combustion_synthesisof advanced materials
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CHEMISTRY RESEARCH AND APPLICATIONS SERIES
COMBUSTION SYNTHESIS OF
ADVANCED MATERIALS
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CHEMISTRY RESEARCH AND APPLICATIONS SERIES
COMBUSTION SYNTHESIS OFADVANCED MATERIALS
B.B.KHINA
Nova Science Publishers, Inc.
New York
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LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA
Khina, B. B. (Boris B.)
Combustion synthesis of advanced materials / author, B.B. Khina.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-61324-254-4 (eBook)
1. Self-propagating high-temperature synthesis. 2. Refractory
materials--Heat treatment. 3. Refractory materials--Mathematical models.
I. Title.
TP363.K46 2010620.1'43--dc22
2009052731
Published by Nova Science Publishers, Inc. New York
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DEDICATION
To the memory of Professor Zinoviy P. Shulman (1924-2007) and Professor
Leonid G. Voroshnin (1936-2006) who had taught me the scientific meaning of
old Russian proverb, trust, but verify.
The important thing in science is not so much to obtain new facts as to
discover new ways of thinking about them.
Sir William Bragg
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CONTENTS
Preface xi
Chapter 1 Advances and Challenges in Modeling Combustion
Synthesis 1
Chapter 2 Modeling Diffusion-Controlled Formation
of TiC in the Conditions of CS 13Chapter 3 Modeling Interaction Kinetics in the CS
of Nickel Monoaluminide 39
Chapter 4 Analysis of the Effect of Mechanical
Activation on Combustion Synthesis 75
References 93
Index 105
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PREFACE
Self-propagating high-temperature synthesis (SHS), or combustion synthesis
(CS) is a phenomenon of wave-like localization of chemical reactions in
condensed media which permits efficiently synthesizing a wide range of
refractory compounds (carbides, borides, intermetallics, etc.) and advanced
composite materials. CS, where complex heterogeneous reactions proceed in
substantially non-isothermal conditions, brings about fine-grained structure and
novel properties of the target products and is characterized by fast
accomplishment interaction, within ~0.1-1 s, whereas traditional furnace synthesis
of the same compounds in close-to-isothermal conditions may take several hours
for the same particle size and close final temperature. Uncommon, non-
equilibrium phase formation routes inherent of SHS, which have been revealed
experimentally, are the main subject of this book.
The main goal of this book is to describe basic approaches to modeling non-
isothermal interaction kinetics during CS of advanced materials and reveal the
existing controversies and apparent contradictions between different theories, on
one hand, and between theory and experimental data, on the other hand, and to
develop criteria for a transition from traditional solid-state diffusion-controlled
phase formation kinetics (a slow, quasi-equilibrium interaction pathway) to
non-equilibrium, fast dissolution-precipitation route.
Features:
analysis of the physicochemical background of modeling approaches toCS;
modeling of phase formation kinetics for two typical SHS reactions,Ti+CTiC (CS of an interstitial compound) and Ni+AlNiAl (CS of an
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B. B. Khinaxii
intermetallic compound), in strongly non-isothermal conditions using the
diffusion approach and experimentally known values of the diffusion
parameters;
novel criteria for the changeover of interaction routes in these systemsand phase-formation mechanism maps;
analysis of the physicochemical mechanism of the experimentally knownstrong influence of preliminary mechanical activation of solid reactant
particles on SHS in metal-based systems.
It is anticipated that the book will serve the scientists, engineers, graduate and
post-graduate students in Solid-State Physics and Chemistry, Heterogeneous
Combustion, Materials Science and related areas, who are involved in the research
and development of CS-related methods for the synthesis of novel advanced
materials.
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Chapter 1
ADVANCES AND CHALLENGES IN MODELING
COMBUSTION SYNTHESIS
1.1.APPROACHES TO MODELING NON-ISOTHERMAL
INTERACTION KINETICS DURING CS
Combustion synthesis (CS), or self-propagating high-temperature synthesis
(SHS), also known as solid flame, is a versatile, cost and energy efficient method
for producing refractory compounds (carbides, borides, nitrides, intermetallics,
complex oxides etc.) and advanced composite materials possessing fine-grain
structure and superior properties. Extensive research in this area was initiated by
A.G.Merzhanov in Chernogolovka, Moscow district, Russia, in mid 1960es [1,2],
who is internationally recognized as a pioneer of SHS. The advantages of CS
include short processing time, low energy consumption, high product purity due
to volatilization of impurities, and unique structure and properties of the final
products. Besides, CS can be combined with pressing, extrusion, casting and other
processes to produce near-net-shape articles [3-10]. Despite vast literature
available in this area, CS is still a subject of extensive experimental and
theoretical investigation.
Combustion synthesis can be carried out in the wave propagation mode, or
true SHS, and in the thermal explosion (TE) mode. In the former case, a
compact reactive powder mixture is ignited at one end to initiate an exothermic
reaction which propagates through the specimen as a combustion wave leaving
behind a hot final product [3-10]. In the latter case, a pellet is heated up at aprescribed rate (typically 1-100 K/s) until at a certain temperature called the
ignition point, Tign, an exothermal reaction becomes self-sustaining and the
temperature rises to its final value, TCS, almost uniformly throughout the sample.
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B. B. Khina2
Typically, the value of Tign is close to the melting point of a lower-melting
reactant or to the eutectic temperature.
Examples of CS products are listed in Table 1.1, and characteristics of SHS
reactions in certain systems are presented in Table 1.2.
Table 1.1. Examples of compounds and materials produced by combustion
synthesis [3-14]
Type of material Compounds and adiabatic combustion temperature, K (in brackets)
Borides TiB2 (3190), TiB (3350), ZrB2 (3310), HfB2 (3320), VB2 (2670),
VB (2520), NbB2 (2400), NbB, TaB2 (2700), TaB, CrB2 (2470),CrB, MoB2 (1500), MoB (1800), WB (1700), LaB6 (2800)Carbides TiC (3210), ZrC (3400), HfC (3900), VC (2400), Nb2C (2600),
NbC (2800), Ta2C (2600), TaC (2700), SiC (1800), WC, B4C,
Cr3C2, Cr7C3, Mo2C, Al4C3
Aluminides Ni3Al, NiAl, Ni2Al3, TiAl, CoAl, Nb3Al, Cu3Al, CuAl, FeAl
Silicides Ti5Si3 (2500), TiSi (2000), TiSi2 (1800), Zr5Si3 (2800), ZrSi
(2700), ZrSi2 (2100), WSi, Cr5Si3 (1700), CrSi2 (1800), Nb5Si3
(3340), NbSi2 (1900), MoSi2 (1900), V5Si3 (2260), TaSi2 (1800)
Intermetallics NbGe, TiCo, NiTi
Sulfides and selenides MgS, MnS (3000), MoS2 (2900), WS2, TiSe2, NbSe2, TaSe2,
MoSe2, WSe2Hydrides TiH2, ZrH2, NbH2, CsH2
Nitrides TiN (4900), ZrN (4900), VN (3500), HfN, Nb2N (2670), NbN
(3500), Ta2N (3000), TaN (3360), Mg3N2 (2900), Si3N4 (4300),
BN (3700), AlN (2900)
Carbonitrides TiC-TiN, NbC-NbN, TaC-TaN, ZrC-ZrN
Complex oxides Aluminates (YAlO2, MgAl2O4), niobates (NaNbO3, BaNb2O6,
LiNbO3), garnets (Y3Al5O12, Y3Fe5O12), ferrites (CoFe2O4,
BaFe2O4, Li2Fe2O4), titanates (BaTiO3, PbTiO3), molybdates
(BiMoO6, PbMoO4), high-temperature superconductors
(YBa2Cu3O7-x, LaBa2Cu3O7-x, Bi-Sr-Ca-Cu-O)Ternary solid solutions
based on refractory
compounds
TiB2-MoB2, TiB2-CrB2, ZrB2-CrB2, TiC-WC, TiN-ZrN, MoS2-
NbS2, WS2-NbS2
MAX phases Ti2AlC, Ti3AlC2, Ti3SiC2
Cermets TiC-Ni, TiC-Cr, TiC-Co, TiC-Ni-Cr, TiC-Ni-Mo, TiC-Fe-Cr, TiC-
Cr3C2-Ni, TiC-Cr3C2-Ni-Cr, Cr3C2-Ni-Mo, TiB-Ti, WC-Co, TiC-
TiN-NiAl-Mo2C-Cr
Composites and
functionally-graded
materials
TiC-TiB2, TiB2-Al2O3, TiC-Al2O3 (2300), TiN-Al2O3, B4C-Al2O3,
MoSi2-Al2O3 (3300), MoB-Al2O3 (4000), Cr3C2-Al2O3, 6VN-
5Al2O3 (4800), ZrO2-Al2O3-2Nb, AlN-BN, AlN-SiC, AlN-TiB2,
Si3N4-TiN-SiC, sialons (SiAlOxNy)
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Advances and Challenges in Modeling Combustion Synthesis 3
Table 1.2. Features of combustion synthesis waves for certain typical
reactions [3-18]
Type of interaction Reaction Experimental
combustion
temperature, C
Combustion wave
velocity, cm/s
Solid-solid (formation of
a carbide) via a transient
liquid phase (melting of a
metallic reactant) [17,18]
Ti (solidliquid) + C(solid) TiC (solid)
2500 3-4
Solid-solid (formation of
a complex oxide) via atransient liquid phase
with participation of an
oxidizing gas
3Cu (solidliquid) +
2BaO2 (solid) +(1/2)Y2O3 (solid) + O2
(gas) YBa2Cu3O7-x(solid)
1000 0.2-0.5
Solid-gas with or without
melting of a metallic
reactant [13]
Ti (solidliquid) +(1/2)N2 (gas) TiN(solid)
1600-2000 0.1-0.2
Solid-solid (formation of
a carbide) via
intermediate gas-transportreactions [12]
Ta (solid) + C (solid)
TaC (solid)2600 0.5-2
Liquid-liquid in organic
systems with the
formation of a solid
product [15]
C4H10N2 (liquid,
piperazine) + C3H4O4
(liquid, malonic acid)
C7H14N2O4 (solid,salt)
155 0.06-0.15
The unique features of the obtained products, e.g., high purity, small and
uniform grain size, etc., are ascribed to extreme conditions inherent in CS, whichmay bring about unusual reaction routes: (i) high temperature, up to 3500 C, (ii)a high rate of self-heating, up to 10
6K/s, (iii) steep temperature gradient in SHS
waves, up to 105
K/cm, (iv) rapid cooling after synthesis, up to 100 K/s, and (v)
fast accomplishment of conversion, from about 1 s to the maximum of 10 s [3-6].
