Solid–solid transformations via nanoscale intermediate interfacial phase: Multiple structures, scale and mechanics effects

Abstract

Solid–solid (SS)(SS) phase transformations via nanometer-size intermediate melts (IMs)(IMs) within the SS interface, hundreds of degrees below melting temperature, were predicted thermodynamically and are consistent with experiments for various materials. A necessary condition for the appearance of IMs, using a sharp interface approach, was that the ratio of the energies of SS and solid–melt (SM)(SM) interfaces, kEkE, were >2. Here, an advanced phase-field approach coupled with mechanics is developed that reveals various new scale and interaction effects and phenomena. Various types of IM are found: (i) continuous and reversible premelting and melting; (ii) jump-like barrierless transformation to IMs, which can be kept at much lower temperature even for kE<2kE<2; (iii) unstable IMs, i.e. a critical nucleus between the SS interface and the IM. A surprising scale effect related to the ratio of widths of SS and SM interfaces is found: it suppresses barrierless IMs but allows IMs to be kept at much lower temperatures even for kE<2kE<2. Relaxation of elastic stresses strongly promotes IMs, which can appear even at kE<2kE<2 and be retained at kE=1kE=1. The theory developed here can be tailored for diffusive phase transformations, formation of intergranular and interfacial phases, and surface-induced phase transformations.

Full text

1
2Solid–solid transformations via nanoscale intermediate
3interfacial phase: Multiple structures, scale and mechanics effects
4Valery I. Levitas
a,b,c,⇑
,Kasra Momeni
a
5
a
Department of Aerospace Engineering, Iowa State University, Ames, IA 50011, USA
6
b
Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA
7
c
Material Science and Engineering, Iowa State University, Ames, IA 50011, USA
8Received 2 October 2013; accepted 18 November 2013
9
10 Abstract
11 Solid–solid ðSSÞphase transformations via nanometer-size intermediate melts ðIMsÞwithin the SS interface, hundreds of degrees
12 below melting temperature, were predicted thermodynamically and are consistent with experiments for various materials. A necessary
13 condition for the appearance of IMs, using a sharp interface approach, was that the ratio of the energies of SS and solid–melt ðSMÞ
14 interfaces, kE, were >2. Here, an advanced phase-field approach coupled with mechanics is developed that reveals various new scale
15 and interaction effects and phenomena. Various types of IM are found: (i) continuous and reversible premelting and melting; (ii)
16 jump-like barrierless transformation to IMs, which can be kept at much lower temperature even for kE<2; (iii) unstable IMs, i.e. a crit-
17 ical nucleus between the SS interface and the IM. A surprising scale effect related to the ratio of widths of SS and SM interfaces is found:
18 it suppresses barrierless IMs but allows IMs to be kept at much lower temperatures even for kE<2. Relaxation of elastic stresses strongly
19 promotes IMs, which can appear even at kE<2 and be retained at kE¼1. The theory developed here can be tailored for diffusive phase
20 transformations, formation of intergranular and interfacial phases, and surface-induced phase transformations.
21 Ó2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
22 Keywords: Intermediate melt; Phase field; Phase transformation; Ginzburg–Landau
23
24 1. Introduction
25 Recently, phase transformations between two solid
26 phases through a nanometer-size molten layer, hundreds
27 of degrees below melting temperature, were predicted ther-
28 modynamically [1–4] and confirmed in experiments for b–d
29 phase transformation in HMX energetic crystals [1–3],
30 PbTiO3nanowires [4] (Fig. 1), and for amorphization in
31 avandia [5] and materials which exhibit a reduced melting
32 temperature under pressure (e.g. ice, Si, Ge, geological
33 and other materials) [3]. As a moving solid 1–melt–solid
34 2ðS1MS2Þinterface propagates through the sample, one
35 solid phase melts and resolidifies into another phase. The
36thermodynamic condition for the formation of an interme-
37diate melt ðIMÞis: [4]:38
E21 E10 E20 Ee>ðG0GsÞd;ð1Þ4040
41
where E10;E20 and E21 are the energies of the S1M;S2Mand
42SS interfaces, Eeis the elastic energy of the coherent SS
43interface, and G0and Gsare the bulk thermal energies of
44the melt and solid phase with the lower melting tempera-
45ture hm
e. Thus, melting significantly below hm
ecan be
46brought about by a reduction in total interface energy
47and a relaxation of the elastic energy.
48For E10 ¼E20 ¼Es0, neglecting elastic energy, and close
49to hm
e(i.e. for ðG0’GsÞ), the necessary thermodynamic
50condition for the formation of an IM reduces to
51kE¼E21=Es0>2. It is clear that for hundreds of degrees
52below hm
e;kEshould significantly exceed 2. However, ther-
53modynamic treatment with sharp interfaces [1–4,6] is an
1359-6454/$36.00 Ó2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.actamat.2013.11.051
⇑
Corresponding author at: Department of Aerospace Engineering, Iowa
State University, Ames, IA 50011, USA.
E-mail address: vlevitas@iastate.edu (V.I. Levitas).
Q2
Q1
www.elsevier.com/locate/actamat
Available online at www.sciencedirect.com
ScienceDirect
Acta Materialia xxx (2013) xxx–xxx
AM 11061 No. of Pages 9, Model 5+
12 December 2013
Please cite this article in press as: Levitas VI, Momeni K. Solid–solid transformations via nanoscale intermediate interfacial phase:
Multiple structures, scale and mechanics effects. Acta Mater (2013), http://dx.doi.org/10.1016/j.actamat.2013.11.051

