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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