It should be noted that traditional furnace synthesis of refractory compounds
requires a much longer time, ~1-10 h, for the same initial composition, particle
size and close final temperature. It has been demonstrated experimentally [16-23]
that in many systems phase and structure formation during CS proceeds viauncommon interaction mechanisms from the point of view of the classical
Physical Metallurgy [24,25].
Modeling and simulation traditionally play in important part in the
development of CS and CS-related technologies (see reviews [3-5,11,26-29] and
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B. B. Khina4
references cited therein). An adequate mathematical model is supposed to
describe both heat transfer in a heterogeneous reactive medium and the interaction
kinetics, which is responsible for heat release during CS.
In modeling CS, a quasi-homogeneous, or continual model [30,31], which is
based on classical combustion theory, is widely used. Heat transfer, which is
considered on the volume-averaged basis, and the reaction rate in a sample are
described as follows:
cp T/t = (T) + Q/t (1.1)
/t = (1)nexp(m)kexp(E/RT), (1.2)
where T is temperature, is density, cp is heat capacity, is thermal conductivity,Q is the heat release of exothermal reactions, is the degree of chemicalconversion (from 0 in the unreacted state to 1 for complete conversion), R=8.314
Jmol1
K1
is the universal gas constant, n (the reaction order), k (preexponential
factor) and m are formal parameters and E is the activation energy; term Q/tdenotes the heat release rate.
The thermal structure of a combustion wave according to Zeldovich and
Frank-Kamenetskiy [32] is shown schematically in Figure 1.1. Typically, three
zones are distinguished: (i) the preheating zone where almost no reaction occurs
and the main processes are heat and mass transfer accompanied with evaporation
of volatile impurities; in Russian literature it is often termed as the Michelson
zone after V.A.Michelson (1860-1927) who described the temperature profile
ahead of the moving combustion front [32], (ii) zone of thermal reaction where
the conversion degree sharply increases and the heat release rate reaches itsmaximum and starts decreasing while the temperature almost reaches theadiabatic value, and (iii) the after-burn, or post-reaction zone where the
interaction terminates. The latter zone is characterized by a slow increase in both
conversion degree and temperature, which finally attain their maximal values =1and T=Tad, and the heat release rate, Q/t, falls down to zero. The temperatureof the reaction front, Tf, corresponds to the onset of fast thermal reaction. In
regard to combustion synthesis of materials, it is believed that complex
heterogeneous reactions, which may proceed via uncommon (fast) mechanisms
and are responsible for major heat release, occur in the thermal reaction zonewhile the after-burn zone, where the heat release rate is minor, is dominated by
the processes bringing about the formation of final structure of the product, such
as Ostwald ripening, recrystallization etc.
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Advances and Challenges in Modeling Combustion Synthesis 5
Figure 1.1. Schematic of the thermal structure of a combustion wave.
The approach formulated in Eqs. (1.1) and (1.2) permitted modeling dynamic
regimes of SHS, e.g., oscillating [30] and spin combustion [33,34]. It was also
used for studying the effect of intrinsic stochasticity of heterogeneous reactions,which can be attributed to a difference in the surface morphology, impurity
content and hence reactivity of solid reactant particles, on the dynamic behavior
of a solid flame for a one-stage [35] and multi-stage reaction [36] employing the
cellular automata method.
It should be outlined that this model is not linked to any process-specific
phase formation mechanism and hence is referred to as a formal one. When
applying this approach to modeling CS in a particular system, the value of the
most important model parameter, viz. activation energy E, is supposed tocorrespond to the apparent activation energy of the CS as a whole. The latter is
determined from experimental graphs the combustion wave velocity vs.
temperature plotted in the Arrhenius form, and in its physical meaning
corresponds to a real rate-limiting stage of phase formation during CS, which may
be different in different temperature ranges. For example, below the melting
temperature, Tm, of a metallic reactant E always refers to solid-state diffusion in
the product while at T>Tm it can refer to processes in the melt (diffusion or
crystallization) [37]. This method for choosing the E value was used when
studying numerically the conditions of arresting a high-temperature state of
substances in the SHS wave by fast cooling for the cases of a one-stage [38] and
two-stage exothermal reaction [39].
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B. B. Khina6
In recent papers [40,41], this formal model [see Eqs. (1.1) and (1.2)] was
employed for studying the SHS of a NiTi shape memory alloy. The activation
energy used in calculations was E=113 kJ/kg, which is equivalent to 12.05 kJ/mol
(because the molar mass of NiTi is 106.6 g/mol). This is an extraordinary low
value for a reaction in a condensed system and can correspond only to diffusion in
a transient melt formed in the CS wave. However, according to reference data
[42], the activation energy for diffusion in some pure liquid metals is the
following: Li, E=12 kJ/mol; Sn, E=11.2 kJ/mol; Zn, E=21.3 kJ/mol; Cu, E=40.7
kJ/mol; Fe, E=51.2 kJ/mol. Thus the value of E used for calculations in [40,41] is
close to that for diffusion in low-melting metals such as Li or Sn, and is by the
factor of 4 lower than for iron whose melting point, Tm, lies between Tm of Ni andTi (the activation energy for diffusion in liquid metals is known to be proportional
to Tm [43]). All the more, this E value is incomparably lower than a typical
activation energy for diffusion in intermetallic compounds. Hence in this case the
most important parameter of the formal model, E, appears to be physically
meaningless.
Recently, new features of SHS were observed experimentally [44-47]. First,
microscopic high-speed video recording [44,45] and photographing [46]
demonstrated micro-heterogeneous nature of SHS which revealed itself in theroughness of the combustion wave front, chaotic oscillations of the local flame
propagation rate and new dynamic behaviors such as relay-race, scintillation and
quasi-homogeneous patterns. Second, the formation of non-equilibrium structure
and composition of SHS products was examined experimentally and interpreted
qualitatively in terms of relationships between characteristic times of reaction tr,
structuring ts and cooling tc [47]. These features were attributed to two main
factors: inhomogeneous heat transfer in the charge mixture and a specific reaction
mechanism [46].These results gave rise to new, heterogeneous models [48-51] involving heat
transfer on the particle-to-particle basis [48-50] and percolation phenomena in a
system of chaotically distributed reactive and inert particles [51]. However, in
these models the traditional formal kinetics for a thermal reaction [Eq. (1.2)] was
employed. Thus, an urgent and still unresolved problem in CS is an adequate
description of fast interaction kinetics in a unit reaction cell containing particles or
layers of dissimilar reactants whose composition corresponds to the average
composition of a charge mixture.
