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Acta Materialia 75 (2014) 337–347

Phase-field modeling of martensitic microstructure in NiTishape memory alloys

Yuan Zhong a, Ting Zhu a,b,⇑

a Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USAb School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA

Received 26 July 2013; received in revised form 6 April 2014; accepted 7 April 2014

Abstract

A phase-field model is developed to study the cubic to monoclinic martensitic phase transformation in nickel–titanium (NiTi) shapememory alloys. Three-dimensional phase-field simulations show the nucleation and growth of monoclinic B190 multivariants that form apolytwinned martensitic microstructure. Parametric studies demonstrate that mechanical constraints and crystallographic orientationgovern the patterning of martensitic twin variants in the formation of strain-accommodating microstructures. Pairing of twin variantsis studied by comparing the phase-field simulation results with the crystallographic solutions of compatible twins. The phase-field methoddeveloped in this work is generally applicable to simulate the dynamic microstructure evolution of metals and alloys that produce low-symmetry phases through martensitic transformation.� 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Phase-field simulation; Shape memory alloys; Polytwinned microstructure

1. Introduction

Nickel–Titanium (NiTi) is the most widely used shapememory alloy [1–4]. Its shape memory effect is primarilygoverned by the martensitic phase transformation betweenthe cubic B2 (austenite) and the monoclinic B190 phase(martensite) [5]. Understanding the mechanism of martens-itic transformation in NiTi is essential to controlling andoptimizing its shape memory behavior.

The martensitic phase transformation in NiTi has beenstudied by a variety of modeling methods. The continuummodels are usually focused on the crystallography andcompatibility of the phase transformation and twinmicrostructure [6–8]. First-principles calculations are wellsuited to investigating the atomic-level structures and their

http://dx.doi.org/10.1016/j.actamat.2014.04.013

1359-6454/� 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights r

⇑ Corresponding author at: Woodruff School of Mechanical Engineer-ing, Georgia Institute of Technology, Atlanta, GA 30332, USA.

E-mail address: ting.zhu@me.gatech.edu (T. Zhu).

stabilities, as well as phase-transformation pathways[9–13]. Atomistic studies based on empirical interatomicpotentials can explore the phase transformation andmartensitic microstructure in systems larger than thoseaccessible by first-principles methods [14–18]. However,both first-principles and interatomic-potential-based stud-ies are severely limited by the spatial and temporal scalesthat are achievable. Such limitations can be alleviated byuse of the phase-field model, which is particularly suitedto studying the dynamic evolution of martensitic micro-structures [19–24].

The NiTi system generally involves a variety of metasta-ble phases (B2, B19, B190, R, etc.), martensite variants (e.g.12 variants in the B190 phase) and twin structures (such astype I, II and compound twins). Such phase and structurecomplexities pose significant challenges to computationalmodeling. Nevertheless, progress has been recently madein the development of phase-field models for the NiTi sys-tem. For example, Shu and Yen developed a multivariant

eserved.

338 Y. Zhong, T. Zhu / Acta Materialia 75 (2014) 337–347

model to study the martensitic microstructure of the Rphase [25]. Yang and Dayal proposed a simple energy func-tion to describe the B190 multivariants [26]. Both modelsassumed the penalty-based energy function and wereapplied to two-dimensional (2-D) phase-field simulations.However, the phase-field model with a physics-basedLandau-type energy function is still not available. TheLandau-type polynomial energy function is favored in thephase-field model, because it can facilitate a direct linkbetween the model parameters and physical properties suchas undercooling temperature [19] and defect energy [27]. Itis, however, non-trivial to construct a Landau-type polyno-mial energy function to characterize the martensitic trans-form to low-symmetry phases. In the case of NiTi, this isdue to the difficulty of representing the 13 coexisting meta-stable energy wells on the system energy landscape, whichrespectively correspond to the cubic B2 phase and 12monoclinic B190 variants. Meanwhile, other metastableenergy wells should be eliminated to avoid the interferenceof physically irrelevant states. In this work, we construct aneffective Landau-type polynomial energy function andperform 3-D phase-field simulations of the B2–B190 phasetransformation. The results reveal the nucleation andgrowth of polytwinned morphology of martensitic micro-structures. The effects of mechanical constraints and crys-tallographic orientation on the patterning of multivariantsin the formation of strain-accommodating microstructuresare investigated.

2. Phase-field model

We take a single crystal of B2 austenite as the startingconfiguration. This parent austenitic phase can transformto B190 martensite when the temperature is reduced belowthe martensite start temperature. The phase-field modelprovides the solutions of the temporal evolution of phasesand microstructures by numerically solving the time-dependent partial differential equations of field variables.Twelve continuous field variables {g1, . . . , g12} between 0and 1 are defined to describe the B2–B190 transformation,which involves one cubic B2 phase and 12 monoclinicB190 variants. The austenitic B2 phase corresponds to thecase of g1 = � � � = g12 = 0. Variant i of the martensiticB190 phase is represented by gi = 1 and gj = 0 for all j – i.

The stress-free strain, which arises from local phasetransformation, can be described as:

e�ðxÞ ¼X12

i¼1

giðxÞe0i ; ð1Þ

where e� (x) is the stress-free strain at a spatial position x,and e0

i is the strain of the complete transformation from B2to B190 for variant i.

