trans., 3, 2889 (1972). 6. M. Morinaga, N. Yukawa, H. Ezaki, H. Adaehi: Philos. Mag. A, 51, 223 (1985). 7. Y. Murata, S. Miyazaki, M. Morinaga, R. Hashizume: Superalloys 1996, Proeeed

ings of the 8th International Symposium on Superalloys (The Metallurgieal Soeiety), p.61.

8. R. L.Dreshfield: Met. Trans., 2, 1341 (1971). 9. D. E. Mongeau, W. Wallaee: Ser. Metall., 9, 1185 (1975).

10. H. Harada, M. Yamazaki: Tetsu to Hagane (in Japanese), 65, 1059 (1979). 11. H. Harada, M. Yamazaki, Y. Koizumi, N. Sakuma, N. Furuya, H. Kamiya: Pro

eeedings of a Conferenee on High Temperature Alloys for Gas Turbines 1982, Liege, Belgium, 4-6 Oetober 1982 (D. Reidel Publishing Company), p.721.

12. H. Harada, K. Ohno, T. Yamagata, T. Yokokawa, M. Yamazaki: Superalloys 1988, Proeeedings of the 6th International Symposium on Superalloys (The Metallurgieal Soeiety), p. 733.

13. L. Kaufmann and H. Nesor: Metall. Trans., 5,1617 (1974); ibid., p.1623; 6A, 2115 (1975); ibid., p.2123.

14. N. Saunders: Superalloys 1996, Proeeedings of the 8th International Symposium on Superalloys (The Minerals, Metals & Materials Soeiety), p.101.


15. R. Kikuehi: Phys. Rev. 81, 998 (1951). 16. J.M. Sanehez, J.R. Barefoot, R.N. Jarrett, J.K. Tien: Acta Metall., 32, 1519 (1984). 17. M. Enomoto, H. Harada: Metall. Trans. A 20, 649 (1989). 18. M. Enomoto, H. Harada, M.Yamazaki: CALPHAD 15, 143 (1991). 19. Y. Saito: Mater. Sei. Eng. A 223, 10 (1997). 20. H. Harada, T. Yokokawa, K. Ohno, T. Yamagata, M.Yamazaki: Proeeedings of a

Conferenee on High Temperature Materials for Power Eng. 1990, Liege, Belgium, 24-27 Sept. 1990, p.1319.

21. H. Harada, T. Yamagata, T. Yokokawa, K. Ohno, M. Yamazaki: Proeeedings of the 5th International Conferenee on Creep and Fraeture of Eng. Materials and Struetures, Swansea, 28th March-2nd April, 1993 (The Institute of Materials), p.255.

22. M. Yamazaki: Alloy Design of Nickel-base Superalloys and Titanium Alloys, Prog. Powder Metall., 41, 531 (1989).

23. K. Kobayashi, Y. Koizumi, S. Nakazawa, T. Yamagata, H. Harada: Proeeedings of the 4th International Charles Parsons Turbine Conferenee, Neweastle upon Tyne, U.K., 4th-6th November, 1997 (The Institute of Materials), p.766.

24. M. Enomoto, H. Harada, H. Murakami: Tetsu to Hagane (in Japanese), 80, 487 (1994).

25. H. Harada, A. Ishida, H. Murakami, H.K.D.H. Bhadeshia, M. Yamazaki: Appl. Surf. Sei. 67, 299 (1993).

26. H. Murakami, H. Harada, H.K.D.H. Bhadeshia: Appl. Surf. Sei., 76/77, 177 (1994). 27. H. Murakami, Y. Koizumi, T. Yokokawa, Y. Yamabe-Mitarai, T. Yamagata, H.

Harada: Mater. Sei. Eng. A 250, 109 (1998). 28. Y. Saito, H. Harada: Mater. Sei. Eng. A 223, 1 (1997). 29. H. Murakami, Y. Saito, H. Harada: Superalloys 1996, Proeeedings of the 8th Inter

national Symposium on Superalloys (The Minerals, Metals & Materials Soeiety), p.249.

30. K. Kawasaki: Phys. Rev., 145, 224 (1966). 31. M. Fujita, T. Yokokawa, T. Noda: Proeeedings of a Conferenee on Intelligent Soft

ware Systems in Inspeetion and Life Management of Power and Proeess Plants, SMiRT Post Conferenee Seminar No. 13, Paris, Aug. 25-27, 1997, p.33.

Design of Titanium Alloys, Intermetallic Compounds and Heat Resistant Ferritic Steels

Hidehiro Onodera

Computational Materials Science Division, National Research Institute for Metals 1-2-1 Sengen, Tsukuba-shi, Ibaraki 305, Japan.

Abstract. This chapter describes alloy design technologies for o+ß type superplastic titanium alloys and 0<+0<2 type heat resistant titanium alloys based on the prediction of microstructure by the empirical and/or thermodynamic calculation of phase equilibrium and the prediction of mechanical properties through various kinds of microstructural and compositional parameters. Improvements in the description of configurational thermodynamics have enabled appropriate evaluations of atomic configurations in the ordered and disordered phases. These advanced models are necessary for the better understanding of ordered and disordered alloys and the improvement of accuracy of alloy design. The latter half of this chapter describes an analysis of the preferential substitution site of the third element in Ti3 AI and TiAI intermetallic compounds based on CVM and an analysis of the effects of short range ordering in the bcc ferrite phase on the long term creep strength of carbon steels based on CAM.

3.1 Introduction

Titanium alloys are noted for their high strength-density ratio and they have been used in the aerospace industry [1]. A common practice [2], in the treatment of oo-ß titanium alloys consists of 00 solution treatment at higher temperature mainly in the 00 - ß range, and ageing subsequently at lower temperatures for strengthening. In the solutioned-and-aged condition, the strength of oo-ß titanium alloys is closely related to the compositional and microstructural parameters such as volume fraction and degree of solid-solution strengthening for the primary 00 (hcp ) phase and electron-atom ratio for the prior ß(bcc) phase [3]. The use of the ordered 002 phase (Ti3 AI) to strengthen the 00 phase is one possible approach to develop new heat resistant titanium alloys [4]. In order to control these parameters, it is necessary to calculate phase equilibria between oo-ß and 00-002 at a given solutioning temperature in multi-component titanium alloys.

For the calculation of oo-ß phase equilibrium, the present author [5] developed the alloy design method A based on Ti-X (X = Al, V, etc.) binary phase diagrams. However, method A gives only a rough approximation because it does not take account of interaction among solute elements into consideration. Several methods [6] have been proposed for the calculation of ,-,' phase equilibrium in multi-component nickel-base superalloys based on the analyzed compositions

Springer Series in Materials Science Volume 34, Ed. by T. Saito © Springer-Verlag Beflin Heidelberg 1999

72 Design of Ti Alloys, Intermetallic Compounds and Ferritic Steels

of'f and 'I' phases. These methods seem to give more reliable results due to the more proper partition ratio of each element given as a function of the ~/ phase composition. Thus, the new method B was developed based on the analyzed compositions of 0: and ß phases in multi-component titanium alloys by modifying the method by Harada and Yamazaki [7].

The recent progresses [8] in the field of thermodynamics will have a big impact on developing and understanding complex alloys as shown in Chaps. 4 and 5 of this book. The analytical representations that have been shown to be adequate for binary systems have also been shown to represent high er order systems adequately. Thus, the subregular solution model of Hillert and Walderstrom [9] was applied to the calculation of o:-ß phase equilibrium (method C). The sublattice model developed by Hillert and Staffansson [10] can be used in a thermodynamic analysis of the system containing components dissolved interstitially and/or ordered phases. The sublattice model was applied to the analysis of the effect of oxygen on the o:-ß phase equilibrium and the calculation of 0:-0:2

phase equilibrium in the multi-component titanium alloys [11, 12]. The prediction of mechanical properties is a necessary step in the alloy design

procedure. In the author's group, effects of microstructural and compositional factors on mechanical properties such as tensile creep properties have been examined by the multiple regression analysis [3] and the regression equations obtained combined with the above mentioned thermodynamic calculations of phase equilibrium have been used for designing o:+ß superplastic titanium alloys having improved strength to density ratios [13] and 0:+0:2 heat resistant titanium alloys [4]. The first half of this chapter describes these design technologies as weIl as the mechanical properties of developed alloys.

The regular solution model has been used widely and successfully for the analysis of solid solution phases, however, as long as random mixing is assumed in that model, the atom configuration may be quite different from reality. Atomic configurations such as long range ordering (LRO) and short range ordering (SRO) not only playa significant role in relative phase equilibrium but also affect physical properties of alloys. Improvements in the descriptian of the canfigurational thermodynamics such as the cluster variation method (CVM) [14] and the central atoms model (CAM) [15] have enabled appropriate evaluations of atomic configurations in the ordered and disordered phases. These advanced models are necessary for the better understanding of ordered and disordered alloys and the improvement of accuracy of alloy design. The latter half of this chapter describes an analysis of preferential substitution site of the third element in Ti3AI and Ti Al intermetallic compounds based on CVM and an analysis of the effects of short range ordering in the bcc ferrite phase on the lang term creep strength of carbon steels based on CAM.

3.2 Design of Superplastic Titanium Alloys

Superplastic forming is one of the effective methods to reduce fabrication costs of titanium alloys [1]. It is weIl known that optimum superplastic behavior is

Hidehiro Onodera 73

achieved by a fine equiaxed microstructure with approximately equal fractions of two phases, a and ß phases, at the forming temperature [16]. Figure 1 depicts a typical microstucture of a+ß titanium alloys showing superplasticity. As for the strength of titanium alloys, Khorev [17] has reported that a combined alloying of several ß-stabilizing elements is more effective in strengthening titanium alloys than a single alloying of one b-stabilizer. Therefore, alloy developments have been done on multi-component titanium alloys, Ti-AI-Sn-Zr-V-Mo-Cr-Fe system. The following methods, A, Band C have been developed [5] and applied for the calculation of a-ß phase equilibrium at a given temperature in a multi-component titanium alloys.

Fig. 1. Back-scattered electron micrograph of GT- 51 alloy solution treated at 1123 K for 1 hand quenched in water.

3.2.1 Outlines of Methods for Calculation of o.-{3 Phase Equilibrium in Multi-component Titanium Alloys

Method A. Method A is based on the data of Ti-X (X = Al, Sn, Zr, V, Mo, Cr and Fe) binary phase diagrams. The outline of this method is as folIows:

Step 1 The composition of the ß phase, except Al, is set within the solubility limit of each element.

Step 2 The concentration of Al in the b phase is calculated from the b-surface equation, (1), at a given temperature, Te,

Te = 1155 + aAI(X~I)2 + bAI(X~I) + L {aj(X:)2 + bj(X:)}, (1) j

where, X: : atomic fraction of element j in the ß phase, aj, bj : constants in Table 1.

Step 3 The concentration of each element in aphase, which is in equilibrium with the ß phase set in Steps 1 and 2, is calculated from (3) by using the respective partition ratios in (2).

GAI = 1.426 - 0.0148X~Jl Gv = 0.731 - 0.263Xe, GZ r = 0.914 - 0.032Xgr ,


Table 1. Constants in (1).

Al Sn Zr v Mo Cr Fe

aj -0.300 0.646 0.152 0.226 0.093 0.630 0.340

bj 18.241 -9.893 -3.109 -17.057 -19.047 -24.430 -18.048

CSn = 0.9289, CMa = 0.078, CCr = 0.081, CFe = 0.089,

Xj = Cj · Xj. (2)

(3)

a and ß phase compositions are calculated from the alloy compositions, Xi, by using (1) to (3) and the following relationships:

(4)

(5)

where fa is the mole fraction of the aphase.

Method B. Method A gives only a rough approximation and a considerable error is observed in the calculated volume fraction of the a phase in some cases. Thus, method B, similar to the one developed by Harada and Yamazaki [7] for ," two phase Ni-base superalloys, was constructed on the basis of compositions of a and ß phases measured by X-ray microanalyses for 20 multi-component titanium alloys equilibrated at 1173 K.

In method B, the ß-surface equation at 1173 K was expressed by Al concentration in the ß phase as a function of the concentration of other elements, excluding Al and Ti, in the ß phase. The equation was derived from ß phase compositions for 20 titanium alloys by the multiple regression analysis, as shown in Table 2. The partition ratio of each element between a and ß phases was expressed as a function of ß phase compositions, also by the multiple regression analysis as shown in Table 3.

The a - ß phase equilibrium is calculated by method B in the same way as by method A. Because the analyzed data of compositions in a and ß phases are affected by interactions among solute elements, effects of such interactions are reflected in the ß-surface equation and the partition ratio of each element in method B. Therefore, it is expected that method B would give more reliable results than Method A.

Method C (Thermodynamic representation by subregular solution model). The free energy, C, of a solution can be represented by Hillert's subregular solution model [9] as

Hidehiro Onodera 75

Table 2. ß-surface equation obtained by regression analysis using ß phase compositions in 20 alloys equilibrated at 1173 K.

XA1(at.%) = 2.192 + 0.468Xzr+ 1.017Xv + 2.078XMo + 1.891Xcr + 1.271XFe

a* 0.11 0.01 0.00 0.08 1.56

Multivariable correlation coefficient for 20 alloys 0.946

* The level of significance for each coefficient (%)

Table 3. Partition ratios obtained by regression analysis.

CA1= 1.363 + 0.012XA1+ 0.016Xsn + 0.009Xv + 0.030Xcr

a* 1.6 16.6 8.8 1.5

M.C.C. for 12 alloys 0.788

CSn= 0.595 + 0.077Xsn + 0.048Xv + 0.065XMo + 0.186XFe

a* 2.62 0.68 1.44 0.13

M.C.C. for 14 alloys 0.834

CZr = 0.686 + 0.045XMo + 0.124Xcr

a* 7.6 0.85

M. C. C. for 13 alloys 0.721

Cv = 0.475 + 0.021XA1+ 0.068Xv + 0.018Xzr + 0.039XMo

a* 9.8 0.01 13.3 11.1

M. C. C. for 18 alloys 0.875

CMo = 0.08, CCr = 0.09, CFe = 0.10

* The level of significance for each coefficient (%)

M. C. C. : multivariable correlation coefficient

where, Gi is the free energy of the i-th component having the structure of the solution relative to the standard state of this component. The atom fractions are Xi and X j , and the parameters Aij and B ij come from the representations of the i-j-th binary system. The second sum is the ideal entropy of mixing. The last two sums represent the excess free energy for i-j-th binary system. The chemical potential of the i-th component, /-Li, is given as

/-Li = Gi + RT ln Xi + 2: A ij (1 - Xi)Xj - 2: AjkXjXk

+ 2: B ij [2Xi X j (Xi - X j - 1) + XJl- 22: BjkXjXk(Xj - X k), (7)

where the i-th component is present in the i-j-th system but not in the j-k-th system.


Table 4. Interaction parameters in (6) taken from the works ofKaufman and Bernstein [18] and Kaufman and Nesor [19-21]. (Jjmol)

i-j ß(bcc) a(hcp)

Ti-Al A ij = -107780 + 38.49T, B ij = 0

Bij = 0

B ij = 0

B ij = 0

Bij = 0

Bij = 6276

A ij = -111629 + 38.49T, B ij = 0

Ti-Sn

Ti-Zr

Ti-V

Ti-Mo

Ti-Cr

Ti-Fe

Al-V

Al-Sn

A ij = -87780,

A ij = 3740,

A ij = 11115,

A ij = 5192,

A ij = 18828,

A ij = 34518,

A ij = 52647.1,

Aj = 11704,

A ij = -87780,

A ij = 7600,

A ij = 11115,

Aj = 15359,

Aij = 45689.5,

Bij = 11506 Aij = 0,

B ij = 0 A ij = 56827.1,

Al-Mo A ij = -7524, B ij = 0

Aj = 11704,

Aij = -7524,

Aij = -60250,

A ij = 0,

AI-Cr Aij = -60250, Bij = 0

Al-Fe A ij = -117040 + 28.424T,

B ij = 0

B ij = 0

B ij = 0

B ij = 0

B ij = 6276.5

B ij = 0

B ij = 0

Bij = 0

B ij = 0

B ij = 0

B ij = 0

B ij = 58520 - 42.636T

V-Mo Aij = -22990, A ij = -22990, B ij = 0

Mo-Cr A ij = 27797 - 8.569T,

B ij = 6497 - 2.717T

Mo-Fe A ij = 32813 - 4.18T,

B ij = -627 - 4.18T

Cr-Fe A ij = 25104 - 1O.46T,

A ij = 27797 - 8.569T,

B ij = 6497 - 2.717T

A ij = 32813 - 4.18T,

B ij = -627 - 4.18T

Aij = 25104 - 10.46T,

Table 5. Interaction parameters in (6) determined from equilibrium composition data in the binary phase diagram by Hansen. (J mol-I)

i-j ß(bcc) a(hcp)

Al-Zr A ij = 74660, B ij = 0 A ij = -76366, B ij = 0

V-Zr A ij = 41601.8, B ij = 0 A ij = 76136, B ij = 0

Sn-Zr A ij = -48400, B ij = 0 A ij = -55970, B ij = 0

Zr-Mo A ij = 27383.2, B ij = 0 A ij = 37540.6, B ij = 0

Zr-Cr A ij = -85476.5, B ij = 0 A ij = -80651.1, B ij = 0

Zr-Fe A ij = 27889.1, B ij = 0 A ij = 66766.9, B ij = 0

Of the 28 binary systems required in the Ti-Al-Sn-Zr-V-Mo-Cr-Fe system, the parameters for 17 binary systems (Table 4) were taken from the works of Kaufman and Bernstein [18], and Kaufman and Nesor [19-21]. Für 6 systems (Table 5), the parameters were determined from equilibrium composition data of a and ß phases in the binary phase diagrams by Hansen [22] on the assumption of a regular solution model. For the other 5 binary systems, V-Sn, V-Fe, Sn-Mo, Sn-Cr and Sn-Fe, since no parameters were reported and both phases did not appear in the binary phase diagram, the parameters were set to zero as the zero

Hidehiro Onodera 77

Table 6. Free energy differences between oe and ß phases for the pure elements taken from the works of Kaufman and Bernstein [18] and Kaufman [19].

Element Cu - Cß (J/mol)

Ti -4351 + 3.7656T

Al -4602.4 + 3.0125T

V 6270 + 3.344T

Sn -418 + 0.418T

Zr -4305.5 + 3.7656T

Mo 8368

Cr 8368

Fe 11124.6 - 3.4618T - 0.7472 x 10-2 x T 2 + 5.1254 X 10-6 x T 3

order approximation. Free energy difference between oe and ß phases for each pure element (Table 6) was given by Kaufman and Bernstein [18] and Kaufman [19].

Titanium alloys usually contain considerable amounts of oxygen, and it affects the relative phase stability in alloys as weIl as the mechanical property of alloys. For improving the accuracy of thermodynamic calculation, it is desirable to take the effect of oxygen into consideration. The sublattice model developed by Hillert and Staffansson [10] on the basis of the regular solution apploximation can be applied to the thermodynamic analysis of the alloy system containing components dissolved interstitially and/or ordered phases. For the Ti-O system, the chemical potentials for 0 and Ti are defined as

/-Lo = üCo + RTln[Yo/(l - YO)] - (2/c) . Yo . L,

/-LTi = üCTi + (c/a) . RTln[l - Yo]- (l/a) . Y5 . L,

(8)

(9)

where L is a regular-solution type interaction parameter. Yo is the measure of the oxygen content defined as

Yo = (l/a) . [xo/(l - xo)], (10)

where Xo is the ordinary mole fraction of oxygen, a and c are constants describing the number of lattice sites and interstitial sites, respectively. For oe phase (hcp) a = c = 1 and for ß phase (bcc) a = 1 and c = 3.

The Volume Fraction of 0: Phase and the Partitioning of Each Element Between 0: and ß Phases. The volume fraction of oe phase (V 0,) observed after quenching from 1173 K and those calculated by methods A, Band C are summarized in Table 9, for 7 alloys, GT-1 to 3 and 8 to 11 (Table 7) which were used to construct the method B, and furthermore for 8 alloys, GT-5, 6, and 12 to 17 (Table 8) [5]. The calculated mole fractions of oe phase were converted to the volume fractions of oe phase by using the lattice parameter data for a and ß phases in some titanium alloys. For the alloys, GT-33, 45 and 46 (Table 8),


Table 7. Chemical compositions of alloys (mass%).

Alloy Al Sn

GT-1 4.431.79

GT-2 3.81 4.87

Zr V Mo Cr Fe 0 Ti

4.28 -

2.99 -

- 0.11 baI.

- 0.11 baI.

GT-3 4.50 - 11.032.15 - - 0.11 baI.

GT-8 5.85 1.35 3.45 0.560.94 1.180.980.12 baI.

GT-9 5.70 1.40 3.86 0.50 0.98 1.29 0.98 0.12 baI.

GT-10 5.03 1.20 9.10 0.71 0.53 0.60 0.65 0.11 baI.

GT-11 5.22 2.73 5.51 0.480.550.61 1.240.12 baI.

Table 8. Chemical compositions of alloys (mass%).

Alloy Al Sn Zr V Mo Cr Fe 0 Ti

GT-5 3.62 6.34 12.07 - 1.11 - - 0.09 baI.

GT-6 4.895.12 6.79 - 2.48 - 0.050.10 baI.

GT-12 5.38 1.71 6.95 0.61 0.580.690.860.11 baI.

GT-13 4.44 1.13 6.84 0.51 0.55 0.61 0.750.07 baI.

GT-14 4.57 1.10 5.19 0.650.51 0.500.970.09 baI.

GT-15 4.77 2.01 3.43 0.780.660.610.860.08 baI.

GT-17 5.971.17 8.43 0.860.650.650.890.09 baI.

GT-16 5.93 1.19 10.120.690.530.52 1.01 0.08 baI.

GT-33 6.54 1.36 1.03 1.432.892.11 1.650.11 baI.

GT-45 6.35 0.88 1.05 2.852.482.54 1.590.11 baI.

GT-46 5.72 0.94 5.06 1.482.453.492.610.16 baI.

measurements and calculations of Va were performed after quenching from 1123, 1073 and 1023 K, respectively.

The Va values measured and calculated by the method A are not in good agreement. It is assumed that these discrepancies ariginated from the ß-surface equation, (1), in which effects of interaction among solute elements were not taken into consideration. The Va calculated by method Band the observed ones are in good agreement except for the alloy GT-5 which has a high concentration of Sn, 6.3 mass%, exceeding the concentration range of Sn, 0 to 4.9 mass% far the alloys used to construct method B. The agreement between the observed and the calculated Va in the case of method C is as good as in method B. Furthermore, method C gives a good estimations of Va for the alloys GT-33, 45 and 46 at the respective temperatures of 1123, 1073 and 1023 K. When the effect of oxygen on the phase equilibrium was taken into consideration by the sublattice model with the following parameters [23], the accuracy of calculations was furt her improved as shown by method C' [24].

°Gö - °G~ = 31200 - 42.6T,

La = 99400 - 88.7T, Lß = 24.9T.

(11)

(12)

Hidehiro Onodera 79

Table 9. The volume fraction of Q phase at 1173 K.

Alloy Va,Observed Va, Calculated

A B C Cf

GT-1 0.30 0.42 0.25 0.23 0.26

GT-2 0.41 0.43 0.36 0.36 0.39

GT-3 0.16 0.51 0.21 0.08 0.15

GT-5 0.30 0.52 0.004 0.26 0.34

GT-6 0.46 0.45 0.36 0.61 0.64

GT-8 0.32 0.48 0.31 0.36 0.38

GT-9 0.37 0.49 0.30 0.26 0.30

GT-11 0.33 0.47 0.34 0.31 0.35

GT-12 0.41 0.50 0.32 0.34 0.37

GT-13 0.29 0.42 0.19 0.25 0.27

GT-14 0.31 0.47 0.27 0.28 0.32

GT-15 0.37 0.50 0.31 0.35 0.37

GT-16 0.25 0.51 0.37 0.31 0.33

GT-17 0.36 0.53 0.37 0.33 0.35

GT-33* 0.37 0.36 0.26 0.27

GT-45*2 0.31 0.51 0.34 0.34

GT-46*3 0.33 0.47 0.31 0.31

d 0.13 0.08 0.0590.045

(0.06)*4

sd 0.087 0.068 0.0360.042

(0.034)*4

A: The method based on Ti-X binary phase diagrams

B: The method based on composition data of Q and ß phases

C: Thermodynamic calculation excluding oxygen Cf: Thermodynamic calculation including oxygen

d: Average difference between observed and calculated Va;

d = (~= IVa,Observed - Va,Calculatedl)/n

sd: Sampie standard deviation of d

*: at 1123 K *2 : at 1073 K *3 : at 1023 K

*4: The GT-5 alloy was excluded from the calculation.

The experimental partition ratios of each element between Cl: and ß phases at 1173 Kare compared to the calculations by methods A to C in Figs. 2 to 4. For the partition ratios of Al, Sn, Zr, and V at 1173 K, excellent agreements were observed between the observed and calculated values by method B. Because Mo, Cr, and Fe scarcely dissolved in the Cl: phase, the analysis of compositions included a fairly large error and this resulted in large scatters in Figs. 2 to 4. Despite these scatters, the partition ratios of these elements in methods A and


2.0

1.0

----.~ 0.5 '-<S

~ o 0.2 oE

0.1 s:: o

',= . .= .... o:!

Q., ----~

-'-0 ---0

0.0 1 ~G~T;:-:;:::i=:;::::::L::L::L:::i:-:::L ........ -'-....L-0I~0I:-:0I~T~Z7_L-I--'--'-........... c.J 1 2358910112425272829./:..1:1:13 45331452463

Alloy NOI

Fig. 2. The partition ratio of Al, V and Mo observed and calculated by methods A to C.

2.0

____ 1.0

~0.7 .~ 0.2

.~ 0.1

s:: o :~ o:!

Q.,

0.01 GT- 01 ~GT. * * 1 2 5 8 9 10 11 2425272829 1:.4 33 145 2463

NOI Alloy

Fig. 3. The partition ratio of Sn and Fe observed and calculated by methods A to C.

B seem to be relatively satisfactory approximations. Excellent agreements were obtained for the partition ratios of Al and Sn by method C. Method C gave higher values far partition ratios of Mo than the observed values, and lower values for Cr and Fe. From these results, it is concluded that method B gives a much more accurate estimation than methods A and C for the cy - ß phase equilibrium. However, the inapplicability to the alloy GT-5 suggests that method B is applicable only within the composition range of Sn, 0 to 4.9 mass%, for the alloys used to construct method Band at the temperature of 1173 K.

Hidehiro Onodera 81

2.0 at 1173K

1.0 ~

.-«l.

~ 0.5 .-<:> :s .~

0.2

0.1 :: 0

.. -A

:~ '" t:l-.

0.02

MI GT_ ~~GT N N '-I '2 '3

3 5 8 9 10 11 24 25 27 28 29 t3 <5; 33 45 46 Alloy

Fig.4. The partition ratio of Zr and Cr observed and calculated by methods A to C.

The thermodynamic calculations by method C predicted very well the tendency of partitioning of each element between a and ß phases that Al dissolved more in the a phase and Mo, V, Cr and Fe dissolved more in the ß phase, and further, Zr and Sn partitioned fairly uniformly between both phases. For the partition ratios of Al and Sn, excellent agreements were seen between the observed and calculated values. These results suggest that the thermodynamic calculations are reasonably adequate for multi-component titanium alloys having 8 elements.

3.2.2 Predictions of Mechanical Properties

Superplastic forming is one of the effective methods to reduce fabrication costs of titanium alloys especially for large complex parts such as a compressor disc of jet engines for aircraft. Increasing attention is being given to a + ß titanium alloys such as a Ti-6AI-4V alloy, because they are suitable for superplastic forming. Thus extensive studies [3-5, 13, 24] have been done in order to develop a + ß titanium alloys having improved mechanical properties at about 300°C (573 K), the working temperature of these parts. For the predictions of these mechanical properties, the regression equations have been reported by the author's group [3, 25] from the multiple regression analyses on the relationships between mechanical properties and microstructural and compositional parameters, as shown in the following paragraphs.

Superplastic Properties. It is well known that superplasticity is achieved by a fine equiaxed microstructure with approximately equal fractions of the two phases at the forming temperature. For titanium alloys, the microstructure consisting of a and ß phases shown in Fig. 1 is suitable for superplastic forming. On the other hand, a lower superplastic forming temperature would be desirable


900

~ 700

§

'! 500 §

5:l 300

f; = 6.7 x 10-4 s-l

1100 Temperature (K)

1200

Fig. 5. Infiuence of temperature on elongation for alloys GT-9, 33, 45 and 46.

for a number of reasons, such as reduced oxidation, lower cost die materials, and increased die life. Wert and Paton [26] have reported that the optimum superplastic forming temperature of the Ti-6Al-4V alloy can be lowered by the additions of Ni, Co, and Fe without an increase in flow stress. However, they have not shown the limit for the addition of these ß stabilizing elements.

For obtaining the optimum design conditions for superplastic titanium alloys, it is necessary to make clear the optimum forming temperature and the optimum volume fractions of two phases at the temperature. For this purpose, the effects of test temperature and volume fractions of the 0: and ß phases on superplastic properties were examined by using four titanium alloys, GT-9, 33, 45 and 46 in Tables 7 and 8 [25]. These alloys were designed to have approximately equal volume fractions of 0: and ß phases at 1173, 1123, 1073 and 1023 K, respectively.

Tensile tests were performed at various temperatures ranging from 1023 K to 1223 K on an Instron testing machine equipped with an infrared heater. All specimens were tested after being heated at the test temperature for 10 min in an argon atmosphere with astrain rate of 6.7 x 10-4 s-l. The total elongation of each alloy was more than 400% in the temperature range between 1023 K and 1123 K as shown in Fig. 5. On the other hand, the maximum flow stress increased with decreasing temperature rapidly in the temperature range lower than 1073 K as shown in Fig. 6. Since a low flow stress is desirable for the superplastic forming, the temperature range between 1073 K and 1123 K was considered to be suitable for the superplastic forming of these alloys at the strain rate of 6.7 x 10-4 s-l.

The m-value, the representative parameter for superplastic properties, is well known to be a function of temperature, strain rate, strain and microstructural parameters such as volume fractions of the two phases and grain size and shape. The effects of test temperature and volume fraction of the 0: phase on the initial m-value at astrain of ab out 30% were examined by means of multiple regression

Hidehiro Onodera

80 ~ ~ 70 ~

~ 60 <I)

!:l '" 50 ~ 0

:;:::: 40 S S ._ 30 ~

::;s 20

~ i: = 6.7 x 10-4 8- 1

.- - GT-9 0- GT-33 /::;,.- - GT-45 ~GT-46

1100 Temperature (K)

1200

83

Fig.6. Infiuence of temperature on the maximum fiow stress for alloys GT-9, 33, 45 and 46.

analysis. The following best regression equation was obtained from the analysis on 14 data sets.

m = 0.422 - 17.128(T - 1.095)2 - 0.514(Va - 0.426)2, (13)

where T is the test temperature (K x 10-3 ) and Va is the volume fraction of the a phase. There was a good correlation (multivariable correlation coefficient = 0.816) between the estimated and observed m-values. Equation (13) suggests that titanium alloys having a volume fraction of the aphase of 0.426 at 1095 K give the best superplastic properties at the strain rate of 6.7 x 10-4 s-l.

Tensile properties at 573 K. A common practice [2J in the treatment of a+ß titanium alloys consists of a solution treatment at high temperature mainly in the a + ß range, and ageing subsequently at lower temperatures for strengthening. Since structural changes, decomposition of martensite or retained ß phase, and mechanical properties of these alloys depends on ß stabilizer contents in the prior ß phase, the mechanical properties of a + ß titanium alloys should be correlated to the microstructural and compositional parameters, such as volume fraction, grain diameter and degree of solid-solution strengthening for the primary a phase, and electron-atom ratio for the prior ß phase.

Onodera et al. [3J examined the effects of these microstructural and compositional parameters on the tensile properties at 573 K by using 9 a + ß titanium alloys GT-5, 9, 10, 11, 15, 16, 33, 45 and 46 in Tables 7 and 8. From the multiple regression analysis on the relationship between the tensile properties of these alloys at 573 K and the parameters in Table 10, the following equations were obtained as the best regression equations:

au(MPa) = 2543.59 - 3627.57Va + 1170.41VadDEa

+1744(1- Va)(e/aß - 4) -186(1- Va)Tag /100, (14)


Table 10. Microstructural and compositional parameters for alloys quenched from respective solution temperatures.

Alloy Va da dDEa e/aß

GT-5 0.30 3.0 2.08 3.96

GT-9 0.292.6 1.92 4.02

GT-10 0.29 2.6 1.94 3.98

GT-11 0.26 3.5 1.97 4.01

GT-15 0.32 3.2 1.74 4.00

GT-16 0.36 3.2 2.11 4.00

GT-33 0.37 1.3 2.28 4.06

GT-45 0.31 1.1 2.21 4.12

GT-46 0.33 1.1 2.25 4.16

Va: Volume fraction of primary Cl: phase

da: Grain diameter (J.l.m) of primary Cl: phase

dDEa: Parameter showing degree of solid solution strengthening for primary Cl: phase

e/aß : Electron-atom ratio of prior ß phase due to group number

(M.C.C. = 0.923) ,

El.(%) = -27.91 - 9.28V" + 13.08dDE" + 2.95(Tag /100), (15)

(M.C.C. = 0.789) ,

where O"u and El. are the tensile strength and the total elongation, respectively, and Tag is the ageing temperature (K).

The degree of solid-solution strengthening for the primary a phase was estimated by the following parameter, dD E", obtained by applying the result reported by Sasano and Kimura [27] on binary titanium alloys to multi-component alloys:

where i : transition metal element,

j : non-transition metal element,

Xi,X j : at.% ofsolute element

Di , DTi : atomic diameter,

iJ.Pj : electrical resistivity increasing rate with alloying.

Parameters in (16) are shown in Table 11.

3.2.3 Development of Superplastic Titanium Alloys

(16)

The author's group was in charge of designing superplastic titanium alloys having improved strength-to-density ratio in the national project, "Advanced Alloys with Controlled Crystalline Structures" (1982-1988) [28] sponsored by the

Hidehiro Onodera 85

Table 11. Parameters in (16).

Transition metal (i) Non-transition (j) Parameters

V Zr Cr Mo Fe Al Sn

I(D i - DTi)/DTil· 100 8.939.69 13.365.71 14.15

L1pj(D· cm/at.%) 14.40 17.61

Ageney of Industrial Seienee and Teehnology (AIST) of MIT!. In the course of the projeet, at first 17 alloys (Tables 7 and 8) having an al ß volume fraction ratio of about unity at 1173 K were designed by the alloy design method A deseribed in the Seet. 3.2.1 [29]. These alloys were are melted as 2 kg double melt ingots. Tensile speeimens were maehined from bars whieh were rolled to about 85% reduetion at 1173 K. Eaeh alloy was solution treated at 1173 K for 1 hand quenehed in water, and aged at various temperatures ranging from 773 K to 873 K for 4 h. Tensile properties of GT-9 alloy at 573 Kare shown in Fig. 7. The tensile strength varied depending on the ageing temperature, but the target properties of this projeet were almost realized by GT-9 alloy.

20

0

0 A. ...--. 15 0 ~ I AI-8V-5Fe GT-9 A. '-' c

.Q 0 ~

t>JJ 10 c 0

0 ii3 13Y- l ler-3AI 0 0 6-2-4-6

6AI-4V 6-2-4-6 5 0 0

lAI-8V-5Fe 6-2-4-6

0 20 22 24 26 2 30 32

Strength to den ity ratio (104 Pa kg)

Fig.7. Tensile properties of developed alloys and commercial alloys at 573 K with a strain rate of 3 x 1O-4 S-1[13] .

As the next step of the research proj ect , alloys GT-33, 45 and 46 (Table 8) were designed to have approximately equal volume fractions of a and ß phases at 1123 K, 1073 K and 1023 K, respectively. The design temperature, where the al ß volume fraction ratio be comes unity, can be decreased by increasing the contents of ß stabilizing elements such as V, Mo, Cr and Fe. Tensile properties of these


alloys exceed those of commercial alloys and the target properties in this project as shown in Fig. 7. Superplastic properties of these alloys are shown in Figs. 5 and 6. Tensile properties of alloys GT-45 and 46 at 573 Kare excellent, however, these alloys are not good for superplastic forming because of the high maximum flow stress in the temperature region where large elongation is obtained.

The results in Sect. 3.2.2 suggest that titanium alloys having a/ ß volume fraction ratio of about unity at 1123 K give the best superplastic properties. Thus, a research program was set up to improve tensile properties of superplastic titanium alloys having a/ ß volume fraction ratio of about unity at 1123 K by optimizing microstructural and compositional parameters on the basis of results described in Sect. 3.2.2. Alloys GT-60 and 61 were developed successfully by this alloy design, and they showed tensile properties exceeding those of formerly developed alloys GT-33, 45 and 46 and the target properties in this project [13]. The GT-61 alloy showed tensile elongation of 660% and the maximum flow stress of 13.5 MPa in the superplastic condition, at 1123 K with astrain rate of 6.7 x 1O-4 s-1 .

Yamauchi et al. [30] successfully formed a near net-shape turbine disk of the developed GT-33 alloy with a diameter of 150 mm by superplastic forging as part of the national project (Fig. 8).

Fig. 8. A ne ar net-shape turbine disk of the GT -33 alloy with a diameter of 150 mm formed by superplastic forging [29].

3.3 Design of Heat Resistant a + a2 Titanium Alloys

The use of a2 phase (Ti3AI) to strengthen a phase titanium alloys is one possible approach in the development of new heat resistant titanium alloys because the high temperature strength of a2 phase is much higher than that of a phase. Aluminum additions of larger than 8 mass% cause precipitations of ordered a2

phase resulting in an embrittlement of titanium alloys. The extent of embrittlement depends on the amount, size and shape of a2 phase [31]. In order to design such alloys, it is necessary to control microstructural and compositional

Hidehiro Onodera 87

parameters such as the volume fraction of the 0:2 phase and degree of solid solution strengthening of the 0: phase. This section will describe a thermodynamic representation for 0: - 0:2 phase equilibrium in multi-component titanium alloys and the best design condition for this alloy system [11,12].

3.3.1 The Thermodynamic representation for 0: - 0:2 Phase Equilibrium in Multi-component Titanium Alloys

The free energy of the 0: phase in the Ti-AI-Sn-Zr system is represented by the regular solution model by (6). The free energy of ordered 0:2 phase (D019 ) is represented by the sublattice model as (Ti, Zrh(Ti, Al, Sn) with Gibbs energy expression,

e0<2 = L IY?}je~j + 3RT L (l}i In nl}i) + L e}i In n 2}i) i

i,j,k i,j,k

where s}i: the site fraction of i element in the sublattices,

en: the Gibbs energy of the pure compound i3j,

L~j,k: the interaction parameter, a comma separates the

elements on the same sublattice whereas a colon separates

the elements on the other sublattice.

(17)

The 0: - 0:2 phase equilibrium can be calculated using the parameters shown in Tables 4, 5 and 12.

Table 12. Thermodynamic parameters for the 002 phase in a Ti-Al-Sn-Zr system evaluated by Onodera et al.

°GTi:Ti = 4oG~i

°Gzr:Ti = 3°G~r + °G~i °GTi:AI = 3oG~i + °GAI - 104034 + 25.82T

°GTi:Sn = 3oG~i + °GSn - 95000

°GZr:AI = 3°G~r + °GAI - 79800

°Gzr:Sn = 3°G~r + °GSn - 78200

LTi,Zr:Ti = LTi,Zr:AI = LTi,Zr:Sn = 7600

LTi:Ti,AI = LZr:Ti,AI = -11918 + 45.029T - 2.248(T 110?

LTi:Ti,Sn = LZr:Ti,Sn = 17000

LTi:AI,Sn = LZr:AI,Sn = 1170


Table 13. Chemical compositions (mass %) and calculated Vc>2 at 873 K of designed alloys.

Alloy Al Sn Zr Ti Va2

GT-786.41 5.74 baI. 0.0

GT-796.13 2.47 6.89 baI. 0.10

GT-80 5.95 3.65 3.21 baI. 0.09

GT-816.20 4.34 5.64 baI. 0.19

GT-827.05 4.13 5.06 baI. 0.29

GT-83 7.81 3.95 6.76 baI. 0.41

GT-848.15 3.63 9.81 baI. 0.48

GT-856.58 2.14 10.99 baI. 0.19

GT-86 2.57 11.00 5.56 baI. 0.04

GT-874.55 6.64 5.60 baI. 0.08

Fig. 9. Optical microstructures for (a) furnace cooled and (b) water quenched specimens of the G T -79 alloy.

3.3.2 Optimum Design Condition for 0: - 0:2 Titanium Alloys

Ten Ti-Al-Sn-Zr alloys (Table 13) having various values of Va2 were designed by the thermodynamic calculations [11]. These alloys were are melted as 2 kg double melt ingots. After forging to 30 mm diameter, they were rolled to 14 mm diameter at 1273 K. All speeimens were ß solution treated for 1 hand furnaee eooled (FC) or water quenched (WQ) and aged at 873 K for 48 h (STA). Creep tests were performed at 823 K with a stress of 274 MPa using ereep speeimens with the gauge length and diameter of 30 and 6 mm, respectively. Room temperature tensile tests were performed using speeimens with the gauge lcngth and diameter of 25 and 5 mm, respeetively.

Optieal mierostruetures for FC and WQ speeimens of thc GT-79 alloy are shown in Fig. 9. The mierosructure of FC material consists of a basket-weave type Widmanstatten a in a ß matrix with thick a platelets and grain boundary a. The microstructure of WQ material eonsists of fine aeieular martensite. The

Hidehiro Onodera

20~--------------------------~

19

17

~ ~

o

,

• LMP =16.74 +3.81 Va 2

LMP = 10-3T(20+log I )

I :time (h) to 0.2% creep strain at 274 MPa

T: test tem erature K

16~~--~~--~~~~~~~~~

0.0 0.1 0.2 0.3 0.4 0.5 Calculated Va2

89

Fig.10. Correlation between Larson-Miller parameter (LMP) für time to 0.2% creep strain and the caIculated Va2 at 873 K.

formation of 0:2 phase in each specimen was examined in TEM. In alloys GT-82, 83 and 84, strong 0:2 reflections were observed in the selected area diffraction pattern for both heat treatments, FC and STA. In alloys GT-79, 80, 81 and 85, weak 0:2 reflections were observed. The intensity of 0:2 reflections increases with increasing amount of designed Va2 suggesting that the present thermodynamic calculation can predict very weIl the tendency of the 0:2 formation in Ti-AI-Sn-Zr alloys.

The creep strength of designed alloys increases with increasing Va2 for both heat treatments FC and STA, as shown in Fig. 10. A good correlation is observed between the creep strength and Va2 . Following equations were obtained from the regression analysis.

LM P(FC) = 17.00 + 4.97Va2

LMP(STA) = 16.73 + 3.85Va2

(C.C. = 0.96),

(C.C. = 0.93),

(18)

(19)

where LM P = 1O-3T[20+log(t)], and C. C . is the correlation coefficient. t is the time to 0.2% creep strain at the stress of 274 MPa and T is the test temperature in K.

Room temperature tensile properties of designed alloys are correlated with calculated Va2 in Fig. 11. In the FC condition, the tensile strength decreases with increasing Va2 • A remarkable decrease in the elongation is observed at around Va2 of 0.1 and premature fracture occurs in alloys GT-83 and 84 containing Va2

of larger than 0.4. On the other hand, ST sam pies do not show such premature fracture. The 0.2% proof stress and the tensile strength in ST sampies increase


1100 • (J u (ST) ... "o.Z<ST) • o (J (FC) f:j. (J0.2(FC) • 1000 • Er. (ST) o EI. FC

03 ... ......

0.- 900 ~

::> 800 b

N ci

b

Fig. 11. Room temperature tensile properties of designed alloys in the furnace cooled (FC) and the solution treated and quenched (ST) conditions.

1

j r~l 1

~ k

Fig. 12. Tetrahedron cluster for Ll0 structure used in the CVM approximation.

with increasing Va2 , while the elongation decreases gradually with increasing Va2 • Rodes et al. [32] reported that water quenching could avoid 0:2 formation in a Ti-8Al-5Nb-5Zr-0.2Si alloy while large amounts of 0:2 phase were observed in the solution treated and aged condition.

From the results shown in Figs. 11 and 12, it can be concluded that Va2 of about 0.1 at 873 K seems to be the best condition for designing 0: + 0:2 type heat resistant titanium alloys having a good combination of creep properties and room temperature tensile ductility in the Fe condition [11]. In the commercial alloys such as IMI834 [33] and IMI685 [34], 0:2 formation was reported after prolonged creep exposure. The IMI834 alloy has been reported to give the highest creep resistance among commercial titanium alloys. The Va2 amounts to be about 0.1 at 873 K in the IMI834 alloy according to the present thermodynamic calculation neglecting ß stabilizing elements such as Mo and Nb. These observations, and

Hidehiro Onodera 91

the calculation, seem to support the present suggestion for the optimum design condition.

3.4 Alloy Design Based on the Prediction of Atomic Configuration

Atomic configurations such as long range ordering (LRO) and short range ordering (SRO) playa significant role in the relative phase equilibrium but also affect physical properties of alloys. Improvements in the description of the configurational thermodynamics such as the cluster variation method (CVM) [14] and the central atoms model (CAM) [15] have enabled appropriate evaluations of atomic configurations in the ordered and disordered phases. This section describes an analysis of preferential substitution sites of the third element in Ti3AI and TiAI intermetallic compounds based on CVM and an analysis of effects of short range ordering in the bcc ferrite phase on the long term creep strength of carbon steels and Cr-Mo steels based on CAM.

3.4.1 Design Basis for Intermetallic Compounds, TiAl and Ti3 Al

There has recently been an increase in research activities [35] on the mechanical properties of intermetallic compounds, ')'TiAI(Llo) and (12Ti3AI, because of its potential application as high temperature structural materials. However, the poor ductility of those compounds at room temperature has prevented their practical use. Recently, alloying additions of V, Mn, or Cr into ')'TiAI have been reported to result in increased ductility. For (12 Ti3AI phase, Nb is known to improve ductility. In order to understand the mechanism of the improvement, it seems to be very important to reveal the substitution site of the third element, the long range order parameter and the energy of defects such as an antiphase boundary in these compounds. The cluster variation method (CVM) developed by Kikuchi [14] has been used widely and successfully for the analysis of orderdisorder phenomena on phase equilibrium in cubic structures such as fee and bcc [36,37] and hcp structure [38]. The CVM can calculate the probabilities of atom configurations affected by the interactions between atoms in the same and different sublattices in a most reliable manner [14]. This section describes an outline of the CVM for intermetallic titanium aluminides, ')'TiAI and (12 Ti3AI, and the analysis on the preferential substitution site of ternary additions in these compounds by the CVM with Lennard-Jones pair potential.

Calculation of atomic configuration in ordered structures, Ll0 and D019 , by the cluster variation method. The equiatomic TiAI compound has the Ll0 type fct superlattice, as shown in Fig. 12. The configurational entropy of the system, 8(Llo), is written as


ijkl ij

(20)

where the superscripts a, b, I, and s show the Al and Ti sublattices, the second and first nearest neighbor atomic distances, respectively. Z, Y, X and kB

are tetrahedron cluster, pair, and point probabilities and Boltzman constant, respectively, and L(X) = X In(X) - X.

For the ordered Ti3 AI (D019 ) structure in Fig. 13, the average entropy per lattice point is obtained by Kikuchi [39] as

S = -kB {(1/2) fu; [L( za1j;1) + 3L(zbi;!:r)]

-(1/4) L [L(tafJ%) + 3L(tbfj~) - L(ua~J%) - 3L(ubfj~)] ijk

-(3/2) L [L(Syafj) + L("ybfJ) + LeyC~;)] ij

ij

+(5/4) ~ [L(xf) + 3L(xf)] } , (21)

where probabilities are for the basic clusters shown in Fig. 13.

1:><---------0. ~~ Ir l ,,': ••••• Q,": ••••• C 4 I

?-.-.T.~d)'r'".-.r:.-b>'?12-- --3 5- --6 3 , "0- - -,- . - ~ - - :0 ' b 1 14 1 1 : 31 1 C ~za ~Zb ~ : :: , ,2 , : : 2 7 I 5 ye I .1. I I I I ~ I .o.r --. --T :<), I I 6 8""-----"'" : "I .... ~.. ' ~ .... :.. .. : 2 yd 4 0'--------·, • ua ub 0--:----0

~: 8 ~W15ye6 5 , '. 0

a2 7 ta 3 tb 6 xa xb

Fig. 13. D019 structure with four simple hexagonal sublattices a, b, c, and d and basic clusters used in the CVM approximation.

Hidehiro Onodera 93

The enthalpy, H, is given by the sum of the energies of first and seeond nearest neighbor pairs using the Lennard~Jones (L~J) pair potential of the form,

(22)

The L~J potential parameters, °eij and rij, were evaluated in prineiple in the same manner as used by Sanehez et al. [36]. Parameters reported for the CVM ealculation of Lb strueture [37] were modified to reproduee the lattiee parameters, al and Cl of TiAI eompounds. The L~J parameters for Ti3AI (DOI9 ) were determined from experimental eohesive energies and lattiee parameters of Ti and Al, and from the energy of formation and lattiee parameters of stoiehiometrie Ti3 Al. The data on enthalpy and lattiee parameter of metals, alloys, and eompounds are taken from the eompilation by Murray [40].

In order to obtain the most probable atomie eonfiguration at equilibrium, the grand potential, [2, of the system defined as

[2 = H - T S + PV - L MiXi (23)

is minimized with respect to both tetrahedron cluster probability and the atomie volume, V, at a eonstant temperature, T, pressure, P, and the ehemieal potential of the i atom Mi under the normalization eondition,

for LID, LZijkl = 1, ijkl

LZaijkl = 1, ijkl

L zbijk1 = 1, ijkl

L Uaijk = 1 and L ubijk = 1. ijk ijk

(24)

Calculation of the a./a.2 Phase Equilibrium in the Ti-Al System. A region ofpresent interest in the Ti~AI system is the 0:+0:2 two phase field whieh is the basis of the 0:2 preeipitates strengthened titanium base alloys [38]. The 0:

phase is the disordered hep titanium solid solution and the 0:2 phase is the DO Ig

ordered Ti3Al. The initial attempts to fit the experimental 0:/0:2 phase boundaries, whieh were assessed by Murray [40], resulted in eonsiderable deviation from them as shown in Fig. 14. The values of °eTiAl and rTiAl were used as thc variables to fit the experimental diagram. In the proeedure, the Ti~Ti and Al-Al parameters were kept eonstant and equal to the values listed in Table 14. Varying ° eTiAl shifts the boundaries along the temperature axis and varying rTiAl ehanges the width of the two phase regions. The parameters for the Ti~AI pair potential providing the best fit to the experimental diagram are listed in Table 14. The eorresponding equilibrium boundaries are eompared to the experimental results in Fig. 14. A ealculated 0:/0:2 phase boundaries in the Ti-rieh region was in gooel agreement with the experimental diagram within the experimental seatters. In


Table 14. Lennard-Jones potential parameters used in the CVM calculation for Ll 0

and D019 structures.