54 oversimplification and cannot be considered as a strict
55 proof of the phenomena. Melting may not be complete
56 (i.e. premelting) during the SS phase transformation and
57 the width of the IM,d, is comparable with the widths
58 d21 and ds0of SS and SM interfaces. Furthermore, kinetic
59 effects and alternative nanostructures may play a key role
60 in the formation of IMs. Therefore, in this paper a
61 phase-field approach is developed, which led to a much
62 more precise proof of the existence of IMs and provided
63 the corresponding conditions, as well as revealing new
64 phases, effects and phenomena. We are not aware of any
65 previous phase-field approaches to solid–solid phase trans-
66 formations via IMs. The most similar phase-field ap-
67 proaches reported are for premelting in grain boundaries
68 [7–9], but due to different phenomena they are quite differ-
69 ent from what we suggest below and do not include
70 mechanical stresses. Premelting at the external surface
71 [10,11] and surface-induced martensitic phase transforma-
72 tion [12,13] include mechanics but utilize two phases only
73 and do not include a moving SS interface. In addition,
74 the premelting discussed in Refs. [7,8,10,11,14] occurs a
75 few degrees below hm
e, while the formation of IM observed
76 experimentally [1–5] takes places several hundreds of
77 degrees below hm
e; prior to this, there was no evidence
78 that the phase-field approach could reproduce these
79 phenomena.
80 One of the advantages of the approach developed
81 here is that, in contrast to multiphase models
82 [12,13,15,16], each of three phase transformations is
83 described by a single-order parameter, without addi-
84 tional constraints on the order parameters. One of the
85nontrivial points in the theory is the introduction of
86the gradient energy for an SS interface within an IM
87(governed by a parameter a0), which describes the interac-
88tion between two SM interfaces. The effect of two unex-
89plored parameters, namely a0and the ratio kd¼d21=ds0
90of the widths of the SS and SM interfaces, along with
91the ratio kE¼E21=Es0, temperature h, and elastic stresses,
92is examined in detail. Several types of IMs are found: (i)
93continuous and reversible premelting and melting for small
94kd; (ii) jump-like (i.e. first-order) barrierless transformation
95to IMs for larger kd; (iii) persistence of IMs as a metastable
96phase at lower temperatures without barrierless resolidifi-
97cation, even for kE<2, i.e. when the necessary condition
98for thermodynamically equilibrated IMs is not satisfied
99even for hm
e; and (iv) an unstable IM, which represents a
100critical nucleus between the SS interface and IM and pro-
101vides a thermally activated transition between them. A sur-
102prising scale effect related to kdis revealed: increasing kd
103suppresses barrierless IMs but allows one to maintain
104IMs at much lower temperatures and even for kE<2.
105For 2 <kE62:5, a nontrivial nonmonotonous effect of
106kdis revealed, including an IM gap (i.e. lack of intermediate
107melting) for a certain range of kd. Crossover from the ex-
108pected increasing dependence of the width of the IM d
109on parameter a0to decreasing dependence at very low tem-
110perature is revealed. The velocity of the SMS interface is
111lower than the velocity of the SS interface. Elastic stresses
112at the coherent SS interface further reduce temperature as
113well as the ratio kEfor barrierless IM nucleation (even be-
114low kE¼2) and for retaining such nucleation (even at
115kE¼1).
2V.I. Levitas, K. Momeni / Acta Materialia xxx (2013) xxx–xxx
AM 11061 No. of Pages 9, Model 5+
12 December 2013
Please cite this article in press as: Levitas VI, Momeni K. Solid–solid transformations via nanoscale intermediate interfacial phase:
Multiple structures, scale and mechanics effects. Acta Mater (2013), http://dx.doi.org/10.1016/j.actamat.2013.11.051
melt (disordered layer) at the moving perovskite–pre-perovskite interface in PbTiO3nanowire [4].
Fig. 1. Experimental evidence of the existence for nanosize intermediate interphase phases: (a) intergranular amorphous films in bismuth-doped zinc oxide
[17]; (b) grain boundary premelting in nickel-doped tungsten [18]; (c) intergranular films at copper–alumina interface [19]; and (d) solidified intermediate

