A phase-field approach to solid–solid phase ...in Ref. (Momeni and Levitas, 2014) but without mechanics. The interaction between two SM interfaces through melt are modeled using

International Journal of Solids and Structures 71 (2015) 39–56

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International Journal of Solids and Structures

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A phase-field approach to solid–solid phase transformations viaintermediate interfacial phases under stress tensor

http://dx.doi.org/10.1016/j.ijsolstr.2015.05.0270020-7683/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Department of Aerospace Engineering, Iowa StateUniversity, Ames, IA 50011, USA.

E-mail address: [email protected] (V.I. Levitas).

Kasra Momeni a, Valery I. Levitas a,b,c,⇑a Department of Aerospace Engineering, Iowa State University, Ames, IA 50011, USAb Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USAc Material Science and Engineering, Iowa State University, Ames, IA 50011, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 April 2015Received in revised form 21 May 2015Available online 3 June 2015

Keywords:Phase transformationPhase-fieldInterfacial phasesInternal stressesIntermediate meltNucleation

A thermodynamically consistent phase-field (PF) theory for phase transformations (PTs) between threedifferent 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 thermody-namic potential which satisfies all these conditions is introduced using polar order parameters.Propagation of a solid–solid (SS) interface containing nanometer-sized intermediate disordered interfa-cial phases (IP) and particularly an interfacial intermediate melt (IM) is studied for an HMX energeticmaterial using the developed PF model. The scale effects (the ratio of widths of SS to solid-melt (SM)interfaces, kd), the effect of the energy ratio of SS to SM interfaces (kE), and the temperature on the for-mation and stability of IM are investigated. An interaction between two SM interfaces via an IM, whichplays a key role in defining a well-posed problem and mesh-independent solution, is captured using aspecial gradient energy term. The influence of the elastic energy on the formation and retainment ofIM and its structure, hundreds of degrees below the melting temperature, is investigated. Elastic energypromotes barrierless IM in terms of an increasing degree of disordering, interface velocity, and width ofIM, but it surprisingly increases nucleation temperature for the IM. The key effect, however, is the drasticreduction (by more than an order of magnitude) of the energy of the critical nucleus of the IM within theSS interface, which is caused by the elastic energy. The developed PF model is applicable for the generalcase of PT between three phases and can be applied (adjusted) to other physical phenomena, such aspremelting/disordering at grain boundaries, martensitic PTs, surface-induced premelting and PTs, anddeveloping the interfacial phase diagrams.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

The formation and stabilization of IP (e.g., surfacial amorphousfilms and intergranular amorphous films) have many importantapplications, e.g., in the electronic industry (Luo and Chiang,2008). Interfacial phases also play an important role in sintering,alloying, and strengthening ceramic materials (Luo et al., 1999;Becher et al., 2000). In addition, the properties of IP are differentfrom the constituent homogeneous phases. A sample with a largevolume fraction of IP shows different overall properties (e.g. creep,superplasticity, and electrical conductivity), compared to the sam-ple of homogeneous phases (Cantwell et al., 2014). Therefore, any

advancement in the theoretical and computational modeling of IPcan have a significant scientific and industrial impact.

Interfaces between different phases can undergo PTs similar tobulk materials and can be treated as quasi-two-dimensionalphases. The chemical and structural properties of the interfaceschange abruptly or continuously during their PT during a variationof the corresponding driving forces. Although the interfaces can beanalyzed using equilibrium thermodynamics, the equilibrium IPshave varying properties. The Gibbs definition of a phase cannotbe applied because of its inhomogeneous structure and composi-tion (Cantwell et al., 2014). The presence of IPs can change thermo-dynamic properties of materials and can lead, for example, toliquid-metal embrittlement (Luo et al., 2011). It also alters thekinetics of processes, e.g., PTs and grain growth (Dillon et al.,2007). Interfacial phases emerge in a wide range of processes suchas premelting and prewetting, surface-induced premelting and PT(Levitas and Samani, 2011a,b; Levitas and Javanbakht, 2011,

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40 K. Momeni, V.I. Levitas / International Journal of Solids and Structures 71 (2015) 39–56

2010), intrinsic PT that occurs in pure materials, and extrinsic PTthat happens in non-pure materials involving an adsorption of animpurity or dopant (Frolov et al., 2013; Frolov et al., 2013).

Solid–solid PT via a nanometer-sized intermediate disordered IPcalled intermediate melt, IM, at temperatures much below meltingtemperature (hm

e ) has been predicted thermodynamically (Levitaset al., 2004, 2006; Levitas, 2005; Levitas et al., 2012) and confirmedexperimentally for an HMX energetic material (Smilowitz et al.,2002; Henson et al., 2002; Levitas et al., 2004, 2006), PbTiO3 piezo-electric nanowires (Levitas et al., 2012), and for amorphization inan insulin enhancer pharmaceutical substance called avandia(Randzio and Kutner, 2008). The PT via formation of IM was alsosuggested as the mechanism for crystal-crystal PT and amorphiza-tion for materials with reducing melting temperature under pres-sure (e.g., ice, Si, and Ge) (Levitas, 2005). Furthermore, stressesmay also be due to an applied external loading. Formation ofIMunder a high strain rate loading was predicted thermodynami-cally and confirmed using MD simulations for aluminum and cop-per (Levitas and Ravelo, 2012).

The thermodynamic condition for the appearance of an IMbetween solid-1 and solid-2 (S1MS2 interface) isE21 � E10 � E20 � Ee > ðG0 � GsÞd� (See Ref. (Levitas and Momeni,2014)). Here, E10; E20, and E21 are the energies of the S1M; S2M,and SS interfaces, respectively; Ee is the elastic energy of the coher-ent SS interface; d� is the width of the IM, and G0 and Gs are the bulkthermal energies of melt and the solid phase with a smaller melt-ing temperature hm

e , respectively. Reduction of the total interfaceenergy and relaxation of the elastic energy are the driving forcesfor the formation of melt significantly below hm

e . Theoretical mod-els based on the sharp-interface approach (Levitas et al., 2012,2006; Levitas, 2005; Levitas et al., 2004) have neglected the changein interface energy during melting as well as the interfacial inter-actions. The main driving force for the formation of IM in thesemodels was the relaxation of internal stresses due to large volu-metric transformation strains. This driving force vanishes as soonas melt forms which results in a supercooled melt and immediateresolidification of melt to the stable solid phase. This intermediatetransient melt is a special form of IMwhich is called virtual melt(Levitas et al., 2004, 2006). Formation of the virtual melt is pre-dicted in materials with large volumetric transformations that gen-erate large elastic energy when other stress relaxing mechanisms,such as plastic deformation and twinning, are suppressed; e.g., inmaterials with complex molecular or atomic structures.

Different techniques are utilized for characterizing and model-ing the IP. The developed models, based on the sharp-interfaceapproach which considers zero-thickness interfaces, are oversim-plified because the IPhas a width in the range of a few nanometers,which is comparable with the width of a SS interface. Furthermore,a sharp-interface approach considers a bulk phase sandwichedbetween two interfaces of zero-thickness and cannot capture for-mation of the IPwith partial melting. Molecular dynamics (MD)simulations have been utilized to investigate interfacial PTs in purematerials (Frolov et al., 2013). However, investigating such PTs inmultiphase materials using the MD technique is limited by aninevitable long simulation time for producing equilibrated compo-sition profiles. Properties of IP have also been studied using ab initiotechniques (Painter et al., 2002; Rulis et al., 2005; Shibata et al.,2004). A phase-field crystal model (Elder and Grant, 2004) was alsoused to investigate the effects of IP, such as grain boundarypremelting (Mellenthin et al., 2008).

An advanced PF approach to formation of the IM was developedin Ref. (Momeni and Levitas, 2014) but without mechanics. Theinteraction between two SM interfaces through melt are modeledusing an SS interface energy contribution within complete melt.A new force-balance model was introduced that could match the

results of PF simulations up to temperatures significantly belowthe melting temperature. Shifting from a jump-like first-order typeto a continuous second-order type PT that are separated by anIM-free region was captured during IM-formation as kd reduced.The presence of three solutions associated with stable and meta-stable IM, as well as a critical nucleus (CN) of IM were revealed.A retainment of IM at temperatures significantly below meltingtemperature for kE < 2:0 was also illustrated. The dependence ofthe interface energy on a0 and kd in the presence of IM was demon-strated using the numerical simulations.

The goal of this paper is to generalize a model (Momeni andLevitas, 2014) for the case when elastic energy and stresses areimportant. An advanced PF approach to PTs with a strong focuson the effect of the stress and transformation strain tensors hasbeen developed for martensitic PT (Levitas et al., 2003; Levitasand Preston, 2002a,b; Levitas et al., 2009; Cho et al., 2012;Levitas, 2013), surface-induced PT in solids (Levitas andJavanbakht, 2011, 2010), melting (Slutsker et al., 2006; Levitasand Samani, 2011a,b), dislocation evolution (Levitas andJavanbakht, 2013, 2015b), and the interaction between PT and dis-locations (Levitas and Javanbakht, 2012, 2014, 2015a,b, Javanbakhtand Levitas, 2015). Here, a thermodynamically consistent phasefield approach for IM at the SS interface under a general stress ten-sor will be developed, combining our PF approaches to PT in solids,melting, and the model (Momeni and Levitas, 2014) withoutmechanics. Some preliminary simulations with mechanics havebeen presented in short communications in Ref. Levitas andMomeni (2014) and Momeni et al. (2015).

The remainder of this paper is organized as follows. In Section 2,a strict thermodynamic approach is developed for the free energypotential that depends on the polar order parameters and theirgradients, as well as elastic strain and temperature. The constitu-tive equations are derived for the general case of a three-phasematerial under a stress tensor. Specific expressions for theHelmholtz free energy and thermal and transformation strainsare presented, which (as we will show in Section 4) satisfy allthe formulated thermodynamic equilibrium and stability condi-tions for homogeneous states. The time-dependent GL equationsare derived in Section 3 for a nonequilibrium propagating interface.In Section 4 thermodynamic stability conditions are derived in thegeneral form and for our specific model. A detailed description ofthe numerical implementations of the model, including the initialconditions used to study the kinetics and capture the CN, aredescribed in Appendix. The developed model is then specified forthe properties of an HMX energetic material, and the effects of dif-ferent parameters on the formation of IM and SS PT are studied inSection 5. In Section 6, formation and structure of the CN of IMwithin the S1S2 interface and CN of the S1S2 interface within theS1MS2 interface are studied in detail for the models with and with-out mechanics. Finally, the results of this study are summarized inSection 7.

We have designated the contraction of tensors A and B over oneand two indices with A � B and A : B, respectively. The subscripts sand a label the symmetric and the skew-symmetric part of asecond-rank tensor. The unit tensor and Kronecker delta are desig-nated by I and dij, respectively; $ is the gradient operator.

2. Thermodynamic theory

2.1. Polar order parameters

Following Refs. Levitas et al. (2003) and Momeni and Levitas(2014), the polar order parameters, the radial ! and the angular#, are introduced in a plane (Fig. 1). Geometrically, p#=2, is theangle between the radius vector !, and the axis 1. The origin of

Fig. 1. Plot of the modified Gibbs potential eGðr; h;!; #Þ for HMX at h ¼ h21e and 1 MPa hydrostatic compressive stress for different hmd

c ¼ h10c and hmb

c ¼ h20c values, in the polar

system of the order parameters ! and #. 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).