L 10

°eij(kJ/mol) °rij(nm)

X Ti-X AI-X X-X Ti-X Al-X X-X

Mn 63.80 59.52* 46.75* 0.2473 0.2590* 0.2663*

Nb 97.48 92.08* 117.48* 0.2840 0.2845* 0.2869*

V 81.25 83.11 83.50 0.2786 0.2580 0.2634

Ni 98.12* 70.94* 70.94* 0.2515* 0.2558* 0.2430*

Co 88.54 64.56* 70.80* 0.2580 0.2452* 0.2444*

Cr 77.26 53.92* 63.99* 0.2624 0.2607* 0.2507*

Mo 94.78 73.43* 96.20* 0.2809 0.2787* 0.2661*

Ti 77.68* 76.07* 77.68* 0.2852* 0.2855* 0.2852*

Al 76.07* 54.83* 54.83* 0.2855* 0.2798* 0.2798*

° eij (kJ /mol)

X Ti-X Al-X X-X Ti-X Al-X X-X

Mn 63.80 59.52* 45.55 0.2875 0.2852 0.2723

Nb 97.48 92.08* 117.93 0.2907 0.2912 0.2937

V 81.25 83.11 83.95 0.2852 0.2641 0.2696

Ni 98.12* 70.94* 70.77 0.2572 0.26170.2485

Co 88.54 64.56* 70.88 0.2637 0.2507 0.2500

Cr 77.26 53.92* 64.34 0.2684 0.2667 0.2565

Mo 94.78 73.43* 96.55 0.2809 0.2787 0.2661

Ti 78.30 75.30 78.30 0.2922 0.2831 0.2922

Al 78.30 53.90 53.90 0.2831 0.2864 0.2864

* after Enomoto and Harada [37]

the Al concentration range greater than 25%, relatively large discrepancies are observed between calculated and experimental a/a2 phase boundaries including the congruent transformation point. The transformation point does not move along the concentration axis with changes in values of ° eTiAl and TTiAl, suggesting the limit at ions of the phenomenological L-J potentials.

Analysis of substitution site of ternary additions in TiAI and Ti3 Al intermetallic compounds. The calculated concentration at 1573 K of the third element, X, in the Al sublattice for Ti(50-y)AI(49+y)X1 alloys are shown in Fig. 15a as a function of bulk Ti content [41]. The preferential substitution site of the third element in Ti(50_y)AI(49+y)X1 alloys varied depending on bulk Ti content. Especially, for Mn, Nb, and V, the preferential substitution site is

Hidehiro Onodera

1500 ,...----------::-0....."....--""""7", Ti-Al _ -:--.:i--}.x... ./

1300

1100

900

ß .-_- x-~ -~ ..... _-..... ; --

_*,# .... __ #,..- .f

• • . . , •

cx I .... "":, ...

IJ ~ / jS'»----"""":'M"':'urr..i....a- y.....,.,[3'"'9'"']--.1

/, - CVM (Best fit ,h ~ CX2 ___ • CVM (Initial)

: ': 0 ,: \J X

H + Experimentally : d c determined ; .. ,,, l:i..

700 L---L.!..L-J.....J~..:.LL.L-.!:::::J::=====:::::!I 5 15 25

At% Al 35 45

95

Fig. 14. The 0;/0;2 phase boundaries in the Ti~AI system calculated by the CVM using the tetrahedron approximation with Lennard~Jones pair potential. Solid lines and fine dashed lines show the best fit CVM results and the initial ones, respectively. Large dashed lines and symbols show experimentally determined boundaries [38].

(a) (b)

~ Ti(50-y)AI(49+y)Xl alloy ~ Ti (75-y) Al (24+y) X 1 alloy

CiS 2.0 CiS 4 <i <i u u

'.:;J oE CiS 1.5 ~3 :c ::: ::: vo vo at 973 K ~ .S 1.0 IJ Mo = = ~ Ni 0 0 h.p 0.g 0.g o 0 tl

0.5 tl

• Nb = = (\) (\)

• Mn u u = at 1573 K = • V 0 0 u u

~ 0.0 ~O 42 44 46 48 50 52 54 56 66 68 70 72 74 76 78

Bulk Ti content, at % Bulk Ti content, at %

Fig. 15. Variation of alloy element concentration in the Al sublattice with the bulk Ti content (a) in Ti(50~y)AI(49+y)Xl alloys at 1573 K [41] and (b) Ti(75~y)AI(24+y)Xl alloys at 973 K [47]. The bulk alloy element concentration is 1 at.%.


changed remarkably on both sides of stoichiometry. In Fig. 15a, if all X atoms were substituted for Al, the concentration of the third element would amount to 2 at.%. In the Ti-rich region, Co, Cr, Ni and Mo mainly occupy the Al sublattice, and Mn, Nb and V occupy both sublattices. In the Al-rich region, Mn, Nb and V mainly occupy the Ti sublattice, and Co, Mo, Ni and Cr occupy both sublattices. Thus, the substitution behavior is classified into two groups; Cr, Co, Mo, and Ni, which are preferentially substituted for Al, and Nb, Mn and V, which are preferentially substituted for Ti. These calculated results are in good agreement with the experimentally determined occupancy fractions of Ni [42], Nb [42], V [43,44], Mn [44-46] and Cr [44, 45] in each sublattice, which were measured by means of electron channeling (ALCHEMI) or X-ray diffraction methods. Especially for Mn, V, and Cr, the present calculation is able to predict the change of site preference on both si des of stoichiometry [41].

The substitution characteristics of ternary additions in Ti(75-y)AI(24+y)X1

alloys are almost the same as those in TiAI intermetallic compound, as shown in Fig. 15b. In Fig. 15b, if all X atoms were substituted for Al, the concentration of the third element would amount to 4 at.%. In the Ti-rich region, Co, Cr, Ni and Mo mainly occupy the Al sublattice, and Mn, Nb and V occupy both sublattices. In the Al-rich region, Mn, Nb and V mainly occupy the Ti sublattice, and Co, Mo, Ni and Cr occupy both sublattices. Thus, the substitution behavior is classified into the same two groups as in TiAI intermetallic compound.

The observed difference of substitution behavior can be interpreted from the relative magnitude of the potential of each atom pair. The interaction parameters, iPA-B, defined by (5) were calculated for the first (rs) and the second (rl) nearest neighbor atom pairs in the alloys Ti(50-y)AI(49+y)X1 and Ti(75-y)AI(24+Y) Xl.

iPA-B = eAB - (eAA + eBB)/2, (25)

where eAB, eAA and eBB are potential energies of A-B, A-A and B-B pairs, respectively.

Figure 16 shows the correlation between the calculated iPTi - X and iP Al-X which are normalized by iPTi-Al. The iPTi- X is larger than iP Al-X for Co, Cr, Mo and Ni which are preferentially substituted for Al in both alloy systems. This suggests that these elements have stronger bonds with Ti atoms than with Al atoms. On the other hand, the opposite tendency is observed for Mn, Nb and V, which are preferentially substituted for Ti. The element having large bonding energy with Ti atom is preferentially substituted for Al. The strength of tendency to occupy each sublattice is proportional to the distance from thc solid line in Fig. 16 which shows the equal values of iPTi- X and iP Al-X' Especially, the slight difference of nearest neighbor interaction between V and Nb in Ti3AI can explain why the tendency of V to occupy Ti sublattice is slightly stronger than that of Nb in Ti3 AI [47].

Hidehiro Onodera

2.0

1.5

::;: ~ 1.0

~--< >9< :>< 0.5

• rl) TiAI ... rs

o rl) Ti 3AI ll. rs

0.0 0.5

<I> /<1>

1.0

X-Ti Ti-Al

97

1.5 2.0

Fig. 16. Correlations between preferential substitution site and interaction parameters, PA-B, TS and Tl represent the first and the second nearest neighbor atom pairs, respectively.

3.4.2 Design Basis for the Long Term Creep Strength of Ferritic Steels

The creep strength of metallic materials decreases with increasing creep time due to the change of microstructure. Recently, Kimura et al. [48] proposed a new concept of the inherent creep strength determining the long term creep strength of ferritic steels from the analysis of the NRIM creep data sheets of steels. According to their analysis, the creep strength of ferritic steels decreases and converges to the constant creep strength level after a long time exposure and the strength level depends mainly on the minute contents of Mo and C [49].

The solubility of substitutional alloying elements is quite limited in the ferrite phase of steels containing carbon. However, substitutional elements such as Mn and Mo are easy to form atomic pairs with carbon in the ferrite phase, and these atomic pairs suppress the recrystallization or recovery process of ferritic steels by strong interaction with dislocations. Thus, these atomic pairs seem to affect the creep properties too when dislocation climb is the rate controlling process. This section describes analysis on the effect of solute elements in the ferrite matrix on the long term creep strength of carbon steels from a viewpoint of atomic configurations such as atomic pairs [50].

Calculation of atomic configuration in the disordered ferrite. The equilibrium concentrations of solute elements in the ferrite matrix were estimated by

98

Ca)

Design of Ti Alloys, Intermetallic Compounds and Ferritic Steels

o ,

,

x o x

J :Central atom

x x

o 'Z=2 X z'=4

Fig. 17. Geometry of the bcc lattice. 0 and x respresent substitutional and interstitial sites, respectively. Z and z: designate, respectively, the numbers of substitutional and interstitial nearest neighbors to a substitutional site (a), while Z' and z' designate respectively these numbers to an interstitial site (h).

thermodynamie ealculations with the sublattiee model (Thermo-Calc. [51]). The eoneentrations of atomie pairs were estimated by the eentral atoms model [15] whieh can calculate the atom eonfigurations in the nearest neighbor atom shell. In the ease of bec ferrite, the nearest neighbor atom shell is shown in Fig. 17. The Wager interaction coeffieients reported for Fe--Mo-C [52] and Fe-Cr-C [53] systems were used in the CAM calculation. The parameter for Mn was estimated from the binding energy for Mn-C reported by Abe et al. [54].

The probability for the configuration in the nearest neighbor atom shell for a central atom of C is obtained as

pe; = (Z'!z'!) tJ ·1 ·I(Z' _ ·)I( , _ ·)1 Z.J. Z • Z J.

where i,j: the number of substitutional (M) and interstitial (C) solute

atom in the nearest neighbor atom shell, respeetively,

YM, Yc: the ratio of the number of atoms M and C to the number of

solvent atoms and the vacant interstitial site, respeetively,

Z', z': the number of substitutional and interstitial nearest neighbors

to an interstitial site,

AM-C, AC-C: interaction parameters,

ACM = 1 + YCAM-C - Y~AM-C(1 - AM-C) + YCYMX~_c +higher order terms,

Acc = 1 + YCAC-C - Y~Ac-c(1- 2Ac_d+higher order terms.

Ridehiro Onodera

Table 15. Chemical compositions of carbon steels (mass%).

Alloy Reat C Si Mn Cr Mo N

CAA 0.20 0.31 0.59 0.046 0.011 0.006

CAB 0.20 0.28 0.60 0.046 0.010 0.007

CAC 0.20 0.29 0.55 0.054 0.012 0.006

CAG 0.21 0.21 0.62 0.046 0.019 0.003 STB410 CAR 0.240.240.640.0740.0190.006

CAJ 0.200.200.470.0560.0100.011

CAL 0.21 0.32 0.48 0.017 0.006 0.004

CAM 0.20 0.31 0.50 0.017 0.006 0.004

CaC 0.28 0.30 0.70 0.07 0.02 0.010

CaD 0.20 0.22 0.68 0.11 0.03 0.008

CaE 0.340.220.700.10 0.03 0.011

CaF 0.280.190.800.06 0.22 0.007 SB480

CaG 0.29 0.20 0.93 0.03 0.36 0.005

CaR 0.29 0.22 1.00 0.07 0.34 0.004

CaM 0.22 0.22 0.82 0.06 0.18 0.008

CaN 0.260.230.740.04 0.15 0.011

99

Effect of solute elements and atomic pairs on the long term creep strength of carbon steels. Effects of solute elements in the ferrite matrix on the long term creep strength were examined [50] for sixteen heat treatments of 0.2 and 0.3 mass% C steels having minute and different contents of Mo, Mn, and Cr (Table 15). In the analysis, the creep rupture values at 773 K and 88 MPa were taken from the NRIM Creep Data Sheets [55]. The equilibrium composition of ferrite matrix at 773 K, which is the temperature of creep test, are calculated by Thermo-Calc. for sixteen carbon steels. Si shows the maximum solubility in the ferrite, ranging from 0.4 to 0.7 at. %, and the solubility decreases in the order of Mn, Cr, Mo, C, and N. Figure 18 shows correlations between the long term creep rupture life at 773 K and 88 MPa and the concentration of solute element in the ferrite matrix. The creep strength is proportional to concentrations of Mn and Mo and inversely proportional to concentrations of C and Si in solution.

The concentration of each atomic pair in the ferrite matrix was calculated by CAM. The relationships between the creep strength and the concentration of these atomic pairs were examined by the multiple regression analysis. In this analysis, the concentration of Cr-C pairs was not 1% significant, and the best regression equation of (27) was obtained by the concentrations of Mn-C and Mo--C atomic pairs.

Log(tR) = l.11XMn- c + 13.60XMo-c + 1.45, (27)

tR: the time to rupture (h) at 773 K and 88 MPa,

X Mn- C , XMo-c: concentration of Mn-C and Mo-C pairs (at ppm).


, r= 0.79 /

/g/ 0 r= 0.95 0

:2 5.0 0 0 .0 -.. 0

(!) / 0 ",

.... 0 0/ ","'0 B / ",60 0 0.- 60/ 2 ",

/ ",

.8 4.0 Ci ", 0 ",

(!) / () 0",6 a /

e /ßCCO ", 0

",

00 Mn ~"'O Mo 0

....l / ",

3.0 /

0.1 0.2 0.3 0.4 0.5 0 0.002 0.004 0.006 0.008 At. % Mn in ferrite At. % Mo in ferrite

~, 0 r= - 0.82 ,

0 r=-0.73 , , 0 p '~ 0 'QO :2 5.0 -..

B 4.0

0, 0 ' -..00 , ,8' ,

0 '~ 0

" , , ,

0', ,

,0 0> , 0' 'Q.

'0 c 0 , 80

, Si 08' , ,

I 13

'.0

~ 3.0

0.0030 0.0035 0.0040 0.0045 0.3 0.4 0.5 0.6 0.7

At. % C in ferrite At. % Si in ferrite

Fig. 18. Correlations between the creep rupture life at 773 K and 88 MPa and the concentration of solute element in the ferrit matrix.

A good correlation was observed between the creep rupture life predicted by this regression equation and the experimental ones (see Fig. 19). The rupture life of 16 steels estimated by the regression equation increases with increasing Mo content, but it saturates at a very minute content of about 0.03 mass% Mo as shown in Fig. 20. This can explain experimental results [49] very weIl. Thus, it is concluded that the long term creep strength of carbon steels is controlled by Mn-C and Mo-C atomic pairs. These M-C atomic pairs seem to have a strong interaction with dislocations. If a M-C pair is trapped at a jog of dislocation as shown in Fig. 21, the absorption of a vacancy at a jog will be prevented due to the strong binding energy of an M-C pair with dislocation. Ushioda and Hutchinson [56] analyzed the effect of solute atom, Mn, and Mn-C pairs on the climb rate of dislocations assuming that the climb rate is controlled by the rate at which vacancies diffuse to or from the dislocation. They reported that the climb rate of dislocations in the ferrite containing Mn-C pairs is about 10 times slower than that in pure Fe. Thus, the M-C pair seems to be very effective to reduce the climb velo city of dislocations.

Hidehiro Onodera

6~----~------~----~

log(tR ) = 1.11MnC + 13.6MoC+ 1.45

MCC = 0.928

tR:time to rupture /h 773K,88MPa

4 5

Observed log (tR)

6

101

Fig. 19. Correlation between estimated and observed values of the creep rupture life [50].

5.0 0 ~ ____ -9----------" !f CO---- 0

~ ""0 B4.0 I (\j

E '.0

CI)

W

0.1 0.2 0.3 Bulk Mo content (mass %)

0.4

Fig.20. Variation of the estimated creep rupture life with the bulk Mo content [50].

Extra half plane

000000 000000 000000

ICM)":O 0 Jog ~ L.....::: ..... r---

earbon

Fig. 21. Schematic of an extra half plane with a jog in an edge dislocation. The climbing of dislocation proceeds by the absorption of vacancies at jogs.


3.5 Summary

In this chapter, materials design methods are described for multi-component titanium alloys based on the prediction of microstructure by the thermodynamic calculation of phase equilibrium and the prediction of mechanical properties through various kinds of microstructural and compositional parameters. For the prediction of 0: + ß and 0: + 0:2 two phase structure, empirical methods were developed based on Ti-X (X = Al, V, etc.) binary phase diagrams, or based on analyzed composition data of 0: and ß phases in multi-component titanium alloys. The regular solution model also has been applied to design microstructure of these alloys. Superplastic titanium alloys having improved strength-to-density ratio have been developed successfully by optimizing such parameters as volume fraction, degree of solid-solution strengthening for the primary 0: phase and electron-atom ratio for the prior ß phase.

Improvements in the description of configurational thermodynamics have enabled appropriate evaluations of atomic configurations in the ordered and disordered phases. These advanced models are necessary for the better understanding of ordered and disordered alloys and the improvement of accuracy of alloy design. The latter half of this chapter describes an analysis of the preferential substitution si te of the third element in Ti3AI and Ti Al intermetallic compounds based on CVM and an analysis of effects of short range ordering in the bcc ferrite phase on the long term creep strength of carbon steels based on CAM.

The preferred substitution site of the third elements (Cr, Co, Ni, Mn, Mo, Nb and V) in the TiAI (Llo) and TiAI (D019 ) intermetallic compounds was investigated by the cluster variation method with a Lennard-Jones pair potential. The calculated substitution behaviors for Ni, Nb, V, Mn and Cr are in good agreement with those evaluated by ALCHEMI and X-ray diffraction methods.

Effects of solute elements in the ferrite matrix on the long term creep strength of carbon steels were studied from a viewpoint of atomic configurations such as atomic pairs. The equilibrium concentrations of solute elements and atomic pairs in the ferrite matrix were estimated by thermodynamic calculations. A good correlation is observed between the long term creep strength and the concentrations of Mn-C and Mo-C atomic pairs suggesting that the long term creep strength of carbon steels is controlled by Mn-C and Mo-C atomic pairs.

Acknowledgments

The original studies described here are the products of enjoyable collaborations with Mr. T. Abe, Dr. M. Yamazaki, Dr. T. Yamagata, Dr. Y. Ro, Mr. S. Nakazawa, Dr. K. Ohno, Dr. K. Kimura, Dr. C. Tanaka. I am very grateful to all of them. Some parts of the work described here were done in relation to the national project, "Advanced Alloys with Controlled Crystalline Structures" (1982-1988), sponsored by AIST and MIT!. Other parts were founded by the NRIM under grants No. 01-21-21, No. 09-00-07 and No. 08-51-01.

Hidehiro Onodera 103

References

1. C.H. Hamilton, G.E. Stacher: Met. Prog., 109, 34 (1976). 2. M. Murakami: Proc. of 4th Int. Gonf. on Titanium, 153 (1980). 3. H. Onodera, K. Ohno, T. Yamagata, M. Yamazaki: Tetsu-to-Hagane 72 284 (1986)

(in Japanese). 4. H. Onodera, S. Nakazawa, K. Ohno, T. Yamagata, M. Yamazaki: Gomputer Aided

Innovation of New Materials, Proc. of GAMBE, 835 (North-Holland, Amsterdam, 1991).

5. H. Onodera, K. Ohno, T. Yamagata, M. Yamazaki: Trans. ISIJ 28 803 (1988). 6. R.F. Decker: Proc. Symposium on Steel Strengthening Mechanics, (Climax Molyb-

denum Co., Connecticut, 1969). 7. H. Harada, M. Yamazaki: Tetsu-to-Hagane 65 1059 (1979) (in Japanese). 8. T. Nishizawa, M. Hasabe: Tetsu-to-Hagane 67 1887 (1981) (in Japanese). 9. M. Hillert, M. Waldenstrom: Scand. J. Met. 6211 (1977).

10. M. Hillert, L.I. Staffansson: Acta Chem. Scan. 24 3618 (1970). 11. H. Onodera, S. Nakazawa, K. Ohno, T. Yamagata, M. Yamazaki: ISIJ Int. 31 875

(1991). 12. H. Onodera, T. Abe, S. Nakazawa, T. Yamagata, M. Yamazaki, T. Tsujimoto: ISIJ

Int. 33 793 (1991). 13. H. Onodera, K. Ohno, T. Yamagata, M. Yamazaki: Proc. 6th World Conf. on

Titanium, 1191 (Les Editions de Physique, Paris, 1989). 14. R. Kikuchi: Phys. Rev. 81 998 (1951). 15. E-H. Foo, C.H.P. Lupis: Acta Metall. 21 1409 (1973). 16. O. Izumi, M. Kobayashi: Bull. Jpn. Inst. Met. 25 8 (1986) (in Japanese). 17. A.I. Khorev: Proc. 3rd Int. Conf. on Titanium, 2111 (AlME, Plenum Press, New

York and London, 1976). 18. L. Kaufman, H. Bernstein: Computer Calculation of Phase Diagrams (Acadernie

Press, New York and London, 1970). 19. L. Kaufman: CALPHAD 1 7 (1977), CALPHAD 2 117 (1978), and CALPHAD 3

45 (1979). 20. L. Kaufman, H. Nesor: CALPHAD 2 55 (1978), and CALPHAD 2 82 (1978). 21. L. Kaufman, H. Nesor: CALPHAD 2 295 (1978), and CALPHAD 2 325 (1978). 22. M. Hansen: Constitution of Binary Alloys (McGraw-Hill, New York, Toronto and

London, 1958) 23. H. Onodera, T. Yokokawa: ScrL Metall. Mater. 24 1119 (1990). 24. H. Onodera, M.Yamazaki: Tetsu-to-Hagane 76 307 (1990) (in Japanese). 25. H. Onodera, K. Ohno, T. Yamagata, T. Ohkoshi, M. Yamazaki: Tetsu-to-Hagane

74123 (1988) (in Japanese). 26. J. A. Wert, N. E. Paton: Metall. Trans. A 14 2535 (1983). 27. H. Sasano, H. Kimura: Proc. 4th Int. Conf. on Titanium, Kyoto, Japan, 1147

(1980). 28. M.Yamazaki: Prog. Powder Metall. 41 531 (1986). 29. H. Onodera, Y. Ro, T. Yamagata, M. Yamazaki: in Titanium, Bcience and Tech

nology, 5th World Conf. on Titanium 1983 (Deutsche Gesellschaft für Metallkunde, 1984).

30. T. Yamauchi, T. Kimura, Y. Nishino: Proc. 6th World Conf. on Titanium, 1439 (Les Editions de Physique, Paris, 1989).

31. M.J. Blackburn, J.C. Williams: Trans. Am. Soc. Mech. Erg. 62 299 (1969).


32. C.G. Rodes, N.E. Paton, M.W. Mahoney in Titanium, Science and Technology, 5th World Conf. on Titanium 2355 (Deutsche Gesellschaft fur Metallkunde, 1984).

33. D.F. Neal: Proc. 6th World Conf. on Titanium, 253 (Les Editions de Physique, Paris, 1989).

34. C. Ramachandra, V. Singh: Metall. Trans. A 13 771 (1982). 35. F.H. Froes, C. Suryanarayana, D. Eliezer: ISIJ Int. 31 1235 (1991). 36. J.M. Sanchez, J.R. Barefoot, R.N. Jarrett, J.K. Tien: Acta. Metall. 32 1519 (1984). 37. M. Enomoto, H. Harada: Metall. Trans. A 20 649 (1989). 38. H. Onodera, T. Abe, T. Yokokawa: Acta Metall. Mater. 42 887 (1994). 39. R. Kikuchi: Final Report, No. NB80NAAE0188, Hughes Research Labs, Calif., 31

(1981). 40. J.L. Murray: ASM Int., Materials Park, Ohio, 1987. 41. T. Abe, T. Yokokawa, H. Onodera: Proc. Int. Conf. on Computer-assisted Materi-

als, Design and Process Simulation, 308 (ISIJ, Tokyo, 1993). 42. H. Doi, K. Hashimoto, K. Kasahara, T. Tsujimoto: J. Jpn Inst. Met. 56232 (1992). 43. E. Mohandes, P.A. Beaven: Scri. Metall. Mater. 25 2023 (1991). 44. S.C. Huang, E.L. Hall: Acta. Metall. Mater. 39 1053 (1991). 45. H. Doi, K. Hashimoto, K. Kasahara, T. Tsujimoto: J. Jpn Inst. Met. 53 1089

(1989). 46. Y. Hotta, T. Sana, M. Nemoto: Abstracts of the Japan Institute of Metals 112

350 (1993). 47. H. Onodera, T. Abe: in Titanium '95, Science and Technology, Proc. 8th World

Conf. on Titanium, 80 (The Institute of Materials, The University Press, Cambridge, 1995).

48. K. Kimura, H. Kusima, K. Yagi, C. Tanaka: Tetsu-to-Hagane 77 667 (1991) (in Japanese).

49. K. Kimura, H. Kusima, K. Yagi, C. Tanaka: Tetsu-to-Hagane 81 757 (1995) (in Japanese).

50. H. Onodera, T. Abe, M. Ohnuma, K. Kimura, M. Fujita, C. Tanaka: Tetsu-to-Hagane 81 821 (1995) (in Japanese).

51. B. Sundman, B. Jansson, J.-O. Andersson: CALPHAD 9 153 (1985). 52. Harue Wada: Metall. Trans. A 17391 (1986). 53. Harue Wada: Metall. Trans. A 16 1479 (1985). 54. H. Abe, T. Suzuki, S. Okada: Trans. Jpn. Inst. Met. 25 215 (1984). 55. NRIM Creep Data Sheet 7B, (1992) and 17B, (1994). 56. K. Ushioda, N.B. Hutchinson: Report of Research Committee on Low Carbon Steel

Sheets, Iron and Steel Institute of Japan, 64 (1987) (in Japanese).


Hiroshi Ohtani

Center for Interdisciplinary Research, Tohoku University, Send ai 980-8578, Japan

Abstract. Abrief outline of the CALPHAD method, which has been developed to alleviate the difficulty in obtaining phase diagrams by experiment alone, is presented. This method enables calculation of stable and metastable phase equilibria, as weil as thermodynamic properties such as activity, enthalpy, driving force for precipitation etc., on thermodynamic grounds. Some results using the CALPHAD approach are illustrated, taking examples from compound semiconductors and solder materials.

4.1 Outline of the CALPHAD Method

The various functions of a material are closely related to the phases and the structures of the material composition. Therefore, to develop an improved material with a maximum level of desired functions, it is essential to undertake design of the structure in advance. Phase diagrams offer the most basic and important information for the design of such new materials. As seen in Fig.l, research concerning phase diagrams of metals is thought to have originated in observation of the microstructures by Sorby in the latter half of the 1800s [1]. Subsequently, epochal findings and inventions such as the idea of phase rule

" GI c ·E

4000

~ 3000 GI (/)

E GI Cii &; 2000

Ö Gi oe E i 1000 experimental phase diagrams

O~ ____ ~~~~ ________ -L __________ ~ __ __

1850 1900 1950

Year Fig. 1. History of CALPHAD.

2000

Springer Series in Materials Science Volume 34, Ed. by T. Saito © Springer-Verlag Berlin Heidelberg 1999

106 CALPHAD Approach to Materials Design

or thermal analysis with a thermoelectric couple, etc., were achieved. Research on experimental phase diagrams was actively pursued during this period. It is well known that Hansen [2] published a compilation of binary phase diagrams in 1936 as a summary of this research. Investigation of the main binary phase diagrams was practically completed, considering the fact that more than 800 binary systems were compiled by Hansen in his book. Thereafter, studies on phase diagrams tended to be done outside the field of metallography. One of the main reasons for this could be that considerable time and labor are required in construction of even a partial region of a phase diagram, because practical materials, unlike ternary systems, are composed of multicomponent alloys.

To overcome this difficult situation, a method of calculation of phase diagrams was advocated, and an international research group, CALPHAD (CALculation of the PHAse Diagrams), was organized in 1973. CALPHAD was originally the name of this research group; recently, however, it has co me to indicate the technique by which a phase diagram is calculated on the basis of thermodynamics. In this method, a variety of experimental values concerning the phase boundaries and the thermodynamic properties are analyzed according to an appropriate thermodynamic model, and the interaction energies between atoms are evaluated [1]. By using this technique, phase diagrams outside the experimental range can be calculated based on thermodynamic proof. Difficulty in extension of the calculated results to higher-order systems is much less than in the case of experimental work, since the essence of the calculation does not change so much between a binary system and a higher-order system. It is also advantageous that the metastable phase equilibria or the important thermodynamic factors such as driving force and chemical potential, etc. can be obtained. However, it should be noted that this technique cannot be used to forecast the appearance of a phase of which the thermodynamic function has not been evaluated.

4.2 Procedures of the CALPHAD Method

4.2.1 Thermodynamic Modeling of Alloy Systems

The free energy of each phase appearing in an alloy system is often approximated by the regular solution model. The superiority of this model lies in its description of the thermodynamic properties even in a considerably complex system because of its simple formalism. The regular solution model has most frequently been adopted in the CALPHAD method for this reason [3]. Besides this thermodynamic model, the sublattice model [4], in which sublattices are introduced into the regular solution approximation, also has been used. This model is characterized by its good description of thermodynamic properties, especially of the ordered structure. However, some inaccuracy is still unavoidable as to the entropy term in these models, mainly due to its assumption of the random atomic arrangement in the lattice points. Thus, the cluster variation method (CVM), in which an accuracy of the entropy term is greatly improved by using an atomic cluster, was proposed [5], and the application of this model was attempted especially in the order-disorder transition of metals.

Hiroshi Ohtani 107

4.2.2 Regular Solution Model

The selection of a thermodynamic model by which the Gibbs free energy of an aIloy system is described is the most important factor when using the CALPHAD method. In a system in which interaction between aIloying elements is not so strong, the regular solution model is weIl known to describe the thermodynamic properties of the aIloys comparatively weIl. For instance, the free energy of the A-B-C ternary aIloy is expressed as

Gm = XA °GA + XB °GB + Xc °Gc

+RT(xAlnxA + xBlnxB + xclnxc)

+XAxBLAB + XBxCLBC + xAxcLAC + XAXBXcLABC,

(1)

where °Gi is the Gibbs energy ofthe pure component i in the standard state, and Xi is the mole fraction of i. The symbol L shows the inter action energy and LAB, for instance, denotes that for A atoms and B atoms. Then, from a qualitative point of view, the first term of (1) represents the energies of mechanical mixing, the second term is the ideal mixing entropy, and the third stands for the excess energy which shows deviation from the ideal solution. These thermodynamic parameters are evaluated so as to fit the various experimental information, i.e. that regarding the phase boundary, specific heat, activity, heat of formation, and so on. The chemical potential G is obtained by using

GA = °GA + RTlnxA + (1- XA)xBLAB + (1 - xA)xcLAC

-xBxcLBC + (1 - 2XA)XBXc L ABC

GB = °GB + RTlnxB + xA(1- xB)LAB - xAxcLAC

+(1 - xB)xcLBC + xA(l - 2XB)Xc L ABC

Gc = °Gc + RTlnxc - XAxBLAB + xA(l- XC)LAC

+XB(1- xc)LBc + xAxB(l - 2xc)LABC .

(2)

(3)

In the calculation of the phase equilibrium, it is convenient to use these chemical potentials as the basic equations. For instance, to obtain equilibrium between phase A and phase B, a combination of the compositions, which satisfy (4), should be calculated by using the numerical analytic method.

(4)

where G~ represents the chemical potential of element i in phase j.


4.2.3 Free Energy Change due to Second Order Transition

In real alloy systems, there are some seeond order transitions that signifieantly affeet the phase equilibria. The magnetie transition is a typieal example. In the CALPHAD method, the Gibbs energy of the magnetie phase is ordinarily divided into two terms, non-magnetic and magnetic, as follows [6,7J:

c= Cm+Cmg . (5)

The term C mg in (5) is the magnetie eontribution to the Gibbs energy whieh is deseribed by the following equation:

cmg = RTln(ß + 1) . f(7)

f( ) -1 [797-1 474(1 1)(73 7 9 7 15 )J/ 7 - - 140p + 497 P - 6"" + 135 + 600 a

f(7) = _(71~5 + 73~~5 + ~;~~)/a for 7< 1,

for 7> 1,

(6)

(7)

where a = 151~85 + ~;~~~ (~ - 1), and p is a constant whieh is given as 0.28 for fee met als and 0.40 for bee metals. 7 is defined as T /TCurie, where TCurie is the Curie temperature for ferromagnetie ordering, and is represented as (8) in the A-B-C ternary system:

TCurie = °TA . XA + °TB . XB + °Tc . Xc

+TAB . XAXB + TAC . XAXC + TBC' XBxC + TABC' XAXBXC, (8)

where °Ti is the Curie temperature of element i. The symbol T ij denotes the eoneentration dependenee of the Curie point in the i-j binary system, while TABc is that in the ternary system. The same expression is applied to the Bohr magneton number, ß, as folIows:

ß = °ßA' XA + °ßB' XB + °ßc' Xc

+ßAB . XAXB + ßAC . XAXC + ßBC . XBXC + ßABC . XAXBXC (9)

4.2.4 N umerical Calculation of Phase Equilibria

The Newton-Raphson method, a numerieal eomputational method for solving simultaneous equations, is often used to obtain phase equilibria. The prineiple of this method ean be explained using an example from the function y = f (x) in whieh only one variable is eontained. Consider the ease where this function has the solution f(a) = 0, as shown in Fig.2. The tangent to the eurve whieh passes through the point (xo, f(xo)) and has a slope of f'(xo) is deseribed as

y = f(xo) + f'(xo)(x - xo) .

The intereept on the abscissa, Xl, is straightforwardly given by

f(xo) Xl = Xo - f'(xo)

(10)

(11)

Hiroshi Ohtani 109

y y = f(x)

Fig. 2. Principle of Newton-Raphson method.

This value is the first approximation to the equation f(x) = 0, and it is used as the new initial value for the subsequent iteration. Consequently, the nth order approximation is obtained from the following equation:

f(x n ) xn+l = X n - f'(xn ) (n = 1,2",,) (12)

xn+l is considered to be the solution of f(a) = 0, provided that the value I Ptx:\ I settles within the given range of accuracy, E.

This principle is applicable to equations having more than two variables in a similar way. For the sake of simplicity, consider the case of equilibrium between the a and ß phases in the A-B binary system. Provided that the Gibbs energy for each phase is described by the regular solution model, the chemie al potentials for A and Bare derived from (2) as folIows:

(13)

For simplicity, the interaction parameter LAB does not depend on the composition, but merely on the temperature. Considering the rest raint condition given by (14), there are two independent variables in the equilibrium, for instance, xB and x~.

XA + XB = 1

The equilibrium condition is expressed by

(14)

(15)


Two independent variables, x'B and x~, are determined unequivocally by solving (15) at the constant temperature T. The objective function f(X) is then defined in the form of a matrix as follows:

(16)

where

G~ - G~ = (oGA - °G~) + RTln(l - x'B)/(l - x~) + (x'B)2 LAB - (X~)2 L~B G~ - G~ = (oG'B - °G~) + RTlnx'B/x~ + (1 - x'B? LAB - (1 - X~)2 L~B .

(17) The tangent of the function j'(X) is also defined as

(18)

where

(19)

If the relative difference between the iterative values is defined as (20), the nth order approximation of the equilibrium compositions is expressed by (21).

where

( -(G~ - G~)) -(G~ - G~) ,

(20)

(21)

The equilibrium compositions are obtained provided that the value C converges within the given limit.

Hiroshi Ohtani 111

4.2.5 Experimental Data Used for Obtaining Parameters

In the CALPHAD method, the thermodynamic parameters used in the expression of Gibbs energy are determined on the basis of the various experimental data such as phase boundaries, activities, specific heats or enthalpies. Some examples of the derivation of such parameters from experimental data will be shown in this section.

Specific Heat. Experimental data on specific heat yields the lattice stability of a pure element. Consider the case that the specific heats of solid (s) and liquid (R) for an element J are given at constant press ure as folIows:

CJ = AS + BS. T

C5 = AR +BR·T

The enthalpy change between sand R at T (K) is given as

L1H:r-+R = L1HJ (Tm) + {L L1CJ-+RdT

= L1HJ (Tm) + (AR - AS) . (T - Tm) + Bl;B S • (T2 - T;,),

(22)

(23)

where Tm and L1HJ (Tm) are the melting temperature and latent heat of fusion of the element, respectively, and L1CJ-+R equals c5 - CJ. The entropy change and free energy change are also given as

L1GT-+l' = L1H:r-+l' - T· L1S:r-+R . (25)

For instance, the specific heats of fee (j Pb and liquid Pb are approximated by the following equations [8]:

Ctb = 23.55 + 9.74 x 1O-3T (J mol-1 K- 1)

C~b = 32.49 - 3.09 x 1O-3T (J mol-1 K-1) (26)

Utilizing the thermodynamic data on the melting temperature Tm (600.6 K) and the latent heat offusion L1Hpb (Tm) ofPb (4870Jmol-1) [8], the changes offree energy, enthalpy, and entropy between the liquid and solid states are given as folIows.

L1GT-+R = 1815 + 50.32T + 5.965 x 1O-3T2 - 8.94T In T

L1H:r-+R = 1815 + 8.94T - 6.415 x 1O-3T2

L1S:r-+l' = -41.38 - 1.238 x 1O-2T + 8.941n T

(J mol-I)

(J mol-I)

(Jmol-1K-1)

(27)


Table 1. Experimental activity values of As in the As-Ga binary system.

Activity. The interaction parameters included in the free energy expression can be directly evaluated from the experimental activity values. For example, the activities of As over the As-Ga binary liquid have been determined as shown in Table 1 [9]. The liquid state of the elements is adopted as its standard. If the composition dependency of the interaction parameter is expressed as (28), the chemical potential of As in the binary system can be derived as (29).

(28)

-e _ ° e e GAs - GAs + RT In aAs

= oG),s + RT In x),s + (1 - x),s)2[O LAsGa + (4x),s - 1)1 LAsGa] (29)

This is straightforwardly arranged as

(30)

Therefore, if the left-hand side of the equation is calculated from the activity data in this table and each value is plot ted against x),s, the interaction parameters ° LAsGa, 1 LAsGa can be obtained, respectively, from the intercept of the ordinate and the slope of the straight line as follows:

(31)

4.3 Phase Diagrams of the Compound Semiconductors

The group Ill-V compound semiconductors and their solid solutions are important materials for optoelectronic and high-speed electronic device applications, because the mobility of the electrons in the semiconductors is several to hundreds of times larger than that of ordinary Si. Furthermore, the band-gap energy spreads widely enough to produce optoelectronic devices, covering a wide range of wavelengths from the visible region to the far infrared area. The compound semiconductors in practical use are composed of the same amount of column III elements (Al, Ga, In) and column V elements (P, As, Sb). An application of the II-VI compounds to electronic devices has been also attempted.

To form a multilayer structure by accurately controlling the composition and electronic properties of compound semiconductors, several techniques are employed such as vapor phase epitaxial (VPE) growth, molecular beam epitaxial (MBE) growth, and liquid phase epitaxial (LPE) growth. Especially in the LPE and VPE methods, phase diagrams play a key role in controlling the chemical compositions of the growing crystals. When bulk single crystal is grown by the

Hiroshi Ohtani 113

LPE method, information on the equilibrium between the compound and the liquid phase is quite useful. A schematic diagram of LPE growth is shown in Fig. 3. It is clear from the figure that a liquid alloy with composition Y is necessary to develop a semiconductor crystal with composition x at temperature Ta on a substrate. In this section, recent results of research on phase equilibria in compound semiconductors will be introduced.

(a) (b)

~ = -Liquid y

Liquid

~ -ompound x ~ To .. : ............................... .

8 I ~ r ~: __________ ----

E-o 1-1 Compound

y

Fig. 3. Schematic diagram of LPE growth.

4.3.1 Description of the Gibbs Energy of the Compound Semiconductors

x AB

The Gibbs energy of the liquid phase is described by the regular solution model. On the other hand, the compound semiconductors generally have a zincblende type of crystal structure. The structure is divided into two fee sublattices: one is occupied by group III or II atoms and the other is occupied by group V or VI atoms. The sublattice model is often used to describe the Gibbs free energy of such an ordered structure. In this model, it is assumed that the crystal lattice is separated into several sublattices with an equivalent relation and that the atoms are arranged at random on each sublattice. For instance, consider the case where atoms A, Band C occupy one sublattice while atom X occupies the other sublattice. The chemical formula of this ternary solution is (A,B,C)X, provided that the size of each sublattice is equivalent. The Gibbs free energy of this solution per mole is given as:

GS = YA °G AX + YB °G BX + Yc °GCX

+RT (YA lnYA + YB lnYB + Yc lnyc)

+YAYBLA,B:X + YAYCLA,C:X + YBYcLB,c:x

(32)

° Gij denotes the formation energy of the ij compound, while Li,j:k is the atomic interaction energy between i atoms and j atoms, provided that all of the sites of one sublattice are completely occupied by the k atoms. Yi shows the mole fraction


of component i on the first sublattice and is converted by using ordinary mole fraction Xi as Yi = xi/2.

The Gibbs energy of the quaternary solution phase (A,B)(X,Y) is given in a similar way as:

GS = YAYX °GAX + YAYY °GAY + YBYX °GBX + YBYY °GBy

+RT (YA InYA + YB InYB + Yx lnyx + YY Inyy)

+YAYBYxLA,B:X + YAYBYyLA,B:Y + YAYxYyLA:X,Y + YBYxyyLB:x,Y, (33)

where Yi shows the mole fraction of the component i, and it is related to Xi in the following equations:

YA = XA/(XA + XB)

YB = XB/(XA + XB)

Yx = XX/(XX +xy)

YY = xy/(XX +xy)

(34)

4.3.2 Solid/Liquid Equilibria in 111-V Compound Semiconductors

Figure 4a,b show phase diagrams of Ill-V-V and III-III-V pseudobinary systems [10]. It can be seen from these figures that a peritectic type with a large

(a) (b)

:r~b-==H "., AlP AlAs AlSb AlP

r:r==~~ ~500~ f- l000G~aP~--~rc'-'A-C-~S--_-----''-; rG~S~b'----_~_lG P

.. I ~ I~nP~--~IunA~s--~I~nS~b--~InP

Fig.4. Phase diagrams of 18 pseudobinary systems: (a) III-V-V and (b) III-III-V systems.

miscibility gap is formed in the AIP-AISb, GaP-GaSb, InP-InSb and GaAs-GaSb systems. Figure 5 shows the liquidus isotherms of the 18 ternary systems. The numerals on the diagrams denote the temperatures of the calculated sections. Calculation of the liquidus and sol idus surfaces of the 15 quaternary systems are shown in Fig.6a,b, respectively. It can be seen from Fig.6b that a compound phase with various compositions can be obtained without difficulty from

Hiroshi Ohtani 115

Fig. 5. Liquidus surfaces of 18 ternary systems, where numerals indicate temperature (x 100 oe).

AlP

Fig. 6. Liquid/solid phase equilibria of 15 quaternary systems, where numerals indicate temperature (xI00°e): (a) liquidus surfaces; (b) solidus surfaces.

the III-III-P-As, III-III-III-Sb, and III-III-III-As systems due to the 1arge solubility range of the compound. On the other hand, Fig. 6b suggests that the compound phase grown by the LPE method is 1imited to a quite narrow range in the III-III-P-Sb, III-III-As-Sb, III-P-As-Sb, and A1-Ga-In-P systems.


4.3.3 Miscibility Gap in the III-V and lI-VI Compounds

The growth of semiconductors by the LPE method seems easy in the alloy systems in wh ich a homogeneous solution forms. However, phase separation in the solid phase is often observed in the compound semiconductors, and this prevents stable growth of alloys. An attempt is made to investigate the features of the compound semiconductors in this section.

The parameters needed to calculate the phase separation in the 111-V compound systems are weIl established [10], while those necessary for the li-VI compound semiconductors are unknown. Therefore, they are estimated in the following way. The Gibbs energies of formation for the binary ij compounds with the zincblende structure are evaluated from the literat ure data. Since the compounds CdS and CdSe have the wurtzite structure, the hypothetical Gibbs energies of formation for the zincblende structure are estimated from the experimental phase boundaries [11]. On the other hand, information on the interaction energies between these component compounds is limited. The miscibility gap in the ZnS-ZnTe and CdS-CdTe systems has been experimentally investigated, and the interaction parameters can be evaluated. For the other pseudobinary systems where experimental information is not available, the interaction energies are estimated from the difference between the lattice constants of the binary compounds. Several models have been developed to estimate the interaction parameter between ij and ik compounds using lattice constants [12-14], and they all lead to the following relation:

L.. = K (L1aij _ ik )2 t.J,k - ,

aij-ik (35)

where K is a coefficient depending on the modulus of elasticity, L1aij-ik is the lattice mismatch, and aij-ik is the average value of the lattice constants in the ij-ik complex compound. Combining all the average values obtained for the interaction parameters either from experiments or by estimation, the value of K is determined to be 1.97 x 106 J mol-I, as shown in Fig. 7 [11].

Calculated isothermal sections of the miscibility gap in the 111-V and li-VI compound semiconductor systems are shown in Fig.8a,b [11]. The critical solution temperatures of the miscibility gaps in the (A,B,C)X and (A,B)(X,Y) systems are also, respectively, indicated by angle brackets and square brackets in the figure. The origin of the miscibility gap in the (A,B,C)X-type compound is mainly attributed to the difference in lattice parameters of the component compounds. In the (A,B)(X,Y) compounds, miscibility gaps arise from a difference in relative stabilities of the binary compounds, as described below.

A schematic ofthe Gibbs energy surface ofthe (A,B)(X,Y) system is shown in Fig. 9. An unstable region does not exist in this alloy system, as long as the energy surface is downwardly convex throughout all the composition area. However, if part of the energy surface is upwardly convex as shown in the figure, the area is thermodynamically unstable. The condition of stability of the solid solution accompanying small compositional perturbations of the system is expressed as

Hiroshi Ohtani

40.-------,--------.--------,--------.

... . '0 E 30

-... ~ E 01 tu 20

A-c .2 Ü f! S .5 10

• Kisker & Zawadzki Tomashik et al.

... Obata et al.

Hg(S,Te)

Zn(S,Te) ~

... Cd(S,Te)

Zn(Se,Te) (Zn Cd)S Cd(Se,Te) .(Zn,H'g)Se

Hg(Se,Te) • (Zn Cd)Se Hg(S,Se (Zn,Hg)Te

(Zn,Hg)Te Zn(S,Se)

o 4.0 8.0 2 3

(t.a/a)x10

12.0 16.0

117

Fig.7. Correlation between interaction parameter and misfit factor in the lI-VI alloy semicond uctors.

AlP CdS

HgSe

Fig.8. Calculated miscibility gap in the (a) TII-V and (b) TI-VI compound semiconductor systems.

follows [15J:

(Pe ä2e äy~ . äy'i - (36)


G

AX BX

Fig.9. Schematic of the Gibbs energy surface of the (A,B)(X,Y) system.

Therefore, the boundary between the metastable and unstable phase regions, i.e. the spinodal curve, is given by the condition

ä 2G ä2G ( ä 2G )2 äy~ . äYk - äYAäyx = 0

(37)

The Gibbs energy for the solid solution is represented by (33). For the sake of simplicity, we introduce the following assumption for the interaction parameters.

LA,B:X = LA,B:Y = LAB, LA:X,y = LB:X,y = L xy . (38)

Considering the conditions of YB = 1 - YA and yy = 1 - Yx, the following equations can be derived readily.

ä2GS (1 1 ) RT -- =RT -- + - -2LAB = -- -2LAB, äy~ 1 - YA YA YAYB

ä2 Gs (1 1 ) RT -2- = RT --- + - - 2Lxy = -- - 2Lxy , äyx 1 - Yx Yx YxYy

(39)

ä2 Gs DG DG DG DG - AG ä ä = AX - AY - BX + BY = L.l •

YA Yx

If we further assume an extreme case such as LAB = Lxy = 0, the combination of (37) and (39) yields the critical solution temperature as shown by

LlG 1/2 Tc = R . (YAYBYXYY) . (40)

The values of LlG for the quaternary II-VI systems are illustrated in Fig.lO [11]. With some exceptions, it is generally true that, the larger the absolute value of LlG, the higher the critical temperature of the miscibility gap in the compound. For instance, the absolute values of LlG are quite large in (Zn, Cd)(8, Te) and (Zn, Hg) (8, Te) systems and so also are their critical temperatures.

Hiroshi Ohtani

0

... I

'0 -20 E -..,

~

CJ <l

-40 -

-60 -

(Cd,Hg)(S,Se)

(Zn,Cd)(Se,Te)

1000 2000

Temperature I K

3000

119

Fig. 10. Variation of LlG for the quaternary lI-VI systems with temperature.

4.3.4 Phase Equilibria in Thin Film of Semiconductors Grown on Substrate

In the case of epitaxial growth on a substrate, it is generally believed that the composition of the grown phase usually deviates from that of the equilibrium in the bulk crystal. This is due to the effect of strain energy arising from the lattice mismatch between the thin film and substrate. A thermodynamic treatment of such a non-equilibrium calculation will be shown in this section, taking an example from the growth of the Ga(As, Sb) thin film on some substrates [16].

We consider the growth of the compound phase, of which the lattice constant a is greater than that of the substrate ao. While the thickness of the growing crystal is small, the epitaxial layer grows coherently to the substrate, as shown in Fig. lla. The lattice mismatch yields strain energy in the growing layer, which increases as the layer thickens. Beyond a certain critical thickness, amisfit dislocation is introduced at the interface of the layer and substrate, and the accumulated strain energy is released as seen in Fig. llb,c. The strain energy is defined by the following equations according to the layer thickness:

( 41)

In the above equations, f.L is the shear modulus, v the Poisson's ratio, Vm the molar volume, hc the criticallayer thickness, and h the layer thickness of the film. The value f is the misfit factor between the epitaxial layer and the substrate,

120

rilical -Iayer lhickness


( ) c :J

----;-

.x x

.x x - -(b)" .... f-e1 ..

(a) Thin film

),

u b lmle

Sirain energy

Fig. 11. Schematic change of strain energy accumulated in thin films.

and is given as a - ao f= -- . (42)

ao The lattice constant of the growing layer, a, is assumed to vary linearly with the composition of the compound. This treatment is generally called Vegard's law. For instance, the lattice constant of the complex compound (A,B)X is expressed by (43).

( 43)

where YBX is the atomic fraction of BX in the compound. According to (41) through (43), the strain energy accumulated in a thin film is described as a function of the layer thickness and the composition. The strain energy in a GaAsSb film grown on a GaAs(100) substrate is calculated as shown in Fig. 12a in case of the thickness of 0.1 !-lm and 1 !-lm.

The criticallayer thickness is given by Yokogawa et al. [17] as follows:

Ll I-v 1 hc =-'--'-

2f-L 1 + v J2 (44)

The value Ll is the energy barrier, assumed to be 0.8Jm-2 . A comparison between the calculated critical layer thickness and the measured values is shown in Fig.12b.