116 The theory developed here can be tailored to describe dif-
117 fusive phase transformations with evolving concentration
118 [20–22], premelting/disordering at grain boundaries [7,8],
119 wetting of the external and internal surfaces [20–24], surface
120 premelting and melting [10,11,14] at a moving solid–melt–
121 gas interface, surface-induced SS phase transformation
122 [12,13],SS phase transformation via surface or grain-bound-
123 ary-induced premelting [4], formation of intergranular and
124 interface amorphous or crystalline phases (complexions)
125 [7,8,19–29] in metallic and ceramic systems and developing
126 corresponding interfacial phase diagrams [20–22], amorph-
127 ization via IMs [3], and austenite nucleation at martensite–
128 martensite or twin interfaces [12,13]. Examples of some
129 intergranular and interface phases are collected in Fig. 1.
130 2. Mathematical model
131 The relationship between the strain tensor
132 e¼1=3e0Iþe(where e0and eare the volumetric and devi-
133 atoric contributions) and the displacement vector u, the
134 decomposition of strain into elastic eel and transforma-
135 tional etparts, and the equilibrium equation are given by:
136
e¼ð$uÞs;e¼eel þet;$r¼0:ð2Þ
138138
139 Here, ris the stress tensor, $and $are the gradient and
140 divergence operators, Iis the unit tensor, and subscript s
141 means symmetrization. Two order parameters are intro-
142 duced using the polar system in the plane of the order
143 parameters to develop the model for phase transformations
144 between three phases: radial !and angular #parameters,
145 where p#=2 is the angle between the radius vector !and
146 the positive first axis. The value !¼0 for any #represents
147 the reference phase, which in this paper will be considered
148 as the melt M(but in general it also can be solid). Solid
149 phases correspond to !¼1; phase S1is described by
150 #¼0, and phase S2is described by #¼1(Fig. 2). Each
151 of the phase transformations: M$S1, corresponding to
152 variation in !between 0 and 1 at #¼0;M$S2, corre-
153 sponding to variation in !between 0 and 1 at
154 #¼1;S1$S2, corresponding to variation in #¼0be-
155 tween 0 and 1 at !¼1, is described theoretically by the po-
156 tential in Refs. [15,16]. Thus, the desired potential should
157 be reduced to known potentials for each of the phase trans-
158 formations and have some noncontradictory representa-
159 tion for simultaneous variation of !and #. The
160 Helmholtz energy per unit volume consists of elastic we,
161 thermal whand gradient wrparts, as well as the term w
^h
162 describing double-well barriers between phases:
163
w¼weþw
^hþwhþwr;ð3Þ
wh¼DGhðh;#Þqð!;0Þ;ð4Þ
we¼0:5Kð!;#Þe2
0el þ2lð!;#Þjeelj2

