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.

The Strong Inﬂuence of Internal Stresses on the Nucleation of a

Nanosized, Deeply Undercooled Melt at a Solid−Solid Phase

Interface

Kasra Momeni,

†

Valery I. Levitas,*

,†,‡,§,∥

and James A. Warren

∥

†

Department of Aerospace Engineering,

‡

Department of Mechanical Engineering,

§

Material Science and Engineering, Iowa State

University, Ames, Iowa 50011, United States

∥

Materials Science and Engineering Division, Material Measurement Laboratory, National Institute of Standards and Technology,

Gaithersburg, Maryland 20899, United States

*

SSupporting Information

ABSTRACT: The eﬀect of elastic energy on nucleation and

disappearance of a nanometer size intermediate melt (IM)

region at a solid−solid (S1S2) phase interface at temperatures

120 K below the melting temperature is studied using a phase-

ﬁeld approach. Results are obtained for broad range of the

ratios of S1S2to solid−melt interface energies, kE, and widths,

kδ. It is found that internal stresses only slightly promote

barrierless IM nucleation but qualitatively alter the system

behavior, allowing for the appearance of the IM when kE<2

(thermodynamically impossible without mechanics) and

elimination of what we termed the IM-free gap. Remarkably, when mechanics is included within this framework, there is a

drastic (16 times for HMX energetic crystals) reduction in the activation energy of IM critical nucleus. After this inclusion, a

kinetic nucleation criterion is met, and thermally activated melting occurs under conditions consistent with experiments for

HMX, elucidating what had been to date mysterious behavior. Similar eﬀects are expected to occur for other material systems

where S1S2phase transformations via IM take place, including electronic, geological, pharmaceutical, ferroelectric, colloidal, and

superhard materials.

KEYWORDS: Intermediate melt, phase ﬁeld approach, solid−melt−solid interface, nucleation, internal stresses

In this study, we investigate the appearance of phases at a

solid−solid (S1S2) boundary, detailing the inﬂuence of

processes within few nanometer thick phase interface, including

its structure and stress state. It is found that the S1S2interface

tends to reduce its energy via elastic stress relaxation and

restructuring. Speciﬁcally, restructuring can occur via the

nucleation of a nanometer-scale intermediate melt (IM) at

the S1S2boundary at temperatures well below the bulk melting

temperature. This mechanism was proposed for β↔δphase

transformations (PTs) in energetic organic HMX crystals

1,2

at

undercoolings of 120 K in order to explain puzzling

experimental data in refs 3 and 4. The appearance of the IM

at these temperatures allowed for a relaxation of elastic energy

at the S1S2phase interface, making the transition energetically

favorable. This mechanism explained, both quantitatively and

qualitatively, 16 nontrivial experimental phenomena.

2

In

addition to stress relaxation and elimination of interface

coherency, the IM eliminates athermal friction and alters the

interface mobility. Along related lines, the mechanism of

crystal−crystal and crystal−amorphous PTs via intermediate

(or virtual) melting for materials (like water) where increasing

the pressure leads to a reduction in the melting temperature

was suggested in ref 5. Amorphization via virtual melting was

claimed in experiments for Avandia (Rosiglitazone), an

antidiabetic pharmaceutical, in ref 6. Also, solid−solid PT via

IM and surface-induced IM in PbTiO3nanoﬁbers was observed

experimentally and treated thermodynamically in ref 7. In this

case, melting within the S1S2interface was caused by reduction

in the total interface energy and relaxation of internal elastic

stresses. And in subsequent investigations, it was found that

relaxation of external deviatoric stresses under very high strain

rate conditions could cause melting at undercoolings of 4000

K.

8

The important role of these phenomena in the relaxation of

stress in crystalline systems is given in ref 9. Most recently, the

transition between square and triangular lattices of colloidal

ﬁlms of microspheres via an IM was directly observed in ref 10.