Calculated pseudo-binary phase diagrams ofthe GaSb-GaAs compound growIl on the GaAs(lOO) substrate are shown in Fig. 13a, provided that the layer thickness is 0.1 !-lm and 1 !-lm, respectively. As seen in Fig.12a, the strain energy

Hiroshi Ohtani

(a) GaAs- GaSb/GaAs

1500

°OL--L--L-~--~~O.-5~~-L--~-L~

GaAs fraction Y GaA,

(b)

10'

Yokogawa et a1. Matthews People et a1.

o "(In,Ga,_, )As/GaAs • "Ga(AsO.sPO.5)/GaAs

" Ga(AS0.4PO.6)/GaP *Si1_xGe,/Si

~ 100 "Si/GaP ~ "SilGaAs .[S 10-' ~. "ZnSe/GaAs ":: \ '"Experimental data

'.

!:~:: \ ...... ,." .. ~",~""", d~:8:: m~ ~ o

121

~ 10-4 O~~~~2~~~~4~~~-6L-~~

Misfit f I %

Fig.12. (a) Calculated strain energy in GaAsSb film on GaAs(lOO) substrate, and (b) comparison between calculated critical layer thickness and measured values.

Lattice matching

(a) composition (b)

1200 1200

1000

1000 matching composition

800

P600 U BOO 0 - -1!! 1!!

-=400 :::I

I! 1;; 8,1200

~ 1200 0-

E E GI GI I- 1000 I-

1000

800

600 800

- Calculated (Thin film)\ - Calculated (Bulk)

400 L-~~ __ ~~ __ ~-L __ ~-L~~W

GaSb 20 40 60 80 GaAs GaAs I atomic %

GaSb 20 40 60 80

GaAs I atomic % GaAs

Fig.13. Calculated pseudo-binary phase diagrams of GaSb--GaAs grown on (a) GaAs(lOO) substrate and (b) InP(100) substrate.

increases as the GaAs fraction in the compound and thickness of the thin film decreases. Thus, the phase boundary on the GaSb side deviates toward the GaSb side. As the thickness decreases, the peritectic temperature becomes accordingly lower. Figure 13b shows the phase equilibria in the GaSb-GaAs thin film on an InP(lOO) substrate, where the postulated thickness of the film is 0.1 f.!m and


1 !-lm. The lattice constant of the pure InP is 0.58688 nm, almost the average value oft hose for GaAs (0.56533nm) and GaSb (0.60959nm). The strain energy disappears at YGaAs >:::; 0.512, where the lattice constant of InP equals that of the compound. The single phase region of the zincblende-type compound forms in the vicinity of this composition, since homogeneous growth is not prevented by the lattice mismatch. The tendency is more remarkable when the thickness of the film is small. Formation of a new homogeneous phase was observed by Quillec et al. [18] when the 49 mol % GaAs compound was grown using the LPE method. Calculated results are in good agreement with the experimental data.

4.4 Phase Diagrams of the Solder Alloys

Soldering is defined as a metallurgical method of joining using solder with a melting temperature below about 425°C. In the electronics industry, Pb-Sn eutectic alloy has been chiefly used as the microsolder material in high density mounting of devices. This alloy is characterized by a low melting temperature as weIl as its excellent wettability and connectability. Therefore, the influence of heat on electronic devices can be limited to a quite narrow region. However, various methods of connecting the semiconductor element have been recently developed. Therefore, to deal with new connection technologies such as the controlled collapse chip connection (C4), it is insufficient to use only this binary alloy. Thus, the development of multicomponent alloys employing combinations of new elements and having special functions is urgently required.

In addition to these technical aspects, familiarity with such subjects as environmental problems and product liability are recently required in the electronics industry. This situation extends to solder materials used in the assembly of electronic circuits, the restriction of Pb usage in solder material being a typical example. Pb is a useful metal, being used for batteries, ammunition, etc., at a rate of about 6000000 tons a year. On the other hand, its toxicity is extremely high and pollution of the underground water supply by Pb, originating in industrial wastes, is becoming a serious problem. To resolve this dilemma, efforts to develop new solders equal in quality to the commercial Pb-Sn alloys have been accelerated in recent years.

When a new solder material is designed, it is necessary to analyze it from various points of view including melting temperature, range of melting zone, wettability, fatigue properties, deformability, mechanical properties, and so on. Since the melting temperature, the range of the melting zone, and the wettability are closely related to the thermodynamic properties of alloys, accurate information on these properties can be obtained from phase diagrams. In research on the phase diagrams of various alloys, the CALPHAD method has been widely employed. In this chapter, the phase diagrams of Pb-bearing and Pb-free solders calculated using this method are introduced.

Hiroshi Ohtani 123

4.4.1 Phase Diagrams of Pb-bearing Solders

Alloys of the Pb-Sn based system form a well-known group of low-melting point, high-fluidity alloys that have been used as solders. Pb greatly enhanees the properties of solder. It is eeonomical and improves the eorr~sion resistivity of solder by the formation of a stable oxide skin. Moreover, Pb ean effeetively prevent the so-ealled tin pest, even if the amount added is less than 0.1 mass %. In addition, it deereases the surfaee tension of solder. Minimizing the surfaee tension improves the solderability of the Ie substrate and suppresses the generation of bridges and skips. Therefore, this material will eontinue to be used in the foreseeable future. At present, there is no material eapable of replacing eonventional Pb-Sn based solders eompletely.

Pb-Sn Binary System. The Pb-Sn binary system is eomposed of two solid phases, I (bet strueture) and 8 (fee strueture), and a liquid phase C. In addition to these, the existenee of a high-pressure phase has been reported.

(a) 1400 (b) 1.0

0.9 ., 1200 ~

'0 0.8

E 1000 ~ f 0.7 ~ ... - ~ I ..

CI .?:' 0.6 c 800 • • s<

~ 'S: m

.. 'E :g 0.5 x v ~ ~

'ö 600 x « ~ 0.4 >- ~ * c. iä 400

A Klappa 0.3 o x Atarashlya el al.

~ ~ Bros el al. v Hawkins et 81. '" '" Yazawa el al. 0.2 x + • Das, Ghosh

W x Zivkovic x Sivaramakrishnan 200 0.1 .. Sommer el al.

o Sugimolo el al.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Sn 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Sn Sn I atomic fraction Sn I atomic fraction

Fig.14. Calculated (a) enthalpy of mixing and (h) activity in the liquid phase in the Pb-Sb system.

The ealculated values of the enthalpy of mixing and aetivities in the liquid phase, shown in Fig. 14a,b respeetively, essentially agree with the experimental data [19]. The ealculated phase diagram is also eompared with the experimental phase boundary data in Fig. 15a.

In a thermodynamie analysis of the phase diagram, information on the phase boundary of metastable systems is also an important objeet of analysis. The metastable equilibrium phase diagram between the liquid and the 8 phase in the Pb-Sn binary system is shown in Fig. 15b as an example. These results were obtained by Mori et al. [20] and Feeht and Perepezko [21]. The formation of the metastable two-phase separation in the 8 phase ean be foreeast by observing


(a) 350

.P -I!! ::J

i! B-E ~

300 L

250

200

150

100

"-50

" o~~-.-,.-.-,,-.-.--.-,\~+ Pb 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Sn

Sn I atomic fraction

.P -I!! ::J

I Y E ~

• Morletal .

300 ~ Facht, Perapezko

250 L

200 li /

/ /

150 / /

I 100 /

I I

I 50 I

I I

O~,,-.-,.-.-,,-.-.--.-.--+ Pb 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Sn

Sn I atomlc fraction

Fig.15. (a) Stable phase diagram and (b) metastable liquid/6 phase equilibria in the Pb-Sn binary system.

the shape of this phase boundary. Aetually, an almost symmetrie two-phase separation is ealculated as shown by the ehain line in Fig. ISa.

To investigate the phase equilibria at higher pressures, an additional change in Gibbs energy of elements is expressed as follows:

LlG'f:1-+<P2 (X T P) = LlG'!'I -+<P2 (X T) + (P - R ) . Ll V<Pl -+<P2 't , , 't' 0 't , (45)

where LlGTl-+<P2 (X, T) is the Gibbs energy differenee between the <PI and <P2 phases at P = Po. Ll V/l -+<P2 is the differenee in molar volume of the element i between the <PI and <P2 phases, and P and Po are the aetual pressure and the atmospherie pressure, respeetively, at whieh the ehanges are ealculated.

Figure 16a gives the phase diagram at 2.5 GPa, in whieh the appearanee of a stable area of a high-pressure E phase is shown in the neighborhood of 80 atom % Sn. Figure 16b shows the pressure-temperature diagram. It ean be understood from the results that the E phase is stabilized under a high pressure of about 1.2 GPa or more.

Pb-Sn-Sb Ternary System. Antimony is an important additive for Pb-Sn solders to prevent embrittlement eaused by the low temperature transformation of Sn. In the Pb-Sn-Sb ternary system, Sb-rieh a (rhombohedral), intermetallic eompound ß (NaCl), Sn-rieh'Y (bet), Pb-rieh 0 (fee), stoiehiometrie eompound Sb2Sn3, and the liquid phases appear to be stable. There is no peeuliar phase in this ternary system.

Figures 17a and bare ternary phase diagrams, the amount of Sn and Pb being fixed respeetively [19]. The experimental data for the phase boundaries from differential seanning ealorimetry (DSC) analysis are also shown in this figure. Information obtained by thermal analysis measurement is limited only to that eoneerning the phase transformation points. Thus, for an equilibrium

Hiroshi Ohtani 125

(a) 600 (b) 320

[!] Tonkov, Apteber

300 (85.5aI%5n) 500 ~ Ozaki, Saito L+E

(80.4al%5n) ~

.P .P 280 400 -f f

~ :::J 260 A E

CI) 300 Ö CI)

a. a. E E 240

Y CI)

~ A I-200 220

ö+y ~ ö+y <> 100 200 ~

~

0 180

Pb 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Sn 0 0.5 1.0 1.5 2.0 2.5

Sn I atomic fraction Pressure I GPa

Fig.16. (a) Pb-Sn binary phase diagram at 2.5 GPa, and (b) effect of press ure on phase boundaries at 85.5 atom%Sn.

(a) 700 +-----'----'_L--'----'-_'----'---'--+

I ... 1 O"!~Sn Presinl WClrk I 600 L._~_ Kogan et al.

100 y+ö

L

ö+a+ß

O~L,~,-_,--_r--,--r--.__.--+ o 10 20 30 40 50 60 70 80 90

Sb I atomic %

(b) 600

500

0° i 400 L L+a

:::J

E A

8. 300 A

+ ß ~ L+ß L+ß+a

/ I-200

y+ö

100 ß+y+ö ß+ö ß+a+ö

O+---,---,---,L--,-L-,---,---~ o 10 20 30 40 50 60 70

Sb I atomic %

Fig. 17. Vertical section diagrams of the Pb-Sn-Sb system at (a) 10 atom % Sn, and (b) 30 atom % Pb.

in which a number of phases are involved, it is necessary to further analyze the constituents of phase regions by using other experimental means such as X-ray diffraction, etc. Definite information about the possible phase regions can be obtained by applying phase diagram calculation to these cases, even if the calculation uses only the results of the binary systems.

Figure 18 shows the calculated results of the liquidus surface of this ternary system. Three kinds of invariant reactions are respectively shown with EI, E2

(both eutectic reactions), and VI (monotectic reaction). The compositions of the practical Pb-Sn based solder are concentrated in the neighborhood of the ternary eutectic point E2 , whose melting temperature is the lowest in this ternary system.


5b

Ternary Eutectic (E2) 179°C 39.6%Pb, 56.3%5n

Ternary Monotectic (U1) Ternary Eutectic (E1) 30 .:-.:- -~' ~- - 243°C 2.2%Pb, 90.0%5n 237'C 84.3%Pb, 3.8%5n - - y y ~----.::=-=~--" 5b2Sn3

20 v v ß

Pb 10 20 30 40 50 60 70 80 90 Sn

Snlmass%

Fig. 18. Liquidus surface in the Pb-Sn-Sb system.

Pb-Bi-Sn, Pb-Bi-Sb, and Pb-Sn-Bi-Sb Systems. Figure 19a,b show the results of calculation ofthe liquidus surface ofthe Pb-Bi-Sn and Pb-Bi-Sb ternary systems. The eutectic point of the Pb-Bi-Sn system is denoted by EI in Fig. 19a. The melting temperature is 92 oe, and from this fact, the alloy system is classified as soft solder in practical applications. The alloy compositions of commercial solders are almost entirely concentrated in the neighborhood of this point.

Bi

(a) Pb

10 20 30 40 50 60 70 80 90 Sn

Sn I mass %

Bi

(b) Pb

0(.1 .'

,t."

10 20 30 40 50 60 70 80 90 Sb

Sb I mass %

Fig. 19. Calculated liquidus surface of the (a) Pb-Bi-Sn, and (b) Pb-Bi-Sb systems.

In the Pb-Bi-Sb ternary system in Fig. 19b, most of the primary crystal surface is covered with the Sb-rich Ct phase of the rhombohedral structure. This is because the Ct phase is completely miscible in the Bi-Sb system, and it pre-

Hiroshi Ohtani 127

cipitates in a wide range in the Pb-Sb system due to a large deviation of the eutectic point to the pure Pb side. Since there is no ternary eutectic reaction, the lowest melting point is located at the binary eutectic point of the Pb-Bi system (125 oe).

There is little information on the phase diagrams and thermodynamic properties of the Pb-Sn-Bi-Sb quaternary system, and therefore, the phase boundaries were determined by DSC in the author's group. The experimental results for the Pb-Sn-20 mass %Bi-lO mass %Sb section are shown in Fig.20a with the calculated phase boundaries. The quaternary inter action parameters are not introduced into this calculation, and the phase diagram is obtained by using only the parameters of four kinds of ternary systems composing this alloy system. Nevertheless, the correspondence of the experimental data and the results of calculation are excellent. This fact shows that the CALPHAD method is extremely effective in the construction of a complex multicomponent phase diagram. Figure 20b shows the change in the liquidus surface when a small amount of Sb is added to the Pb-Bi-Sn system. The primary crystal field of the ß phase appears in the neighborhood of the Pb-Bi-Sn ternary eutectic point, and therefore the melting temperature of the alloy rises remarkably. This is because the addition of Sb promotes the formation of the intermetallic compound ß phase by coexistence with Sn. Precipitation of such a high-temperature phase in solder results in the degradation of the fluidity and the embrittlement of solder. This knowledge was obtained from merely empirical observation; however, it can be quantitatively explained from the phase diagrams.

(a) (b)

300

L Pb

2SO L +Ct

JJ 200 -GI

:; L +Q+3 1ii 150 .. X-E GI I- 100

50

o+-L-~~ __ ~ __ -. __ -. __ .-__ + o 10 20 30 40 50 60 70 BI

Sn Imass% Sn/m ... '"

Fig. 20. (a) Calculated vertical section diagram of the Pb-Sn-20 mass %Bi-10 mass %Sb alloy, and (b) effect of Sb addition on the Pb-Bi-Sn liquidus surface.


300 300

(a) (b) 250 250

P (J L - 200 0 200

f L+a a-:::I f e 150 .a 150

01 GI 8. y Q. E ~ 100 E 100

~-lnSn ~ y-ln5n

50 50 (In)

0.2 0.4 0.6 0.8 Bi 0.2 0.4 0.6 0.8 In

Bi I atomic fraction In I atomic fraction

500 1000

450 (c) 900 (d) 400 L 800

P 350 P 700 - 300 - 600 L f f :::I 250 L + (Zn)

:::I e e 500

~ 200 ~ 400 E Y E GI 150 ~ 300 I- (Zn) .

100 Y + (Zn) 200 y. Ag3Sn 50 100

0 0 Sn 0.2 0.4 0.6 0.8 Zn Sn 0.2 0.4 0.6 0.8 Ag

Zn I atomic fraction Ag I atomic fraction

Fig. 21. Calculated binary phase diagrams for the (a) Sn-Bi, (b) Sn-In, (c) Sn-Zn, and (d) Sn-Ag systems.

4.4.2 Potential Alloy System for Pb-Free Solder

Considering melting temperature, toxicity, and supply, the alloy system for Pbfree solder is limited to that of the Sn base [22]. Sn is abundant and poses no problems for the environment. Moreover, it is excellent in electric conduction. However, as it is inferior in melting temperature and mechanical properties compared with Pb-Sn solders; other elements must be added.

Alloying elements which might potentially alleviate these problems are Bi, In, Zn, Ag, etc. Binary phase diagrams of these elements and Sn are shown in Fig.21. Bi is excellent in strength; however, its elongation and thermal fatigue properties must be improved by alloying with other elements. In is good with regard to wettability and thermal fatigue, but its use in large quantities is difficult due to its high cost. Zn is suitable as an alloying element for Pb-free solders on account of its low cost and low melting temperature, although it has extremely poor wett ability. Due to its fine microstructure, Ag is excellent in mechanical

Hiroshi Ohtani 129

properties, although disadvantageous in melting point. Therefore, the Sn-Bi-X, Sn-Ag-X, and Sn-Zn-X systems might be considered as candidates for new Pbfree solder .

Sn-Bi-Sb Ternary System. Figure 22a shows the calculated results for the liquidus surface of the Sn-Bi-Sb ternary system [23]. The primary crystal surface is composed of CI! (rhombohedral Sb), intermetallic compound ß (NaCl type), 'Y (bct Sn), and stoichiometric compound Sb2Sn3; two kinds of monotectic-type invariant reactions (VI and V 2 in the figure) occur in this system. According to the vertical section diagram at 80 atom % Sn given in Fig. 22b, the melting temperature in the region with a small amount of Sb is almost equal to that in the Pb-Sn system, and the solidus temperature is about 150°C, indicating this system can be used at higher temperatures. In view of these points, this alloy system would appear to be a strong candidate for use as Pb-free solder. However, elose attention should be given to the amount of Sb added because the melting temperature rises significantly with even a small amount.

(a) Sb

(b) 80%Sn 350

L + Sb2Sn3 0 300 0 L -CI) 250 L+ß ... :::J -ca 200 ... CI)

L+ß+y Cl c.. 400 E 150

{!!. (HY ß+y 350°C 100

300 .. ~ a+ß+y

.. 2!i0 ... 50

·.200 0

Sn y 20 U2 40 60 80 Bi 0 0.05 0.10 0.15 0.20

Bi/atomic% Sb I atomic fraction

Fig. 22. (a) Liquidus surface, and (b) vertical seetion at 80 atom % Sn of the Sn-Bi-Sb ternary system.

Sn-Bi-In Ternary System. The most widely used Pb-free solder is the group of Sn-Bi-In based alloys. This offers very low melting points and forms excellent bonds with substrates such as Cu, Ni or Au, in addition to Pb not being ineluded. Therefore, this alloy system is attractive for use in a system that requires low melting processing, such as optoelectronic devices. The calculated liquidus surface of the ternary system is shown in Fig.23 [24]. There are two ternary eutectic points in the system, namely EI and E2. The lowest melting temperature is estimated to be 58°C.

130

(a)

10 1; Sn


In

30

20

Ternary Eutectic (E2) 58°C 16%Sn, 30%ln

Biln Ternary Eutectic (EI) 81°C 16%Sn, 62%ln

Sn 10 20 30 40 50 60 70 80 90 Bi Bi I mass %

Fig.23. Calculated liquidus surface of the Sn-Bi-In ternary system.

Zn (b) 216

212 "-"-

"-208 "-

0 °

204 "- Bi '--... - "-

GI 200 "-... "-::l 196 -ca In ... 192 GI

Cl. E 188

{!!. 184

'.' .. ' (Ag) 180

176

20 Ag3Sn 40 60 (~Ag)80 Ag 0 2 4 6 8 10 12 14

Ag/mass% Content I mass%

"-"-

"-

16 18 20

Fig.24. (a) Calculated liquidus surface, and (b) effect of alloying elements on the ternary eutectic point of the Sn-Ag-Zn system.

Sn-Ag-Zn Ternary System. This alloy system is also a promising candidate for use as a Pb-free solder. Due to a lack of experimental information, the phase boundaries were measured in this study by the technique of thermal analysis over the whole composition range, and the results were analyzed thermodynamically. Figure 24a shows the calculated results for the liquidus surface [25]. The phase equilibrium of this alloy system is rat her complicated, and it is difficult to clarify the whole phase diagram only by the customary experimental means. In this sense, the phase diagram calculation based on the CALPHAD method can be an extremely efficient approach to material development. The ternary eutectic point of this system falls at Sn-4 mass %Ag-1 mass %Zn, and the melting point is 216°C. This melting point should be reduced by alloying, since it is a little high

Hiroshi Ohtani 131

to be used &'l an alternative to Pb-Sn solder. Figure 24b is a calculation of the effect of the addition of Bi and In on the ternary eutectic point of the Sn-Ag-Zn system. The Sn-Ag base solder is an excellent alloy with regard to mechanical properties, thermal fatigue, and corrosion resistance. Therefore, the adjustment of the melting point by the addition of these elements enables determination of the appropriate composition of a new Pb-free solder.

4.4.3 Application of Thermodynamic Parameters to Estimate Surface Tension

Since solders should possess many properties, needless to say, we cannot evaluate all of them only from phase diagram data as described here. However, in the CALPHAD method, thermodynamic properties of alloys are evaluated as numerical values. Some other important properties of solders can also be estimated by using these parameters. An example is thermodynamic data being applied to the analysis of surface tension, as illustrated in this section.

The theoretical treatment of the surface energy of a molten metal has been carried out by some groups [26-28]. Butler [28] derived the surface energy of an i-j binary liquid as follows by considering the balance of the free energy between the surface monolayer and bulk:

where 'Yk (k = i, j) denotes the surface energy of pure metal k, x~ and x~ denote

the molar fractions of kin the bulk phase and the surface monolayer, exc: and

eXG; represent for the partial molar excess energy of k in the bulk and the surface monolayer, and Sk is the molar free surface area of pure metal k.

The surface tension of the Sn-Ag-X liquid is given by expanding (46) into the ternary system as in the following equation:

(1 S S) L RT -XAg-XX "I = 'YSn + -S In 1 _ B _ B

Sn XAg Xx

1 { 3 eXGB (T S S ) ex GB (T B B ) } + Ssn"4 Sn ,XAg'XX - Sn ,XAg'XX

RT X Ag 1 3 ex-B S S ex -B B B ( S ) { } = "lAg + SAg In xfg + SAg "4 GAg(T,XAg'XX) - GAg(T,XAg'XX)

RT I ( x~ ) 1 { 3 eXGB (T S S ) ex GB (T B B )} = "Ix + Sx n x~ + Sx"4 X ,XAg'XX - X ,XAg'XX .

( 47)

eXG~ is obtained by using (3) as follows.

eXG~ = GI - °Gl - RT In Xl (l = Sn, Ag,X) . (48)


Bi (320)

0\0 70

.~ 60 ['J ['J ['J

~ 370 377 385

~ 50 ['J ['J ['J ['J ,tri 384 392 400 409

~ 40 ['J ['J ['J ['J ['J 400 409 418 428 438

30 ['J ['J ['J ['J ['J ['J 418 428 439 450 462 475

20 ['J ['J ['J ['J ['J ['J ['J 440 451 464 477 491 507 524

10 ['J ['J ['J ['J ['J ['J ['J ['J 465 479 494 511 529 549 573 600

Sn 10 20 80 90

(482 mN/rn)

30 40 50 60 70

Ag I atomic% Ag (940

Fig.25. Calculated surface energy of the Sn-Ag-Bi ternary system.

Furthermore, for a system in which the interaction between atoms is comparatively weak, the following expression holds [29]:

(49)

Si is given from the molar volume Vi as

(50)

Since the values of Vi for elements are already given [30], the surface tension of liquid ,L can be calculated by determining xig and x:k from the simultaneous equations (47). The result for the Sn-Ag-Bi ternary system is shown in Fig.25 [31]. The remarkable reduction of the surface tension of Sn by the addition of Bi corresponds weIl with the experimental results. The surface tension of molten solder plays an important role in determining its wetting behavior; however, very little is known concerning this property. One of the great advantages of the CALPHAD method is that it can be used to obtain information on various thermodynamic properties of a material by analyzing its phase diagram.

Acknowledgment. The author is greatly indebted to Professor Kiyohito Ishida at Tohoku University for advice and criticism during this work. The author would like to thank Dr. K. Inoue and Mr. M. Miyashita for their assistance in carrying out this work.

Hiroshi Ohtani 133

References

1. T. Nishizawa: Progress ofCALPHAD. Mater. Trans. Jpn. Inst. Met. 33, 713 (1992). 2. M. Hansen: "Der Aufbau der Zweistoffiegierungen" (Springer, 1936). 3. L. Kaufman, H. Bernstein: "Computer Calculation of Phase Diagrams" (Academic

Press, NY, 1970). 4. M. Hillert, L.-I. Staffansson: The regular solution model for stoichiometric phases

and ionic solutions. Acta Chem. Scand. 24, 3618 (1970). 5. R. Kikuchi: A theory of cooperative phenomena. Phys. Rev. 81, 988 (1951). 6. G. Inden: Report on the Project Meeting Calphad V, 21-25 June 1976, Max-Planck

Inst. Eisenforsch. Düsseldorf (1976), III, 4-1. 7. M. Hillert, M. Jarl: A model for alloying effects in ferromagnetic metals. CALPHAD

2, 227 (1978). 8. "Kinzoku Data Book", 2nd ed. (Maruzen, Tokyo, 1984) (in Japanese). 9. J.R. Arthur: Vapor pressures and phase equilibria in the Ga-As system. J. Phys.

Chem. Solids 28, 2257 (1967). 10. K Ishida, H. Tokunaga, H.Ohtani, T. Nishizawa: Data base for calculating phase

diagrams of III-V alloy semiconductors. J. Cryst. Growth 98, 140 (1989). 11. H.Ohtani, K Kojima, K Ishida, T. Nishizawa: Miscibility gap in lI-VI alloy semi

conductor systems. J. Alloys Compd. 182, 103 (1992). 12. K Osamura, K Nakajima, Y. Murakami: Calculation of 111-V quasi-binary phase

diagrams and theoretical analysis of the excess free energies for their solid solutions. J. Jpn. Inst. Met., 36, 744 (1972) (in Japanese).

13. G.B. Stringfellow: Calculation of ternary and quaternary III-V phase diagrams. J. Cryst. Growth 27, 21 (1974).

14. D.W. Kisker, A.G. Zawadzki: Estimation of solid-vapor distribution coeffieients in organometallic vapor phase epitaxy of lI-VI semiconductors. J. Cryst. Growth 89, 378 (1988).

15. E. Rudy: Boundary phase stability and critical phenomena in higher order solid solution systems. J. Less-Common Met. 33, 43 (1973).

16. K Kobayashi, H.Ohtani, K Ishida: Thermodynamic analysis on phase equilibria of thin film in 111-V alloy semiconductors. Proceedings of the 14th Electronic Materials Symposium, Izu-Nagaoka, Japan (1995) 141.

17. T. Yokogawa, H. Sato, M. Ogura: Dependence of elastic strain on thickness for ZnSe films grown on lattice-mismatched materials. Appl. Phys. Lett. 52, 1678 (1988).

18. M. Quillec, H. Launois, M.C. Joncour: Liquid phase epitaxy of unstable alloys: Substrate-induced stabilization and connected effects. J. Vac. Sei. Technol. BI (2), 238 (1983).

19. H.Ohtani, K Okuda, K Ishida: Thermodynamic study of phase equilibria in the Pb-Sn-Sb system. J. Phase Equilibria 16, 416 (1995).

20. K Mori, KN.Ishihara, P.H. Shingu: Metastable phase diagram of the Pb-Sn system. Mater. Sei. Eng. 78, 157 (1986).

21. H.J. Fecht, J.H. Perepezko: Metastable phase equilibria in the lead-tin alloy system: Part I. Experimental. Metall. Trans. A 20, 785 (1989).

22. J. Glazer: Metallurgy of low temperature Pb-free solders for electronic assembly. Int. Mater. Rev. 40, 65 (1995).

23. H.Ohtani, K Ishida: A thermodynamic study of phase equilibria in the Bi-Sn-Sb system. J. Electron. Mater. 23, 747 (1994).

24. S.Ishihara, H. Ohtani, K Ishida, T. Saitoh: Thermodynamic assessment of the SnBi-In system. Abstracts of the Japan Institute of Metals 121, 386 (1997) (in Japanese).


25. M. Miyashita, H. Ohtani, K. Ishida: Thermodynamic study of the phase equilibria in the Ag-Sn-Zn ternary system. Abstracts 117th meeting of JIM, International Symposia on Advanced Materials and Technology for the 21st Century (1995) 1262, p305.

26. E.A. Guggenheim: Statistical thermodynamics of the surface of a regular solutions. Trans. Faraday Soc. 41, 150 (1945).

27. L Prigogine, R. Defay, A. Bellemans, D.H. Everett: "Surface Tension and Absorption" (John Wiley & Sons, 1966).

28. J.A.V. Butler: The thermodynamics of the surface of solutions. Proc. R. Soc. Lond.A 135, 348 (1932).

29. T. Tanaka, T. Iida: Application of a thermodynamic database to the calculation of surface tension for iron-base liquid alloys. Steel Res. 65, 21 (1994).

30. T. Iida, R.LL. Guthrie: "The Physical Properties of Liquid Metals" (Clarendon Press, Oxford, 1988).

31. M. Miyashita: Thermodynamic analysis on the phase equilibria in the Sn-Ag based microsolder alloys. Master Thesis, Tohoku University, Japan (1998) (in Japanese).

Phase Equilibria and Microstructural Control in Iron-base Alloys*

Kiyohito Ishida

Department of Materials Science, Graduate School of Engineering, Tohoku University, Aramaki Aoba, Send ai 980-8579, Japan

Abstract. The utilization of phase diagrams on the microstructural control in iron-base alloys are presented focusing on solidification processes, low alloy steels, stainless steels and functional steels. It is shown that the recent progress in the computational approach for thermodynamic properties and phase diagrams have contributed significantly to progress in alloy design and microstructural control.

5.1 Introduction

Phase diagrams playa key role in the selection and control of microstructures for realizing optimum material properties in alloys chosen for specific applications. Even though, over the years, several compilations of experimental binary and ternary phase diagrams have become available [1-3], there still exists a need for information on phase diagrams that involve more than three components. Since detailed experimental determination of such multi-component pha'le diagrams is impossible in terms of the effort required and the available time, there has been a concerted attempt to develop a computational approach for the determination of phase equilibria in metallurgical systems. The last decade has seen significant strides in this approach to metallurgical alloy design. Computational techniques such as first principle calculations and Monte Carlo simulation have been increasingly utilized in the calculation of phase diagrams [4]. The CALPHAD (calculation of phase diagrams ) method has become increasingly popular [5] as the method for predicting the phase equilibria in multicomponent systems. Several databases for calculating phase diagrams of commercial alloy systems using the CALPHAD approach have also been developed, as listed in Table 1. The advantage of the CALPHAD approach lies in the fact that not only it is possible to calculate phase diagrams using this method but also that it is possible to extract a wealth of information concerning the various aspects of phase diagrams. Examples of such useful information relate to the following: stable and metastable phase equilibria, the position of the To line, volume fractions of the phase constituents under specified conditions, values of various thermodynamic quantities such as activity, mixing enthalpy, Gibbs energy of formation, driving

* A major part of this article was originally published in the "161st and 162nd Nishiyama Memorial Seminar", The Iron and Steel Institute of Japan, (1996) 31 (in Japanese).

Springer Series in Materials Science Volume 34. Ed. by T. Saito © Springer-Verlag Berlin Heidelberg 1999

136 Phase Equilibria and Microstructural Control in Iron-base Alloys

Table 1. Date base of calculated phase diagrams of commercial alloys.

Alloys Element

Fe base alloys

Low alloy steels Fe-C-N-Si-Mn-Cr-Mo-Ni

-Co-Al-Nb-V-Ti-W

Microalloying steels Fe-C-N-S-Mn-Si-Al-Cr

-Ti-Nb-V

Tool steels Fe-C-Cr-V -W -Mo-Co

Stainless steels Fe-C-N-Si-Cr-Ni-Mn

-Mo-Al

Phase

L, 00, "(,

carbide, nitride

L, 00, "(, carbide,

nitride, sulfide

L, 00, "(, cabide

L, 00, "(, carbide,

nitride

Ni base alloys

Superalloys Ni-Al-Ti-Cr-Mo-Co-Ta-Nb L, 00, "(, "(t, ß -Zr-W-Hf-B-C TCP(u, Laves, p),

boride, carbide

Ti base alloys

Al base alloys

Ti-Al-V -Mo-Cr-Si-Fe-Nb

-Sn-Ta-Zr-B-C-N-O

Al-Cr-Cu-Fe-Mg-Mn-Ni

-Si-Ti-V-Zn-Zr

Alloy semiconductors Al-Ga-In-P-As-Sb

Microsoldering alloys Pb-Bi-Sb-Ag-Zn-Cu

00, ß, compound,

boride, carbide

L, 00, compound

L, compound

L,oo,ß,"(,8

forces for phase transformation, etc., as illustrated in Fig. 1 [6,7]. In this article the state of progress achieved in applying the computational techniques of phase diagram calculation to microstructural design in steel is reviewed.

5.2 Solidification Process

5.2.1 Solid/Liquid Phase Equilibria

Many of the significant aspects of solidification are quantitatively analyzed in terms of the solute partitioning process. A ratio k~ltermed the solute distribu

tion or partition coefficient and defined as k~l=X~/ X~ where X~ and X~ are the concentrations of solute in the solid and liquid phases, respectively, is frequently used in such analyses. The distribution coefficient is actually a measure of the degree of segregation of a solute and the quantity 1 - k~l, defined as the segregation coefficient can be utilized to quantify the rejection of solute at the interface during the solidification process [8]. Changes observed in the relative positions of the liquidus and solidus lines, as a result of solute additions, are also directly related to the segregation coefficients. Thermodynamic calculations

Kiyohito Ishida

Driving force Chemical potential

Therrnodynamic properties such as activity, enthalpy etc.

CALPHAD

Fig. 1. Scheme of CALPHAD method.

137

have been utilized in calculating these coefficients for several solutes and such calculations have also been extended for estimating the partition coefficients of ')'-loop forming elements, even though there exists no direct phase equilibrium between austenite and liquid in steel. Table 2 lists some of the equilibrium distribution coefficients of alloying elements between austenite and liquid or ferrite and liquid phases in steel [7]. An immediate observation is that solute elements

such as S, P and C in steel, which have low values of k,;JI, have a strong tendency to segregate during solidification. Table 2 also lists the effect of adding 1 mass % alloying element on the variation in the position of the liquidus and the sol idus lines in steel.

The addition of alloying elements also affects the characteristics of the peritectic reaction and thereby the solidification behavior, which are to a great extent dependent on whether the primary solidifying phase is austenite or ferrite. Thermodynamic calculations have been performed to predict the effect of alloying elements on the temperature and composition of the peritectic reaction in the Fe-C system. Figure 2a, b shows the results of such thermodynamic calculations [7].

5.2.2 Control of Inclusions Morphology

Since non-metallic inclusions in steels are generally detrimental to their material properties, considerable efforts have been directed towards the development of alloys and processes to reduce their presence to aminimum. However, of late, the utilization of inclusions in steels as a means of controlling microstructures and thereby improving their mechanical properties, is becoming a subject of considerable interest [9]. In this sense, an understanding of the formation of different morphologies and distributions is essential for arriving at the conditions


Table 2. Effect of alloying elements on solid/liquid phase equillibria in iron base alloys.

Partition Change in liquids and solidus

coefficient by adding 1 mass % element Element

Liquidus solidus k~!L kilL

[LlTL]O [LlTLP [LlTS]O [LlTSp

C 0.15 0.31 75.0 51.5 493 169

N 0.38 0.47 47.3 34.0 125 72.7

Al 0.99 0.80 0.2 6.6 0.2 8.3

Si 0.65 0.61 13.8 12.8 21.3 21.0

P 0.14 0.11 30.4 26.3 223 236

S 0.05 0.02 32.2 27.9 596 1300

Ti 0.46 0.32 12.3 13.0 26.8 41.0

V 0.78 0.62 4.7 6.8 6.0 10.9

Cr 0.90 0.88 2.1 2.2 2.3 2.5

Mn 0.69 0.70 6.3 5.0 9.2 7.1

Co 0.87 0.91 2.4 1.4 2.7 1.6

Ni 0.81 0.87 3.6 2.1 4.5 2.4

Cu 0.70 0.72 5.2 4.1 7.4 5.6

Nb 0.28 0.20 8.6 8.0 31.0 41.0

Mo 0.79 0.61 2.4 3.7 3.0 6.1

W 0.86 0.70 0.8 1.5 1.0 2.2

for aehieving the optimum morphology and distribution, eommensurate with the required properties in a given steel. An example of such a study earried out on MnS inclusions is a ease in point.

The morphology of well-known inclusions of MnS ean be broadly classified into three types, i.e. (i) randomly dispersed globular sulfides, (ii) rod-like fine sulfides and (iii) angular sulfides [10]. The formation meehanism of the different MnS morphologies has been diseussed in detail on the basis of phase diagram information [11]. Figure 3 shows the stable and metastable phase diagrams of the Fe-MnS pseudo-binary systems. Aeeording to the stable diagram, any Fe-rieh melt (Ld, on eooling from the liquid state, would be expected to follow a solidifieation path that would result in the solidifieation of the primary Fe phase first, followed by euteetie solidifieation aeeording to the euteetie reaetion (Li -+ Fe( s)

Kiyohito Ishida

0.4 ~ <Il <Il

'" Ei -0.3 C ~ 0 u <= 0.2 0

~ U

0.1

(a)

A

Nb p

S

0 0.5 1.0

Alloying Content / mass %

1420

(b)