;ð5Þ
wr¼0:5½bs0ð#Þjr!j2þb21/ð!;a/;a0Þjr#j2;ð6Þ
w
^h¼As0ðh;#Þ!2ð1!Þ2þA21ðhÞ#2ð1#Þ2qð!;0Þ:ð7Þ
165165
166Here, Kand lare the bulk and shear moduli; bs0and b21
167are the SM and SS gradient energy coefficients, respec-
168tively; DGhand As0are the difference in thermal energy
169and energy barrier between Sand M;A21 is the SS energy
170barrier; qand /are the interpolating functions, and awith
171various subscripts are parameters in these functions. The
172transformation strain is: 173
et¼½et1þðet2et1Þqð#; at#Þqð!;at!Þ:ð8Þ175175
176
All functions of !(or #) smoothly interpolate properties of
177solid–melt (or two solid) phases, e.g. the difference between
178the thermal energy of solids and melt: 179
DGhð#Þ¼DGh
10 þðDGh
20 DGh
10Þqð#; 0Þ;ð9Þ181181
182
the barrier between solid and melt: 183
As0ðh;#Þ¼A10ðhÞþðA20ðhÞA10 ðhÞÞqð#; a#Þ;ð10Þ185185
186
the gradient energy coefficient: 187
bmsð#Þ¼bs1mþðbs2mbs1mÞqð#; amsÞ;ð11Þ189189
190
the bulk modulus: 191
Kð!;#Þ¼K0þðKsð#ÞK0Þqð!;aKÞ;ð12Þ
Ksð#Þ¼Ks1þðKs2Ks1Þqð#; aks Þ;ð13Þ193193
194
and the shear modulus: 195
lð!;#Þ¼l0þðlsð#Þl0Þqð!;alÞ;ð14Þ
lsð#Þ¼ls1þðls2ls1Þqð#; alsÞ:ð15Þ197197
198
Interpolating function qðx;aÞ¼ax22ða2Þx3þ
199ða3Þx4varies between 0 and 1 when xvaries between 0
200and 1 and has zero x-derivative at x¼0 and x¼1. Func-
201tion /ð!;a/;a0Þ¼a/!22ða/2ð1a0ÞÞ!3þða/3
202ð1a0ÞÞ!4þa0differs from qin that it is equal to a0
203(rather than 0) at !¼0. Thus, we penalized the gradient
204energy for the solid–solid interface even within the com-
205plete IMð!¼0Þ. The difference between the thermal energy
206of Sand Mis: 207
DGh
s0¼Dss0ðhhs0
eÞ;s¼1;2;ð16Þ209209
210
where hs0
eis the equilibrium temperature between solid
211phase Ssand M, and Dss0is the jump in entropy between
212Ssand M. The energy barriers between Ssand Mare
213
As0ðhÞ¼As0
chhs0
c

with hs0
cfor the critical temperature
214of the loss of stability of the phase Sstowards the melt;
215
the barrier between solid phases is A21 ðhÞ¼e
A21
cþ
216
A21
chh21
c

with h21
cfor the critical temperature of the loss
217
of stability of the phase S2toward S1(when e
A21
c¼0).
218A schematic of the developed thermodynamic potential
219is shown for three different temperatures in Fig. 2. Apply-
220ing the first and second laws of thermodynamics to the sys-
221tem with a non-local free energy, and assuming a linear
222relation between thermodynamic forces and fluxes, we
223obtain the elasticity rule: 224
r¼Kð!;#Þe0elIþ2lð!;#Þeel ;ð17Þ226226
227
and the Ginzburg–Landau equations:
V.I. Levitas, K. Momeni / Acta Materialia xxx (2013) xxx–xxx 3
AM 11061 No. of Pages 9, Model 5+
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Please cite this article in press as: Levitas VI, Momeni K. Solid–solid transformations via nanoscale intermediate interfacial phase:
Multiple structures, scale and mechanics effects. Acta Mater (2013), http://dx.doi.org/10.1016/j.actamat.2013.11.051

228
_
!¼L!q@w
@!þb21
2
@/
@!j$#j2þ$ðbs0$!Þ

;ð18Þ
_
#¼L#q@w
@# þ$ðb21 /ð!;a/;a0Þ$#Þ

:ð19Þ
230230
231 Here L!and L#are the kinetic coefficients and derivatives
232 of ware evaluated at e¼const.
233 For the S1MS2diffuse interface, !varies from 1 to 0 and
234 back to 1, and #varies from 0 to 1 (Fig. 3). It is logical to
235 assume that in Mð!¼0Þ, variation of #is irrelevant, in
236 particular wrshould be independent of $#, i.e. a0¼0. In
237 particular, a similar condition is satisfied for melting within
238 the grain boundary [7,8]. However, our simulations dem-
239 onstrate that with a0¼0, the problem formulation is ill-
240 posed and results in zero width of the SS interface within
241 the melt and a strongly mesh-dependent solution. Indeed,
242 a solution with a sharp S1S2interface at the point with
243 !¼0 is an energy-minimizing solution, and with an
244 increasing number of finite elements, the numerical solu-
245 tion tends to this minimum. Thus, one must introduce
246 a0>0 to account for the interaction between S1Mand
247 S2Minterfaces via $#in the melt. At elevated tempera-
248 tures, when the width of IM is large and !¼0 in the melt,
249the energy of the IM can be approximated as
250E¼ðG0GsÞdþb21a0=ð2dÞ. With d!1 the effect
251of a0disappears, which confirms that we do not introduce
252any contradictions for a homogeneous melt or when the
253distance between the SM interfaces is large.
2543. Analytical solutions and verification
255One of the advantages of the model developed here is that,
256in contrast to multiphase models [13,15,16], each of three
257phase transformations is described by a single order param-
258eter, without additional constraints on the order parameters.
259An analytical solution for each interface between phases i
260and j, propagating along the y-direction is: [16,30] 261
gij ¼1=1þepðyvijtÞ=dij
hi
;ð20Þ
dij ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
bij.2Aij ðhÞ3DGh
ijðhÞ
hi
r;ð21Þ
vij ¼6Lijdij DGh
ijðhÞ=p;ð22Þ
Eij ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2bij AijðhÞ3DGh
ijðhÞ