The free energy of the system, F, can be described by thevolume integral of three free energy densities, includinglocal (chemical) energy density flocal, gradient energy den-sity fgrad, and elastic strain energy density fel:

F ¼Z

Vðflocal þ fgrad þ felÞdV : ð2Þ

2.1. Local free energy

The local free energy density flocal is governed by thebulk thermodynamic properties of the system. A Landau-type polynomial is used to represent flocal as follows:

flocal ¼ f0 þ Df ðT Þ 1

2AX12

i¼1

g2i

!� 1

3BX12

i¼1

g3i

!(

þ 1

4CX12

i¼1

g2i

!2

þ 1

4DX12

i¼1

g4i

!9=;; ð3Þ

where f0 is the free energy density of the austenitic phaseand can be set as zero, Df(T) is the difference of the freeenergy density between austenite and martensite thatdepends on temperature T, and A, B, C and D are con-stants characterizing the shape of the free energy densityfunction flocal. Note that parameters A, B, C and D cannotbe arbitrarily assigned, and three constraints must be satis-fied. Firstly, the partial derivative of flocal with respect tothe field variables {g1, . . ., g12} must be zero wheng1 = � � � = g12 = 0, or gi = 1 and gj = 0 for all j – i, suchthat the austenitic and martensitic phases correspond tolocal energy minima. Secondly, the difference in the freeenergy density between austenite and martensite shouldbe Df(T). These two requirements lead to:

�Aþ B� C � D ¼ 012A� 1

3Bþ 1

4C þ 1

4D ¼ �1

(: ð4Þ

In this work we choose to represent flocal with the followingset of parameters A = 1, B = 15, C = 7 and D = 7, whichprovide physically reasonable phase-field results for mar-tensitic microstructure evolution. The driving force on eachfield variable gi associated with flocal is given by:

F locali ¼ � @flocal

@gi

¼ Df ðT Þ �Agi þ Bg2i � Cgi

X12

j¼1

g2j

!� Dg3

i

( ): ð5Þ

Thirdly, the B2 phase should be a thermodynamicallymetastable state. This requirement is related to the choiceof Df(T) and will be examined later.

2.2. Gradient energy

The gradient energy density fgrad is the nonlocal part ofthe chemical free energy density, which characterizes theenergy of the interface between neighboring twin variants.We express fgrad in terms of the gradient of field variables:

fgrad ¼1

2

X12

p¼1

bijðpÞ@gp

@xi

@gp

@xj; ð6Þ

Y. Zhong, T. Zhu / Acta Materialia 75 (2014) 337–347 339

where the Einstein summation convention is applied onlyfor index i and j. In Eq. (6), the coefficients bij(p) are thecomponents of a semipositive definite tensor; they are notnecessarily the same among different field variables gp,and may be anisotropic depending on the direction of thegradient given by partial derivatives with respect to spatialcoordinates xi and xj. In order to capture the essentialphysical effects of phase transformation and microstructureevolution with reduced numerical complexity, isotropicgradient energy is assumed in this work. Namely, we takebij(p) = bdij, where dij is the Kronecker delta. It follows thatEq. (6) is reduced to:

fgrad ¼1

2

X12

p¼1

bjrgpj2: ð7Þ

The driving force on each field variable gi associated withfgrad is [25]:

F gradi ¼ br2gi: ð8Þ

2.3. Elastic energy

The elastic energy density fel is given by:

fel ¼1

2ðe� e�ÞTCðe� e�Þ; ð9Þ

where e = (e11, e22, e33, 2e23, 2e31, 2e12)T is the total strain,and e� ¼ ðe�11; e

�22; e

�33; 2e�23; 2e�31; 2e�12Þ

T is the stress-free strainof phase transformation, and C is the 6 � 6 stiffness matrixin the Voigt notation. The constitutive equation can bederived from Eq. (9) with the stress given by:

r ¼ ðr11; r22; r33; r23; r31; r12ÞT ¼ Cðe� e�Þ: ð10ÞThe driving force acting on each field variable gi associatedwith fel is:

F eli ¼ �

@fel

@gi¼ ½Cðe� e�Þ�T @e

�

@gi¼ rT @e

�

@gi: ð11Þ

In this work, we study the boundary conditions undereither applied strain and/or zero applied stress (ra = 0),such that the term involving the applied traction vanishes.Both the total strain e and stress r can be solved using thetechnique of Fourier transform, and the detailed formulasare derived in the Appendix A.

2.4. Phase-field equation

The evolution of the phase field is governed by the time-dependent Ginzburg–Landau (TDGL) equation, which is astochastic phase-field kinetic equation based on theassumption that the rate of change of field variables is pro-portional to the thermodynamic driving force:

@gi

@t¼X12

j¼1

bLij F localj þ F grad

j þ F elj

� �h iþ niðx; tÞ; ð12Þ

where bLij is the matrix of kinetic coefficients, and ni(x, t) isthe Langevin noise term which follows the normal

distribution and is mutually independent at different loca-tions and times. To satisfy the requirement of the fluctua-tion–dissipation theorem [19], the correlation of ni(x, t) isgiven by:

hniðx; tÞnjðx0; t0Þi ¼ 2kBT bLijdijdðx� x0Þdðt � t0Þ; ð13Þ

where kB is the Boltzmann constant and d is the Dirac deltafunction. For simplicity, the kinetic coefficient bLij isassumed diagonal, i.e. bLij ¼ Ldij, with the assumption thatthe driving force on each field variable gi does not affect theevolution of field variable gj when i – j. Substitution ofEqs. (5) and (8) into Eq. (12) yields:

@gi

@t¼ L Df ðT Þ �Agi þ Bg2

i � Cgi

X12

j¼1

g2j

!� Dg3

i

" #(

þ br2gi þ F eli

)þ niðx; tÞ: ð14Þ

2.5. Numerical simulation

The phase-field simulations are performed in a 3-Dcubic cell, subjected to periodic boundary conditions inall three directions. We discretize the 3-D cell into uniformgrids and the time into equal steps. All of the field variablesat time step n are represented in the form of gn

i ðx; nDtÞ fori = 1, . . ., 12, where Dt denotes the time step size. It is con-venient to normalize the length and time scales, therebyeliminating the unnecessary parameters. We define thedimensionless space coordinate ex1 ¼ x1=l0, ex2 ¼ x2=l0 andex3 ¼ x3=l0, where l0 is the size of the grid cell, and thedimensionless time et ¼ tLDf ðT Þ and Det ¼ DtLDf ðT Þ. It fol-lows that Eq. (14) is normalized as:

@gi

@et ¼ �Agi þ Bg2i � Cgi

X12

j¼1

g2j

!� Dg3

i

" #þ eb er2gi

þ F eli

Df ðT Þ þeniðex; tÞ; ð15Þ

where er2 ¼ @2

@ex21

þ @2

@ex22

þ @2

@ex23

, eb ¼ bDf ðT Þl2

0

and eniðex; tÞ¼ 1

LDf ðT Þ niðex; tÞ. The random variable eniðex; tÞ, which is

mutually independent at different space coordinates andtime steps, follows the normal distribution with the mean

of zero and the variance of 2kBTDf ðT Þl3

0

.

To solve a partial differential equation of the heat-equation type by numerical integration, one can typicallyapply the forward Euler (explicit) method, the backwardEuler (implicit) method or the Crank–Nicolson (implicit)method. The nonlinear terms, i.e. the first and third termsin Eq. (15), pose a computational challenge to either thebackward Euler or the Crank–Nicolson method. On theother hand, the stability condition of the forward Eulermethod due to the Laplace operator limits the time stepsize Det � Dex2. To overcome these difficulties, we use thesemi-implicit Fourier-spectral method proposed by Chenand Shen [21] which provides an efficient and accurate

340 Y. Zhong, T. Zhu / Acta Materialia 75 (2014) 337–347

solution for the TDGL equation. The key of this semi-implicit method is to calculate the Laplace term implicitlyand the nonlinear terms explicitly, such that Eq. (15) canbe discretized as:

gnþ1i � gn

i

Det ¼ F locali ðgnðx; nDetÞÞ

Df ðT Þ þ eb er2gnþ1i ðx; ðnþ 1ÞDetÞ�

þ F eli ðgnðx; nDetÞÞ

Df ðT Þ

�þ eniðex; ðnþ 1ÞDetÞ: ð16Þ

It is computationally efficient to solve Eq. (16) in theFourier space, so as to avoid the inverse Fourier transfor-mation of stress. To this end, the Laplace operator istransformed to �4pðs2

1 þ s22 þ s2

3Þ in Fourier space, wheres = (s1, s2, s3)T is the coordinate in Fourier space. Eq.(16) can be transformed to:

dgnþ1i ¼ 1

1þ 4pebðs21 þ s2

2 þ s23ÞDet bgn

i þDet

Df ðT ÞdF local

i ðgnÞ�

þ DetDf ðT Þ

dF eli ðgnÞ

�ð17Þ

2.6. Model parameters

In Eq. (1), the stress-free strains of the B2–B190 transfor-mation are described by 12 field variables {g1, . . ., g12} thatrepresent 12 B190 variants, respectively. When the globalCartesian coordinate system is aligned with the cubic axesof the parent B2 phase, the 12 transformation strain ten-sors e0

i (i = 1, . . ., 12) in Eq. (1) are given by:

e01 ¼

h q q

q r s

q s r

0B@1CA; e0

2 ¼h �q �q

�q r s

�q s r

0B@1CA; e0

3 ¼h �q q

�q r �s

q �s r

0B@1CA;

e04 ¼

h q �q

q r �s

�q �s r

0B@1CA; e0

5 ¼r q s

q h q

s q r

0B@1CA; e0

6 ¼r �q s

�q h �q

s �q r

0B@1CA;

e07 ¼

r �q �s

�q h q

�s q r

0B@1CA; e0

8 ¼r q �s

q h �q

�s �q r

0B@1CA; e0

9 ¼r s q

s r q

q q h

0B@1CA;

e010 ¼

r s �q

s r �q

�q �q h

0B@1CA; e0

11 ¼r �s q

�s r �q

q �q h

0B@1CA;

e012 ¼

r �s �q

�s r q

�q q h

0B@1CA: ð18Þ

The components of e0i in Eq. (18) have been calculated by

Hane and Shield [7] using the lattice constants given byOtsuka et al. [5], i.e. h = �0.0437, r = 0.0243,q = �0.0427 and s = 0.0580. When the global Cartesiancoordinate system is not aligned with the cubic axes ofthe parent B2 phase, a rotation operation of Re0

i RT isrequired, where R is the rotation matrix.

