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Cite this: Phys. Chem. Chem. Phys.,

2016, 18,12183

A phase-field approach to nonequilibrium phase

transformations in elastic solids via an intermediate

phase (melt) allowing for interface stresses†

Kasra Momeni

a

and Valery I. Levitas*

b

A phase-field approach for phase transformations (PTs) between three diﬀerent phases at nonequilibrium

temperatures is developed. It includes advanced mechanics, thermodynamically consistent interfacial

stresses, and interface interactions. A thermodynamic Landau–Ginzburg potential developed in terms of

polar order parameters satisfies the desired instability and equilibrium conditions for homogeneous phases.

The interfacial stresses were introduced with some terms from large-strain formulation even though the

small-strain assumption was utilized. The developed model is applied to study the PTs between two solid

phases via a highly disordered intermediate phase (IP) or an intermediate melt (IM) hundreds of degrees

below the melting temperature. In particular, the b2dPTs in HMX energetic crystals via IM are analyzed.

The effects of various parameters (temperature, ratios of widths and energies of solid–solid (SS) to solid–

melt (SM) interfaces, elastic energy, and interfacial stresses) on the formation, stability, and structure of the

IM within a propagating SS interface are studied. Interfacial and elastic stresses within a SS interphase and

their relaxation and redistribution with the appearance of a partial or complete IM are analyzed. The

energy and structure of the critical nucleus (CN) of the IM are studied as well. In particular, the interfacial

stresses increase the aspect-ratio of the CN. Although including elastic energy can drastically reduce the

energy of the CN of the IM, the activation energy of the CN of the IM within the SS interface increases

when interfacial tension is taken into account. The developed thermodynamic potential can also be

modified to model other multiphase physical phenomena, such as multi-variant martensitic PTs, grain

boundary and surface-induced pre-melting and PTs, as well as developing phase diagrams for IPs.

1 Introduction

Experimental phenomena

Interest in solid–solid PTs via intermediate nanoscale phases,

in particular, melt, has appeared relatively recently. This idea

was initially suggested in ref. 1 and elaborated in ref. 2 and 3 in

order to describe numerous counterintuitive phenomena for

reconstructive b–dPTs in energetic HMX organic crystals,

which were found experimentally in ref. 4 and 5 and were

considered major puzzles in HMX polymorphism for decades.

In particular, the absence of temperature hysteresis for b2d

PTs (despite the large volumetric strain, e

0

= 0.08), the absence

of b-aPT, where it is thermodynamically preferable with

respect to b-dPT (despite the two times smaller e

0

C0.04),

and a large hysteresis for d2aPTs (e

0

C0.04) could not be

explained by any known theories. It was suggested that the

internal stresses generated by volumetric strain for b–dPTs are

responsible for melting far below the bulk melting temperature

because they relax during melting and produce an extra thermo-

dynamic driving force for melting. The stresses relax as the melt

forms, and then, the stress-free melt re-solidifies into the

thermodynamically stable phase. Such a melt has been called

the ‘‘virtual melt’’ and it was considered an intermediate,

transitional, short-lived state, still having all the properties of

the bulk melt. Thus, PTs between two solids occurred through

propagation of the solid 1-melt–solid 2 (S

1

MS

2

)interface,witha

few-nanometer thick molten layer within the solid 1–solid 2

(S

1

S

2

)interface.

4–7

The thermodynamic treatment was based on

the sharp interface approach, change in surface energy during

melting was neglected, and that is why the width of the virtual

melt was not defined. The virtual melting was considered as the

mechanism of relaxation of internal stresses, plasticity, and

elimination of the athermal interface friction, and it can be

activated when traditional relaxation mechanisms (dislocation

plasticity and twinning) are suppressed.

8,9

Reduction in the

a

Department of Materials Science and Engineering, Pennsylvania State University,

University Park, Pennsylvania 16802, USA

b

Department of Aerospace Engineering, Department of Mechanical Engineering,

Department of Material Science and Engineering, Iowa State University, Ames,

Iowa 50011, USA. E-mail: vlevitas@iastate.edu

†Electronic supplementary information (ESI) available. See DOI: 10.1039/

c6cp00943c

Received 11th February 2016,

Accepted 15th March 2016

DOI: 10.1039/c6cp00943c

www.rsc.org/pccp

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melting temperature of a laser-heated Al nanolayer by 35 (Kelvin)

due to internal stresses caused by a biaxial constraint was

predicted thermodynamically and confirmed in phase field

simulations in ref. 10. However, this was not related to processes

within the SS interface. More details on the PT via the IM and its

applications are presented in Section S1 of the ESI.†

Phase field approach

This is a continuum-based method intended to study the structure

of nonequilibrium interfaces during structural changes, like

martensitic and reconstructive PTs,

11–18

twinning,

19–22

melting,

10,23–27

and evolution of multigrain structures.

28

This

method is computationally less expensive compared to atomistic

simulations and includes the scale effects in contrast to the

classical continuum model of PTs with sharp interfaces. More

details about the phase field approach can be found in Section S2

of the ESI.†The main contribution to the PFA for the IM was

developed with the help of polar order parameters.

29–32

The starting

point was the local potential in hyperspherical coordinates

developed in ref. 33, in which austenite was at the center of the

sphere, and all martensitic variants were located at the hypersphere

of radius U= 1. Each variant belonged to the corresponding

coordinate axis and was characterized by the angular order para-

meter W

i

, which is an angle between radius-vector !and the

corresponding coordinate axis, normalized by p/2. Belonging to

the hypersphere represents a nonlinear constraint for orientational

order parameters. Because of impossibility to obtain consistent

PT criteria from the thermodynamic instability conditions, the

nonlinear constraint for the hyperspherical order parameters

was substituted in ref. 22 with the linear constraint (plane),

which does not include austenite. Nevertheless, PT criteria could

not be derived in a completely consistent way for more than

three phases. For three phases the constraint is linear for both

models ref. 22 and 33. In polar order parameters, with single

radial Uand orientational Worder parameters, this theory is

completely consistent with the two-phase theory and produces

correct PT criteria. This theory was generalized for three arbitrary

phases and utilized for studying the solid–solid PT via the IM in

ref. 30 for stress-free cases and in ref. 29, 31 and 32 with elastic

stresses. However, interface stresses were not introduced in the

theory for the IM yet, which is the topic of the current paper.

Designations

Contraction of tensors Aand Bover one index is designated as

AB, and contraction over two indices is represented by A:B.

Elastic, thermal, and transformational parts of the strain tensor

are designated with subscripts e,y, and t, respectively. Sub-

scripts sand adesignate symmetric and antisymmetric parts of

tensors; := is ‘‘equal by definition’’, Iis the unit tensor, and #

is a dyadic product of vectors. The gradient operator in the

undeformed state is r

, and in the deformed state is r.

Interface stresses

It is well known

34–37

that any phase interface is under biaxial

interface stresses of magnitude T(N m

1

), which equals the

surface energy gfor a hydrostatic medium (liquid–liquid and

liquid–gas interfaces). Interfaces that involve at least one solid

phase, e.g., solid–liquid, solid–gas interfaces, and interfaces

between solid phases, grain and twin boundaries, are subjected

to extra surface stresses as they can also elastically deform.

There is significant literature that includes deriving of balance laws

and constitutive equations for elastic interfaces (see ref. 38–45 and

reviews in ref. 46–48). Existing problems are related to the fact

that: (a) detailed information about the interface properties,

e.g. elastic moduli, is unknown; (b) strongly heterogeneous

fields of properties, transformation and total strains across

an interface; and (c) simple constitutive equations may not be

derived for stresses, even including gradient type theories.

49

These theories are used, in particular, for the determination of

the effect of the interface stresses on the effective properties of

nanomaterials, see e.g., ref. 50.

A review of the literature on interface stresses in the PFA is

presented in ref. 51 and will not be repeated here. For a sharp

interface, when the surface energy gis defined per unit of the

deformed area, the magnitude of the surface stresses is deter-

mined by the Shuttleworth equation

s

S

=g+qg/qe

i

=

s

st

+

s

S

e

,

where e

i

is the mean interface strain.

47,52

A bar above the

symbol sis used for these ‘‘stresses’’, because they are singular

at the sharp interface and have dimensions of force divided by

the interface length rather than the area (as usual stresses).