r.6;ð23Þ263263
Fig. 2. Plot of the local Landau potential w¼whþw
^neglecting the elastic energy (Eqs. (1) and (4)) at different temperatures in the polar system of the
order parameters !and #. (a) 3-D plot of the potential surface at h¼h21
e; (b) contour plot of the potential at h<h21
ewhen the S2phase is stable and the
other two are metastable; (c) contour plot of the potential at h¼h21
e, when both solids are in equilibrium, and (d) contour plot of the potential at h>h21
e.
4V.I. Levitas, K. Momeni / Acta Materialia xxx (2013) xxx–xxx
AM 11061 No. of Pages 9, Model 5+
12 December 2013
Please cite this article in press as: Levitas VI, Momeni K. Solid–solid transformations via nanoscale intermediate interfacial phase:
Multiple structures, scale and mechanics effects. Acta Mater (2013), http://dx.doi.org/10.1016/j.actamat.2013.11.051

264 where p¼2:415[16],g10 ¼!at #¼0;g20 ¼!at #¼1,
265 and g21 ¼#at !¼1;vij is the interface velocity. These
266 equations allow us to calibrate material parameters
267 bij;Aij ;hij
cand Lij when the temperature dependence of
268 the interface energy, width and velocity are known. The en-
269 ergy of the IM;E, is defined as an excess energy, with re-
270 spect to S1for points with #60:5 and with respect to S2
271 for points with #>0:5. Numerical solutions show good
272 agreement with the analytical solutions for the interface
273 profile and velocity. Note that for !¼1 our model reduces
274 to the melting model in Refs. [10,11], which describes well
275 the experimental results on the size dependence of the melt-
276 ing temperature and the temperature dependence of the
277 width of surface melt for Al. For #¼0 or 1, our model re-
278 duces to the model for solid–solid transformation in Refs.
279 [12,13,16], which has been widely used for studying mar-
280 tensitic transformations in bulk and at the surface and
281 for predicting various scale and mechanics effects. When
282 phases 1 and 2 are the same but in twin relation to each
283 other, our model reduces to that in Ref. [31], which de-
284 scribes very well the different nontrivial nanostructures that
285 form during multivariant martensitic phase transforma-
286 tions and twinning. All these results verify the correctness
287 of our model for the nontrivial particular cases.
288 4. Results and discussion
289 The SM interface is considered to be coherent with van-
290 ishing shear modulus for melt, l0¼0. For simplicity, all
291 transformation strains are purely volumetric. Properties
292 of melt, dphase ðS1Þand bphase ðS2Þof the energetic mate-
293 rial HMX ðC4H8N8O8Þwill be used [1,2] (when available),
294 for which an IM was considered. It is assumed that all
295 a¼3; all phases have K¼15 GPa; solid phases possess
296 l¼7 GPa and bs0ð#Þ¼const;L!¼2L#¼2596:5m
2=
297ðNsÞ;Ds10 ¼793:79 kJ=m3K, Ds20 ¼935:45 kJ=m3K,
298melting temperatures h10
e¼550 K and h20
e¼532:14 K;
299h21
e¼432 K;e10
0t¼0:067, e20
0t¼0:147 (i.e. e21
0t¼0:08);
300
e
A21
c¼0;Aij
c¼3Dsij (this choice corresponds to the tem-
301perature-independent interface energies and widths [30]);
302hij
c¼hij
eþpEij=ðDsij dij Þand bij ¼6Eijdij =p[16];
303E21 ¼1J=m2and d21 ¼1 nm. A 24 8nm
2rectangular
304sample with a roller boundary condition on the left side
305and a fixed lower left point is modeled. Two initial condi-
306tions are considered: (i) an equilibrium SS interface; and
307(ii) equilibrium S1Mand MS2interfaces with quite a broad
308melt ð!¼0Þregion between the two solids.
309Stress-free IM First, we will consider the case without
310mechanics. In Fig. 3, the minimum value of !is plotted
311for h¼h21
e¼432 K (i.e. 100 K below hm
e), a0¼0, and dif-
312ferent kE;kd, and initial conditions. Note that the effect of
313temperature is similar to the effect of kd. For small kd, there
314is only a single stationary solution independent of the ini-
315tial conditions, corresponding to barrierless premelting
316and melting within the SS interface. The degree of melting
317(disordering) continuously increases with increasing kEand
318temperature. There is no hysteresis with increasing/
319decreasing temperature. However, for kEP2:7 an increase
320in kdpromotes the formation of an IM (reduces !min ); for
321kE62:5, the dependence of !min ðkdÞis surprisingly nonmo-
322notonous, exhibiting a disappearance of the IM above crit-
323ical kd. In contrast, for larger kd, different initial conditions
324result in two different stationary nanostructures. For the
325SS initial condition, premelting does not start up to some
326quite large critical value kE(e.g. kE¼3:39 for kd¼1),
327above which jump-like (i.e. first-order) premelting or com-
328plete melting occurs. For SMS initial conditions, almost
329complete melt is stabilized at kE¼1:94 (for kd¼1), i.e.
330even below the critical value kE¼2 for premelting at hm
e.
331For kEP2:7, however, an increase in kdpromotes the
Fig. 3. Intermediate melt formation at h21
e¼432 K (100 K below melting temperature) and a0¼0. (a) Minimum stationary value of !min within an SS
V.I. Levitas, K. Momeni / Acta Materialia xxx (2013) xxx–xxx 5
AM 11061 No. of Pages 9, Model 5+
12 December 2013
Please cite this article in press as: Levitas VI, Momeni K. Solid–solid transformations via nanoscale intermediate interfacial phase:
Multiple structures, scale and mechanics effects. Acta Mater (2013), http://dx.doi.org/10.1016/j.actamat.2013.11.051
interface as a function of kE, starting from SS (black lines and triangles) and SMS (red lines and circles) initial states for several kd. Inset shows distribution
of !and #, when almost complete melt (black solid line) and premelt (dotted red line) are formed. (b) Minimum stationary value of !min within the SS
interface vs. kd, starting from SS (dots) and SMS (solid lines) initial states for several kE. Coincidence of solutions for different initial conditions
corresponds to continuous reversible premelting and melting. When solutions do not coincide, a jump-like transformation to and from IM occurs,
exhibiting significant hysteresis. It is noteworthy that nonmonotonous !minðkdÞand an IM gap for small kd are observed. (For interpretation of the
references to color in this figure legend, the reader is referred to the web version of this article.)