The most widely used kinetic model, which is connected to a particular phase
forming mechanism, is a solid-state diffusion-controlled growth concept first
applied to CS in [52] for planar symmetry and in [53] for spherical symmetry of
an elementary diffusion couple. As in a charge mixture there are contacts of
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Advances and Challenges in Modeling Combustion Synthesis 7
dissimilar particles, a layer of an intermediate or final solid product forms upon
heating thus separating the initial reactants. The growth rate of the reaction
product and associated heat release necessary for sustaining combustion is
controlled by solid-state diffusion through this layer. Then, the diffusion-type
Stefan problem is formulated instead of Eq. (1.2). However, as demonstrated
below in more detail, in most cases modeling was performed not with real
diffusion data, which are known for many refractory compounds, but using either
dimensionless coefficients varied in a certain range or fitting parameters chosen to
match the calculated and measured results of the SHS temperature profile and
velocity. It should be emphasized that Diffusion in Materials is a well-developed
cross-disciplinary area within Materials Science and Solid State Physics, and thediffusion parameters for many of the phases produced by CS (carbides, nitrides,
intermetallics etc.) have been measured experimentally at different temperatures,
and these data are supposed to be used in modeling. Besides, in most of the CS-
systems fast interaction begins after fusion of a lower-melting-point reactant [3-
5,31] but within this approach melting does not alter the phase layer sequence in
an elementary diffusion couple [52,53].
A number of experimental results obtained by the combustion-wave arresting
technique in metal-nonmetal (Ti-C [17,18], (Ti+Ni+Mo)-C [19], Mo-Si [20]) andmetal-metal (Ni-Al [21,23]) systems gave rise to an qualitative notion of a non-
traditional phase formation route. It involves dissolution of a higher-melting-point
reactant (metal or non-metal) in the melt of a lower-melting-point reactant and
crystallization of a final product from the saturated liquid.
Besides, there is much controversy over the presence of an intermediate solid
phase in the dissolution-precipitation route. In [21] it is concluded that during
SHS in the Ni-Al system, solid Ni dissolves in liquid Al through a solid interlayer
separating aluminum from nickel, which agrees with the phase diagram. In thiscase, the rate-limiting stage is solid-state diffusion across this layer. But in [23]
for the same system it is found that above 854 C a solid interlayer between nickeland molten Al is absent; then the overall interaction during CS is controlled by
either diffusion in the melt or crystallization kinetics.
Such a situation is considered in recent models [54-59], where a solid reactant
(nickel [54-56] or carbon [57-59]) dissolves directly in the liquid based on a
lower-melting component (Al and Ti, respectively) and product grains (NiAl and
TiC, correspondingly) precipitate from the melt; the rate-limiting stage is liquid-phase diffusion [54-56] or crystallization kinetics [57-59]. However, within these
approaches the fundamental problem of the existence of a thin solid-phase
interlayer at the solid/liquid interface is not discussed nor a criterion is obtained
for transition between the solid-state diffusion-controlled mechanism and the
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B. B. Khina8
dissolution-precipitation route with or without a thin interlayer. Hence, the
applicability limits of the existing modeling approaches have not been clearly
determined so far. The role of high heating rates, which are intrinsic in CS, in
most of the models is not accounted for in an explicit form.
Thus, adequate description of the interaction kinetics in condensed
heterogeneous systems in non-isothermal conditions of CS is an urgent problem in
this area of science and technology, and the absence of a comprehensive model
hinders the development of new CS-based processes and novel advanced
materials.
Hereinafter the situation where a reaction between condensed reactants
proceeds through a solid layer, i.e. solid reactant (C for the Ti-C system or Ni forthe Ni-Al system)/solid final or intermediate product (TiC or one of intermetallics
of the Ni-Al system, respectively)/liquid (Ti or Al melt), will be provisionally
called solid-solid-liquid mechanism since the interaction occurs at both
solid/solid and solid/liquid interface. This term will be used both for the solid-
state diffusion-controlled growth pattern where the product layer is growing and
for dissolution-precipitation route where the interlayer remains very thin. As the
diffusion coefficient in a melt is much higher than in solids, the rate-limiting stage
in this mechanism is diffusion across the solid interlayer. The second route, viz.dissolution-precipitation without an interlayer, can be referred to as solid-liquid
mechanism since the interaction of condensed reactants (solid C or Ni with
molten Ti or Al, respectively) occurs at the solid/liquid interface while the product
(TiC or NiAl) crystallizes from the melt. However, up to now the solid-liquid
mechanism has not been validated theoretically, nor the applicability limits of the
solid-solid-liquid mechanism based on solid-state diffusion kinetics have ever
been determined with respect to strongly non-isothermal conditions typical of CS.
Thus, the main goal of this work is to develop a system of relatively simpleestimates and evaluate the applicability limits of the solid-solid-liquid
mechanism approach to modeling CS and determine criteria for a change of
interaction routes basing the calculations on experimental data to a maximum
possible extent [60,61]. Below, a brief discussion of the diffusion concept of CS is
presented. Then, calculations for particulars system, viz. Ti-C and Ni-Al, are
performed using available experimental data on both the diffusion coefficients in
the growing phase and thermal characteristics of CS.
The choice of these binary systems for a modeling study is motivated by thefollowing reasons. First, those are typical SHS systems which have been a subject
of extensive experimental investigation (see reviews [3-11,16] and references
cited therein). Second, the synthesis products, viz. TiC and NiAl, have a wide
industrial application because of their unique physical and mechanical properties.
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Advances and Challenges in Modeling Combustion Synthesis 9
Hence a large number of parameters needed for numerical calculations can be
found in literature. Third, both of these substances are typical representatives of
wide classes of chemical compounds that have different properties connected with
their intrinsic structural features. Titanium carbide is a typical interstitial
compound (like many carbides, nitrides and certain borides) wherein the
diffusivities of constituent atoms, Ti and C, differ substantially. Hence the growth
of TiC in an elementary diffusion couple Ti/TiC/C during CS is dominated by the
diffusion of carbon atoms in the TiC layer and proceeds mainly at the Ti/TiC
interface. The experimentally measured parameters such as the chemical diffusion
coefficient or the parabolic growth-rate constant for TiC are connected with the
partial diffusion coefficient of carbon in this compound. Nickel monoaluminide isa typical substitutional compound with an ordered crystalline structure (like many
intermetallics) where the rates of diffusion of Ni and Al atoms are comparable.
Thus its growth during CS occurs at both sides of a NiAl layer and can be
characterized by a single parameter, namely the interdiffusion coefficient, which
is measured experimentally.
For the Ti-C system, different situations are considered that can arise during
CS within the frame of the above concept and, wherever possible, a quantitative
and/or qualitative comparison between the outcome of calculations andexperimental results is drawn. Emphasis is made on the structural characteristics
of the CS product, titanium carbide, that emerge from this approach. The
conditions for a change of the geometry of a unit reaction cell in the SHS wave
due to melting of a metallic reactant (titanium) are analyzed and a
micromechanistic criterion for the changeover of interaction pathways is derived.
For the Ni-Al system, calculations within the frame of the diffusion-controlled
growth kinetics are performed taking into account both the growth of the product
phase, NiAl, and its dissolution in the parent phases (solid or liquid Ni and moltenAl) due to variation of solubility limits with temperature according to the
equilibrium phase diagram. Finally, the solid-liquid mechanism concept for CS
is justified and phase-formation mechanism maps for these two systems in
strongly non-isothermal conditions are plotted.
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1.2.BRIEF REVIEW OF DIFFUSION-BASED
KINETIC MODELS OF CS
The interaction kinetics controlled by solid-state diffusion was used for
numerical [52,53,62-69] and analytical [70] study of CS for the case of planar
diffusion couples (alternating lamellae of reactants) [52,63,64,66,70] and
spherical symmetry (growth of a product layer on the surface of a spherical
reactant particle) [53,65,67-69].
Inherent in this concept are two basic assumptions: (i) the phase composition
of the diffusion zone between parent phases corresponds to the isothermal cross-section of an equilibrium phase diagram, i.e. the nucleation of product phases
occurs instantaneously over all contact surfaces and (ii) the interfacial
concentrations are equal to equilibrium values. This results in the parabolic law of
phase layer growth [71-73].
It should be noted that in many diffusion experiments the phase layer
sequence deviates from equilibrium: the absence of certain phases was observed
in solid-state thin-film interdiffusion [74,75] and in the interaction of a solid and a
liquid metal (e.g., Al) [76,77]. These phenomena were ascribed to a reaction
barrier at the interface of contacting phases [78] without considering the
nucleation rate of a new phase. The effect of a nucleation barrier was examined
theoretically using the thermodynamics of nucleation [79,80] and the kinetic
mechanism of phase formation in the diffusion zone [81], and it was shown that in
the field of a steep concentration gradient the formation of an intermediate phase
is suppressed [79-81]. This effect has never been considered in the diffusion
models of CS. As in the theory of diffusion-controlled interaction in solids the
nucleation kinetics is not included and it is assumed that critical nuclei of missing
phases continuously form and dissolve [72,73], this qualitative concept issometimes used in interpreting the results of CS [21].
