1

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.

5

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.