332 formation of an IM (similar to the SS initial state), for
333 kE62:5, the dependence !minðkdÞis nontrivial, with an
334 IM gap (i.e. lack of IM) for a certain range of kd, which in-
335 creases with decreasing kE. Outside the IM gap, with
336 increasing/decreasing temperature, discontinuous first-or-
337 der phase transformations to and from the melt occur at
338 different temperatures, exhibiting significant hysteresis.
339 An increase in kdincreases the value of kEfor melting from
340 the SS state and reduces critical kEfor keeping melt from
341 SMS state. Thus, at kd¼1:2, almost complete melt can
342 be kept within the SS interface for kE¼1:58. Increasing
343 kdincreases the width of the hysteresis loop and also shifts
344 melting to higher temperatures.
345 The presence of two stable stationary nanostructures
346 indicates that there is another unstable nanostructure
347 between them, which represents a critical nucleus. If the
348 difference between the energy of the critical nucleus and
349 that of the SS (or SMS) interfaces is smaller than
350 40 80kBh, where kBis the Boltzmann constant, then melt-
351 ing (or solidification) within the SS interface will occur due
352 to thermal fluctuations. Determining the critical nucleus
353 and kinetic studies will be performed elsewhere.
354 The effect of temperature and the parameter a0on the
355 minimum stationary value of !min, the interface energy,
356the interface width and the velocity were studied for
357kE¼4 and kd¼1(Fig. 4). The width of the IM,d,is
358defined as the difference between locations of two SM
359interfaces where !¼0:5. Note that almost complete melt
360
of the width exceeding 1nm exists at 0:65h21
e, i.e. 240K
361below the melting temperature. For any a0, increasing tem-
362perature promotes melting, i.e. reduces !min and the SMS
363interface energy, and increases dand the interface velocity
364
(for h=h21
e>1). When the temperature approaches the
365melting temperature of the bphase, !min ’0 and the width
366of the IM is determined by @E=@d¼0, which results in
367
d¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0:5b21a0=ðG0GsÞ
q;ddiverges when h!hm
eand
368G0!Gs. The energy of the IM tends to the energy of
369two SM interfaces, which is 0:5E21 for our case. The veloc-
370ity of the SMS interface is less than the velocity of the SS
371
interface (even for L!¼500L#), is zero at h21
e, and varies
372
linearly with deviation from h21
ewith some acceleration
373close to the melting temperature. At very small a0, the
374interface velocity tends to zero. Since for very small a0,
375the SS interface width within the melt tends to zero, a very
376large number of finite elements is required to obtain mesh-
377independent results.
Fig. 4. Effect of temperature on the formation, energy and width of the IM, as well as the interface velocity in the presence of IM: (a) minimum value of
!min indicating formation of IM; (b, c) normalized SMS interface energy and width; (d) SMS interface velocity vs. normalized temperature h=h21
e. Solid
line in (d) is the velocity for the SS interface obtained both analytically, Eq. (22), and numerically. Simulations are performed for kE¼4;kd¼1, and
different values of a0, starting with the SS interface.
6V.I. Levitas, K. Momeni / Acta Materialia xxx (2013) xxx–xxx
AM 11061 No. of Pages 9, Model 5+
12 December 2013
Please cite this article in press as: Levitas VI, Momeni K. Solid–solid transformations via nanoscale intermediate interfacial phase:
Multiple structures, scale and mechanics effects. Acta Mater (2013), http://dx.doi.org/10.1016/j.actamat.2013.11.051

378 The energy of the SMS interface monotonously
379 increases with an increase in a0, as expected. The value of
380 !min also increases monotonously with an increase in a0,
381 excluding temperatures close to h20