Consequently, the interface stress consists of two parts, one,

s

st

, is the same as for an interface in a hydrostatic medium,

and another, s

S

e

, is due to the interface’s elastic deformation.

Subscript st means the structural part of the interface stresses,

for which we will also use synonyms interfacial stresses or

interface tension. Our main point in ref. 51, 53–55 for martensitic

PTs and in ref. 26 and 27 for melting is that the elastic part of the

interface stresses s

S

e

is fully defined from the solution of the

coupled Ginzburg–Landau and mechanics equations for a PT

problem. It is present in all previous studies, which did not focus

on interface stresses at all, e.g.,inref.11–13,18,33,56and57.

The elastic contribution of the interface stresses exists also for

the complex SMS interfaces in ref. 29, 31 and 32. However, the

structural part of the interfacial stresses, i.e.,

rst ¼sstðIkkÞ;k¼rU=jrUjor k¼rW=jrWj;

T:¼ð1

1

sstdB¼g;

(1)

should be strictly introduced into PFA. Here s

st

is the magnitude

of the biaxial interface stresses (tension), kis the unit normal to

the finite-width interface, either S

s

M(s= 1 or 2) or S

1

S

2

,in

the deformed configuration, Bis the coordinate along the k,and

Ik#kis the two-dimensional unit tensor within an interface.

It was shown in ref. 51, 53–55 that it is a nontrivial problem even

for interfaces between two phases. It was determined that, for

introducing the interface stresses, s

st

, in the form of biaxial tension

with the magnitude equal to the nonequilibrium interface energy,

some geometrically nonlinear terms should be introduced or kept

in the thermodynamic potential even if strains are infinitesimal.

Namely, gradient of the order parameters should be evaluated in

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the deformed configuration and some terms should be multi-

plied by the Jacobian of the deformation gradient J=r

0

/rC1+

e

0

, where r

0

and rare the mass densities in the reference

(nondeformed) and current (deformed) configurations. These

terms are the gradient energy, double well barrier, and some

part of the term proportional to the thermal driving force for

the PT. The main challenge is to determine the last term. For

this purpose, the closed-form solution to the Ginzburg–Landau

equation for a moving interface is utilized, and the condition

under which interface stresses reduce to a biaxial tension

determined the desired term. For an equilibrium interface,

integration of the interface stresses in eqn (1) resulted in the

equilibrium interface energy, as required. For a nonequilibrium

interface, even the definition of the interface energy was

not well-formulated, because it involves the concept of an

arbitrary Gibbsian dividing surface. This problem was resolved

in ref. 51, 53–55, and fully consistent expressions for the

interface energy and stresses for the two-phase system were

obtained.

It was specifically mentioned in ref. 51 and 53 that interface

stresses for variant-variant (twin) interfaces cannot be strictly

defined within the existing theories because these interfaces

are described with the help of two order parameters, an

analytical solution for such an interface does not exist, and

interface energy could be determined analytically. Still, some

intelligent guesses have been made and interface stresses have

been calculated numerically for some cases in ref. 58. Theory in

ref. 22 formally included interface stresses for multivariant

martensitic PTs, but they were not justified. Previous multi-

phase theories

59–69

did not consider interface stresses at all. In

ref. 70, interface stresses were included in the new version of

the multiphase theory, without justification; the results have

been applied to multivariant martensitic PTs only. Thus, there

is currently no PFA for S–S PT via the IM or any other complex

interfaces that include interfacial stresses. Note that even for

two-phase interfaces, there were no studies of the effect of extra

terms due to interface stresses in the thermodynamic potential

on instability conditions.

Theobjectivesofthispaperare:(a)todevelopaPFAto

PTs between three phases (in particular, for S–S PT via the

IM), which takes into account both elastic and interfacial

stresses and (b) to study their effects as well as the effect of

material parameters and temperature on the local thermo-

dynamic instability conditions, barrierless formation and the

disappearanceoftheIM,structureandparametersoftheIM

(SMS interface energy, width, and mobility), and activation

energy for the thermally activated IM nucleation and dis-

appearance. Our starting points will be the model

31

for the

IM with mechanics but neglected interface stresses and PFAs for

two-phase interfaces with interfacial stresses.

51,53,55

The reminder of this paper is organized as follows. The polar

order parameters and kinematic relationships are described in

Section 2. The global first and second laws of thermodynamics

are utilized to derive the local dissipation inequality in Section 3.

Constitutive equations are derived in Section 4. The thermo-

dynamic potential that satisfies all the equilibrium and

instability conditions for all three homogeneous phases,

as well as includes interfacial stresses, is introduced in

Section 5. Explicit expression for interfacial stresses is derived

in Section 6. The GL evolution equations are derived in

Section 7 by assuming a linear relation between the rates of

order parameters, _

Uand _

W, and the conjugate thermodynamic

forces. The thermodynamic conditions for the loss of stability

of each homogeneous phase to any of the other two homo-

geneous phases are elaborated in Section 8. Details for setting

up the computational models for studying the barrierless

and thermally-activated phase transformations, as well as

the structure of critical nuclei and activation energies, are

described in Section 9. The results for the analysis of stability

and barrierless formation of the IM are presented in Section 10

for diﬀerent system parameters for two samples with and

without surface tension. We revealed ranges of parameters for

which the IM forms in a barrierless way or via a thermally

activated process that is separated by an IM-free gap. Relaxation

of the elastic stresses due to the formation of the IM and

the eﬀect of surface tension on the interface velocity have been

studied in this section as well. The eﬀect of elastic energy

and interfacial tension on the formation and structure of

CN and its energy is studied in Section 11, where we have

shown that while elastic energy significantly promotes the

appearance of the IM CN, surface tension can increase its

energy. Finally, the results are summarized in Section 12

along with concluding remarks, potential applications, and future

research directions.

2 Polar order parameters and

kinematics

Polar order parameters are utilized here,

29–31,33

where the origin

of the polar coordinate system is considered as the reference

phase 0 and phases 1 and 2 are located on two orthogonal basis

vectors. Variation in the radial order parameter Udescribes the

PT between phase 0 and any of phases 1 or 2 along the

corresponding basis vectors. The angular order parameter W,

where pW/2 is the angle between the radius !and the positive

first basis vector, describes the PT between phases 1 and 2.

Examples of the thermodynamic potentials in terms of the polar

order parameters are presented in Fig. 1.

In this paper, we will model the SS phase transformation via

an IM for the energetic material HMX. It is convenient to

choose the reference phase 0 to be melt M. However, depending

on the problem specifications, any phase can be chosen as the

phase 0 – e.g., austenite for multivariant martensitic phase

PTs.

19,33

We also choose that the HMX d-phase corresponds to

(U= 1 and W= 0) and refer to it as solid phase 1 (S

1

). Phase 2 is

the HMX b-phase, which will be referred to as solid phase 2 (S

2

),

and corresponds to (U= 1 and W= 1). This model will be

equivalent to the model for two-variant martensitic PT by

assuming the parent phase 0 to be austenite, and S

1

and S

2

are two martensitic variants with equal thermal energies.

22,33

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Using small-strain approximation and the additive decomposi-

tion, the strain tensor can be described as the sum of elastic e

e

,

transformational e

t

, and thermal e

y

contributions:

e¼r

u

s

¼eeþetðU;WÞþeyðU;W;yÞ¼1=3e0Iþe;

r0=r¼1þe0¼1þI:e;

(2)

where e

0

and eare the volumetric and deviatoric parts of the

strain tensor, respectively; uis the displacement vector.

As it was found in ref. 51, 53–55 and discussed in the

Introduction, interface stresses are introduced by including

some geometric nonlinearities even when strains are small.

Thus, a strict approach should be based on fully large-strain

formulation (like in ref. 51) followed by simplification of

the final equations utilizing small strain approximation. We

followed this method and obtained strict final results for small

strains with interfacial stresses. However, in order to make

our derivations simpler and accessible to those who are not

familiar with strict large-strain formulations, we will work

within a small strain approach and will keep only those

finite-strain terms, which we obtained by simplifying the strict

geometrically linear formulation.