Ni

~~~~§~~co 1"- Cu -=~;::::---I Mn

p

W,Cr V Mo Al

1380'-------'-----' o 0.5 1.0

Alloying Content / mass %

139

Fig. 2. Effect of alloying elements on the peritectic reaction of Fe-C alloy; (a) peritectic composition and (b) peritectic temperature.

( a ) -Stable ( b ) Content of S / mass % - - - - Metastable 0 0.5 1.0 1.5 2.0 a 2.5

I 1600 /

LI LI+L2 L2 LIlFe LI /

vi 1550 /

/ '--

M2: I

J L]+MnS , 1500

g , _.:'------- _______ J

/ / LIlMnS(s) MI öT<3K

,

Fe+MnS 1400 /~.LIIL2

// 1350 b /d

Fe MnS 0 0.02 0.04 0.06 0.08 Fraction of MnS Mole Fraction of MnS , z

Fig.3. Phase diagrams of Fe-MnS pseudo-binary in stable and metastable systems; (a) schematic and (b) calculated one in Fe-rich portion.

+ MnS( s)). This would preclude the formation of any globular or droplet type MnS because the droplet type MnS would result only when the solidification path involves the monotectic reaction (LI --+Fe(s) + L2 ). However, in practice, the globular or drop let type MnS has always been observed in Fe-Mn-S ternary alloys cooled from the liquid state in the normal way. It is now possible to explain the reason for this by referring to the calculated metastable equilibrium diagram represented by the broken lines in Fig. 3a. In Fig. 3b are shown the actual phase diagram and the calculated stable and metastable phase equilibria in the Fe-rich portion of the diagram. The calculated difference between the eutectic temperature (e) and the monotectic temperature (m) is so small (only


3°C) that a metastable reaction leading to the formation of the globular MnS becomes a distinct possibility. The deciding factor that tips the balance in favor of the globular MnS is the magnitude of interfacial energy between Fe(L1 ) /Mns(L2 )

boundary which is estimated to be lower than that ofthe Fe(Ld/MnS(s) boundary. Since the energy barrier for MnS(L2 ) nucleation is thus smaller than that for MnS(s), the nucleation of MnS(L2 ) is energetically more favored and the metastable monotectic reaction predominates over the stable eutectic reaction.

Fig. 4. Morphology of MnS in steel; (a) spherical, (b) dendritic, (c) angular, (cl) monotectic, (e) rod-like eutectic, (f) irregular eutectic.

The morphology of MnS inclusions is also strongly affected by the alloying element [11,12]. Figure 4 shows typical micrographs of the various shapes associated with the MnS inclusions found in steel. The changes in the morphology of MnS inclusions brought about by the additions of C, Si, Al and Ti in hypo-and hyper-eutectic Fe-MnS specimens respectively, are summarized in Figs. 5 and 6. TiN crystallites act as nucleation catalysts for MnS(s) in the Fe-MnS alloys containing Ti (melted under N2 ) and Al20 3 crystallites act as nuclei in the FeMnS alloys containing Al (melted under Ar), promoting primary crystallization of the MnS(s) phase and the eutectic reaction associated with the stable sys-

Kiyohito Ishida

Mass% -- 0.05 0.1 0.2 0.3 0.4 0.5 Element

1.0

c * Si *

AI*

:: I U!1J _ ~----------------_._-------------, .. .. : .. .. : .. .. :

3.0 5.0

[::-llri-:i:-llri-----------------------·

~--- -----------. r----------------, •• .<>*IDIf Ti * •• .<> •• .<> I _________________________ J

•• •• •• . rod-like

monotectic eutectic irregular eutectic

* melted under Ar ** melted under N 2 *** eutectic TiS+monotectic MnS

141

Fig. 5. Morphology of MnS in hypo-eutectic (or monotectic) specimens containing C, Si, and Ti.

tem. When the same alloys containing Ti are melted under an Ar atmosphere only oxides Ti02 and Si02 are formed which tend to react with MnO forming compounds with lower melting temperatures, although its own melting temperatures is about lOO°C higher than Fe. As a result, liquid oxide droplets are formed first, in the Ti containing alloys melted under Ar which act as nucleants for MnS(L2), thereby favoring the metastable monotectic reaction leading to the monotectic morphology of MnS. On the other hand, the eutectic, dendritic and angular morphology of MnS observed in the specimens containing C or Si is due to the fact that C and Si lower the melting point of iron and raise the activity of sulfur. The resulting expansion of the 2-liquid phase region and the increase in the slope of the liquidus line for iron, leads to an increase in the temperature difference between the metastable monotectic and the stable eutectic points, as shown in Fig. 7. This difference in temperature is estimated to be about lOoC in the Fe-l %Mn-O.3%S-1 %C alloy, which could explain why the stable eutectic reaction develops more easily in preference to the metastable monotectic re action in these alloys.

5.2.3 Simulation of Composition Change in Non-metallic Inclusions During Solidification

Computer simulation for predicting the solidification path and the composition of the non-metallic inclusions in steels has been developed by combining the

142 Phase Equilibria and Microstructural Contral in Iran-base Alloys

~=..:,:'10.05 0.1 0.2 0.3 0.4 0.5 0.7 1.0 2.0 3.0 4.0 5.0

Si * • .----.---1 * • .-----------------~-----~--r_____-~ ------------~

.---1 '::--.-------- • * • spherical dendritic angular ,

c*

Ti * Ti*

* melted under Ar * * melted under N2

Fig.6. Morphology of primary MnS in hyper-eutectic (or monotectic) specimens containing C, Si, and Ti.

-- Fe - Mn - S stable Fe - Mn - S metastable Fe - Mn - S + C(Si) stable Fe - Mn - S + C(Si) metastable

Fig. 7. Changes in phase equilibria in the pseudo-binary system by alloying C or Si.

kinetic model of micro-segregation and the database for calculating phase diagrams [13, 14J. This simulation is carried out based on the following assumptions: (i) local equilibrium exists at the solid/liquid interface, (ii) complete mixing of solutes and homogeneous distribution of inclusions occurs in the liquid, and (iii) astate of chemical equilibrium exists between the residual liquid and the inclu-

Kiyohito Ishida 143

sions. In the simulation procedure, the segregation ratio at the solid/liquid interface is determined using the Clyne-Kurz micro--segregation model [15], which corresponds to the Scheil micro--segregation model. Then the temperature is decreased by a small amount (.1T) and the phase equilibrium at the new temperature is calculated. After that, the solidification phase is removed out of the system in which phase equilibrium is calculated in the next solidification step. This sequence is repeated until the temperature coincides with the solidus temperature. An example of such a simulation where the precipitation behavior of oxides in a Fe-21 %Cr-11 %Ni-O.l %Si-O.02%AI-O.002%Ca-0.005%O stainless steel is calculated during solidification is shown in Fig. 8.

800

700

~ 600 v Q) .... ':' 500 .:::: 01) '-' Q)

"Cl 400 .;< 0 ..... 0 .... 300 s:: :l 0

200 S <t:

100

0 0

total amount

average of total amount

121.3( g I T) A1203 Si02

0.2 0.4 0.6

Solid Fraction

~

0.8 1.0

Fig. 8. Changes of the amounts of non-metallic inclusions in the residual liquid during solidification of Fe-21 %Cr-l1 %Ni-O.l %Si-O.02%AI-O.002%Ca-O.005%O stainless steel.

5.3 Low Alloy Steels

5.3.1 Surface Fissure in Cu-containing Steels

From the point of view of energy and resource conservation, there is a growing emphasis being placed on finding more efficient ways of recycling sc rap iron in


steel making. One of the problems often encountered in steels made using scrap iron is the presence of residual Cu that comes from the ferrouH scrap, which causes the weIl known "surface fissure" defects during the hot-rolling process. The origin of this defect can be traced to the weaker oxidation tendency of Cu as compared to that of Fe in steel. Cu is not oxidized and taken into the oxide phase during the heating stage before hot working and remains as Cu in the Fe matrix. When the concentration of Cu in the surface layer exceeds the solubility of Cu in solid Fe, liquid Cu forms at the interface of steel and the oxide. This liquid penetrates the grain boundary of the Fe matrix during hot rolling and generates surface cracks. Figure 9 shows the experimental results obtained for the composition of liquid phase formed at the interface of matrix and oxide, in specimens containing Cu and Sn heated at 1200°C in air, superimposed on the calculated phase diagram of the Fe-Cu-Sn system [16]. It should be noted that in the experimental specimens the liquid phase is in equilibrium with the austenite and the alloy compositions are located on the equilibrium boundary of liquid phase. It is therefore evident that the surface fissure is strongly re la ted to the austenite/liquid phase equilibrium in the Fe-Cu base systems. When the , + L two-phase field of the Fe-Cu binary system is enlarged by the addition of an alloying element, a liquid phase is easily formed and the tendency for surface cracking is increased. Figure 10, where the solid lines represent the calculated ones [17], shows the expected effect of alloying elements on the solubility of Cu in the solid phase. It is seen that Ni and Co increase the solubility of Cu while Al decreases up to rv 2.5 mass % and then steeply increases it in ferrite. All the other alloying elements such as Si, V, Mn, Cr, and Sn decrease the solubility of Cu. In particular, the addition of Sn drastically decreases the solubility of Cu. The calculated trend of the effect of alloying elements on the solubility of Cu can weIl explain the origin of surface cracking due to liquid embrittlement in steels made using scrap iron, which is suppressed by Ni and promoted by Sn.

5.3.2 Effect of Alloying Elements on the Relative Stabilities of Ferrite and Austenite

Various types of high-strength low-alloy steels with good ductility have been developed by thermo-mechanical controlled processing. One typical example is that of a dual phase steel obtained by employing an intercritical treatment which consists of heating the steel to a temperature in the (0+,) region. Another exampIe is one that relates to the so-called TRIP (transformation induced plasticity) steels, where the alloy composition and heat treatments are controlled in such a way as to deliberately contain a certain amount of retained austenite in the final product. Arecent entrant into this group of steels is a group of C-Mn-Si steels designated as low alloy TRIP-type steels [18]. The basic thermo-mechanical treatment sequence for all these steels consists of firstly an intercritical annealing treatment followed by an austempering treatment to transform the resulting bainite. In the second stage of austempering, Si supresses thc formation of cementite in bainite and prornotes carbon enrichment of austenite, which in turn suppresses the Ms temperature of the , phase. In the case of both dual phase steels and

Kiyohito Ishida 145

o 0.2 0.4 0.6 0.8 1.0 Mole fraction Cu

Fig. 9. Composition of liquid phase fomed at the interface of matrix and oxide in the Fe-Cu-Sn system projecting the calculated phase diagrams at 1200°C.

5 10 15 20 o Alloying Element I mass %

5 10 15

Alloying Element I mass %

20

I::. V I!I Cr ~ Mn X Co v Ni + Al • Si x Sn

Fig. 10. Effect of alloying elements on the solubility of Cu in solid phases.

146 Phase Equilibria and Micrastructural Contral in Iran-base Alloys

?- 8.0 .t: 6.0 ö >< 4.0 11

?:::... Cl former ö ..:c

E 0 'u <C --0 0

U Y former s:: 0

".g 0.2

~ 0.1 C/J

600 800 1000 1200 1400 1600 a Temperature (Oe)

Fig. 11. Tepperature dependence of the distribution coefficients of alloying elements between a and 'Y phases.

TRIP type steels, the intercritical treatment in the a+, dual phase region is the most important step in controlling the final microstructure. The critical parameters that have to be identified for a successful intercritical treatment, such as the extent of the dual phase fields, volume fractions and equilibrium compositions of a and "! phases, Ae3 temperatures, etc. can now be accurately predicted using information from thermodynamic databases shown in Table 1. The distribution coefficient k:;;!' =X" I X, of alloying elements between a and "! phases is one of the most important factors that determines the al"! phase equilibrium. This is directly related to the change in partial molar Gibbs energy at infinite dilute solution of the alloying element X accompanying the ah transformation [19]. Figure 11 shows the calculated distribution coefficients of alloying elements as a function of temperature [20]. It is to be noted that k';;!' for most of the elements exhibits a strong temperature dependence. This arises mainly from the magnetic transition in the aFe phase. The thermodynamic treatments enable us to predict more accurately the nature of the al"! transformation in the these steels. The validity and potential of the calculation approach is again illustrated in Fig. 12 where it is seen that there is a significant agreement between the calculated and observed Ae3 temperatures [21]. Another example of a successful application of computational approach to thermodynamic predictions in steel systems is the calculation of the effect of various solute elements on the M s temperature [22]. This has been accomplished by dividing the free energy change accompanying the martensitic transformation into its chemical and non-chemical parts. The chemical free energy change is estimated from the change in partial molar Gibbs energy associated with the "!----'ta transformation. The non-chemical part arises mainly as a result of the change in friction stress required to move dislocations, which in turn is related to solid solution hardening in the parent austenite.

Kiyohito Ishida 147

850 • •

U 0 800

r"l <\)

<:(

"0 0 .§ 750 ::l

..S! <.: U

700 700 750 800 850

Observed Ae3 ( oe )

Fig. 12. Comparision of calculated results on Ae3temperature and observed ones.

The calculated chemical and mechanical contributions of alloying elements to M s temperature are shown in Fig. 13. Such calculations have enabled the formulation of the following equation that relates M s temperature to the alloy composition in low alloy steels: Ms (OC, mass%) = 545-330C+2AI+7Co-14Cr-13Cu-23Mn-5Mo-4Nb-13Ni-7Si +3Ti +4V +OW.

+5

0 C

-, ,-... t;>:: -5 C;; '-'

U - 10 0

'" ::E <l - I

hemical effecI

-20 Mechanical effect

-25 o Fig. 13. Effect of alloying elements on the M s temperature.

148 Phase Equilibria and Microstructural Control in Iran-base Alloys

5.3.3 Carbides and Nitrides in low Micro-alloyed Steels

It is weIl known that Nb, Ti and V have a strong affinity for C and N. A small addition of these elements to steels brings ab out a significant improvement in their mechanical properties by forming fine precipitates of carbides or nitrides of these elements. Because of the crucial role played by these precipitates in infiuencing the mechanical properties, information on their phase constitution and solubility in steels is of fundamental importance for exercising microstructural control in micro-alloyed steels. For instance, a precise estimate of the content of interstitial elements in the ferrite matrix is required for controlling the strength by bake hardening in the recently developed group of IF (interstitial free) steels. Since the solubilities of both C and N in ferrite are significantly low and not easily determined by experiments, the estimation of the amount of soluble C and N from the computed phase diagrams offers a much easier and faster method of exercising this control.

Temperature (Oe) 150014001300 1 ZOO 1100 1000 900 SOO

......... Tie,.. ..yc .... ; Nb ......... !

~ - 2 "·'::~:: ••••••••• r"""" ~ ! S -4 '-'

y

-10

Fig. 14. Solubility products of carbides,nitrides and sulfides in ,and 0'.

Figure 14 shows the solubility products of various carbides, nitrides and sulfides in austenite and ferrite. It can be seen that the overall solubilities of nitrides are smaller than those of carbides, while both compounds are more soluble in austenite than in ferrite. Figure 15 shows the effect of micro-alloying elements on the austenite region in Fe-C alloys [23]; very small amounts significantly change the extent of the phase field. Some of these micro-alloying elements such as Ti,

Kiyohito Ishida 149

(a)Nb I~r-------------,

(b)TI (c) V

~ O.S

Fe O.S 1.0 l.S 2.0 Fe 1.0 I.S 2.0 Fe 1.0 I.S 2.0 M'lW: M'lW: M'lW:

I~ (d)P (e) S (f)B

"- I --. e 5 I I! 11. ~ I

800 17t!.

~ r,tlle2 P 1l+Pe, C,B)

0.5 ~ Fe 1.0 1.5 2.0 Fe 0.5 1.0 1.5 2.0 Fe 0.5 1.0 1.5 2.0

M'lW: M'lW: M'lW:

Fig. 15, Effect of micro-alloying elements on the austenite region of Fe-C alloy.

V and Nb also form complex carbonitrides in steels. Figure 16 shows the calculated phase equilibria between austenite and (Nb, Ti) (C, N) at 1200°C, in a steel where the total Nb+Ti content in austenite is kept at 0.02 mass % [24]. When the content of Ti is extremely smalI, an alloy containing carbonitrides heat treated in the 'Y + carbonitride two phase field would have a homogeneous distribution of carbides as expected from Fig. 16a. However, an increase in Ti would result in the phase separation of carbonitride and the three-phase field 'Y+TiN+NbC is formed, as seen in Fig. 16b. This is due to the formation of a closed miscibility gap in the Nb-Ti-C-N system. Figure 17 shows the calculated miscibility gap contours in NaCl-type carbonitrides consisting of Ti, Nb and V. The miscibility gap in the NbC-VC pseudo-binary system is formed due to the large difference in lattice parameters between NbC and VC, while the miscibility gap islands formed in the double-pseudo-binary system of (Nb, Ti) (C, N), (Nb, V) (C, N) and (V, Ti) (C, N) are caused by the differences in chemical stability of the carbides and nitrides, that is, the difference in the Gibbs energy of formation of the compounds [25]. This kind of miscibility gap is also formed in the Ill-V alloy semiconductors [26]. It is apparent that such calculated phase diagrams would be of great use in predicting the phase separation of carbonitri des in the austenite and ferrite phases and would play an important role in the precise control of microstructure in micro-alloyed steels.


(a) (mass %Ti)y = 0.0000 I TiN ~~~,-~~="-~-,~,-,

mass %C

Z 0.004 I

t;I' I

~ 0.002 f-U.!...U..l..l....\_

y

Fe mass %C

---- ........... " NbN

, , \ \ \ \ I

Fig.16. Phase equilibria between I and (Nb, Ti) (C, N) at 1200°C keeping (Nb+Tih=0.02 mass %.

TiN

TiN NbN 1000

1300

1800 Q 1000

TiC 1300

TiC

Fig. 17. Miscibility gaps of complex carbonitrides.

Kiyohito Ishida 151

5.4 High Alloy Steels

5.4.1 Hard Metal Cemented Carbides

Alloys from the WC-Co system are classed as hard alloys commercially and their microstructure consists of a hard tungsten carbide phase which is uniformly distributed in a Co matrix. An important requirement for their successful utilization is that the carbon composition range should be precisely controlled so that the formation of graphite or any other carbide in the final product is precluded. Because it is a strategic material and the supply of cobalt is subject to the vagaries of the world political situation or marketing strategies, attempts to develop cheaper alternatives to Co as a binder are in progress [27,28]. Calculated phase diagrams have an important role to play in the developmental work related to this system. For example, Fig. 18 shows the computed section diagram of the W-Fe-C-Co-Ni system at 5 mass % Fe, 2.5 mass % Ni and 2.5 mass % Co. This diagram indicates that the alloys with a carbon content between "a" and "b" could be expected to have the WC+fcc phase constitution [27].

1500

U 140 0

~ ::l 1300

~ ~ 1200 Q)

ro 110

1000

1 Owt. %(Co+Fe+Ni)

L+~C+WC

fcc+M6 C+WC

M6C+WC

4.0 4.5 5.0

mass % C

5.5 6.0

L+grnph.+WC

fcc+grnph. +wc

Fig. 18. Calculated phase diagram of W-5%Fe-2.5%Ni-2.5%Co-C system.

5.4.2 High Speed Steels

Recent additions of data to the thermodynamic databases shown in Table 1 permit calculation of phase constitutions in high speed steels, where strengthening is achieved by the presence of fine precipitates of carbides such as of MC, M2 C, M 6C etc. Figure 19 shows the calculated vertical section diagrams for the typical high speed steel, A1S1 M2 [29]. The calculated transition temperatures between different phase regions, as set out at the edges of the figures for the


~ ....... e 160 i ISO

Cl 140

o 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Carbon concentration / mass %

y+C

~ 130~~~~~--,-~--~-r~ o 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Carbon concentration / mass %

Fig.19. Vertical seetion diagram of AISI M2 high-speed steel, (a) calculated and (b) observed.

technically relevant alloy with a carbon level of 1 % are in good agreement with the actually observed phase transition temperatures. It is also now possible to predict phase compositions, volume fractions of the matrix and of the carbides using these databases. Such information allows one to map the solidification sequence, choose appropriate alloy compositions and heat treatments for achieving secondary hardening, etc.

One of the problems often encountered in high carbon tool steels is the pronounced tendency of the carbides to segregate during solidification, resulting in coarse primary carbides, formation of carbide networks, banded carbides and non-uniform microstructures. Powder metallurgy techniques have been tried and developed to avoid the formation of such carbide clusters. Yet another process termed the carbide dispersion carburizing (CDC) method has also been developed for producing high speed steels with fine and uniformly distributed carbides [30,31]. Figure 20 illustrates the microstructures obtained in the different carburized compositions ofFe-C-M ternary system, where M is the carbide forming element. In the conventional carburization process only the surface layer is enriched in carbon as shown in (a), whereas both carbon enrichment of the sur-

Kiyohito Ishida

y +

C FI:JC

Fe M

(a)

(b)

(e)

(d)

Surfaee Interior

Fig.20. Schematic illustration of carburization in Fe-M-C ternary alloy.

153

Fig. 21. Microstructure of Fe-Mo-V-C alloys obtained from (a) CDC and (b) melting process.

face layer and the formation of carbide particles in the matrix are achieved by the CDC process as shown in (b)- (d). The resulting two-phase structure made up of"f (martensite after quenching) + carbide obtained by the CDC process has high hardness and excellent wear resistance. Microstructures from Fe-Mo-V alloys obtained by the CDC process and by the conventional melting process are compared in Fig. 21 [31]. The alloy produced by the CDC process is characterized by the presence of hyperfine grains and a profusion of fine carbide particles. A slow rate of carbide growth, leading to the presence, in particular, of very finely dispersed NaCl-type extremely hard carbides in great quantity is the main reason for such improvement in properties in the alloy produced by the CDC process.


5.4.3 Stainless Steels

Stainless steels are classified under various headings based on their phase constitutions as martensitic, ferritic, austenitic and ferrite/austenite duplex stainless steels. The Schaeffler diagram is a weH known practical guide, often used in welding for predicting phase constitutions in Cr-Ni stainless steels [32]. Although this diagram is very useful for estimating the compositional limits at room temperature of austenite, ferrite and martensite in terms of Ni and Cr equivalents, it is difficult to use this diagram for predicting high temperature equilibria. However, it is now possible to make use of the latest databases and perform computer calculations to obtain phase relations in commercial stainless steels. One such example is illustrated in Fig. 22 which shows the calculated phase diagram that is associated with a typical ferrite/austenitic duplex stainless steel of composition Fe-25Cr-7Ni-3Mo-O.5Si-lMn-O.15N-C.

1600

L

1400 a+L

~

E (Hy+L

~ 1200 ;:::l ..... ~ Q)

a+y+M23C6 0. E Q)

1000 f-<

a+y+ M23C6+Cr2N 800

a+y+M23C6+Cr2N+0"

600 0 0.1 0.2 0.5

a+y+M23C6+Cr2N+0"+Laves

mass % C

Fig.22. Calculated phase diagram of Fe-25%Cr-7%Ni-3%Mo-O.5%Mn-O.15%N-C duplex stainless steels.

Kiyohito Ishida 155

80 70

~ 60

2:l 50 '2 40 • 2 00 , ;::l 30 • -< o Calculation • 20

• • Measured • 10

0 600 800 1000 1200

Temperature (OC)

Fig. 23. The variation of volume fraction of austenite in commercial duplex stainless steels.

Another example of a calculational approach applied to stainless steels is shown in Fig. 23 where the volume fraction of austenite in commercial duplex stainless steels is plotted as a function of temperature [33].

Various other Cr-Mn based stainless steels have also been developed for various special applications. They come under the following categories: (i) high strength alloys (ii) non-magnetic stainless steels (iii) alloys for cryogenic service (iv) shape memory alloys (v) alloys resistant to void swelling under irradiation and (vi) alloys for medical applications with built-in Ni allergy triggers etc. One of the characteristic features of Cr-Mn base alloys is that they enable the introduction of higher contents of nitrogen as compared to conventional stainless steels because nitrogen solubility is much higher in Cr-Mn steels. Figure 24 shows the computed phase diagram for a high nitrogen Cr-Mn stainless steel of composition Fe-18% Cr-O.4%N-Mn [34].

5.5 Functional Iron Alloys

5.5.1 Magnetic Alloys

The development of precipitation hardened magnetic alloys with high coercivity began in the 1930s and varieties of precipitation hardened magnets have been produced. One of the important characteristics of these materials is that their high coercivity is a result of magnetic hardening due to spinodal decomposition, which leads to a microstructure composed of very fine precipitates of one phase embedded in the other.

Typical examples of phase diagrams associated with spinodal magnets are shown in Fig. 25. Figure 25a shows a vertical section diagram of a Fe-Ni-AI-Co


1600 L L+bcc

1400 u L+bcc cc 0 bcc+fcc 1200 ~ bcc+fcc+h+cr ::l .<:: ......

1000 + o:s .... '" v + 0.. u

S u

fcc+cr -v t-<

bcc1 600

5 10 15 20 25 30 mass % Mn

Fig.24. Calculated phase diagram of Fe-18%Cr-0.4%N-Mn stainless steels.

alloy from the basic system of Alnico magnet alloys [35J. It can be seen that Co addition narrows the miscibility gap between (}:1 and (}:2 bcc phases, shifts it to the Fe-rich corner and also shifts the order-disorder transition li ne to the Fe-rich side. It should also be noted that the miscibility gap develops a ridge along the order-disorder transition line [36J. This particular feature of the phase diagram can be used to advantage in obtaining microstructures with desired magnetic properties. Good magnetic properties in the precipitation hardened alloys are obtained under the following conditions; (i) the ferromagnetic (}:1 phase is present as isolated particles embedded in a weakly magnetic or nonferromagnetic matrix, (ii) these particles are highly ferromagnetic and uniaxially alligned, and (iii) the volume fraction of this phase is in the range 0.4-0.6. The first and third factors are strongly infiuenced by the composition of alloys and the shape of the miscibility gap. Since the addition of Co shifts the summit of the miscibility gap to the Fe-rich side, the volume fraction of the ferromagnetic phase increases and the ferromagnetism of the matrix phase is decreased.

Another group of alloys that come under the heading of spinodal hard magnet alloys is the group of Fe-Cr-Co alloys, which have good ductility and excellent magnetic properties [37J. Figure 25b shows the shape of the miscibility gap in the Fe-Cr-Co system. A variety of microstructures that can be obtained with different heat treatments in the case of two alloys, A and B on either side of the peak in the miscibility gap are also shown in the same figure. Ageing alloy A at any temperature will produce an undesirable microstructure in which the (}:2

phase will be embedded in the ferromagnetic (}:1 phase. On the other hand, alloy B would be suitable for obtaining high coercivity because the (}:2 phase would be formed as an isolated minor phase.

Figure 26 shows schematic illustrations of phase diagrams of various spinodal magnets [38, 39J. In the stable microstructures of the so-called Käster magnet al-

Kiyohito Ishida

20Co

140C p • "'"' ___ a l / .....

~ 120~ \ -'j.... U "-' Y ,.y+ u1 .. /\ ~ g 1 000 ----7/----~~ ~ - Pt>.. "- ::s ~ , 800 ;;rf,/s.. ----.j:l Co ~ Q.) - ........ l;" " ~', Q)

Q. . 'i ',20 Co .... Q. S H-- - - ---"'c- - - -. S ~ 600:! a f + a P '- \ ~ S OT03 f-----++<===-------=:==--------\

a '! 1 2 '- U; 97.S ÜOCo

a''-' _____ -', b'

77.SFe +20Co +2.SAl

OFe +20Co +SOAl

(a) Fe-Ni-Al-Co

40 SO 60 70

wt%.Cr

(b) Fe-Cr-Co

Fig.25. Miscibility gaps in Fe-AI-Ni-Co and Fe-Cr-Co permanent magnets.

157

loys Fe-Mo-Co and Fe-W -Co, precipitates of interemetallic compounds are found in a ferrite matrix. However, the ca1culated diagrams shown in Fig. 26a and Fig. 26a' indicate that there exists a metastable phase diagram showing a miscibility gap between Fe-rich bcc a and Mo- or W-rich bcc a' phases. Since the matrix phase in this case is the ferromagnetic a phase, the coercivity is due to the pinning of domain walls by the fine precipitates of non-magnetic Fe lean a' phase. Another group of famous spinodal magnets are from the systems Cu-Ni-Fe and Cu-Ni-Co. High coercivity in these alloys is a result of the impeding action of the matrix on the rotation of the magnetic particles, resulting in large shape anisotropy. The composition of these alloys are chosen to lie in the Cu-rich side as shown in Fig. 26b and Fig. 26b', to enable the precipitation of the ferro magnetic "( phase in the Cu-rich "(' matrix. It is difficult to obtain a proper volume fraction of the ferromagnetic phase in these alloys because the miscibility gap is nearly symmetricaL

The role of miscibility gap in the Alnico and Fe-Cr-Co alloys has already been discussed. Figure 26B again shows the shape of the miscibility gaps in the pseudobinary and ternary sections of these alloy systems. The difference between these alloys and the Alnico magnets lies in the fact that the miscibility gap in Alnico magnets is formed along the order-disorder transition line, while that of the FeCr-Co alloy is expanded along the Curie temperature. This peculiar feature of the asymmetrie miscibility gap plays a significant role not only in explaining the magnetic properties associattzd with these alloys but also in guiding the choice of suitable heat treatments to achieve the best microstructure for optimum magnetic properties. It is also worthwhile mentioning here that these spinodal alloys exhibit giant magnetoresistance (GMR) and the microstructure control of GMR is very similar to that of hard magnets [40].

158

(A)

NI

(8)

Phase Equilibria and Microstructural Control in Iron-base Alloys

(a) IsothcrmaI ScctiOll a,

(b) lsotbennal SectiOll

Co(Fe)

(a) IsothcrmaI Scction

Co

(b) botbermal Scction Ni

(., Vertical Section

(b') Vertical Sectioo

(a' ) Vertic:al Scction

Fe-Co

(b') Vertical Section

er

(a' ~ Microstruc~

, ,,' " " ' ,,'l'J.','" ,,', ", , , , , "

" , " , , (c) Microstruc~

Mapetic fleld

Fig. 26. Illustration of phase diagrams and mircrostructure of spinodal type magnets.

Kiyohito Ishida

CI5

~ ..... ~

9

8

7

6

5

4

3

2

1

f Donner and Hombogen D Murakami et al. 300K o Ghosh et al. l::. Otsuka et al. V Sade et al.

400K l::. 293

263V V 295

o +---T---,-Lo~-+---.--.---.--.---+ 20 22 24 26 28 30 32 34 36 38

mass % Mn

159

Fig. 27. Calculated and observed M s temperatures of Fe-Mn-Si shape memory alloys.

5.5.2 Shape Memory Alloys

The last decade has seen the development of ferrous shape memory alloys based on the Fe-Mn-Si system. The shape memory behavior observed in this system is based on the /,( fcc) / E(hcp) martensitic transformation. Thermodynamic assessment of the Fe-Mn-Si system has been made in order to develop furt her shape memory alloys based on Fe-Mn-Si [41]. Thermodynamic parameters necessary to calculate the phase diagram have been obtained which facilitate the estimation of parameters such as driving force of martensitic transformation, Ta temperature, M sand As temperatures etc. in this system. Figure 27 shows a comparison between calculated and experimental M s temperatures in this system.

5.6 Concluding Remarks

Recent progress in the development of computational approach to phase diagrams and creation of thermodynamic databases have contributed significantly to the progress in alloy design and microstructural control. In this paper, ex amples have been drawn from a whole gamut of high and low alloy steels to illustrate how phase diagrams and their calculations could be utilized to effect significant improvements in alloy design and processing. It is emphasized that such phase


diagram calculations are not only of academic interest but of practical use in design of commercial multicomponent alloys. It is pointed out that the thermodynamic database created for phase diagram calculations can also be used to advantage when combined with other computer simulation packages concerned with phenomena such as diffusion reactions in multicomponent steels [42].

Acknowledgements

The author would like to thank Dr. L. Chandrasekaran of DRA, UK for his critical reading of the manuscript. This work was supported by the Grants-inaid for Scientific Research from the Ministry of Education, Science, Sports and Culture, Japan and also by Proposal-Based Advanced Industrial Technology R & D Program.

References

1. Binary Alloy Phase Diagrams, T.B. Mssalski (ed.), 2nd ed., (ASM International, 1990).

2. Ternary Alloys, G. Petzow, Effenberg (eds.), (VCH. Vol. 1, 1988 - Vol. 11, 1994). 3. Handbook of Ternary Alloy Phase Diagrams, P. Villars et al. (eds.) (ASM Inter

national, Vol. I~Vol. 10, 1995). 4. Computer Aided Innovation of New Materials, M. Doyama et al. (eds.), (North-

Holland, 1993). 5. T. Nishizawa: Mater. Trans. Jpn. Inst. Met. 33, 713 (1992). 6. H. Ohtani, K. Ishida: Thermochim. Acta, 314, 69 (1998). 7. K. Ishida: Fundamentals and Application of Phase Diagrams, Seminar text, The

Japan Institute of Metals, 49 (1987). 8. J. Chipman: Basic Open Hearth Steel Making, (AlME, New York, 1964) p. 640. 9. J. Takamura, S. Mizoguchi: Proc. 6th Int. lron and Steel Cong. Vol.1, ISIJ, Tokyo,

591 (1990). 10. C.E. Sims, F.B. Dahle: Trans. Am. Foundarymen's. Ass. 46, 65 (1938). 11. K. Oikawa, H. Ohtani, K. Ishida, T. Nishizawa: ISIJ Int. 35, 402 (1995). 12. K. Oikawa, K. Ishida, T. Nishizawa: ISIJ Int. 37, 332 (1997). 13. W. Yamada, T. Matsusmiya, B. Sundman: Computer Aided Innovation of New

Materials, Ed. M. Doyama et al. (North-Holland, Amsterdam, 1991) p. 587. 14. T. Matsumiya, W. Yamada, T. Koseki: Computer Aided Innovation of New Ma

terials, Ed. M. Doyama et al. (Elsevier Science Publishers B.V., 1993) p. 1723. 15. T.W. Clyne, W. Kurz: Metall. Trans. A 12, 965 (1981). 16. S. Akamatsu, T. Senuma, O. Akisue: Effects of Cu and Other Tramp Elements on

Steel Properties, (The lron and Steel Institute of Japan, 1997) p. 126. 17. H. Ohtani, H. Suda, K. Ishida: ISIJ Int. 37, 207 (1997). 18. 1. Sawai, S. Uchida, Y. Sakuma, 0, Kono: BuH. Jpn. Inst. Met. 29, 376 (1990). 19. M. Hillert, T. Wada, H. Wada: J. Iran Steel Inst. 205, 539 (1967). 20. K. Ishida, H. Wakakuwa, T. Nishizawa: HSLA Steels, Metallurgy and Applicatiolls,

J.M. Gray et al. (eds), (ASM International, 1987) p. 851. 21. K. Hashiguchi, J.S. Kirkaldy, T. Fukuzumi, V. Pavaskar: CALPHAD 8,173 (1984). 22. K. Ishida: J. AHoys Compd. 220, 126 (1995).

Kiyohito Ishida 161

23. H. Ohtani, K. Ishida, T. Nishizawa: Trans. Mater. Res. Soc. Jpn. 9,199 (1992). 24. K. Inoue, H. Ohtani, K. Ishida: CALPHAD XXVI Florida, (1997) p. 21. 25. E. Rudy: J. Less-Common Met. 33, 43 (1973). 26. K. Ishida, H. Tokunaga, H. Ohtani, T. Nishizawa: J. Cryst. Growth 98, 140 (1989). 27. B. Uhrenius: User Aspects of Phase Diagrams, F. H. Hayes, (ed.) (The Institute

of Metals, 1991) p. 1. 28. B. Uhrenius, K. Forsen, B.-O. Haglund, 1. Andersson: J. Phase Equilibria 16, 430

(1995). 29. J. Golczewski, H.F. Fishmeister: Steel Res. 63, 354 (1992). 30. K. Ishida, Z. Ze-Gue, T. Kokubun, R. Kainuma, T. Nishizawa: Proc. 13th Inter

national Phansee-Seminar, Vol. 2, 343 (1993). 31. 1. Ohnuma, M. Mizumori, H. Ohtani, K. Ishida: Presented at International Sym

posia on Advanced Materials and Technology for the 21st Century, Honolulu, (1995).

32. A.C. Schaeffier: Met. Prog. 56, 620 (1949). 33. F.H. Hayes: J. Less-Common Met. 114,89 (1985). 34. C. Qiu: Metall. Trans. A, 24, 2393 (1993). 35. S.M. Hao, T. Takayama, K. Ishida, T. Nishizawa: Metall. Trans. A, 15, 1819 (1984). 36. T. Nishizawa, S.M. Hao, M. Hasebe, K. Ishida: Acta Metall. 31, 1403 (1983). 37. M. Okada, M. Homma: Recent Magnets for Electronics, Japan Annual Reviews in

Electronics, Computer & Telecommunication, (Ohshima and North Holland, 15, 1984) p. 231.

38. S. M. Hao, K. Ishida, T. Nishizawa: Metall. Trans. A 16, 179 (1985). 39. K. Ishida, T. Nishizawa: User Aspects of Phase Diagrams, F. H. Hayes ed., (The

Institute of Metals, 1991) p. 185. 40. H. Takeda, N. Kataoka, K Fukamichi, Y. Shimada: Jpn. J. Appl. Phys., 33, Ll02

(1994). 41. A. Forsbery, J. Ägren: Mater. Res. Soc. Symp. Proc. 246, 289 (1992). 42. J. Ägren: Proceedings ofInternational Conference on Computer-Assisted Materials

Design and Process Simulation, (The Iron and Steel Inst. Jpn., 1993) p. 284.

Computational Approach to the Fusion Reactor Materials

Tetsuji Noda and Johsei Nagakawa

Advanced Nuclear Materials Group, National Research Institute for Metals, 1-2-1, Sengen, Tsukuba, Ibaraki 305, Japan, E-mail:[email protected]@nrim.go.jp

Abstract. Collision cascade processes using the binary collision approximation and molecular dynamics are introduced. The combination of these approaches makes it possible to simulate the generation of cascade by neutrons with a wide energy. Various models of the irradiation creep mechanism as a macroscopic phenomena of radiation damage were proposed and the simulation was supported by the experimental result. Based on the radiation deformation mechanism, radiation induced stress relaxation in a fusion reactor was predicted. Simulation calculation of neutron spectrum, displacement of atoms, transmutation, and induced activity for candidate structural materials were made; Transmutation of so me elements like Wand V are influenced by the neutron spectrum. Simulation methodology for selecting optimum materials from the viewpoints of nuclear properties is presented.

6.1 Introduction

A fusion reactor is considered as an ultimate energy source and conceptual design studies have been made for more than 30 years. Among various engineering and technological problems, materials issues are directly connected not only with economics but also with the life and reliability of the structural components. In designing and prediction of the life time of fusion reactor structures, the irradiation of high energy neutrons especiaHy 14 MeV neutrons produced by the D-T reaction is considered as the most substantial issue. Although there is knowledge on radiation damage of materials in fission reactors, structural materials are considered to be exposed to the irradiation of 14 MeV neutrons which causes 3-7 times the amount of damage produced by neutrons of fission reactors. The incident neutrons scatter the lattice atoms in the materialleading to displacement radiation damage and, moreover, react with atoms resulting in compositional change, accumulation of helium and hydrogen gases, and induced radioactivity. At present, it is, however, not possible to conduct the material testing under the fusion neutron irradiation, with a fluence which the structural materials might meet, since the fusion reactor has not been realized.

Computational approaches therefore take a very important role in understanding the materials behaviors under fusion neutron irradiation and developing materials as weH as extrapolation studies on radiation damage using accelerators

Springer Series in Materials Science Valurne 34, Ed. by T. Saita © Springer-Verlag Berlin HeideJberg 1999

164 Computational Approach to the Fusion Reactor Materials

and fission reactors. In this chapter, at first, atomic sc ale damage processes in materials and macroscopic phenomena focusing on irradiation creep and stress relaxation in fusion reactors using simulation models are described. Transrnutation and induced activity of structural materials are then simulated where mainly neutron spectrum and fiuence effects are discussed.

6.2 Radiation Damage in Fusion Reactor Materials

6.2.1 Computational Study on Atomistic Phenomena

Many important physical properties of so lids are determined by the presence of lattice defects such as interstitial atoms and lattice vacancies. It is, therefore, clear that any process which alters their concentration affects the physical properties. Radiation damage triggered by atomic displacement from the crystallographic lattice sites is such a process. Neutrons with a very high kinetic energy of 14 MeV from the D-T fusion reaction can cause very severe radiation damage. These will occasionally collide with the nuclei of lattice atoms and transfer kinetic energy. If the transferred energy is large enough to displace the atom from its lattice site, this knocked-out atom (PKA: primary knock-on atom) will move rapidly through the crystal, colliding with other atoms and, if the transferred energy is still high, displacing them from their sites. A cascade of atomic displacements is thus generated by the incidence of a single fusion neutron.

The difficulty in the study of radiation damage lies in the fact that it consists of various physical processes vastly ranging in terms of time, size and energy. Most of the phenomena induced by the irradiation result from the atomistic events, for instance, atomic displacement and subsequent diffusion of interstitials and vacancies. It is, therefore, important to determine the behavior of individual atoms when the radiation damage problems in the fusion reactor materials are to be investigated. However, it is almost impossible, at least for the present, to study experimentally the atomistic processes of the radiation damage which completes within 10-15_10- 12 s. This leads to the necessity of computational studies based on atomistic models. In this section, computer simulation studies on the non-equilibrium atomic displacement events, induding the displacement cascade process during the radiation damage, will be reviewed. Two major computational study methods are to be discussed: (1) binary collision approximation and (2) molecular dynamics.

The rough scheme of the collision cascade is as shown in Fig. 1. A fusion neutron can produce the primary recoil atom leaving its lattice site at high energy, generating a trail of weIl separated Frenkel pairs (pairs of interstitial atom and vacant lattice site). The distance between the displacement collisions becomes shorter as the energy falls, until it is dose to the interatomic distance. Then, the region around the trajectory is violently disturbed, every atom is fiung outwards, and it is called a displacement spike. Each of the energetic higher-order recoil atoms will imitate this scheme in miniature.

The invention of computers made it possible to calculate the movement of the atoms involved in a collision cascade in detail. The computer calculation has the


o : Displaced atom (lnterstitial atom)

D : Vacant lattice site (Vacancy)

Incident energetic particle

Primary knock-on atom

(PKA)

Fig. 1. Schematic illustration of the collision cascade.

165

great advantages, over the analytical one, that any type of interatomic potential can be used and the crystallographic lattice can be introduced. The binary collision approximation (BCA) approach deals with only the individual collision between the traveling recoil atom and its nearest neighbor. This is treated by calculating the angle of scattering and energy transfer from the interatomic potential and the impact parameter, i.e. the separation normal to the incident track between the trajectile and the lattice atom. As described above, this approach gives a fair picture of the cascade at higher energy.

Beeler and Besco [1,2] developed a computer code, CASCADEjCLUSTER, in which the cascade process is regarded as a sequence of binary collisions independent from each other. The lattice atoms are displaced if they receive kinetic energy larger than the threshold energy Ed characteristic of the target material and, to be more exact, a function of the crystallographic direction. Calculation of a collision is carried out in the order of traveling speed of the knock-on atom during each calculation step, thus the interaction between cascade branches is taken into account. Figure 2 shows their result for the cascade in Fe produced by a 10 ke V Fe atom. The track of the first energetic atom termin at es rather swiftly, transferring its energy to secondaries, tertiaries, etc. In fact the original 10 ke V Fe atom (primary) was slowed principally by collisions with two primary knock-on atoms (PKA-l and PKA-2). In a statistical survey of several hundred cascades with 5- 20keV Beeler and Besco concluded that 50% cascades had at least one cluster of more than 10 vacancies when the PKA energy is larger than 15keV. It can be noticed in the figure that interstitials tend to form in an outer zone with their centroid ahead of that of the vacancies. However, one must be cautious of the magnitude of the interstitial-vacancy distances since the lower energy phenomena in the cascade process are excluded in this BCA cal~ulation.


150 Z COORDNATES

EVEN ODD 0 • VACANCY

140 0 • INTERSTITIAL ATOM

PKA PRIMARY KNOCK-ONATOM SKA SECONDARY

eeo '" 0 • TKA TERTIARY

130 CI)

!z <t 120 ... CI)

Z 0 (,)

110 w (,)

i= <t 100 ..J

90

• STARTING POINT ~~ FOR 10-IeVFe 0

0 SKA ~ . 0.8. o~~. CXl 0 .

• •• ct-.a • v

PRN~~ t~t • 0 ·l04°i~ ~o 80 o 0Qe oe o 0 • o 0

~ p-~~J~ l~. <b o~ • • oo·~ rf'.r:? 0 o ~ I- TKÄ ~ 0

~. ...;:,." h 9.08 0 0 0

~. V ? • . v. • • 0

• • .:f · 0 pt •• o • 0 •

• PKA·2 • 0 .. •

80 0

90 100 110 120 130 140 150 160 170

LATTICE CONSTANTS

Fig.2. Projection of a lOkeV Fe atom collision cascade in Fe onto a (001) plane (from [2]).

MARLOWE code developed by Robinson and Torrens [3] is still the most widely used among the BCA codes. It involves a scheme basically the same as the CASCADE/CLUSTER code, but it uses the Moliere approximation to the Thomas-Fermi screening function which is more accurate, particularly in the lower energy region. A principal advantage of BCA is that it requires only a small amount of information to describe the target crystal, hence a shorter computer run-time. In the earlier BCA code the translational symmetry of the crystal is used. The scheme used in MARLOWE is a slight modification of this and a list is made of the positions of several atoms, using one of the lattice sites as the origin of coordinates. The necessary atomic positions anywhere in the crystal may then be generated by using this list in conjunction with the position of a lattice site elose to any moving cascade atom. Since the BCA cannot deal directly with defect stability and interactions, it is necessary to inelude a model describing the with conditions under which atoms may become permanently displaced from lattice sites. MARLOWE allows considerable flexibility in defining this model and makes it possible to study its influence on the calculated results.

When the collision energies are elose to the energy binding the atom to its lattice, mutual interaction of collisions is no longer negligible and a crystallographic nature of the target material must be correctly introduced. The molecular dynamics (MD) approach for radiation damage study was first started by workers


• ... • • s····

• • C~ ! . ' •

:. ... .. o· . . . · .. . .. " . . .

A'" ...• '. . .. . • i.. . . .. • • • . ~ ... ": . ' .

. " • K'~ ~!· •.. ~ •..•.. •.. ~ .• ·0 ·0 . X. ' . • • 0

.K.·.··~~' -. ~ .. • • .. .•.. . . . . ., . '. .. . (al

. . . :. :::.: . . . . . . . . . . .... . . . . . . .-. :.:::.:.:.:~:.:.: .. ....... . . ........... . . .. 00 ......

••••••••••••• • •• • 0 • • • • • • • • • • • • • 0 • • • • •• •• 0 •••••

• • . 00 • • • • • • 0 ••••• · . . . . . . . . · ...... . · . . . . . . . . · . . . . . 0.

• •••••• 0 • • •••••• 0 · ...... . .

· . . · .. • ••• · . . · . . · . . . · . . • •••• · .. . · . .. . · .. . · . . . • 0 ....... : ..... . .. - .. . ..... . . .

( hl

167

Fig.3. Shot in (100) plane at 400eV, 10° away from [011] (from [4]). Set used in (a) indicated by rectangle.

at Brookhaven National Laboratory in the early 1960s [4,5]. They constructed a crystallite of about 1000 atoms which interact with realistic forces. Atoms on the surface of the crystallite are supplied with extra forces simulating the reaction of atoms outside, as though the crystallite is embedded in an infinite crystalline matrix. One atom is initially endowed with arbitrary kinetic energy and direction of motion, as if it had just been struck by a bombarding particle. A computer then integrates the classical equation of motion for the rest of atoms, showing the knock-on atom transfers energy to neighboring atoms, how the dynamic stages evolve, and how the kinetic energy finally dies away and the atoms in the crystallite come to rest in a damaged configuration. Figure 3a shows a collision cascade started at 400 e V in the {100} plane of Cu, while Fig. 3b shows the resulting array of defects. The knock-on atom started at K and moves to K'. At the end of the calculation run collision chains A, B, ... H are still active. It can be noticed that the interstitials have a dumbbell configuration (the extra atom is accommodated by sharing a site with another atom) and a total of 11 Frenkel pairs have been produced as a result of estimated 39 replacement events. The configuration of defects indicates several important features. Early recoils created no interstitial but aspace vacated by atoms they have struck.

Such "replacement" events greatly outnumber the number of permanent displacements. Secondly, each interstitial lies at the end of a sequence of replacement collisions which are along simple lattice directions. Also, there occurs a great number of (110) sequences which do not involve replacement but simply transport energy.

When the application of MD to radiation damage was started, the possible initial impact energy was only several tens of eV. This energy is far lower compared with the value expected for the PKA in the fusion reactor first-wall and blanket materials, i.e. several tens or hundreds of keV. Recent tremendous improvement of computer capability and development of reliable interatomic po-


300.-------~----~==~~==n

(I.u.)

200

100 -.. ::.:___ MARLOWE \ , -~ _I-

100 keV Si - ;> (lOOlSi

\" Iany-body forces,

MD [OW energies

I ' Damage srructure

I' Binary collisions, high ene rgy

I" Fast development of cascade tree structure

60 120 (I.u. )

Fig.4. A 100keV Si+ into (100) Si cascade calculated by MARLOWE with a 15eV cutoff energy, some branches of which were fed to the MD simulator (from [10]).

tentials [6,7] have made it possible to simulate the displacement cascades with 10-100 ke V [8,9]. On the other hand, in order to bypass the computer limitation in MD and to extend to the higher energy cascade, a linkage technique between MARLOWE and MD has recently been proposed [10]. The developed interface makes it possible to generate a high energy cascade, using MARLOWE to a certain minimum cut off energy, and then it feeds a portion of that cascade into the MD simulator to carry on the simulation at the lower energy end, where BCA is no longer valid, as shown in Fig. 4.

6.2.2 Computational Study on Macroscopic Phenomena: Radiation-Induced Deformation

Irradiation by high energy neutrons produced in the fusion reaction induces various property changes in the component materials. One of the most important alterations is the change in mechanical properties including dimensional stability, which results in the deterioration of performance and endurance of the fusion reactor components. Dimensional instability due to the irradiation originates from two types of macroscopic phenomena involving microscopic rearrangements of radiation induced point defects. One is void swelling, which is an expansion of the material as a consequence of agglomeration of atomic vacancies. The other is the radiation-induced deformation (irradiation creep and radiation-induced stress relaxation, depending on the external constraint) resulting from the interaction of the strain field of point defects (mainly interstitial atoms) with the externally applied stress field. In this section, a computational

Tetsuji Noda and Johsei Nagakawa 169

study of the radiation-induced deformation will be described for an example of the application of computational methods to the macroscopic property changes in materials associated with the radiation damage in fusion reactor materials.

Mechanisms of the radiation induced deformation are clearly different from those of the ordinary deformation in the unirradiated conditions. When the material is not irradiated, deformation proceeds by stress-driven motion of dislocations overcoming certain rate-controlling process with the help of thermal activation in a low or medium temperature range. At very high temperatures a high density of thermal equilibrium vacancies move under stress and contribute to the strain production. Under irradiation, on the other hand, inequilibrium densities of both interstitial atoms and lattice vacancies are continuously introduced in the material. Those point defects, especially the interstitial atom which has astronger strain field and thus a stronger interaction with the stress field, actively move in alignment with the applied stress and contribute to the strain production. Such a difference in the deformation mechanism is refiected in the dependence of creep (i.e. deformation under constant load) rate as shown in Fig. 5 [11]. Without irradiation, the plastic deformation rate is very sensitive to the stress and it is expressed by the so-called power law, i.e. dSp/dt = G(Jn

where G is a constant, with the stress exponent n of about 5. For the creep rate under irradiation the stress exponent n is only one until the thermal creep overwhelms at very high stress. This means that the radiation induced deformation is not only very active but also less sensitive to the stress level. This behavior leads to another important aspect of the radiation-induced deformation, namely radiation-induced stress relaxation. As shown in Fig. 6, the power law creep data in Fig. 5 give stress relaxation curves for both with and without irradiation. With n > 1, higher initial stress (Ja gives larger initial relaxation. However, the relaxation slows regardless of (Ja. Under irradiation, owing to the proportionality of the plastic deformation rate, the relaxation curve is independent of (Ja and the stress is steadily reduced to zero. This illustrates the significance and severity of the radiation-induced stress relaxation.

Modeling and Calculation Procedure. Mechanisms for the radiation induced deformation have been under active investigation for the last few decades and a number of theoretical models have been proposed. The latest overview of these models has been presented by Mathews and Finnis [12]. The models can be categorized into three major mechanisms, i.e. SIPN (stress induced preferred nucleation of planar agglomerates of point defects), SIPA (stress induced preferred absorption of point defects by network dislocations or dislocation loops), and Enhanced CG (climb-controlled glide of network dislocations enhanced by point defects), which are shown schematically in Fig. 7. None of these three mechanisms are exclusive of each other and it is quite natural to presume that all of these mechanisms operate at the same time competing for the point defects. In the computational study to be explained in the following, computer calculation of dynamic defect reactions, i.e. continuous production, absorption at various

