a phase-field approach to solid-solid phase ... · extended for diffusive, electric, and magnetic...
TRANSCRIPT
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Findings
One of the main goals of the project is to develop and verify nanoscale phase field
approach (PFA) to phase transformations (PTs) between multiple phases, dislocation
nucleation and motions, and interaction between PT and dislocational plasticity and
twinning. The following breakthrough results are obtained in these directions.
1. Phase-Field Approach to Solid-Solid Phase Transformations via Intermediate
Interfacial Phases under Stress Tensor [1]
A thermodynamically consistent phase-field (PF) theory for phase transformations
(PTs) between three different phases is developed with emphases on the effect of a stress
tensor and interface interactions. The phase equilibrium and stability conditions for
homogeneous phases are derived and a thermodynamic potential, which satisfies all these
conditions, is introduced using polar order parameters (ϒ, ϑ), where ϒ and ϑ are the
radial and angular order parameters, respectively. Here we assumed disordered phase
(melt or amorphous solid) to be located in the origin of coordinate system and two
crystalline solid phases on the circumference of the unit circle (Fig. 1). Therefore, each
crystal phase is determined by the angular order parameter ϑ and disordering of each
crystal phase is determined by the radial order parameter ϒ. Propagation of a crystal-
crystal (CC) interface containing nanometer-sized intermediate disordered interfacial
phases (IP) and particularly an interfacial intermediate melt (IM) is studied for an HMX
energetic crystal using the developed PF model. The scale effects (the ratio of widths of
CC to crystal-melt (CM) interfaces, kδ), the effect of the energy ratio of CC to CM
interfaces (kE), and the temperature on the formation and stability of IM are investigated.
An interaction between two CM interfaces via an IM, which plays a key role in defining a
well-posed problem and mesh-independent solution, is captured using a special gradient
energy term. It is shown that the elastic energy promotes the formation and retaining of
IM, hundreds of degrees below the melting temperature, and also increases the interface
velocity and width of IM. However, it surprisingly increases nucleation temperature for
the IM and drastically reduces (by 16 times for HMX energetic crystals!) the energy of
the critical nucleus of the IM within the CC interface. Because of last result, our paper
was published in Nano Letters [2].
The developed PF model is applicable for the general case of PTs between three
phases and can be applied (adjusted) to other physical phenomena. The current model is
applicable specifically for the transformation between two solid phases via an unstable
IM, known also as a virtual melt, far below the thermodynamic melting temperature. This
model is also applicable to interfacial phases that emerge in a wide range of processes
such as premelting and prewetting, surface-induced premelting and PT, intrinsic PT that
occurs in pure materials, and extrinsic PT that happens in non-pure materials involving an
adsorption of an impurity or dopant. Further applications encompass
premelting/disordering at grain boundaries, martensitic PTs, and developing the
interfacial phase diagrams. However, theory cannot be generalized noncontradictory for
more than three phases and new theory (see item 2) is developed.
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Fig. 1. Plot of the developed Gibbs potential for HMX at CC equilibrium temperature and
hydrostatic pressure of 1 MPa, at different critical values for the loss of stability of each
crystal phase C1 and C2. The 3D plot of the potential surface is shown in (e) along with
its contour plot, for the same critical temperatures as plotted in (a). (g,h) Effect of kE and
kδ on the formation and retention of the IM. The stationary minimum values of ϒ, i.e.,
ϒmin, are plotted at e=432K and a0=0.01, for different kδ values for a model without
mechanics (g) and with mechanics (h).
2. Multiphase phase field theory for temperature- and stress-induced phase
transformations [2]
Thermodynamic Ginzburg-Landau potential for temperature- and stress-induced
phase transformations (PTs) between n phases is developed. It describes each of the PTs
with a single order parameter without an explicit constraint equation, which allows one to
use an analytical solution to calibrate each interface energy, width, and mobility;
reproduces the desired PT criteria via instability conditions; introduces interface stresses,
and allows for a controlling presence of the third phase at the interface between the two
other phases. A finite-element approach is developed and utilized to solve the problem of
nanostructure formation for multivariant martensitic PTs. Results are in a quantitative
agreement with the experiments (Fig. 2d). The developed approach is applicable to
various PTs between multiple solid and liquid phases and grain evolution and can be
extended for diffusive, electric, and magnetic PTs.