There are two main equations, where we need to keep

geometrically nonlinear terms during derivations and can

neglect them in the final results only. First, some terms in

the free energy will be multiplied by r

0

/r=1+e

0

in order

to obtain proper contribution to the interface stresses after

diﬀerentiation with respect to strain. Indeed, even for negligible

e

0

, according to eqn (2) d(r

0

/r)/de=I, which will result in the

desired contribution to the mean interface stress. In the case of

neglecting the small strain term in the initial equation, i.e.,

utilizing JC1, one obtains dJ/de=0, and could not obtain

correct interface stresses.

Furthermore, we will need to specifically take into account

that the gradient of the order parameters, rUand rW, are

defined in the current configuration, when we evaluate their

time derivative. Thus, we will use

drUðÞ

dt¼r

_

UrUrv;dðrWÞ

dt¼r

_

WrWrv;(3)

where vis the velocity, rv=_

e+xis the velocity gradient tensor and

xis the spin tensor with respect to current configuration. Should

the gradient be defined in the undeformed state, the second

(convective) terms in eqn (3) would disappear and we will not be

able to obtain one more contribution to the interface stresses.

After these two equations are utilized in the derivations, in

all final equations we can implement rCr

0

and rCr

0

,

which we will do without extra reminders.

3 Balance of energy and the entropy

production rate

Let us consider a continuum sub-domain of a three-phase body

with the volume Venclosed by the surface boundary S. The

first law of thermodynamics for this sub-domain is

ðS

pvhnðÞdSþðS

QU_

UþQW_

W

ndS

þðV

rðfvþrÞdV¼d

dtðV

rUþ0:5vvðÞdV:

(4)

Here, Uis the internal energy per unit mass, p=rnis the

traction vector, ris the stress tensor, his the heat flux, fis the

body force per unit mass, ris the specific volumetric heat

supply rate per unit mass, and nis the surface outer unit

normal (see Fig. 2). It should be noted that the surface energy is

not included because Sis an arbitrary internal surface. The

generalized surface forces Q

U

and Q

W

balance the terms, which

will appear due to the dependence of the thermodynamic

potential on rUand rW, respectively. Ignoring these surface

forces (as it was always done in physical literature) prevents the

volume Vin eqn (4) to be arbitrarily chosen and the global

energy equation cannot be transformed into its local form

Fig. 1 Gibbs potential G

˜(r,y,U,W) for HMX at y=y

21

e

and principal stresses

of r= {1, 2, 4} MPa for diﬀerent y

md

c

=y

10

c

and y

mb

c

=y

20

c

values, in the

polar system of the order parameters Uand W(a–d, f). 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).

Fig. 2 Schematic plot of thermodynamic forces and phases in a con-

tinuum sub-domain with volume Venclosed by a surface S. Surface

traction, heat flux, and surface unit normal are designated by p,h,andn,

respectively. The body force and specific volumetric heat supply rate per

unit mass are represented by fand r, respectively. Generalized thermo-

dynamic forces Q

U

and Q

W

are shown near the phase interface.

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without additional assumptions. The global second law of

thermodynamics in the form of the Clausius–Duhem inequality

is presented as

S¼d

dtðV

rsdVðV

rr

ydVþðS

h

yndS0;(5)

where Sis the total entropy production rate and sis the

entropy per unit mass. Using the Gauss theorem, relationship

rv=_

e+x, conservation of mass, drVðÞ=dt¼0, and symmetry

of the stress tensor r, after some well-known mathematical

manipulation we obtain:

ðV

r:_

er_

Urhþrrþr QU_

UþQW_

W

dV

þðV

rrþrfr_

!

_

vdV¼0;

(6)

S¼ðV

r_

srr

yþrh

y

dV0:(7)

The energy balance should be invariant with respect to any

rigid-body motion. Therefore, substituting vwith vv0, where

v0is the superposed rigid body velocity, should not change

eqn (6). This is only possible if rr+rf=r:

v, which is the

conservation of linear momentum law. It indicates that the

local momentum balance is not aﬀected by the thermodynamic

forces Q

U

and Q

W

. Localizing the global eqn (6) and (7), we have:

r:_

er_

Urhþrrþr QU_

U

þr QW_

W

¼0;(8)

r~

S¼r_

srr

yþrh

y¼r_

srr

yþ1

yrhry

y2h0;(9)

where ~

Sis the local entropy production rate per unit mass.

Substituting rhrrfrom eqn (8) in eqn (9), multiplied by y,

and defining the local dissipation rate D:¼y~

S, we obtain

rD:¼rDry

yh0

rD:¼r:_

er_

Uþry_

sþr QU_

U

þr QW_

W

:

(10)

Assuming thermomechanical processes to be mutually inde-

pendent, the inequality (10) splits into rD0and the Fourier

inequality, ry

yh0. Introducing the Helmholtz free energy

function c=Uys, inequality rD0can be rearranged as

rD¼r:_

er_

crs_

yþrðQU_

UÞþr QW_

W

0:(11)

4 Structure of the constitutive

equations

Exploring dissipation inequality

We assume c=c(e

e

,U,W,y,rU,rW). Expressing the Helmholtz

free energy cin terms of total strain e, rather than elastic strain

e

e

, makes further derivations more compact. Therefore, we

introduce

cby equation

c=c(e

e

,U,W,y,rU,rW)

=c(ee

t

(U,W)e

y

(y,U,W),U,W,y,rU,rW)

=

c(e,U,W,y,rU,rW), (12)

where e

t

(U,W) and e

y

(y,U,W) are included in

cin terms of their

arguments U,W, and y. Using eqn (3), the last two terms in

eqn (11) can be written as

r QU_

U

¼rQU

ðÞ

_

UþQUr_

U¼rQU

ðÞ

_

UþQU rU

þQUrU:rv;(13)

r QW_

W

¼rQW

ðÞ

_

WþQWr_

W¼rQW

ðÞ

_

WþQW rW

þQWrW:rv:(14)

Rewriting eqn (11) using eqn (13) and (14), and expanding _

c

we have:

rD¼rr@

c

@eþQUrUþQWrWðÞ

s

:_

e

þQUrUþQWrWðÞ

a:xrsþr@

c

@y

_

y

r@

c

@UrQU

_

UþQUr@

c

@rU

rU

r@

c

@WrQW

_

WþQWr@

c

@rW

rW

0:

(15)

Assuming that Dis independent of _

y,rU

, and rW

leads to

s¼

@

c

@y;(16)

QU¼r@

c

@rU;QW¼r@

c

@rW:(17)

Eqn (17) defines the generalized thermodynamic forces that

have been introduced in the first law of thermodynamics. It

follows from eqn (17) that, without them, free energy cannot be

dependent on the gradient of the order parameters. Substituting

eqn (17) back into eqn (15) and assuming the dissipative rate

to be independent of _

e, the constitutive equation for the stress

tensor is

r¼r@

c

@errU@

c

@rUþrW@

c

@rW

s

:(18)

Dissipative stresses can also be introduced into eqn (18), like in

ref. 51 and 53.

Isotropic interface energy

Now we will prove that the antisymmetric tensors in eqn (15)

vanish for an isotropic interface energy, i.e., when

cis an

isotropic function of the vectors f

U

:=rUand f

W

:=rW. In this

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case

c=

c(...,n

U

n

U

,n

W

n

W

,n

U

n

W

). We have

QUrU¼r@

c

@fU

fU¼r@

c

@fUfW

ðÞ

@fUfW

ðÞ

@fU

þ@

c

@fUfU

ðÞ

@fUfU

ðÞ

@fUfU

¼r@

c

@fUfW

ðÞ

fWfU

ðÞþ2@

c

@fUfU

ðÞ

fUfU

ðÞ

:

(19)

Similarly, we have

QWrW¼r@

c

@fW

fW¼r@

c

@fWfU

ðÞ

fUfW

ðÞ

þ2@

c

@fWfW

ðÞ

fWfW

ðÞ

:

(20)

Therefore

QUfUþQWfW¼r@

c

@fUfW

ðÞ

fWfU

ðÞþfUfW

ðÞ½

þ2@

c

@fUfU

ðÞ

fUfU

ðÞþ

@

c

@fWfW

ðÞ

fWfW

ðÞ

;

(21)

which is a symmetric tensor. Thus, (Q

U

#rU+Q

W

#rW)

a

=0

and the second term in eqn (15) vanishes. The reminder of

eqn (15) leads to the dissipative inequality for the order para-

meters,

rD¼XU_

UþXW_

W0;(22)

where the generalized forces, X

U

and X

W

, are defined as

XU¼r@

c

@Uþr r@

c

@rU

;(23)

XW¼r@

c

@Wþr r@

c

@rW

:(24)

Boundary conditions for order parameters

Among all possible boundary conditions associated with the

order parameters we will consider two conditions: prescribed

constant value for the order parameter, usually 0 or 1,

U=0or1; W= 0 or 1, (25)

or prescribed (usually zero) normal component of the thermo-

dynamic forces Q

U

and Q

U

:

nQU¼n@c

@rU¼0;nQW¼n@c

@rW¼0:(26)

The first type of boundary conditions corresponds to a specified

phase at the boundary. The second type of boundary conditions

corresponds to the constant surface energy during PTs. If the

surface energy changes, it is easy to take this into account, like

in ref. 26, 54 and 71.