K. Momeni, V.I. Levitas / International Journal of Solids and Structures 71 (2015) 39–56 41

the coordinate system, described by ! ¼ 0 for any #, correspondsto the reference phase 0. In the current paper, it is a melt M; how-ever, it can be any phase for a general three-phase system, in par-ticular, austenite for multivariant martensitic transformations(Levitas and Preston, 2002b; Levitas et al., 2003, 2013; Levitasand Roy, 2015). Points (! ¼ 1 and # ¼ 0) and (! ¼ 1 and # ¼ 1)correspond to solid phases S1 and S2, respectively.Transformations Ss $ M are described by varying ! between 0and 1 at # ¼ 0 or 1; Transformations S1 $ S2 occur by variationof # between 0 and 1 at ! ¼ 1. In our applications, phases S1 andS2 correspond to the HMX d and b phases, respectively.

2.2. Kinematics

In the small-strain approximation the additive decompositionof the strain tensor is

e ¼ ð$uÞs ¼ ee þ etð!; #Þ þ ehð!; #; hÞ; ð1Þ

where u is the displacement vector, h is the temperature, ee; et , andeh are the elastic, transformational, and thermal strains,respectively.

2.3. Laws of thermodynamics

The first law of thermodynamics for an arbitrary volume V of athree-phase material with a boundary S isZ

Sp � v � h � nð ÞdSþ

ZS

Q!_!þ Q #

_#� �

� ndSþZ

Vq f � v þ rð ÞdV

¼ ddt

ZVq U þ 0:5v � vð ÞdV ; ð2Þ

where p is the traction vector, v is the particle velocity, h, is the heatflux, n, is the outward unit normal to S, d

dt is the time derivative, U isthe internal energy per unit mass, f represents the body forces perunit mass, and r is the specific volumetric heat supply rate per unitmass, and q is the mass density, which will be considered as a con-stant within small-strain approximation. The introduced general-ized forces Q! and Q# originate from the dependence of thethermodynamic potential on $! and $#, respectively. These ther-modynamic forces allow us to write the energy equation for an arbi-trary volume and consequently transform the global energy balanceto its local form.

Applying the second law of thermodynamics in the global formof the entropy balance along with the Clausius–Duhem inequalitygives

ddt

ZVqsdV �

ZVq

rh

dV þZ

S

hh� ndS P 0; ð3Þ

where s is the entropy per unit mass. Utilizing the Gauss theoremand conservation of mass (dðqVÞ=dt ¼ 0), both thermodynamicslaws transform to integrals over the volume after some mathemat-ical manipulationsZ

Vr : _e� q _U � $ � hþ qr þ $ � ðQ!

_!þ Q #_#Þ

� �dV

þZ

V$ � rþ qf � q _vð Þ � vdV ¼ 0; ð4ÞZ

Vq_s� q

rhþ $ � h

h

� �dV P 0: ð5Þ

The constitutive equations must be objective and independent ofthe motion of the observer. Therefore Eq. (4) should be invariantwith respect to the superposition of the rigid body motion. In par-ticular, substituting the velocity vector v with a vector v � v 0,where v 0 is the constant velocity, should not alter Eq. (4). This canonly be achieved if the term in parenthesis of the second integralin Eq. (4) is identically zero,

$ � rþ qf ¼ q _v : ð6Þ

Thus, introducing thermodynamic forces Q! and Q# does not affectthe local momentum balance Eq. (6). Since Eqs. (4) and (5) are validfor an arbitrary volume, they have to be valid for all individualpoints in that volume:

r : _e� q _U � $ � hþ qr þ $ � ðQ!_!Þ þ $ � ðQ #

_#Þ ¼ 0; ð7Þ

q~S ¼ q_s� qrhþ $ � h

h¼ q_s� q

rhþ 1

h$ � h� $h

h2 � h P 0; ð8Þ

where ~S is local total entropy production rate per unit mass.Defining the local dissipation rate as D ¼ h~S, multiplying Eq. (8)by h gives qD. Eliminating qr � $ � h from Eqs. (7) and (8) resultsin the local dissipation inequality,

qD :¼ r : _e� q _U þ qh_sþ $ � ðQ!_!Þ þ $ � ðQ#

_#Þ � $hh� h P 0: ð9Þ

Assuming independence of the heat conduction and other thermo-dynamic processes, Eq. (9) splits into the Fourier’s inequality,� $h

h � h P 0 and the remaining inequality without the last term(for which we keep using the D designation):

qD ¼ r : _e� q _w� qs _hþ $ � ðQ!_!Þ þ $ � ðQ#

_#ÞP 0; ð10Þ


where the Helmholtz free energy, w ¼ U � hs, was introduced.

2.4. Constitutive equations

We postulate that w ¼ wðee;!; #; h;$!;$#Þ. The derivation pro-cess would be more concise if the Helmholtz free energy wwasexpressed in terms of e rather than ee,

w ¼ wðee;!; #; h;$!;$#Þ¼ wðe� etð!; #Þ � ehðh;!; #Þ;!; #; h;$!;$#Þ¼ �wðe;!; #; h;$!;$#Þ: ð11Þ

The last two terms in Eq. (10) can be expanded as $ � ðQ!_!Þ ¼

ð$ � Q!Þ _!þ Q! � $ _! and $ � ðQ#_#Þ ¼ ð$ � Q#Þ _#þ Q# � $ _#. Also,

d $!ð Þ=dt ¼ $ _! and d $#ð Þ=dt ¼ $ _# due to small strain approxima-tion. Then, inserting Eq. (11) in Eq. (10), we have:

qD ¼ r� q@�w@e

� �: _e� qsþ q

@�w@h

� �_h� q

@�w@!� $ � Q!

� �_!

þ Q! � q@�w@$!

� �� $ _!� q

@�w@#� $ � Q #

� �_#

þ Q# � q@�w@$#

� �� $ _# P 0; ð12Þ

Assuming the dissipation rate to be independent of _e; _h; $ _!, and$ _# leads to the elasticity rule, the equation for the entropy, andthe definition of thermodynamic forces Q! and Q#:

r ¼ q@�w@e

; s ¼ � @�w@h

; ð13Þ

Q! ¼ q@�w@$!

; Q # ¼ q@�w@$#

: ð14Þ

Dissipative (viscous) stresses can be introduced in a standard way(see, e.g., Ref. Levitas (2013)). Substituting Eq. (14) back in Eq.(12), we obtain the residual dissipative inequality

qD ¼ X!_!þ X#

_# P 0; ð15Þ

with the definition of generalized thermodynamic forces

X! ¼ �q@�w@!þ $ � q

@�w@$!

� �; ð16Þ

X# ¼ �q@�w@#þ $ � q

@�w@$#

� �: ð17Þ

To satisfy the inequality Eq. (15), one has to prescribe the proper

kinetic equations X! ¼ X!ð _!Þ and X# ¼ X#ð _#Þ, which together withdefinitions Eqs. (16) and (17) result in the generalized Ginzburg–Landau equations for the evolution of the order parameters.Onsager’s cross effects will be neglected for simplicity.

2.4.1. Boundary conditions for the order parametersOne of the boundary conditions that we will use is ! ¼ 0 or 1

and #ðtÞ ¼ 0 or 1, which corresponds to the prescribed phases atthe boundaries. Alternatively, we can prescribe the normal compo-nent of the introduced generalized forces Q! and Q# (similar to thetraction, heat or mass flux):

n � Q! ¼ n � q @w@$!

¼ H!; n � Q # ¼ n � q @w@$#

¼ H#; ð18Þ

where H! and H# are given functions. Also, one can prescribe peri-odic boundary conditions for the order parameters, which is alwaysthe case when spectral methods of solutions of the boundary valueproblems are used (Wang et al., 2001; Chen, 2002). Functions H!

and H# can be expressed in terms of th the variation of the surface

energy during a phase transformation under study (Levitas andSamani, 2014, 2011a,b; Lipowsky, 1984; Pluis et al., 1990; Levitasand Javanbakht, 2010). In this study we are not interested insurface-induced phenomena and will use H! ¼ H# ¼ 0, which corre-sponds to the unchanged surface energy during any PTs.

2.5. Helmholtz free energy for SS phase transformation via IM

The Helmholtz energy function per unit mass is defined as

w ¼ we þ wh þ �wh þ wr ¼ wl þ wr; ð19Þ

with the elastic energy

qwe ¼ 0:5ee : Cð!; #Þ : ee; ð20Þ

Cð!; #Þ ¼ C0 þ CSð#Þ � C0½ �qð!; aCÞ; ð21Þ

CSð#Þ ¼ C1 þ C2 � C1ð Þqð#; acSÞ; ð22Þ

thermal driving force

wh ¼ Gh0ðhÞ þ DGhðh; #Þqð!;0Þ; ð23Þ

DGhðh; #Þ ¼ DGh10 þ DGh

21qð#;3Þ; ð24Þ

triple-well barrier

�wh ¼ AS0ðh; #Þmð!Þ þ A21ðhÞqð!; aAÞmð#Þ; ð25Þ

AS0ðh; #Þ ¼ A10ðhÞ þ A20ðhÞ � A10ðhÞh i

qð#; a#Þ ð26Þ

and the gradient energy

wr ¼ 12q

bS0ð#Þ $!j j2 þ b21/ð!; a/; a0Þ $#j j2h i

; ð27Þ

bS0ð#Þ ¼ b10 þ b20 � b10� �qð#; abÞ; ð28Þ

where wl is the local energy. Elastic energy has the simplest formcorresponding to the linear elasticity rule. All other terms are justi-fied in Ref. Momeni and Levitas (2014). They reduce to the equa-tions in Ref. Levitas (2013) for any two phases.

In Eqs. (19)–(28), C i is the elastic modulus tensor of phase i, Gh0

is the thermal energy of melt; DGh is the change in thermalenergy between solid and melt; DGh

s0 (s ¼ 1;2) is the difference

in thermal energy between the solid Ss and M; DGh21 is the

difference in thermal energy between solid S2 and S1; AS0 and

A21 are the SM and SS energy barriers, respectively; bS0 and b21

are SM and SS gradient energy coefficients, respectively. Whilethe capital S in the super- and subscripts refers to solid andusually designates some function of #, small s (s ¼ 1;2), desig-nates a specific solid Ss; the value of the phase indicator i is0for M, 1 for S1, and 2 for S2.

The monotonous interpolating functions connecting propertiesof phases are justified in Ref. Momeni and Levitas (2014):

q y; að Þ ¼ ay2 � 2ða� 2Þy3 þ ða� 3Þy4

¼ y2 aðy� 1Þ2 þ ð4� 3yÞyh i

; ð29Þ

/ !;a/;a0� �

¼ a/!2 � 2½a/ � 2ð1� a0Þ�!3 þ ½a/ � 3ð1� a0Þ�!4 þ a0;

ð30Þ

which have the following derivatives

@qðy; aÞ=@y ¼ 2yðy� 1Þ 2ða� 3Þy� a½ �; ð31Þ

@/ð!; a/; a0Þ=@! ¼ 2!ð!� 1Þ 2 a/ þ 3ða0 � 1Þ�

!� a/

�: ð32Þ


The double-well function mðyÞ that participates in the triple-wellfunction, Eq. (25), is defined as

mðyÞ ¼ y2ð1� yÞ2 ð33Þ

and has the following derivative

dmðyÞ=dy ¼ 2yðy� 1Þð2y� 1Þ: ð34Þ

The derivatives in Eqs. (31), (32), and (34) will appear in the GLequations (see Section 3). In the interpolation function qðy; aÞ fory ¼ # or !, parameters a (a#; ab, . . .) are different for different mate-rials as well as for different material properties. These functions sat-isfy the following conditions:

q 0; að Þ ¼ 0; q 1; að Þ ¼ 1; @qð0; aÞ=@y ¼ @qð1; aÞ=@y ¼ 0;

0 6 a 6 6; ð35Þ/ 0; a/; a0� �

¼ a0; / 1; a/; a0� �

¼ 1; @/ 0; a/; a0� �

=@!