170

I .... .s (I)


-4,-------------~----~------~

321 Stainless steel 500°C 1.2 x10- 6dpa·s-1

'§ -5 • c. (I)

I!! Ü Cl

..Q

without irradiation

slope = 5

-6~ __________ ~ __________ ~

10 100 Stress (MPa)

1000

Fig. 5. Stress dependence of creep rate with and without irradiation (from [11]).

b o ~ CI) CI)

~ U5

Displacements per atom o 1

321 Stainless steel 500°C, 1.2 x 10- 6 dpa·s-1

from thermal creep data

- - _ _ _ initial stress (Jo

------- 100 MPa

----- ----- 200 ------------- 300

from irradiation creep data (J 0 = 100, 200, 300M Pa

OL----L--~~--~--~--~

o 10 20

Time (day)

Fig. 6. Stress relaxation with and without irradiation (calculated from the creep data of [11]).

sinks, mutual recombination, and agglomeration of migrating interstitials and vacancies under the influence of external stress, has been carried out.

Chemical rate equations in the form of simultaneous differential equations were solved for the following defect concentrations;


• • • •

Ca)

• • •

1

: ..l.. • , , , ' D \ D , ,

-+-' . a : • , D ,

I , . (h)

• , , Ce)

Fig. 7. Schematic illustration ofmajor irradiation creep mechanisms Ca: SIPN, b: SIPA, c: Enhanced CG).

1) single interstitials and vacancies

dCi/dt = G - ki-vC;Cv - 2 (ki- iA + 2ki- iN ) C; - (kdA- i + 2kdN- i ) Ci

- (kileA-iCileA + 2kileN-iCileN) Ci - (kilA-iCilA + 2kilN- iCiIN) Ci

(1)

dCv/dt = G - ki-vCiCv - kd-vCv - (kileA-vCileA + 2kileNCileN) Cv

+ (kileA-vCileA + 2kileN-vCileN) Cv , (2)

2) aligned and non-aligned interstitialloop precursors dCileA/dt = ki-iAC; - kileA-vCileACv - KileA-iCileACi (3)

dCileN/dt = ki_iNCi2 - kileN-vCileNCv - KileN-iCileNCi, (4)

3) growing interstitialloops on aligned and non-aligned planes dCilA/dt = KileA-iCileACi (5)

dCilN/dt = KileN-iCileNCi, (6)

4) net interstitials absorbed by aligned and non-aligned network dislocations dCdA;/dt = kdA-iCi - kdA - vCv/3 (7)

(8)

5) net interstitials absorbed by growing aligned and non-aligned dislocation loops

dCilAi/dt = kileA-iCileACi + kilA-iCilACi - KilA-vCilACv (9)

dCilAi/dt = kileA-iCileACi + kilA-iCilACi - KilA-vCilACv, (10)


Table 1. Main parameter values used in the calculation.

Parameter

Defect migration energy

interstitial

Value

0.92eV

vacancy 1.15eV

Defect migration pre-exponent

interstitial 8.0 x 10-7 m2s- 1

vacancy

Lattice constant

Atomic volume

Strength of Burgers vector

Young's modulus

Shear modulus

Poisson's ratio

Dislocation density

annealed

cold-worked

Defect relocation volume

1.4 x 1O-6 m2s- 1

3.524 x 1O-10m 1.1 x 1O-29m3

2.1 x 10-10 MPa

2.6 x 105 MPa

1.0 x 105 MPa

0.3

interstitial + 1.4 x atomic volume

vacancy 0.46 x atomic volume

Difference in shear modulus

between defect and matrix

interstitial

vacancy

1.0 X 105 MPa

OM}?a

where km-n is a rate constant for each reaction between m and n. Deduction and discription of each rate constant requires a very extensive discussion and is beyond the space of this section. Readers are referred to [13] for furt her information.

In the calculation, point defect migration energies were taken from the evaluation by Dimitrov and Dimitrov [14], that is, O.92eV for interstitials and 1.15eV for vacancies. Important parameter values used in the calculation are listed in Table 1. Important parameters which indicate the strength of interstitial absorption, Zi (stress-free interstitial bias) and L1Zi (stress-induced bias) for loops and network dislocations, were calculated using the equations given by Wolfer and Ashkin [15] and Heald and Speight [16], respectively. As for a vacancy flux, values of Zv and L1Zv given in [15,16] were used in the calculation, but unlike L1Z;, L1Zv is too small to affect the radiation-induced deformation. At each numerical iteration step, the loop size was re-averaged and Zi,v and L1Zi,v were re-evaluated. In the interstitialloop nucleation process, a di-interstitial was provided to be aprecursor and its formation was considered to be affected by the external stress following the SIPN model proposed by Brailsford and Bullough [17].


In the calculation, SIPN (stress induced preferential nucleation of interstitial loops) [17,18] and loop growth driven by SIPA (stress induced preferred absorption of point defects) [16,19] are taken into account, as weIl as PA (SIPA climb) and PAG (glide enabled by SIPA climb) contributions [20] by network dislocations. From the calculated derivatives of defect concentrations, the plastic strain rate produced by each mechanism was evaluated at every iteration step using the foIlowing equations:

1) PA (SIPA climb creep by network dislocations)

. 2 CPA = 3' (dCileAjdt + dCileNjdt) ,

2) PAG (dislocation glide induced by SIPA climb)

EPAG = (4eVC7l'jLd)j3b) (dCdAddt - dCdNddt) ,

3) SAIL (SIPA climb creep by growing interstitialloops)

. 2 CSAIL = 3' (dCileAjdt + dCileNjdt) ,

4) SIPN (creep by stress-induced preferentialloop nucleation)

. 4 CSIPN = 3' (dCileAjdt + dCileNjdt) ,

(11)

(12)

(13)

(14)

where e is the elastic deflection (a JE ; a is the external stress, E is Young's modulus), Ld is the network dislocation density, and b is the strength of the Burgers vector. In the calculation and results, a nominal dpa, not the value multiplied by the damage efficiency, is used to describe the damage and the damage rate unless otherwise indicated.

Comparison with Experiments. Figure 8 shows the calculated irradiation creep rate contribution from each mechanism at 300°C, 1 x 10-6 dpa·s- 1 and 100 MPa for both solution annealed (SA) and cold-worked (CW) 316 stainless steel [13]. The contribution from the SIPA-driven loop growth (SAIL) was much lower than the others. SA material shows a region of high creep rate prior to the steady state. This is due to the transient high flux of interstitials that continues until a high concentration of slowly moving vacancies builds up in the matrix and increases the vacancy flux to a comparable level. Such a transient is also predicted in CW material at lower temperatures and a high defect production rate as shown later. Figure 9 shows the temperature dependence of the irradiation creep rate by each mechanism after reaching the steady state for SA and CW material. For SA material, in which mutual recombination of interstitial and vacancy is significant, aIl mechanisms predict a rather strong temperature dependence. The same tendency can be seen in CW material below about 200°C where the predominant mechanism changes from PA to SIPN. Also in the figures are the light-ion irradiation creep data which show rather good coincidence with the calculation.

174

I" ~ <1l

~ a. <1l 22 u c

.Q co '6 ~ Ol

.Q


-6

-8

-1C

-12

-16

-18

-20

-22 -6

316 Stainless steel 300'C, 100 MPa

~SIPN ' ----------------·PA

, '----------------SIP~ ,/ --------- -------- PAG ,/ // i // ~"Gl ,/ PA

- SA

--- CW

~ j

-4 -2 0 2 4 6 8

log Time (5)

Fig.8. Calculated irradiation creep rates vs. time (from [20]).

Low Temperature Irradiation Creep. Recently, very high creep strain has been reported for austenitic stainless steels, both SA and CW, irradiated in ORNL ORR reactor at 60°C [21]. These experiments were carried out in the course of irradiation studies for ITER (International Thermonuclear Experimental Reactor). Accumulated creep strains of pressurized tubes measured after the irradiation were comparable to, or higher than, those at 330°C and 400°C. This result is of direct relevance and importance to the design of ITER which will be cooled with low temperature water.

As has been shown in Fig. 9, the irradiation creep rate at the steady state decreases as the temperature is lowered since the bulk recombination of interstitials and vacancies becomes more prominent. However, this enhanced creep deformation at 60°C can be regarded as the consequence of a significant transient in the point defect kinetics at low temperature [13, 21]. At 60°C a very high creep rate region is predicted by the calculation, even in CW material which does not show such a transient at 300°C [22]. This is accompanied by a changeover of the dominant mechanism to SIPN. This appears to correspond to the microstructure dominated by a very dense population of "black-spot" loops observed in the 25% cold worked PCA samples irradiated to 7.4dpa in ORNLjORR at 60°C [24].

Grossbeck and Mansur [21] tried to explain the aforementioned low temperature irradiation creep in CW material by a mechanism based on a climbcontrolled dislocation glide. It is similar to the "I-creep" proposed by Gittus [25], although the origin of the excess interstitial flux is not the surplus of void swelling as for the I-creep but the transient of point defect kinetics. Such a mechanism can operate simultaneously with the present four mechanisms and it has been


Temperature (0C)

_4rr5~00~~~0~2~0~0 __ ~10~0~6~O~-.

-6 ., ~ -8 Q)

~ -10 Q.

~ -12 Q)

~ -14

~ <11 -16

ä5 01 -18 .Q

316 Stainless steel- SA 1 x 10~dpa·s-l 100 MPa

PAG

PA

-201~--~--2~--~~3~~

1 I T (10-3K-l)

Temperature (0C)

_4~5~00~~~0~2~0~0~~1~OO~60~-.

-6 'j'

~ -8 Q)

~ -10 Q.

~ '" -12 ~ u; -14 >.

~ -16 .S! cn 01 -18 .Q

316 Stainless Sieel - CW 1 x 10-6dpa.s-1 100 MPa

-201~~~~2~~--~3~~

1 I T (10-3K-l)

175

Fig. 9. Comparison of calculated irradiation creep rates with experiments (from [20]).

included in the evaluation using the following equation, analogous to that of the I-creep,

(15)

Figure 10 shows the calculated creep strain for both SA and CW material as a function ofdpa at 60°C. Contribution by the "transient I-creep (TIC)" in CW material is notable and makes the creep strain comparable between SA and CW while that in SA material is rat her insignificant [23J. The stress dependence of the low temperature irradiation creep at 60°C is a little stronger than linear but still very weak. Although the number of data points is rather limited in the experiment [21], the calculated stress dependence appears to coincide with the experimental results. This low dependence on stress is believed to cause a striking stress relaxation, as will be discussed in the following.

Radiation Induced Stress Relaxation in Experimental Fusion Reactor. The principle of the stress relaxation is given by

Ct = Ce + cp = (J / E + cp = const., (16)

where Ct is the total strain, Ce ( = (J / E ) is the elastic strain and cp is the plastic strain. The elastic strain necessary to reduce the elastic stress from (Jo to (J is given by

(17)

In most steels the elastic modulus E is around 2 x 105 MPa between room temperature and about 500°C so that only a very minute plastic strain of the

176

;? ~ c .~

Ci) a. Q) Q) ~

0 c 0

~ '5 ~


0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0

316 Stainless steel

1x10- 6dpa·s- 1,100MPa

300°C

--SA ----- CW

2 4 6 8 10 12

log Time (s)

Fig.l0. Calculated irradiation creep curves for 60°C and 300°C.

order of 0.1 % is required to completely relax the initial stress of 200 MPa. This small strain is yet quite difficult to be produced in the unirradiated condition, but is rather easily induced under irradiation as previously discussed in Fig. 6.

When the creep rate is expressed in the form of apower law, the relaxation equation can be obtained analytically as

(18)

!!..... = exp [ __ I_ln {I + (n - 1) ECO'~-l}] , 0'0 n-l

(19)

where C and n are parameters for the power law creep, dO'p/dt = cO'n. At low temperatures, as can be seen in Fig. 7, the plastic deformation rate depends on not only stress but also time (fluence) until the point defect kinetics reaches the steady state. Such transient behavior excludes these simple analytical equations at low temperatures. Instead, stress relaxation is calculated directly by computer simulation as follows. The amount of stress reduction L\O' by the total plastic strain L\c:p produced by all four mechanisms during each iteration step was evaluated using the equation

dO' dc: p - = -E- -+ L\O' = -EL\c: . & & p (20)

All kinetic parameters for the next iteration step are recalculated with the newly evaluated stress value [22J.


The stress relaxation behavior at 60°C is summarized in Fig. 11 for three different defect production rates. The most probable portion of ITER to be irradiated at low temperatures is the rear side of the blanket structure that encircles the fusion plasma, where the displacement rate should be two orders of magnitude lower than 1 x 10-6 dpa·s- 1 of the front portion. When the displacement rate is high, the stress relaxes at an extremely high rate, particularly in the initial stage of irradiation. This is clearly visible when compared with the relaxation at 300°C. Even when the displacement rate is very low (1 x 10-8 dpa·s- 1 ), like at the rear portion ofthe blanket structure, initial relaxation at 60°C is considerable and exceeds that at 300°C with 1 x 10-6 dpa·s- 1 . These results indicated the importance of the radiation-induced stress relaxation in ITER with the watercooled stainless steel blanket. Figure 12 shows the blanket structure proposed during the early conceptual design activity [26]. Blanket modules were planned to be connected to each other with bolts and the coolant water temperature was 60°C in the CDA design. In the following engineering design activity the coolant water temperature has been increased (110°C at the inlet and 140°C at the outlet ) and the mechanical joints have been principally replaced by welding because of the low temperature enhancement of radiation-induced deformation.

o ~ 0.5 (JJ (JJ

~ U5

\ , ,

316 Stainless steel solution annealed

60°C, (J =300MPa

, ,

o

',300°C 1Q.6dpa·s·1 '\ '\ , "' ...

0L-~~==~------~ o 2

Displacement of atoms (dpa)

Fig.11. Calculated radiation-induced stress relaxation curves for 60°C and 300°C (from [22]).


Beclric Insularor

Fig. 12. Early CDA design of the ITER blanket structure (from [26]).

6.3 Simulation of Transmutation and Induced Activity of Structural Materials

6.3.1 Reactions Between Materials and Neutrons

Transmutation of materials occurs as a result of reactions with energetic part icles such as neutrons, ions, electrons, etc. Especially, high-energy neutrons generated by the D-T fusion reaction interact with materials to form various kinds of nuclides through reactions as (n, 'Y), (n, 2n), (n, 3n), (n, p), (n, d), (n, t), (n, 0:), (n, 3He) and so on, in addition to producing displacements of atoms by knocking on. In Fig. 13, the nuclear reactions of 60Ni are schematically shown where primary reactions with neutrons and decay schemes of radioactive nuclides are indicated. The concentration of 60Ni at a moment during the neutron irradiation is expressed as

d (60 Ni) dt = -O"totalcP (60 Ni) + O"n,2ncP (61 Ni)

+A60=co (60mco) + A60co (60 Co ) , (21)

where (60Ni), (61 Ni), (60mco) and (60 Co) represent the number of 6oNi, 61Ni, 60mco and 60Co, respectively. O"total indicates total neutron cross-sections of 6oNi, cP the neutron flux, and A the decay constant of a radioactive nuclide. Equation (21) is given by (22) in a more general form:

(22)

where Ni: number of atoms of nuclide i at time t, Ai: decay constant of nuclide i, Aij: decay constant of nuclide j producing nuclide i, O"i : total cross-section for


~ ~O.47m

(n~ß- (n. '1)

.):Ei, ß _ ~rl'-~

~(n. p)

~ ,;n. nf>.<n.d) rsclF 44.S~d ~ 7SX1~ ....-S-9N'---. ...., ~ ß- ~~ I

~ EC.ß+ .II!f"'" ~n.3JIe) /

rsclFe 70.78d ~~(n,t) ~ "EC,ß+~

~~a)

fSCI~na) L....!!...J

Fig. 13. Nuclear reaction of 6°Ni.

179

nuclide i, O"ij: cross-section of nuclide j producing nuclide i and 4;: the neutron flux with energy spectrum. The concentration change after neutron irradiation is given when 4; =0 in (22).

Compositional change, and He and hydrogen gas production lead to the degradation of materials and the induced radioactivation and decay he at of materials should be reduced from the viewpoint of reactor and environment al safety [27,28].

The degree of transmutation is determined not only by irradiation time but by the neutron spectrum which depends on the configuration and composition of reactor structures. Then it is necessary to know the design of the fusion reactor, material composition, methodology of simulation calculation and nuclear data to quantitatively estimate the transmutation.

6.3.2 Reactor Structures and Neutron Spectra

Figure 15 shows a one-dimensional model of the mid-plane of the experimental thermonuclear reactor (ITER) [29]. The main structures are composed of carbon armor of 2 cm, first wall of 1.5 cm, blanket/shield of around 150 cm, SUS 316 vacuum vessel, super-conducting magnet, and liquid helium vessel. Several structural and blanket materials have been proposed. Table 2 shows candidate


Table 2. Material composition of several first wall, blanket and shield materials.

Region Thickness Outer Composition

(ern) radius (ern)

SUS First wall 1.5 198.5 SS 0.7+H20 0.3

Blanket 19.5 218.0 SS 0.7+LhO 0.1575+Be 0.4725+H20 0.05+He 0.1

Shield 1 15.0 235.0 SS 0.9+H20 0.1

Shield 2 13.0 248.0 SS 0.95+H20 0.05

RAF First wall 1.5 198.5 9Cr2W 0.7+H20 0.3

Blanket 19.5 218.0 9Cr2W 0.7+LhO 0.1575+ Be 0.4725+H20 0.05+He 0.1

Shield 1 15.0 235.0 9Cr2W 0.9+H20 0.1

Shield 2 13.0 248.0 9Cr2W 0.95+H20 0.05

V First wall 1.5 198.5 V Alloy (V5Cr5Ti)

Blanket 19.5 218.0 Li+Be

Shield 1 15.0 235.0 V Alloy+Li

Shield 2 13.0 248.0 V Alloy+Li

SiC First wall 1.5 198.5 SiC

Blanket 19.5 218.0 SiC 0.25+Li2Zr03 0.14+Be 0.56+He 0.05

Shield 1 15.0 235.0 SiC 0.56+B4 C 0.24+He 0.20

Shield 2 13.0 248.0 SiC 0.665+B4 C 0.285+He 0.05

combinations ofblanketjshield materials [30]. ITER originally adopted the blanket (SUS blanket) made of stainless steel, Li2ü with condensed 6Li, Be and water, and the shield materials made of stainless steel and water, respectively. In the reduced activation ferritic steel blanket (RAF), SUS316 was replaced by 9Cr-2W steel. A V blanket was one of the options for ITER where V-5Cr-5Ti was considered as the first wall and liquid lithium is used as breeder and coolant. The composition ofthe SiC blanket was presented in the ARIES-1 blanket model [31]. The blanket is composed of SiC, Li2Zrü3, Be and He pressurized gas. In the shield, B4 C is added to SiC and He gas to improve the shielding efficiency.

The neutron spectrum along the axis from plasma to the liquid helium can is calculated by transport theory. In the case of Fig. 1, the spectrum is obtained by solving Boltzman transport equation for the one-dimensional cylindrical model of a fusion reactor structure [32].

The Boltzman equation for neutron transport in the material is expressed as

[r\! <p (r, fl, E) = qe (r, fl, E) + qs (r, fl, E) + Ut (r, E) 4> (r, fl, E) , (23)

where r: position, fl: flow vector, E: energy, 4>: angular flux, qe: number of source neutrons, qs: number of neutrons produced by scattering, and Ut : total crosssection. Equation (23) is rewritten in words as


0.0 180.0 198.5

• rI'. rI\, rI'. ... ... ...

• rI'. "'" """ ... ... ... • rf4. I\, "

Shlcld2 h h h • h ,. "'" .... ... ....

220.0 235.0 250.0 290.4 334.0

Scl1lp<-orr La)"«

Annor

Gap

360.0

Vacuum Vessel ( S 0.86 + Vacuum)

181

550.0 cm

S M (SS 0.33 + Cu 0.34 + He<Lj 0.23 + Epo,y 0.1)

InsuLalor

He V .... I ( 0.86 + Vecuum)

Fig. 14. One-dimensional inboard model of a fusion reactor based on ITER design.

(Change due to neutron flow) = (Scattering in from other angles and energies)

+ (Source of neutrons)

- (Removal of neutrons). (24)

The solution gives steady-state neutron fluxes with different energies at each position in the direction of thickness for the structure from the neutron source of the fusion plasma. Equation (23) is numerically solved using one-dimensional code, ANISN [33] or two-dimensional code, DOT3.5 [34]. The Monte Carlo transport method is generally applied for three-dimensional calculation [35]. Nuclear data such as VITAMIN-C [36] and FUSION-J3 [32], are prepared for neutron transport calculation.

Figure 15 shows the typical neutron spectra at the first wall, the first shield and the vacuum wall for SUS blanket/shield system. The neutron flux decreases with the distance from the plasma. The flux almost linearly decreases with distance from the armor and the decay slope of 14 MeV neutron flux was around one order of magnitude per 15-17 cm independently of the blanket/shield compositions [30].

Figure 16 shows the neutron spectra at the position of the first wall for various blanket/shield compositions. There is a large difference in neutron flux at the low energy region between blanket/shield materials. In particular, in the V blanket/shield system where liquid Li is used as a coolant, the sharp decrease of the neutron flux is observed with decreasing neutron energy.

Neutron fluence, the product of wallioading and time, is often given as a dpa (displacement per atom). The dpa is also a measure of the radiation damage of materials due to displacements of atoms described in Sect. 6.2.


>C ::I [L g 1011

::I

Z

Irst WaU ----Irst Shield

."l. •

" ~

,,' r"'- -- ...

-- -- -- -,- Vaccum Wall

1010 -'--.---.-.,....--.----.-.,....--.----.-.,....--.---1 I ()-03 10.01 I ()+OI I ()+OJ I ()+OS 10.07

Neutron Energy (eV)

Fig. 15. Neutron spectra at first wall, shield and vacuum wall position for SUS blanket/shield system under 1 MW m- 2 wallloading.

1016

First Wall >-co il ~ 1014

'e ::I

& SUS316

'" 1012 9Cf2W '"i E " C

>C 1010 ::I

[L Cl

g VSCrSTi il Z

lOS 10'()3 10-01 10+01 10+03 10+05 1()+<l1

eutron nergy (eV)

Fig. 16. Comparison of neutron spectra at first wall for various blanket/shield compositions.

Table 3_ DPA of several candidate materials at the first wall for SUS and V blankets

DPA / year Material

SUS blanket V blanket

SUS316 7.9 9.1

9Cr2W steel 7.8 9.0

V5Cr5Ti 8.8 10.2

SiC 5.4 9.2


The dpa, D, of the material is expressed as

I G I

D = L L Ni (a dpakg4>gj L Ni , (25) i=1 g=1 i=1

where Ni is the number of nuclide i, (a dpa)i,g the dpa cross-section of nuclide i at neutron energy group g, 4>g the neutron flux at 9 energy group. land Gare, respectively, the number of nuclides in a material and neutron energy group.

The dpa cross-section, a dpa, is expressed as a function of incident neutron energy En [37J.

a dpa (En) = 2:d Lx J 1 + K:~ 9 (Ep ) ax (En, E p ) dEp , (26)

where K: displacement efficiency, Ep : knock on energy, En : incident neutron energy, Cd: displacement energy

(Z : atomic number, A : mass number)

9 (Ep ) = C + 0.40244c3 / 4 + 3.4008c1/ 6

and c = 0.0115Ep jZ2/7 ,

(27)

(28)

(29)

ax : neutron cross-sections of (n, ,), (n,2n), (n, p), (n, a), (n, np) and so on. Table 3 shows the dpa per 1 MW y m -2 for various materials at the first wall

of SUS and V blankets, where Cd was assumed as 40 eV. The dpa of all alloys for the V blanket is larger than for the SUS blanket. Since the neutron flux at an energy above 1 MeV for the V blanket is more than that for the SUS blanket, it is said that the dpa is mainly controlled by the high energy neutron fluxes

6.3.3 Compositional Change of Materials

The compositional change of 9Cr-2W steel for the RAF blanketjshield system with neutron fluence is shown in Fig. 17. Tungsten transmutes Re and Os leading to the decrease in concentration with the neutron fluence. Such change affects the phase stability of ferritic steels.

The effect of neutron spectrum on the compositional change of 9Cr-2W steel, V-5Cr-5Ti and SiC were examined for various blanketjshield systems by changing the first wall materials.

Tables 4 and 5 show the concentration change of some elements in 9Cr-2W steel, V-5Cr-5Ti, and SiC for different blanketjshield systems after 10 MW y m-2 irradiation. In 9Cr-2W steel, W is transmuted to Re and Os and reduces the concentration to below half of the initial value in SUS and RAF systems, while the transmutation is suppressed in V blanketjshield system. The concentrations of chromium and manganese are not affected by the neutron spectrum, although the amount of manganese slightly increases due to the transmutation of Fe and Cr.


Fe

+----------------------Cr

§ 10-02 W Mn .- I------'--------==---.r H C g li § u '§ 10-04

~

Si Re

Os Ta

10-06 +-------r''-----"'4------,--------1 10-02 10-01 10+00 10+01 10+02

F1uence (MW Y ffi-2 )

Fig.17. Compositional change of 9Cr-2W steel for RAF blanket/shield system as a function of neutron fluence.

If V-5Cr-5Ti is used as a first wall in SUS or RAF blanket/shield system, the Cr content increases by about 40% in contrast to the V blanket / shield system where only several% of the increase is predicted. The titanium concentration is not afIected by the neutron spectrum as seen in Table 4.

Regarding SiC, any compositional change, except gaseous products, is not obvious. However, substantial He formation occurs in SiC for any blanket/shield systems. Helium of about 0.7-0.8 at.% is produced after 10 MW y m-2 irradiation.

The results in Table 5 indicate that the amounts of He and hydrogen produced in the present materials are not much changed by the neutron spectrum, although there is a tendency for more He to be produced in V and SiC blanket/shield systems where fluxes with energies of 100 keV-1 MeV are rather higher than in other systems.

The W depletion in 9Cr-2W steel and Cr formation from V in V-5Cr-5Ti are afIected by the neutron spectrum. In particular, significant transrnutation of W and V occurs when the neutron flux with low energy is considerably higher in such case of SUS blanket / shield system.

Among several metallic elements, refractory met als such as W, Re, Hf and V have been noted to show high transrnutation rates [27,28] since stable isotopes of W have a large cross-section for the (n, , ) reaction in low energy region of neutrons. W is composed ofO.135% 180W, 26.4% 182W, 14.4% 183W, 30.6% 184W and 28.4% 186W. Re and Os productions mainly occur through the following paths:

184W (n, ,)185 W (75.1d) -+ 185Re (n, ,)186 Re (90.64h) -+186 Os (30)

184W (n, ,)185 W -+ 185Re (n, ,)186 Re (n, ,)187 Re (n, ,)188 Re (16.98h) -+186 Os

(31)


Table 4. Concentration change of some elements for candidate first wall materials after 10 MW y m -2 irradiation in various blanket/shield 'systems (at %).

Blanket / shield Material

Before SUS RAF V SiC

9Cr-2W steel

W 0.61 0.270.27 0.48 0.32

Re + Os 0 0.170.16 0.06 0.14

Cr 9.71 9.8 9.8 9.81 9.81

Mn 0.46 0.81 0.81 0.84 0.81

V-5Cr-5Ti

Cr 4.89 6.776.47 5.13 5.35

Ti 5.31 5.36 5.36 5.4 5.39

SiC

Si 50 49.649.6 49.5 49.5

C 50 49.249.2 49.1 49.2

Table 5. Helium and hydrogen production for candidate first wall materials after 10MWym-2 irradiation in various blanket/shield systems (atppm).

Blanket / shield Material

SUS RAF V SiC

9Cr-2W steel

He 1510 1510 1710 1630

H 0 0.17 0.16 0.06

V-5Cr-5Ti

He 1510 1510 1710 1630

H 0 0.17 0.16 0.06

SiC

He 1510 1510 1710 1630

H 0 0.17 0.16 0.06

186W (n, ')')187 W (23.9h) -+187 Re (n, ')')188 Re -+188 Os (32)

186W (n, ')')187 W (n, ')')188 W -+188 Re -+188 Os . (33)

Arnong the above reactions, (32) covers about 90% of Os production. Neutron capture is also significant for the transrnutation ofvanadiurn. Chrorniurn

is forrned according to the following reaction:

(34)


As shown in Table 4, W depletion and Cr accumulation, caused by the transmutation, are much affected by the neutron spectrum. In particular, neutron fluxes with energies below rv 100 e V determine the degree of concentration change of these elements.

Returning to the neutron spectra of various blanket/shield compositions, it is necessary to examine the factors controlling spectrum changes. In Fig. 16, the V blanket/shield system showed the lowest flux at low energies of neutrons. In order to clarify the compositional dependence of the spectrum, V-5Cr-5Ti, structural material of V blanket/shield system, was changed to 9Cr-2W steel and the neutron spectrum was calculated.

Figure 18 shows the comparison of neutron spectra for RAF blanket/shield systems with different coolant and breeder. It is clear that H2 0 softens the neutron spectrum resulting in the production of more neutrons with low energies than Li. The neutron spectra in this figure suggest that W depletion in 9Cr-2W steel is minimized

Considering the real first wall/blanket/shield system, the combination of V alloy /Li20 /H20 is not necessary realistic because vanadium is not stable in water. Hence Cr accumulation in V alloys will not be significant so far as V alloy /Li/Be system is considered.

1016

First Wall; Fe-9Cr-2W Sieel

>-Oll t<I

~ 1014

'2 :I

t Cl. ~ 1012

";' S " = >< :I 1010 Li+Be li: = 0 l:l :I

" Z 108

10.03 10-01 10+01 10+03 10+05 10+07

Neutron Energy (e V)

Fig.18. Neutron spectra at the first wall of 9Cr-2W /LbO /H2 0 and 9Cr-2W /Li/Be blanket/ shield systems.


6.3.4 Induced Radioactivity

When nuclide i is a radioactive nuclide which is transmuted to other nuclides radiating particles as (x, ß, and"( rays, the disintegration rate is expressed as

dN --' = A·N· dt " . (35)

The radioactivity of material 1a is a sum of (35) for all radioactive nuclides as

N

1a (Bq) = L AiNi (36) i=l

and "( ray dose rate 19 is given as

N

19 (Sv h-1) = L AiNiri , (37) i=l

where r i is the exposure rate constant of radioactive nuclide i. The decay heat, D H is deduced from the flux and energy of "( and ß rays as

D H (W) = L AiNi (Eßi + XiE,i) x 1.6021 x 10-13 , (38)

where Eßi is the average energy of the ß-ray, Xi the "( -ray intensity, and E,i the "( -ray energy.

The decay behaviors of air-shielded "( -ray dose rates and decay heats for various candidate materials for the first wall of the SUS blanket of the fusion reactor are, respectively, shown in Figs. 19 and 20. In the calculations, the materials were assumed to have ideal compositions without impurities and to be irradiated for lOMWym-2 at the first wall positions of SUS and V blanketjshield systems. The radioactive nuclides controlling the dose rates for longer cooling times, are 94 Nb, 60Co, 42 Ar( 42 K), and 26 Al, respectively, for SUS316, 9Cr-2W steel, V-5Cr-5Ti and SiC. Similar "( -ray decay behaviors were observed for both neutron spectra shown in Fig. 17. That is, the induced radioactivity of SUS316, 9Cr-2W steel, V alloy and SiC are not mainly determined by the fluxes of thermal neutrons but by those of fast neutrons. The neutron fluxes with energies more than 1 MeV are almost independent of the blanketjshield compositions [30] of fusion reactors as seen in Fig. 16. The decay heat also shows the same tendency as the dose rate as seen in Fig. 20.

Induced radioactivity of fusion reactor components, particularly the blanket structural materials, is one of the most important problems from the viewpoints of reactor safety, contact maintenance, waste management, and environment al aspects. Although the high radioactivity of fusion reactors is limited to plasma facing materials and blanket structural components, and no fission products are produced, the activity level of commercial reactors just after the shutdown is assumed to be about 105-106 GBq kw- 1 (th). The reduction of the induced

188

Cl

J:: > ~ Q)

~ ::J 0 Ul Q)

:5 E ,g E

Cii Q)

"§ Q) Ul 0

Cl


SUS316/Li2 O/H 20 Blanket

V/Be/Li Blanket

10 -1 '---'------'------'----L.-L-L----'------'-L.J.....LJL......JL--L--L..>...L.::""--J

10 0 10 2 10 4 10 6 1081010101210141016

Time (5)

Fig. 19. Decay behaviors of ')'-ray dose rate of various materials after irradiation for 10 MW y m -2 at the first wall positions of SUS and V blanket/shield systems.

-- SUS316/Li 201H20 Blanket

10 0 - - - V/Be/Li Blanket

Fig. 20. Decay heats of various materials after irradiation for 10 MW y m -2 at the first wall positions of SUS and V blanket/shield systems.

activity of materials is therefore, considered as one of the key issues in assuring clean energy from fusion reactors.

The evaluation of the induced radioactivity of elements has been carried out by many authors for different first wall irradiation conditions [38-42]. The definition of low activation is not simple because the safety level varies within different categories such as maintenance, accidental safety, waste disposal and recycling. Among the criteria, quantitative analyses to limit the concentration of elements were made for waste disposal and hands-on recycling [38-40], which


..... :~ c o ~ C Q) (.) c o Ü -5 Cl

.3

Cr,Ti,Fe,Ta W,Y,As

AI (26AI) N (14C)

Ni (59Ni)

Cu (63Ni) Mo (99Tc Nb (94Nb)

Ag (108mAg)

Tb (158Tb)

~ 0 2 4 Log Fluence(MW y m-2)

189

Fig. 21. Concentration limit of elements satisfying 10CFR61 Class C criterion as a function of neutron fluence.

pro duces the indices to design low and/or reduced activation materials. The regulation for shallow land waste disposal referred to as 10CFR61 Class C of the US NRC specifies the radioactivity limits for all radioactive nuclides [38,40]. The concentration limits for elements are given for the fixed irradiation conditions such as neutron spectrum and fluence on the basis of the specific radioactivity limits. Under irradiation of 20MWym-2 , concentrations of Mo, Ag, Tb, Nb, Gd, Ho, Ir, and Bi have been reported to be limited to 0.1 to 10 appm. Major alloying elements such as Al, Ni, and Zr were limited to 0.1 to 10% of content. No concentration limits were applied to C, 0, Ti, V, Cr, Mn and Fe. The same tendency in concentration limits for the hands-on recycling criterion [39] was reported for elements after 12.5 MW y m-2 irradiation, although more restriction was given to the concentrations of Ag, Ho, Tb, Nb and Dy. In this criterion, the acceptable concentration was assumed to satisfy 25 IlSv h -1 of hands-on dose rate after 100 y cooling [39].

Since radioactive nuclides accumulate with neutron fluence, which causes the increase of radioactivity, the acceptable concentrations of elements might vary with the fluence. Referring to the criteria of lOCFR61 Class C and the hands-on recycling, the relation between concentration limits and the neutron fluence were examined.

Figures 21 and 22 show the concentrations of some elements satisfying the criteria as a function of neutron fluence under the irradiation condition foreseen for the first wall. The concentration is expressed as a relative ratio to that of pure material and the value of 1 indicates no concentration limitation. In Fig. 22, the dose rate was evaluated for the activation volume, corresponding to 1 m2

in area and 1.5 cm in thickness. Nuclides in parentheses are the main radioactive


Cr,Ti,Fe,Ta O~~----~~----

W,Y,As .... :~ c o ~ C ~ -4 c o Ü Cl

~ -6

AI

Mo

Nb

Bi

Ag

Tb -8~ __ ~ __ ~ __ ~ __ ~ __ ~ __ ~

-2 0 2 Log Fluence(MW y m-2)

Fig.22. Concentration limit of elements satisfying hands-on recycling at 25~ Sv h- 1

after 100 y cooling as a function of neutron fluence.

nuclides controlling the radioactivity levels. Among elements, Tb, Ag, Nb and Mo are severely limited in concentration. The acceptable levels of these elements are more restricted with increasing fluence. For fluences higher than 10 MW y m-2 , a very strict limitation is required to these elements because of the accumulation of long-lived radioactive nuclides such as 158Tb, 108m Ag, 94Nb and 99Tc. Considering the low fluence assumed in ITER, below 1 MW y m-2 , the concentration of Al is not limited and milder limitation in concentration will be applied for impurities than the values [38]- [40] reported for 10-25 MW y m-2

irradiation. From the above results, it is concluded that neutron fluence should be con

sidered in determining the concentration limits of elements for low induced radioactivation.

As described above, impurity concentrations in the materials are important factors in the evaluation of induced radioactivity. High-purity structural materials are required to realize low and/or reduced activation. The impurity levels and their control on commercial steels and vanadium alloys have been already examined [43,44]. The main points producing high-purity alloys can be summarized as:


- using high-purity raw materials and

191

- adopting proper refining processes without contamination.

For Cr-W steels, ingots carefully controlled in impurities were manufactured by vacuum-melting, with Nb and Co contents respectively 0.7 and 50 ppm [43]. SiC composites, of which metallic impurity concentration of less than 50 ppm, are also produced by the CVI method [45].

Start 1

Selection of the most suitable ---I fusion material

"- using material database t

I Candidate materials t On the basis of

Conceptual design for engineering of reactor

fusion reactor - design

1 Input data

Neutron transport Structure of reactor Composition of each parts

calculation by ANISN Neutron wallloading t Nuclear data

Simulation of compositional ).- Input data change and DPA by IRAC Materials composition

t Nuclear data

~ Evaluation of candidate materials I t - t Experimental evaluation I I I

End

Fig. 23. Flow of material design of fusion reactor materials from viewpoint of compositional change by transmutation.

6.3.5 Material Design für First Wall and Blanket Structural Materials

In the previous chapter, it was shown that the transmutation and dpa of materials depend on the neutron spectrum. However, the neutron spectrum itself is determined by the composition of materials and vi ce versa. On the other hand,


the induced activity is not so sensitive to neutron flux although it increases with neutron fluence.

Figure 23 shows the flow diagram of the material design process for fusion reactor materials from the viewpoints of transmutation, dpa and induced activity [4]. It is natural to consider the material properties including mechanical strength and compatibility with plasma, coolant and breeder materials in addition to the present nuclear properties.

References

1. J.R. Beeler, D.G. Besco: J. Appl. Phys. 34, 2873 (1963). 2. J.R. Beeler: J. Appl. Phys. 35, 2226 (1964). 3. M.T. Robinson, I.M. Torrens: Phys. Rev. B 9, 5008 (1974). 4. J.B. Gibson, A.N. Goland, M. Milgram, G.R. Vineyard: Phys. Rev. 120, 1229

(1960). 5. G.R. Vineyard: Discuss. Farad. Soc. 31, 7 (1961). 6. M.S. Daw, M.1. Baskes: Phys. Rev. B 29, 6443 (1984). 7. M.W. Finnis, J.E. Sinclair: Philos. Mag. A 50, 45 (1984). 8. T. Diaz de la Rubia, M.W. Guinan: J. Nucl. Mater. 174, 151 (1990). 9. K. Morishita, T. Diaz de la Rubia: Mater. Res. Soc. Symp. Proc. 396, 39 (1996).

10. M. Jaraiz, G.R. Gilmer, D.M. Stock, T. Diaz de la Rubia: Nucl. Instrum. Methods 102, 180 (1995).

11. J.A. Rudson et al.: J. Nucl. Mater. 65, 279 (1977). 12. J.R. Matthews, M.W. Finnis: J. Nucl. Mater. 159, 257 (1988). 13. J. Nagakawa et al.: J. Nucl. Mater. 179-181,986 (1991). 14. C. Dimitrov, O. Dimitrov: J. Phys. F 14, 793 (1984). 15. W.G. Wolfer, M. Ashkin: J. Appl. Phys. 46, 547 (1975); 46, 4108 (1975). 16. P.T. Reald, M.V. Speight: Philos. Mag. 29, 1075 (1974). 17. A.D. Brailsford, R. Bullough: Philos. Mag. 27, 49 (1973). 18. R.V. Resketh: Philos. Mag. 7, 1417 (1962). 19. R. Bullough, J.R. Willis: Philos. Mag. 31, 855 (1975). 20. L.K. Mansur: Philos. Mag. A 39, 497 (1979). 21. M.L. Grossbeck, L.K. Mansur: J. Nucl. Mater. 179-181, 130 (1991). 22. J. Nagakawa: J. Nucl. Mater. 212-215, 541 (1994). 23. J. Nagakawa: J. Nucl. Mater. 225, 1 (1995). 24. P.J. Maziasz: J. Nucl. Mater. 191-194,701 (1992). 25. J.R. Gittus: Philos. Mag. 25, 345 (1972). 26. ITER Conceptual Design Report (IAEA, Vienna, 1991), p.165. 27. L.R. Greenwood, F.A. Garner: J. Nucl. Mater. 212-215, 634 (1994). 28. C.B.A. Forty, G.J. Butterworth, J.-Ch. Sublet: J. Nucl. Mater. 212-215, 640

(1994). 29. K. Maki, R. Takatsu, T. Kuroda, Y. Seki, M. Kajiura, N. Tachikawa, R. Saito,

R. Kawasaki: Shielding Design of Reactor Core Region in Fusion Experimental Reactor, JAERI-M 91-017, (1991).

30. T. Noda, M.Fujita: J.Nucl. Mater. 233-237, 1491 (1996). 31. S. Sharafat, C.P.C. Wong, E.E. Reis: Fusion Technol. 19, 901 (1991). 32. K. Maki, K. Kosako, Y. Seki, R. Kawasaki: Nuclear Group Constant Set FUSION

J3 for Fusion Reactor Nuclear Calculations Based on JENDL-3, JAERI-M 91-072, (1991).


33. W.W. Engle: "A user's manual for ANISN, A one-dimensional discrete ordinate transport code with anisotropie scattering", K-1693, (1976).

34. W.A. Rhoades, F.R. Mynatt: "The DOT-III Two Dimensional Discrete Ordinates Transport Code", ORNL-TM-4280, (1973).

35. E.A. Straker: "The MORSE Code - A Multigroup Neutron and Gamma ray Monte Carlo Transport Code", ORNL-TM-4585, (1970).

36. Y. Gohar, M.A. Abdou: DLC-60, ORNL, (1978). 37. M.T. Robinson: "Energy Dependence of Neutron Irradiation Damage in Solids",

Proc. BNES Nuclear Fusion Reactor, Conf., British Nuclear Energy Society, London, (1970), p.364.

38. F.M. Mann: Fusion Techno!. 6, 273 (1984). 39. G.J. Butterworth, L. Giancarli: J. Nucl. Mater. 155-151,575 (1988). 40. S. Fetter, E.T. Cheng, F.M. Mann: Fusion Eng. Des. 13, 239 (1990). 41. P. Rocco, M. Zucchetti: Fusion Eng. Des. 15, 235 (1992). 42. C.B.A. Forty, R.A. Forrest, D.J. Compton, C. Rayner: "Handbook of Fusion Ac-

tivation Data", AEA FUS 189, (1993). 43. D. Murphy, G.J. Butterworth: J. Nuc!. Mater. 191-194, 1444 (1992). 44. N. Yamanouchi et a!.: J. Nuc!. Mater. 191-194,822 (1992). 45. T. Noda: J. Nuc!. Mater. 233-231, 1475 (1996).

Modeling of Microstructural Evolution in Alloys

Yoshiyuki Saito

Department of Materials Science and Engineering, Waseda University, Tokyo, 169 Japan

Abstract. This chapter deals with theories of first order phase transitions and their application to the simulation of microstructural evolutions including such phenomena as nucleation, diffusion-controlled growth, phase separation and interface migration. A theoretical background for modeling of microstructure is first introduced, including a summary of the thermodynamics of phase transformations. This is followed by a treatment of homogenous nucleation theory including classical nucleation theory, which provides the basis for all subsequent nucleation theory. The next section deals with the interface motion. This also contains macroscopic models for microstructural evolution: equations of diffusioncontrolled growth in multicomponent alloy systems and of phase separation of alloys. Computer simulations of mesoscopic and microscopic systems will be introduced. Some applications of macroscopic models to microstructural evolutions in steels during thermomechanical processing are presented as examples of materials modeling.

7.1 Introduction

Computational methods are now used widely in materials science and engineering for the design of structural materials. However, more accurate materials design is required for efficient development of materials with superior properties. The modeling of microstructure on the basis of statistical physics and phase transition is essential for this purpose. This chapter presents the fundamental theories for modeling microstructural evolution and their application to computer simulation of metallurgical phenomena such as phase transformation, spinodal decomposition and grain boundary migration.

We will consider the typical example that a binary alloy system is quenched from a single-phase state of equilibrium to one where transformation is forwarded. The formation of the new phase (nucleation) requires the occurrence of the spatial fiuctuations of composition and structure of a certain critical size (the critical nucleus). A certain activation energy is required for the formation of such critical nuclei. Cluster of atoms which are larger than the critical size will grow with time, while smaller clusters will shrink. This dynamical evolution of clusters is nucleation theory, which has been a subject of research for at least sixty years.

A theoretical background for modeling of microstructure is first introduced, including a summary of the thermodynamics of phase transformations. This is

Springer Series in Materials Science Volume 34, Ed. by T. Saito © Springer-Verlag Berlin Heidelberg 1999

196 Modeling of Microstructural Evolution in Alloys

followed by a treatment of homogenous nucleation theory including classical nucleation theory, which provides the basis for all subsequent nucleation theory. A generalized treatment of overall transformation kinetics is also included.

This also includes heterogeneous nucleation theory and equations of diffusioncontrolled growth in multicomponent alloy systems and of grain growth. Some applications of macroscopic models to microstructural evolutions in steels during thermomechanical processing are presented as examples of materials modelling. The Monte Carlo method and its technical problems are described together with applications to grain size and shape evolution. Ordering kinetics in Ni base alloys is a further example of Monte Carlo simulation.

7.2 Theoretical Background

7.2.1 Thermodynamical Consideration

Let us consider a binary alloy consisting of NA A-atoms and NB = N - NA B-atoms. The order parameter of the system is the concentration of B-atoms, which will be denoted by eB = NB/N. The basic process of interest is that decomposition of a supersaturated single phase alloy into a two-phase state occurs at constant temperature and press ure P and thus promoted by a possible re duction of Gibbs free energy, G = E + PV - TB. Thermodynamic equilibrium is attained when the Gibbs free energy has a minimum. In solids the term PdV is negligible and thus Gibbs free energy G can be approximated by the HeImholtz free energy, F = E - TB, which will be used as the themodynamic function. Equilibrium is achieved if F is minimized:

8FT,v = dE - TdB = 0 , (1)

if dni(i = A, B) atoms of the component i are added, the variation of Fis

dF = -PdV - BdT + L /.Lidni = 0, (2)

where the chemical potential of the component i(i = A, B), /.Li, is defined as:

/.Li = (;~) T,v,ni (3)

The equilibrium condition (1) is then given by

L /.Lidni = O. (4) i=A,B

In a system of A and B atoms with the phase 0: and ß in equilibrium at constant temperature and volume, the change in F can be described as the sum of the change in both phases. Thus, in equilibrium

8FT ,v = dE - TdB = 0 (5)

with dn" - -dnß A - A'

(6)

The equilibrium conditions are given by

Yoshiyuki Saito 197

and (7)

or

( 8FQ) (8Fß)

8nA T,V,nB = 8nA T,V,nB (8)

( 8FQ) (8Fß) 8nB TVn = 8nB TVn , , A , , A

(9)

At equilibrium, the chemical potential of the component i (A or B) in two phases are identical and the two phases have a common tangent to the free energy curve.

Figure la shows a schematic illustration of the phase diagram of a binary alloy with a two-phase region at lower temperatures. the associated free energy versus composition, F(CB), curve is show in Fig. Ib. In the one-phase region of Arich alloy, F first decreases with increasing composition of B atoms, CB. Similarly, in the B-rich one-phase region, F decreases with increase in composition of A atoms, CA = 1- CB. In the two-phase coexistence region in equilibrium, F varies linearly with c. This reflects that the amounts of the two coexisting phases (with the concentrations of c~ and cß) change linearly with C according to the lever rule.

(a)

(!X.+13) t wo phase regi on

Composition, C B

LL

::n Cl) L 0> C

'" '" 0> L LL

(b)

d :<f I dc2=0 : d2 F/dCZ >0 d2 F/dc2>0:

Composition, cB

Fig.1. (a) Phase diagram of a binary model alloy (b) schematic free energy versus composition at a temperature Tl below melting point.

This hypothetical free energy now allows a furt her distinction of two different types of instability which initiate phase transformation, corresponding to two different types of statistical fluctuation, "heterophase fluctuations" and "homophase fluctuations". The first is an instability against finite amplitude, localized (droplet like) fluctuations which leads to the initial decay of a metastable system. The rate of formation of such drop lets is described by homogeneous nucleation theory. The second is an instability against infinitesimal amplitude,


non-Iocalized (long wavelength) fluctuations which lead to the initial decay of an unstable system. This instability is termed spinodal decomposition. States where

( ~) T > 0 correspond to metastable states and states where (~) T < 0 are

unstable states. The locus of inflection points in the (T, CB) plane,

(fJ2F) -2- =0 aCB T

(10)

defines the spinodal curve CB = CS (T). This distinction is now linked to the two transformation mechanisms described above.

Let us consider an M -component system. Helmholtz free energy, F, is a homogenous first order function of the extensive parameters, NI, .. N i , .. , NM, where Ni is the number of atoms of component i of the system. That is, if all the extensive parameters of the system are multiplied by a constant a, F is multiplied by this same constant,

F(T, V, aN!, ... , aNM) = aF(T, V, NI, ... , NM). (11)

Differentiating (11) with respect to a, we obtain M ",aF(T,V,aN1 , ... ,aNM)N. =F(T VN N) (12) W a( aN.) J , , 1, ... , M· j=1 J

For a = 1

(13)

Substituting (13) into (2), we have a relation known as the Gibbs-Duhem relation

M

PdV + SdT + L dJ-LjNj = O. j=1

7.2.2 Nucleation Kinetics

(14)

Let us consider a homogenized solid solution quenched into the metastable region. The system will survive for some time as a metastable solid solution; however, it will eventually reach a thermodynamically equilibrium state. Some microclusters at one of the equilibrium concentrations will form in the matrix. The theory of nucleation aims to calculate the rate of formation of such nucleating microclusters, the nucleation rate, which we shall denote by J.

In this section we will only consider the case of homogeneous nucleation: nucleation which takes place in a completely homogeneous phase. In most met als and alloys, nucleation occurs preferentially at grain boudaries, dislocations, and so on (heterogeneous nucleation). Thus, the following discussion is not directly applicable to nucleation phenomena which dominate such a solid-solid phase transformation. Nevertheless, the fundamental concept of homogeneous nucleat ion is very useful in understanding microstructural evolutions controlled by nucleation and growth mechanism as will be explained in the following chapters.

Yoshiyuki Saito 199

Droplet Model. We will first consider the droplet model in the framework of equilibrium statistical thermodynamics. This model provides an insight into the mechanism of decay of a metastable state.

To take a special example, consider the precipitation of a B-rich ß phase from a supersaturated 0: solid solution. The drop let model pictures this system as a "gas" of noninteracting droplets. The number of clusters of size I is given by the Boltzman factor

(15)

where c is the free energy of formation of a cluster of size l. The crucial problem is then to determine c. In classical theories, the free energy change associated with the formation of the cluster will have the following contribution (here the effect of strain energy is not taken into consideration, simplifying the problem).

1. At a temperature where the ß phase is stable, the formation of a cluster of size l causes a free energy reduction of 1Of-L, where Of-L is the difference in free energy per atom between the matrix and the precipitated phase.

2. The creation of an 0:/ ß interface will give a free energy increase of al2j3 ,

where a is a constant proportional to the surface energy.

Thus, we have (16)

Let us consider the behavior of nl as a function of the drop let size l. For Of-L < 0, c(l) is a monotonically increasing function of l, and nl decreases rapidly with increasing l. On the other hand, for Of-L > 0, the situation is different. There is competition between the bulk and surface terms, with the surface term dominating for small land the bulk term dominating for large l. As a consequence

there is a critical droplet lc [lc = (2a /30f-L)3] such that droplets for which l > lc

are energetically favored and grow. These droplets thus provide the nucleation mechanism by which metastable states decay.

This also implies that at a condensation point Of-L = ° and T < Tc, the free energy has an essential singularity. This singularity results from the droplet contribution to the free energy. The free energy, F(Of-L), is given by

1 ~ ~ (-Of-Ll + ad ) F(Of-L) = N t:-t NI (Of-L) = t:-t exp - kBT

and which can be written as a Mayer-like cluster expansion 1 00

F(Of-L) = N l:b1z1, 1=1

where

z = exp (k~T) and b1 = exp (- :~; ) .

(17)

(18)

(19)

It is clear that the free energy for the metastable phase in this model should be obtained by truncating the sum in (17) at lc:


O~------'-~---------l

I c

Fig. 2. The distribution of cluster size according to the droplet model.

(20)

The remaining contribution of the droplets in (17) gives rise to an essential singularity in F(ofJ) (see Fig. 2). For small!ofJ!, lc will be very large and F will be determined only by smalll values in the sum in (17). The function F(ofJ) should not be sensitive to the precise way in which this sum is cut off near lc.

Becker-Döring Theory. Classical nucleation theory is based on the idea that metastable clusters are formed by fluctuations and that only clusters with sizes which exceed the critical size by chance grow continuously. The interesting problems are the mechanism of the formation of stable clusters and the growth of domains which exceed the critical size.

To obtain a rough idea of what J is, let us consider the size l* for which the free energy E(l) of a cluster is a maximum. For l > l* the cluster lowers its energy by growing. Conversely, a cluster with size l < l* dissolves into the matrix. Therefore, E(l*) appears as an energy barrier which gives the activation energy for the process. It is natural to consider that the nucleation rate has to be proportional to exp[-E(l*)/kBT].

A more detailed analysis was done by Becker and Döring [1]. They formulated the kinetics of cluster formation. The starting point of the Becker-Döring theory is the kinetic equation for a time dependent nl(t), where nl(t) is the average number of drop lets of size l at time t. The basic assumption of their theory is that the time evolution of nl(t) is due only to an evaporation-absorption mechanism, in which a droplet of size l gains or loses a single atom (monomer). Therefore, effects such as the coagulation of two droplets are not considered. The rate per unit volume at which a drop let of size l - 1 grows to l, J(l), is given by

J(l) = a(l- l)n(l- 1, t) - b(l)n(l, t), (21)

Yoshiyuki Saito 201

where a(l - 1) is the rate at which monomers are absorbed by a cluster of size, l - 1, and b(l) is the rate at which monomers are evaporated from a cluster of size l. The rate equation for nl(t) can be written as

8n~; t) = J(l) _ J(l + 1). (22)

Equation (22) does not hold for single particle clusters, since such clusters are not constrained to events involving other one-particle clusters. Becker and Döring essentially assumed that nl remains constant.

The coefficients a(l - 1) and b(l) in (21) can be obtained by invoking the principle of detailed balance: their ratio must be such as to drive the system towards thermal equilibrium.

a(l - l)no(l- 1) = b(l)no(l). (23)

The equilibrium droplet distribution function no(l) is given by the Boltzmann factor,

no(l) = n(l) exp [_ :~l~] , (24)

where E(l) is the free energy of formation of a droplet of size l. The classical assumption is that E(l) is the sum of a bulk term and a surface term. The bulk term corresponds to the driving force for droplet formation and the surface term expresses the energy associated with surface tension. Thus

E(l) = ao(l- 1)2/3 - bo(l - 1), (25)

where ao is a constant proportional to the surface energy and bo is the difference in chemical energy per atom between the matrix and the precipitated phase. We have chosen the form of E(l) in (25) so that both terms on the right hand side vanish for the case of a monomer. The flux J(l) is obtained from (21) and (25):