The elastic constant matrix C is taken from the densityfunctional calculations by Hatcher et al. [10], i.e.C11 = 183 GPa, C12 = 146 GPa and C44 = 46 GPa. Thetypical strain energy density, which scales with e0T

1 Ce01, is

�4.403 � 108 J m�3. We take Df(T) to be 10% of the strainenergy, i.e. Df(T) = 4.403 � 107 J m�3. A simple linear rela-tion between Df(T) and the undercooling temperature DT isassumed [19]:

Df ðT Þ ¼ QDTT 0

: ð19Þ

In Eq. (19), the latent heat is taken as Q = 110 MJ m�3 andthe equilibrium temperature as T0 = 271 K [28], so that theundercooling temperature is about DT = 108 K given theabove assigned value of Df(T).

In Section 2.1, we note that the local free energy densityflocal should satisfy the physical requirement that the B2phase is a thermodynamically metastable state. From Eq.

(3), one obtains @2flocal

@g2i¼ Df ðT ÞðA� 2Bgi þ 3Cg2

i þ 3Dg2i Þ

when gj = 0 for all j – i. To represent the B2 phase, gi

has to be zero as well, such that @2flocal

@g2i

���gi¼0¼ Df ðT ÞA. Since

both Df(T) and A are taken as positive values, it follows

that @2flocal

@g2i

���gi¼0

> 0, thereby showing that the B2 phase is a

thermodynamically metastable state in our phase-fieldmodel.

The interfacial energy density c (i.e. interfacial energyper unit area) is related to the coefficient b in Eq. (7)

according to c ¼ 4ffiffi2p

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibDf ðT Þ

p[29]. We use the normalized

interfacial energy coefficient eb in Eq. (15). It follows that

the grid size becomes l0 ¼ 3c

4Df ðT Þffiffiffiffi2ebp , where c is taken as

the interfacial energy density of type I twin, 187 mJ m�2

[13].The time step needs to be carefully chosen to ensure the

convergence of numerical integration. By trial and error,we find that Det ¼ 0:01 provides reasonable accuracy and effi-ciency. The numerical integration is performed up to 12,500time steps (i.e.et ¼ 125), and selected simulations with longertimes show no further change in the microstructures.

3. Results and discussion

3.1. Microstructure evolution

To study the nucleation and growth of martensiticmicrostructures, we first conduct a phase-field simulationunder a mixed loading condition: zero in-plane strain andzero out-of-plane stress, i.e. ea

1 ¼ ea2 ¼ ea

6 ¼ 0 andra

3 ¼ ra4 ¼ ra

5 ¼ 0. Such boundary conditions correspondto a thin film subjected to rigid constraints from the sub-strate. The global Cartesian coordinate system is alignedwith the cubic axes of the parent B2 phase, as shown inFig. 1. The system contains 64 � 64 � 64 mesh grids. We

set eb ¼ 2, so that l0 ¼ 3c

4Df ðT Þffiffiffiffi2ebp ¼ 1:6 nm and accordingly

the simulation cell has a side length of �102.4 nm.Given the positive value of Df(T), the martensitic trans-

formation of B2!B190 is energetically favored. However,

B2100

B2010

B2001

010

B2

B2

100

B2001

B2010

1x2x3x

8~ =t 32~ =t 60~ =t

85~ =t 90~ =t 125~ =t

Fig. 1. A 3-D phase-field simulation result of martensitic transformation from B2 to B190, showing the formation of polytwinned microstructure. (a)Time-lapse snapshots showing the nucleation and growth of twinned B190 structures. The mesh grids are colored by the value of

P12i¼1igiðxÞ, showing

different twin variants. (b) 2-D projections of the 3-D microstructure at et ¼ 125. The actual simulation cell, as shown in (a), is periodically doubled in thetwo in-plane directions in (b), in order to provide a clear visualization of the polytwinned structure. (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this article.)

Y. Zhong, T. Zhu / Acta Materialia 75 (2014) 337–347 341

owing to the metastable state of the B2 phase, the martens-itic transformation would not occur spontaneously, thusrequiring thermal fluctuations to assist the nucleation ofmartensite. The Langevin noise term ni(x, t) in Eq. (12)plays a role in thermal fluctuations. This stochastic termis independent of each other at different time steps or spa-tial locations, and thus does not provide any constraint onthe phase-transformation process. It is turned off after 3000simulation steps when et ¼ 30.

Fig. 1a shows the nucleation and growth of martensite,resulting in a polytwinned microstructure. At et ¼ 8, themartensite precursors form, driven by the positive Df and

facilitated by thermal fluctuations. Colors represent differ-ent values of field variables gi. Appearance of multiple col-ors indicates the nucleation of multivariants. At this stage,none of the field variables gi is close to 1 and thus no B190

variant is fully formed. However, the field variables alsodeviate from all-zero values (the metastable B2 phase).These martensite precursors cause lattice distortion, as evi-denced by the increase of local elastic energy. They formand disappear during this early stage. The microstructurefurther evolves from the existing lattice distortion pro-moted by thermal fluctuations, when the stochastic noiseterms are turned off at et ¼ 30. At et ¼ 32, shortly after

Table 1Pairing of twin variants from crystallography theory and the phase-fieldsimulation in Fig. 1.