¼ @/ 1; a/; a0� �

=@! ¼ 0; 0 6 a/ 6 6: ð36Þ

The first two conditions ensure that each of the materials parame-ters changes from its known value in one phase to that in otherphase; the second two conditions guarantee fulfillment of the ther-modynamic equilibrium conditions for each phase for all tempera-tures and stress tensors, see Section 3, and the last inequalityguarantees monotonous behavior of the connecting functions. Fora0 ¼ 0 both functions coincide, /ðy; a;0Þ ¼ qðy; aÞ. The parametera0 is included for gradient energy only (Eq. (27)) and penalizessolid–solid gradient energy even in melt (! ¼ 0), which is propor-tional to a0; wrð! ¼ 0Þ ¼ b21a0 $#j j2. Otherwise, the width of theIM is zero and the problem is ill-posed (see Ref. Momeni andLevitas (2014)). The parameter a0 controls the interaction betweenS1M � S2M interfaces, which significantly affects the final structureof IM. Also, the above equations satisfy the desired instability con-ditions, which represent PT criteria (see Section 4). Thedouble-well connecting function mðyÞ has a zero value and deriva-tive at y ¼ 0 and 1, to satisfy the thermodynamic equilibrium con-ditions and also to avoid introducing any contribution to the energyfunction of homogeneous phases.

The transformation strain et and thermal strain eh tensors aredefined as

etð!; #Þ ¼ et1 þ et2 � et1ð Þqð#; at#Þ½ �qð!; at!Þ; ð37Þehðh;!; #Þ ¼ eh0 þ eh1 � eh0 þ eh2 � eh1ð Þqð#; ah#Þ½ �qð!; ah!Þ;ehi ¼ aiðh� h0Þ; ð38Þ

where ai is the tensor of the linear thermal expansion coefficientof phase i and h0 is the reference temperature for all three phasesfor both thermal expansion and transformation strain. If h0

belongs to the region of metastability of all three phases, thentransformation strain represents an actual jump in thestress-free strain (interpolated with the help of thermal expansioncoefficients) at h ¼ h0. If some of the phases do not exist at h ¼ h0,then transformation strain is defined as the jump in stress-freestrain extrapolated (utilizing thermal expansion coefficients) toh ¼ h0. Because the splitting of the total inelastic strain intothermal and transformational parts depends on the choice of thereference temperature h0 and is therefore somewhat arbitrary,the same extrapolating functions will be used for them, i.e.,ah# ¼ at# and ah! ¼ at!.

It is clear that for melt the thermal expansion is isotropic,a0 ¼ a0I. Usually, transformation strain for solidification/meltingis also considered isotropic and is determined by the change inmass density. In Refs. Levitas and Samani (2011a,b) the deviatoricpart of the transformation strain for melting/solidification wasintroduced, which is proportional to the deviatoric stresses anddescribes their relaxation within a solid-melt interface. For sim-plicity, we will neglect this type of deviatoric transformation strain

here. However, because we need to distinguish between crystallattices of two solid phases and because transformation strainbetween these lattices, et2 � et1, is well defined, we have to definetensorial transformation strains et1 and et2 for the solidification ofeach phase. The transformation strain of cubic lattices from meltis isotropic, ec

t ¼ 1=3et0I, where et0 is the volumetric transformationstrain. If et0 ¼ 0, then there is no difference from the continuumpoint of view between the melt and cubic lattice. Thus, for the def-inition of tensorial transformation strain for solidification/melting,we will consider melt as the cubic lattice with the same volumeand define eti with respect to it.

The effect of stresses can be illustrated using the local part ofthe Gibbs potential, Gðr; h;!; #Þ ¼ wðee; h;!; #Þ � 1

q r : e, where the

stress tensor is an independent variable. For simplicity, i.e., to focuson transformation strain rather than on elastic compliances, we will

use in Figs. 1 and 2 the modified Gibbs potential eGðr; h;!; #Þ ¼Gðr; h;!; #Þ þ 1

2q r : C : r ¼ � 1q r : etð!; #Þ þ wh þ �w for pure hydro-

static and shear loading, respectively. Each phase is designatedand has contour plots for different material parameters.

Using the constitutive Eq. (13)1 and the developed Helmholtzfree energy, Eqs. (19)–(30), the relation for the stress tensor is

r ¼ q@�w@e¼ q

@�w@ee

:@ee

@e¼ q

@�w@ee¼ q

@we

@ee¼ Cð!; #Þ : ee: ð39Þ

For a two-phase system the developed potential can be simpli-fied significantly by substituting # ¼ 0 (or 1) in Eqs. (19)–(30) forM–Ss PT and ! ¼ 1 for S1–S2 PT, respectively. This potential func-tion for M—Ss PT is

ws0 ¼ 12q

ee : Cð!; s� 1Þ : ee þ DGhs0ðhÞqð!;0Þ þ As0ðhÞ!2ð1�!Þ2

þ bs0

2q$!j j2;

ee ¼ e� eh0 � ðets þ ehs � eh0Þqð!; at!Þ ð40Þ

and for S1—S2 PT is

w21 ¼ 12q

ee : Cð1; #Þ : ee þ DGh10 þ DGh

21qð#;0Þ þ A21ðhÞ#2ð1� #Þ2

þ b21

2q$#j j2;

ee ¼ e� et1 þ eh1 þ et2 þ eh2 � et1 � eh1ð Þqð#; at#Þ½ �; ð41Þ

which coincide with Ref. Levitas et al. (2003). Eqs. (40) and (41) areequivalent (except an unimportant constant shift DGh

10), and areone-to-one with the models proposed in Ref. Levitas and Samani(2011a) for melting and in Ref. Levitas and Preston (2002a) foraustenite–martensite PT.

3. Ginzburg–Landau equations

The governing equations for the evolution of order parameters(GL equations) can be obtained by assuming a linear relationbetween the time derivative of the order parameters and their con-jugate dissipative driving forces:

_!L!¼ �q

@�w@!þ $ � q

@�w@$!

� �¼ �q

@�wl

@!

��e

þ b21

2@/ð!; a/; a0Þ

@!j$#j2

þ $ � bS0ð#Þ$!� �

; ð42Þ

_#

L#¼ �q

@�w@#þ $ � q

@�w@$#

� �¼ �q

@�wl

@#

��e

þ b20 � b10

2@qð#; abÞ

@#j$!j2

þ $ � b21/ð!; a/; a0Þ$#�

; ð43Þ

Fig. 2. Plot of the modified Gibbs potential eGðr; h;!; #Þ for HMX at h ¼ h21e and principal stress of r ¼ 1;�1;�3f gMPa for different hmd

c ¼ h10c and hmb

c ¼ h20c values, in the polar

system of the order parameters ! and #. 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).


where subscript e emphases that the derivative is evaluated at fixede, and L! and L# are the kinetic coefficients. To obtain a more explicitexpression for the local driving force, we utilize the definition inEqs. (11) and (39) to elaborate:

@�wl

@!

��e

¼�1qr :@etð!;#Þ@!

�1qr :@ehðh;!;#Þ

@!þ@w

l

@!

��ee

¼�1qr :@et

@!�1

qr :@eh

@!þ 1

2q@qð!;aCÞ

@!ee : CSð#Þ�C0½ � : ee

þ@wh

@!þ@

�wh

@!¼�1

qr : et1þeh1�eh0þðet2�et1þeh2�eh1Þ½

qð#;at#Þ�@qð!;at!Þ

@!þ 1

2q@qð!;aCÞ

@!ee : CSð#Þ�C0½ � : ee

þDGhðh;#Þ@qð!;0Þ@!

þAS0ðh;#Þ@mð!Þ@!

þA21ðhÞ@qð!;aAÞ@!

mð#Þ:

ð44ÞFollowing the same approach,

@�wl

@#

��e

¼�1qr :@etð!;#Þ

@#�1

qr :@ehðh;!;#Þ

@#þ@w

l

@#

��ee

¼�1qr :@et

@#�1

qr :@eh

@#þ 1

2q@qð#;acSÞ

@#qð!;aCÞee : C2�C1½ � :ee

þ@wh

@#þ@

�wh

@#¼�1

qr :ðet2�et1þeh2�eh1Þ

@qð#;at#Þ@#

qð!;at!Þ

þ 12q

@qð#;acSÞ@#

qð!;aCÞee : C2�C1½ � :eeþDGh21@qð#;3Þ@#

qð!;0Þ

þ A20ðhÞ�A10ðhÞh i@qð#;a#Þ

@#mð!ÞþA21ðhÞqð!;aAÞ

@mð#Þ@#

: ð45Þ

Substituting Eqs. (44) and (45) in Eqs. (42) and (43), respec-tively, results in a more detailed form of GL equations:

1L!

@!@t¼ r : et1 þ eh1 � eh0 þ ðet2 � et1 þ eh2 � eh1Þqð#; at#Þ½ � @qð!; at!Þ

@!

� 0:5@qð!; aCÞ

@!ee : CSð#Þ � C0½ � : ee � qDGhðh; #Þ @qð!;0Þ

@!

� qAS0ðh; #Þ @mð!Þ@!

� qA21ðhÞ @qð!; aAÞ@!

mð#Þ

� b21

2@/ð!; a/; a0Þ

@!j$#j2

þ $ � b10 þ b20 � b10� �qð#; abÞ$!

� ; ð46Þ

1L#

@#

@t¼ r : ðet2 � et1 þ eh2 � eh1Þ

@qð#;at#Þ@#

qð!;at!Þ

� 0:5@qð#;acSÞ

@#qð!;aCÞ � ee : C2 � C1½ � : ee

�qDGh21@qð#;3Þ@#

qð!;0Þ �q A20ðhÞ �A10ðhÞh i@qð#;a#Þ

@#mð!Þ

�qA21ðhÞqð!;aAÞ@mð#Þ@#

þ 0:5ðb20 � b10Þ@qð#;abÞ@#

j$!j2

þ$ � b21/ð!;a/;a0Þ$#�

: ð47Þ

3.1. Local thermodynamic equilibrium conditions

It is evident that each of three homogeneous phases,M � f! ¼ 0 and any #g; S1 � f! ¼ 1 and # ¼ 0g and S2 � f! ¼ 1and # ¼ 1g, make the thermodynamic driving forces for a changein ! and # (i.e., the right-hand side of Eqs. (44) and (45)) equal tozero for any stress and temperature; i.e., they correspond to thelocal thermodynamic equilibrium. This was one of the mainrequirements to the thermodynamic potential, which in particularimposed zero derivatives of approximating functions in Eqs. (35),(36) for each of three phases. Note that since melt is defined as

! ¼ 0 for any #, then the disappearance of the term with j$#j2

in Eqs. (42) for melt requires @/ð0; a/; a0Þ=@! ¼ 0, which doesnot have a counterpart in the previous theories (Levitas et al.,2003; Levitas and Preston, 2002a,b). At the same time, forhomogeneous solid phases all gradients should be zero and thecondition @/ð1; a/; a0Þ=@! ¼ 0 in Eq. (36) does not follow fromthe condition of thermodynamic equilibrium. It is just a conve-nient condition that provides q y; að Þ ¼ / y; a; 0ð Þ. The /ðy; a/; a0Þfunction describes the energy of the interaction between twoSM interfaces versus the distance between them (Momeni andLevitas, 2014). While more general functions can be used, thechosen /ðy; a/; a0Þ, Eq. (30), results in the interaction energybetween two SM interfaces versus the distance between them,which matches with the available sharp interface models (seeRef. Momeni and Levitas (2014)).