Advantages:
1. Theory allows describing each of the PTs with a single order parameter, in contrast
to all known theories for multivariant martensitic transformations and multiple twinning.
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This allows one to use analytical solution to calibrate each interface energy, width, and
mobility.
2. In contrast to all theories for multiphase materials, this is achieved without explicit
constraint equation. As it was demonstrated, imposing explicit constraint produces
significant problems in the theory, in particular, does not allow introducing the desired
transformation criteria via thermodynamic instability conditions.
3. The problem is resolved by combining our previous theory for multivariant
martensitic transformations with the terms that penalize deviation of the trajectory in the
order parameter space from the desired straight lines connecting each of two phases. It is
demonstrated that this approximately (but with controlled accuracy) reproduces all the
desired constraints.
4. The developed theory satisfies all the desired conditions. It introduces the desired
phase transformation criteria via thermodynamic instability conditions.
5. It allows for the first time for a multiphase system to include consistent expression
for interface stresses for each interface.
6. It allows controlling presence of the third phase at the interface between two other
phases.
Fig. 2. Initial conditions (a) and stationary solution for two-variant martensitic
nanostructure exhibiting bending and splitting martensitic tips based on the current theory
(c); experimental nanostructure from Boullay et al., J. de Physique IV 11, 23, 2001 (d).
Green color is for austenite, blue and red are for martensitic variants P1 and P2.
Results of the current simulations resemble the experimental nanostructure (Boullay
et al., J. de Physique IV 11, 23, 2001) and quantitatively reproduce the bending angle
(Fig. 2d).
3. Advanced Phase-Field Approach to Dislocation Evolution [3]
A qualitatively new, thermodynamically consistent, large strain PFA to dislocation
nucleation and evolution at the nanoscale is developed. Each dislocation is defined by an
order parameter, which determines the magnitude of the Burgers vector for the given slip
planes and directions. The kinematics is based on the multiplicative decomposition of the
deformation gradient into elastic and plastic contributions. The relationship between the
rates of the plastic deformation gradient and the order parameters is consistent with
phenomenological crystal plasticity. Thermodynamic and stability conditions for
homogeneous states are formulated and satisfied by the proper choice of the Helmholtz
free energy and the order parameter dependence of the Burgers vector. They allow us to
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reproduce desired lattice instability conditions and a stress-order parameter curve, as well
as to obtain a stress-independent equilibrium Burgers vector and to avoid artificial
dissipation during elastic deformation. The Ginzburg-Landau equations are obtained as
the linear kinetic relations between the rate of change of the order parameters and the
conjugate thermodynamic driving forces. A crystalline energy coefficient for dislocations
is defined as a periodic step-wise function of the coordinate along the normal to the slip
plane, which provides an energy barrier normal to the slip plane and determines the
desired, mesh-independent height of the dislocation bands for any slip system orientation.
Gradient energy contains an additional term, which excludes the localization of a
dislocation within a height smaller than the prescribed height, but it does not produce
artificial interface energy. An additional energy term is introduced that penalizes the
interaction of different dislocations at the same point. Non-periodic boundary conditions
for dislocations are introduced, which include the change of the surface energy due to the
exit of dislocations from the crystal. Obtained kinematics, thermodynamics, and kinetics
of dislocations at large strains are simplified for small strains and rotations as well. Finite
element solutions to some physically important problems are found (see, e.g., Fig. 3) and
interpreted. This problem represents the first step of the problem on promotion of PTs
under compression and shear in rotational diamond anvil cell (RDAC).
Fig. 3. Schematics of a sample with two nanograins under compression and shear with a
stationary dislocation nanostructure at prescribed shear of 0.2 with stationary values of
pressure p = 4.3 GPa and shear stress τ = 2.6 GPa in the left grain, and p = 5.3 GPa and τ
= 5.5 GPa in the right grain. Evolution of dislocations in both grains is presented as well.
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A similar approach can be developed for partial dislocations and expended for
dislocation reactions.