5 Specific model for a three-phase

system with interfacial stresses

Helmholtz energy

We will introduce the Helmholtz energy function per unit mass

for a model with interface stresses in the form

c¼ceþcyþr0

r

cyþr0

rcr¼clþr0

rcr:(27)

Here, the elastic energy is

r

0

c

e

= 0.5e

e

:C(U,W):e

e

; (28)

C(U,W)=C

0

+[C

S

(W)C

0

]q(U,a

C

); (29)

C

S

(W)=C

1

+(C

2

C

1

)q(W,a

cS

), (30)

and C

i

is the elastic modulus tensor of phase i(= 0 is the melt,

i=1issolidS

1

,andi=2isS

2

). Below, the capital S in super-

and subscripts means solid and usually designates some

function of W;thesmalls= 1, 2 designates specific solid

phase S

s

. The thermal energy related to the driving force

for PTs is

c

y

=G

y

0

(y)+DG

y

S

(y,W)q(U,3); DG

y

(y,W)=DG

y

10

+DG

y

21

q(W,3);

DG

y

s0

=Ds

s0

(yy

s0

e

), (s= 1, 2), (31)

where G

y

i

(y) is the thermal energy of phase i,Ds

s0

and DG

y

s0

are

the jumps in the entropy and thermal energy between solid

phase S

s

and melt M, DG

y

21

is the diﬀerence in thermal energy

between solid S

2

and S

1

,andy

s0

e

is the thermodynamic equilibrium

melting temperature of S

s

. Eqn (31) corresponds to the same

specific heat for all phases.

The local thermal energy that contributes to the interface

stresses is

cy¼AS0ðy;WÞ3DGy

Sðy;WÞ

mðUÞ

þA21ðyÞqU;aA

ðÞ3DGy

21qðU;3Þ

mðWÞ;

(32)

A

S0

(y,W)=A

10

(y)+[A

20

(y)A

10

(y)]q(W,a

W

); m(y)=y

2

(1 y)

2

.

(33)

Here, m(y) describes a double-well potential, and A

S0

and A

21

are SM and SS energy barriers, respectively. The local inter-

facial energy is chosen such that it reproduces the desired

interface stresses, which for each of the interfaces between

any two of three phases represents a biaxial tension with the

magnitude equal to the nonequilibrium interface energy. For

r=r

0

,thesumcyþ

cyreduces to the expression for the total

thermal energy in ref. 29 and 30 for negligible interface

stresses. The simplest linear temperature dependence is

usually accepted:

A

s0

(y)=A

s0

c

(yy

s0

c

); A

21

(y)=A

21

c

(yy

21

c

), (34)

where y

s0

c

and y

21

c

are the critical temperatures at which the

stress-free melt and S

1

lose their stability toward S

s

and S

2

(this

follows from eqn (66) and (68)) and A

s0

c

and A

21

c

are corres-

ponding proportionality factors. For the three diﬀerent phases

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these critical temperatures exist; this is not the case when some

phases have equal thermal energies, like martensitic variants

(see ref. 30 and 31).

The gradient energy is

r

0

c

r

= 0.5[b

S0

(W)|rU|

2

+b

21

f(U,a

f

,a

0

)|rW|

2

]; (35)

b

S0

(W)=b

10

+(b

20

b

10

)q(W,a

b

), (36)

where b

S0

and b

21

are SM and SS gradient energy coeﬃcients,

respectively. A detailed description of the temperature dependent

parameters can be found in ref. 30.

Transformation strain e

t

and thermal strain e

y

are

e

t

(U,W)=[e

t1

+(e

t2

e

t1

)q(W,a

tW

)]q(U,a

tU

); (37)

e

y

(y,U,W)=e

y0

+[e

y1

e

y0

+(e

y2

e

y1

)q(W,a

yW

)]q(U,a

yU

); e

yi

=a

i

(yy

0

),

(38)

where a

i

is the tensor of the linear thermal expansion of phase i.

Both the transformation and thermal strains are measured with

respect to the reference temperature y

0

for all three phases.

However the choice of y

0

is arbitrary, and therefore, decomposing

the total inelastic strain into thermal and transformational parts,

which depends on the choice of y

0

, is also somewhat arbitrary.

Therefore, we will consider the same interpolating function for e

t

and e

y

–i.e.,a

yU

=a

tU

and a

yW

=a

tW

.

Monotonous interpolating functions that connect properties

of mutual phases are

q(y,a)=ay

2

2(a2)y

3

+(a3)y

4

;qq(0,a)/qy=qq(1,a)/qy=0;

0oao6; (39)

f(U,a

f

,a

0

)=a

f

U

2

2[a

f

2(1 a

0

)]U

3

+[a

f

3(1 a

0

)]U

4

+a

0

;a

0

40;

qf(0,a

f

,a

0

)/qU=qf(1,a

f

,a

0

)/qU=0; 0oa

f

o6 (40)

In the interpolation function q(y,a), y=Wor U, and a

(a

W

,a

b

,...) is the material parameter. The interpolating

function fdiﬀers from qin a way that f(0,a

f

,a

0

)=a

0

40.

This means that when Wvaries from 0 to 1 at U=0,i.e., one

solid transforms into another through the IM, r

0

c

r

=

0.5b

21

a

0

|rW|

2

40. While such a penalization seems, from

the first glance, counterintuitive, it controls the interaction

between S

1

M–S

2

M interfaces, eliminates ill-posedness of the

problem, eliminates zero IM width, and significantly affects the

final structure of the IM when its width is comparable to the SM

interface width.

29–31

Zero derivatives of both interpolating functions as well as for

the double-well potential m(y) for homogeneous bulk M and S

s

phases follow from the conditions that the bulk phase must

satisfy thermodynamic equilibrium equations for the order

parameters,

29–31

i.e.,X

U

(M) = X

U

(S

s

) = 0 and X

W

(M) = X

W

(S

s

)=0,

see Section 7.

6 Elastic and interfacial stresses

Using the constitutive eqn (18) and the developed Helmholtz

free energy, eqn (27)–(40), the relation for the stress tensor is,

r¼r@

c

@er@

c

@rUrUþ@

c

@rWrW

¼r@ce

@ee

þr

cyþcr

IbS0rUrU

(41)

b

21

f(U,a

f

,a

0

)=W#=W. (42)

We took into account: (a) that

cdepends on strain ethrough

elastic energy c

e

and multiplier r

0

/r=1+e

0

; (b) the relation

@ce

@e¼@ce

@ee

:@ee

@e¼@

c

@ee

, and (c) relation d(1 + e

0

)/de= d(1 + I:e)/

de=I. The stress tensor rcan be decomposed into elastic, r

e

,and

the structural stress, or interface stresses (surface tension), r

st

:

r¼reþrst;re¼r@ce

@ee

’r0

@ce

@ee

¼CðU;WÞ:ee;

rst ¼r

cyþcr

IbS0rUrUb21 fU;af;a0

rWrW:

(43)

The elastic stress tensor has the same expression as for bulk

phases. Defining the unit normal vector to the SM interface,

n

U

=rU/|rU|, and the SS interface, n

W

=rW/|rW|, the interfacial

stress can be transformed to

rst ¼r0

cyþbS0ðWÞ

2rU

jj

2þb21

2fU;af;a0

rW

jj

2

I

bS0ðWÞrUrUb21 fU;af;a0

rWrW

¼r0

cybS0ðWÞ

2rU

jj

2b21

2fU;af;a0

rW

jj

2

I

þbS0ðWÞrU

jj

2InUnU

ðÞ

þb21fU;af;a0

rU

jj

2InWnW

ðÞ:

(44)

As required, the interfacial stresses vanish in the bulk where the

order parameters are constant, because in bulk rU=rW=0and

cy¼0due to the double-well potential in eqn (32).