Solving @wl=@!��e¼ 0 for # ¼ 0 and # ¼ 1 gives three roots corre-

sponding to the extremum points along the SsM phase transforma-tion path:

!I ¼ 0; !II ¼ 1; ð48Þ


!III ¼ 0:5 As0ðhÞ � at!=qr : ets þ ehs � eh0ð Þn

� 0:5aCee : ðC0 � CsÞ : ee=qo

As0ðhÞ � 3DGhs0

n� at! � 3Þ=qr : ets þ ehs � eh0ð Þð �0:5ðaC � 3Þee :

ðC0 � CsÞ : ee=qo: ð49Þ

Two of these roots are associated with the local minima (stable/me-tastable solid or melt); the third root corresponds to the maximum(unstable equilibrium state) of the potential. The Gibbs energy bar-

riers for the M $ Ss PTs are Gð!IIIÞ � Gð!IÞ and Gð!IIIÞ � Gð!IIÞ.Following the same procedure for the path between two solids,

! ¼ 1 and 0 6 # � 1, the roots of equation @wl=@#��e¼ 0 are:

#I ¼ 0; #II ¼ 1; ð50Þ

#III ¼ 0:5 A21ðhÞ þ at#=qr : et1 þ eh1 � et2 � eh2ð Þ þ 0:5acSee :h

ðC2 � C1Þ : ee=qi= A21ðhÞ � 3DGh

21 þ ðat# � 3Þ=qr :h

et1 þ eh1 � et2 � eh2ð Þ�0:5ðacS � 3Þee : ðC1 � C2Þ : ee=qi: ð51Þ

The Gibbs energy barriers for the S1 $ S2 PTs are Gð#IIIÞ � Gð#IÞ andGð#IIIÞ � Gð#IIÞ.

When !III ¼ !I or !III ¼ !II , an energy barrier disappears andthe corresponding phase loses its thermodynamic stability andtransforms into an alternative phase under the combination of rand h that follows from these equalities. For the stress-free casethis happens at the corresponding instability temperature (see afew lines below). The same is valid when #III ¼ #I or #III ¼ #II .Instability conditions will be considered in more detail in Section 4.

3.2. Some specifications

Further specification of the temperature-dependent parametersis outlined below and described in more detail in Ref. Momeni andLevitas (2014). Thus, we assume DGh

s0 ¼ �Dss0ðh� hs0e Þ (s ¼ 1;2),

where Dss0 is the jump in entropy between solid phase Ss and meltM, and hs0

e is the thermodynamic equilibrium melting temperature

of Ss. Then for the SStransformation, DGh21 ¼ DGh

20 � DGh10 ¼ �Ds21

ðh� h21e Þ with Ds21 ¼ Ds20 � Ds10 and h21

e ¼ ðDs20h20e � Ds10h

10e Þ=Ds21.

For parameters defining the height of the energy barriersbetween equilibrium phases we assume a linear temperature

dependence: As0ðhÞ ¼ As0c ðh� hs0

c Þ and A21ðhÞ ¼ eA21c ðhÞ þ A21

c

ðh� h21c Þ, where hs0

c (and h21c ) is the critical temperature at which

stress-free M(and S1) loses its stability toward Ss (and S2) (whichfollows directly from Eqs. (66)–(69) below). For the PT betweentwo phases with the same thermal properties, such as two variantsof martensite or twins, critical temperature h21

c does not exist and

A21ðhÞ ¼ eA21c ðhÞ. For all other phases with different thermal proper-

ties, one has to set eAc ¼ 0 in order to make the stress-free instabil-ity condition, inequality (67), below satisfied first at h ¼ h21

c (Ref.Momeni and Levitas (2014)). While a more general temperaturedependence of the energy barriers height is possible, the simplestlinear relationship is generally accepted in literature and allowsone to describe experiments on size-dependence of the meltingof nanoparticles and surface melting in a broad range of tempera-tures (Levitas and Samani, 2011a,b).

3.3. Interface profile, energy, width, and velocity

Solving the GL equations without mechanics for PT betweentwo phases lead to the following relations for the interface profile,energy (E), width (d), velocity (v) (Levitas et al., 2003; Levitas et al.,2010; Momeni and Levitas, 2014):

#ðxÞ ¼ 1= 1þ e�pðx�v21tÞ=d21h i

; !ðxÞ ¼ 1= 1þ e�pðx�vs0tÞ=ds0h i

; ð52Þ

E21 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2b21 A21ðhÞ � 3DGh

21ðhÞh ir

=6; Es0

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2bs0 As0ðhÞ � 3DGh

s0ðhÞh ir

=6; ð53Þ

d21 ¼ dq½#ðxÞ;3�=dxf g�1max ¼ p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib21= 2 A21ðhÞ � 3DGh

21ðhÞh in or

;

ds0 ¼ dq½!ðxÞ;3�=dxf g�1max ¼ p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibs0= 2 As0ðhÞ � 3DGh

s0ðhÞh in or

; ð54Þ

v21 ¼ 6L21d21DGh

21ðhÞ=p; v s0 ¼ 6Ls0ds0DGh

s0ðhÞ=p; ð55Þ

where p ¼ 2:415 (Levitas et al., 2003). The main parameters of thesystem that determine formation and stability of IM are the energyratio (kE) and width ratio (kd) of the SS to SM interfaces. Theseparameters are determined using Eqs. (53) and (54), which arefunctions of temperature (Levitas and Momeni, 2014). For a partic-

ular case Aijc ¼ �3Dsij, energy and widths of interfaces are becoming

temperature-independent and we obtain:

kE ¼E21

Es0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib21

bs0

Ds21ðh21c � h21

e ÞDss0ðhs0

c � hs0e Þ

s; kd ¼

d21

ds0

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib21

bs0

Dss0ðhs0c � hs0

e ÞDs21ðh21

c � h21e Þ

s: ð56Þ

In the numerical simulations, we have considered an SS interfacewidth of 1 nm, Porter (1981) then the width of the SM interfaceto be equal to 1=kd nm. The energy of the interface with IM isdefined as Levitas (2013)

E� ¼Z x#¼0:5

�1qðw� ws1Þdxþ

Z 1

x#¼0:5

qðw� ws2Þdx; ð57Þ

where x# is the location of the sharp interface (Gibbsian dividingsurface), which should be determined using a static equivalenceapproach (Levitas, 2014; Levitas, 2014). For the interface withoutIM and 2–3–4 potential that is symmetric with respect to # ¼ 0:5,the location of the corresponding sharp interface is x# ¼ 0:5. Here,we assume the same position of the dividing surface for the casewith IM and will treat this problem in a stricter way in the future.

4. Thermodynamic stability conditions

The PT criteria between different phases in the phase-field the-ory are derived using the condition for a loss of stability of thehomogeneous crystal lattice under spontaneous variation of theorder parameters. Instability analyses of the crystal lattice havebeen used for determining the ultimate strength of perfect crystals(Milstein and Hill, 1979; Hill and Milstein, 1977; Hill, 1975; Bornand Fürth, 1940; Born and Misra, 1940; Misra, 1940; Born, 1940),melting (Born, 1939; Wang et al., 1993; Ida, 1969), andcrystal-to-amorphous transition (Wang et al., 1993; Li andJohnson, 1993). Investigating the stability of a loaded crystalrequires a specification of the loading type – i.e., conservative,extrinsic, or intrinsic loadings. The extrinsic loading accounts forthe rotation of the specimen during the loading process as in


conventional mechanical tests. The intrinsic loading assumes thatthe stresses follow the material boundaries. The stability of a crys-talline lattice under conservative external loading has been studiedusing the continuum mechanics approach and large deformationformulation by applying the Lagrange stability criteria (Milsteinand Hill, 1979; Hill and Milstein, 1977; Hill, 1975). Stability ofthe crystalline systems under a special type of nonconservativeloading was studied in Refs. Wang et al. (1995) and Wang et al.(1993). A strict thermodynamic approach for a stability analysisof the homogeneous crystalline material under external loadingand finite rotations is developed in Refs. Levitas and Preston(2005) and Levitas (2013).

Here, we limit ourselves to small strain formulation and specifyan approach from Ref. Levitas (2013) to determine conditions forthe instability of homogeneous crystalline phases Ss, and extendthis approach to describe the instability of homogeneous melttoward solidification. We formulate the following thermodynamicdefinition of instability under the prescribed stress tensor r: equi-librium phases (#eq ¼ s� 1 for Ss and !eq ¼ 0 or 1 ) are unstableunder prescribed stress tensor r if a spontaneous deviation oforder parameters results in a non-negative dissipation rate – i.e.,DP 0:

X!ðr; ee þ Dee;!eq þ D!; #eq þ D#; hÞ _!þ X#ðr; ee þ Dee;

!eq þ D!; #eq þ D#; hÞ _# P 0: ð58Þ

Here, Dee is the change in elastic strain due to changing elastic mod-uli that depend on the order parameters. Using Taylor series expan-sion around the equilibrium phases and imposing X! ¼ X# ¼ 0 foreach equilibrium phase, Eq. (58) can be approximated by

@X!ðr; ee;!eq; #eqÞ@!

_!2 þ @X!ðr; ee;!eq; #eqÞ@#

_# _!

þ @X#ðr; ee;!eq; #eqÞ@!

_# _!þ @X#ðr; ee;!eq; #eqÞ@#

_#2 P 0; ð59Þ

where h was not shown for compactness. In Eq. (59), all the deriva-tives are calculated at constant stress. For homogeneous states, theexpressions for X! and X# are given in Eqs. (16), (17). Utilizing them,Eq. (59) expands to

r :@2ðetð!eq;#eqÞþehð!eq;#eqÞÞ

@!2 �q@2wlðee;!eq;#eqÞ


@ee@!:@ee

@!

!_!2

þ 2r :@2ðetð!eq;#eqÞþehð!eq;#eqÞÞ

@#@!�2q

@2wlðee;!eq;#eqÞ@#@!

�q@2wlðee;!eq;#eqÞ

@ee@!:@ee

@#�q

@2wlðee;!eq;#eqÞ@ee@#

:@ee

@!

!_# _!