J(l) = -a(l _ l)no(l- 1)~ [n(l- 1, t)] . az no(l- 1)

Using (26), (22) is rewritten as:

8n(l, t) = ~ {(l) (l)~ [n(l, t)] } 8t az a no az no(l) .

(26)

(27)

This is a Fokker-Planck equation with a l dependent diffusion coefficient a(l). For small l, the initial distribution, n(l,O), is nearly equal to no(l) and not

dependent upon the supercooling of the system. If the above condition is fulfilled for t > 0, after some relaxation time n(l, t) approaches a steady state distribution n s (l) which satisfies the following equation

8 [n(l,t)] a(l)no(l) az no(l) = -J = const, (28)

where J is the steady state nucleation rate. Becker and Döring used the following choice of boundary conditions to obtain a time-independent solution:

if l ---+ 00 (29)


ns(l) no(l) -+ 1, if l -+ O. (30)

The steady state solution of (27) is

1 J = roo dl .

Jo a(l)no(l)

(31)

The range of integration in (31) includes the peak at l = l* where E(l) is a maximum and the integral can be evaluated by the saddle point approximation. The integrand no (l) -1 has a rather sharp maximum at l*. Thus, the integral (31) can be evaluated by expanding the integrand around l* . We have

(l) = (l*) (l - l*)2 f)2 E(l) I E E + 2 >ll2 . u l=l*

(32)

Thus the steady state nucleation rate, J, can be written as follows:

( a2,(l) ) 1/2 *

J - (l*) - ----;w-Il=l* (l*) _ T [_ E(l )] - a 21rkBT no - JO exp kBT· (33)

This is the Becker-Döring result for the nucleation rate. This equation indicates that nucleation is a thermally activated process and Jo is the so-called nucleation prefactor. It should be noted that the steady state nucleation rate depends very strongly on the exponential term. The quantity E(l*) is an activation energy (the energy for a forming critical nucleus). The factor (_f)2E(l)/f)[2Il=l*/21rkBT)1/2 on the right hand side of (33) is known as the Zeldvich factor (Z). It should be noted that the major contribution to the integral in Eq.(31) comes from the integration in the range of Il - l* 1 < ~, where ~ is a correlation length and is given by

[I

f)2 (l) 1 ] -1/2 ~ ~ - ----;.- /21rkBT

f)[ l=l* (34)

Thus the steady state nucleation rate, J, is proportional to the number of droplets which reach the critical size times the rate which they cross. The Zeldvich factor accounts for the fact that not all droplets which reach size l* actually continue to grow.

Time-dependent N ucleation Rate. Steady state will be attained once the clusters have attained sizes, le, for which the probability of dissolution is negligibly small. Feder et al. [2] evaluated the size le

le = l* + ~Z. (35)

As the gradient of E(l) within the region of Il -l*1 < l/Z is rather small, the cluster will move across this region predominantly by random walk with the jump frequency a(l*). The time required for a cluster to diffuse a distance l/Z by a random walk is given by the time lag. Applying the random walk theory of diffusion, the time lag is evaluated as

Yoshiyuki Saito 203

1 tlag = 2a(l*)Z2' (36)

Direct interpretation of the above-mentioned time lag estimation is not easy and the mathematical treatment is not exact (the drift term is neglected). Straight-forward formulation of time lag evaluation was proposed by Stauffer and coworkers [3,4]. We will now discuss methods for solving the Fokker-Planck equation.

First, we will rewrite (27) by introducing a new variable

(l ) = n(l, t) - ns(l) U ,t no(l) (37)

as au a [ au] no(l) at = 8l a(l)no(l)8i . (38)

The solution of (38) can be represented as

00

u(l, t) = L Ck exp( -Akt)Uk(l) (39) k=O

with

LUk(l) = :l [a(l)nO(l) :lUk(l)] + AknO(l)uk(l) = 0, (40)

where uk(l) are the eigenfuctions and the Ak the eigenvalues of the Liouville operator L. Equation(40) is called the Sturm-Liouville equation [5]. The boundary conditions of (40) are given by

uk(l) = 0,

uk(l) = 0,

l ~ 0,

l ~ 00. ( 41)

The problem to find the upper and lower bounds of eigenvalues is called the Sturm-Liouville boundary value problem. Defining a functional I[u] by

(42)

A suitable trial function u% is

U% = sin [~(k + 1) ( 1 - 2Js 1: a(l)~O(l)) ]. (43)

We rewrite the integral in (42) by introducing a new variable

11 dl' x = 2Js 1* a(l')n(l') (44)


as 2Js t (äuUx») dx

I[ t] = -1 äx uk 1 .

2~s LI no(x)2a(x)ut(x)dx

The boundary condition in (41) is described as

ut(x = -1) = 0,

ut(x=I)=O.

( 45)

( 46)

The boundary conditions in (46) have a physical explanation: for x = -1, i.e. l --+ 0, very small droplets are always in equilibrium, and for x = 1, i.e. l --+ 00,

there are very few large droplets. It can be shown that

(47)

holds for any function ut(l) which satisfies the boundary conditions and is a solution of (40). For the lowest eigenvalue, the following relation is obtained:

A lower bound is obtained from the relation [5]

)..k ~ min cf;()..) , -1:Sx:Sl

ft [a(l)n(l) ftu'~ (l)] cf;()..) = n(l)u't(l) ,

( 48)

( 49)

where u'~ is any function which satisfies the boundary conditions (41) and does not change sign in the interval -1 :s; x :s; 1. Taking u'~ = uL we have a lower bound for the lowest eigenvalue:

(50)

Thus, we have the asymptotic behavior of u(l, t) as

u(l, t) '" exp( -)..ot)Js ,

1 < )..0 < n 2

- 4a(l*)Z2 - 4 . (51)

7.2.3 Interface Dynamics

In this section, we will discuss the dynamics of an unstable interface. One interesting example is the motion of curved antiphase boundaries (APB), which arise in order-disorder transitions. An APB is an interface separating domains with identical properties in systems having long range order. Another example is grain growth. Grain growth in pure met als requires the rearrangement of atoms

Yoshiyuki Saito 205

in grain boundary regions, thus requiring diffusion over distances of the order of an atomic spacing.

A general Ginzburg-Landau model without the noise term ofthe free energy functional is given by:

(52)

The local rate of displacement of the order parameter, fJTJ(x)/8t, is linearly proportional to the local thermodynamic force, JF/JTJ(x):

8TJ(x) _ r JF Eit - - JTJ(X) , (53)

where r is the response coefficient which defines the time scale for the system. Thus

8TJ(x) = -r [-K'v2TJ + d(.::1f)] . 8t dTJ

(54)

The total time derivation of the free energy is

(55)

Thus F is a strictly non-increasing function of time. This excludes any activated processes. First, let us consider the case of a planar interface. For the equlibrium, 8TJ/8t = 0,

(56)

Equation (56) is analogous to the equation of motion of a point mass in a doubleweH potential with the foHowing change of variables: z -+ time, TJ -+ displacement, K -+ mass and V(TJ) -+ -.::1f (see Fig. 3). In the absence of friction, the total energy is a constant and equal to -.::1(TJe), and the kinetic energy vanishes at the top of the hill. Therefore:

K (dTJ )2 2" dz - .::1f(TJ) = -.::1f(TJe). (57)

The surface energy (J" associated with the interface is calculated by substituting the bulk term from the total energy:

Next, consider a spherical drop let immersed in a solid solution. The energy required to form this drop let consists of a bulk term and a surface term

(59)


(8) (b)

v v

o

~

TI o TI

Fig. 3. Mechanical analogs corresponding to (a) static interface and (b) moving interface.

This energy is maximized for

(60)

The kinetic equation (54) may be written as

ory(x) = -r [-K (d2ry + ~ dry) + d(.:1f)] . ot dr2 r dr dry

(61)

If we look for a solution of the type ry = ry'(r - R(t)) then

ory(x) = _ dry' dR = -r [-K (d2ry, + ~ dry') + d(.:1f)] . (62) ot dr dt dr2 r dr dry

In the vicinity of the interface r = R, we may write (62) as

K d2ry' + (2K +~) dry' _ d(.:1f) _ 0 dr2 r r dr dry - , (63)

where v =dR/dt. By integrating (63) we find

J dr [Kd2ry' _ d(.:1f)] dry = _ (2K +~) Jdr (dry')2 (64) dr2 dry dr r r dr

Equation (64) may be rewritten as

J dr [Kd2ry' _ d(.:1 f )] dry = Jdr~ [K (~)2 -d(.:1f)] dr2 dry dr dr rmdr

= ; (2: + 3f) (65)

Yoshiyuki Saito 207

and thus

8ry = - ; (2: + f) . (66)

From (66) we may obtain the velo city of the moving interface:

v = _ 2Kr _ 8ryrK = 2Kr (_ 8ry __ 1_) = 2Kr (~ __ 1_). (67) R (J 2(J R(t) R* R(t)

We may recall the results of classical nucleation theory: the drop let collapses for R < R* and grows for R > R*. The unstable stationary situation, v = 0, occurs when R = R*, as expected according to the Gibbs-Thompson relationship.

The preceeding discussion may be generalized to apply to curved interfaces which are not spherical (Allen and Cahn [6]).

Completion Formula. Let us consider the late stage of phase transformation which is governed by nucleation and growth. In this situation, the second phase will eventually consume the sampie. An important quantity to calculate is the fraction of the volume at a time t after quenching [7-9]. We derive a formula for the time evolution of the volume fraction of the transformed phase at t, f(t) [10,11]. The volume fraction of the metastable phase is given by

g(t) = 1 - f(t) . (68)

Take a point A at random. We calculate the rate of the fraction transformed at the point A within the period T to T+dT

(69)

Let v(P, t) be the growth rate at time t of a droplet which nucleated at a point P at a time t'. The distance between the point P and a point Q, which is the intersection of the line AP and the surface of the drop let (see Fig. 4), at time t" is given by

t"

PQ = Z(P,t",t') = 1 v(P,t)dt. t'

(70)

The surface of a droplet formed at a point P' at moment t' reaches the point concerned within the period T to T+dT only if the distance between A and P' satisfies the following condition:

1T 1T+dT v(P',t)dt < P'A < v(P',t)dt.

t' t' (71)

Thus the point A is reached by droplets formed at time t' within the period between T and T+dT only if nucleation occurs at points within a closed domain of volume

VA = In J Z(P', T, t')v(P', T)dxdydT, (72)

where D is the surface on which each point is situated at a distance Z(P'(x, y, z), t, t') from the point A. The prob ability of nucleation in this domain within the period t' to t' +dt' is equal to


Fig.4. Condition for a point A reached by a drop let at moment t' within the period between T and dT.

PAdt' = 10 1 Z(P', T, t')v(P', T)J(t')dxdydt', (73)

where J(t') is the nucleation rate per unit volume at time t'. Then the probability that droplets formed at a randomly selected point during the period of time between t' = 0 and t' = t reach the point A is

PA = l T 10 1 Z(P', T, t')v(P', T)J(t')dxdydt'. (74)

The point A transforms to a stable phase from the metastable phase only if the point has not transformed previously. Thus, the rate of transformation is

df(T) = -dg(T) = g(T)PA. (75)

The solution of (75) is given by

In ;~~~ = -lt dT lT 10 1 Z(P', T, t')v(P', t)J(t')dxdydt'. (76)

Since the integral of the right hand side of (76) has a finite value, we can change the sequence of integration. Then

In ;~~~ = -lt J(t')dt' Lt 10 1 Z(P', T, t')v(P', t)dxdydT. (77)

The volume at a time t of a droplet which formed at a time t' is

v(t,t') = Lt 101 Z(P',T,t')v(P',t)dxdydT. (78)

At t = 0, g(O) = 1, we obtain the weH known Kolmogorov-Johnson-MehlA vrami type formula

f(t) = 1 - exp [-l t J(t')V(t, t')dt] . (79)

Yoshiyuki Saito 209

7.3 Macroscopic Modeling of Microstructure

7.3.1 Diffusion Controlled Growth of Precipitate

Growth of an Isolated Precipitate. Let us consider diffusion al growth of a precipitate. Growth of an isolated precipitate in an infinite matrix is described by a diffusion equation

with conditions

G(r = R,t) = Gr ,

G(r = 00, t) = GM,

G(r,t = 0) = GM,

0< t < 00,

0< t < 00,

r > R,

(80)

(81)

where r = R at the precipitatejmatrix interface, Gr is the concentration in the matrix at the precipitatejmatrix interface and GM is the bulk composition of the alloy. It is necessary to satisfy the independent flux balance:

dR dGI (Gp-Gr)-=D- , dt dr r=R

(82)

where D is the diffusion coefficient and Gp is the composition of the precipitate. For a sphere, (80) takes the form:

8G =D [~+~8G] 8t 8r2 r 8r .

(83)

This equation may be transformed by making the change of variable

(84)

to yield the ordinary differential equation.

1] dG D d ( 2 dG) ---=-- 1]-2 d1] 1]2 d1] d1]'

(85)

The solution of (85) has the form:

100 exp( _1]'2 j4D)d1]' G = k1 + k2 12

'T] 1]

1 [JDi ( _r2 ) 1 (r) 1 = k1 + k2 -r- exp 4Dt - "2V1ferfc 2(Dt)1/2 ' (86)

where k1 , k2 , k~ are constants and the error function complement erfc( x) is defined as


erfc(x) = Jn 100 exp(-e)d~ = Jn [1 00 exp(-e)d~ -lx exp(-e)d~] 2 r = 1 - .j1f Ja exp( -e)d~ = 1 - erf(x), (87)

where the error function, erf(x), is

2 r erf(x) = 1 - .j1f Ja exp( -e)d~. (88)

The exact solution to this equation, subject to conditions of (81) and (82) is given by [12, 13]

2>'(Gr - GM) G(r, t) = GM + exp( _>.2) _ >..j1ferfc(>.)

x [VDt exp (_~) - .j1f erfc (_r )] (89) r 4Dt 2 2VDt'

where R = 2>'V75t (90)

and >. is given by

(91)

where k = 2( GM - Gr) .

Gp -Gr (92)

Diffusional Growth in Multicomponent Systems. The simple treatment described above is not always applicable to the problem of diffusional growth of precipitates in practical alloys. Diffusion coefficients of several components have to be known and the interdependence of fluxes also has to be take into account. Kirkaldy and Young [14] proposed a mathematical method for describing diffusion in multicomponent systems. Kirkaldy applied the theory of linear irreversible thermodynamics [15-17] based on a pioneering paper by Onsager [18] on the Onsager reciprocity theorem, to the multicomponent diffusion problem.

It is reasonable to assume a linear homogenous relation between the thermodynamic flux J and the thermodynamic force X for conditions near equilibrium

n

Ji=~LijXj (i=I,2, ... ,n). j=l

(93)

The coefficients Lij are called phenomenological coefficients. The Onsager recipro city theorem expresses that if the flux J i , corresponding to the irreversible process i, is influenced by the force X j of the irreversible process j, then the flux

Yoshiyuki Saito 211

J j is also influenced by the force Xi through the same interference coefficient L ij

Lij=Lji (i,j=I,2 ... ,n). (94)

Onsager recognized that the symmetry law for the irreversible processes, such as flow of mass, heat and electricity, derives from the time reversal invariance of the mechanical law of particle motion.

Let us consider isothermal diffusion in n-components. The thermodynamic forces of the system are the chemical potential gradients

Xi = -grad!-Li' (95)

The thermodynamic forces are related by the Gibbs-Duhem relation.

n

LNiXi =0. (96) i=1

The component n (conventionally n is chosen as the solvent) can be eliminated by using the Gibbs-Duhem relation,

n-1

X __ '"' NiXi n- ~ N .

i=1 n

(97)

We define the chemical potential gradient of component i relative to that of component n

, Vi ( ) X i = Xi - ~Xn i = 1, .... ,n -1 , Vn

(98)

where Vk(k = 1, ... , n) is the partial molar volume of component k. Thus the flux equations yield

n

J i = LLijX~j (i = 1, ... ,n -1). j=1

(99)

To obtain Fick's type equations, we must transform the chemical potential gradients to the concentration gradients. The chemical potential is expressed as a function of molar concentration:

(100)

Hence - n-1

grad (!-Li - ~ !-Ln) = L gradej ,

Vn j=1

(101)

where

(102)


Substituting (101) into (98), (99) yields

where

n-1

J i = - L DijgradCj ,

i=1

n-1

D ij = - L LikJ-lkj.

k=1

Hence, we obtain the n-1 independent partial differential equations

When the diffusion constants Dij are constants, we obtain

(103)

(104)

(105)

(106)

If the boundary conditions of all components are symbolically the same, thcn we may construct the multicomponent soluti6ns as linear combinations of the solutions of the binary equations (.Ak yet to be defined)

of the form n-1

Cj = ajO + L ajkCk .

j=1

Substitution of (108) into (106) yields

or

n-1

.Akaik = L Dikajk

j=1

(107)

(108)

(109)

(110)

where the ak are column vectors of the coefficient matrix. It is clear that (110) is the characteristic equation of the matrix D ij and .Ai (i = 1, ... , n) are eigenvalues of the matrix D ij .

The solutions of the binary equation have the form

(111)

where K 1 and K 2 are constants. Coefficients aik are determined by boundary conditions.

Yoshiyuki Saito 213

7.3.2 Decomposition of Alloys

The dynamical behavior of an unstable alloy is described by spinodal theories. The spinodal theories are based on the ass um pt ion that the free energy of a non-equilibrium solid solution is defined in terms of compositional fluctuations in the early stages of a decomposition reaction.

The Cahn-Hillard Equation. Let us consider a binary alloy consisting of A and B atoms quenched from a single-phase into an unstable two-phase region. The kinetics of decomposition of the supersturated single-alloy into the twophase state is described by the Cahn-Hilliard equation.

In a homogenous alloy the difference in the chemical composition is proportional to the quantity ö f / öc. In the presence of composition fluctuations the quantity which is proportional to the chemical potential difference is given by the variation derivative of the free energy with respect to composition.

(112)

where c is the composition of the alloy: c = CB.

Cahn and Hilliard [19] showed that a non-uniform environment of an atom may be accounted for by adding a single gradient energy term (V'C)2 to the local free energy fo (C). The free energy of an inhomogenous system is given as

F = i[1o(C) + K(V'C)2 + .. ·]dV. (113)

Substituting (113) into (112) we obtain:

öf 2 J-L = öC - 2KV' c. (114)

According to the theory of linear irreversible thermodynamics the net flux of B atoms is proportional to the thermodynamic force given by the chemical potential gradient.

J= -MV'J-L, (115)

where M is a mobility. The time-dependent concentration field C(x, t) satisfies a continuity equa

tion: öC 8t + V'. J = O. (116)

Substitution of (114) and (115) into (116) yields

öC = MV'2 (ö fo _ 2KV'2C) . öt öt

(117)

Here we obtained the Cahn-Hilliard non-linear diffusion equation [20].


0.9- --- \-0 0.8 0.25 h

0.7 0.5 h

06 "1 -·-- 1h ~ . • - 2h ..... .; " i'\ "

:; 0.5 " 5 h eil ~~' ~ • ~ ~ IV' . '" I 0" 0.4 :. i I ~ ~ ~ ~v~ ~ ~\ ~ tl tw; \ I - 1 0 h

0.3 - 20h

0.2 V v v V \! V ...... 50 h v -·_·· 100 h

0.1 - 200h O L-----~------~----~------~----~

o 5 10 15 20 25 Diaplacement , nm

Fig. 5. Temporal evolution of Cr-concentration in Fe-40Cr alloy at 723 K.

The atomic scale microstructure resulting from heat treatment of a Fe-Cr binary alloy was investigated by a numerical solution of the Cahn - Hilliard equation (Fig. 5). The Cr composition profile at 10000 MCS shows a modulated structure with the wavelength of 1-3 nm. These result are consistent with those of the Mössbauer measurement and atom-probe FIM analyses.

7.4 Mesoscopic Modeling of Microstructure

Computer simulation models based on classical nucleation and growth theories have been successfully applied to microstructural control and design of materials. However, advances in microscopy and microanalysis have been extended the study of microstructure to the mesoscopic or microscopic region where the properties of materials depend on variations in chemical composition andjor microstructure at the nanoscale level.

7.4.1 Simulation of Interface Motion

As shown in previous sections, microstructures simulated by chemical thermodynamics and phase transformation theory are in good agreement with those observed. However, the inhomogeneity of microstructure of TMCP steels affects the mechanical properties of structural materials. For this reason, the incorporation of factors which characterize the spatial and temporal distributions of metallurgical factors is particularly important.

The Monte Carlo computer simulation technique proposed by the Exxon group [21-23] is one of the most promising methods to obtain detailed information on the topology and kinetics of microstructure evolution. Morphology of grains and kinetics of grain growth and recrystallization are reproduced very well

Yoshiyuki Saito 215

by these models. Further, the effect of second phase particles on the kinetics of grain growth was investigated by the Potts model. However, most of these simulations were performed at temperatures near 0 K. The effect of the anisotropy of grain boundary energy, the temporal and spatial distribution of precipitates on grain growth at higher temperatures must be considered in the simulation of microstructural evolution of steel during TMCP [24].

In this section, temporal evolution of grain structure will be described by a model which incorporates the effects of anisotropy of grain boundary energy and temporal and spatial distribution of precipitates.

Method of Simulation. The Monte Carlo simulation of interface motion can be performed by the Metropolis algorithm [25]. The procedure of the simulation is as follows:

1. The crystallographic orientation of the constituent grain as expressed by a spin variable, from 1 to Q, is assigned to each lattice point representing a small volume of the system. As an initial microstructure, an orientation between 1 to Q was assigned to each lattice site at random.

2. The evolution of microstructure is tracked by the change of spins on each lattice (spin flip). (a) One lattice site is selected at random. (b) A new orientation of the lattice is generated. ( c) The change in energy, iJ.E, associated with the change of spin variables

is calculated. (d) The re-orientation trial is accepted if iJ.E is less than or equal to zero.

If the value iJ.E is greater than zero, the re-orientation is accepted with prob ability, w.

(118)

where kB is the Bolzmann constant and T is the temperature. If the system size is N, N re-orientation attempts are referred to 1 Monte Carlo step (MCS).

A segment of boundary, therefore, moves with velocity, Vi, related to the local free energy, iJ.G, or chemical potential difference, iJ.p" that drives the atoms across the boundary:

(119)

The prefactor in (119) constitutes a boundary mobility, and reflects the symmetry of the mapped lattice. In a pure met al iJ.G and iJ.p, are identical and given by

(120)

where "f is the grain boundary energy, Vm is the molar volume of a material and R is the radius of a curved boundary. Equation (119) is equivalent to the boundary velo city derived from classical rate theory.


The change of interfacial energy accompanying re-orientation is a driving force of interface migration. The interfacial energy is related to the interaction energy between nearest neighbor sites. The interfacial energy is a function of the grain misorientation:

Eo = - L Msis j ,

<ij>

(121)

where Si is a spin variable which takes a value from 1 to Q. The sum is taken over nearest neighbor sites. The matrix M ij is given by [21]

(122)

where J is a positive constant which sets the scale of the interfacial energy and 6ij is the Kronecker delta function.

The anisotropy of the interfacial energy is incorporated in the following manner: The energies of grain boundaries are largely divided into two groups; the higher energy group may correspond to general high angle boundaries and the lower group, twin or high coincidence boundaries. The parameter representing anisotropy of interfacial energy, r, is defined as the ratio of the energies of the two boundary groups (0 < r < 1 ). In the case of isotropie interfacial energy (r = 1). Then the matrix Mij is given by [24]

(123)

It is assumed that the interface of relative misorientation k is the lower energy interface.

Condition of the Simulation. Simulations were performed on a 2-dimensional tri angular lattice of size N = 10002 . N re-orient at ion attempts are referred to 1 Monte Carlo step (MCS). All the simulations were performed on the lattice systems with periodic boundary condition.

In order to prevent the impingement of grains like orientation too frequently, a large value of Q is chosen, typically Q = 32 or 64. The simulation results do not depend significantly on the value of Q, when Q is larger than 32. The interfacial energy, ,,(, is related to the value Jas:

"( = ZN' J (124)

where Z is the nearest neighbor co ordination number of site, N' is the number of atoms per unit areaofinterface. In austenitic steel(fcc), N' is about 1 x 1015cm- 2 ,

Z' for 2-dimensional triangular lattice is 2, "( for a general grain boundary is about 800 erg cm-2 and the Bolzmann constant, kB , is 1.381 x 1O- 16 ergcm-2 .

Consequently, the value J/kBT in the temperature range from 1073 to 1473 K is about 2.0 to 2.7. In the following simulation, the value J / kBT is set to 2.25 corresponds to T = 1273 K.

Yoshiyuki Saito 217

Data Analysis of the Monte Carlo Simulation Result. Temporal evolutions of the configuration of spin variables are stored on computer disks as an ASCII type data file. With use of the cluster analysis method, variations of parameters deseribing mierostructural evolutions of the system, such as the exponent of growth law, the size and the number of edges distributions and the distribution of orientation are obtained. The effieient cluster analysis algorithm was proposed by Hoshen and Kopelman [26] and modified by Sakamoto and Yonezawa [27].

Grain Growth of Pure Metal. As an initial mierostructure, an orientation between 1 and Q was assigned to eaeh lattiee site at random. Figure 6a shows an example of the temporal evolution of mierostrueture of a material with isotropie grain boundary energy. The formation of grain strueture is deteeted in the early stage of the simulation. The eoarsening of large grains by absorbing small grains is observed. The uniform and isotropie grain structure is obtained. Figure 6b shows the temporal evolutions of mierostrueture of materials with anisotropie grain boundary energy. Mixed grain strueture is obtained in the ease of the anisotropie grain boundary energy.

I • 1000 MCS I -2000 l1CS t • 4000 I1CS JII'!1 J . ,

f. -

.-.\J . . ;f"~'" r"I

11'«· )-

Ä ~ ~ •

,-(5 . ) A ~

Fig. 6. Example of temporal evolution of boundary for Q = 32 on a tri angular lattiee. (a) material with isotropie boundary energy and (b) material with anisotropie boundary energy.

The average area A, against time t, for simulations by the Potts type model on a 2-dimensional triangular lattiee of size of 10002 is shown in Fig.7. The average area is found to be proportional to time. In the ease of an anisotropie boundary, growth is suppressed. The average grain size, R, is deseribed as a power-law kineties:

R = Bto.5 , (125)


where B is a eonstant whieh is temperature dependent. The exponent is the same as that from the analytieal model by Hillert [28].

Figure 8 shows the variation of the sealed grain size distribution for the simulated mierostruetures. The grain size distribution becomes broad in the ease of anisotropie grain boundary energy. Saito and Enomoto [24] showed that the oeeurrenee of the wetting phenomena is responsible for the broadness of the grain size distribution in the structure with anisotropie grain boundary energy .

. .... ................. ,

70 0 Isotropie /~ 60 • Anisotropie ....

/ ,,-(tI 50 . /

" Q) y ~ 40 /--"

30 ." JI" /

/

20 , /

10 .-0

0 2000 4000 6000 8000 10000 Time, MCS

Fig.7. The average area A, against time t for simulations on a 2-dimensional triangular lattice of size of 10002 .

C co .... " O . C ::I ... c co o. .... ::I ... 'C O. .... ., :;;

'" O.

N

in 0

0 1

RIR

Isotropie

Ani sotropi c

2 3

Fig. 8. Variation of the scale grain size distribution for simulated microstructure.

To evaluate the profile of the grain size distribution, a parameter S, is defined as:

(126)

where gi is the value of the normalized grain size, 9 = R/ R, in the i-th size group of R/ R. From the analogy to the entropy of an ideal mix, S, is ealled mierostruetural entropy [29]. When the size distribution has a narrow peak, the value of S is expeeted to be small. On the other hand, S passes through an extremum when the distribution beeomes flat or widely seattered. The variation of S with time is shown in Fig. 9. The value of S is larger in the grain strueture with anisotropie grain boundary energy. This indieates that the eharaeteristies of the size distribution profile may be represented by S .

The distributions of the number of edges Ne for individual grains of the simulated microstructures are shown in Fig. 10. The distribution beeomes timeinvariant at longer times. The frequeney inereases rapidly for a small number of edges, peaks at a value of five or six and then deeays quiekly. The distribution of Ne for grain strueture with anisotropie grain boundary energy is broader than that for the isotropie ease.

Yoshiyuki Saito

30r-------------------------,

2

19OIrOpic

--<:I- 4000 l1es ~ 6000l1es .... <>... 0000 Mes

-- 10000l1es AniscIToplc(f>-O.33~

...• .. 4000 MCS --+- 6000l1eS ... -.... 8000 MCS -- ... -- 10000 Mes

OL-L-L-L-L-L-L-~~~~~ 3 4 S 6 7 0 9 10 11 12 13 14 IS

HIInber ot slclel

Fig. 9. Distributions of the number of edges Ne for individual grains of simulated mierostrueture.

219

08

~ -<>- ISOlroplc e

06 • AnIsOIropie 1i i! a ..,

04 2 8 b :i 0.2

0.0 0 2000 4000 6000 8000 10000

nne,MCS

Fig. 10. Variation of mierostructural entropy of grain strueture with anisotropie grain boundary energy with that of isotropie grain boundary.

7.4.2 Simulation of Atomic Arrangement

Alloying elements have an important role in controlling the high temperature performance of Ni-base superalloys. Hence, understanding the alloying effects upon microstructure is vitally important for the development of Ni-base superalloys.

Many experimental and theoretical studies have been made to investigate "(' /"( equilibrium in Ni-base alloys. One of the most appropriate methods may be the cluster variation method with Lennard-Jones pair potential, which was first used for phase analysis in the Ni-Al binary alloys by Sanchez et al. [30] and applied to ternary and multi-component Ni-base alloys by Enomoto et al. [31,32]. It has been demonstrated that the predicted "(' /"( phase equilibrium and the site occupancies of alloying elements in sublattices of "(' phase are in good agreement with those observed.

In recent years, the observation of the atomic arrangement of Ni-base alloys has become possible by atom probe FIM analysis. If this method can be combined with theoretical studies of atomic arrangement, we will be able to obtain a very powerful tool of materials design.

Method of Simulation. The initial structure can be generated by assigning numbers which are distinct by atom species on lattice sites at random according to the composition of the met al. The procedure for the simulation of ordering kinetics is similar to that for the simulation of grain growth except that the elementary process is controlled by direct exchange of a randomly selected single atom with one of its neighboring atoms (Kawasaki dynamics [33]) and that the


transition probability is given by the symmetrieal solution:

W(s s') _ exp(-iJ.HjkBT) , - 1 +exp(-iJ.HjkBT)

(127)

The study of ordering in binary and multi-eomponent alloys by the Monte Carlo method requires accurate modeling of the atomie interaction. At present, the Monte Carlo simulation of ordering in multi-eomponent alloys must rely on sem iempirieal models, although the interactions of atoms are eomplex. In order to simplify the numerieal ealculation the phenomenologieal pair potential was utilized. The pair interaction are expressed by the Lennard-Jones potentials of the form,

_ 0 [(rij)miJ mij (rij)miJ] edr) - e· - - - -J tJ r nij r '

(128)

where i and j stand for the different atomie species. It was shown by Sanehez et al. [30] that the best values for the exponents

nij, mij of the attractive and repulsive potential of metals were 4 and 8, respeetively, rat her than the original values of Lennard-Jones 6 and 12. The values of the Lennard-Jones potential parameters for fee met als are determined by the knowledge of experimental eohesive energies EM and lattiee parameters o,i of pure elements and heat of formation Ec and lattiee parameter o,c of a stoiehiometrie ordered eompound C with Lb strueture. In the systems, in whieh no Lb has been reported to exist, parameters are determined by fitting the enthalpy and lattiee parameter for the fee solid solutions [31,32].

The simulation was made to investigate the temporal evolution of atomie arrangement in a Ni-Al binary alloy system, three kinds ofternary Ni-AI-X (X=Co, Ti and Cr) systems and a multi-eomponent eommereial Ni-base superalloy. The simulations were performed on the fee lattiee of size of 323 (131072 atoms). Periodie boundary eonditions were used throughout. The temperature of the system was set to 1273 K.

With use of the cluster analysis method, the volume fraction and size distribution of ,,/ phase and eomposition of "'( and "'(' phases are determined.

Simulation of Atomic Arrangement in Ni-base Superalloys [34]. Figure 11 shows the temporal evolution of atomie arrangement in a eommereial CMSX-4 superalloy, the chemie al eomposition whieh is shown in Table 1. The ealculated phase eomposition of "'(' and "'( phases of the CMSX-4 superalloy by the Monte Carlo simulation is shown in Fig. 12. The enriehment of Al, Ti and Ta in the "'(' phase and Co in the "'( phase is observed. Cr is partitioning into both "'(' and "'( phases. Figure 13 also shows a eomparison of the Monte Carlo simulation results with those observed by atom probe FIM analysis and that ealculatecl by cluster variation method. The present Monte Carlo simulation result is in good agreement with those from atom probe FIM analysis and the cluster variation method [32].

Yoshiyuki Saito

.. . ' .. . , i.i" ..

1000 MCS l0000MCS

221

AI

• Ti

• Cr

• Co

Mo

Re

Fig. 11. Temporal evolution of atomic arrangement in superalloy CMSX-4.

100 100

7 ' 60 7

60 M • MC M MC r:.' • Exp. r:.' co • CVM co - 60 - 60 u u CI t t E 40 E 40 CI CI .- :( '"

20 20

0 0 NI AI TI Cr Co Mo Ta W Re NI AI TI Cr Co Mo Ta W Re

Fig. 12. Comparison of Monte Carlo simulation results of phase composition of " and , phases in CMSX-4 with those observed by atom probe FIM analysis and those calculated by the cluster variation method.

100 '00

Ni site "Isie 111

~ c' 60

oB ~

4D ~

• 111 MC • MC • Exp. ~ • Exp. • CVM ",' • CVM

oB 60

~ ~ 40 -<

-<

20

NI AI TI C r Co Mo Ta W Re NI Al TI C r Co Mo Ta W Re

Fig. 13. Comparison of Monte Carlo simulation results of alloying element in Ni and Al sublattices " phase in CMSX-4 with those observed by atom probe FIM analysis and those calculated by the cluster variation method.


Table 1. Chemical composition of CMSX-4 superalloy.

Ni Al Ti Cr Co Mo Hf Ta W Re C

63.0 12.6 1.3 7.6 9.8 0.38 0.03 2.2 2.1 0.980.01

7.5 Industrial Applications

Now we consider an industrial application of the models. Let us consider materials design of structural steels as an example. These steels have been produced by a thermomechanical control process (TMCP), which is an essential manufacturing process for producing high strength low alloy (HSLA) steels with good low temperature toughness and low weId cracking susceptibility [35]. Mechanical properties of steel plates produced by TMCP are highly influenced by manufacturing conditions. Consequently, control of microstructure through an optimization of chemistry and manufacturing process is important for the best use of TMCP and the improvement of mechanical properties of HSLA steels. For the adoption of the computer simulation model to the design of chemistry and manufacturing process, it is necessary to describe evolutions of these metallurgical phenomena in terms of alloying elements and processing variables, such as reheating, rolling and cooling conditions. The synergetic effects of the two, or three, phenomena must be taken into consideration in the modeling.

7.5.1 Modeling of Metallurgical Phenomena [36]

As we have seen in Sect. 7.1, rigorous treatment of nucleation on the basis of statistical mechanics of a first order phase transformation is very complicated. From the industrial point of view, the most prominent approach for the calculation of the nucleation rate, J(t), may be to use the classical nucleation theory [37]. From the discussion in Sect.l, the nucleation rate, J(t), is given by

(129)

where N (t) is number of nucleation sites per unit volume, ß* is the rate at which single atoms join the critical nucleus, Z is the Zeldvitch non-equilibrium factor, LlG* is the free energy of activation for formation of the critical nucleus, T is the so-called incubation time, T is the absolute temperature and ks is the Bolzmann factor. The above parameters depend on the morphology of the nucleus, grain structure and chemistry of a steel. Unknown parameters in (129) are number of nucleation sites, N(t), interfacial energy, !Jaß' and the shape factors of nucleus, K and L. These parameters are determined so that good agreement between the computed and observed results are obtained.

Growth of a stable nucleus is predicted by the diffusion al equation with boundary conditons which describe the phase equilibrium of a newly created precipitate and the matrix (see Sect. 7.4). The overall reaction is given by the

Yoshiyuki Saito 223

generalized Kolmogorov-Johnson and Mehl-Avrami (KJMA) equation as we have seen in Sect 7.2.3.

The kinetics of phase transformation is predicted with use of the above models in the following manner: The init ial condition of austenite is predicted by the models of austenite recrystallization and grain growth, the accumulated strain and carbonitride precipitation. The equilibrium temperature, Teq , is calculated from steel chemistry. The cooling process is divided into infinitesimal time steps. The temperature of the steel at each step is calculated by the numerical solution of the Fourier equation of he at conduction. If the temperature of steel is lower than Teq , the following calculation of ferrite transformation is executed at each time step. The changes in free energy associated with the nucleation and the eutectiod carbon concentration are calculated. The nucleation rate at the time step is predicted by (129). The growth of the nucleus is estimated by the diffusional equations. The fraction transformed is calculated by the KJMA equation. The above calculation is repeated until the carbon concentration in austenite reaches the eutectoid carbon concentration. Pearlite is formed after the carbon concentration in austenite reaches the eutectoid carbon concentration. If the value of the free energy associated with nucleation of the phase with the same composition as austenite exceeds 400 J mol-I, bainite is formed [38J .

The computed continuous-transformation-time (CCT) diagram of 0.08C-1.50Mn- 0.035 Nb steel is shown in Fig.14. The computed CCT diagram is in good agreement with that observed.

800

700

P 1])-

600 L :>

500 ;0 '-I]) a. 400 E I])

>- 300

200

100 1 10

CalCI.Ilated

100

Time 5

übserved

~i ,

1000 10000

Fig. 14. Computed continous-transformation-time diagram of O.08C- 1.5Mn-O.035 Nb steel compared with those observed.


7.5.2 Effect of Processing Variables on Microstructure of Steel Produced by Thermomechanical Control Process

Accelerated cooling after controlled rolling is a very useful technology for producing high strength steel plates. The most important point in accelerated cooling is to control the transformed microstructure.

~ 25

~ 10

0) E

'" 20 :i.

<J s::; 2nd phBse 0) Cl. N

"0 15 "00 c N Ferrite grain size 5 c .... ";;;

10 ::: ............ <> ......... '-0 """ .................... <~

Cl

ci ~ '" '- 5 "C .... '-

'0 Q)

LL

> °6 160 B 10 12 14

Cooling rate, Cis

Fig. 15. Influence of cooling rate on the volume fraction of second phase and the ferrite grain size of O.lC-1.5MnO.035Nb steel.

~

0) ifj

" s::; Cl.

"0 c N .... 0

~

U

" .:: '0 >

20 ".,Fn~"i"~~"~r"~"i"~"~i"~"7:,

15

E 15 B B B B :i.

~ 10~-2nd phase ifj

C 10 ";;; L

5 Cl

5 2 C L Q) LL

°6 B 10 12 14 16 ° Cooling rate, Cis

Fig. 16. Influence of cooling rate on the volume fraction of second phase and the ferrite grain size of O.06C-1.25MnO.035Nb steel.

Influences of the cooling rate and cooling start temperature on the transformation microstructures of two Nb steels: O.IC-1.5Mn-O.035Nb and O.06C-1.25Mn-O.035Nb steels (called high C-high Mn and low C-low Mn steels, respectively) can be predicted by the model. Figure 16 shows the influence of the cooling rate after rolling on the volume fraction of second phase and the ferrite grain size of high C-high Mn steel. At lower cooling rates, the transformed microstructure consists of ferrite and pearlite phases and the volume fraction of the second phase is small. The retained C concentration in austenite is higher in high C-high Mn steel than that of low C-low Mn steel at the same fraction transformed. The formation of pearlite is easier in high C-high Mn steel. With the increase of the cooling rate, the volume fraction increases. The transformed microstructure at higher cooling rates is changed to ferrite-bainite structure. The influences of the cooling rate on the volume fraction of second phase and the ferrite grain of low C-low Mn steel is shown in Fig.16. The transformed microstructure consists of ferrite and bainite phases even at lower cooling rates. The effect of the cooling rate on the volume fraction of second phase is smaller than that of high C-high Mn steel. The overall transformation kinetics is controlled by the transformation behavior prior to accelerated cooling. Hence, the influence of the accelerated cooling rate on the transformed microstructure is very small.

The computer simulation model can be applied to the design of chemistry and the manufacturing condition of steel plates according to the demands. For exam-

Yoshiyuki Saito 225

pIe, the influence of the cooling rate on the transformed microstructure in low C-Iow Mn steel are sm aller than those in high C-high Mn steel so that one can easily find the manufacturing condition for producing steel plates with uniform microstructure and mechanical properties. On the other hand, the transformed microstructure of high C-high Mn steel is apparently influenced by processing variables in the thermomechanical treatment. The manufacturing condition for the best use of the thermomechanical control process can be determined with use of high C-high Mn steel.

References

1. R. Becker, W. Döring: Ann. Phys. 24, 719 (1935). 2. J. Feder, K.C. Russel, J. Lothe, G.M. Pound: Adv. Phys. 15, 117 (1966). 3. K. Binder, D. Stauffer: Adv. Phys. 25, 343 (1976). 4. I. Kanne-Dannetschenk, D. Stauffer: J. Aerosol. Sei. 12, 105 (1981). 5. R. Courant, D. Hilbert: "Methoden der Mathematischen Physik", Band I

(Springer, Berlin, 1931). 6. S.M. Allen, J.W. Cahn: Acta Metall. 27, 1085 (1979). 7. A.N. Kolmogorov: Izv. Akad. Nauk SSSR, Sero Matern. 3, 355 (1937). 8. W.A. Johnson, R.F. Mehl: Trans. AlME 135, 416 (1939). 9. M. Avrami: J. Chem. Phys. 7, 1103 (1939).

10. A.A. Chernov: "Modern Crystallography III, Crystal Growth" (Springer Verlag, Berlin, Heidelberg, 1984).

11. Y. Saito: Thesis, Kyoto University, Kyoto, Japan (1986). 12. H.S. Carlslaw, J.C. Jaeger: "Conduction of Heat in Solids", 2nd. ed. (Oxford Uni

versity Press, Oxford, 1959). 13. H.B. Aaron, D. Fainstein, G.R. Kotler: J. Appl. Phys. 41, 4404 (1970). 14. J.S. Kirkaldy, D. Young: "Diffusion in the Condensed State" (The Institute of

Metals, London, 1985). 15. L.D. Landau, Lifshitz: "Statistical Physics", 2nd. ed. (Pergamon Press, Oxford,

1969). 16. P. Glansdorf, I. Prigogine: "Thermodynamic Theory of Structure, Stability and

Fluctuations" (John Wiley and Sons, New York, 1974). 17. H. Callen: "Thermodynamics and Introduction to Thermostatistics", 2nd. ed.

(John Wiley and Sons, New Yok, 1985). 18. L. Onsager: Phys.Rev. 37, 405; 38, 2265 (1931). 19. J.W. Cahn, J.E. Hilliard: J. Chem. Phys. 28, 258 (1958). 20. J.W. Cahn: Acta Metall. 9, 795 (1961). 21. M.P. Anderson, D.J. Srolovitz, G.S. Grest, P.S. Sahni: Acta Metall. 32, 783 (1984). 22. D.J. Srolovitz: (ed.) "Computer Simulation of Microstructural Evolution" (TMS-

AlME, Warrendale, PA, USA, 1985). 23. D.J. Srolovitz, G.S. Grest, M.P. Anderson: Acta Metall. 34, 783 (1986). 24. Y. Saito, M. Enomoto: ISIJ International 32, 809 (1992). 25. N. Metropolis, A.W. Rosenbluth, N.N. Rosenbluth, A.H. Teller, E. Teller: J. Chem.

Phys. 21, 1098 (1953). 26. J. Hoshen, R. Kopelman: Phys. Rev. B 14, 3438 (1976). 27. S. Sakamoto, F.Yonezawa: Kotai Butsuri (Solid State Physics) 24, 219 (1989). 28. M. Hillert: Acta Metall. 13, 227 (1965).

226 Modeling of Mierostruetural Evolution in Alloys

29. H.K.D.H. Bhadeshia, A.A.B. Sugden: "Reeent Trends in Welding Seienee and Teeh-nology" Eds. S.A.David and J.M.Vitek, (ASM, Ohio, 1989), p.745.

30. J.M. Sanchez, J.R. Barefoot, R.N. Jarret, J.K. Tien: Aeta Metall. 33, 219 (1984). 31. M. Enomoto, H. Harada: Metall. Trans. 20 A, 649 (1989). 32. M. Enomoto, H. Harada, M. Yamazaki: CALPHAD 15, 143 (1991). 33. K. Kawasaki: Phys. Rev. 145, 224 (1966). 34. Y. Saito, H. Harada: Mater. Sei. and Eng. A 223, 1 (1997). 35. I. Tamura, H. Sekine, T. Tanaka, C. Ouehi: "Thermomechanieal Proeessing of High

Strength Low Alloy Steel" (Butterworth, London, 1988). 36. Y. Saito: Mater. Sei. and Eng. A223, 134 (1997). 37. H.1. Aaronson, J.K. Lee, "Lecture on the Theory of Phase Transformation", Ed.

H.1. Aaronson, (Ameriean Institute of Mining, Metallurgy, and Petroleum Engineering, New York, 1975), p.83.

38. H.K.D.H. Bhadeshia: Progress Mater. Sei. 29, 321 (1985).

Finite Element Analysis of the Deformation in Materials Containing Voids

H. Shiraishi

Computational Materials Science Division, National Research Institute for Metals 1-2-1 Sengen, Tsukuba-shi, Ibaraki 305, Japan.

Abstract. The stress-strain relationship of material which contains voids in its matrix is analyzed by the large-deformation two-dimensional finite-element method. Continuum mechanics, Mises yield criterion, J2 flow theory, the n-th power work hardening law, and cyclic boundary condition are assumed. It is possible to apply this concept to evaluation of grain boundary bubble embrittlement.

8.1 Introduction

Engineering alloys usually contain defects such as voids and bubbles. For exampIe, many bubbles can exist in the interface between the base metal and the weId metal. In powder metallurgy, the sintered material is slightly less dense than the theoretical one because of the presence of pores. In ductile material the fracture surface consists of the so-called dirn pIe structure, which is considered to be produced by a mechanism in which the fracture of nonmetallic inclusions trigger the formation of dimpIes. Also, in nuclear reactor core materials irradiation by neutrons produce vacancies and helium in the matrix. The vacancies condense to voids in the matrix and the helium atoms diffuse to the grain boundary to precipitate there as bubbles. These voids and bubbles are experimentally well known to degrade the material strength severely, but not enough quantitative analysis of the effect has been done.

Finite element analysis goes back to the beginning of the twentieth century and was established as a powerful method of structural design in the 1950s. In pace with the progress of computer hardware, it is widely used today, for example in the field of airplane and automobile manufacture. This method can be useful in materials science. Strength and fracture are the most interesting problems for the scientist and engineer. These problems have been attacked from two different viewpoints. One is continuum mechanics theory and the other is crystalline dislocation theory. Recently, a more direct method, that is molecular dynamics has been applied in this field. In real materials, the strength and fracture are controlled by such crystalline defects as grain boundaries, strengthening se co nd particles, voids, bubbles, etc. It is considered that the sc ale of continuum theory is too large to treat these defects and conversely the atomic scale is too small. The recent remarkable increase in computer speed makes it possible to increase the degrees of freedom of systems in the finite element method. The present

Springer Series in Materials Science Velume 34, Ed. by T. Saite © Springer-Verlag Berlin Heidelberg 1999

228 Finite Element Analysis of the Deformation in Materials Containing Voids

paper tries to apply the finite element method based on a continuum body to analyze the stress-strain relationship of materials containing voids.

In Sect.8.2, the outline of the formulation of the finite element method is given. This program was produced by the author for the analysis of large plastic deformation. The finite element method is a popular form of analysis for various problems, and many instruction books are now published [1].

In Sect. 8.3 the void lattice is defined and the results of a parameter survey such as the unit cell size of the void lattice, the aspect ratio of the void lattice, the size of voids, and work hardenability are reported. In reality, the three-dimensional void distribution is approximated by the distribution of holes in a plate and the plane stress condition is assumed. Section 8.4 shows the result of the preliminary application of the void lattice to the grain boundary bubble problem. This is a very rough approximation. The difference of crystalline orientation between neighboring grains and the existence of the tri pIe point of the grain boundary are not taken into account. The role of the grain boundary is only to produce a special bubble arrangement. The internal pressure of the bubbles is neglected. The effect is considered to be not so large for sub micron bubbles. Section 8.5 compares the calculated results with the experimental evidence. Here, the calculation is for the two-dimensional problem and the correspondence is very qualitative in nature. The last section states the conclusion and the direction of future development.

8.2 Formulation of the Large Plastic Deformation Finite Element Method

In the finite element method, the whole body is subdivided into many sm all parts in which the analysis of a process, in this chapter plastic deformation, is easily undertaken. The equation of analysis is derived for each element and finally all the equations are assembled. The overall system of equations is then solved to reveal the behavior of the whole body.

Figure 1 shows a plate of unit thickness which has a circular hole in the center. This plate is discretized by radial lines such as AB and circular lines such as FHK. The principle of discretization is as follows. This mesh consists of two kinds. One is a coarse mesh which is expressed by bold solid lines and the other is a fine mesh which is depicted by fine dotted lines. The length AB is divided in such a way that

AB: AF: AG: AX = 1 : (1/2)1 : (1/2)2 : (1/2)nrc. In circular division, the angle of 90° is divided equally by 2ntc . This is the

principle of formation of the coarse mesh. The method of subdivision into the fine mesh is slightly different. One coarse element GFHJG is divided equally by nr j in the radial direction and equally by 2ntf in circular direction. Here, nrc, ntc, nr j and ntj are all integers and the whole mesh division is defined by these four parameters. In the finite element method, the computational time is roughly proportional to DOF2.5 , where DOF means degrees of freedom of the system, i.e. twice the number of no des for the two-dimensional case. So, the

H. Shiraishi 229

m

,~ L~ ____ ~ __ .-__ .--,D __ -r __ -r ________ ~(

Fig. 1. Illustration of mesh formation.

most efficient method of mesh construction must be determined, which means forming the finest mesh in the area in which the highest localization of strain is expected. Each reet angular element is divided into fOUf triangular elements, as illustrated in the upper part of Fig. l.

As a starting point, consider the equilibrium condition between the extern al forces Fij , Fjk and Fki which are operative on the cross-section ij, j k and ki of the tri angular element ijk and the internal stress state (o-x, o-y,TXY )' as illustrated in Fig. 2. For example, the x and y components of the force Fij are related to the stress by the following equations:

Fijx = (o-xl/x + Txyl/y)Sij (1)

(2)

where I/x, I/y are the direction eosines of the normal to the cross-section ij and Sij is the cross-sectional area of the side ij. The other components of Fjk and Fki are expressed in the same way.

In the next calculational step, the apices i, j and k of the triangular move by infinitesimal quantities (dUi,dvi), (duj,dvj) and (dUk,dvk)' The incremental form of (1) is as folIows.

dFijx = (I/xdo-x + l/ydTxy)Sij + (o-xdl/x + Txydl/y)Sij + (o-xl/x + Txyl/y)dSij (3)

The other five components are expressed in a similar form.


k (xI+du., y.+ dv.) FJ,,+dF Jh

L f, .. +df u ,

F ... +dF I .. next position J

J F'j,+dP ,J.

F iJl +df!1I

(x,+dUt, y,+dv,)

k (x •. y,)

present position

i (x" y,)

Fig. 2. Illustration of the shift of the triangular element and the change of the nodal forces between two consecutive calculational steps.

Hereafter, to have a compact expression, we use a matrix style. In this style, the six increments of force which operate on three sides of the tri angular element ijk and the six increments of displacement defined at the three apices of this triangular are expressed concisely as follows:

dFijx

dFijy

dFjkx

dFjky

dFkix

dFkiy

dFi~x + dFi~x + dFi~x dFi~y + dFi~y + dFi~y dFAx + dFAx + dF}kx

dFAy + dF}ky + dF}ky

dFfix + dFfix + dF}ky

dFfiy + dFfiy + dF~iY

(4)

H. Shiraishi

dUi

dVi

dUj

dVj

dUk

dVk

231

(5)

where the first, second and third terms in (4) correspond to each term of the right side of (3), respectively.

Next, the increments dax , day , dTxy , dvx , dvy , dSij , dSjk , and dSki must be derived as functions of the six infinitesimal increments of displacement. For example, consider the increments of the stress components. It is assumed that the increments of the displacement vector du within the tri angular element ijk is given by the linear combination of coordinates x and Y

dUi

dVi

dUj

dVj

dUk

dVk

where N is a 2 x 6 matrix which is called a displacement function.

(6)

The displacement function can be determined from the above assumptions and from the requirement that it must take the given values at each apex of the triangle ij k.

Ni = (ai+bix+Ciy)/2S, N j = (aj+bj x+cjY)/2S, Nk = (ak+bkx+Cky)/2S, (7)

where ai = XjYk - xkYj,bi = Yj - Yk,Ci = Xk - Xj

aj = XkXi - XiXko bj = Yk - Yi, Cj = Xi - Xk

ak = XiYj - XjXi, bk = Yi - Yj, Ck = Xj - Xi

1 x· y. 1 ' , S = - det 1 x· y. 2 J J

1 Xk Yk

where S is an area of the tri angular element ijk. The three components of the strain vector are defined by the following three

equations: OU

OX

e = [:: 1 ov (8)

oY 'Ixy OU OV -+-

oY OX


From (6) and (8), the increments of the strain vector can be expressed by the linear combination of the increments of the displacements

de = Bdue (9)

where B is 3 x 6 matrix and is expressed as follows using (5),(6),(7) and (8):

[b 0 b 0 bk 0 1 1 2 J

B = 28 0 Ci 0 Cj 0 Ck .

Ci bi Cj bj Ck bk

(10)

In elastic deformation, Hook's law holds and is written as follows for the plane stress condition:

where 1vO

D = C ~v2) vI 0

I-v 00--

2

(11)

From (9) and (11), the increments of the stress vector are given by the linear combination of the increments of the displacement vector

du = Dei Bdue . (12)

Now, the first term of (3) is written as follows: sdFe1 = tSKe1,eldue

K e1 ,el = B T Dei B ,

(13)

where Kel,el is a 6 x 6 matrix called the stiffness matrix and t is the thickness of the tri angular element ijk. B T is the transpose of matrix B.