For a two-phase system, the developed potential and expres-

sion for the transformation strain can be simplified signifi-

cantly by substituting W= 0 (or 1) in eqn (27)–(40) for the M–S

s

PT and U= 1 for the S

1

–S

2

PT, respectively. They reduce to

cs0¼1

2r0

ee:CðU;sÞ:eeþDGy

s0ðyÞqðU;3Þ

þr0

rAs0ðyÞ3DGy

s0ðy;sÞ

U2ð1UÞ2þbs0

2rrU

jj

2;

(45)

e

e

=ee

y0

(e

ts

+e

ys

e

y0

)q(U,a

tU

) (46)

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for the M–S

s

PT and to

c21 ¼1

2r0

ee:Cð1;WÞ:eeþDGy

10 þDGy

21qðW;3Þ

þr0

rA21ðyÞ3DGy

21ðy;1Þ

W2ð1WÞ2þb21

2rrW

jj

2;

ee¼eet1 þey1þet2 et1 þey2ey1

ðÞqW;atW

ðÞ½;

(47)

for the S

1

–S

2

PT. Eqn (45) and (47) are equivalent, except

for a constant shift DG

y

10

, and correspond with the models proposed

in ref. 27 for melting and in ref. 53 for the austenite-martensite PT.

Interface stresses for two-phase systems reduce to

r

21

st

=(r

0

[A

21

(y)3DG

y

21

(y,1)]W

2

(1 W)

2

0.5b

21

|rW|

2

)I

+b

21

|rW|

2

(In

W

#n

W

);

r

s0

st

=(r

0

[A

s0

(y)3DG

y

s0

(y,s)]U

2

(1 U)

2

0.5b

s0

|rU|

2

)I

+b

s0

(W)|rU|

2

(In

U

#n

U

). (48)

These expressions coincide with equations for the interface

stresses justified in ref. 53 for two-phase systems, which was

the main reason for the choice of functions

cyand c

y

in

eqn (31)–(33). With such a choice, it is proven analytically that

for the propagating interface the mean stress in eqn (48)

disappears at each point, interface stresses represent biaxial

tension, and the resultant interface force is equal to the non-

equilibrium interface energy, as required. Eqn (31)–(33)

represent the simplest noncontradictory generalization for the

three-phase system, which allows us to determine interface

stresses in a complex SMS interface with complete or incom-

plete melt.

7 Ginzburg–Landau equation

Assuming the simplest linear relation between generalized

rates, _

Uand _

W, and work-conjugate generalized thermodynamic

forces, X

U

and X

W

, with neglected cross eﬀects, we can write GL

equations as

_

U

LU

¼XU¼r@

c

@Uþr r@

c

@rU

¼r@

cl

@Ue

þb21

2

@fU;af;a0

@UrWjj

2þr bS0ðWÞrU

;

(49)

_

W

LW

¼XW¼r@

c

@Wþr r@

c

@rW

¼r@

cl

@We

þb20 b10

2

@qW;ab

@WrU

jj

2;

r@ce

@Uee

r@cy

@Ur0

@

cy

@U;

(50)

where |

e

indicates that the derivatives are calculated at constant

e;L

U

and L

W

are the kinetic coeﬃcients. Note that because

of terms r

0

/r, the local energy

c

l

depends not only on e

e

through c

e

, but on eas well. Since eis constant while evaluating

all other partial derivatives in eqn (49) and (50), ris constant

as well, and this dependence will not produce new terms.

Using eqn (12) and (43) we have for the local thermodynamic

driving force

Xl

U:¼r@

cl

@Ue

¼r@ce

@ee:@eetðU;WÞeyðy;U;WÞ

ðÞ

@Ue

r@ce

@Uee

þ@cy

@Ur0

@

cy

@U¼r

r0

re:@etðU;WÞþeyðy;U;WÞðÞ

@U

r@ce

@Uee

r@cy

@Ur0

@

cy

@U;(51)

and similar

Xl

W:¼r@

cl

@We

¼r

r0

re:@etðU;WÞþeyðy;U;WÞðÞ

@W

r@ce

@Wee

r@cy

@Wr0

@

cy

@W:

(52)

In more detail,

@ce

@Uee

þ@cy

@Uþr0

r

@

cy

@U¼1

2r0

@qU;aC

ðÞ

@Uee:CSðWÞC0

½:ee

þDGyðy;WÞ@qðU;3Þ

@Uþr0

rAS0ðy;WÞ@mðUÞ

@UþA21ðyÞ@qU;aA

ðÞ

@U

:

(53)

Using the same procedure we obtain

@ce

@Wee

þ@cy

@Wþr0

r

@

cy

@W¼1

2r0

@qW;acS

ðÞ

@WqU;aC

ðÞee:C2C1

½:ee

þDGy

21

@qðW;3Þ

@WqðU;0Þþ r0

rA20ðyÞA10 ðyÞ

@qW;aW

ðÞ

@WmðUÞ

þA21ðyÞqU;aA

ðÞ

@mðWÞ

@W:(54)

The expanded GL eqn (49) and (50) are

1

LU

@U

@t¼XU¼r

r0

re:et1 þey1ey0þet2 et1 þey2ey1

ðÞqW;atW

ðÞ½

@qU;atU

ðÞ

@Uþ0:5r

r0

@qU;aC

ðÞ

@Uee:CSðWÞC0

½:ee

rDGyðy;WÞ@qðU;3Þ

@Ur0AS0ðy;WÞ@mðUÞ

@UþA21ðyÞ@qU;aA

ðÞ

@U

þ0:5b20 b10

@qW;ab

@WrUjj

2þr b21fU;af;a0

rW

;

(55)

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1

LW

@W

@t¼XW¼r

r0

re:et2et1 þey2ey1

ðÞ

@qW;atW

ðÞ

@WqU;atU

ðÞ

0:5r

r0

@qW;acS

ðÞ

@WqU;aC

ðÞee:C2C1

½:ee

rDGy

21

@qðW;3Þ

@WqU;0ðÞ

r0A20ðyÞA10 ðyÞ

@qW;aW

ðÞ

@WmðUÞþA21ðyÞqU;aA

ðÞ

@mðWÞ

@W

b21

2

fU;af;a0

@UrW

jj

2þr b10 þb20 b10

qW;ab

rU

:

(56)

The important point is that the interface stresses r

st

do not

directly contribute the Ginzburg–Landau equations. However,

since they participate in equilibrium equations and boundary

conditions, they contribute indirectly the Ginzburg–Landau

equations by changing elastic stresses r

e

.

Equilibrium conditions for homogeneous phases

According to one of the main requirements formulated in

ref. 19, 33 and 72, the thermodynamic driving force in the

Ginzburg–Landau eqn (49) and (50), X

U

and X

W

,shouldbezerofor

all homogeneous bulk phases (i.e.,forW=0or1andU=0or1)for

any temperature and stress tensor. Otherwise, the equilibrium

order parameters will depend on the temperature and stress tensor,

which will lead to artificial dependence of all material properties of

phases on the temperature and stress tensor through interpolation

functions in eqn (27)–(40). Due to the properties of interpolating

functions, namely, zero derivatives for W=0or1andU=0or1,

these requirements are satisfied identically.

The Gibbs energy per unit mass is defined as G:= cr

01

r:eand is more convenient in the analysis of the thermo-

dynamic equilibrium under stress tensor r. Solution for

qG/qU= 0 and qG/qW= 0 for homogeneous states gives three

roots corresponding to extrema points along the SM and SS

phase transformation paths, respectively. Two of them corre-

spond to the local minima of the potential discussed above and

are designated as W

eq

=s1 and U

eq

= 0 or 1. The third roots

correspond to the maximum along the PT path. Modified Gibbs

energy G

˜(without c

e

) is plotted in Fig. 2. While we cannot prove

this strictly analytically, numerous numerical studies under

various stress rshow that the developed potential does not

possess unwanted minima for 0 oW

eq

o1 and 0 oU

eq

o1

and corresponding unphysical phases.