þ r :@2ðetð!eq;#eqÞþehð!eq;#eqÞÞ

@#2 �q@2wlðee;!eq;#eqÞ

@#2

�q@2wlðee;!eq;#eqÞ

@ee@#:@ee

@#

!_#2 P 0: ð60Þ

Note that since in Eq. (60) q @wl

@ee¼ r ¼ const, all terms that involve

these derivatives disappear:

r :@2ðetð!eq;#eqÞþehð!eq;#eqÞÞ


@!2

!_!2

þ 2r :@2ðetð!eq;#eqÞþehð!eq;#eqÞÞ

@#@!�2q

@2wlðee;!eq;#eqÞ@#@!

!_# _!

þ r :@2ðetð!eq;#eqÞþehð!eq;#eqÞÞ

@#2 �q@2wlðee;!eq;#eqÞ

@#2

!_#2 P 0; ð61Þ

Utilizing Sylvester’s criterion for a positive-definite matrix one caneasily find the explicit instability conditions, which are, however,complicated and bulky. It is very difficult to design a thermody-namic potential w and transformation and thermal strains for

which such sophisticated conditions reduce to simple conditionsfor each of three PTs, when they are considered separately.Similar to the sharp-interface approach, it is natural to assume thatPT criteria between two phases are independent of the third phase.This can be satisfied if and only if all mixed derivatives in Eq. (61)vanish:

@2etð!eq; #eqÞ@#@!

¼ 0;@2ehð!eq; #eqÞ

@#@!¼ 0;

@2wlðee;!eq; #eqÞ@#@!

¼ 0:

ð62Þ

Note that these constraints are imposed for equilibrium phasesonly; for intermediate values of ! and # there are no restrictionson the mixed derivatives. Eqs. (19)–(30) have been formulated insuch a way that conditions of Eq. set (62) are satisfied. Therefore,the conditions for the loss of stability of melt and solid reduce to

@X!ðr; ee;!eq; #eqÞ@!

¼ r :@2ðetð!eq; #eqÞ þ ehð!eq; #eqÞÞ

@!2

� q@2wlðee;!eq; #eqÞ

@!2 P 0; ð63Þ

@X#ðr; ee;!eq; #eqÞ@#

¼ r :@2ðetð!eq; #eqÞ þ ehð!eq; #eqÞÞ

@#2

� q@2wlðee;!eq; #eqÞ

@#2 P 0: ð64Þ

Elaborating Eqs. (63), (64) for the accepted Eqs. (19)–(30), the con-ditions for each phase to lose its stability toward any of the otherphases (i.e., phase transformation criteria) are:

M ! Ss : @X!ð! ¼ 0; # ¼ s� 1Þ=@! P 0! qAs0ðhÞ 6 at!r

: ðets þ ehs � eh0Þ þ 0:5aCee : ðC0 � CsÞ : ee; ð65Þ

Ss ! M : @X!ð! ¼ 1; # ¼ s� 1Þ=@! P 0

! 6DGhs0 � As0ðhÞ

h iq P ð6� at#Þr : ðets þ ehs � eh0Þ

þ 0:5ð6� aCÞee : ðC0 � CsÞ : ee; ð66Þ

S1 ! S2 : @X#ð! ¼ 1; # ¼ 0Þ=@# P 0! qA21ðhÞ 6 at#r

: ðet2 þ eh2 � et1 � eh1Þ � 0:5acSee : ðC2 � C1Þ : ee; ð67Þ

S2 ! S1 : @X#ð! ¼ 1; # ¼ 1Þ=@# P 0! 6DGh21 � A21ðhÞ

h iq

P ðat# � 6Þr : et2 þ eh2 � et1 � eh1ð Þ � 0:5ð6� acSÞee

: ðC2 � C1Þ : ee: ð68Þ

Formally, Eqs. (65)–(68) are similar to the instability conditions inRefs. Levitas et al. (2003), Levitas and Preston (2002a,b). For zerostresses they reduce to the instability conditions in Ref. Momeniand Levitas (2014). An essential difference should be elaboratedfor the melt instability condition (65), because applied stressesreduce to the hydrostatic pressure p0 ¼ �1=3I : r and elastic strainreduces to volumetric strain ev

e . Therefore, Eq. (65) can be rewrittenas

M ! Ss : As0ðhÞq

6 �at!p0ðevts þ ev

hs � evh0Þ þ 0:5aCðev

e Þ2=ðK0 � KsÞ; ð69Þ

where evts; ev

hs; evh0 are the volumetric transformation and thermal

strains; and K is the bulk modulus. Phase transformation criteria(66)–(69) have the desired form: transformation work exceeds thethermal threshold plus the threshold related to jump in elasticcompliances.

We will remind that the stability conditions (65)–(68) representPT criteria for a PT between two homogeneous phases; they are


independent of a third phase and are necessary for the formulationof a consistent local part of the thermodynamic potential.However, our main problem in this paper is to find conditions forinstability of an S1S2 interface with respect to the formation ofIM. Since both initial and final states in this problem are heteroge-neous, this problem (i.e., finding the conditions for the formation ofIM) will be studied here numerically using the developed PF model.During the formation of the S1MS2 diffuse interface, ! reduces from1 to some smaller value (in particular, to 0 for complete melt) andincreases back to 1, while # varies from 0 to 1 inside the IM.

5. Barrierless nucleation and disappearance of IM

Effects of system parameters on the formation and stability ofIM within a vertical interface in a rectangular sample of 40 nmheight and 300 nm width (in the reference liquid state) are inves-tigated for a wide range of temperatures. Various PT processes arestudied in this section such as jump-like first-order type PTs andcontinuous second-order type PTs, both barrierless. Two differentinitial conditions are considered for each simulation – i.e., a per-turbed SS interface and a pre-existing melt confined betweentwo solids S1MS2. Problem formulation, material properties, andnumerical approach are summarized in the Appendix. The pre-sented results revealed multiple phases, which have differentproperties and can play a key role in designing advanced materialswith new applications.

Effect of kE and kd— The effect of two main dimensionlessparameters will be studied: (i) the ratio of SS interface energy tothe energy of the SM interface (kE) and (ii) the ratio of SS interfacewidth to SM interface width (kd). The first parameter characterizesthe change in interface energy during IM formation, and the secondone characterizes the change in energy distribution that is respon-sible for the scale effects.

The effect of kE on the formation and stability of IM is studiedfor different values of kd at a fixed temperature h ¼ h21

e ¼ 432 K.The results for melting and solidification processes for models withand without mechanics are shown in Fig. 3. The results show thatincreasing kE promotes formation and retainment of disordered IM.Decreasing kd changes a first-order jump-like PT to a continuousreversible PT. Furthermore, the range of stability and retention ofIM decreases (i.e., width of IM hysteresis reduces) with decreasingkd (Fig. 3). While IM with 0 < ! < 0:15 is captured when2:5 < kE < 2:8 for kd ¼ 0:7, this range is broadened to1:6 < kE < 3:4 with 0 < ! < 0:08 for kd ¼ 1:1. Although the sharpinterface approach imposes a kE > 2 condition as a requirementfor the formation of IM, our phase field simulations indicate theemergence and retention of IM for kE < 2 by appropriate choiceof kd values. A comparison of the results presented in Fig. 3b withthe results for the system with no elastic energy (Fig. 3a) demon-strates a promotion of the IMformation in the presence of elasticenergy: all lines are shifted to smaller kE. Distribution of the orderparameters ! and # at the SMS interface, which indicates the struc-ture of IM, is shown for cases without and with mechanics in Fig. 4.

Distribution of elastic stress components r ye and r z

e are plottedin Fig. 5 for S1S2 and S1MS2 interfaces. Stresses in bulk which aredetermined by thermal and transformation strains in a clamp inthe z-direction sample do not change during formation of the IM.However, stresses in melt relax which provides an additional driv-ing force for the IM appearance.

Formation and retainment of IM as a result of the scale effect isstudied in detail by performing simulations for a wide range of kd

values at a fixed temperature (h ¼ h21e ¼ 432 K) for different kE val-

ues. The results are presented in Fig. 6 for the problem formulationwithout and with mechanics. In both cases, for small kd, after theinitial jump of ! from 1 down to some value, continuous change

in !min with change in kd in both directions is observed. For largekdIM cannot nucleate from ! ¼ 1; however, for pre-existing IMthere is a jump from small ! up to ! ¼ 1. For intermediate kd anIM-free gap is revealed which separates these two types of behav-ior. Including the elastic energy reduces the width of the IM-freegap and causes the IM for smaller kE compared to the case withoutmechanics. As kE increased the width of this IM-free gap reducedand finally disappeared for large enough kE values. Comparingthe results of simulations with and without elastic energy, a drasticchange in trend of !min for small kd is found. Without mechanics,the variation of !min as a function of kd is nonmonotonic indicatingthe existence of a kd with a maximum disordered IM. However,!min increases monotonically in the presence of elastic energy withincreasing kd.

Effect of the thermal driving force — The effect of the thermaldriving force on the energy E� and width d� of IM, as well as theinterface velocity is studied for two different parameters of theinterfacial interactions – namely, a0 ¼ 0:01 and 0.1. Simulationsare performed for kd ¼ 1:0 and kE ¼ 4:0 to ensure formation of IMin a wide range of temperatures, independent of initial conditions.Two different cases without and with elastic energy are consid-ered. Formation of an IM (!min � 0:07) with a width of � 1:4 nmis captured at � 0:8h21

e , of which the is almost 185 K below themelting temperature. The width of IM is comparable to the widthof SS and SM interfaces that are 1 nm for kd ¼ 1:0. This is consistentwith the simplified thermodynamic predictions and experimentalevidences (Levitas et al., 2004; Levitas et al., 2006). Phase-fieldsimulations demonstrate the enhancing effect of temperature onthe formation of IM, in which !min and E� reduce while d� increasesfor all case studies (Figs. 7–9). All S1MS2 interface velocities are cal-culated with respect to the fixed frame for a steady-state movinginterface, of which the profile does not change during the interfacepropagation.

Elastic energy plays a dual role: it reduces !min promoting IMbut surprisingly increases nucleation temperature for the IM, i.e.,suppresses IM (Fig. 7). An increase in nucleation temperature forIM is caused by the extra elastic energy that needs to be overcomeduring the initial stage of IM-formation before it can be compen-sated by the relaxation of the elastic energy during melting.

The interface profile and velocity during the PT between twophases with negligible elastic energy are studied numerically fora steady-state moving interface in the fixed frame. The results ofnumerical simulations quantitatively match the analytical solu-tions, Momeni and Levitas (2014) which indicates the validity ofthe developed numerical model. This model is then utilized to per-form the simulations for the samples with elastic energy, for whichan analytical solution does not exist. The simulation results indi-cate that the interface velocity increases with increasing tempera-ture and interfacial interaction coefficient a0. The simulations forthe model without elastic energy indicate a zero S1MS2 interfacevelocity at equilibrium temperature h ¼ h21

e and its magnitudeincreases with deviation from h21

e in both directions. However,the elastic energy increases the S1MS2 interface velocity frommetastable to stable solid phases and produces nonzero interfacevelocities at h21

e (Fig. 10). With elastic energy, S1MS2 interfacevelocity is positive only, because IM does not appear at a lowertemperature that may cause negative velocity.

6. Nucleation and disappearance of IM via thermally-activatedprocesses and critical nuclei

The results on the effect of different system parameters(Section 5) revealed the presence of two different steady solutionsfor IM corresponding to the local energy minima, which are

Fig. 3. Effect of kE and kd on the formation and retention of the IM. The stationary minimum values of !, i.e., !min , are plotted at h ¼ h21e ¼ 432 K and a0 ¼ 0:01, for different kd

values for a model without mechanics (a) and with mechanics (b).