Twin pair Theoretical solution [7] Phase field

{2:6} {110} type I {110}{1:8} {110} type I {110}{1:2} {110} compound {110}{6:8} {100} type I {110}

342 Y. Zhong, T. Zhu / Acta Materialia 75 (2014) 337–347

the turn-off of stochastic noise terms, nuclei with sizesaround tens of nanometers are generated. The boundariesof nuclei are curved and no obvious laminate twin structureis visible. The growth and elimination of these nuclei aredriven by the dynamic changes of local free energy, gradi-ent energy and elastic energy that collectively lower thetotal energy of the system. The laminate twin pattern firstemerges at et ¼ 60, when several nuclei, colored red, blueand green, grow into different twin variants. Other nucleidisappear at et ¼ 85. Soon after, the twin pattern becomesstable at et ¼ 90, while the twin boundaries are still not flat.Finally, at et ¼ 125, the 3-D twin microstructure consistingof four different B190 variants forms with the respectivetransformation strain of e0

1; e02; e

06 and e0

8. Fig. 1b showsthe top and left-side view of the 3-D polytwinned micro-structure at et ¼ 125 .

Given the above polytwinned microstructure, it is neces-sary to examine whether the multivariants are geometri-cally compatible with the prescribed boundary condition,as well as satisfy the geometric compatibility at each indi-vidual twin boundary. From a consideration of the energet-ics, the transformation from austenite to martensite lowersthe local (chemical) energy. Regarding the strain energy,we note that the total strain e, transformation strain e*

and stress r can be decomposed into the homogeneousand inhomogeneous components, as defined in theAppendix A. It follows that the homogeneous part of the strainenergy can be minimized by adjusting the volume fractionsof multivariants to accommodate the applied boundarycondition. On the other hand, the nucleated martensitevariants form twins, such that the heterogeneous part ofthe strain energy can be minimized if the compatibilityrequirements are fully satisfied at twin boundaries. Tocheck the compatibility with the prescribed boundary con-dition in phase-field simulations, we estimate the averagetransformation strain of the final state atet ¼ 125 by assum-ing equal volume fractions of the four variants:

ðe01 þ e0

2 þ e06 þ e0

8Þ=4 ¼�0:01 0 0

0 �0:01 0:0504

0 0:0504 0:0243

0B@1CA:ð20Þ

Eq. (20) indicates that each in-plane component of theaverage transformation strain is small with �1% compres-sive normal strain and zero shear strain, and thus satisfiesapproximately the applied boundary condition of zero in-plane strain: ea

1 ¼ ea2 ¼ ea

6 ¼ 0.We next examine the geometrical compatibility of vari-

ants at twin boundaries by comparing the phase-fieldresults with the theoretical crystallographic solutions tabu-lated by Hane and Shield [7]. In Fig. 1 the observed twinplanes between each pair of variants {2:6}, {1:8}, {1:2}and {6:8} are all of the {110} type, as summarized inTable 1. Three of the four pairs are consistent with thecrystallographic solutions of type I twin in NiTi, exceptthe {6:8} pair whose violation of compatibility at the twin

interface increases the heterogeneous strain energy. How-ever, the emergence of such an incompatible twin pair isgeometrically necessary for accommodating the other threetwin pairs, so as to reduce the overall energy of the system.Generally, theoretical solutions based on the principle ofgeometric compatibility at twin interfaces [7] provide aguide for selecting twin pairs when the applied stress iszero. However, under the boundary conditions such asapplied strain/displacement, there might be no “perfect”solution which is fully compatible at all the twin interfacesand hence minimizes the local (chemical) free energy andthe homogeneous and heterogeneous strain energies simul-taneously. In other words, the compatibility condition isnot always guaranteed at every twin interface in realisticmicrostructures [5], as shown in our phase-fieldsimulations.

The martensitic phase transformation in a complex sys-tem such as NiTi could yield a non-unique final product.We repeat the phase-field simulation with all the parame-ters unchanged to explore different possible microstruc-tures. Different polytwinned structures arise due to thestochastic effect of the Langevin noise term ni(x, t) in Eq.(12). Fig. 2a shows another phase-field simulation resultof martensite nucleation and microstructure evolution,which contrasts with those in Fig. 1a. As seen fromFig. 2a, the precursors of martensite initially form and dis-appear at et ¼ 20, similar to Fig. 1a. At et ¼ 50, martensiticnuclei form and merge with others containing the sametype of twin variant, and finally form a polytwinned struc-ture at et ¼ 125. This final state consists of four differentB190 variants with the basis transformation strain e0

2; e04; e

05

and e06, respectively. In this case, the twin interfaces are par-

allel to the side faces of the simulation cell, in contrast tothe inclined ones in Fig. 1a. The top view of the 3-D poly-twinned structure is shown in Fig. 2b. For the {2:6} and{4:5} pairs, their twin planes are of the {110} type, whilethe other two pairs have twin planes of the {100} type,as summarized in Table 2. Similar to the result in Fig. 1,compatibility is not satisfied for the {5:6} pair, while theother three pairs are geometrically compatible at therespective twin interface. Likewise, the average transforma-tion strain of variants 2, 4, 5 and 6 at et ¼ 125 is estimatedby assuming the equal volume fractions of four variants:

ðe02þ e0

4þ e05þ e0

6Þ=4¼�0:01 0 0:0504

0 �0:01 0

0:0504 0 0:0243

0B@1CA: ð21Þ

B2100

B2010

B2001

B2010

B2100

20~ =t 50~ =t

90~ =t 125~ =t

Fig. 2. A different phase-field simulation result of polytwinned microstructure under the same boundary condition as Fig. 1. (a) Time-lapse snapshotsshowing the nucleation and growth of twinned martensite. (b) 2-D projection of the 3-D microstructure at et ¼ 125. The same schemes of coloring and 2-Dperiodic doubling are used as in Fig. 1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of thisarticle.)

Table 2Pairing of twin variants from crystallography theory and the phase-fieldsimulation in Fig. 2.