It will be fair to say that deviation of phase-boundary concentrations from
equilibrium due a reaction barrier was examined qualitatively for SHS [64] in the
case of planar geometry. This effect is noticeable only in the low-temperature part
of the SHS wave, and at high temperatures a strong barrier can only slightly
decrease the combustion velocity [64]. Also, the influence of such barrier on self-
ignition in the Ni-Al system at low heating rates, dT/dt
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Advances and Challenges in Modeling Combustion Synthesis 11
formed below the melting point of aluminum Tm(Al)=660 C [67]. At a thickNiAl3 layer (low heating rates) the reaction barrier is of little significance, but it
can slow down the interaction for thin layers (higher heating rates) [83,67]: e.g., at
dT/dt > 35 K/min the formation of the primary product can be suppressed [67].
But, as noted in [83,67], these results refer not to the SHS itself but only to a
preliminary stage (i.e. the preheating zone of the SHS wave) because fast
interaction begins at T>Tm(Al), the combustion temperature reaches 1400 C andthe final product is NiAl [67].
It should be outlined that in many works using the diffusion model of CS the
calculations were performed with dimensionless (relative) parameters varied in a
certain range. A known or estimated value of the activation energy for diffusion in
one of the phases was used only as a scaling factor and thus the results obtained
revealed only qualitative characteristics of the process [52,53,62,66]. Besides,
many of the modeling attempts [52,53] did not account for a change in the spatial
configuration of reacting particles due to melting and spreading of a metallic
reactant. The effect of melting was reduced to a change of interfacial
concentrations and the ratio of diffusion coefficients in contacting phases [62].
In more recent papers [67,68], the parameter values (the activation energy E
and preexponent D0) used for calculating the diffusion coefficient in a growingphase were presented. However, those were not the real values measured in
independent works on solid-state diffusion but merely fitting parameters
calculated from the characteristics of CS. For example, the formation of NiAl
above 640 C was modeled using D0=4.8102
cm2/s and E=171 kJ/mol [67]. As
noted in [67], this E value was the experimentally determined activation energy
for the CS process as a whole. Then the diffusion coefficient in NiAl at T=1273 K
is D = D0exp(E/RT) = 4.6109
cm2/s. Lets compare it with experimental data
on reaction diffusion in the Ni-Al system. For NiAl, D=(2.53.6)1010 cm2/s atT=1273 K [84]. The parameters for interdiffusion in this phase are E=230 kJ/mol
and D0=1.5 cm2/s [85], hence at T=1273 K D=5.41010 cm2/s. Thus, the
diffusion coefficient used in modeling SHS exceeds the experimental value by an
order of magnitude.
SHS wave in the Ti-Al system with the Ti-to-Al molar ratio of 1:3 in the
charge mixture was modeled using E=200 kJ/mol and D0=4.39 cm2/s for phase
TiAl3 [68]. This E value was obtained from experiments on combustion synthesis
using Arrhenius plots, and D0 was chosen to match the calculated and measuredresults of the propagation speed. Again, these values refer to the SHS wave as a
whole but not to interdiffusion in TiAl3. However, experimental data on SHS of
TiAl3 for the same starting composition, which were analyzed using the classical
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B. B. Khina12
combustion model [see Eqs. (1.1),(1.2)], gave a substantially higher activation
energy: E=483 kJ/mol [86]. If solid-state diffusion in the TiAl3 layer is really the
rate-limiting stage of the process, then the values of apparent activation energy
ought to agree (within an experimental error) regardless of the particular form of a
model.
Diffusion coefficients are measured experimentally within a rather wide
margin of error using a variety of techniques, and typically various methods yield
different values. But since diffusion parameters for many refractory compounds,
which can be produced by combustion synthesis, can be found in literature, it
appears possible to verify the validity of the diffusion-based kinetic model of SHS
employing a somewhat opposite approach: estimating the product layer growthand heat release using the experimental characteristics of SHS and independent
diffusion data. The models, parameter values and results of simulations for two
classical CS-systems, viz. Ti-C and Ni-Al, will be considered in more detail in the
subsequent chapters.
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Chapter 2
MODELING DIFFUSION-CONTROLLED
FORMATION OF TIC IN THE CONDITIONS OF
CS
2.1.INTRODUCTION
CS in the Ti-C system was a subject of extensive theoretical and experimentalstudies [53,17,18,62,69] because of industrial significance of the product, titanium
carbide, which is used for a wide range of applications because of its high melting
point, hardness and chemical stability. It is a suitable candidate for theoretical
investigation for the following reasons: (i) the Ti-C phase diagram [87] (see
Figure 2.1) contains only one binary compound TiC whose melting temperature
Tm(TiC)=3423 K exceeds the experimental SHS temperature TCS=3083 K [88]
and (ii) numerous diffusion data for titanium carbide are available in literature
[89-91]. We consider the case of spherical symmetry which better fits a typicalconfiguration of reacting particles in CS. With respect to the phase diagram, here
the solid-solid-liquid mechanism [situation C(solid)/TiC(solid)/ Ti(liquid)] is
quasi-equilibrium and the solid-liquid mechanism [situation C(solid)/Ti(liquid)] is
truly non-equilibrium.
Lets consider solid-state diffusion-controlled formation of the product,
titanium carbide, during heating of the Ti+C charge mixture in the SHS wave.
Typical particle radii are 5 to 100 m for Ti, about 0.1 m for carbon black and 1
to 30 m for milled graphite [17,18,69,88]. Two scenarios with a different
geometry of a unit reaction cell are examined: (1) a solid Ti particle surrounded
by carbon particles in a stoichiometric mass ratio at temperatures below the Ti
melting point, Tm(Ti)=1940 K [Figure 2.2 (a and d)], and (2) a solid carbon
particle surrounded by liquid titanium at T>Tm(Ti) [Figure 2.2 (c and e)].
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B. B. Khina14
Figure 2.1. The equilibrium Ti-C phase diagram [87] and experimental SHS temperature.
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Modeling Diffusion-Controlled Formation of TiC 15
Figure 2.2. Schematic of an elementary reaction cell in the SHS wave in the Ti-C system
(a and c) and corresponding concentration profiles for solid-state diffusion (d and e) [60]:
(a and d) growth of the TiC layer on the surface of a titanium particle at T
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B. B. Khina16
, (2.1)
,(2.2)
where D is the chemical diffusion coefficient in TiC, which is usually associated
with the diffusion coefficient of carbon in the carbide layer, cC is the massconcentration of carbon, R1(t) is the current position of the TiC/Ti interface, R2 is
the outer radius of the Ti particle, and c01, c
021 and c
023 are the interface
concentrations [Figure 2.2 (d)] according to the equilibrium phase diagram. The
boundary (at r=R2) and initial conditions to Eq. (2.1) are
cC(t, R2) = c023, cC(t, r=0) = c
01, cC[t, R1(t)] = c
021, R1(t=0) = R2. (2.3)
In [62,63] the Stefan-type boundary condition, Eq. (2.2), was posed at both
Ti/TiC and TiC/C interfaces. We should outline that in interstitial compounds
such as nitrides, carbides and many borides, the partial diffusion coefficient of
nonmetal species exceeds that of metal atoms by several orders of magnitude,
which is due to the interstitial diffusion mechanism. Hence, the growth of
titanium carbide occurs at the Ti/TiC interface and is controlled by the diffusion
of C atoms across the TiC layer. But at the C/TiC interface the growth of TiC at
the expense of graphite, which requires the supply of Ti atoms, cannot occur.
Thus, the first-kind boundary condition, cC(t, R2)=c0
23 [see Eq. (2.3)] is used for
the C/TiC interface, which actually denotes an ideal diffusion contact of carbon
particles with the outer surface of the growing TiC layer due to fast surface
diffusion of the C atoms from the C/TiC contact spots.
2.3.SCENARIO 2:GROWTH OF A TICLAYER ON THE
SURFACE OF SOLID CARBON PARTICLES
The physical background for scenario 2 [Figure 2.2 (c)] is the following.
Spreading of molten titanium towards solid carbon in the SHS wave was observed
experimentally [92,93]. Since it is accompanied with chemical interaction, for a
sufficiently small C particle size the spreading velocity is not the rate-limiting
=
r
cr
rr
D[T(t)]
t
cC2
2
C
(t)R
C10
1
0
21
1
r
cD[T(t)]
dt
dR)cc(
=
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Modeling Diffusion-Controlled Formation of TiC 17
stage [93,94]. Hence we consider that at TTm(Ti) the carbon particles arecompletely enveloped with liquid titanium, and a thin TiC layer forms at the Ti/C
interface separating the parent phases. The product growth occurs at the
Ti(melt)/TiC interface, i.e. outwards, due to diffusion of carbon atoms across the
TiC layer [Figure 2.2 (e)]. Since the diffusion coefficient of carbon in the melt is
at least an order of magnitude higher than in the carbide (Table 2.1), it is
reasonable to presume that the titanium melt is saturated with carbon (otherwise
the TiC layer will be dissolving). Then the boundary condition at r=R0 to
diffusion equation (2.1) and initial conditions to Eqs. (2.1),(2.2) look as
cC(t, R0) = c023, cC(t, r>R1) = c01, R1(t=0) = R0, (2.4)
where R0=const is the initial radius of the carbon particle.