Fig. 4. Scale effects on the structure of the IM. Effect of the scale parameter kd on the distribution of ! and # at the SMS interface is studied for a0 ¼ 0:01; kE ¼ 2:3, anddifferent kd ¼ 0:1; 0:2; 0:3; 1:0, and 1.2 at h ¼ h21

e ¼ 432K , i.e., for a stationary interface in a fixed frame. The origin is considered to be the location where ! is minimum.Structure of the IM is shown for partially disordered phases obtained barrierlessly from the perturbed SS interface (top row) and retained complete IM obtained from abroader pre-existing IM (bottom row). The results are plotted for the model without mechanics (a and c) and with mechanics (b and d). Allowing for mechanics significantlyincreases the degree of disordering the IM (a vs. b), e.g., !min drops from 0.92 to 0.5 for kd ¼ 0:1. The structure of IM is no longer symmetric with respect to !min for the modelwith mechanics, due to different transformation strains of each phase.


separated by an energy barrier. This barrier corresponds to thethird solution or saddle point of the energy functional and repre-sents the CN. The presence of multiple minima emphasizes theimportance of thermally-activated processes and kinetics on thefinal structure of an interface. The method to study CN for the IM

without mechanics was developed in Ref. Momeni and Levitas(2014). In this section, a generalized version of this method forthe case with mechanics is utilized for finding CN of the IM withinan S1S2 interface, and study the morphology and energetics of theCN during a heterogeneous nucleation at the horizontal interface

Fig. 5. S1S2 interface profile (a) and corresponding stress distributions (c) and S1MS2 interface profile (b) and corresponding stress distributions (d). Stress distribution isconsidered in the middle of the sample (y ¼ 20 nm). The results indicate relaxation of elastic stresses when melt is formed. Simulations are performed at h ¼ he ¼ 432 K fora0 ¼ 0:01; kE ¼ 4:0, and kd ¼ 1:0.

Fig. 6. Scale effect and formation of IM. Scale parameter kd was varied to study the formation and persistence of IM for h ¼ h21e ¼ 432K and a0 ¼ 0:01 at different kE values.

Results are obtained using models without mechanics (a) and with mechanics (b).


between two solid phases in an infinitely long cylindrical sample ofradius 20 nm (see the Appendix for details of problem formulationand numerical procedure).

Two main solutions are found: CN1 at the center of a sample andCN2 at the surface. The structures of the CN1 and CN2 are shown inFig. 11 for the solution without mechanics. Structures of these twocritical nuclei for the problem with mechanics are presented in

Figs. 12 and 13, along with the distribution of stresses throughoutthe sample.

The value of !min for the CN1 is 0.24 for the model withoutmechanics, and 0.30 for the model with mechanics. This is logical,because in the model with mechanics the elastic energy providesan additional driving force for the formation of IM and results ina CN with higher !min. An oval-shaped CN1 of IM at the center

Fig. 7. Effect of mechanics on temperature dependence of !min forkE ¼ 4:0; kd ¼ 1:0, and two a0 values. Square points designate temperatures belowwhich the IM cannot nucleate barrierlessly within the SS interface. Elastic energyplays a dual role: it increases nucleation temperature of the IM but reduces !min .


and a half-oval-shaped CN2 of the IM at the surface are formed. TheCN of the IM for the model without any elastic energy has the smal-ler aspect ratio. The width of the S1S2 interface narrows downinside the CN1 and CN2 of the IM, with and without elastic energy.Although the S1S2 interface is a plane for the model without anelastic energy, it is curved when the effect of elastic energy is con-sidered. This is due to complete relaxation of the radial stress at thesurface due to the stress-free boundary condition, while it cannotrelax in the central part of the cylindrical sample. Also, phase S2

in an upper part of the sample has a larger contraction comparedto the phase S1 in the bottom part.

We utilize the following definitions of the energies. Energy ofthe ground states with stationary S1S2 and S1MS2 interfaces aredefined as Wss ¼

RqwssdV and Wsms ¼

RqwsmsdV , respectively,

where wss and wsms are distributions of the local energy for thesesolutions. The energy for CN1 is WCN1 ¼

RqwCN1 dV , and energy of

the CN2 is WCN2 ¼RqwCN2 dV . We also define effective energies for

the stationary S1S2 interface, E21 ¼ Wss=Aint , and solid-melt-solidinterface, ESMS ¼ Wsms=Aint . Here, the interface area Aint is definedas Aint ¼

RdS#¼0:5, where S#¼0:5 is the level set of the points with

# ¼ 0:5.The values of all the above energies are collected in Tables 1 and

2, along with contributions to each of them due to the thermalenergy wh þ �wh, gradient energy wr, and elastic energy we, whichare designated with Wh; Wr, and We, respectively;W ¼ Wh þWr þWe. It is evident that without mechanics and for

Fig. 8. Effect of mechanics on temperature-dependence of the normalized energy ofIM; E� , for kE ¼ 4:0; kd ¼ 1:0, and two different a0 values. Square points designatetemperatures below which the IM cannot nucleate barrierlessly within the SSinterface. Internal stresses do not affect IM energy for smaller a0 and reduce it forlarger a0. Also, stresses increase the nucleation temperature of the IM. The IMenergy is almost a linear function of temperature.

all solutions with mechanics but S1MS2, the difference betweenthermal Wh and gradient Wr energies is smaller than 1%. Note thatfor one-dimensional solutions for a plane interface and for aone-dimensional CN, both involving two and three phases, the con-dition Wh ¼ Wr is proven analytically (see Ref. Momeni and Levitas(2014) for three phases). Consequently, deviations are due totwo-dimensional correction (without mechanics) and elastic stres-ses (with mechanics). Despite this small difference, the energy ofthe S1S2 interface without mechanics is equal to its analyticallyprescribed value of 1 J=m2. For both cases (without and withmechanics), the energy of S1MS2 is smaller than that of the S1S2

interface. The contribution of elastic energy to the total energy isrelatively small for all cases; for the S1MS2 interface it is almosttwo times smaller than for the S1S2 interface. While elastic energyincreases total energies of both S1S2 and S1MS2 interfaces, itreduces the energies per unit area because the area of the interfacethat is curved due to elastic energy is larger than the area of initialplane interfaces.

The activation energy for the CN1 of S1MS2 within the S1S2 inter-face is DECN1

sms ¼ WCN1 �Wss and the activation energy for the CN2 of

S1MS2 within the S1S2 interface is DECN2sms ¼ WCN2 �Wss. Similarly, we

define the activation energy for the CN1 and the CN2 of the S1S2

within the S1MS2 interface as DECN1ss ¼ WCN1 �Wsms and

DECN2ss ¼ WCN2 �Wsms, respectively.To experimentally detect a thermally-activated process within a

reasonable time the kinetic nucleation criterion, DE 6 ð40� 80ÞkBh,should be satisfied (Ref. Porter (1981)), where kB is the Boltzmannconstant and DE is an activation energy. After forming the CN, ther-mal fluctuations may lead to further growth of the CN and forma-tion of the alternative stable steady nanostructure. A more detailedmodel should include a stochastic term within the Langevindynamics that satisfies the fluctuation–dissipation theorem.However, computationally it is more time consuming for our pur-poses. At h ¼ he ¼ 432 K, the energy of thermal fluctuations is80kBh ¼ 0:48 10�18 J. The calculated activation energies and theircomponents are listed in Table 3. The minimum activation energyis obtained for the CN1 of the S1MS2 at the center of the samplewithin the S1S2 interface, DECN1

sms , when mechanics is included. It issmaller than the magnitude of thermal fluctuations at simulationtemperature. Therefore, a S1S2 ! S1MS2 phase transformation canoccur as a thermally-activated process when mechanics isincluded.

Remarkably, the elastic energy for the CN1 of S1MS2 makes a lar-

ger contribution to the activation energy DECN1sms than two other

Fig. 9. Effect of mechanics on temperature dependence of the normalized width ofIM; d� , for kE ¼ 4:0; kd ¼ 1:0, and for two different a0. Square points designatetemperatures below which the IM cannot nucleate barrierlessly within the SSinterface. Elastic stresses increase the width of IM; d� .

Fig. 10. Effect of IM formation on the interface velocity. Square points designatetemperatures below which the IM cannot nucleate barrierlessly within the SSinterface. Black dashed line is the velocity of the SS interface when elastic energy isnegligible, which is obtained using an analytical solution. The upper line is thevelocity of the SS interface while allowing for mechanics. All other lines with dots(indicating nucleation temperature of the IM) are for velocities of the S1MS2

interfaces. Simulations are performed for kE ¼ 4:0 and kd ¼ 1:0. For these param-eters, the magnitude of the velocity of the SS interface is larger than velocity of theS1MS2 interface. Allowing for mechanics increases the magnitude of all interfacevelocities.


contributions together. However, it slightly changes the distribu-tion of the order parameters and geometry of the CN in comparisonwith the case without mechanics which, surprisingly, drasticallyreduces contributions of Wh and Wr to the activation energies:by a factor of 46 and 31, respectively. The total activation energyof CN1 reduces due to mechanics by a factor of �16, which makesthis nucleation kinetically possible.

Such a strong effect of mechanics seems to contradict the smallchanges in the structure and geometry of the CN. To understandthis one has to look carefully at the data in Table 1. Elastic energyincreases the energy of both S1S2 interface, Wss, and the CN1; WCN1 ,slightly, but the energy of the ground state Wss is increased slightlymore than WCN1 . However, a small change in two large values, Wss

and WCN1 , drastically reduces their small difference,DECN1

ss ¼ WCN1 �Wsms! This explains the obtained paradoxical result.

Fig. 11. The 2D contour plot of distribution of !ðr; zÞ (a, c) and #ðr; zÞ (b, d) whenelastic energy is neglected, all mapped back into initial undeformed state. Theresults are plotted for CN1 of the IM within the S1S2 interface at the center of thesample (a, b) and for CN2 at the outer surface of the sample (c, d). The dotted lines in(b) and (d) show the part of the sample that contains the CN and zoomed in theinsets (a) and (c). Note a different scale in the (a, b) and (c, d): the narrowing of theS1S2 interface at the bottom occurs within CN of IM only. Simulations are performedat h ¼ h21

e ¼ 432 K, for kd ¼ 0:7, and kE ¼ 2:6. The value of !min is 0.24 for CN1.

A similar drastic reduction of the activation energy due to elasticstresses is obtained for CN2 of the IM; DECN2

ss , due to a similar rea-son. However, it is still too large to satisfy nucleation criterion. Aninteresting point is that the elastic energy makes a negative contri-bution to the activation energy DECN2

ss . Still, with a reduction insample size, the volume and energy of CN2 reduces and it mayappear below some critical sample radius. At the same time theactivation energy of CN1 is independent of the sample radius fora much smaller radii. Another important point is that the activa-tion energy for a S1MS2 ! S1S2 phase transformation is much largerthan the energy of thermal fluctuations for CN1 (and CN2); thus, theformed IM is metastable and retains.