4. Phase field approach to interaction of phase transformation and dislocation
evolution: general theory [4,6]
The first thermodynamically consistent PFA for coupled multivariant martensitic PTs,
including cyclic PTs, variant-variant transformations (i.e., twinning), and dislocation
evolution is developed at large strains. One of the key points is in justification of the
multiplicative decomposition of the deformation gradient into elastic, transformational,
and plastic parts, in which plastic part includes four mechanisms: dislocation motion in
martensite along slip systems of martensite and slip systems of austenite inherited during
PT and dislocation motion in austenite along slip systems of austenite and slip systems of
martensite inherited during reverse PT. Plastic part of the velocity gradient for all these
mechanisms is defined in the crystal lattice of the austenite utilizing just slip systems of
austenite and inherited slip systems of martensite and just two corresponding types of the
order parameters. The explicit expressions for the Helmholtz free energy, transformation
and plastic deformation gradients are presented that satisfy the formulated conditions
related to homogeneous thermodynamic equilibrium states of crystal lattice and their
instabilities. In particular, they result in constant (i.e., stress- and temperature-
independent) transformation deformation gradient and Burgers vectors. Thermodynamic
treatment resulted in the determination of the driving forces for change of the order
parameters for PTs and dislocations as well as in the boundary conditions for the order
parameters that include variation of the surface energy during PT and exit of dislocations.
Ginzburg-Landau equations for dislocations include variation of properties during PTs,
which in turn produces additional contributions from dislocations to the Ginzburg-
Landau equations for PTs. Similar theory can be developed for PFA to dislocations and
other PTs, like reconstructive PTs and diffusive PTs described by the Cahn-Hilliard
equation, as well as twinning and grain boundaries evolution.
5. Phase field approach to interaction of phase transformation and dislocation
evolution: finite element simulations [5,6]
The complete system of phase field equations for coupled martensitic transformations,
dislocation evolution, and mechanics at large strains is presented. Finite element
approach is utilized to solve this system for two important problems.
(a) The first one is related to simulation of shear strain-induced phase transformation
at the evolving dislocation pile ups in a nanosized bicrystal. This problem is used to
explain the drastic reduction in PT pressure for strain-induced PT under high pressure in
RDAC in comparison with PT under hydrostatic conditions. Plasticity plays a dual part in
interaction with PT. Dislocation pile ups produce strong stress tensor concentrators that
lead to barrierless martensite nucleation. On the other hand, plasticity in the transforming
grain relaxes these stress concentrators suppressing PT. The final stationary martensite
morphology is governed by the local thermodynamic equilibrium, either at the interfaces
or in terms of stresses averaged over the martensitic region or entire grain. This is very
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surprising because of strong heterogeneity of stress fields and is in contrast to previous
statements that phase equilibrium conditions do not enter the description of strain-
induced transformations. This result will be utilized for the scaling up our simulations by
producing nano-to-microscale transition.
Fig. 4. Schematics of the sample under simple shear. Slip systems in the left and right
grains are shown.
Fig. 5. Stationary martensitic and dislocational microstructure in the right grain for four
different cases. The left and right columns for each case correspond to PT without and
with plasticity in the right grain, respectively. Dislocation structures in the left grain are
shown above the right grain.
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(b) The second problem is devoted to martensitic plate propagation through bicrystal
during temperature-induced transformation. For elastic growth (without dislocations) and
large thermal driving force, a complex transformation path with plate branching and
direct and reverse transformations is observed, which still ends with the same stationary
nanostructure as for a smaller driving force and traditional transformation path. Sharp
grain boundary arrests plate growth at relatively small driving force, exhibiting an
athermal friction. For elastoplastic growth, the generation of dislocations produces
athermal friction and arrests the plate below some critical driving force, leading to a
morphological transition from plate to lath martensite. The width of the martensitic plate
increases in comparison with elastic growth due to internal stress relaxation. Plate growth
is accompanied by nucleation of dislocations within martensite and remaining them in
martensite, nucleation of dislocations at the tip of a plate and spreading them in austenite,
and passing some of dislocations through martensite, martensite-austenite interface, and
then in austenite. Due to existence for each temperature of a stationary equilibrium
martensite microstructure and concentration, for large enough observation time one
observes athermal, rate- and time-independent kinetics, even while local kinetics is rate
dependent. In the final structure, most of dislocations are in martensite despite the three
times larger yield strength than for austenite, which is consistent with experiments.
The interaction between phase transformations and dislocations drastically changes
transformation thermodynamics, kinetics, and microstructure and is the most important
basic and applied problem in the study of martensite nucleation and growth. Numerous
applications include heat and thermomechanical treatment of materials to obtain desired
structure and properties; transformation-induced plasticity; synthesis of materials under
high pressure and high pressure with large plastic deformations, e.g., during ball milling
and in rotational diamond anvil cell; and phase transformations during friction,
indentation, surface treatment, and projectile penetration.