2.4.DIFFUSION DATA FOR TIC
The parameters for calculating the diffusion coefficient in TiC in the
Arrhenius form
D = D0 exp[E/RT(t)] (2.5)
are listed in Table 2.1, wherein the experimental data available in literature [95-
103] for different temperature intervals T are collected. It is seen that differentdata sources give substantially different values of both activation energy and
preexponential factor, thus it seems necessary to select the parameters values
suitable for numerical calculations. Since the extrapolation of D to the wholetemperature range of SHS may bring about overestimated values, the diffusion
coefficients in TiC calculated at T=Tm(Ti) and TCS must be compared with the
diffusion coefficient in molten titanium: it is obvious that the value of D in a solid
metal-base refractory compound is at least an order of magnitude lower than in a
melt of the corresponding metal.
Because of the absence of experimental data, the diffusion coefficient of C in
molten Ti is estimated by a simple Stokes-Einstein (or Sutherland-Einstein)
formula, which was used for assessing the diffusion parameters of C, N, O and Hin liquid metals (Fe, Co, Ni, etc.) [104,105]
Di(m)
= kBT/(nai), (2.6)
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where numerical factor n=4 for substantially differing atomic radii of the melt
components and n=6 for close radii, ai is the atomic radius of i-th diffusing
species in the melt, =m is the dynamic viscosity, is the kinematic viscosityand m is the liquid-phase density. For carbon atoms, the covalent radius isaC=0.077 nm [106]. The density of molten Ti is m=4.11 g/cm
3[107]. A typical
value of the kinematic viscosity for such liquid metals as Al, Fe, Co, Ni et al. near
the melting point is (0.5-1)102 cm2/s [106]. For liquid titanium saturated withcarbon, =0.94102 cm2/s at T=Tm(Ti) [107], then DC
(m)(Tm) (4.87.2)10
5
cm2/s. For higher temperatures, the value =1.03102 cm2/s at T=2220 K is
known [107]; using it at T=TCS gives DC(m)
(TCS) (6.910.4)105
cm
2
/s. Itshould be noted that since Eq. (2.6) doesnt account for chemical interaction in the
melt, which may be substantial for the Ti-C system, these DC(m)
values are upper
estimates. Then the values of diffusion coefficients in TiC, which are close to or
higher than the upper estimate of DC(m)
(TCS), are excluded from consideration
(lines 10 to 14 in Table 2.1).
Table 2.1. Diffusion data for titanium carbide
Species No. D0, cm /s E,kJ/mol
T, K D[Tm(Ti)],cm
2/s
D(TCS),cm
2/s
Refs. Note
C
1 5102 235.6 2073-2973 2.3108 5.1106 [90,91,95,101]2 6.98 398.7 1723-2973 1.31010 1.23106 [89,95,96,98]
(TiC0.97)3 10 438.9 1873-2573 1.51011 3.7107 [95,96] (TiC0.9)4 45.44 447.3 1723-2553 4.11011 1.2106 [96,98]
(TiC0.887)
5 114 460.2 2018-2353 4.61011 1.8106 [96,99](TiC0.67)
6 0.1 259.4 1553-17731.010
8
4.0106
[90,91]7 6.5102 269.9 1673-1973 3.5109 1.7106 [90,102]8 4.2102 307.1 1983-2573 2.31010 2.6107 [90,91] (TiC0.9)9 0.48 328.42 1473-2023 6.91010 1.3106 [103] (TiC1.0)10 99.48 328.42 1473-2023 1.4107 2.7104 [103] (TiC0.5) D(TCS) >
DC(m)
(TCS)11 77.8 338.9 1473-1673 5.8108 1.4104 [89] D(TCS) >
DC(m)
(TCS)12 220 405.8 2200-2600 2.6109 2.9105 [100] (TiCx,
x=0.86-0.91)D(TCS) DC
(m)(TCS)
13 370 410.0 2200-2600 3.4109 4.1105 [100] (TiCx,
x=0.86-0.91)
D(TCS)
DC(m)
(TCS)14 1.31103 347.3 1173-1473 5.8107 1.7103 [90] D(TCS) >>
DC(m)
(TCS)
Ti 4.36104 736.4 2193-2488 6.51016 1.5108 [90,97] (TiCx,x=0.67-0.97)
DTi
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Modeling Diffusion-Controlled Formation of TiC 19
2.5.TEMPERATURE OF THE
REACTION CELL IN THE SHSWAVE
Self-heating from ambient temperature, T0, to TCS during the combustion
synthesis is due to the adiabatic heat release of chemical reactions which are
almost accomplished when maximal temperature is reached, and in the after-burn
zone (at TTCS) only coalescence and sintering of the product particles occur withminor heat release [3-5]. Hence calculations of the product layer thickness and
relevant heat release should be done in the time interval [0, tCS] corresponding to
the attainment of TCS.To perform calculations, we have to know the time dependence of
temperature in the reaction cell, T(t). We consider a steady-state combustion
regime. For a low-temperature portion of the SHS wave, T0
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B. B. Khina20
2.3, x=0 corresponds to the melting point of Ti, consequently the heating time
from Tm(Ti) to TCS is tCS = tCStm = x(TCS)/vSHS.
Figure 2.3. Temperature profile of the SHS wave in the Ti-C system [60]: 1, analytical
solution for a steady-state SHS wave [Eq. (2.7)] for TTm(Ti); 2, cubic-splineapproximation of experimental curve [88] in the range Tm(Ti)TTCS.
2.6.ADIABATIC HEAT RELEASE IN THE REACTION CELL
Having the heating law of the reaction cell, we can calculate the heat release
due to diffusion-controlled phase layer growth in non-isothermal conditions and
thus the maximal temperature attained, and then compare it with experimental
TCS. In adiabatic conditions, a heat balance equation for the formation of
stoichiometric TiC1.0 is written as:
H0298(TiC1.0)mTiC(t) = mTiC(t) + mC(t) +
mTi(t) , (2.9)
dT)TiC(cadT
298
p dT)C(cadT
298
p
+ )Ti(H)]Ti(TT[IdT)Ti(c mmadT
298p
ad
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Modeling Diffusion-Controlled Formation of TiC 21
where Tad is the adiabatic combustion temperature, cp(i) is heat capacity, mi(t) is a
current mass of i-th substance, H0298(TiC1.0) = 3.077 kJ/g is the standardenthalpy of TiC1.0 [110], Hm(Ti) = 0.305 kJ/g is the heat of fusion of Ti [110]and I[TadTm(Ti)] is the Heaviside unit-step function. The masses of all thesubstances are determined using a solution of the Stefan problem for particular
geometry of the reaction cell, and then Tad is calculated from Eq. (2.9).
2.7.MODELING OF TICLAYER GROWTH ON THE TITANIUM
PARTICLE SURFACE
2.7.1. Analytical Solution to Scenario 1
Problem (2.1)-(2.3),(2.5) is non-linear and in a general case can be solved
only numerically. However, for a similar linear problem (with D=const) an
asymptotic solution for the growth of a spherical phase layer, which is valid for a
small layer thickness h=R2R1
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Modeling Diffusion-Controlled Formation of TiC 23
Figure 2.4. Thickness of the TiC layer formed on the surface of a titanium particle by the
time of attainment of Tm(Ti) (a) and TCS (c), and relevant adiabatic heating (b and d) [60].
Numbers at curves correspond to diffusion data sets in Table 2.1.
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B. B. Khina24
The corresponding adiabatic heating is only Tad=1064 K for the Ti particleradius of 10 m and sharply drops with increasing R2 [Figure 2.4 (d)]. Thus, heat
release due to product growth is insufficient to sustain the SHS wave ( i.e. to reach
TCS=3083 K).
The obtained result, viz. a small thickness of TiC grown in the temperature
range below Tm(Ti), qualitatively agrees with experimental data [17,18]: in
rapidly cooled samples almost no interaction was observed in the so-called
preheating zone of the SHS wave.
However, at the attainment of T=Tm(Ti) the melting of titanium can bring
about the rupture of the primary TiC shell and the spreading of the metallic melt.
It should be noted that in [69] the diffusion-controlled TiC formation was
assessed using 6 different sets of the diffusion data, but only an isothermal
situation below the titanium melting point was examined. Besides, the TiC layer
growth was considered on the surface of a carbon particle whereas, as mentioned
above, the initial TiC film at T
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Modeling Diffusion-Controlled Formation of TiC 25
where urrand u are the radial and shear strain, correspondingly, and A and B are
constants which are determined from boundary conditions (2.16).