In plastic deformation the matrix Dei must be replaced by the corresponding plastic matrix Dpl. Assuming the Mises yield criterion, J2 flow theory, and an n-th power work hardening law, this matrix is expressed for the plane stress condition as follows:

r E 8~ j (1-v 2 ) - 5 2 symmetry vE 8 1 8 2 E 8 2

(1-v 2 ) - -S- (1-v 2 ) - 5 ' _ 8 1 8 6 _ 8 2 8 6 _E _ _ 8~

8 8 2(1+v) 8

(14)

where

8 1 = (~) (O"~ + vO"~) , I-v

H. Shiraishi 233

86 = C ! LI) Txy ,

where Hf = da / dE'P is the work hardening coefficient, a, EP are the equivalent stress and the equivalent strain, respectively, a~, a~ are the incremental stresses.

The second and the third terms of (3) can be derived by a rat her lengthy algebra but here the procedure is omitted.

Next, the equivalent nodal forces must be determined and this is done so that half of the cross-sectional force dFij contributes to each of the nodal forces dFi and dFj . The situation is the same for the other two sectional forces dFjk

and dFki . So, the nodal forces at the three apices of the triangular element can be determined as follows:

dFix dFijx + dFikx

dFiy dFijy + dFiky

ndFe = dFjx 1 dFjkx + dFjix

(15) -dFjy 2 dFjky + dFjiy

dFkx dFkix + dFkjx

dFky dFkiy + dFkiy

Next, all the increments of nodal force at the apex i are summed and this sum is the linear combination of the increments of displacement at the apices of the triangles which belong to the apex i. Finally, all these equations form a large ensemble of the linear equations relating the increments of nodal force to the increments of displacement.

dFIx

dFIy

dF2x

dF= dF2y

(16)

dFnx

dFny

Ku K I2 K I3 Kl4 · . KI,n-1 KIn dUI

K 2I K 22 K 23 K 24 · . K 2,n-1 K 2n dVI

K 3I K 32 K 33 K 34 · . K 3,n-1 K 3n dU2

K 4I K 42 K 43 K 44 · . K 4 ,n-1 K 4n dV2

K 2n - I ,1 K 2n - I ,2 K 2n - I ,3 K 2n- I ,4 .. K 2n - I ,2n-1 K 2n - I ,2n dUn

K 2nI K 2n2 K 2n3 K 2n4 .. K 2n,2n-1 K 2n2n dVn

where n is the number of nodes.


I lfesh forllation I 1

I Material properties given I t

Boundary condition/lnitial displacement givenl I

Add infinitesimal I displacellent I Stiffness matrix calculation I

J

I Solution of line ar equations 1 Yes I r--1 Too large yield stress ? I

I No Correction Yes I

--{ Too large plastic st rain? 1 I No

1

I Renewal of yie Id point I I

No l ?I Unloading elements :1

I Yes

I Rest ore to elastic state 1 No I

: O. 8* (maxilLu conventional stress)attained ?I J Yes

I End

Fig. 3. Calculation procedure for the large deformation finite element method.

The number of equations and unknowns is the same and we can solve this ensemble, taking into account the boundary conditions. Figure 3 shows the procedures of the finite element method. To the y components of nodal points on boundary CD, the small constant increment oUy is added at each calculation step. The y components on the boundary AB must be zero because of symme-

H. Shiraishi 235

try and the x components on the boundary DE also must be zero for the same reason. For the boundary BC, also the x components are set to zero because of the assumed periodicity of the void lattice. For the components in which the displacements are fixed or given, the increments of the forces are unknown factors. For all the other components, the force components are zero and the increments of displacement are unknown. The sum of each calculation step yields the stress-strain relationship.

In the present work, irradiated and unirradiated austenitic stainless steels are the main concern. The following material properties are assumed: Young's modulus. 200000 MPa; proportional limit, 200 MPa; tensile strength, 550-600 MPa, work hardening exponent, 0.03-0.3.

Several points require attention. One is the treatment of the yielded element. For the given increment of displacement, the elastic element extends beyond the yield stress and must be corrected so that the modified equivalent stress drops to between ayi and ayi+6ayi. In the present paper, 6aYi is set to 0.0001 MPa or less. When whole triangular elements yield, the maximum equivalent strain increment must be suppressed under some allowable strain increment. This level of strain was set to be 0.001 or 0.002. Some troubles occurred when some elements unload during the work hardening stage. In this case the element was reset to the elastic state and the yield point was reset to the attained flow stress. This unloading and yielding was repeated many times.

The validity of this program is assured by the following observations.

1. The type of work hardening law (j = c(co + cp)n holds experimentally. The numerically calculated stress-strain relations hip and that calculated from this program coincided weIl.

2. In the different triangular elements, the stress states are different because the positions of those elements are differently related to the voids. But the relations hip between the equivalent stress and the equivalent strain should drop onto the same master curve. The results computed from this program showed that this was true.

3. The sum of the total nodal forces was shown to be zero. The sum of the y components on boundary AB and CD and the sum of the x components on boundary BC and DE were also zero.

4. The mesh of Fig. 1 is wholly calculated and the obtained strain distribution had fourth-axis symmetry.

8.3 Analysis of the Effect of Void Lattice on Stress-Strain Relationship

Figure 4 defines the void lattice, and the void spacings Lx,Ly and the void diameter Dx , D y are changeable. Because of symmetry, calculation is only necessary for the part ABCDEA of the rectangle CLMNC in Fig. 1. Figure 5 is an example of the mesh used. The region near the x axis is more finely discretized because strain localization is expected to occur there. The ratio of the void diameter to


D.

Fig. 4. Definition of a two-dimensional void lattice

the void spacing changes widely from 0.0001 to 0.8. So, the extremely fine mesh is necessary near the void. Figure 5b is an enlargement of Fig. 5a.

Figure 6 shows the effect of the void diameter on the stress-strain relationship. The effects can be divided into two categories. With increase of the void diameter from 0.001 to 0.1 !J.m, the effect of the void is mild and the work hardening behavior is the same. One point is noteworthy. With no voids, the uniform elongation is about 30% because the work hardening exponent is 0.3. All the calculated uniform elongations are about half of the no void case. A very tiny void of diameter of 0.001 !J.m can cause an observable reduction of elongation. Further increase of the void diameter causes a decrease of flow stress, but the effect on total elongation is quite small. This result is reasonable because the decrease of the net cross-section between two voids causes a reduction of the ability to sustain the external force.

We consider now the details of deformation process. Figures 7-9 are examples of the case in wh ich the void diameter is 0.01 !J.m and the void spacing is 1.0 !J.m. At first, the deformation begins at the right-hand side of the void and proceeds along the x axis. Note that Fig.7a is an enlargement by a factor of twenty to show the details of the strain distribution near the void. The vertically striped area is considered to be elastic. The pattern of the initial deformation varies somewhat, depending on the void morphology. The shadow of the void is clearly

H. Shiraishi 237

(a>

(b)

Fig.5. (a) An example of a computed mesh. Void diameter: 0.01 !-Lm; void spacing: 1.0 !-Lm; Void lattice aspect ratio: 1.0; nre: 11; nte: 3; nrf: 2; ntf: 3,1,1,1,1,1,1,1; No. of nodes: 1103; No. of elements: 2112. (b) An enlargement of (a) to show detailed neighborhood of void.

observed, as illustrated in Fig. 7b. The strain-concentrated area is limited to the neighborhood of the void until plastic instability is attained. Figure 8a shows the strain distribution at this point. Beyond the point of plastic instability, the strain-concentrated area develops along the x axis, that is along the line which connects two neighboring voids, as illustrated in Fig. 8b. Figure 9a shows the final state of the strain distribution. All the plots are based on the original mesh. All the calculations were interrupted when the conventional stress dropped to 80% of the tensile strength, and the attained strain is defined as the total elongation. Figure 9b shows the detail of the strain distribution in the neighborhood of the void. A large strain concentration greater than 2.0 develops along the x axis.


'" 600.0 0... ::0;

" 0'

UJ UJ 480.0 (l)

I-t ..... UJ

(l) ...... ·rl UJ ~ (l)

360.0 ..... ...... '" ~ 0

.rl ..... ~ (l)

:> 240.0 ~ 0 0

~ 0 .... ..... 0 (l)

I-t 120.0 .... "(j

UJ .... ><

'" >--

0.0 0.0

No "oid

L.

Void parallleter of void lattice (L,.=L.) : Void spacing : 10.0 ILIl

Void diameter: <>:0.001 @>:0.01 0:0.1 *:1.0 ILm .:2.0 .: 5.0 .: 8.0

10.0 20.0 30.0 40.0 50.0

Y-axis-direction conventional tensile strain , c:,,(%)

Fig. 6. Effect of void diameter on stress-strain relationship. Work hardening exponent: 0.3.

Here, the given strain is the natural strain. These regions are divided into two parts, so each origin is considered to start at a different point. This band of most concentrated strain is located at the void surface slightly above the x axis. Behind the void, a basin of strain distribution is found. In the smallest void, the pattern of strain concentration is similar, and this is the reason why the stress-strain curves are almost the same.

Figures 10-12 show the unloading behavior. Unloading begins at two different points. One is the back point of the void. The other is an inner point of void lattice. Two unloading areas coalesce into one and the plastic state region reduces to the x axis, as illustrated in Fig. 11. Figure 12 shows the final plastic region distribution. The plastic region is compressed to the x axis but a somewhat broad plastic region exists in the center part of the two voids. It is interesting

H. Shiraishi

Total elonlatioll : O.0810n

0.0335 ,. ..

(a)

Total elontalloll : O.l389 ~

[JJJ] ~ < 0 _0001 - 0 .001

c=J ~ - 0 01 - 0 _01 = ~ ... lO ..... Z.O

0.25 /1.- •

~OOI ~001 ~I _ EJ ~ ~ 0 .1 - 0 ! - 0 S

~ (b)

239

..

Fig.7a,b. Development of the distribution of the equivalent true strain around the void with a total elongation less than 0.1389%. Work hardening exponent: 0_3; void spacing: 1.0 ~m; void diameter: 0.01 ~m; void lattice aspect ratio: 1.0.


EI] ~ 0 . 2

(b)

Fig. 8a,b. Development of the distribution pattern of the equivalent true strain around the void in the intermediate strain range. Calculation parameters are the same as in Fig.7.

H. Shiraishi

[IJ]

.. jij~~6~i ~~~~~~ ~j~~~ H L~ ~~~~ ~ H

..

.... ............................. ......... -..................... . .... ............................. ....... ...... , ........ .... ..... , . ................................. ...... -.. -., .................... . .... .. ... ..... ........ .... ..... .. .. . ..... -........ -..................... .. .... . .................. ............ .............. .. ....... -......... . ... -.................... . , ................................ .......... . ...... -- .. .... ... -- ...... ..... -.... .. ......... . .... , ................................... ...... . .... , ................................... , ..... . ............. .................................. :::::::::::::::::::::=;:::::::::::::::::;::::::

.. .......

................................................. .. .... .................. ......................... ... ......... ............ . ................... ............... .. ....................................................... ::::::::::::::::::::: :: :: :::::::::::::::::::::::::::::: ....................................................... : : : : : : : : : : : : : : : : : ~ : : : : : : : : : : : : : : : : : : : : : : : : : : : ; : : ; : : : ; : : :: : ~ : : : : : : : : : ::: : : :: : : : :: : : :: : :: : : : : :: : : : : : : : : : : : : :: : : : ::::::::::::::::::::::::::::::::::::::::::::::::::::::: ............ .... ........ ............................... .. ......... ........................... ................. ::::::::::::::::::::::::::: :::: :::::::: ::: :::::::: :: ::: ..................... ... ... .. ....... .. ................. ..................... ...... .... ....... .. ............... .. , ........................................ .... ...... . .......................................................

0.5 p.m

(a) · . . . . . . . . . . . . .............. . ....... . ........ .. .... . ............ .. . . ................. ..... . .......... .. ..... ..... . · . . . . . . . . . . . . . . .. ...... . .. . .. . ................ ...... . . .. . . · . . . . . . . , . . . . . . .. ............ . :: ::::::::::::=:::: ~~~~! :: ~~~n. ~~ .tion 15.04\1 · . . . . . . . . , . . . . . . . ................. .. ..... .. ..... ... . . · . . . . . . . . . . . . . , . . ................. · . . . . . . . . . . . . . , . . .. . .. . ............ . .. ......... .... ............... ......... .. .................... .... ........ ......... ....... . . ................. ...................... ..... ..... . .. ................. .. ............... .. ............. . .. ................. ......................... .. ..... . · . . . . . . . . . . . . . . . . ..... ... .. ... .... · . . . . . . . . . . . . . . . . ................... ................. ... ...... ........ .............. ... .. ... , ..... . .. , . .. . .... ............. .... ....... ...... .... , .......... .. .. ......... . ...... . · . . . , . . . . . . . . . . . . ... ... ...... .. ... ................. .. ................ ,

~

............. ........ .. ........ .. . ..................... ... ...... .. . . ..... , ........................ . .. . .. ............ , ........ .. .... ... . .................. .... .. ....... , . ...... .. ................... .. ....

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...... ...... , ................... . ..... ............................ ........ .................. ....... : : : : : : ; : : : : : : : : ; : : : : ; : : : : : ~ : : :: : : ..... .. ........ ...... ............ :::::::::::::::;:::::::::::: ::::: .......... ................. ...... ............................. .... . . ................ , .... , : :::::::::::::::::::::::::::::::: .... ... ... ... ... .. .... .. ... . .. ... ........................... ...... ........... ...... ........ .. ...... ............................ .. ... . . ............ .. ..... ... ..... ...... ................... .. .... .. ...... ........ ... , .................... . ... .. ........ .................. .. ..... ......... .... .......... .... . . . .............. ... ........ ....... . ..... ..................... .. ..... .... .......... ................... ................................. . . ................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................. ............................... .. ......... ................

~ ~ f:ffil < 0 .0001 ~ 0.00 1 -0 _001 ~ 0 _005 - 0 .01

E2l . . . ~ - EIl ...... ~ ,/."." . ., ...... ~ 0.01 ~ 0 _0 5 ~ 0 .1 ~ 0 .2 - 0 .5

~ ~ ... ßß -1.0 ~ 2.0 - 5.0

(b)

241

Fig. 9. (a) Final distribution pattern of the equivalent true strain around the void. (b) is an enlargement of (a). Calculation parameters are the same as in Fig. 7.

242 Finite Element Analysis of the Deformation in Materials Containing Voids - [[[[]] Elastic slate Plastic slate Unloading slate

0.5 pm

Fig. 10. Distribution of elastic, plastic, and unloading elements with a total elongation of 13.20%. Work hardening exponent: 0.3; void diameter: 0.01 Il-m; void spacing: 1.0 Il-m; void lattice aspect ratio: 1.0.

~ I I I I I I I I Elastic slate Plastic slate Unloading state

0.5 ,tJm

Fig. 11. Distribution of elastic, plastic, and unloading elements with a total elongation of 14.06%. Calculation parameters are the same as in Fig. 10.

H. Shiraishi

Elastic state - [[[[[] Plastic state Unloading state

I , ' . . ~~~ j~ ..

0.5.um

243


that several discontinuous unloaded areas exist on the x axis. The final plastic region pattern also varies, depending on the void lattice morphology.

The void grows slowly initially, as illustrated in Fig. 13. There is a critical point when the void growth mode changes to a very high growth rate, and this strain corresponds to the plastic instability point. The void shape stays slightly elongated,but beyond the critical point elongation along the load axis is more pronounced. At the present time, it is not clear what determines the critical point. It is known that triaxiality has an infiuential effect on the void growth.

Figures 14a,b show the case of a larger void with a diameter of 5 !-Lm and a spacing of 10 !-Lm.ln contrast with the small void case, the area of uniform strain distribution is not observed, as illustrated in Fig. 14a. The highest strain region is formed in the center part of two neighboring voids, as shown in Fig. 14b. There is a region of sm aller strain than its neighborhood at the right-hand side of the void. The smallest strain area forms behind the void slightly away from the void and at the inner part from the y axis. The severe strain localization which appears in the small void case is not observed here. This is the reason why the total elongation does not decrease, despite the large decrease of strength. Figure 15 shows that two unloaded regions are divided by the plastic strain band and this band does not coincide with the x axis.

When the void volume fraction is the same, the same stress-strain relationship results in different combinations of void size and void spacing, as shown in Fig. 16. To avoid complexity, only two examples are shown in Fig. 16, but in the


1.0 2.0 ~ : Yoid aspect ro t i o( D,/D.)

0 : Woid volu • .e 1.8

0 0.8 1.6

u ci ......

-a e '" 0.6 1.2 ......

0

-.... -., 1.0

os e ... " 0 -0.4 0 > .,

0.

:;: ., .. 0 > ~

'0 0.2 >

0.0 0.0 4.0 8.0 12.0 16.0 20.0

Y a xis direction conventional te ns i 1 e s tr ain,I,,1 " Fig. 13. Void growth and void aspect ratio as a function of conventional strain. Calculation parameters are the same as in Fig. 10.

wide range of void size of 0.001 to 1.0 IJ-m and also in the range of void spacing of 0.1 to 10 IJ-m this result is confirmed. So it is unnecessary to evaluate the total elongation as a function of both the void size and void spacing. This conclusion holds only when the void lattice aspect ratio is one. The total elongation is affected by the void lattice aspect ratio and the same void volume does not give the same result.

The reduction of the work hardening exponent from 0.3 to 0.03 causes a severe reduction of the total elongation, as illustrated in Fig.17. Figure ISa shows that this is caused by extreme strain localization along the x axis and this is also known as internal necking between two voids. The strain is less than 1 % over a large part of the material. Figure ISb is an enlargement of Fig. ISa. Near the void and along the x axis, the strain distribution has a fine structure. The two highest strain regions exist separately and the third tiny one suggests that generation occurs between these two regions. Unexpectedly, the strain increases along the x axis as we move from the void surface to the center of the two voids. The maximum strain point on the void surface is not the intersection of the void surface and the x axis. The small plastic area appears just behind the void. Figure 19 gives the distribution of the elastic, plastic, and unloaded elements near the voids at the final stage. The periodic appearance of unloaded regions along the x axis, which are also observed in Fig. 12, is interesting. The zigzag lines reflect the insufficiently fine element discretization.

The decrease of the void lattice aspect ratio increases the total elongation, as shown in Fig. 20.This result is reasonable. As the void lattice aspect ratio de-

H. Shiraishi

Total eloncation : 0.836811

5 ,. m (a)

Total elon,ation : 11.<16 11

~001 r.:':'l l...:....:....:

~.,.~~ ~ '" 0 .00 2 '" 0 .00) '" 0 .0 I - EJ r,,!\·?,~ , .... ".

'" 0 .02 ..... 0 .1 '" O. l '" O. S

Im) - 1.0

Im '" S.O

(b)

245

Fig.14a,b. Development of the distribution of the equivalent true strain around the void. Work hardening exponent: 0.3; void diameter: 5.0 /l-m; void spacing: 10.0 flm; void lattice aspect ratio: 1.0.


bd ~ ITIIJ Elastic state Plastic state Unloading state

. 111"0 s",m


creases, the interaction between neighboring parts becomes smaller and the deformation proceeds more independently. This behavior can easily be understood when Ly / Lx ---+ 0.0 and the two parts are separated perfectly. It is concluded that the elongation is a function of both the void volume fraction and the void lattice aspect ratio. The similarity law which is implied by Fig. 16 only holds when L y / Lx = 1.0.

8.4 Application to Helium Bubble Grain Boundary Embrittlement

8.4.1 Engineering Importance of Helium Embrittlement

In the structural design of a fusion reactor, helium embrittlement is one of the greatest concerns since it determines the upper limit of the operating temperature of the material. Rewelding of structures for repair is now thought to be a big problem in this context in fusion technology [2]. It is essential for material researchers to elucidate the mechanism of helium embrittlement and to find how to suppress it. It is thought that this phenomenon is caused by the existence of helium bubbles on grain boundaries. But it has not been possible as yet to predict the ductility of material which contains bubbles on grain boundaries. If

H. Shiraishi

'" 600.0 r-~-~------~-~-~-~-, 0..

'" '1 '" ~ 480.0 ... '"

c :: 360.0

~ ~ 240.0 o

" " o

" " .~ 120.0 -0

'" " '"

0.0

L.

Void parameter of void lattice (L,/L."1.O) Void distance Void diamet er *: 10.0~. 1.0~.

o : 1.0 0.1

0.0 10.0 20.0 30.0 40.0

Y axis direction conventional tensile strain , lyl"(~)

50.0

247

Fig. 16. Effect of void spacing and void diameter on the stress-strain relationship. Work hardening exponent: 0.3.

this becomes possible, the alloy design will be easier and more reasonable. This section is a preliminary report of the effect of bubble size on helium embrittlement. In this calculation, the internal pressure of the bubble is not considered. So, in reality, a void is treated, but it is referred to as a bubble because the major concern here is helium embrittlement.

8.4.2 Formulation of Finite Element Method for Helium Embrittlement Analysis

It is necessary to simplify the problem to take the present performance of computers into consideration. As illustrated in Fig. 21, the helium embrittlement is assumed to be caused by the strain localization near the grain boundary on which the maximum normal tensile stress is in operation.There are many bubbles on the grain boundary, but the present calculation is done only for the rectangular region ABCDA, which contains one bubble. The case of a grain boundary wh ich contains many bubbles is approximated by periodic boundary conditions. It is assumed that the dis placements of the nodes on AD are equal to those on the corresponding side BC. The y-axis components of displacement of no des on AB are set to zero because of symmetry. Here, it is assumed that the shape of the grains is hexagonal, and they all have the same size and the same mechanical property.


600.0 ~

'" 0-

'" . . '" "' 480.0 "' " H +'

'" tJ

.--< .... "' c 360.0 " ~

'" c 0 .... +' C tJ 240.0 > c 0 ()

c 0 .... +' ()

" 120.0 H .... '0

"' .... " "

"...

0.0 0.0

L.

Void parameter of void lattice (L,(L,=1.0) Vo id sp acing : 1.0 ~ ..

4.0

Void dialleter: 0.01 ~'" Workhardening exponent : o : 0.3 -I:!: 0.2 0: 0.1 ß: 0.03

8.0 12.0 16.0

Y axis direction conventional tensile strain , I rr (%)

20.0

Fig. 17. Effect of work hardening exponent on the stress-strain relationship in material containing voids.

The role of the grain boundary is only to arrange the bubbles on the plane. A small increment of y-axis-direction displacement is given to nodes on CD and accumulated. The calculation is interrupted when the conventional y-axis tensile stress drops to 80% of the tensile strength, i.e. the plastic instability point, and the attained elongation is called the total elongation for convenience. The grain size and the bubble spacing are BC and AB, respectively. The grain radius is set at 10 Il-m . The bubble size was varied for the initial calculation conditions. As the results of calculations showed that the periodic boundary condition is the same as the y-axis symmetry boundary condition, most of the calculations were done under this condition for calculational efficiency. In the present program, the plane stress condition is assumed. The number of triangular elements is 639 and the number of nodes is 349. The stress-strain relationship is nearly that for austenitic stainless steel. The proportional limit, the tensile strength and the work hardening exponent are 200 MPa, 550 MPa and 0.3, respectively. The y-axis conventional stress is defined as the sum of the y component forces of all the nodes on CD divided by the original cross-section.

H. Shiraishi 249

(.)

0.005JJ"

[]]]TI ~ ~ [11 mI < 0.0001 - 0.00 I - 0.002 - 0.005 - 0.01

[] ..... ~ • D .,':;~, ~Jrtm . . ?} - 0.02 - 0.05 - 0.1 - 0.2 - 0.5

m lmD - 1.0 - 2.0 - 5.0

(b)

Fig.18. (a) Effect of reduction of work hardening exponent on distribution of equivalent true strain around void. (b) is enlargement of (a). Work hardening exponent: 0.03; void diameter: 0.01 ~m; void spacing: 1.0 ~m; void lattice aspect ratio: 1.0.


B ~ Elaslic state Plaslic stale

[]TI Unloading state

, I 1.'

ii 0669 pm

Fig.19. Elastic, plastic, and unloading element distribution for a total elongation of 0.4919 %. Work hardening exponent: 0.03; void diameter: 0.01 ~m; void spacing: 1.0 ~m; void lattice aspect ratio: 1.0.

8.4.3 Calculation Results

Figure 22 shows the effect of bubble size on the stress-strain relationship. In the smallest void case, there does not appear the influence of the void. The total elongation is not strongly dependent on bubble size for intermediate bubble sizes of 0.002- 0.05 ~m. When the bubble size exceeds 0.1 ~m, the total elongation decreases sharply. For all bubble sizes, the stress-strain relations hip in the work hardening stage is nearly the same till plastic instability is attained.

The analysis of this section is a special case of high void-lattice aspect-ratio. Here, the void-lattice aspect-ratio is given as the ratio of Ly / Lx, where L y is the grain size, fixed as 20.0 ~m, and Lx is the reciprocal of the void density. When the void-lattice aspect-ratio is high, i.e. L y / Lx » 1.0, the grain interior is divided into two parts. One is the neighborhood of the void in which the deformation is affected by the existence of the void and the other is the area remote from the void in which the void has no effect. The difference between Fig. 6 and Fig. 22 can be understood from this viewpoint.

H. Shiraishi

600.0 ,--~-~-....,...-~-~-~-~~-~--,

'" 480.0 '" " f.o

~ 360.0 " ~ ~

c o .~

C ~ 240.0 c o

" c o

" '" 120.0 ... 'Ö

'" .~

" " ... 0.0

0.0

L.

Aspect ratio of void lattice (L,/L.) :

o : 0.2 -4 : 0.5 0 : 1.0 © : 2.0 0 : 5.0

10.0 20.0 30.0 40.0 50.0

Y axis direction conventional tensile strain , t. T7 (%)

251

Fig. 20. Effect of aspect ratio of void lattice on stress-strain relationship. Work hardening exponent: 0.3; void diameter: 0.01 IJ.m; void spacing: 1.0 IJ.m.

8.5 Discussion

As this calculation assurnes a plane stress condition and treats exactly only the two-dimensional hole distribution in the plate, the comparison with the experimental results remains qualitative in nature. In addition, detailed experimental data on both the tensile tests and the void morphology in the same material is limited.

In ductile fracture which is known as dimpIe fracture, fracture proceeds in three stages: void nucleation, void growth, and void coalescence [3]. In this chapter, we ass urne the pre-existence of voids. In real met als and alloys the voids nucleate by the fracture of secondary particles such as non metallic inclusions and precipitation-hardening intermetallic compounds or decohesion at the interface between such particles and the matrix. It is not unreasonable to ass urne the pre-existence of voids when their nucleation occurs in the early stages of deformation.

A void of diameter of 1 nm can reduce the uniform elongation to half of the no-void case. The uniform elongation in austenitic stainless steel is usually about 30% and this may mean that such defects do not exist. It is reported that for pores containing copper alloys produced by powder metallurgy the pore volume


Tensi I e stress

1

D (

A B

1 Tensi I e stress

Fig. 21. Modeling of grain boundary helium embrittlement by the finite element method.

fraction sharply reduces the total elongation by 1-10% [4]. The results of the present work shown in Fig. 6 show little dependence of elongation on void size. The reason is not clear, but the approximation of a cyclic boundary condition might be an oversimplification. In real materials the scatter in void morphology can cause a larger scale pattern of strain concentration than the void spacing. The reduction of the stress in Fig. 6 predicts that the decrease of toughness begins at a hole areal fraction, which is the same here as the void volume fraction of 0.01-1.0%.

The propagation of a crack ahead of which a row of voids exists is analyzed by the finite element method [5]. In this work several assumptions are made: (1) average stress-strain relationship in a cell containing the void, (2) void growth rate as a function of strain and (3) cell extinction at the critical void volume fraction. The present work shows that these assumptions are not necessary and that the plastic instability occurs automatically.

The highest strength and the lowest total elongation were observed in austenitic stainless steels which were irradiated by neutrons at a temperature of around 300°C [6,7]. The reason for the low ductility is not clear. The results of this chapter suggest a possible mechanism. One factor is a very low work hardenability. Explicit evidence for this has not been seen, but the very high proof stress in irradiated material usually means a low work hardening exponent. The

H. Shiraishi

; 480.0

~ 360.0 ~

~

c o

~ 240.0 , § ü

§

~ 120.0 :;;

Bubbi. ,iz. ( ~. ): <:. : 0.001 o : 0.002 @ : 0.005 0 : 0.01 0: 0.02 /:i : 0.05 \l: 0.1 +: 0.2 x; 0.5

Y axis directicm conventional tensile strain . tu (%)

253

Fig. 22. Effect of helium bubble size on stress-strain relation ship in grain boundary helium embrittlement. Work hardening exponent: 0.3; bubble density: 1.0 iJ-m -1; grain size: 20.0 iJ-m.

second factor is the possibility of the existence of small voids or vacancy clusters at this irradiation temperature. The combination of low work hardenability and the existence of voids is most deleterious to ductility, as shown in Fig. 17.

When the experimental stress-strain relationship in pure silver which was internally oxidized and contained water bubbles is compared with Fig. 22, a similar behavior is seen. In the case of silver, the bubble morphology was varied by changing the temperature of the internaioxidation [8]. In this case, not only grain-boundary bubbles but also grain-interior bubbles existed and the characterization of the grain boundary bubbles was not reported. The behavior that total elongation decreases with an increase of bubble size and that the stress-strain relationship in the work hardening stage is almost the same strikingly resemble each other. The reduction of the total elongation to values in the range 20 to 5% due to the presence of helium was reported in tensile tests of austenitic stainless steels which were carried out in the temperature range of 923-1123 K [9, 10]. These values are nearly equal to those predicted in this chapter. It is said that the resistance to helium embrittlement of ferritic-martensitic steel is superior to that of austenitic stainless steel [11,12]. It is believed that this is due to a finer helium morphology in the former than in the latter, and this explanation is also consistent with the predictions of these calculations. Remarkable helium embrittlement was reported in heat resistant Fe-Ni-Cr super alloys which were strengthened by intermetallic compound precipitation [13]. Generally, the high


strength means low work hardenability, and this result can be explained by the present calculations. Helium embrittlement is often said to occur in high temperature tensile tests. The high temperature is a necessary condition for bubble growth but the high temperature also reduces the work hardening exponent.

8.6 Conclusion

The stress-strain relationship of materials which contain voids in the matrix or helium bubbles on the grain boundaries was analyzed by the two-dimensional finite element method. Continuum mechanics, the Mises yield criterion, J2 flow theory, the n-th power work hardening law, and cyclic boundary conditions were assumed.

1. The efIect of void size on total elongation in material containing a square void lattice is small and a total elongation of about 15% was obtained over a wide range of void diameter of 0.001-1.0 Il-m when the void spacing was 10 Il-m and the work hardening exponent was 0.3.

2. The tensile strength begins to drop at a void volume fraction of around 1%. 3. The reduction of work hardenability has aremarkable impact on the strain

concentration and thus causes a sharp reduction of the total elongation to less than 1 %.

4. With a decrease of the aspect ratio of void lattice, the total elongation increases gradually.

5. For the square void lattice, the elongation is a function of void volume fraction, i.e. the similarity law holds. In general, the total elongation is a function of both the void volume fraction and the void-lattice aspect-ratio.

6. For the grain-boundary helium-embrittlement case, a reduction of the total elongation was clearly observed with increase of the bubble size.

In future work, the present program should be expanded to treat the threedimensional void distribution and more than one bubble to allow for the statistical scatter of the void size and the void spacing.

References

1. O.C. Zienkiewicz, R.L. Taylor: The Finite Element Method, Vol. 1 (McGraw-Hill, London, fourth edition, 1994).

2. C.A. Wang, M.L. Grossbeck, N.B. Potluri, B.A. Chin: J. Nucl. Mater. 233-237, 213 (1996).

3. R.H. Van Stone, T.B. Cox, J.R. Low, J.A. Psioda: Int. Met. Rev. 30, 157 (1985). 4. B.1. Edelson, W.M.Baldwin Jr: Trans. Am. Soc. Metals, 55, 230 (1962). 5. L. Xia, C.F. Shih: J. Mech. Phys. Solids 43, 233 (1995). 6. J.E. Pawel, A.F. Rowcliife, D.J. Alexander, M.L. Grossbeck, K.Shiba: J. Nucl.

Mater.: 233-237, 202 (1996). 7. G.E. Lucas, M. Billone, J.E. Pawel, M.L.Hamilton: J.Nucl.Mater.233-237, 207

(1996).

H. Shiraishi 255

8. J.C. Gibeling: Scri. Metall. 23, 167 (1989). 9. D. Kramer, K.R. Garr, C.G. Rhodes, A.G.Pard: J.Iron & Steel Inst. 207, 1141

(1969). 10. K. Matumoto, T. Kataoka, M. Terasawa, M. Shimada, S. Nakahigasi, H. Sakairi,

E. Yagi: J. Nucl. Mater. 67, 97 (1977). 11. H. Shiraishi, N. Yamamoto, A. Hasegawa: J. Nucl. Mater. 169, 198 (1989). 12. A. Hasegawa, H. Shiraisi: J. Nucl. Mater. 191-194, 910 (1992). 13. H. Shiraishi, N. Yamamoto, H. Shinno, H. Yoshida, H. Kamitsubo, I. Kohno, T.

Shikata: J. Nucl. Mater. 118, 179 (1983).

N umerical Analysis of the Interface Problem in Continuous Fiber Ceramic Composites

Y. Kagawa1 and C. Masuda2

1 Institute of Industrial Science, The University of Tokyo 7-22-1, Roppongi, Minato-ku, Tokyo,106, Japan

E-mail: [email protected] 2 National Research Institute of Metals

1-2, Sengen, Tsukuba, Ibaraki, 305, Japan E-mail: [email protected]

Abstract. Finite element analysis (FEA) and pushout tests have been used to discuss several key elements of interface debonding in continuous fiber ceramic matrix composites. This study includes (i) the interaction between fiber debonding and a matrix crack, (ii) the interface shearstress distribution, and (iii) the interface de-bonding criterion. The interface debonding process is simulated including a pre-existing crack in the matrix. This simulation shows that the stability of the pre-existing crack may be affected by interface debonding ahead of it. The interfacial shear-stress distribution during the pushout test is also computed and gives useful information about the interface debonding criterion. Moreover, the numerical results also give a rational explanation far the experimentally observed trends, including the effect of thermally induced stress on the interfacial debonding.

9.1 Introduction

Recent developments in the field of continuous fiber ceramic composites (CFCC) make it possible to obtain high fracture resistance composites which are difficult to achieve in monolithic ceramics. The composites possesses superior damage tolerance up to elevated temperatures achieved by cumulative microfracture processes. It has been recognized that the interface mechanical properties of CFCCs play an important role in controlling the microfracture process. Recent technical developments permit the control of the interface conditions, both physical and chemical. Various approaches have been proposed for the control of the interface mechanical conditions. Among them, fiber coating is the most widely used method; the fibers are coated before incorporation in the matrix. This coating is usually made uniform consistened by a nanometer-order amorphous carbon and boron nitride layer.

A good interface must satisfy two major conditions: it should enhance (i) interface debonding and (ii) post-debond friction. Interface debonding prevents fiber breakage by deflecting approaching cracks at the interface. Interfaces produced using eoating technology usually fulfill this condition. However, the postdebond frictional resistance also has to be high to delay final fracture.

Springer Series in Materials Science Volume 34. Ed. by T. Saito © Springer-Verlag Berlin Heidelberg 1999

258 Numerical Analysis of the Interface Problem

This property is affected by various factors, such as interface roughness, thermal mismatch stress, and fiber~matrix properties. Much research has been carried out on the above properties. However, the published studies are not directly applicable to practical problems. In this chapter we will discuss the crack-fiber inter action and the role of the interface, load transfer at the interface and evaluation of the interface mechanical properties using the pushout test though application of finite element analysis (FEA) [1~3].

9.2 Crack-Fiber Interaction Problem

9.2.1 Crack-Fiber Interaction in CFCC

As stated above, incorporation of ceramic fibers into brittle materials pro duces a complete change of the mechanical response of the brittle materials. Such materials are currently obtained as fiber-reinforced ceramic matrix composites (CFCCs). The change in the mechanical response is believed to be the result of a complex micro-failure process. The understanding of a crack~fiber interaction process is important. However, many theoretical and experimental research studies have focused on the matrix cracking behavior under the assumption of long matrix cracks [4~12]. On the other hand, the initial matrix cracking process, including the crack~fiber interaction mechanism before formation of a long crack, has not been clearly explained.

So far, very few analyses have been reported for the initial matrix cracking behavior. It was reported that the initial cracking stress ofthe composite depends on the constraint effect of the stiff fiber [13, 14] or the existence of an energy dispersion mechanism near the crack tip [15,16]. These reports show only the trends of the cracking behavior, but no quantitative analysis has been carried out on the initial cracking behavior and long crack formation in continuous fiber-reinforced ceramic matrix composites. Recently, Goto and Kagawa [1,16] reported some numerical analysis of the crack~fiber interaction problem and classified the interaction mechanism.

9.2.2 Analysis Technique for Crack-Fiber Interaction

The fiber~matrix crack interaction problem has been solved using finite element analysis (FEA) [14]. A typical example oft he two-dimensional plane strain layerby-Iayer model used in the study is shown in Fig. 1; the model can describe the effects of incorporation of fiber on the local stress fields near the defect tip. A pre-existing small crack, which was assumed as a source of stress concentration, is located in the matrix phase without touching the fibers, under the perfect bonding condition of the fiber~matrix interface. The size of the defect was smaller than the fiber spacing. The pre-existing matrix crack was assumed to be a slitlike through-crack and the crack opening was assumed to be zero. The effects of thermally induced stress were neglected because the major attention of the analysis was focused on the effects of the Young's modulus and the interfacial conditions on the crack growth from a pre-existing small crack.

Fig. 1. (a) Modeling of the composite for FEA analysis and (b) detail of the geometrical relation for the analysis. The hatched area shows where the analysis was used in this study.

Microstructural level FEA, which includes separate phases of fiber and matrix, was developed to investigate the stress fields near the pre-existing matrix crack. A two-dimensional layer-by-layer tensile test specimen was modified for numerical simulation. Four kinds of fine mesh were used, according to the fiber diameter. The fiber diameter was assumed to be 100, 140,200, and 280 Il-m. The width of the fiber phase in the model was chosen to be the same as the diameter of the fiber. In all the models, the fiber volume fraction was fixed at 0.5 for the applied modellayer composites. The volume fraction of the fiber phase, f, of the layer-by-layer composite model is defined as

Vi = Wf , Wf+Wrn

(1)

where Wf is the width of fiber phase and W rn is the width of the matrix phase. Figure 2 shows an example of the geometry and computational mesh used

in this analysis for the ease of a fiber diameter of 140 Il-m. Due to symmetry eonsiderations of the model, only a quarter-seetion of the eomposite was modeled. The fine mesh near the defeet eomprised 252 elements and 285 nodes, and the eoarse mesh eomprised 378 elements and 418 nodes. A four-nodal isoparametrie element with elastie and thermally isotropie properties was used in the meshes. Far from the defeet the fiber and the matrix phase in the longitudinal direction were assumed to be deformed under an equal strain eondition

Ef = Ern = Ec , (2)


I' Matrix. I • Fiber • I. Matrix • I

z

Fig. 2. Finite element mesh used to simulate the stress distribution near a defect tip.

where cr is the strain of the fiber, Crn is the strain of the matrix, and Ce is the strain of the composite. Under this condition, the stresses in the fiber, af' , and in the matrix, a~ , far from the pre-existing matrix crack are given by

(3)

(4)

where E c is the Young's modulus of the composite (Ec = ErVf + E rn (1- Vf)) and a'("' is the stress of the composite applied far away from the crack. Frictional elements were added to the interface between fiber and matrix, and the elements included the effect of interfacial sliding friction after the onset of interfacial debonding. The numerical computation was carried out under a plain strain condition using the MARC (MARC-MENTAT) program.

The pre-existing matrix crack size, 2a, was obtained knowing the critical stress intensity factor, K rc , of the monolithic matrix and the tensile strength of the matrix, (ar;) . The defect size was calculated from [17]

(5)

The calculated crack size, a, for the glass matrix from equation (5) is ~40 ~m. It was recommended that the computation was best carried out under the condition of no crack growth at the matrix crack tipi the defect tip stress intensity factor, Kfocal, is always smaller than the critical stress intensity factor of the monolithic matrix, K 1c '

9.2.3 Interface Shear Stress Distribution ahead of Matrix Crack Tip

The material system used in the study was ceramic-fiber reinforced glass with completely linear elastic properties up to fracture. For Er = 200-400 GPa, the

Y. Kagawa and C. Masuda 261

Table 1. Properties of fiber and matrix.

Properties SiC . Glass

fibera matrixb

Young's modulus (GPa) 406 70

Poisson's ratio 0.15 0.2

Tensile strength (MPa) 3450 60

K lc (MPa1/ 2 ) 0.8

aTextron Corp., SCS-6

bCorning Gleass Works, Code #7740 (Pyrex glass).

com-puted results show the conventional SiC-fiber reinforced glass matrix co mposite. To examine the effects of Young's modulus mismatch on the fracture behavior, the ratio of the Young's modulus of the fiber to that of the matrix, Er! Ern, was varied from 1 to 10 by changing the Young's modulus of the fiber. The properties of fiber and matrix used are listed in Table 1. Typical ex am pIes of the contours of the stress distributions az , ax and T xz near a crack tip are shown in Fig. 3a, band c, respectively, together with the computed contour levels. The difference between a z and a x obtained from well-known linear elastic fracture mechanics is the stress step which occurs at the interface because of the difference of the Young's modul i of the fiber and matrix. The step ratio in the tensile stress, a z, at the interface is approximately 6, which agrees well with the Ef / Ern ratio. In the transverse direction, the tensile stress and compressive stress are slightly enhanced at the interface. In addition, as a result of the modul i difference, shear stress is generated along the interface for z > 0 and z < O.

Figure 4 shows the relation between the normalized stress intensity factor Kioea1 / a~z and the distance between the pre-existing matrix crack tip and fibermatrix interface as a function of E f / Ern ratio. Here, it should be assumed that all the computation conditions satisfy the condition Kioea1 < K Ie , i.e. the crack growth from the crack tip never occurs under the simulation. The crack tip local stress intensity factor of the composite, Kioea1 , is obtained assuming that the stress in the z direction near the defect, a z, is as follows [17]:

K10eal _ I

az - V21fX '

where x is the distance from the defect tip with the constraint that x « a.

(6)

For the same Ef / Ern ratio, as the defect tip nears the interface, the local stress intensity factor at the crack tip decreases. This tendency becomes more pronounced with the increase of the Ef / Ern ratio. This result indicates that the distance between the pre-existing crack tip and the fiber-matrix interface strongly affects the local stress intensity factor. The high modulus ratio of fiber to matrix effectively reduces the local stress intensity factor at the defect tip compared with the applied stress intensity factor. Thus the efficiency of the crack tip shielding depends on the Young's modulus ratio and on the distance

262

a)

b)

c)

Contour Level (MPa)

1 -1.714 2 10.22 3 22.15 4 34.08 5 46.02

~atrix

Numerical Analysis of the Interface Problem

Fiber ~atrix ..

6 57.95 7 69.89 8 81.82 9 93.76

10 105.6 11 117.6 12 129.5

Z 2

tl~~_4x3~ __ ~ ______ ~ ~ --- Interface I so o.m I

Contour Level (MPa)

1 -19.62 2 -15.00 3 -10.38 4 -5.758 5 -1.134 6 3.488

~atrix Fiber ~atrix

jIn~z~s S66

11 26.60 """"'7!..1..,;;;----'-________ -:::-_---'

12 31.22 X --- Interface I so o.m I

~atrix Fiber ~atrix

Contour Level (MPa)

1 -14.39 2 -12.52 3 -10.65 4 -3.783 5 -6_913 6 -5_043 7 -3_176 z

10 2.436 Q

..

: ~~;6~5~S 11 4.306 ""-.Zl9--'--""=-___________ --' 12 6.176 X --- Interface I 50o.m I

Fig_ 3_ Typical example of the computed contours of the stress components, (a) I7 z ,

(b) I7x , and (c) T xy for the gross applied tensile stress at 20 MPa.

between the defect tip and the fiber-matrix interface, respectively. This result gives a guideline for obtaining higher crack stability in fiber-reinforced ceramic matrix composites.

9.2.4 Possible Interface Debonding Process

A typical ex am pie of stress distribution near a defect tip along the interface is shown in Fig.5 for the composite with a fiber diameter of 140 IJ.m and an applied tensile stress of 20 MPa. The tensile stress normal to the interface has


0.13r------------,

0.12 1-~~~rI

0.11

--- A= 1.0 --0--- A = 2.0

0.10 ------- A = 3.0 -{S-- A= 5.8

------- A = 10 0.09L..':'~~_'___~====:.J

o 20 40 60 80 100 120

Distance from defect s (11m)

Fig.4. Effect of Young's modulus ratio on the relation between normalized stress intensity factor and distance from the tip of a defect.

/ ItJll't'I':.ll't.'

, , t - ---"

/ " "I • : . I •

• • A a(:~

; . i • : . I • : . 1I

/1 / 2~ I:;

20\11';,

• • • • •

11 .~

~.n·,,' alul1~ itt'lTf:u'(' (\IPa>

IHIl .§-

o

Fig. 5. (a) Schematic illustration of the position of the maximum tensile stress (afax) and maximum shear stress (T;max) at the interface, and (b) example of the stress distributions along the interface (applied tensile stress = 20 MPa, fiber diameter = 140 11m).

a maximum value at the front of the crack tip. However, the shear stress has a maximum value above and below the crack tip. The result shown in Fig. 5 indicates that the interfacial failure behavior depends on the interfacial tensile and shear strength of the composite. The possible types of interfacial fracture at the fiber-matrix interface are:

(i) tensile fracture of interface at z = 0, (ii) shear fracture of interface at z > 0 and z < O.

The interfacial fracture behavior of the composite depends on the ratio of interfacial tensile debonding strength and shear debonding strength. The interfacial fracture process is modeled as shown schematically in Fig. 6. There are three


Ia 11

Ia I

t t t t t t

• •

Fig. 6. Schematic drawing of the possible interfacial fracture process in fiber-reinforced ceramics. (i) interface shear failure, (ii) tensile failure, (iii) without interface failure.

types of fracture behavior und er a given appropriate criterion. At stage I the composite containing the cracking matrix is loaded, both the tensile and shear stresses at the interface increase while the defect tip stress intensity factor remains below its critical value, K Ic . If the interface is sufficiently strong, the crack from the matrix crack tip propagates and reaches the interface without interfacial failure. On the other hand, two types of interfacial failure are considered from the interfacial fracture criteria. This stage is represented as stage H. The possible processes are (i) symmetrical interfacial debonding by shear force and (ii) well-known Cook-Gordon mechanisms [18]. The origin of type (i) fracture is the generation of shear stress by the constraint effect of the fibers as the closure force against the defect tip. Type (ii) is the debonding ahead of the defect by the tensile force at the interface. For the case of no interface debonding, the crack which reaches the interface should split along the interface to avoid fracture of the fiber.

Figure 7 shows the condition for interface fracture behavior as a function of the distance between the defect tip and the fiber for various Young's modulus ratios of fiber to matrix Er! Ern. Above the curves, the interface shear fracture is more likely and below the li ne the interface tensile fracture is more likely. This figure shows the dependence of the interfacial fracture mode on the distance between the defect tip and the interface; however, the fracture mode is less dependent on the Young's modulus between the fiber and matrix. If we can specify the interfacial stresses, the failure behavior can be determined from Fig. 7. Summarizing the results, the failure process is modeled into three different mechanisms. These are (i) without interfacial failure, (ii) interfacial shear sliding, and (iii) interfacial debonding. If we can specify the interfacial stress, both


tensile and shear, the interfacial failure process can be understood using these results (Fig. 7). The FEA results provide valuable insights into the dependence of the defect stability on the interfacial mechanical properties. It is expected, however, that many of the conclusions from the two-dimensional analysis may also apply to the three-dimensional model as weIl. The details of the comparison between the experimental and theoretical data are reported elsewhere [12].

L '" -

Int ..... ra

8 40J.Lm 8 : Defect len th ~). J

),-2

'i Er/E.-).

). ~ SJI .. ),-10 ....

). - I

Ten ile debonding area

o~--~~~~--~~----~

o 40 80 120

Di tlIßC from d fact (r..lm)

u, -Fiber

Fig. 7. Mapping of interfacial failure modes (>.. is the ratio of Young's modulus of fiber to matrix).

9.3 Evaluation of Interface Shear Stress: FEA Analysis

9.3.1 Background of Analysis

The crack-fiber interaction process is strongly affected by the interface mechanical properties; the shear properties are especially important for the control of the post-debond interface load transfer potential. Therefore, it is important to evaluate the interfacial shear mechanical properties of the composites. The interface mechanical properties have been measured by various techniques such as the pushin test [19,20], the pushout test [21-24], and the protrusion test [25,26]. Among the techniques, the thin specimen pushout test is regarded as most useful because of the relative simplicity of the specimen preparation procedure and the relatively simple experimental process.

To analyze the pushout process, it is important to know the distribution of stress in the specimen before and during the test, including the condition of the frictional shear sliding event. The shear stress distribution during this process has been estimated in many studies, which have included different types and levels


of approximations such as modeling, shear-lag analysis, and interface sliding conditions. Early studies ofthe pushout process [21,22] used constant shear stress along the bonded and sliding interface, while other studies dealt with the load transfer including shear friction at the interface. This model assumes Coulomb's friction law as the basis of the interfacial shear stress transfer mechanism, and has been used to obtain the distribution of the shear stress during the process [23-30]. Although a more realistic analysis incorporating thermally induced stresses in the calculation is not the pushout analysis, it can easily be extended to the pushout problem, as was reported by Gao et al. [31], Hutchinson and Jensen [32], McCarteny [33], and Hsueh [34] for a pullout situation.

Another type of analysis to determine the stress distributions before and during the pushout process is finite element analysis (FEA), which has been applied to the pullout and pushout problem in fiber-reinforced brittle matrix composites. Faber et al. [35], Grande et al. [36], and Ballarini et al. [37] analyzed both frictionally bonded and chemically bonded interfaces using FEA models of single fiber surrounded by a coaxial cylinder of a matrix. The study of Ballarini et al. [37] included the thermally induced residual stress; however, a constant stress was added to all the elements and the effect of the distribution of the thermally induced stress was not considered in the computation. Povirk and Needleman [38] reported FEA simulation of the pullout process wh ich included thermally induced stress and a frictional sliding process of fiber-reinforced plastics (FRP). Recently, Honda and Kagawa [2,3] reported details of FEA analysis on the pushout problem. The analysis included the effects of an actual thermally induced stress field at the interface in this process. Hereafter, the distribution of the stress field on the pushout process and Coulomb friction sliding at the interface using FEA, which includes realistic experimental conditions, are discussed.

9.3.2 Modeling for FEA Analysis of Pushout Problem