Pair Theoretical solution [7] Phase field

{2:4} {100} type I {100}{2:6} {110} type I {110}{4:5} {110} type I {110}{5:6} {110} compound {100}

Y. Zhong, T. Zhu / Acta Materialia 75 (2014) 337–347 343

Eq. (21) indicates that each in-plane component of theaverage transformation strain is small with �1%compressive normal strain and zero shear strain, and thusapproximately satisfies the applied boundary condition ofzero in-plane strain ea

1 ¼ ea2 ¼ ea

6 ¼ 0.It is interesting to note that repeated simulations with

the same applied boundary condition always result in theselection of four variants from the candidate variants 1–8, but not 9–12. This is because the in-plane transformationstrains among variants 1–8 can render one tensile and onecompressive component, as shown in Eq. (18); they canceleach other to reduce the averaged in-plane transformationstrain, so as to lower the strain energy, as shown by Eqs.(20) and (21).

3.2. Loading effect

The mechanical loading dictates the patterning of multi-variants in the formation of strain-accommodating micro-structures. We next explore different combinations ofboundary constraint. Here the cubic simulation cell

contains 32 � 32 � 32 mesh grids, eb is adjusted to be 0.5

and l0 ¼ 3c

4Df ðT Þffiffiffiffi2ebp ¼ 3:2 nm. Correspondingly, the side

length of the cell is 102.4 nm.Fig. 3a and b show two possible microstructures when

the in-plane biaxial tensile strain is applied, i.e.ea

1 ¼ ea2 ¼ 1% , ea

6 ¼ 0 and ra3 ¼ ra

4 ¼ ra5 ¼ 0. In Fig. 3a,

variants 9 and 12 form the {10 0} twin; in Fig. 3b, variants10 and 11 form the {11 0} twin, as listed in Table 3. Thetransformation strain tensors associated with variants 9,10, 11 and 12 belong to the group that shares the same nor-mal components, but involves both in-plane tension andout-of-plane compression, as shown in Eq. (18). This groupof transformation strain tensors can better match theimposed biaxial tensile strain. Pairing of variants in thisgroup yields the self-accommodating twin structuresthrough a complete cancellation of in-plane shear strains.Repeated simulations invariably select the variants in thegroup containing variants 9, 10, 11 and 12. In most cases,twin compatibility is satisfied through, for example, forma-tion of the {100} twin between variants 9 and 12, as shownin Fig. 3a. However, exceptions are observed, such as the{110} twin of variants 10 and 11, as shown in Fig. 3band Table 3. Selection of these twin pairs can also beunderstood in terms of the accommodation of the appliedboundary condition, as discussed earlier. Recall that whenthe zero in-plane strain is applied in Section 3.1, only vari-ants 1–8 can nucleate because the in-plane normal straincomponents contain both tension and compression. Wealso perform simulations when the in-plane biaxial

2B100

2B010

001B2

Fig. 3. Formation of polytwinned martensitic microstructures under different applied boundary conditions. The mesh is colored to distinguish differentB190 variants. (a and b) Two different microstructures form under in-plane biaxial tension ea

1 ¼ ea2 ¼ 1%, ea

6 ¼ 0 and ra3 ¼ ra

4 ¼ ra5 ¼ 0. (c and d) Two

different microstructures form under out-of-plane compression ea3 ¼ �2% and ra

1 ¼ ra2 ¼ ra

4 ¼ ra5 ¼ ra

6 ¼ 0. (For interpretation of the references to colourin this figure legend, the reader is referred to the web version of this article.)

Table 3Pairing of twin variants from crystallography theory and the phase-fieldsimulation in Fig. 3.

Twin pair Theoretical solution [7] Phase field

{9:12} Fig. 3a {100} type I {100}{10:11} Fig. 3b {100} type I {110}{10:12} Fig. 3c {100} type I {100}{9:10} Fig. 3d {110} compound {110}

344 Y. Zhong, T. Zhu / Acta Materialia 75 (2014) 337–347

compression is applied. The resulting microstructures areidentical to those in Figs. 1 and 2. These results can be sim-ilarly understood by the requirement to accommodate theimposed boundary condition.

Fig. 3c and d show the phase-field result when the out-of-plane compressive strains are applied, i.e. ea

3 ¼ �2% andra

1 ¼ ra2 ¼ ra

4 ¼ ra5 ¼ ra

6 ¼ 0. Such loading mode of out-of-plane compression yields the results similar to those underin-plane biaxial tension as shown in Fig. 3a and b. This isbecause the two loading modes essentially differ by ahydrostatic stress that plays a minor role in the selectionof twin variants. As a result, the same group containingvariants 9, 10, 11 and 12 is the optimal choice, producingthe {100} twin between variants 10 and 12, and the{11 0} twin between variants 9 and 10, as listed in Table 3.

3.3. Orientation effect

In this section, we study the orientation effects on theformation of polytwinned martensitic microstructures. Inthe example plotted in Fig. 4, the global Cartesian coordi-nate system is aligned with the h110i, h001i and h1�10idirections of the parent B2 phase. In this case, the rotationof the transformation strain tensor Re0

i RT is required withthe rotation matrix given by:

R ¼0:707 0:707 0

0 0 1

0:707 �0:707 0

264375: ð22Þ

In addition, the elastic modulus tensor given in the cubiccrystal system should be transformed to the global coordi-nate system by rotation [27] and then cast into the 6 � 6stiffness matrix C in the Voigt notation.