e. Over the entire temper-
382 ature range, the magnitude of the velocity of the SMS
383 interface in both directions increases with an increase in
384 a0. The most unexpected finding is the dependence of the
385 IM width on a0. According to the above analytical model,
386 close to the melting temperature, dshould grow when a0
387 grows, which is in agreement with simulation results
388 (Fig. 4c). The same is true down to h=h21
e’0:85, while
389 the effect of a0is getting weaker. At h=h21
e’0:85, the effect
390 of a0disappears and reverses at lower temperature. There
391 are no simplified models or any indications on other plots
392 (!min and Evs. h=h21
e), which could explain such a behav-
393 ior for the IM width. Thus, Fig. 4c helps to explain the
394 development of the theory described here, and but shows
395 that it is necessary to find the reasons for the crossover
396 in temperature dependence of the IM width.
397 If a0is excluded from the dependences Eðh;a0Þand
398 dðh;a0Þ, one obtains the dependence of the energy of the
399 SMS interface vs. its equilibrium width for various temper-
400 atures (Fig. 5). While for high temperatures an expected
401 increasing dependence of the SMS interface energy on its
402width is obtained, at h¼382Kthe interface energy is inde-
403pendent of d, and at lower temperatures a counterintuitive
404reduction in interface energy with growing dis observed.
405Coupling with mechanics — The elastic energy at coher-
406ent SS interfaces strongly promotes the formation of an IM
407(Fig. 6 for kd¼1), and this effect increases with the size of
408the sample. Starting from the SS interface, with energy
409(including the elastic energy) of 1:1J=m2, melt forms at
410kE¼3:3; neglecting the elastic energy, melt forms at
411kE¼3:5. The elastic energy increases the width of the IM
412and also drives an SMS interface at h¼h21
e(when v¼0
413without mechanics), because it changes with interface posi-
414tion in a finite sample. Whereas without mechanics the IM
415is homogeneous along the SS interface, with mechanics it
416starts at the intersection of SS interface and the free sur-
417face. The velocity, width and energy of the SMS interface
418when the elastic energy is considered are 70 m=s, 2:4nm
419and 0:834 J=m2, respectively. For the case without mechan-
Fig. 5. Energy of the IM interface vs. its equilibrium width for various
temperatures, obtained by excluding a0from plots in Fig. 2b and c.
Fig. 6. Effect of mechanics on IM formation. Formation of IM with (bottom row) and without (top row) mechanics starting from the SS interface for
h¼h21
e;kd¼1 and a0¼0:01 are shown. Melt appears at kE¼3:5 when the mechanical energy is negligible and at kE¼3:3 for the case with mechanics.
Fig. 7. The presence of mechanics promotes IM formation. !min is plotted
for different ratios of the energies of the SS and SM interfaces, kE, for
kd¼0:2;a0¼0:01, and h¼h21
e¼432 K, for the case with initial SS
interface (black) and pre-existing melt SMS (red). Results without
mechanics (line) and with mechanics (dots) are presented. In both cases,
continuous and reversible intermediate premelting and melting are
observed. The promotional effect of the relaxation of elastic stresses at
the SS coherent interface on the formation of the IM is evident. (For
interpretation of the references to color in this figure legend, the reader is
referred to the web version of this article.)
V.I. Levitas, K. Momeni / Acta Materialia xxx (2013) xxx–xxx 7
AM 11061 No. of Pages 9, Model 5+
12 December 2013
Please cite this article in press as: Levitas VI, Momeni K. Solid–solid transformations via nanoscale intermediate interfacial phase:
Multiple structures, scale and mechanics effects. Acta Mater (2013), http://dx.doi.org/10.1016/j.actamat.2013.11.051