Hookes law for spherical symmetry looks as
rr= [(1)urr+ 2u], = (u + urr),
(2.18)
where rr and are the radial and shear stress, correspondingly, Y is the elastic
modulus and is the Poissons ratio [113]. Then the solution for is obtainedfrom Eqs. (2.16)-(2.18):
(r) = , f = , fr= ,
= . (2.19)
Rupture of the primary TiC shell occurs when the maximal shear stress in the
spherical layer (at r=R2) exceeds the ultimate tensile stress uts. Then from Eq.(2.19) we obtain a critical thickness, hcr= R2R1, of the TiC layer:
, ,
. (2.20)
The TiC case can burst at hhcr. This is an upper estimate because we donttake into account partial dissolution of TiC in molten titanium due to the eutectic
reaction at 1645 C.To calculate the hcr value, we have to determine the mechanical properties of
TiC at the melting temperature of titanium. The temperature dependencies of the
elastic modulus, Y, and shear modulus, G, for TiC are known in the following
form [96]:
)21)(1(
Y
+ )21)(1(Y
+
f1
ffp
f1
f11
21
Y r0
r
3/1
m
s
+
++
31
32
R2
R3
32
r2
R
+
21
1
=1
Rh 2cr
3/1
0uts
0uts
)p(
p2
+
+=
= 1
21
Y33/1
m
s
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B. B. Khina26
Y(T) = Y0 bYTexp(T0/T), G(T) = G0 bGTexp(T0/T), (2.21)
where T0=320 K, Y0=461 GPa, bY=0.0702 GPa/K, G0=197 GPa and bG=0.0299
GPa/K. Then at Tm(Ti)=1940 K we have Y=346 GPa and G=148 GPa, thus the
Poissons ratio is = Y/(2G)1 = 0.17. As foruts values for TiC at elevatedtemperatures, there are only disembodied data, e.g., uts(T=1073 K) 380 MPa,uts(T=1273 K) 280 MPa [90]. However, available are data on the bendingstrength, b, of titanium carbide over a wide temperature range because it is atypical test for brittle refractory compounds; b has a maximum of approximately
500 MPa around T=2000 K [96, page 233]. Then, using an estimate uts ~ b/2 =250 MPa, from Eq. (2.20) we obtain hcr0.6R2. Since the calculated valueh[T=Tm(Ti)] is very small, for any initial size of Ti particles used in SHS (R2=5 to
100 m) melting of the titanium core will inevitably bring about the rupture of the
primary TiC shell and spreading of the melt. This changes the geometry of a unit
reaction cell as shown in Figure 2.2 (a-c).
2.9.GROWTH OF A TICLAYER ON THE SURFACE OF A SOLIDCARBON PARTICLE
2.9.1. Analytical Solution to Scenario 2
For scenario 2 [Figure 2.2 (c and e)], an asymptotic solution to Eqs.
(2.2),(2.3)-(2.5) with respect to the TiC layer thickness, h, can be obtained
similarly to Eq. (2.11) [25,111,112]:
h() = R1() R0 = 1/2
1/R023/2
/(2R02). (2.22)
Here coefficients , 1 and 2 are defined, as previously, by Eqs. (2.12),(2.13) and is determined according to Eq. (2.10) where integration is performed over thetime range 0ttCS, which corresponds to the temperature range TmTTCS(Figure 2.3).
To calculate adiabatic heating, we turn to Eq. (2.9). For the reaction cell
shown in Figure 2.2 (c), mTiC() = (4/3) (R13
()R03
)TiC, mC() = mC0
0.2mTiC(), mC
0= (4/3)R0
3C and mTi() = 4mC0 0.8mTiC(). Then, ignoring the
temperature dependence of heat capacities and neglecting the melting enthalpy of
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Modeling Diffusion-Controlled Formation of TiC 27
titanium (because Hm(Ti)
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B. B. Khina28
Figure 2.5. Formation of the TiC layer on the surface of a carbon particle in the SHS wave
after spreading of molten titanium (Tm(Ti)TTCS): (a) product layer thickness and (b)corresponding adiabatic heating vs. carbon particle radius [60]. Numbers at curves
correspond to diffusion data sets in Table 2.1.
The results obtained regarding the above concept suggest that fast and
complete conversion of reactants into the final product providing the required heat
release can be achieved via a different route (without diffusion control of the
product formation). For further analysis of the diffusion model it makes sense toestimate a structural parameter of the product, viz. porosity. To do this, it is
necessary to evaluate the displacement of the C/TiC interface.
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Modeling Diffusion-Controlled Formation of TiC 29
2.9.3. Displacement of the C/TiC Interface in the Emptying-Core
Mechanism
Above we have calculated the thickness of a spherical product layer formed
due to interstitial diffusion of C atoms through titanium carbide, i.e. outward
growth of TiC on the surface of carbon particles. Hence the TiC particles formed
after complete conversion of the reactants will be hollow. This pattern of
diffusion-controlled product formation is sometimes called the emptying-core
mechanism [115]. Lets estimate the displacement of the C/TiC interface, i.e.
inward growth of the product layer due to diffusion of Ti atoms across TiC.
From Figure 2.5 (a) it is seen that at R05 m the effect of curvature is minor:raising R0 from 5 to 12.5 m increases the TiC thickness by less than 10%. Thus
the diffusion problem can be considered for a semi-infinite rod. The diffusion
equations are written for both C and Ti atoms
, i C,Ti. (2.25)
The Stefan-type boundary conditions to Eq. (2.25) are formulated at
interfaces Ti(melt)/TiC (r=R1) and C/TiC (r=R0) taking into account that here
R0=R0(t) and cC + cTi =1
, . (2.26)
The initial conditions are
R0(t=0) = R1(t=0) = R00. (2.27)
Here DC and DTi are the partial diffusion coefficients of C and Ti atoms in TiC
(see Table 2.1), r is the radial coordinate and R00 is the initial position of the
C/Ti(melt) interface at which a thin TiC layer originates at t=0. Using substitution
, i C,Ti, (2.28)
2
i
2
ii
r
c)]t(T[D
t
c
=
(t)R
CC
10
1
0
21
1
r
c[T(t)]D
dt
dR)cc(
=
(t)R
CTi
00
32
0
r
c[T(t)]D
dt
dR)c1(
=
= d)][T(D(t)t
0ii
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B. B. Khina30
the non-isothermal problem (2.25)-(2.27) is reduced to an isothermal (linear) case
which has an analytical solution [118] for the displacement of phase boundaries
Ti/TiC (h) and C/TiC ():
h(C) = R1(C) R00 = C C
1/2, (Ti) = R0(Ti) R
00 = Ti Ti
1/2. (2.29)
The coefficients C and Ti are determined from transcendental equations:
1/2(C/2)exp(C/2)2{erf(C/2) + erf[Ti(Ti/C)
1/2/2]} = (c
023c
021)/(c
021c
01)
1/2(Ti/2)exp(Ti/2)2{erf(Ti/2) + erf[C(C/Ti)
1/2/2]} = (c
023c
021)/(1c
023).
(2.30)
The calculated displacement of the C/TiCx interface during interaction in the
SHS wave (at TmTTCS) is negligibly small: =4.7 nm
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Modeling Diffusion-Controlled Formation of TiC 31
[114,116] or by short-term compaction of a hot pellet immediately after the
completion of SHS [3,4 and literature cited therein]. If as-synthesized TiC
particles were hollow, which follows from the above described route, the
attainment of such relative density would require prolonged pressure sintering at a
high temperature.
Thus the structural characteristic of the SHS product emerging from the
diffusion model disagrees with experimental observations, which therefore
supports an idea of a dissolution-precipitation route capable of producing dense
TiC particles, which includes dissolution of carbon in molten titanium and
subsequent crystallization of the product grains.
It should be noted that a conclusion in favor of the diffusion-controlledgrowth of hollow TiC shells on the surface of carbon particles having a size of
2R0=7 m (initial porosity of a sample was 0=0.2) and 20 m (0=0.4) was madein [115] basing solely on the porosity measurements and microstructures of as-
synthesized specimens. Lets analyze these experimental data. For all of the
samples the initial temperature, T0, was 293 K, and the total porosity measured
after SHS was almost the same, t(m)
=0.460.5. As shown above, SHS of TiC viathis mechanism is possible for small-sized carbon particles, R03.5 m, but closed
porosity will be cl=0.33 which greatly exceeds the measured valuecl
(m)=0.060.08 [115]. Besides, total porosity of an as-synthesized sample for the
formation of TiC1.0 is estimated as
t = 1 (10)[TiC(0.8/Ti+0.2/C)]1
(2.31)
implying that the specimen volume doesnt change during SHS, which is true for
strongly compacted green pellets (as in [115]). Here Ti=4.51 g/cm3
[106] is the
density of initial -Ti particles. Then for samples with 7 m diameter carbonparticles the formation of dense TiC grains (TiC=4.91 g/cm
3) yields the total final
porosity t=0.41, which is close to experimental data. But if hollow TiC particlesare formed via the diffusion mechanism, then, substituting into Eq. (2.31) eff=3.3g/cm
3instead ofTiC we obtain t'0.13. In this case t' signifies the fraction of
pores between the hollow particles. But this value is less than 0.1540.005 (theScher-Zallen criterion), which is required by the percolation theory [120,121] for
the existence of open porosity. Thus, the sample will contain only closed pores
(inside the TiC particles and between them) whereas in [115] high open porosity
was observed: op(m)
= t(m)cl
(m) 0.4.