8 Thermodynamic stability conditions

for homogeneous phases

The condition for the loss of stability of a homogeneous phase

has been used for determining the PT criteria for small

strains

19,31,33,72

and large strains.

73

Here, we have limited

ourselves to small strain approximation and pursued the same

approach as in ref. 31 and 73 to analyze the instability of each

homogeneous phase. The thermodynamic instability condition

under prescribed homogeneous stress rdefines that the equilibrium

of a homogeneous phase is unstable if, under spontaneous deviation

of the order parameters from the equilibrium state, the dissipation

rate becomes non-negative, rD0. This condition can be

expressed in the following form:

XUr;eeþDee;e0þDe0;Ueq þDU;Weq þDW;y

_

U

þXWr;eeþDee;e0þDe0;Ueq þDU;Weq þDW;y

_

W0;

(57)

where De

e

is the change in elastic strain due to the dependence

of elastic moduli on order parameters. We also included De

0

because of the dependence of the thermodynamic forces X

U

and

X

W

on the ratio r/r

0

(see eqn (55) and (56)). This term was

absent in ref. 31 and the main question is whether it makes an

additional contribution to the instability conditions. Approximating

eqn (57) using the second-order Taylor series expansion about the

equilibrium phase, X

U

=X

W

= 0, gives

@XUr;ee;e0;Ueq;Weq

@U

_

U2þ@XUr;ee;e0;Ueq;Weq

@W

_

W_

U

þ@XWr;ee;e0;Ueq ;Weq

@U

_

W_

Uþ@XWr;ee;e0;Ueq;Weq

@W

_

W20;

(58)

where all derivatives are calculated at constant rand yis omitted

for conciseness.

First, let us prove that the terms in the driving forces that

include r/r

0

= 1/(1 + e

0

) (or r=r

0

/(1 + e

0

)) and e

e

do not

contribute to the derivatives in eqn (58). Thus

@r=r0

@Ur

¼ 1

1þe0

ðÞ

2

@e0

@Ur

¼ 1

1þe0

ðÞ

2

@e0e þe0t þe0y

ðÞ

@Ur

:(59)

Interpolating functions for e

t

+e

y

are chosen to satisfy conditions

that @etþey

ðÞ

@U¼0for U

eq

= 0 or 1 (see eqn (37)–(39)), thus the last

two terms in eqn (59) disappear. We will work with the elastic strain

tensor e

e

instead of e

0e

,i.e., with more general case, because we will

need this below. Since e

e

=C

1

:r,weneedtoevaluate@C1

@U.Utilizing

C

1

:C=I

4

,whereI

4

is the fourth-rank unit tensor, we obtain

dC

1

:C+C

1

:dC=0anddC

1

=C

1

:dC:C

1

.Accordingto

eqn (29), (30) and (39), for U

eq

=0or1onehasdC=0,which

completes the proof. The same can be repeated for @rr0Ueq

@Wr

.

Summarizing,

@eeUeq ;Weq

@Ur

¼@eeUeq ;Weq

@Wr

¼0;(60)

@rr0Ueq ;Weq

@Ur

¼@rr0Ueq ;Weq

@Wr

¼0;(61)

which means that e

e

and e

0

can be excluded from arguments in

eqn (58). Consequently, we can impose rCr

0

in expressions for the

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driving forces before differentiation in eqn (58), which reduces the

problem to the case in ref. 31 when we neglected interface stresses.

Substituting the relations X

U

and X

W

for homogeneous phases,

eqn (51) and (52), in relation (58) with allowing for eqn (60) and

(61) gives

r:@2etUeq ;Weq

þeyUeq ;Weq

@U2r@2clee;Ueq ;Weq

@U2

_

U2

þ2r:@2etUeq ;Weq

þeyUeq ;Weq

@W@U2r@2clee;Ueq;Weq

@W@U

_

W_

U

þr:@2etUeq ;Weq

þeyUeq ;Weq

@W2r@2clee;Ueq;Weq

@W2

_

W20:

(62)

The condition for the loss of stability can be derived using

Sylvester’s criterion for a positive-definite matrix, but it involves

quite complex off-diagonal terms. When applying this criterion

for the thermodynamic potential for any two-phase systems,

namely, M–S

s

or S

1

–S

2

, we can obtain simple PT criteria.

19,33,72

Our main assumption is that the same criteria for PT between

any of the two phases should be obtained from our general

three-phase potential. This also means that for homogeneous

states PT conditions for any two phases are independent of any

other (in the current case, the third) phases, similar to the sharp

interface approach. For the neglected interface stresses, it is

shown in ref. 19, 33 and 72 for cartesian order parameters and in

ref. 31 for polar order parameters that these conditions can be

satisfied by imposing that the mixed derivatives of the Gibbs

potential vanish for homogeneous phases, which results in

@XUr;Ueq;Weq

@W¼0;and @XWr;Ueq;Weq

@U¼0)

@2etUeq;Weq

@W@U¼0;@2eyUeq;Weq

@W@U¼0;

@2clee;y;Ueq;Weq

@W@U¼0:

(63)

Note that the conditions (63) do not impose any restriction on

the potential for the intermediate values of the order parameters.

The developed thermodynamic potential (27)–(40) satisfies these

conditions. After eliminating off-diagonal terms in Sylvester’s

matrix, the criteria for instability reduce to the non-negativity of

each diagonal terms:

@XUr;ee;Ueq;Weq

@U¼r:@2etUeq;Weq

@U2r@2clee;Ueq;Weq

@U20;

(64)

@XWr;ee;Ueq;Weq

@W¼r:@2etUeq;Weq

@W2r@2clee;Ueq;Weq

@W20:

(65)

Substituting the developed potential eqn (27)–(40) in eqn (64)

and (65), the conditions for the loss of stability of each phase

toward any of the other phases are obtained as

M-S

s

:qX

U

(U=0,W=s1)/qUZ0-rA

s0

(y)ra

tU

r:(e

ts

+e

ys

e

y0

)

+ 0.5a

C

e

e

:(C

0

C

s

):e

e

; (66)

S

s

-M: qX

U

(U=1,W=s1)/qUZ0-[6DG

y

s0

A

s0

(y)]r

Z(6 a

tW

)r:(e

ts

+e

ys

e

y0

) + 0.5(6 a

C

)e

e

:(C

0

C

s

):e

e

;

(67)

S

1

-S

2

:qX

W

(U=1,W= 0)/qWZ0-rA

21

(y)

ra

tW

r:(e

t2

+e

y2

e

t1

e

y1

)0.5a

cS

e

e

:(C

2

C

1

):e

e

;

(68)

&S

2

-S

1

:qX

W

(U=1,W= 1)/qWZ0-[6DG

y

21

A

21

(y)]r

Z(a

tW

6)r:(e

t2

+e

y2

e

t1

e

y1

)0.5(6 a

cS

)e

e

:(C

2

C

1

):e

e

.

(69)

Further specification can be achieved by utilizing eqn (34) for

A

s0

(y) and A

12

(y). The equation for loss of stability of the melt,

eqn (66), can be further simplified as only hydrostatic pressure,

p

0

=1/3I:r, and volumetric elastic strain, e

e0

, are present in

the melt:

M-S

s

:rA

s0

c

(yy

s0

c

)ra

tU

p

0

(e

t0

+e

s

y0

e

m

y0

)+0.5a

C

(e

e0

)

2

/(K

0

K

s

),

(70)

where Kis the bulk modulus.

9 Computational model

The kinematic relations (2), (37) and (38), momentum balance,

the elasticity rule (18), and GL eqn (55) and (56), along with

relations for elastic moduli eqn (29) and (30), thermal energy

change eqn (31), jump in thermal energy eqn (33), and inter-

polating functions and their derivatives (39) and (40) comprise

the complete system of equations. Two types of problems have

been considered: (a) barrierless (i.e., without fluctuation) for-

mation of the SS and SMS interfaces in a rectangular sample

under plane strain conditions (Fig. 2a and b) the formation of a

critical nucleus of the IM within the SS interface in a cylindrical

sample (Fig. 2b).