We have to be sure that the small activation energies obtainedas the difference of two large numbers do not contain large numer-ical errors, which would compromise our conclusions. Away fromthe CN the ground state solution is not disturbed, so we can eval-uate the activation energy as DECNi

sms ¼RqðwCNi � wssÞdV �ði ¼ 1;2Þ,

where V� V is the volume around the CN, outside of which theground state solution is not changed due to the CN. By varyingV�, we found that the calculated activation energy is not sensitiveto the chosen volume V� as long as the boundaries of the V� enclosethe CN and are few nanometers away from the boundaries of CN.The same is true for DECNi

ss .Our results on the CN in Ref. Momeni and Levitas (2014) with-

out mechanics were disappointing, since they led to a conclusionthat thermally-activated nucleation of the IM is impossible. Thiswas in contradiction with the experimentally-observedthermally-activated interface kinetics for a b–dPT in HMX(Levitas et al., 2004, 2006). Now these contradictions are resolved.

In summary, considering mechanics is crucial for the correctdetermination of the activation energy for both critical nuclei,the description of thermally-activated nucleation, and the disap-pearance of the IM, as well as the kinetics of the S1MS2 and S1S2

interface propagation.

7. Concluding remarks and future directions

In this paper, a thermodynamically-consistent PF approach forPTs between three different phases is developed using polar orderparameters. It includes the effect of the stress tensor, interfaceinteractions via IP, and phase equilibrium and stability conditionsfor homogeneous phases. Explicit expressions for the Helmholtzfree energy and transformation and thermal strains, which satisfyall formulated conditions are derived. The GL equations are derivedand coupled to the full system of equations of continuum mechan-ics. They are implemented in the FE package COMSOLMultiphysics. Propagation and equilibrium of the SS interface con-taining nanometer-sized intermediate disordered IP and, particu-larly, an interfacial IM are studied in detail for an HMX energeticmaterial. Melting releases the energy of internal stresses at acoherent SS interface, which provides an additional thermody-namic driving force that promotes IM significantly below the bulkmelting temperature. The main focus of the study is the effect ofmechanics on the structure of the IM and parameters at which itnucleates and disappears, either barrierlessly or via the CN andthermal fluctuations. However, results depend strongly on theratio of widths of SS to SM interfaces, kd, the energy ratio of SS toSM interfaces, kE, temperature, and the parameter a0 that describesthe interaction of two SM interfaces via an IM. Most of the resultsare presented for temperatures that are more than hundreds ofdegrees below the bulk melting temperature. In particular, forma-tion of the IM(!min � 0:07) with a width of � 1:4 nm is captured at� 0:8h21

e ¼ 0:65h02e , which is almost 185 K below the melting tem-

perature. This is even below the 120 K reduction in melting

Fig. 12. The 2D contour plot of !ðr; zÞ (a) and #ðr; zÞ (b) for CN1 placed at the center of the sample. The dotted line in (b) shows the part of the sample that contains the CN andzoomed in the inset (a). Note a different scale in the (a) and (b): the narrowing of the S1S2 interface at the bottom occurs within CN of IM only. Simulations are performed ath ¼ h21

e ¼ 432 K, for kd ¼ 0:7, and kE ¼ 2:6. Distribution of stresses are shown in (c) rrr , (d) rrz , and (e) rzz . The dotted half-ellipse in the inset (e) represents the outline of CN1.

Fig. 13. The 2D contour plot of !ðr; zÞ (a) and #ðr; zÞ (b) for CN2 placed at the outer surface of the sample. The dotted line in (b) shows the part of the sample that contains theCN and zoomed in the inset (a). Simulations are performed at h ¼ h21

e ¼ 432 K, for kd ¼ 0:7, and kE ¼ 2:6. Distribution of stresses are shown in (c) rrr , (d) rrz , and (e) rzz . Thedotted half-ellipse in the inset (c) represents the outline of CN2. Note a different scale in the (a) and (b): the narrowing of the S1S2 interface at the bottom occurs within CN ofIM only.

Table 1Energies of the ground states ð10�18 JÞ with stationary S1S2 (i.e., Wss) and S1MS2(Wsms) interfaces, as well as the energy of S1S2 interface with CN1 (WCN1 ) at the center and CN2

(WCN2 ) at the surface of the sample, are calculated for kd ¼ 0:7; kE ¼ 2:6, and h ¼ he ¼ 432 K.

Without mechanics With mechanics

Wh Wr W Wh Wr We W

Wss 625.3041 631.3359 1256.64 633.8613 635.4197 21.4274 1290.7084Wsms 581.3217 580.9447 1162.2663 607.4991 565.2936 12.7266 1185.5193

WCN1 628.3561 634.3279 1262.684 633.9283 635.5157 21.6346 1291.0786

WCN2 662.3559 660.6505 1323.0063 637.8243 639.4214 17.9696 1295.2153

Table 2Energies per unit area (J/m2) of the ground states with stationary S1S2 (i.e.,E21 ¼ Wss=Aint ) and S1MS2 (i.e., ESMS ¼ Wsms=Aint) interfaces.


Wh=Aint Wr=AintW=Aint Wh=Aint Wr=Aint We=Aint W=Aint

E21 0.4976 0.5024 1 0.4881 0.4893 0.0165 0.994

ESMS 0.4626 0.4623 0.925 0.4678 0.4353 0.0098 0.913


temperature, which is what was predicted with the simplifiedthermodynamic treatment and observed in experiments (Levitaset al., 2004, 2006). Size-dependent melting hysteresis, transitionfrom a first-order type, jump-like, IM to a second-order type, con-tinuous and reversible IM, and formation of an IM-free gap areobtained without and with mechanics. Unexpectedly, mechanicsplays a dual role in barrierless IM nucleation: it promotes IM interms of decreasing the stationary value of !min and energy and

Table 3The activation energy ð10�18JÞ of the CN1 and CN2 within S1S2 interface, DECN1

sms and DECN2sms , and S1MS2 interface, DECN1

ss and DECN2ss .


DWh DWr DW DWh DWr DWe DW

DECN1sms

3.052 2.992 6.044 0.067 0.096 0.2072 0.3702

DECN2sms

37.0518 29.3146 66.3663 3.963 4.0017 �3.4578 4.5069

DECN1ss

47.0344 53.3833 100.4177 26.4292 70.2221 8.908 105.5593

DECN2ss

81.0342 79.7058 160.74 30.0945 73.8966 5.7049 109.696

Table 4Thermophysical properties of homogeneous HMX.

Property Value

Molar mass (M) 0.296 kg/molDensity (q)a 1848:78 kg=m3

K0 ¼ K1 ¼ K2 15 (GPa) (Sewell et al., 2003)l0 0 (GPa)l1 ¼ l2 7 (GPa) (Sewell et al., 2003)

a Calculated for b-HMX at h ¼ h21e and considered as a constant within small-

strain approximations.


increasing interface velocity and width of IM but it increases thetemperature for barrierless IM nucleation. An increase in nucle-ation temperature for IM is caused by the extra elastic energy thatneeds to be overcome during the initial stage of IM-formationbefore it can be compensated by the relaxation of the elasticenergy during melting. Although the sharp interface approach pre-dicts the formation of IM for kE > 2, our PF simulations demon-strated the nucleation and retainment of IM for kE < 2 evenwithout mechanics. With mechanics the critical kE for IM nucle-ation reduces. The difference in results is because thesharp-interface model assumes bulk properties of the melt whilePF approach operates with incomplete intermediate nanostruc-tures and includes the scale effect parameter kd. The results onthe scale effect and presence of an IM-free gap is not limited tothe specific modeled material (i.e., HMX); they are expected for awide range of materials, such as Si, Ge, and other materials withreducing melting temperature as a function of applied pressure(Levitas, 2005).

The most important and surprising result is in the revealedstrong (more than an order of magnitude) reduction of the activa-tion energy of the CN of the S1MS2 within the S1S2 interface due tomechanics in comparison with the model without mechanics.Elastic energy only slightly changes the structure of the CN, thegeometry of the S1S2 interface, and the total energy. However,allowing for mechanics increases energies of the ground S1S2 statemore than the energy of the CN. Since activation energy for theS1MS2 CNi within the S1S2 interface, DECNi

sms, is much smaller than

each of the energies WCNi and Wss, even a small difference in theeffect of mechanics on WCNi and Wss significantly reduces DECNi

sms.For simplicity, simulations have been performed assuming the

same interface energy and width for both SM interfaces, as wellas a linear dependence of the SM interface interactions on gradientenergy of the SS interface inside the melt. Also, pure volumetrictransformation strain tensors and temperature-independentenergy and width of each individual interface were accepted.These limitations can be avoided in more detailed simulationsand when corresponding experimental data is available. Also,relaxation of the elastic stresses at the solid-melt interface canbe introduced, similar to that in Ref. Levitas and Samani (2011a).The effect of IMformation on the elimination of the athermal fric-tion (Levitas et al., 2006) and amorphization via IM formation(Levitas, 2005) will be studied in the future. Note that change ofthe material parameters (kd; kE, and a0) may change some of theconclusions of the current paper.

The developed model can be adjusted and implemented tostudy the PT between any three non-equilibrium phases (Tiadenet al., 1998; Folch and Plapp, 2005), austenite and multiple marten-sitic variants (Levitas et al., 2013, 2003, 2001, 2002), grain bound-ary pre-melting (Lobkovsky and Warren, 2002; Tang et al., 2006;Mishin et al., 2009), and surface induced PT and pre-melting(Levitas and Samani, 2011b; Levitas and Javanbakht, 2010), espe-cially, for the finite width of the external surface (Levitas andJavanbakht, 2011; Levitas and Samani, 2014). It will advance theabove studies by adding the effect of the stress tensor and new

scale parameters, like kd and a0. The diagrams on the effect of sys-tem parameters such as temperature, kd, and kE can be utilized as aguideline to develop new materials and alloys.

Our model can also be used to study the nucleation of solidphases from melt (T ’oth et al., 2011), heterogeneous nucleationat the interface or an external surface (Gránásy et al., 2007), as wellas the effect of mechanics on the morphology and structure ofnuclei and nucleation kinetics. One potential direction of researchwould be extending the current model to capture diffusive PTs inmulti-component systems. Other areas of inquiry could encompassthe study of interfacial and inter-granular phases (also called com-plexions) and their phase diagrams (Cantwell et al., 2014; Tanget al., 2006), as well as a comparison with the results obtainedusing other simulation techniques such as MD (Frolov et al.,2013) and phase-field crystal (Heo et al., 2011, 2013, 2008, 2005).

Acknowledgments

The support of NSF (CMMI-0969143), DARPA (W31P4Q-13-1-0010), ARO (W911NF-12-1-0340), ONR (N00014-12-1-0525),and Iowa State University (Schafer 2050 Challenge Professorship)are gratefully acknowledged.

Appendix A. Numerical model

Equations — The complete system of equations for the descrip-tion of the PT is comprised of kinematic relations Eqs. (1) and(37), (38), momentum balance Eq. (6), elastic coefficients Eqs.(21), (22), change in thermal energy Eq. (24), energy barrier Eq.(26), connecting functions and their derivatives Eqs. (29)–(34),GL Eqs. (46), (47), elasticity rule Eq. (39), and boundary conditionsEq. (18).