Interaction between phase transformations and plasticity is also a key point in
developing materials with high strength and ductility, in particular, utilizing
transformation toughening.
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Fig. 6. Simultaneous evolution of martensitic plate and dislocations in a bicrystal.
6. Large elastoplasticity under megabar pressure: formulation and application to
compression of a sample in a diamond anvil cell
The study of materials behavior under extremely high pressure is of great
fundamental and applied interests. In high pressure research, static high pressure is
produced by compression of a thin sample by two diamonds in a diamond anvil cell
(DAC). This process is accompanied by large plastic deformation (sample thickness is
reduced by a factor of 60), and finite elastic deformation of a sample and diamond. It
involves the geometric and physical nonlinearities. Previous FEM results in literatures
failed to reproduce experimental pressure distribution at the contact surface between
sample and diamond at megabar pressures (Fig. 7). To reproduce and interpret
experimental phenomena and reveal mechanical responses in DAC, a thermodynamically
consistent system of equations for large elastic and plastic deformation of an isotropic
material with a nonlinear elasticity rule and pressure dependent yield condition is
formulated for a sample. The Murnaghan elasticity rule and pressure-dependent J2
plasticity are utilized. The finite-strain third order elasticity rule for cubic crystals is
utilized for diamond. The finite element method (FEM) algorithm is presented with
emphasis on the stress update procedure and derivation of the consistent tangent moduli.
It is implemented as a user material subroutine in the FEM code ABAQUS. All material
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parameters for a rhenium sample and diamond are calibrated based on the experimental
and atomistic simulation results in literature. Evolution of the stress and strain tensor
fields in a sample and diamond is studied up to pressure of 285 GPa in the sample. A
good correspondence between numerical and experimental pressure distributions at the
diamond-sample contact surface is obtained (Fig. 7). The obtained results pave a new
way for understanding the mechanical response in DAC, extraction and interpretation of
materials properties from the heterogeneous fields, design of experiments, and for
optimum design of DAC for reaching the maximum possible pressure in a volume
sufficient for the desired measurements. However, the cupping phenomenon observed in
experiment still cannot be reproduced in our simulations, which will be further studied in
future.
0 50 100 1500
50
100
150
200
250
300 experiments in Hemley et al., 1997
curent simulation results
curent simulation results
curent simulation results
simulations in Metkel et al., 1999
p
(G
Pa)
r (m) (a) (b)
Fig. 7. Pressure distributions at the contact surface (a), and stress zz distribution in the
sample with the growth of applied stresses (b). In (a), lines with symbols are the current
simulation results; black solid lines are experimental data from (Hemley et al., 1997); the
green dashed line is the simulation results from (Merkel et al., 1999). In (b), the zoomed
central part of a sample is shown above the sample.
7. New design of rotational diamond anvil cell
Jointly with the researchers from the Institute for Superhard Materials of the
Ukrainian Academy of Sciences (Kiev, Ukraine), a new automated RDAC for in situ x-
ray studies has been developed and manufactured [7]. The main advantage of the design
is reduction of the size (and weight) of the new cell by almost a factor of 2 with the same
size of an anvil and accessible pressure.
References:
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1. Momeni K., Levitas V. I., and Warren J. A. The Strong Influence of Internal Stresses
on the Nucleation of a Nanosized, Deeply Undercooled Melt at a Solid–Solid Phase
Interface. Nano Letters, 2015, 15, 2298-2303.
2. Levitas V.I. and Roy A. M. Multiple phase field theory for temperature- and stress-
induced phase transformations. Physical Review B, 2015, Vol. 91, 174109.
3. Levitas V.I. and Javanbakht M. Thermodynamically consistent phase field approach to
dislocation evolution at small and large strains. Journal of the Mechanics and Physics of
Solids, 2015, Vol. 82, 345-366.
4. Levitas V.I. and Javanbakht M. Interaction between phase transformations and
dislocations at the nanoscale. Part 1. General phase field approach. Journal of the
Mechanics and Physics of Solids, 2015, 82, 287–319.
5. Javanbakht M. and Levitas V.I. Interaction between phase transformations and
dislocations at the nanoscale. Part 2. Phase field simulation examples. Journal of the
Mechanics and Physics of Solids, 2015, 82, 164-185.
6. Levitas V.I. and Javanbakht M. Interaction of phase transformations and plasticity at
the nanoscale: phase field approach. Materials Today, 2015, DOI:
10.1016/j.matpr.2015.07.334 (in press).