For the case (a), the following boundary conditions have

been applied:

(i) For the order parameters: eqn (26) is applied on all sides.

(ii) For the mechanics problem: on the left side – zero

normal displacement and zero shear stress, as well as a fixed

lower left point; all other sides are stress-free; plane strain

conditions along the zaxis.

For the case (b), axis zis the symmetry axis, and eqn (26) for

order parameters and stress-free conditions for all surfaces are

applied.

The developed model was implemented in the commercial

finite element code COMSOL

74

utilizing the standard PDE and

Structural Mechanics modules. A mapped mesh of quadratic

Lagrange elements was implemented. Mesh density corre-

sponded to five elements per SS interface width, which ensures

mesh independent solution.

58

The relative calculation error is

set to be 10

4

.

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We have chosen HMX energetic crystals as the model

material and studied the formation of the IM at diﬀerent

temperatures for various system parameters. Mechanical and

transformational properties of each three HMX phases are

listed in Tables A.1 and A.2 of the ESI.†Isotropic elasticity is

used with the same bulk modulus Kfor all phases and the same

shear modulus mfor both solid phases. We assumed the inter-

face energy and width ratios to be temperature independent,

which implies A

s0

c

=3Ds

s0

and A

21

c

=3Ds

21

; also, all awith

diﬀerent subscripts are equal to 3. Analytical solutions for a

nonequilibrium interface and parameters of the nonequili-

brium interface between any two phases are presented in the

Appendix, along with an explicit expression for the ratio of the

energy and the width of the SS to SM interfaces: k

E

=E

21

/E

s0

and

k

d

=d

21

/d

s0

. Definition of the energy of the complex SMS

interface is presented in eqn (A.6), ESI.†The eﬀect of main

material parameters on the formation and stability of the IM is

investigated following the procedure in ref. 31. Two diﬀerent

initial conditions have been used:

(a) perturbed equilibrium SS interface (U=0.99and0oWo1)

and

(b) equilibrium S

1

M and MS

2

interfaces with a broad melt

(U= 0) region between two solid phases.

We underline that the initial state for all problems is the

complete melt, and the plane-strain constraint is applied for

the molten state. Next, the melt solidifies in two solid phases

divided by the SS interface. For the plane strain problem,

significant tensile stress s

z

appears during solidification.

Then, diﬀerent initial conditions for the order parameters are

prescribed for diﬀerent problem formulations, which are

described in ref. 31 and in the Appendix for completeness.

Three types of stationary solutions have been found: for SS and

SMS interfaces that correspond to the local energy minima and

for the CN between them corresponding to the minimax,

i.e., maximum with respect to the nucleus size and minimum

with respect to the structure. Since the stationary solution

for the CN cannot be obtained as a result of the solution of

time-dependent Ginzburg–Landau equations, the stationary

GL equations are solved with different initial guesses to find

a configuration close to the CN so that the solution of the

nonlinear solver converges and gives the final structure of

the CN.

We have chosen samples with diﬀerent sizes to be sure of

the size-independence of solutions. A sample size of at least an

order of magnitude larger than the widest interface width is

considered in all simulations. In the case of propagating

interfaces, a sample size is chosen such that when the steady

nanostructure is obtained, the distance between the interface

and sample edges is at least ten times the largest interface

width at the end of simulations. For the barrierless formation

of IP within the SS interface (Fig. 3a), a rectangular sample with

a height of 40 nm and a width of 300 nm was used; all sample

sizes are measured in the initial molten state.

To study the CN and thermally activated processes, an

infinitely long axisymmetric sample, with a radius of 20 nm

(in an initial molten state) was used. This was done by defining

‘‘perfectly matched’’ layers in COMSOL code with the height of

10 nm at both ends of the sample in the z-direction.

10 Barrierless formation and stability

of the intermediate melt

Diﬀerent system parameters are considered to study the for-

mation and stability of the IM. The interface energy ratio and

scale eﬀects are investigated for the perturbed SS interface and

the pre-existing IM. Simulations are performed for a model

with and without interfacial stresses. Two sets of simulations

are performed: (i) at solid–solid equilibrium temperature with

focus on the formation and stability of the IM, and (ii) at other

temperatures to study the kinetics of phase transformation in

the presence of the IM. For the first set of simulations, we have

neglected the interfacial interactions, i.e.,a

0

= 0. For the second

set of simulations, we included the interfacial interactions

by considering a

0

= 0.01 and 0.1 for a wide temperature range.

The effects of interfacial stresses on morphology, structure,

and kinetics of IM, as well as stress distribution in the sample

are studied.

Energetic estimates

Here we will study the formation and stability of equilibrium

IM for diﬀerent system parameters at y=y

21

e

= 432 (Kelvin)

(which is more than 100 (Kelvin) below melting temperature),

with and without interfacial tension.

The necessary condition for the formation of IMs using the

sharp-interface approach in a stress-free sample with neglected

interaction between interfaces is E

21

4E

10

+E

20

+DG*d*,

where DG* is the energy diﬀerence between the melt and solid

phase, which melt substitutes, and d* is the width of the IM.

29

Since current simulations consider E

10

=E

20

, the formation of

the IM is expected for k

E

> 2.0 at the melting temperature when

DG* = 0. For a temperature 100 (Kelvin) below the melting

temperature, it is expected that the formation of the IM occurs

at k

E

c2. The presence of elastic stresses at the coherent

interface promotes the formation of the melt by providing an extra

driving force due to their relaxation after melt formation.

1,2,75

However, the sharp-interface model is oversimplified and does

Fig. 3 Schematics, boundary conditions, and examples of solutions.

(a) Formations of the SMS interface within a rectangular sample under

plane strain conditions along the out-of-plane z-axis. Distribution of U(x,y)

for the S

1

S

2

initial condition at y= 432 (Kelvin) for k

E

= 3.6 and k

d

= 1.0.

(b) An axisymmetric problem on the formation of the critical nucleus of the

IM within the S

1

S

2

interface in a cylindrical sample. Boundary conditions

include eqn (26) for the order parameters and stress-free boundaries,

excluding the left boundary in (a).

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not consider the formation of a partial melt, interfacial tension,

scale, and kinetic eﬀects. Also, it does not include conditions of

existence of metastable phases and structures.

Eﬀects of k

E

for several k

d

The results of simulations in Fig. 4 reveal a much more

sophisticated picture, which strongly depends on the scale

parameter k

d

, which is meaningless in the sharp interface

approach. For small k

d

= 0.3, the stationary nanostructure is

independent of the initial condition. With increasing k

E

, the

degree of melting (disordering) continuously increases; with

decreasing k

E

, the degree of melting continuously decreases

along the same line. However, even at k

E

= 4 one has U

min

=

0.2424 without and U

min

= 0.2625 with interface stresses. Thus,

it is still incomplete melting and interface stresses slightly

increase U

min

.

For larger k

d

, two new features appear in Fig. 4:

(a) two diﬀerent nanostructures appear depending on

whether the initial state is the SS or SMS interface, with clear

hysteresis in k

E

, and

(b) there is a significant jump in U

min

between these nano-

structures when one of them ceases to exist.

One of the nanostructures represents the SS interface and

it persists in a metastable state even for k

E

43 for k

d

Z1,

when the structure with IM is energetically favorable. Interface

stresses slightly reduce the critical value of k

E

at which the SS

interface loses its stability. Another nanostructure is quite close

to the complete IM, especially for large k

d

. It persists in a

metastable state well below k

E

=2.Atk

d

o1, the eﬀect of the

interface stresses on the critical k

E

for the loss of stability of the

IM is weak and suppressing: k

E

= 2.61 for the barrierless

nucleation of the IM without interface tension and k

E

= 2.62

with surface tension; similar numbers for the disappearance of

the IM are 2.18 and 2.19, respectively. For k

d

= 1.1, the eﬀect of

the interface stresses is promoting and strong: without r

st

, the

IM disappears at k

E

= 1.3, while with the interface stress, the IM

is metastable even at k

E

=1.