420 ics and E21 ¼1J=m2, the interface is stationary and the
421 interface width and energy are 1:64 nm and 0:79 J=m2,
422 respectively. For kd¼0:2 with mechanics, jump-like
423 incomplete IM forms for kE¼2:0, in contrast to continu-
424 ous premelting to the same !min ¼0:57 at kE¼3:1 (see
425 Fig. 7). For pre-existing melt, kd¼1anda0¼0:01, the
426 IM remains stable even at kE¼1, while for the case
427 neglecting elastic energy, stationary melt remains at
428 kEP2.
429 5. Concluding remarks
430 A phase-field approach has been developed to describe
431 SS phase transformations via IMs hundreds of degrees
432 below melting temperature. Although quantitative compar-
433 ison with experiment is currently impossible due to the lack
434 of data on interface energies, our results demonstrate the
435 possibility of IM formation under conditions that were
436 not accessible using the sharp interface approach.
437 Detailed parametric simulations revealed surprising
438 scale effects as well the effects of kinetics, elastic stresses
439 and interaction between SM interfaces. In particular, vari-
440 ous types of IMs and scenarios for their appearance and
441 disappearance are found. The latter include continuous
442 and reversible premelting and melting, and jump-like barri-
443 erless transformation to IM. This barrierless transforma-
444 tion exhibits large hysteresis and persistence of IM at
445 lower temperatures than what was expected without barri-
446 erless resolidification. A surprising scale effect related to kd
447 is revealed, indicating that increasing kdsuppresses barrier-
448 less IM but allows retention of IM at much lower temper-
449 atures and even for kE<2. An unexpected
450 nonmonotonous effect of kdis found, which produces an
451 IM gap (i.e. lack of intermediate melting) in some kdand
452 kEranges. For the parameters studied herein, the IM
453 reduces the interface velocity in comparison with that for
454 the velocity of the SS interface. Mechanics (i.e. stress gen-
455 eration by the coherent SS interface and stress relaxation
456 due to the formation of IM) produces an additional pro-
457 moting effect on IMs and further reduces the temperature
458 and ratio kEfor barrierless IM nucleation (even below
459 kE¼2) and for the persistence of the IM (even at
460 kE¼1). The formation of a thermally activated IM via a
461 critical nucleus, which is probably happening in HMX
462 [1,2], will be treated in future studies.
463Our phase-field approach can be adjusted to treat vari-
464ous other experimental problems related to wetting, sur-
465face-induced phenomena, and intergranular and interface
466phases, mentioned in the introduction. If the intermediate
467phase is solid, additional mechanics and size-induced mor-
468phological transitions are expected [12,13].
469Acknowledgements
470This work was supported by NSF, ARO, DARPA,
471ONR, and ISU.
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8V.I. Levitas, K. Momeni / Acta Materialia xxx (2013) xxx–xxx
AM 11061 No. of Pages 9, Model 5+
12 December 2013
Please cite this article in press as: Levitas VI, Momeni K. Solid–solid transformations via nanoscale intermediate interfacial phase:
Multiple structures, scale and mechanics effects. Acta Mater (2013), http://dx.doi.org/10.1016/j.actamat.2013.11.051