Figure 8 shows the three phase axisymmetric cylindrical model used in FEA [1]. This model consists of an anisotropic fiber surrounded by an isotropic matrix and an anisotropic outer composite layer. Such a model has been used in both analytical and finite element analyses of pushout, pushin and pullout problems of fiber composites [39,40]. The finite element model is composed of arbitrary quadrilateral axisymmetric elements with four nodes and four numerical integration points.

Figure 9 shows details of the mesh; R f , R m, Re and L represent the radius of fiber, matrix, composite and thickness of the specimen, respectively. The coordinates, r (radial), e (tangential), and z (axial) are also defined as shown in the Figure. Subscripts or superscripts f, m, and c, are used to distinguish between the fiber, matrix, and composite.

Element sizes are selected by continuously refining the mesh until approximate convergence of the numerical solution is achieved. The FEA computation was done using the MENTAT-MARC computer program. The interface between the matrix and fiber in the initial state was assumed to be perfectly bonded, and was allowed to slide after the onset of interface debonding; this means that

Y. Kagawa and C. Masuda

Fib r

Frictionall -liding

interrace

Bonded Int rrac

Rr: Radiu orriber. Rm : Radiu 01 matrix

R c: Radiu 01 compo it

Fig. 8. Models of the pushout test specimen for FEA analysis.

o Load Interrace Total o. or eIeDIent = 140

~r +mrnT(G~Pnmel~emnren,t)-r,-,,~~Tro~t.~I ~O'_Orr,nO_d~e·~T~-r15,~. _-r z

- Ma!rix-IttI+++-+---I-+-l--+-Composite+-I-H-t--t-H-l .. ~~~~Hffl~~~--I-+-l--+~+-H-+-++-l-~~~

F

6ö

J ~---------------- Rc

Fig. 9. Analytical model of FEA.

L

267

all the debonded interfaces slide. Hereafter, the debonding interface is referred to as the sliding interface to avoid misunderstanding. The radius of fiber, Rf, matrix, R m , and the fiber volume fraction, V[, satisfy

(7)

The outer radius of fiber, Rf, and composite, Re, were fixed at 0.7 and 2.0mm, respectively, and the thickness of the specimen, L, was fixed at 1 mm, which corresponded to a ratio of fiber length to fiber diameter of ",7. This thickness is typical of a pushout specimen in Sie fiber-reinforced glass matrix composites.

The computation requires elastic moduli and thermal expansion coefficients. These values are obtained from a simple rule of mixtures. Elastic moduli of a composite in longitudinal and transverse directions (Ec,z, Ec,r) were obtained

268 Numerical Analysis of thc Interface Problem

from a simple rule of mixtures assuming Vf = Vrn ,

Ec,z = VfEf + (1 - Vf) Ern , (8)

(9)

Longitudinal and transverse coefficients of thermal expansion (Cl:c,z, Cl:c,r) were obtained [40] by

VfEf,zCl:f,z + (1 - Vf) ErnCl:rn Cl:c,z = VfEf,z + (1 - Vf) Ef '

Cl:c,r = Cl:f + (Cl:f - Cl:c,z) Vc - (1 - Vf) (Cl:f - Cl: rn ) (1 + vrn )

= Vf (1 + vr) Cl:f + (1 - Vf) (1 + vrn ) Cl:rn - VcCl:c,z

(10)

(11)

where vc,zr and vc,rz are the Poisson's ratio of the composite under longitudinal and transverse loading, respectively, and given by vc,zr = VfVf + (1- Vf)Vrn and vc,rz = -Er/Ez. Cl: rn , Cl:f are the coefficients of thermal expansion of the matrix and fiber, respectively.

9.3.3 Boundary Conditions and Analytical Procedures

The flow chart of the analytical procedure using FEA is shown in Fig. 10. The calculation consisted of two steps: (i) calculation of thermally induced stress in the pushout specimen, and (ii) calculation of stress distribution in the pushout process using the thermally induced specimen stress. The first step was carried out for the perfectly bonded fiber-matrix interface condition. In the second step, both interfacial debonding and sliding are considered. To determine the stresses at the interface, the stresses of the fiber no des nearest to the interface was used.

In the first step, the given boundary conditions of continuity of displacement, u(r, z), for the model are

u~ (r, L/2) = u; (r, L/2) = u~ (r, L/2) = 0 0:::; r :::; Re , (12)

and for interface continuity at the bonding interface region are:

u~ (Rf , z) = u; (Rf , z) 0:::; z:::; L

u~ (Rf, z) = u~ (Rf , z) O:::;z:::;L

Ug (Rr, z) = u~ (Rf, z) O:::;z:::;L (13)

u~ (Rrn , z) = u~ (Rrn , z) 0:::; z:::; L

u~ (Rrn , z) = u~ (Rm, z) 0:::; z:::; L

Ug (Rm, z) = Ug (Rm, z) 0:::; z:::; L.


Tying (fiber-Imatrix interface)

Calculation of thermally induced stress (1023 K - R.T.=298 K)

Change of fixed dis placement

t

Fig. 10. Flowchart of FEA.

269

The computation of thermally induced stress in the thin specimen was carried out for a temperature change from 1023 K (fabrication temperature ) to 298K (test temperature) [3] under the boundary conditions (12) and (13). In this computation, the thermal expansion coefficients, Young's moduli, and Poisson's ratios were assumed to be independent of the temperature change. The interfacial shear stress for the bonded interface along the z direction at r = Rf, T;r(Rf, z), was defined as the shear stress of the fiber element located nearest to the interface.

After computation of the thermally induced stress in the composite specimen, the boundary conditions of the specimen were changed to calculate the pushout process. The boundary condition in the second step for the fixed surface is

ur: (r, L) = u~ (r, L) = 0 (14)


For the bonding interface region, the displacement continuity boundary conditions are

ur;' (Rf, z) = u~ (Rf, z)

u~ (Rf, z) = u~ (Rf, z)

ue (Rf, z) = u~ (Rf, z) ur;' (Rm , z) = u~ (Rm , z)

u~ (Rm , z) = u~ (Rm , z)

ue (Rm , z) = Uo (Rm , z)

0'5; z '5; L,

0'5; z '5; L,

0'5; z '5; L,

0'5; z '5; L,

0'5; z '5;L

O'5;z'5;L

(15)

and for the sliding interface region, the displacement continuity boundary conditions are

ur;' (Rm , z) = u~ (Rm , z)

u~ (Rm , z) = u~ (Rm , z)

ue (Rm , z) = Uo (Rm , z)

0'5; z '5; L,

0'5; z '5; L,

0'5; z '5; L.

(16)

The top and side surfaces of the model were still free to deform, while the deformation at the back surface of the composite specimen in the z direction for Rf < r '5; Re was fixed. For the computation of the pushout process, the distributed load was applied to the surfaces of the central elements for the z direction with an incrementalload step of 2 N, which corresponded to an applied fiber stress increment of ",500 MPa. The diameter of the central elements was 35 ~m, and was nearly the same diameter as when a hard indenter was used for the experiment [3,24,42]. The pushout process was analyzed using an applied load control condition. The load was applied until complete debonding occurred at the interface, assuming that pushout of the fiber from the back surface occurred immediately thereafter.

As adebonding criterion, a stress-based approach [1-3], [43-47] and an energy-based approach [31-33], [46-59] have been used. In the former, interfacial debonding is assumed to occur once the interfacial shear strength reaches a critical value. In the energy-based approach, the debonded interfacial zone is regarded as a tunnel crack which grows in size once interfacial fracture toughness has been overcome at the debond front. However, there is some confusion in the literature concerning the debond criterion, and the stress-criterion appears to be the most commonly used for the analysis of CFCCs. In our computation, the interface debonding event was assumed predictable from the maximum interfacial shear stress 1 T;r I. Interface debonding occurred when 1 T;r 1 exceeded the critical value, Tir, i.e.

(17)

The debonding shear stress, Tir, was assumed to be greater than the value of the thermally induced shear stress; hereafter, the value of Tir= 150 MPa is used. Interfacial debonding by transverse stress [1,51] was not considered in this analysis. Debonding by transverse tensile stress at the interface was neglected, thus the debonding only occurs when the maximum shear stress at the interface exceeds


a critical value. This critical shear stress value was selected from preliminary experimental results [25]. When I T;r I exceeded the critical value of shear stress at the interface, the interfacial bonding condition was changed to a sliding condition by applying frictional sliding elements after releasing the no des between the fiber and matrix. For a sliding interface, relative sliding at the interface was allowed, however, the sliding interface had a frictional sliding resistance.

The frictional shear sliding stress along the debonded interface in the z direction was given by the product of the friction coefficient, J-l, and the clamping stress, (]"~(Rf, z), according to Coulomb's friction law as

(18)

A gap element was used for the sliding interface with a coefficient of friction J-l = 0.2 [49].

9.3.4 Thermal Stress Distribution in Pushout Specimen

For the computation, a model composite system of 8iC (8C8-6, Textron, Lowell, V.8.A) fiber-reinforced borosilicate glass matrix (Pyrex™, Corning, New York, V.8.A) was selected, because it has been studied in detail for basic understanding ofthe pushout problem. The fiber volume fraction, Vi, was set to 0.1,0.5, and 0.7. The properties of fiber, matrix, and composite are shown in Table 2 [50-52]. The same value of Young's modulus of the fiber was used for the longitudinal and transverse directions because a preliminary computation showed only a slight influence of this modulus on the stress distribution.

Table 2. Properties of fiber and matrix.

Young's

modulus

E (GPa)

SiC fiber(SCS-6) Et =406

Boro-silicate glass Ern = 70

Composite

Vi- = 0.1 Ec,z = 104

Ec,r = 76

Vi- = 0.5 Ec,z = 238

Ec,r = 119

Vi- = 0.7 Ec,z = 305

Ec,r = 166

Poisson's

ratio

l/

l/rn = 0.15

l/rn = 0.2

l/c,zr = 0.20

l/c,rz = 0.15

l/c,zr = 0.18

l/c,rz = 0.09

l/c,zr = 0.17

l/c,rz = 0.09

Coeffieient of

thermal expansion Q (xl0-6 K- 1 )

Qt,z = 5.0

Qt,r = 2.63

Qrn = 3.25

Qc,z = 3.92

Qc,r = 3.05

Qc,z = 4.74

Qc,r = 2.63

Qc,z = 4.88

Qc,r = 2.48

Figure 11 shows the distribution of thermally induced stresses at the interface of the composite specimen as a function of the axial coordinate for three different


(a) Vf;O.l 0

a Uz Ur

= 200 " ~.

~ ~ 400

~ .§ ~ 600 e-S '" .::

800 ~ = ~

~ 1000 ~ ~ ~ ~ ~ ~ ~ ~ ~ '"I

Thermany induced stress, u, r(MPa)

(b) Vr; 0.5

a ;t

) Uz I' Ur / U" "\! ~zr ~ '"

200 '" ~ ~ ~ 400

.~ e- 600 S '" .:: '" '" 800 = !! ä ':>

1000 "- ) :\ Q


(c) Vf;0.7

a ;t ~ Ur U"

\' , 'Czr

~ '"

200

'" ~ ~ ~ 400

S .~

e- 600 S '" .:: 1l 800 = !! ä i i

, ,

"- l~ 1000

Q


Fig. 11. Distribution of thermally induced stresses at the fiber-matrix interface: (a) Vf = 0.1, (b) Vf = 0.5 interface, (e) Vf = 0.7.


:z

Fig. 12. Sign of the interfacial shear stress.

fiber volume fractions. The stresses, ar , ae, az , and T zr indicate the radial, tangential, axial, and shear component, respectively. The signs of an ae, a z

used in this chapter are taken as positive and negative for tensile and compressive stresses, respectively. The sign of the interfacial shear stress is defined in Fig. 12. As expected, the stress distributions were symmetrical about the mid-plane of the thickness in a specimen except in the direction of the shear stress. However, the stresses were not uniform in the axial direction. Differences in the signs of the shear stress were due to the difference in the direction. The shear and axial stress at the interface should be zero at z = 0 and z = L (L = specimen thickness = 1 mm) as a result of the given boundary conditions. However, both stresses at the interface increased sharply at the surface (z = 0 and L) and never returned exactly to zero. However, the effect was of the order of a few elements and it decreased with larger values of r. This tendency was independent of the fiber volume fraction.

The radial stress for Vf = 0.5 and 0.7 showed tension, indicating that the interface would tend to separate. This result was not consistent with the theoretically calculated value using a single fiber surrounded by a coaxial matrix [53]. The difference is due to the boundary condition. In the analysis of Honda and Kagawa [2,3], the order of the coefficient of thermal expansion and Young's modulus in a radial direction is O:m > O:r > O:c and Er > Ec > Ern; thus, the radial movement of the matrix at the interface is constrained by the outer composite layer. However, the interface does not debond because the interface transverse strength is assumed to be greater than that of the thermally induced tensile stress at the interface. After the onset of interface debonding, the radial clamping stress so on recovers to compression because of the increase of radial displacement of the fiber by the applied force. Stress transfer at the debonding interface by the frictional resistance therefore becomes possible.

The peak stresses in the specimen typically originated near the free surface below 10-70 11m (~ Rr ) of both surfaces d ue to the differences in the thermal expans ion coefficient between the fiber and matrix. The distribution of stresses became a constant value at the distance rv Rr from the surface except for Vf = 0.1. For the composite with Vf = 0.5 and Vf = 0.7, an ae, and az were constant


at Rf < Z < L - Rf and (Jr ~ (Je. In the case of Vi = 0.1, the complex stress fields originated in a thin specimen. Within the same mesh model, the absolute value of the thermally induced peak shear stress also depended on the fiber volume fraction. With increasing fiber volume fraction, the peak shear stress tended to decrease, as shown by 62, 39, and 22 MPa for Vi = 0.1, 0.5, and 0.7, respectively. The difference in the value of peak shear stress originates from differences in the potential of deformation in the matrix phase and the distribution of thermally-induced stress before the load is applied, because the shear stress primarily occurs near the interface and is constrained by the composite phase. A thick matrix phase (i.e. low fiber volume fraction) results in a larger free deformation zone and thus pro duces high shear stress concentrations. This result indicates that the thermally-induced shear stress appears to exist in the as-prepared pushout specimen, and the absolute value of the peak shear stress and its location are both affected by the fiber volume fraction. In Sect. 9.4, the comparison between FEA and experimental results will be discussed.

9.3.5 Stress Distribution During Pushout Process

The shear stress distribution along the fiber-matrix interface for a given applied load is shown in Fig. 13. In the commonly used shear-lag analysis, the peak shear stress occurs at the surface of the specimen. However, in this analysis the maximum shear stress occurs below the surface. This seems to be due to the differences in the applied load condition, i.e. the shear-lag analysis assurnes a uniform pressure over a cross-section of the fiber, whereas the present computation uses a smallload area at the center of the fiber [17,33].

The location of the peak shear stress initially originated from the point of the maximum shear stress, which was produced by the thermally induced stress. The negative value of the peak interfacial shear stress near the pushing surface increased with the increase in applied load because the thermally induced shear stress was in the same direction as the shear stress generated by loading. The location of the peak value tended to shift toward the inner part of the interface with increasing applied load. The shifts in position were originated by the superposition of shear stresses at the deformation of the soft matrix phase produced by the applied load and the thermally induced stress. On the other hand, before the initiation of debonding another peak of shear stress located just above the back surface remained unchanged from the initial state, i.e. was completely insensitive to the applied load. In comparison with the same applied load for the perfect bonded stage, the maximum shear stress near the pushing surface tended to decrease with increase of the fiber volume fraction. The initial shear debonding at the interface occurred about 30 IJ.m below the pushing surface because the peak interface shear stress appeared at this position. After the initiation of shear debonding at the interface, the interfacial shear stress increased and the debonding progressed toward both the pushing surface and the back surface.

As shown in Fig. 13, a peak shear stress above 200 MPa appears at the front of the sliding interface for all the fiber volume fractions. This shear stress value, which is greater than the shear debonding criterion, is due to the statistical anal-


(a), Vr=O.l

Shear stress, "zr (MPa)

(b) Vr=O.5


(c) Vr=O.7 Pa=ON Pa=4N Pa""8N Pa=10N P.~24N P.~30N

e ~ .. t 200

'i: 51 i:l 400

.~ ii' 600

g

'\

'" g g '"

'\1 ,\1 1~ ,~ : .5

i ~ .e

..... ..--i '-- ~

..... \ ... r-:-."

.~ _ ... ~+

g g '" g g '" g g '" g g = fg


275

Fig. 13. Change of interfacial shear stress distribution with increase in applied load (Pa): (a) Vf = 0.1, (h) Vf = 0.5, (e) Vf = 0.7.


ysis employed. For the next increment of the applied load, the interface where the shear stress exceeds the debonding criterion will be debonded. In this analysis, the use of a smaller mesh size resulted in a slight change in the peak shear stress. However, the peak shear stress location did not change. During the sliding process of the interface, a peak shear stress similar to that appearing just below the pushing surface prior to debonding still existed. Its location near the pushing surface at the sliding interface was stable after the onset of debonding; however, it tended to increase with increase of applied load. At the sliding interface the shear stress, which was less than the peak shear stress, decayed at z ~ 2Rf from the pushing surface and reached a nearly constant value. This steady shear sliding stress was independent of the applied load and the stress value was insensitive to the fiber volume fraction. The shear stress in the steady-state sliding region was approximately equal to p,CJr (Rf,z). With a furt her increase of the applied stress, two-way debonding [54], which means debonding from both the top and the back surfaces, initiated near the back surface. Before the initiation of two-way debonding near the back surface, the sign of shear stress near the back surface changed from positive to negative. This change of sign required a relative sliding displacement at the interface in the opposite direction, and the change of sliding direction then required a larger applied load. Finally, the sliding regions joined and the whole interface slided.

Figure 14 shows the relation between normalized applied load and normalized interfacial debonding length; the debonding length increases exponentially with increasing load. This tendency was independent of the fiber volume fraction. Following complete debonding over the whole interface, the pushing out of the fiber occurred from the back surface. To illustrate this, the thin specimen pushout process is schematically shown in

Figure 15. The initial debonding at the interface occurs about 30 I-lm from the top surface, and the debonding at the interface subsequently proceeds both towards the top surface and the back surface, with furt her advance of the interface debonding, debonding at the back surface is initiated (initiation of two-way debonding), and the debonding parts join for complete debonding at the interface. Protrusion of the fiber from the back surface occurs just after the complete debonding; however, this behavior is beyond the scope of this paper because of the applied load control condition.

9.4 Interface Debonding Criterion

9.4.1 Interface Debonding Process: Experimental Approach

As discussed in Sect. 9.3, detailed FEA on the pushout test of CFCC showed that the interface shear stress distribution was sensitive to residual stress fields created by the processing of a composite [2,3]. The result also demonstrated a peculiar stress distribution near the surface of the specimen. This peculiar stress field seems to influence the interface mechanical properties, especially in the debonding event, because debonding usually initiates near the pushing surface. In this study, an interfacial debonding process during a thin specimen pushout


1.0 Vr =0.1

0.8 I ,

v 0.6

0.4

"-l 0.2 '-... ~ 0.0 ..r:f .... Vr = 0.5 Oll ::: 0.8 ~ Oll ::: 0.6 :a ::: Q 0.4 ~ "CI "CI 0.2 ... ~ 0.0 01 Ei Vr = 0.7 S 0.8 Z

0.6 -- Top surrace

0.4 -- Back surrace -- Totallengtb

0.2

0.0 0.0 0.2 0.4 0.6 0.8 1.0

Normalized applied load, Pa I P:'3J.

Fig. 14. Relation between normalized applied load and normalized interface debonding length.

test of SiC fiber-reinforced glass matrix composite was studied by experiment and FEA. A stress-based approach and an energy-based approach are currently available to determine debonding behavior, however, the applicability of these criteria is not weH understood since accuracy of the assumption used for the analysis is not known.

Thus, it is important to know the actual interface debonding behavior. However, only few reports are available concerning interface debonding. Recently, direct observation of interface denbonding behavior has been reported by Honda and Kagawa [3]. The SiC fiber (SCS~6) reinforced borosilicate glass matrix (Pyrex) composite was used for the observation of interface debonding. The average radius of the fiber was ",70f..lm with an approximately 3.5 f..lm thick carbon-rich coating [55]. The composite was fabricated by a conventional vacuum hot pressing technique. The large fiber-to-fiber spacing was chosen to avoid the interaction of fibers; thus the composite used was considered an ideal state the same as the single fiber composite.

Thin pushout specimens were prepared by a conventional mechanical cutting procedure and the surfaces of the specimen perpendicular to the fiber axis were polished by a diamond paste. Final polishing of the surfaces was done using

278

Load

s 1 B d

(a) Initial debonding

(d) Debonding at back urface

Numerical Analysis of the Interface Problem

(b) debonding at top surface

(e) Complete debonding

(c) Progress of debonding

t

(f) PliS out of fiber

Fig. 15. Schematic illustration of interface debonding process by pushout test.

mm diamond paste. The specimen was ,...., 4 x 4 mm square and the thickness, L, was 1-2mm, which corresponded to the fiber aspect ratio of,...., 7-14. Figure 16 shows the shape and dimensions of the pushout specimen subjected to the pushout test. It contained a single fiber in the central position and the nominal fiber volume fraction, l/f, was ,....,0.001. The specimen was dried before the test to minimize the effects of water on the results [56].

Figure 17 shows a schematic illustration of the homemade pushout test equipment; the basics have been reported elsewhere [19]. The composite specimen was mounted on a 0.5 mm diameter hole which was machined into a support block, allowing the fiber to be pushed out without resistance from the block. The sampIe support block was put on a load cell holder and the holder was secured to a three-axis (x, y, z) translation table, which allowed positioning of a fiber in the composite beneath the indenter. A hard steel spherical indenter (Young's modulus, E = 550 GPa) with a tip radius of 50 I-lm was used.

Pushout tests were done in air at room temperature (298 K) and a load was applied by moving the z stage upward at a constant displacement rate of


RT

L

a: -4mm b: -4mm L: 1-2mm

Fig. 16. Shape and dimension of the pushout test specimen.

AE[nn~uar--------------~

Fig. 17. Schematic drawing of pushout test equipment.

279

5 X 10-7 m S-1. The applied load was measured by a load ceH which was attached to the holder. The displacement of the fiber surface was continuously monitored by a reflective type laser displacement meter. The reflective target plate was attached to the arm of the indenter. The acoustic emission (AE) event count rate (fuH scale of 50 counts/s) during the pushout experiment was measured. An AE transducer whose resonance frequency was 150 kHz was attached to the load cell holder and the signals from the transducer were amplified and counted. The load, the displacement of the fiber surface and the AE event count rates were stored in a digital memory scope and later transferred to a personal computer for furt her analysis. After the test, the measured displacement was calibrated


from the load dependence of compliance of the pushout equipment, which was obtained from pushing of a sintered Sie plate (E = 400 GPa).

In-situ observation of the interfacial debonding and sliding behavior during the pushout test was carried out using a video microscope, and the change of reflected light at the interface was stored in a video tape recorder. The selected images were processed with image analysis software.

80r---~--.----r---r--~---.--~---.

-210 -200 (!1m) : .. • -t 400 600 800

Distance from specimen surface, z (11m)

Fig. 18. Change of relative intensity of reflected light along the interface with increase of applied load.

Table 3. First AE-detected load and corresponding peak shear stress at the interface.

Specimen First AE detected load

thickness (first debonding load)

L

1.20 1.47 1.72 1.85

1.7 2.0 1.6 1.5

Peak shear stress by FEA

TP(MPa)

89

96 87

87

The results of the image analysis are shown in Fig. 18. The relative intensity of the reflected light is given by the subtraction of the intensity at the corresponding applied load from that before the loading. When the applied load, Pa, reached ,,-,2 N (b), there was a sudden enhancement of reflected light near the top surface. The load was nearly the same as that for the first detection of an AE event


count (Table 3), and we therefore define the load listed in Table 3 as the first debonding load at the interface. The first sharp increase of the reflected light was at the distance from the surface of rv 120 ~m (rv 2Rr), according to the results of the image analysis, and was typical for the tested specimens and independent of the specimen thickness. The debonding at the interface is initiated at an early stage of the loading and the first debonding length is rv 2Rr. After the first rapid debonding, microscopic and local increases in the white part at the tip of the debonding front occurred at a unit length of rv5 ~m (hereafter denotes microdebonding). The unit length of the micro-debondings was nearly the period of the surface roughness of SiC fiber.

With furt her increase of the applied load the white part increased. However, this increase was not continuous but spontaneous, with adebonding step length of rv200 ~m (rv 3Rr), as shown in Fig. 18. Micro-debondings were observed at the front of the rapid debondings during each interval of them. After repeating three or four debonding steps, the entire interface changed to white, i.e. it completely debonded. Settlement of the fiber surface in the matrix was seen by the video microscope after the complete debonding. Even after this complete debonding the applied load still increased up to the maximum. Displacement of the fiber surface for the onset of debonding was rv3 ~m (Pa i'::j 2 N) and for the complete debonding of the whole interface it was rv8 ~m (Pa i'::j 5 N). Most of the AE signal was detected during the load range of rv2 to rv5 N. The change of the light reflected from the interface and the AE signal clearly demonstrated the spontaneous interface debonding over this load range. The spontaneous interface debonding process of SiC-glass composite is also reported in a pullout condition [50].

These results suggest that the debonding does not involve the asperities sliding past one another, and the fiber surface displacement up to the complete debonding is mainly due to the elastic deformation of the fiber and matrix. An interfacial sliding occurs after the complete debonding, and the applied load reaches a maximum value after an interfacial sliding of rv2 ~m, which corresponds to the average surface roughness of the fiber. This seems natural because the debonded interface is rough and frictional force is carried by the rough surface contact at the interface as reported elsewhere [58,59]. Further, from the surface morphology of the fiber, the micro-debondings are believed to occur when the interface between fine grains and the matrix debonds. The spontaneous debondings occur in succession by the interfacial debonding of rv200 ~m in length at nodules and matrix. This indicates how the roughness of the fiber surface affects the debonding process at the interface. However, the explanation of this peculiar debonding behavior needs furt her research, including the effect of shear-frictional resistance behind the crack tip.

Summarizing the experimental results, the interfacial debonding process during the pushout test is illustrated schematically in Fig. 19. An interfacial debonding (debonding length, ld i'::j 2Rr) is initiated at an early stage and the microdebonding takes place at the debonding front. The unit interfacial debonding area (ld i'::j 3Rr) increases spontaneously until the entire interface completely


ad

Fiber ___

I t deboDdlng Ud ... ZRr)

Rapid debondiDg Ud ... 3Rr)

Micro debonding

Rapid deboDding Ud ",3Rr}

(I d: nit leD tb or d bondill event)

Fig. 19. Schematic drawing of the interface spontaneous debonding process during the pushout test (lcl: debonding length).

debonds with the increase in applied load, and micro-debonding occurs during each stage of the macro-debonding. After complete debonding dose to the maximum applied load, all interfacial slides and the applied load continue to increase to the maximum, probably due to interlocking of the asperities [60-63]. Details of the mechanism after the onset of interface debonding will be reported by the authors. In the composite used, interface debonding from the back surface [64,65] is not considered in the analysis.

9.4.2 Effects of Thermal Stress on Interface Debonding

Using FEA, the effects of thermal stress in the specimen could be discussed. The shear stress distribution before and during the pushout test for a specimen with a thickness of L = 1.0 mm were obtained by the FEA method. Figure 20 shows the normalized interfacial shear stress distribution before the pushout test, i.e. due to the thermally induced stress. The thermally induced shear stress is normalized by

(19)


"" 0.0

'" .r ~ 0.2 = '" 5 e 0.4 ·1 ., e Q 0.6 .::

1.0 '-~---'-~--'--====---~---' -0.2 -0.1 0.0 0.1 0.2

Normalized shear stress, r(z)/u,

283

Fig.20. Normalized interfacial shear stress distribution due to the thermally induced stress.

where Tl and To are the processing temperature and room temperature (Tl > To), respectively, and af, a m are the coefficients of thermal expansion (CTE) of the fiber and matrix, respectively, and Ef is the Young's modulus of the fiber. The vertical axis shows the distance from the specimen surface normalized by the specimen thickness, L. As the used materials are linearly elastic, the stresses are proportional to LlaLlT, namely the normalized interfacial shear stress distribution is independent of LlaLlT. The distribution of the thermally induced shear stress in the specimen is completely symmetrie against half of the specimen thickness, and the distribution is non-uniform through the axial direction. The peak shear stress originates rv 40 J.lm below both surfaces of the specimen. A similar shear stress distribution was obtained for the calculated specimen thickness of 1.0-2.0 mm. The origin of the shear stress peak near the free surface is due to differences of displacement at the fiber-matrix interface due to the mi sm at eh of the thermal expansion coefficients between the fiber and the matrix for both radial and longitudinal directions.

Figure 21 shows the normalized interfacial shear stress distribution during the pushout test. The shear stress is normalized by the applied stress, (Ja (= Pa / 1f Rn. The stresses are proportional to the applied load because of the linear elastic materials; thus the normalized shear stress distribution is unaffected by the applied load. The interfacial shear stress has a peak near the pushing surface at almost the same position as the peak position of the thermally induced shear stress. In linear elastic materials, there is a principle of superposition for the interfacial shear stress, T(Z), the thermally induced interfacial shear stress, Tt(Z), and the interfacial shear stress induced by the applied load, Ta(Z) as follows

(20)


0.0

~

" ,; '" 0.2 :! ... = '" = .. .~ 0.4 .. =-'" 8 0.6 = .::: .. '" = ~ 0.8 :a "CI

~ 1.0 0:1

! -0.3 -0.2 -0.1 0.0 = Z Normalized shear stress, r(z) 117.

Fig. 21. Normalized interfacial shear stress distribution due to the applied stress.

If we consider that debonding is initiated when the peak value of T( z) reaches a critical value, T d , at z = zo, the debonding condition is given by

T (zo) = Tt (zo) + Ta (ZO)

~ T d . (21)

With the increase in applied load the absolute value of the peak shear stress near the pushing surface increases, because the distribution of the shear stress due to the applied load is the same as that of the thermally induced shear stress. Another peak of shear stress just above the back surface decreases with the increase in applied load because both directions are reversed. This trend of shear stress distribution was independent of the specimen thickness used in this study.

Equation (20) means that the debonding load depends on both Tt(Z) and Ta(Z). In the composite used, the thermally induced shear stress, Tt(Z), is about 1/3 the value of the total interfacial shear stress, T(Z). Thus the thermally induced shear stress is considered to play an important role in an interfacial debonding.

9.4.3 Interfacial Debonding Criterion

The applied load for the onset of debonding at a bonded interface (first debonding load), P~, for different specimen thicknesses, L, is listed in Table 3. As discussed earlier, the first debonding load occurs in the applied load range from 1.5 to 2.0 N and is independent of specimen thickness. As shown in Fig. 21, the maximum shear stress occurs rv40-50 Il-m below the pushing surface due to the


thermally induced stress, and the peak shear stress increases with the increase of the applied load. The peak shear stress, T P , at the interface at each first debonding load, P~, shown in Table 3 was calculated by FEA. Table 3 indicates that both P~ and T P are independent of specimen thickness and alm ost constant: T P = 85-95 MPa for P~ = 1.5-2.0 N. Thus, it can be assumed that the debonding at the interface is initiated when the peak shear stress at the interface reaches a critical value, T d .

-140 --~ ~

-120 Debonding-initiation range obtained '-' c. from pushout experiment 4 .7; I ~

-100 ~ - ~~ ~

'"' 0: .. -80 ..= I ~ • Experiment ~ 0: -0- L=l.Omm .. I Q.. ~ L=1.4mm ..... 0

-0- L=1.8mm .. .: 0:

> 0 1 2 3 4 Applied load, Pa (N)

Fig. 22. Relationship between peak shear stress and applied load. The solid circles show experimentally obtained debonding stress. This stress is defined as the first AE detected applied load.

Figure 22 shows the relation between the peak shear stress, T P , and the applied load, Pa, with the change of specimen thickness, L. The black-filled circles are plots of P~ versus T P shown in Table 3. The value of T P at Pa = 0 N (TP ~ 65 MPa) is the peak value of the thermally induced shear stress before loading. This value is almost independent of the specimen thickness in the range of computation. Thus, from Fig. 22, a simple correlation represented as the linear function

T P = 15.3Pa - 64.8 (MPa) (22)

is obtained between the peak shear stress and the applied load. The peak shear stress at the initiation of the interfacial debonding (the interfacial debonding shear stress, T d ) can be evaluated, substituting the applied load, P~, at the initiation of the interfacial debonding (obtained from the pushout experiment) for the above equation. In this composite system, T d = 85-95 MPa is obtained using P~ = 1.5-2.0 N by the above described procedure. An interfacial debonding shear stress in a composite taking into consideration the distribution of the thermally induced shear stress can thus be estimated using both a pushout experiment and FEA.


9.4.4 Effect of Thermally Induced Shear Stress on Interfacial Debonding

The interfacial shear stress, T(Z), is given by (22), and the absolute value of T(Z) is affected not only by the absolute value of Tt(Z) and Ta(Z) but by their sign. The sign of the thermally induced shear stress, Tt(Z), changes according to the relation of CTE between the fiber and the matrix, and the sign of the shear stress induced by the applied load, Ta(Z), depends on the applied load direction. Both of these sign factors should thus be considered in evaluating an interfacial debonding criterion. Figure 23 is a schematic illustration of the direction (sign) of Tt(Z)) and Ta(Z) in four cases by combinations of pushout (push-in), pullout and Cl:f,z > Cl:m , Cl:f,z < Cl:m •

Pu bout (Pusb.in)

Pu bout

J t: Dirtdio.n or th.rmall)·-induced sbt:lr ,,' .

• f : D1rtttlon or hm- Iodu«d by appllotllo:wl, "'.

Fig. 23. Schematic drawing of the directions of the stresses due to the thermal mismatch strain and due to the applied load in four combinations of pushout and pullout conditions.

Figure 24 shows the relation between the normalized shear stress of T(Z)j(Jt and T(Z)j(Ja in the same four cases as Fig. 23. In all cases, the peak of the shear stress occurs near the pushing (pulling) surface because the peak of the shear stress due to the applied stress occurs at this position; interfacial debonding is therefore initiated at the same position. In the case of af,z > am , the direction of the shear stress near the pushing (pulling) surface due to the thermally induced stress and the applied load is the same for pushout, and the opposite for pullout. Therefore, the applied load required to produce the critical debonding shear stress in pushout is smaller than that in pullout. This relation is the opposite


ll'r,z > ll'm 0.0 "'-::::=:::;':==-T~~, 0.0 "'-::::===0-==,""",

~ l·····-...i i/i (a)

Pushout (Push-in)

"'0.2 '.: 0.2 :!

i 0.4 \,: j :ä i 0.4 I ~ ~ U ~.' .~ 0.6 :'. -~ '\ .;:, 1\

~ / ' U i \ U ,

i i ...... ;. . . .li i 1.0 L....._--'2.0~--'----"LJ...O ~"--'-...J2.0 1.0 -2.0 0

Normalized shear stress, 'C(z)/Ut" 'C(z)/u. 2.0

0.0 ..--;z=---r~==...,......, 0.0 ,.-~~~"""=...,......,

(b)

Pushout

.... : "-4 ". : ~O.2 \~~ ~ \~ ~ 0.4 \ ;a ~

0.2

0.4

~ ! .~ 0.6 0.6

~ ~ 0.8 0.8

\., ..• \

, .. , .' t // : ! :! :!

{ I

/ // :

/ : 1.~2.0 0 2.0 1.~2.0 0 2.0

Normalized shear stress, 'C(z)/Ut, 'C(z)/u.

-----... Normalized shear stress by thermaDyindused rtress, 't(z)/Ot

-- Normalized shear streu by applied streel, 't(Z)/Oa

287

Fig. 24. Relation between the normalized shear stress due to the thermally induced stress and the normalized stress due to the applied stress in four combinations of pushout and pullout conditions.

for af,z < am, i. e. the load at the onset of the interfacial debonding in pushout is larger than that in pullout. From these considerations, if a thermally induced stress is not taken into account in the case of af,z > am , the interfacial debonding stress is underestimated for pushout and is overestimated for pullout. In the case of af,z < am , in contrast, the interfacial debonding stress is overestimated for pushout and underestimated for pullout.

9.5 Conclusion

The interface mechanical properties in fiber-reinforced ceramic matrix composites play an important role in their mechanical performance. In Sect. 9.1, the interface debonding mode of a continuous fiber ceramic matrix composite (CFCC) has been discussed together with the crack~fiber interaction process. Interface evaluation by the pushout test has been discussed in Sects. 9.3 and 9.4. In Sect. 9.3, details of stress distributions in the composite specimen were presented. The analysis included the processing conditions of the composite specimen. Based on the processing results ofthis section, some important factors which should be


considered were discussed. In Sect. 9.4 details of experiments and analysis were compared. In that section, we discussed interface debonding behavior and the evaluation of interfacial mechanical properties in CFCC.

The interface debonding criterion difIers with the fracture mode; thus it is important to agree on the debond criterion before the discussion. The evaluation of the debonding and post-debond frictional resistance are also important subjects. According to the discussion in this chapter, some guidelines for the prediction of the interface debond criterion by a pushout test have been given and the procedures give important information for a practical application. Thermal stress is another important factor which should be considered for the evaluation of interface mechanical properties. The FEA method is an efIective tool for the analysis of interface problems in CFCCs because the method makes it possible to derive the distribution of stresses in the composite, which is usuallydifficult to obtain analytically.

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Index

accelerated cooling 224 adatom diffusion 26, 32, 34 adiabatic approximation 3 0:+0:2 heat resistant titanium alloys 72 o:-ß phase equilibrium 71-74,80 o:-ß titanium alloys 71 0:/0:2 phase boundaries 93-94 0:2 phase 71, 86-87, 89-90, 93, 156 Al site 58, 60-61, 63-64, 66 Alloy454 47 alloy design program (ADP) 51-52,

59-60, 64 alloying element (Al, Co, Cr, etc.) 39,

41-42, 56, 58, 63-66, 99, 107, 128,130, 137-140, 144-149, 191, 221-222

ANISN 181, 191 anisotropy of interfacial energy 216 antiphase boundary 91 atom-probe field ion microscopy

(APFIM) 39, 59-61, 64, 69 atomic arrangement 26, 35-36, 43, 69,

106, 219-221 atomic configuration 14, 17, 60, 62,

66-67, 72, 91, 93, 97, 102 atomic displacement 166 atomic pairs 10,97-100, 102 atomic volume 57, 93, 172 augmented plane wave (APW) 5

B1900Hf 47 bainite 144, 223-224 Becker-Döring theory 200-202 ß-stabilizer 73, 83 ß-surface equation 73-75, 78 binary alloy 46, 122, 195-197, 213,

219-220

binary collision approximation 163-165 blanket 167, 177-188, 191 Boltzmann constant 56, 62 Boltzmann transport equation 180 bond order (BO) 42 boride(s) (M3 B2 , MsB3 ) 42, 48-49 136 boundary conditions 62, 201, 203-204,

212, 216, 220, 227, 234, 247-248, 252, 254, 268-269, 273

breeder 180, 186, 192 bubble size 247-248, 253-254 burner rig test 50

Cahn-Hilliard equation 213-214 CALPHAD 70, 105-108, 111, 122, 127,

130-132,135-137 carbide(s) (MC, M6 C, M23 C6 ) 42-43,

48-49,51,136,148-149,151-153 carbide dispersion carburizing 157 carbon steels 71-72, 91, 97, 99-100, 102 carbonitride 149-150, 223 central atoms model 72, 91, 96 chemical potential 22--24, 57, 75, 77,

93, 106-107, 109, 112, 137, 196-197, 211, 213, 215

climb velocity of dislocations 100 climbing of dislocation 101 cluster analysis method 217, 220 cluster variation method (CVM) 39,43,

46, 54, 67, 72, 91, 102, 219-221 CM186LC 55 CMSX-(2,4,10) 48-49, 53, 55, 59-62,

64-66, 68, 220-222 coherency 53, 59 collision cascade 163-167 completion formula 207

292

compositional parameters 81, 83~84, 86, 102

compound semiconductors 21, 105, 112~114, 116

computer networking 66 configurational entropy 56, 91 continous-transformation-time (CCT)

diagram 223 continuous fiber ceramic composites

257 conventionally cast (CC) 39, 49 cooling rate 224~225

CoPt3 10, 12 crack-fiber interaction 258, 265 creep properties 72, 90, 97 creep rupture data 48 creep rupture life 41, 49~50, 52~53, 66,

68, 99~101 creep rupture strengths 52 creep rupture temperature capabilities

54~55

creep strength 39, 41, 48, 50~51, 54~55, 71~72, 89, 91, 97, 99~100, 102

critical droplet 199 critical shear stress 271 critical stress intensity factor 260 CrPt3 10~14

crucible test 50

D019 87, 91~94, 102 database 39, 43~44, 48, 51, 54, 66~67,

135, 142, 146, 151 ~ 152, 154, 159~ 160, 191

decomposition of alloys 213 density 1 ~5, 12~ 13, 24~26, 49, 52,

71~72, 84~85, 102, 122 169, 172~173, 250, 253

density functional theory 1 ~5 developed titanium alloys (GT alloys)

81~86, 88~91

diffusion constant 1, 32~35, 212 diffusion controlled growth 195~ 196,

209,213 diffusion equation 209, 213 diffusional growth in multicomponent

system 212 directionally solidified (DS) 39, 54 discontinuous coarsening 53 displacement 8, 29~31, 162, 164, 168,

177~178, 183, 205 230~233, 235,

Index

247~248, 268, 270, 273, 276, 278, 281, 283

distribution coefficient 133, 136, 146 distributions of the number of edges

218 dpa 170, 173~17~ 181~187, 191~192 driving force 105~196, 137, 159,201,

216 droplet model 199~200

dual phase steel 144 duplex stainless steel 154--155

effect of processing variables 224 effect of strain energy 119, 199 electron counting model 21 electron probe microanalysis (EPMA)

47 electron vacancy number (Nv) 42 electron-atom ratio 71, 83-84, 102 electropolishing 59 elongation 82, 84~86, 89~90, 128,

236~237, 239, 242~244, 246, 248, 250~254

enhanced CG 69, 171 enthalpy 56~57, 93, 105, 116, 111, 123,

135, 137, 220 equilibrium atomic configuration 67 equilibrium between "( and "(' 56 equilibrium calculation 43 equilibrium state 57, 59, 198 eutectic reaction 125, 127, 138, 140

F-test 50 Fe4 N 8~10

Fe16N2 1~2, 6~1O

Fe-C System 142 Fe-Cr-Co Alloy 163 Fe-Cu-Sn system 137 Fe-Mn-Si system 159 Fe-MnS pseudo-binary 138-139 Fe-Ni-AI-Co alloy 155 FePt3 1O~12

Fermi-contact terms 9~1O, 97~100, 102 ferrite 9, 71 ~72, 76, 91, 97~ 100, 102,

137, 144, 148~ 149, 154, 157, 223~224 ferrite-bainite structure 224 ferritic steels 71~72, 97, 182~183 finite element analysis 227, 257 finite element method 228, 247, 254 fiow chart of analyzing program 203

Index

flow chart of searching program 54 fluctuation 54 Fokker-Planck equation 201,203 Frenkel pairs 164, 167 frictional shear sliding 265, 271 full-potentiallinearized augmented plane

wave (FLAPW) 3-5,7-8, 11 fusion reactor 163-164, 167-169, 175,

179-181, 187-188, 191-192, 246

(GaAsh-x(Ge2)x 14-16 GaAs(OOl) surfaces 1, 3, 21-23, 26, 32,

34-35 GaAs/Ge hetero-valent superlattices 2,

16-17 "( 1"(' interphase interface 41, 62-63 "( and "(' phase composition(s) 39,

42-43 "(' hypersurface 44, 46 "(' phase 39, 41-46, 49, 53, 56-57, 59,

61-64,66,71-72 "(' phase fraction 41-43 giant magnetic moments 2 Gibbs free energy 107, 113, 196 Gibbs-Duhem relation 198, 211 Gibbs-Thompson relationship 207 Ginzburg-Landau model 205 grain growth 196, 204, 214-215, 219,

223 grain size distribution 218 grand potential 57, 93 growth law 217

hard metal 151 helium embrittlement 246-247, 252,

254 Helmholtz free energy 196, 198 high strength low alloy (HSLA) steels

144, 222 high temperature structural materials

91 high-speed steel 152 hot corrosion resistance 49-51 hydrogen-terminated Si(OOl) surface 1,

3,26 hyperfine fields 6-7, 9-10

I-creep 174-175 IMI(685, 834), 90

293

IN (Inconel) alloy (IN 700, IN 713C, etc.) 47, 49-50

incipient melting temperature 49 incubation time 222 induced radioactivity 163, 187-188,

190, industrial applications 222 inherent creep strength 97 inhomogenous system 213 interaction parameter 76-77, 87, 98,

109, 112, 116-118, 127 interatomic distance 57, 164 interatomic potential 37, 39, 43, 60 interface debonding length 277 interface dynamics 204 interface motion 195, 214-215 interfacial debonding process 276, 281 interfacial failure mode 265 intermetallic compounds 10, 71-72, 91,

94, 96, 102, 127, 129, 251, 253 international collaboration on superalloys

66 interstitial atom 165, 168-169 interstitial bias 172 iron-base alloys 135 irradiation creep 163-164,168,170-171,

173-176 ITER 174,177-181,190,192

jet engine 39, 43, 54, 81

Kawasaki dynamics 219 kinetics of ordering 60 Kolmogorov-Johnson - Mehl-Avrami

type formula 208

L10 90-94, 102 Llz ordered structure 41, 93, 220 lattice misfit 41, 43, 50, 52-53, 55, 58,

60, 62 lattice parameter 39, 43, 49-50, 56-57,

77, 93, 116, 149, 220 layer-by-layer model 258 Lennard-Jones potential 56-58, 60, 67,

91, 93-95, 102, 219-220 linear irreversible thermodynamics 213 liquidus temperature 49 lithium 180 local spin density approximation 3, 5 long range order parameter 91

294

long term creep strength 71-72, 91, 97, 99-100, 102

low temperature irradiation creep 174-175

m-value 82-83 f.l phase 66 Md-electron level 43 MA6000 55 magnetic circular x-ray dichroism (MCD)

2, 6 magnetic multilayer and film systems 2 MarM alloys (MM200, MM247CC, etc.)

55 matrix 18-19, 41, 66, 88, 97, 99-100,

102, 110, 144-145, 148, 151-153, 156-157, 167, 172-173, 198-201, 209, 212, 216, 222, 227, 230-232, 251, 254, 257-268, 271-274, 277, 281, 283, 286, 289

maximum flow stress 82-83, 86 mesh 62, 228-229, 235-237, 259-260,

266, 274, 276 mesoscopic modeling 214 metastable system 123, 139, 197 Method A (for designing multi-

component titanium alloys) 71, 73-74, 78, 85

Method B (for designing multicomponent titanium alloys) 72, 74, 77-80

Method C (for designing multicomponent titanium alloys) 72, 74, 78,80-81

Metropolis algorithm micro-alloying elements microsolder 122, 136

215 148-149

microstructural control 135, 146, 159, 214

microstructural entropy 218 microstructural evolution 37, 60, 69,

195-196, 198, 215, 217 microstructural parameters 43, 51, 82 miscibility gap 114, 116-118, 149-150,

156-157 MM200Hf 40 Mn-C and Mo-C atomic pairs 99-100,

102 MnPt3 10, 12, 14 MnS 138-142

Mo-Re cluster 66-67 Mo-W cluster 66

Index

molecular dynamics 34, 163-164,227 Monte Carlo simulation (MCS) 3, 37,

39,60, 69, 137, 181, 196, 214-215, 217, 220-221

Monte Carlo step 62, 215-216 morphology of MnS 140-141 Ms Temperature 144,146-147,161 muffin-tin approximation 5 multi-component 37, 43-46, 52-53, 58,

60, 66-67, 71-74, 81, 84, 87, 102, 135, 219-220

multiple correlation coefficient 46 multiple regression analysis 72, 74, 83,

99

N80, N80A, N90, etc. 40, 48 NASAIR 100 58, 62 near net-shape turbine disk 86 nearest neighbor atom shell 98 nearest neighbor coordination number

57, 216 needle-like sampIes 59 negative lattice misfit 50, 52-53, 55 neutron irradiation 163, 178-179 neutron spectrum 163-164, 179-180,

183-184, 186, 189, 191 Ni site 58, 61, 64, 66 Ni3AI 41, 62, 64 Ni-Al binary system 219-220 Ni-AI-X ternary system 43, 220 Ni-base superalloys 39-44, 51, 54, 60,

67, 69, 74, 219-220 non-metallic inclusions nuclear data 179 nucleation kinetics 198 nucleation processes 26

137, 141,143

nucleation rate 198, 200-203, 208, 222-223

one-electron approximation 3 Onsager reciprocity theorem 210 open laboratory for materials design

(OLMD) 39,66-69 optical interband transitions 18 orbital-moment and spin-moment sum

rules 2 order parameter 91, 196, 205 order-disorder phenomena 91

Index

oscillator strength 18-20 oxidation 41, 82, 144, 253 oxide dispersion hardening (ODS) 54 oxygen 72, 77-79

partitioning behavior, tendency 64-66 partitioning coefficient 43, 46-48 partitioning ratio 51, 58 Pb-free solder 122, 128-131 pearlite 223-224 periodic boundary condition 62, 216,

247-248 peritectic reaction 137, 139 Phacomp (phase computation) 42, 48 phase boundary 107, 121, 123-124 phase diagram 43, 45-46, 48, 71, 73, 76,

79, 102, 105-106, 112, 114, 120-125, 127-128, 130-133, 135-139, 144-145, 148-149,151,154-160,197

phase equilibrium 43, 51, 71-74, 78, 80, 87, 91, 93, 102, 10~ 130, 137, 143-144 146, 219, 222

phase transformation 124, 136, 195, 197-198, 207, 214, 222-223, 226

phenomenological coefficients 210 Platinum 3d-transition-metal inter-

metallics 2, 10 Poisson's ratio 119, 172, 261, 268 Potts model 215 powder metallurgy processed (PM) 49,

54, 152, 227, 251 precipitate 41-42, 66, 93, 148, 151, 155,

157, 199, 201, 209-210, 215, 222, 227 preferential substitution site 71-72, 91,

94-95, 97 primary a: phase 83-84, 102 primary knock-on atom 164-165 principle of detailed balance 201 prior ß phase 71, 83-84, 102 pseudopotential method 2, 14 pushin test 265 pushout test 257-258, 265, 267, 276,

278-283, 287-288 PWA(1480,1484) 53

quantum wells wires and dots, 2, 14 quantum-confinement effects 17

radiation damage 163-164, 166-167, 171, 192

295

radiation-induced deformation 168-169, 172, 177

rafted structure 50 regression analysis 44, 46, 48, 51, 72,

74-75, 83, 89, 99 regression coefficient 50 regression equation 37, 43-44, 47-48,

54, 67, 72, 81, 83, 99-100 regular solution model 72, 74, 76, 87,

102, 106-107, 109, 113, 133 relaxation effect 62 Rene( 41,80,95,N5,N6) 48-49

second phase 209, 215, 224 shape memory alloys 155, 159 shear stress distribution ahead of matrix

crack tip 260 Si quantum wires 1-2, 14, 17 silicon carbide(SiC) 180, 182-185,

187-188, 191, 261, 267, 271, 277, 280-281

Silicon-based light-emitting devices 17 single crystal (SC) 37, 49, 58-59, 67 SIPA 169, 171, 173 SIPN 169,171-174 site occupation, site occupancy 56, 59,

61-62, 64-66, 219 sliding interface 266-267, 270-271, 274,

276 solid solution 41,47,71-72,83-84,93,

102, 112, 116, 118, 133, 146, 198-199 205, 213, 220

solidification 135-139, 141, 143, 152 solidus temperature 49, 129 solubility (in the ferrite matrix) 73, 97,

99, 144-145, 148, 155 solubility product 148 solute elements 59, 61, 66, 71, 74, 78,

84,97,99-100, 102, 137, 146 solution index (SI) 52 solvus temperature 48, spinodal decomposition 156, 197-198 spinodal magnets 155-157 stainless steel 135-136, 143, 154-156,

170, 173-174, 176-177, 180,235, 251-253

statistical thermodynamics 43, 54, 134, 199

steady state nucleation rate 201-202 strain rate 82-83, 85-86, 173

296

strength to density ratio (strength-todensity ratio) 72, 84-85, 102

stress relaxation 163-164, 168-170, 175-177

stress-strain relationship 227-228, 235, 238, 247-248, 250-254

structural materials 91, 163-164, 178, 186-187,191, 195,214

Sturm - Liouville equation 203 sublattice model 72, 77-78, 87, 98, 106,

113 superplastic forming 72, 81-82, 86 superplastic properties 81-83, 86 surface energy 23-24, 26, 131-132, 199,

201,205 surface fissure 143-144 surface tension 123, 131-132, 134,201

t-test 50 temperature capability 39-40, 53, 55 tensile properties (titanium alloys) 83,

85-86, 89-90 tetrahedron cluster 90, 92-93 tetrahedron nearest neighbor atom

cluster 56 thermal expansion coefficient 267, 269,

283 thermal mismatch strain 286 thermally induced stress 257-258, 266,

268-269, 272, 274, 282-283, 285-287 Thermo-calc. 43, 98-99 thermodynamic database 134, 146, 151,

159 thermodynamic force 205, 210-211, 213 thermodynamic modeling 106 thermodynamics 43, 54, 71-72, 91, 102,

134, 195, 199, 210, 213, 225 thermomechanical control process

(TMCP) 222, 224-225 Ti3Al 71-72, 86, 91-94, 96, 102 Ti-6AI-4Valloy 81-82 TiAI 71-72, 91, 93-94, 96-97, 102 tie-line 42-43 time lag 202-203 time-dependent nucleation rate 204

Index

titanium alloys 70-74,77,81-84,86-90, 102

TM alloys(TM-321, TM-49, TM53, etc.) 40, 47-48, 52-53, 55, 58, 60, 62, 64, 193

transformed microstructure 224-225 transient I-creep (TIC) 175 transmission electron microscopy (TEM)

41 transmutation 163, 178-179, 183--185,

191-192 transport theory 182 turbine blade 39

U500 700, 720, 40

vacancy 21, 42, 100, 165, 172-173, 253 vanadium alloys 190 void diameter 235-239, 242, 245, 247,

249-251, 254 void growth 243-244, 251-252 void lattice 228, 235-239, 242-246,

249-251, 254 void lattice aspect ratioo 237, 239, 242,

244-246, 250, 254 void swelling 155, 168, 174 volume fraction of 00 phase 74, 77, 79,

83 volume fraction of 002 phase 87 VPt3 10,12

Wager interaction coefficients 98 Wannier excitonic effects 19 Waspaloy 48 waste disposal 188-189 work hardening coefficient 233 work hardening exponent 235-236,

238-239, 242, 244-245, 247-250, 252-254

world wide web (WWW) 39, 66, 69

X75 48

Young's modulus 172-173, 258, 260-261, 263-265, 271, 273, 278, 283

Zeldvitch non-equilibrium factor 222

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