Material properties — The energetic crystal cyclotetramethylene-tetranitramine (C4H8N8O8) (also known as 1,3,5,7-tetranitro-octahydro 1,3,5,7-tetrazocine), which is called HMX, is chosen asthe model material for our simulations. Parameters of the devel-oped model were calibrated by the thermodynamic properties ofd- and b-HMX which are designated by S1 and S2, respectively.Mechanical and thermophysical properties of HMX are extensivelystudied in Refs. Levitas et al. (2007), Levitas et al. (2007), andLevitas et al. (2006) and parameters for the PT kinetics aredescribed in Ref. Momeni and Levitas (2014). Here, we will use

Table 5Thermophysical properties of melt (phase 0), d (phase 1), and b (phase 2) HMX.

Ds ðJ=kg � KÞ (Henson et al., 2002) he ðKÞ (McCrone, 1950) Lðl � s=kgÞ hc ðKÞ b ðnJ=mÞ e0t (Menikoff and Sewell, 2002)

d� b (1–2) �76.62 432 1298.3 �16616a 2.4845a �0.08m� d (0–1) �429.36 550 2596.5 f ðkE; kdÞ gðkE; kdÞ �0.067m� b (0–2) �505.98 532.14 2596.5 f ðkE; kdÞ gðkE; kdÞ �0.147

a This value was calculated using Eqs. (53), (54), assuming E21 ¼ 1 J=m2 and d21 ¼ 1 nm (Porter, 1981).

Fig. 14. Distribution of !ðx; yÞ in the sample that is used for studies of barrierlessIM-nucleation at the S1S2 interface (Section 5). This result is obtained for S1S2 initialcondition at h ¼ 432 K for kE ¼ 3:6 and kd ¼ 1:0, when elastic energy is neglected.Axis x is along the width and y-axis is along the height of the sample. The axis z isperpendicular to both x- and y-axes.


an isotropic approximation for thermal expansion ai ¼ aiI and fortensors of elastic moduli:

C i ¼ KiII þ 2liD; ðA:1Þ

where Ki and li are the bulk and shear modulus of phasei; II ¼ fdpjdklg is the volumetric and D ¼ 0:5fdpkdpl þ dpldjk �2=3dpjdklg is the deviatoric part of the forth rank unit tensor(Landau and Lifshitz, 1970). Thermophysical properties of thehomogeneous phases and interfaces between them are listed in

Tables 4 and 5, respectively. We have assumed all a ¼ 3; As0c ¼

�3Dss0; A21c ¼ �3Ds21, and eA21 ¼ 0, which results in temperature-

independent interface energies and widths. However, the energyand width of all interfaces, which are determined by parameters band critical temperatures, are unknown as well as the parametera0 that determines the interaction of two SM interfaces. Weassumed typical values E21 ¼ 1 J=m2 and d21 ¼ 1 nm; Porter (1981)and used Eqs. (53), (54) to calculate b and critical temperaturesfor all phases prescribing values kd and kE, see Table 4. Unknownparameters kd and kE and a0 were varied in our simulations in orderto understand their effect and predict possible microstructures andphenomena. Proper atomistic simulations or experimental studiesare necessary to determine the exact value of these parameters.

The large negative value of h21c indicates high stability of S1 at

h ¼ h21e ¼ 432 K and loss of stability cannot occur in reality. A neg-

ative instability temperature has also been reported for the NiAl byfitting it to the MD simulations (Levitas and Preston, 2002b). Allnon-strain related material parameters are the same as in Ref.Momeni and Levitas (2014).

Problem formulation — Two different problem formulations andmethods of solutions were applied to study barrierless IM nucle-ation and microstructure evolution and for finding the CN of theIM.

(a) The GL evolution equations for barrierless IM nucleation andmicrostructure evolution are solved for a rectangular sample of40 nm height and 300 nm width with a roller boundary condition(i.e., zero normal displacement and zero shear stress) on the leftside and a fixed lower left point. Note that all initial sample sizesare for initial molten state and they are reduced during solidifica-tion into S1S2 state. Axis x is along the width of the rectangularsample and perpendicular to the interface, axis y is along theheight of the sample, and axis z is orthogonal to both of them(see Fig. 14). Plane strain conditions are imposed along the z-axis.Stresses at the faces orthogonal to the x- and y-axis are zero andstresses in the z-direction are due to confinement of the samplein this direction and transformation strains. Sample sizes withdimensions at least an order of magnitude larger than the largestinterface width (equal to the largest 1=kd > 1 in examples understudy) are chosen and simulations are performed for different sam-ple sizes to ensure size-independent results. For propagating verti-cal interface studies, an even larger sample size of up to 50 timeslarger than the largest interface width is considered to get a steadyinterface velocity and profile.

(b) For the problem of finding the CN, an axisymmetric problemformulation is used and an infinitely long cylindrical sample with aradius of 20 nm (in an initial molten state) and free boundaries are

considered. We defined ‘‘perfectly matched’’ layers in COMSOLcode (see below) with the height of 10 nm at both ends of the sam-ple in the z-direction for mimicking the infinitely long sample. Theinterface is orthogonal to the axis of symmetry z. The gradient ofthe order parameters is assumed to be zero at the boundaries,i.e., constant surface energy during PT is assumed.

Numerical approach — Our model is implemented in the finiteelement commercial software package COMSOL. The standardPDE and Structural Mechanics modules are used. The relative calcu-lation error is set to be 10�4 and a0 values larger than this numer-ical zero (a0 ¼ 0:01 and a0 ¼ 0:1) are utilized in the simulations. Amapped mesh of quadratic Lagrange elements with five elementsper SS interface width is used to ensure a mesh-independent solu-tion (Momeni and Levitas, 2014). All the simulations presentedhere are performed without applying any external force, i.e.stress-free boundary conditions are assumed.

The time scale of the simulations is determined by L#=A21 and ison the order of magnitude of 1 ns. A time-dependent solver is uti-lized to study the evolution of microstructures and barrierlessnucleation (problem formulation (a)). Time-dependent GL equa-tions are integrated using an implicit time-stepping method, whichuses variable-order variable-step-size backward differentiable for-mulas. An initial time step on the order of 1 ps and a relative toler-ance of 10�4 are chosen. It is worth noting that the integration timestep will be chosen automatically during the simulations and itreaches larger values (e.g. 0:1 ns) when steady state interfacemotion is reached. Typical simulation time for the problem understudy for simulating up to 1 s is a few hours using a conventionaldesktop computer.

A stationary solver is used for finding the CN associated with themaximum energy of the structure by solving stationary GL equa-tions (problem formulation (b)). A nonlinear solver is used that isan affine invariant form of the damped Newton method.Different initial guesses are utilized to initialize the problem andfind a configuration close to the CN so that the solution of the non-linear solver converges and gives the final structure of the CN.

When the numerical model was calibrated by the properties ofan austenite!martensite PT in NiAl, we could reproduce theresults for surface-induced PT, which were reported in the litera-ture (Levitas and Javanbakht, 2011; Levitas and Javanbakht,


2010), indicating correct implementation of our model. Withoutmechanics numerical simulations perfectly matched the analyticalsolutions, see Ref. Momeni and Levitas (2014). There is an analyt-ical solution for a sample under external homogeneous stresses inRef. Levitas et al. (2010), and our model also describes it well. Butbecause of the homogeneity of the stresses, this is equivalent torescaling the homogeneous temperature and does not complicatethe problem.

Initial conditions for a critical nucleus — The problem of findingCN consists of finding a stationary solution for GL and mechanicsequations corresponding to the saddle point of the energy func-tional. It is equivalent to the maximum of the energy functionalversus size for the given shape of the CN. However, a stationarysolution to the time-dependent GL equations corresponds to thelocal minimum of the energy functional. Thus, time-independentGL equations need to be solved to find the solution correspondingto the saddle point of the free energy functional. We have used amodified Newtonian numerical technique (Deuflhard, 1974) tosolve the stationary GL equations. Two geometries of the CN areconsidered: (i) IMforms at the center of a sample, CN1, and (ii) IMis located near the external surface, CN2.

The convergence of a numerical algorithm is drastically affectedby initial conditions. Therefore, choosing initial conditions close tothe final morphology of the CN is a key step in this process andinvolves many iterations. To introduce the CN1 of melt with differ-ent widths determined by a free parameter r0 within a SS interface,the following function is used in the cylindrical coordinate systemwith the symmetry axis z and radial coordinate r:

!CN1 ðr; zÞ ¼ 1þ exp � z� z0 �W=2ð Þ=d20� ��1

hþ 1þ exp zþ z0 �W=2ð Þ=d10� ��1

iHðr0Þ; ðA:2Þ

where W is the width of the sample, the plane SS interface is orthog-onal to z, and H is the smoothened Heaviside function. The degree ofsmoothness of the Heaviside function depends on the choice ofsystem parameters and should be chosen to avoid numericalinstabilities. To find the configuration of the CN, different initialconditions with different widths need to be considered. It can beachieved by substituting different z0 values. The chosen value ofz0 that results in the solution of the CN strongly depends on the sys-tem parameters, especially kE and kd. However, during the simula-tions we found out that an initial guess of z0 � 0:5d21 commonlyleads to a proper solution for CN1. Furthermore, including mechan-ics makes the system of equations extremely sensitive to the choseninitial conditions. In this case, a final configuration of the CN cannotbe found only by adjusting the z0 and r0 values and a tedious trialprocess of different initial conditions. Faster convergence to thefinal CN configuration can be achieved by first solving the systemwithout mechanics and using this solution as the initial conditionfor the model with mechanics. During this process graduallyincreasing the thermodynamic driving force (i.e., temperature) inorder to equilibrate the contribution due to elastic energy mightbe necessary. The CN2 can be obtained by defining

!CN2 ðz; rÞ ¼ 1�!CN

1 ðz; rÞ, with a large z0 value – e.g., z0 ¼ 8d21.Initial conditions for barrierless SS$ SMS phase transformation —

Formation and stability of IM are studied using two different initialconditions: (i) a perturbed SS interface and (ii) a pre-existing meltconfined between two SM interfaces. The final result is sensitive tothe chosen initial conditions, because of the existence of two dif-ferent stationary solutions corresponding to the local energy min-imum, namely, with and without the IM. This is in contrast to theprevious models for multivariant martensitic PTs, for which thefinal results were not sensitive to the initial conditions becauseone solution without intermediate phases exists only. For a per-turbed SS interface, the order parameter ! is initialized as 0.99 in

the entire sample and the SS interface is modeled by a variationof # according to Eq. (52) for a stationary interface with v21 ¼ 0.Alternatively, using a step function for initializing # in a SS inter-face is easy to implement and would not affect the final resultsfor multivariant martensitic PTs. However, using this functionleads to a large gradient energy at the interface and unphysicalIM (i.e., IM which cannot appear barrierlessly from the SS interface)forms in our simulations.

To study the stability of IM during the solidification process weneed to initialize the system with a pre-existing melt, which can betreated as a specific form of the S1MS2 CN in the S1S2 interface.Thus, we can use a specific form of the CN initializing function

!CN1 ðr; zÞ, Eq. (A.2), as an initial condition. In this case, we choose

a large value for z0–e.g. z0 ¼ 10d21 and substitute the Heavisidefunction H with 1. For the plane strain problem and the rectangularsample, where the x and y axes are respectively along the widthand height of the sample, we need to substitute r with y and z withx in Eq. (A.2).

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Documents

A phase-field approach to solid–solid phase ...in Ref. (Momeni and Levitas, 2014) but without mechanics. The interaction between two SM interfaces through melt are modeled using