7. Novikov N.V., Shvedov L.K., Krivosheya Yu. N., and Levitas, V.I. New Automated
Shear Cell with Diamond Anvils for in situ Studies of Materials Using X-ray Diffraction.
Journal of Superhard Materials, 2015, 37, 1-7.
This work was leveraged by the support from DARPA, ARO, ONR, and PI’s endowment.
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Presentations
Conference Presentations (Talks and Proceedings Abstracts)
1. V. I. Levitas. Interface- and Surface-Induced Phenomena during Phase
Transformations: Phase Field Approach. 51th Annual Meeting Society of
Engineering Science, Lafayette, IN (Invited lecture), November 1-3, 2014.
2. V. I. Levitas, M. Javanbakht. Phase field approach to interaction of phase
transformations and plasticity at large strains. 51th Annual Meeting Society of
Engineering Science, Lafayette, IN, November 1-3, 2014.
3. B. Feng and V. I. Levitas. Strain-induced phase transformation under.
compression and compression and torsion in a diamond anvil cell: simulations of
a sample and gasket. 51th Annual Meeting Society of Engineering Science,
Lafayette, IN, October 1-3, 2014.
4. V. I. Levitas. Phase Transformations: Geometrically Nonlinear Phase Field
Approach with Interface Stresses. ASME International Mechanical Engineering
Congress, Montreal, Canada (Invited lecture), November 16-20, 2014.
5. V. I. Levitas, M. Javanbakht. Interaction between phase transformations and
dislocations at the nanoscale: Phase field approach. Plasticity'15 International
Symposium, Montego Bay, Jamaica (Keynote lecture), January 3-9, 2015.
6. W.A. Goddard, V.I. Levitas, Ma Y. Multiscale Theory and Experiment in Search
for and Synthesis of Novel Nanostructured Phases in BCN Systems. NSF
DMREF Grantee Meeting, Bethesda, MD, January 12-13, 2015.
7. V.I. Levitas and A. Roy, Multiphase Phase Field Approach with Elastic and
Interface Stresses. European Solid Mechanics Conference, Madrid, Spain, July 6-
10, 2015.
Seminars given by the PI
1. Levitas V.I. Ways of characterization of pressure-, stress-, and strain-
induced phase transformations. Geophysical Laboratory, Carnegie
Institution of Washington, 12//13/2014.
2. Levitas V.I. Interaction between phase transformations and dislocations at
the nanoscale: Phase field approach. University Erlangen-Nuernberg,
Nuernberg, Germany, June 23, 2015.
3. Levitas V.I. Interaction between phase transformations and dislocations at
the nanoscale: Phase field approach. Max-Planck-Institute for Steel
Research, Duesseldorf, June 23, 2015.
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Organization of Symposia
Name of conference and
organization
Description of duties
Plasticity'16 International Conference
(Big Island, Hawaii), 01/ 2016
Organization of the Symposium on Phase Transformations
Annual Meeting of the Society of
Engineering Sciences (West
Lafayette, IN), 10/2014
Organization of the Symposium on Coupling Plasticity and
Phase Transformations
Plasticity'15 International Conference
(Montego Bay, Jamaica), 01/2015
Organization of the Symposium on diffusive and displacive
deformation and transformation processes versus plasticity
All these symposia were devoted to interdisciplinary interaction between theoretician,
computational researchers, and experimentalists, in a spirit of MGI philosophy.
PI’s Honors
Who's Who in America, 2014 and 2015
Who's Who in the World, 2014 and 2015
Dictionary of International Biography, 2014
2000 Outstanding Intellectuals of the 21st Century, 2014
Who's Who in Science and Engineering, 2016
Student Award
Biao Feng, PhD student
Iowa State University Research Excellence Award for Fall 2014;
2015 Alexander Lippisch Memorial Scholarship.
Kasra Momeni, PhD student
Iowa State University Teaching Excellence Award for Fall 2014;
Research Award from Graduate and Professional Student Senate of ISU, Spring
2015;
Teaching Award from Graduate and Professional Student Senate of ISU, 2015;
Iowa State University Research Excellence Award for Summer 2015.
Mahdi Javanbakht, post doc
Karas Award for Outstanding Dissertation in the Mathematical and Physical
Sciences, and Engineering discipline at Iowa State University, 2014.