Fig. 4a1 shows the simulated polytwinned microstruc-ture under an in-plane biaxial compressive strain of 0.5%;the system is stress free in the vertical direction. The{110} twin between variants 11 and 12 is formed. Variants11 and 12 cancel the shear transformation strain of eachother, and also provide the in-plane transformation strain

2B110

2B001

2B011

1x2x3x

Fig. 4. Formation of polytwinned martensitic microstructures under different applied boundary conditions. (a1) {110} compound twin forms under an in-plane biaxial compressive strain of 0.5% and the system is stress free in the vertical direction. (a2) Atomic structures of {110} compound twin from aprevious molecular statics simulation [16]. The green unit cells of Ni atoms (blue) exhibit tilting of the monoclinic phase, and obey the mirror-reflectioncondition with respect to the twin boundary (dashed line). In contrast, the white unit cell of Ti atoms (red) straddles the twin plane and retains arectangular shape. (b) Layered twin lamellae form under an in-plane biaxial tensile strain of 1% and the system is stress free in the vertical direction. (c)Polytwinned structures form under a compressive strain of 2% in the vertical direction; all other stress components are zero. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version of this article.)

Y. Zhong, T. Zhu / Acta Materialia 75 (2014) 337–347 345

of biaxial compression, thus accommodating the imposedmechanical load. The {110} twin between variants 11and 12 is the so-called compound twin [7], as the crystallo-graphic directions of both the twin plane and twin shearare rational.

Previously, we have conducted the interatomic-potential-based molecular statics and dynamics simulations tostudy the structure of compound twins [16]. Fig. 4a2 showsan example of the atomic configuration of a compoundtwin in a supercell under periodic boundary conditions. Itis the same type of compound twin as the one inFig. 4a1, with the twin boundary aligned with the{11 0}B2 plane in the B2 basis, which corresponds to theequivalent {010}B19 plane in the B19 basis. This atomisticstudy complements the continuum phase-field simulationby providing atomic-level structural details of the twinboundary. For example, Fig. 4a2 shows that the twinboundary is atomically sharp, and the mirror reflection ofatoms at the twin boundary can be clearly resolved to com-pare with high-resolution transmission electron microscopyimages, as described in detail in Ref. [16]. Furthermore,

from the standpoint of multiscale modeling, one can linkthe atomistic and continuum phase-field simulations. Forexample, the atomic-level geometry such as the thicknessof the atomically sharp phase boundary as revealed inFig. 4a2, in conjunction with the atomistically calculatedtwin boundary energy, can provide a quantitative calibra-tion of the gradient coefficients in the phase-field model,i.e. bij in Eq. (6). Such an atomistically informed phase-fieldapproach is beyond the scope of this work, but warrantsfurther study in future.

To demonstrate the loading effect, Fig. 4b shows thesimulated polytwinned microstructure under an in-planebiaxial tensile strain of 1%; the system is stress free in thevertical direction. Variants 2, 3, 6 and 7 form {100} twins.The corresponding twin pairs are listed in Table 4. Selec-tion of variants 2, 3, 6 and 7 is still dictated by the appliedbiaxial tension, because they are the only four variantswhose in-plane components of transformation strain ten-sors Re0

i RT render the biaxial elongation. Moreover, theshear component of the average transformation strain isalso minimized:

Table 4Pairing of twin variants from crystallography theory and the phase-fieldsimulation of biaxial tension in Fig. 4b.

Twin pair Theoretical solution [7] Phase field

{2:3} {100} type I {100}{3:6} {110} type I {100}{6:7} {100} type I {100}{7:2} {110} type I {100}

Table 5Pairing of twin variants from crystallography theory and the phase-fieldsimulation of biaxial tension in Fig. 4c.

Twin pair Theoretical solution [7] Phase field

{2:3} {100} type I {100}{2:6} {110} type I {110}{3:7} {110} type I {110}{6:7} {100} type I {100}

346 Y. Zhong, T. Zhu / Acta Materialia 75 (2014) 337–347

Rðe02 þ e0

3 þ e06 þ e0

7ÞRT=4 ¼0:033 0 0

0 0:0243 0

0 0 �0:0524

0B@1CA:ð23Þ

In addition, Fig. 4c shows the polytwinned microstructureformed when the applied compressive strain is 2% in thevertical direction and all other stress components are zero.Similarly, variants 2, 3, 6 and 7 form, although the poly-twinned structure is different from the one in Fig. 4b. Assummarized in Table 5, variant pairs {2:3} and {6:7} formthe {100} twin; while {2:6} and {3:7} form the {110} twin.All the compatibility requirements at the twin boundaryare satisfied.

4. Conclusion

We have developed a phase-field model to study thediffusionless cubic to monoclinic martensitic phase trans-formation in NiTi shape memory alloys. A Landau-typefree energy function is constructed to characterize themartensitic transformation from the cubic B2 phase to alow-symmetry phase of monoclinic B190 with 12 variants.3-D phase-field simulations reveal the nucleation andgrowth of B190 multivariants that form the polytwinnedmicrostructures. The simulation results are analyzed interms of the overall accommodation of applied boundaryconditions, which reduces the homogeneous strain energy,as well as the local twin compatibility, which lowers theheterogeneous strain energy. In particular, our simulationsdemonstrate the difficulty of attaining the “perfect”polytwinned microstructure where the theoretical crystallo-graphic solution of twin compatibility is satisfied ideally atevery twin boundary. Further parametric studies of theloading and orientation effects show the versatility/non-uniqueness of the formation of polytwinned micro-structures, where the patterning of martensiticmultivariants can be interpreted in terms of the need to

accommodate the applied strain. The present phase-fieldmodel provides a flexible framework to study the dynamicmicrostructure evolution during the martensitic transfor-mation to low-symmetry phases.

