Content uploaded by Valery I. Levitas

Author content

All content in this area was uploaded by Valery I. Levitas on Dec 14, 2017

Content may be subject to copyright.

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>ðG0GsÞd;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

et¼½et1þðet2et1Þqð#; at#Þqð!;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;#Þ¼A10ðhÞþðA20ðhÞA10 ðhÞÞqð#; a#Þ;ð10Þ185185

186

the gradient energy coeﬃcient: 187

bmsð#Þ¼bs1mþðbs2mbs1mÞqð#; amsÞ;ð11Þ189189

190

the bulk modulus: 191

Kð!;#Þ¼K0þðKsð#ÞK0Þqð!;aKÞ;ð12Þ

Ksð#Þ¼Ks1þðKs2Ks1Þqð#; aks Þ;ð13Þ193193

194

and the shear modulus: 195

lð!;#Þ¼l0þðlsð#Þl0Þqð!;alÞ;ð14Þ

lsð#Þ¼ls1þðls2ls1Þqð#; alsÞ:ð15Þ197197

198

Interpolating function qðx;aÞ¼ax22ða2Þx3þ

199ða3Þx4varies between 0 and 1 when xvaries between 0

200and 1 and has zero x-derivative at x¼0 and x¼1. Func-

201tion /ð!;a/;a0Þ¼a/!22ða/2ð1a0ÞÞ!3þða/3

202ð1a0ÞÞ!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

s0¼Dss0ðhhs0

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

chhs0

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

chh21

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+

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

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

250E¼ðG0GsÞdþb21a0=ð2dÞ. 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=1þepðyvijtÞ=dij

hi

;ð20Þ

dij ¼pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

bij.2Aij ðhÞ3DGh

ijðhÞ

hi

r;ð21Þ

vij ¼6Lijdij DGh

ijðhÞ=p;ð22Þ

Eij ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2bij AijðhÞ3DGh

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

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 ¼793:79 kJ=m3K, Ds20 ¼935:45 kJ=m3K,

298melting temperatures h10

e¼550 K and h20

e¼532:14 K;

299h21

e¼432 K;e10

0t¼0:067, e20

0t¼0:147 (i.e. e21

0t¼0:08);

300

e

A21

c¼0;Aij

c¼3Dsij (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

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 dand 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=@d¼0, which results in

367

d¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

0:5b21a0=ðG0GsÞ

q;ddiverges 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

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, dshould 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 Evs. 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 Eðh;a0Þand

398 dðh;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 dis 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

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.

472References

473

[1] Levitas VI, Henson BF, Smilowitz LB, Asay BW. Phys Rev Lett

474

2004;92:235702.

475

[2] Levitas VI, Henson BF, Smilowitz LB, Asay BW. J Phys Chem B

476

2006;110:10105–19.

477

[3] Levitas VI. Phys Rev Lett 2005;95:075701.

478

[4] Levitas VI, Ren Z, Zeng Y, Zhang Z, Han G. Phys Rev B

479

2012;85:220104.

480

[5] Randzio SL, Kutner A. J Phys Chem B 2008;112:1435–44.

481

[6] Grinfeld MJ. Thermodynamic methods in the theory of heteroge-

482

neous systems. Longman Scientiﬁc and Technical; 1991.

483

[7] Lobkovsky AE, Warren JA. Physica D 2002;164:202–12.

484

[8] Tang M, Carter W, Cannon R. Phys Rev B 2006;73:024102.

485

[9] Bishop CM, Cannon RM, Carter WC. Acta Mater 2005;53:4755–64.

486

[10] Levitas VI, Samani K. Phys Rev B 2011;84:140103. R.

487

[11] Levitas VI, Samani K. Nat Commun 2011;2:284–6.

488

[12] Levitas VI, Javanbakht M. Phys Rev Lett 2010;105:165701.

489

[13] Levitas VI, Javanbakht M. Phys Rev Lett 2011;107:175701.

490

[14] Lipowsky R. Phys Rev Lett 1982;49:1575–8.

491

[15] Tiaden J, Nestler B, Diepers HJ, Steinbach I. Physica D

492

1998;115:73–86.

493

[16] Levitas VI, Preston D, Lee D-W. Phys Rev B 2003;68:134201.

494

[17] Wang H, Chiang Y-M. J Am Ceram Soc 1998;81:89–96.

495

[18] Luo J, Gupta VK, Yoon DH, Meyer HM. Appl Phys Lett

496

2005;87:231902.

497

[19] Avishai A, Scheu C, Kaplan WD. Acta Mater 2005;53:1559–69.

498

[20] Luo J. Crit Rev Solid State 2007;32:67–109.

499

[21] Luo J, Chiang Y-M. Ann Rev Mater Res 2008;38:227–49.

500

[22] Luo J. J Am Ceram Soc 2012;95:2358–71.

501

[23] Cahn JW. J Chem Phys 1977;66:3667–72.

502

[24] Baram M, Chatain D, Kaplan WD. Science 2011;332:206–9.

503

[25] Luo J, Chiang YM. Acta Mater 2000;48:4501–15.

504

[26] Avishai A, Kaplan WD. Acta Mater 2005;53:1571–81.

505

[27] Cantwell PR, Tang M, Dillon SJ, Luo J, Rohrer GS, Harmer MP.

506

Acta Mater 2013.

507

[28] Qian H, Luo J, Chiang Y-M. Acta Mater 2008;56:862–73.

508

[29] Chung S-Y, Kang S-JL. Acta Mater 2003;51:2345–54.

509

[30] Levitas VI. Acta Mater 2013;61:4305–19.

510

[31] Levitas VI, Roy AM, Preston DL. Phys Rev B 2013;88:054113.

511

Q3

8V.I. Levitas, K. Momeni / Acta Materialia xxx (2013) xxx–xxx

AM 11061 No. of Pages 9, Model 5+

12 December 2013