For larger carbon particles (2R0=20 m), as demonstrated above, SHS via the
diffusion mechanism is impossible (for T0298 K) because of low heat release per
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B. B. Khina32
unit reaction cell [Figure 2.5 (b)]. If dense TiC particles are formed, from Eq.
(2.31) for 0=0.4 we have t=0.56 which is close to experimental porosityt
(m)=0.460.5. For the formation of hollow particles (cl=0.33), Eq. (2.31) gives
t'=0.34, and then the total porosity will be t = t' + cl = 0.67, which substantiallyexceeds the measured value. Maximal closed porosity in the experiments was
cl(m)
=0.22 for samples with the particle diameter 17 m (Ti) and 20 m (carbon)[115]. Since SHS was performed under isostatic gas pressure, the origin of closed
pores should be ascribed to partial sintering of dense TiC grains (presumably
precipitated from melt) around voids formed on the sites of outflown titanium
particles. This is supported by the fact that the closed porosity was noticed to
increase with raising the gas pressure (from cl(m)
=0.12 at 1 bar to 0.22 at 70 bar)
while the total porosity remained almost the same, t(m)
=0.470.5 [115].
2.10.ANALYSIS OF THE SHRINKING-COREMECHANISM IN
THE TI-CSYSTEM
Lets discuss a dissolution-precipitation route of the TiC formation, whichcan produce 100% dense particles. According to the idea first proposed in [122]
and used for studying SHS in the Ni-Al [67] and Nb-C [123] systems, as soon as a
metallic melt spreads and engulfs solid particles, a thin film of an intermediate
phase (here TiC) forms around them instantaneously. In this interaction pattern,
the phase layer sequence in the reaction cell corresponds to the equilibrium phase
diagram (Figure 2.1). Then the product particles (TiC) precipitate from the
saturated melt due to diffusion of carbon atoms across this film. The film
thickness remains constant: it is believed that its growth rate at the C/TiCinterface is equal to the dissolution rate at the melt/TiC interface. Thus, the TiC
film shrinks to the center of the carbon particle as the latter dissolves. This pattern
is sometimes called the shrinking-core mechanism. It corresponds to the solid-
solid-liquid mechanism which, for the Ti-C system, is truly quasi-equilibrium.
However, the film thickness has not been previously estimated using realistic
diffusion data.
The concentration profile of carbon in the reaction cell is similar to that
shown in Figure 2.2 (a) but with R0=R0(t); final TiC particles precipitate from the
melt in domain [R1(t), R2]. In a general case, the displacement of the melt/TiC and
C/TiC interfaces is determined by Eqs. (2.25)-(2.27) with the only difference that
Eq. (2.25) should be written in spherical symmetry. But since the TiC film
thickness is small and DC>>DTi, outdiffusion of carbon atoms through the film is
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Modeling Diffusion-Controlled Formation of TiC 33
not the rate-limiting stage. Thus the process is controlled by indiffusion of Ti
atoms across the TiC layer, and radial shrinking of the film is described as
, R0(t=0) = R00, h0 = R1(t)R0(t) = const,
(2.32)
where h0 is the layer thickness. For a thin film, a steady-state concentration profile
can be used to determine the concentration gradient at r=R0(t) in Eq. (2.32):
cC(r) = c023R0/r + (1 R0/r) (R1c
021 R0c
023)/h0, R0(t) r R1(t).
(2.33)
Using Ti defined by Eq. (2.28) and introducing z = R0/R00, from Eqs.
(2.32),(2.33) we obtain:
. (2.34)
By the attainment of the maximal SHS temperature TCS, which corresponds to
time t=tCS, the carbon particle completely dissolves, i.e. R0(tCS)=0. Then,
integrating Eq. (2.34) from 0 to tCS, we receive a non-linear equation linking the
initial radius of the carbon particle R00 with the thickness of the TiC film
. (2.35)
The results of the numerical solution of Eq. (2.35) are shown in Figure 2.6. It
is seen that the thickness of the TiC film for the initial radius of carbon particles
R0
0=0.5 m is close to the crystal lattice period: h0=0.5 nm ~ aTiC=0.4327 nm
[89], and still decreases with increasing R
0
0. Hence the aforesaid quasi-equilibrium solid-solid-liquid mechanism loses its physical meaning: a minimal
thickness of a crystalline phase must be about the size of a critical nucleus which
is typically of the order of 10 lattice periods.
(t)R
CTi
00
32
0
r
c[T(t)]D
dt
dR)c1(
=
20
0
0
23
0
21
0
23
Ti0
0
0))(Rc1(
ccddz
1/hzRz
=+
( )[ ]0
23
0
21
0
23SHSTi0
0
00
0
00c1
cc)(t1hRhRh
=+ /ln
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B. B. Khina34
Figure 2.6. Calculated thickness of a titanium carbide film vs. carbon particle radius fordiffusion-controlled dissolution-precipitation (solid-solid-liquid) mechanism [60].
2.11.PHASE-FORMATION-MECHANISM MAP FOR NON-
ISOTHERMAL INTERACTION IN THE TI-CSYSTEM
The above-presented consistent analysis of the solid-solid-liquid (diffusion-
controlled) mechanism, which was performed using available experimental data
on both solid-state diffusion in the product phase and characteristics of the SHS
wave, has demonstrated that this widely used concept is actually not applicable to
modeling SHS of titanium carbide. This is because the physical meaning of the
results obtained within this approach (e.g., the product structure and density)
disagrees with experimental data.
It is shown that formal calculation of the product-layer thickness and
associated heat release for small particles of a nonmetallic reactant (for scenario
2) can bring about numerical data supporting the diffusion model. Thus the
comparison of theoretical and experimental results should be performed usingstructural characteristics of SHS products, e.g., porosity.
Therefore, only the solid-liquid mechanism, which for the Ti-C system is
truly non-equilibrium, can operate during SHS to produce dense product particles.
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Modeling Diffusion-Controlled Formation of TiC 35
It involves direct contact of solid carbon with molten titanium without the
presence of a continuous TiC interlayer. The product formation occurs via
dissolution of carbon in liquid titanium at the C/Ti(melt) interface and
precipitation of TiC grains. Because of fast diffusion in high-temperature melts
(DC(m)
~105104 cm2/s), diffusion in liquid is not the rate-limiting stage and the
phase-forming process responsible for major heat release is crystallization of the
product particles. Because of the presence of the solid/liquid interface and strong
chemical interaction between the C and Ti atoms, crystallization of the TiC
particles will occur via heterogeneous nucleation at the C/Ti(melt) interface rather
than through homogeneous nucleation in the melt. Besides, the activation energy
for the former is generally lower than for the latter. The nucleated TiC grains must
detach from the carbon particle surface, otherwise a thin TiC layer separating the
parent phases will form and the situation will reduce to the solid-solid-liquid
(quasi-equilibrium) pattern considered above. This process continues until
complete consumption of solid carbon is achieved. Further, in the after-burn zone
of the SHS wave, growth and coalescence of the TiC particles in the metallic melt
can occur. In this case, the final size of product particles will depend on the
conditions of crystallization and subsequent coalescence/sintering but not on the
size of initial reactants. An important factor is the melt lifetime which depends onthe Ti-to-C ratio in the charge, structure of a green pellet determining the melt
spreading conditions and heat exchange with the environment.
This solid-liquid mechanism qualitatively agrees with the results obtained in
experiments on arresting SHS wave in the binary Ti-C [18] and multi-component
Ti-C-Ni-Mo [19] systems where in rapidly quenched samples the formation of
small uniform-sized TiC particles was observed in the molten metal around a
graphite particle, which were apparently detaching from the surface of carbon
particles [18]. This mechanism may also be valid for other interstitial compoundssuch as carbides, borides etc. for which the SHS temperature exceeds the melting
point of a metallic reactant but is below the melting temperatures of the non-metal
and product, and for certain intermetallics.