Acknowledgement

Support by NSF Grant CMMI-0825435 is greatlyacknowledged.

Appendix A. Solving strain and stress in the Fourier space

One can solve the total strain e and stress r through theFourier transform of the equilibrium equation. To this end,we define a differential operator [25]:

A ¼

@@x1

0 0 0 @@x3

@@x2

0 @@x2

0 @@x3

0 @@x1

0 0 @@x3

@@x2

@@x1

0

0BB@1CCA

T

; ðA1Þ

where x1, x2 and x3 are the spatial coordinate. The geomet-ric equation of strain can then be represented as:

e ¼ Au; ðA2Þwhere the displacement vector is u = (u1, u2, u3)T and eachof its component is a function of x1, x2 and x3. It followsthat the equilibrium equation can be written as:

ATr ¼ 0: ðA3ÞIt is non-trivial to obtain the analytical solution of Eqs.(10), (A2), and (A3) for generally prescribed boundary con-ditions. Instead, these equations can be discretized andsolved numerically. Direct numerical solution of Eqs.(10), (A2), and (A3) is computationally inefficient, since aset of partial difference equations for the whole field needsto be satisfied at each time step. However, for the problemwith periodic boundary conditions, the semi-implicit algo-rithm originally developed by Chen and Shen [21] can sig-nificantly improve the computational efficiency by applyingfast Fourier transformation (FFT). In this scheme, the dif-ferential equations are transformed to mutually indepen-dent linear equations at different mesh nodes. In thisAppendix A, we present the detailed formulation and pro-cedure of solving the equilibrium equations with the FFTmethod by partially following the approach introducedby Shu and Yen [25].

The total strain e, transformation strain e* and stress r

are decomposed into homogeneous and inhomogeneouscomponents:

e ¼ hei þ e0; where hei ¼Z

VedV ðA4Þ

and

e� ¼ he�i þ e�0; where he�i ¼Z

Ve� dV : ðA5Þ

One can further decompose the total displacement u intohomogeneous and inhomogeneous components:

Y. Zhong, T. Zhu / Acta Materialia 75 (2014) 337–347 347

u ¼ hui þ u0 satisfying hei ¼ Ahui and e0 ¼ Au0: ðA6ÞNote that the homogeneous component hui is a linear func-tion of x because the homogeneous strain hei is a constant.Although the decomposition in Eq. (A6) is not unique witha constant plus a rigid body rotation, it will not affect theresult in which we are interested because we will only usethe differential form of displacement u0.

Substitution of Eqs. (A4) and (A5) into Eq. (10) yields:

r ¼ Cðe� e�Þ ¼ Cðhei � he�iÞ þ Cðe0 � e�0Þ¼ hri þ r0; ðA7Þ

where the homogeneous stress is defined as:

hri ¼ Cðhei � he�iÞ ðA8Þand the inhomogeneous stress is:

r0 ¼ Cðe0 � e�0Þ: ðA9ÞIn each simulation time step, once the field variables

{g1, . . ., g12} are known, e* can be directly calculated fromEq. (1), so that it is straightforward to calculate he*i fromEq. (A5). The applied boundary conditions can be given byeither hei = ea or hri = ra, or a mixed type. Together withEq. (A8), all the homogeneous components hei and hri canbe obtained.

The equilibrium equation of r0 remains the same form:

ATr0 ¼ 0: ðA10ÞSubstituting Eq. (A6) into Eqs. (A9) and (A10), we obtainthe partial differential equations:

ATCAu0 ¼ ATCe�0: ðA11ÞFor an integral equation f:R3! R, the Fourier transfor-

mation is defined by:

dfðsÞ ¼ Jðf Þ ¼Z Z Z 1

�1f ðxÞe�2pix�sdx; ðA12Þ

where s = (s1, s2, s3)T is the coordinate in reciprocal space.The inverse Fourier transformation is defined by:

f ðxÞ ¼ J�1ðf Þ ¼Z Z Z 1

�1

dfðsÞe2pix�sds: ðA13Þ

It can be shown that the differential operator A in realspace is transformed to the linear operator B in reciprocalspace, which is given by:

B ¼ 2pi

s1 0 0 0 s3 s2

0 s2 0 s3 0 s1

0 0 s3 s2 s1 0

0B@1CA

T

: ðA14Þ

Then u0 is obtained from Eq. (A11) as:

bu0 ¼ Jðu0Þ ¼ ðBTCBÞ�1BTCce�0 : ðA15Þ

From Eq. (A6), the transformed inhomogeneous strain is:be0 ¼ Jðe0Þ ¼ BðBTCBÞ�1BTCce�0 : ðA16Þ

From Eq. (A9), the transformed inhomogeneous stress is:br0 ¼ Jðr0Þ ¼ CBðBTCBÞ�1BTCce�0 � Cbe0 : ðA17Þ

The inhomogeneous stress and strain can be obtained bythe inverse Fourier transformation according to Eq. (A13).Finally, together with the homogeneous stress given by Eq.(A8) and the boundary condition, one can obtain thedriving force associated with elastic energy density givenin Eq. (11). If the kinetic equations in Eq. (16) are alsosolved in the Fourier space, the inverse transformation ofstress is not needed in the phase-field simulation.

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