Scale eﬀects for several k

E

The results for U

min

presented versus k

d

for five values of k

E

(Fig. 5) look even more nontrivial. For small k

d

there is a single

solution independent of the initial condition. With growing k

d

,

the magnitude U

min

reduces for k

E

42, increases for k

E

= 2, and

is equal to one for k

E

= 1.5. For k

E

42 and growing k

d

, at some

critical k

d

(growing with increasing k

E

), the second solution

corresponding to the SS interface appears for SS initial condi-

tions, while the solution for SMS initial conditions continues to

vary smoothly with growing k

d

. For k

E

42 and reducing k

d

,

behavior is qualitatively diﬀerent. While U

min

smoothly

increases and always exists (at least for k

d

Z0.1) for SMS initial

conditions, for SS initial conditions, U

min

= 1 but then ceases to

exist and there is a jump to the single solution with significant

IM, which is independent of the initial conditions. Especially

interesting is the behavior for k

E

= 2: the IM exists in the regions

with small and large k

d

, separated by the region on nonexistence

of the IM, which we called the IM-free gap. For k

E

= 1.5, the SS

interface exists for all k

d

under study, but the IM exists for large

k

d

and disappears at some critical k

d

. The essential eﬀect of

interface stresses is visible in Fig. 5 with reduction in critical k

d

for the disappearance of the IM for k

E

=1.5.

The eﬀect of the scale and interfacial tension on the structure

of the SMS interface is shown in Fig. 6 by plotting distributions

of Uand Wfor k

d

below and above the IM-free gap. Simulations

for SS initial conditions in Fig. 6a and b demonstrate significant

asymmetry of the partial IM for small k

d

and an essential

increase in U

min

for k

d

= 0.3. The results for SMS initial condi-

tions in Fig. 6c and d exhibit relative change in the shape of plots

for Ufor k

d

= 0.1 and 0.2 allowing for interfacial stresses. Also,

without interfacial tension, melting starts from the intersection

of the SS interface and the free external surface (Fig. 7). However,

with interfacial tension, melting starts at the intersection of the

SS interface and the symmetry plane of the sample, i.e., inside

the sample. The reason is the change in elastic stresses at the

free surface: without interfacial tension they are zero, and with

interfacial stresses they are negative interfacial stresses, both

due to zero total stresses and due to boundary conditions.

Stresses in the presence of IM

Simulations are performed at y=y

e

= 432 K, for a

0

= 0.01,

k

E

= 4.0 and k

d

= 1.0. Distributions of total stress along the

y-direction for SS and SMS interfaces are shown in Fig. 8;

distributions of elastic stresses along yand z-directions are

presented in Fig. 9c–f. The distribution of the interface tension

Fig. 4 The eﬀect of k

E

and k

d

on the barrierless formation and disap-

pearance of the IM. The stationary minimum values of U,i.e.,U

min

, are

plotted at y=y

21

e

= 432 (Kelvin) and a

0

= 0.01, for diﬀerent k

d

, for a model

without interfacial stresses (a) and with interfacial stresses (b).

Fig. 5 The eﬀect of the scale parameter k

d

on the formation and persis-

tence of IM is studied for y=y

21

e

= 432 (Kelvin) and diﬀerent k

E

values. The

results are obtained using models (a) without and (b) with interfacial

stresses.

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s

y

st

, which is always localized at the interface, is included as

well. Distributions of order parameters Uand Wfor SS and SMS

interfaces are also plotted in Fig. 9a and b. For the SS interface,

our numerical solution for s

y

st

is in good correspondence with

the analytical solution in ref. 53. The edge effect is avoided by

choosing a sample width of at least an order of magnitude

larger than the interface width. The PF simulations always

result in quite large internal elastic stresses close to the inter-

face, even without the introduction of the interfacial tension,

s

y

st

. The magnitudes of s

y

are equal on both sides of the SS

interface for the model without interfacial tension, which is in

agreement with ref. 54. For the SS interface, the interfacial

tension, s

y

st

, is larger than the total stress s

y

. This is in contrast

to what was reported for multivariant martensitic interfaces,

where s

y

st

{s

y

,

58

indicating the importance of interfacial

tension for HMX.

Obtained distributions of stresses for the SS interface for

both cases illustrate basic problems in definition of the concept

of interface stresses for elastic solids for an equivalent sharp

interface.

37

Indeed, without r

st

, the total force is close to zero,

but a bending moment exists; with r

st

, the total force is not

zero and a force couple exists as well. Also, nonzero stresses

penetrate into bulk phases at a distance equal to several inter-

face widths rather than localized at the interface only. The main

reason for both these phenomena is in the existence of the

transformation strain within the interface, which usually was

not considered in the model for the sharp interface, which

defines interface stresses.

38–44,46–48

Without s

y

st

, the appearance of the melt changes internal

stresses and produces a double-peak compressive stress with

the minimum between them close to zero, and a broader tensile

stress peak at the MS

2

interface, which also spreads into phase

S

2

(Fig. 8). The compressive peak closer to S

2

is larger than

that closer to S

1

. Interface tension distribution represents the

tensile double-peak curve with almost zero s

y

st

between them,

and the peak closer to S

1

is larger than that closer to S

2

. The

resultant stress for the case with s

y

st

is not a sum of s

y

st

and

elastic stresses for the case without s

y

st

because s

y

st

changes the

distribution of the elastic stresses. The resultant s

y

for the case

with interface tension represents a four-peak curve with two

compressive and two tensile stress peaks. Maximum compressive

and tensile peaks are smaller than for the case with s

y

st

=0,and

compressive stresses penetrate more in solid S

1

. The presence of

the melt makes the definition of the interface stresses for an

equivalent sharp interface even more problematic.

The elastic stresses s

y

e

in Fig. 9 for the case with s

y

st

=0

coincide with the total stresses s

y

in Fig. 8 for s

y

st

= 0. They

change sign and spread outside of the interface on both sides

of the interface. Note that in plane strain formulation, always

s

z

st

=0,i.e., elastic and total stresses in the zdirection are the

same. In contrast to s

y

e

, which tends to zero at both ends of the

sample, stress s

z

e

has a nonzero value since the sample is

clamped in the zdirection. These diﬀerent values of s

z

e

at the

ends produce additional problem in the conceptual definition

of the interface stresses, because interface stresses have to be

determined as excess fields with respect to values in bulk. Thus,

one has to define a Gibbsian dividing surface

37,47,48,51,76

equiva-

lent to a sharp interface, similar to that for the definition of an

Fig. 6 Structure of the SMS interfaces for three values of the scale

parameter k

d

is studied at temperature y=y

21

e

= 432 (Kelvin) and k

E

=

2.3. Structures in (a) and (b) for partially disordered IM are obtained

barrierlessly from a perturbed initial SS interface as an initial condition

and in (c) and (d) from a pre-existing complete IM within the SMS interface.

The results are plotted for the model without ((a) and (c)) and with

interfacial stresses ((b) and (d)).

Fig. 7 Eﬀects of the interfacial stresses on the distribution of the order

parameter U. Initial stage of IM-formation (a) without and (b) with inter-

facial stresses from an initial perturbed SS interface for y= 0.8, y

21

e

= 346 K,

k

E

=4,k

d

= 1 and a

0

= 0.01. Melting starts at the intersection of the SS

interface and free surface in (a) and from the intersection of the SS

interface and symmetry plane in (b). Here only upper half of the sample

is shown due to symmetry.

Fig. 8 The stress distributions within the interface is plotted for the SS

interface (a) and SMS interface (b). Total stresses and interfacial structural

stresses are plotted for the models with and without interfacial stresses r

st

.

Simulations are performed at y=y

e

= 432 (Kelvin) for a

0

= 0.01, k

E

= 4.0,

and k

d

= 1.0. The insets show the schematic of the sample with designated

axis.

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energy of a nonequilibrium interface. While for the dividing

surface for the interface energy we were able to find its posi-

tion,

51,76

for total stresses this is still an open question.

Considering a large value of k

E

, the IM with UE0 forms

independently of chosen initial conditions (Fig. 9b). The for-

mation of the melt relaxes all elastic stresses, as the shear

modulus of solid vanishes when the melt forms. Relaxation of

elastic energy, especially due to the relaxation of large elastic

stresses in the