In this situation, the critical thickness, hcr, of the layer of a primary product
formed on the surface of a metal particle before the attainment of the melting
temperature, which is determined by Eq. (2.20), acquires a precise physical
meaning. This is a criterion for the changeover from the solid-solid-liquid
(diffusion-controlled) mechanism (a slow route of product formation) to thesolid-liquid mechanism (a fast rout). As stated above, in the wave propagation
mode this thickness is small (h
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B. B. Khina36
melting reactant may be small, down to ~110 K/min [67,83], and hence asufficiently thick case of the primary product can be formed on a metal-particle
surface to prevent the liquid core from spreading.
Basing on the above, a diagram of interaction routes can be constructed for
non-isothermal heterogeneous interaction in the Ti-C system. Consider linear
heating with rate vT, which corresponds to CS in the TE mode. Then is the time tm
corresponding to the attainment of melting temperature of titanium is determined
as
tm = [Tm(Ti)T0]/vT. (2.36)
The thickness of a primary TiC case (see scenario 1) corresponding to the
changeover of interaction pathways, is written as
hTiC[(tm)] = hcr, (2.37)
where is determined by formula (2.10) and hcr is calculated according to Eq.(2.20).
Solving Eq. (2.37) together with Eqs. (2.10)-(2.13) and (2.36) permits
obtaining the aforesaid criterion for the changeover of interaction mechanisms at
different heating rates vT and initial particle sizes of the metallic reactant
(titanium) R0
0. A typical radius of titanium particles used in CS is 1-50 m; incalculations the R
00 value was varied within 0.5-150 m. For numerical
calculations, we use three data sets, viz. Nos. 1, 11 and 14 from Table 2.1 for the
following considerations. As seen in Figure 2.4 (a), sets No. 1 and 11 give a
reasonable thickness of the TiC layer grown on the titanium particle surface
during heating in the SHS wave while data set No. 14 yields a maximal (probablyoverestimated) value. The results are presented in Figure 2.7 as a map of phase
formation mechanisms in coordinates heating rate and initial radius of a
titanium particle where lines 1-3 refer to different sets of the diffusion
parameters in the titanium carbide.
Parametric domain I, which lies below the line corresponding to the
attainment of the critical thickness, hcr, of the primary refractory product (TiC),
refers to the diffusion-controlled growth mechanism. This is a slow, quasi-
equilibrium route typical of diffusion annealing in weakly non-isothermalconditions. Here, a sufficiently thick shell of a primary refractory product (in our
system, TiC) is grown of the metal particle surface during heating to Tm so that
after melting the metallic reactant remains inside the shell (hTiC>hcr) and further
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Modeling Diffusion-Controlled Formation of TiC 37
interaction (at T>Tm) proceeds slowly since the rate-limiting stage is still the
solid-state diffusion across this spherical layer.
Figure 2.7. Diagram of phase formation mechanisms for synthesis of titanium carbide in
non-isothermal conditions: domain I is the solid-state diffusion-controlled TiC growth, or
slow route typical of furnace synthesis; domain II is the non-equilibrium crystallization
mechanism, or fast route typical of CS. Calculated using different data sets for diffusionin TiC (see Table 2.1): data set No. 1 (line 1), No.11 (line 2) and No. 14 (line 3).
Thus, the models of CS employing this approach [62-66,68,69] are valid inthis range of parameters (the heating rate and metal particle size). From Figure 2.7
it is seen that for small-sized metal particles (R000.5 m) the quasi-equilibrium
diffusion-controlled growth of the refractory product can proceed at high heating
rates typical of the SHS wave, vT~104-10
5K/s. This agrees qualitatively with
certain results observed during SHS in mixtures of small (nanosized) particles and
in mechanically activated SHS [124].
Domain II, in its physical meaning, corresponds to a fast route typical of CS
where the non-equilibrium dissolution-precipitation mechanism operates toprovide fast completion of the reaction. Here, the refractory product layer formed
during heating from T0 to the melting point of the metallic reactant is thin
(hTiC
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B. B. Khina38
direct contact with the solid non-metal particles, which is a necessary condition
for the dissolution-precipitation interaction route to occur. In this parametric
domain, the model [57-59] is applicable.
The diagram clearly demonstrates the difference, in terms of reaction
mechanisms, between CS (domain II), where fast conversion of reactants into the
products takes place, and traditional furnace synthesis (domain I) where the
interaction proceeds slowly because of a low heating rate and/or large particle size
of the metallic reactant.
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Chapter 3
MODELING INTERACTION KINETICS IN THE
CS OF NICKEL MONOALUMINIDE
3.1.INTRODUCTION
The CS of nickel monoaluminide has gained much attention in literature
[21,23,67] because this compound possesses a unique combination of strength
properties and resistance to gas corrosion at high temperatures, and is used in a
variety of applications as a structural material [125,126] and protective coating
[127]. Besides, of substantial interest is CS in multilayer thin-film Ni-Al system
where the layer thickness, h, varies from ~1 m to ~10 nm and in stacked foilswith h~10-100 m [128]. The former process is used for joining of metallicglasses [129-131], welding of a pure NiAl layer to high-strength superalloys
[132], welding/soldering of microscopic objects such as electronic components,
and similar applications [133-135] while the latter can be used for near-net-shape
manufacturing of NiAl articles [136]. Earlier [137], SHS in laminated Ni/Al foils
was used for experimental modeling of the reaction mechanism in Ni-Al powder
mixtures. In both cases, CS in this system is a subject of extensive experimental
and theoretical investigation.
However, an intricate physical mechanism of phase and structure formation
during CS of NiAl is not well understood yet. It has been demonstrated
experimentally that during SHS in powder mixtures [21,23] and in lamellar Ni-Al
systems [136] with h~10-100 m the dissolution-precipitation (DP) mechanism
takes place: at heating above the Al melting temperature solid nickel dissolves inliquid aluminum and NiAl grains crystallize from the supersaturated melt. This
interaction route may have non-equilibrium nature since the Ni-Al phase diagram
[138,139] contains four compound phases: NiAl3 (melts at 1127 K), Ni2Al3
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existing at T
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Modeling Interaction Kinetics in the CS of Nickel Monoaluminide 41
Besides, there is a difference in interaction patterns between SHS in thin (h
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Modeling Interaction Kinetics in the CS of Nickel Monoaluminide 43
substantially below Tad because of heat removal into the substrate, thus the
formation of only solid final product, NiAl, during CS is possible.
As is known [21,23,136,137], fast interaction during SHS in the given system
begins after melt formation, i.e. above the aluminum melting temperature,
Tm(Al)=933 K, or the eutectic temperature Teu(Al-NiAl3)=913 K (Figure 3.1).
According to the Gibbs phase rule, in a binary system the contact of a solid phase
layer (here pure nickel or Ni-base solid solution) with a two-phase region (here
solid NiAl particles dispersed in Al-base melt at T
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B. B. Khina44
Hence, this situation is close to CS in the TE mode. In this case, the NiAl
phase layer growth is due to solid-state interdiffusion across this layer. The role of
interface kinetics, which was studied in [81], is ignored, and the Al concentrations
at phase boundaries 1/2 and 2/3 correspond to equilibrium values.
3.2.2. Phase Composition of the Reaction Zone
The experimental data concerning the phase formation sequence in Ni-Al
thin-film diffusion couples during isothermal annealing or upon heating at low
rates, vT ~ 1 K/s, are contradictory. According to [163,164], the first phase to form
is NiAl3, and the other equilibrium phases of the Ni-Al system can appear only
after the NiAl3 layer exceeds a certain thickness or after complete consumption of
one of the starting metals. The final phase composition corresponds to the initial
Al-to-Ni thickness ratio of the films. However, in [165,166] the growth of B2-
NiAl with a metastable concentration of about 63 at.% Al as the first phase was
observed. In [167], the formation of metastable -phase (Ni2Al9) during annealingof Ni/Al multilayers was detected, which quickly transformed into stable NiAl3
due to interaction with nickel. At high heating rates, which are typical of CS, theconditions for the first phase nucleation at the Ni/Al interface may be different
from those at slow heating.
For constructing the model, the following basic assumption is made. It is
considered that a thin continuous NiAl layer nucleates at the initial Ni/Al interface
at T0 = Teu(Al-NiAl3) = 913 K. During the growth of NiAl compound in the whole
temperature range T0 T Tad=Tm(NiAl), the interlayers of other equilibriumphases of the Al-Ni system (Figure 3.1) [138,139], viz. NiAl3 (Tm=1127 K),
Ni2Al3 (Tm=1406 K) and Ni3Al (Tm=1668 K), are absent, and metastableequilibria at interfaces Al-base melt/solid NiAl and NiAl/Ni(s) are described
by the corresponding equilibrium solubility-limit lines, viz. lines GFE/LK and
HIJ/ABC, respectively (Figure 3.1). The latter presumption is only for the sake of
simplicity, in order to avoid cumbersome calculation of