However, there are some essential inconsistencies in the

thermodynamic and kinetic interpretation of this phenomenon

in ref 10. While it is stated that crystal−crystal transformation

occurs below the bulk melting temperature Tm, the bulk driving

force for melting is considered to be positive, which is possible

above Tmonly. In contrast to the statement in ref 10, crystal−

Received: November 14, 2014

Revised: February 21, 2015

Published: March 19, 2015

Letter

pubs.acs.org/NanoLett

© 2015 American Chemical Society 2298 DOI: 10.1021/nl504380c

Nano Lett. 2015, 15, 2298−2303

crystal transformation via intermediate (virtual) melting have

been discussed for a decade, indeed signiﬁcantly below melting

temperature

1,2,5−9,11,12

and with much more general thermody-

namic and kinetic description.

In the above treatments of this phenomena, the theoretical

approach was limited to simpliﬁed continuum thermodynamics.

Recently, however, we introduced a phase-ﬁeld approach for

the S1S2phase transformation via IM and the formation of

disordered interfacial phases both without

11

and with

12

mechanical eﬀects. This approach yielded a more detailed

picture of the interface, including the appearance of a “partial”

IM and the substantial inﬂuence of the parameter kδ,aneﬀect

necessarily not present in sharp-interface theories. In refs 11

and 12, the eﬀect of relaxation of internal stresses was brieﬂy

investigated for the case of barrierless IM nucleation, and

nucleation via a critical nucleus (CN) was not explored. In fact,

results in ref 11 for CN appeared to eliminate it as a mechanism

for stress relaxation, as the CN had too high an activation

energy to explain observation of macroscopic kinetics of β↔δ

PTs in HMX crystals.

1−4

In this Letter, we employ our phase ﬁeld approach to study

eﬀect of mechanics, that is, internal stresses (for diﬀerent ratios

kEand kδ) on the thermodynamics, kinetics, and structure of

IM within a S1S2interface, describing its appearance and

disappearance due to barrierless and thermally activated

processes 120 K below bulk melting temperature in a model

HMX system. It is found that internal stresses only slightly

promote barrierless IM nucleation but qualitatively alter the

system behavior, allowing for the appearance of the IM when kE

< 2 (thermodynamically impossible without mechanics) and

elimination of what we termed the IM-free gap. Remarkably,

when mechanics is included within this framework, there is a

drastic (16 times for HMX energetic crystals) reduction in the

activation energy of IM critical nucleus. After this inclusion, a

kinetic nucleation criterion is met, and thermally activated

melting occurs under conditions consistent with experiments

for HMX, elucidating what to date had been mysterious

behavior.

1,2

CN at the surface of a sample is also studied.

Model. For description of PTs between three phases, a

phase-ﬁeld model introduced in ref 12 (and presented in

Supporting Information) employs two polar order parameters:

radial Υand angular ϑ, where πϑ/2 is the angle between the

radius vector Υand the positive horizontal axis in the polar

order parameter plane. The melt is represented by Υ= 0 for all

ϑ. Solid phases correspond to Υ= 1; phase S1is described by ϑ

= 0 and phase S2is described by ϑ= 1. This representation of

the three phases sits in contrast to other multiphase

models

13−16

that used three order parameters with a constraint

that they always sum to a constant. Unlike the prior

approaches, the polar variable approach has desirable property

that each of the PTs: M↔S1, corresponding to variation in Υ

between 0 and 1 at ϑ=0;M↔S2, corresponding to variation

in Υbetween 0 and 1 at ϑ= 1; and S1↔S2, corresponding to

variation in ϑbetween 0 and 1 at Υ= 1, is described by single

order parameter with the other ﬁxed, which allowed us to

utilize analytical solutions for each of the nonequilibrium

interfaces and determine their width, energy, and velocity.

11,12

Similar to sharp-interface study of phase transformations in

HMX, which is consistent with experiments,

1,2

we assume that

internal stresses cannot cause nucleation of dislocations. The

model was implemented in the ﬁnite element package

COMSOL.

17

Material parameters have been chosen for organic

HMX energetic crystal (Tables 1 and 2 in Supporting

Information). Problems have been solved for diﬀerent kEand

kδvalues at equilibrium temperature of two HMX solid phases,

θe= 432 K, which is 120K below the melting temperature of

the δphase, which melts and resolidiﬁes into βphase during β

→δPT. Here the values of kEand kδare explored to determine

their inﬂuence, partly as they are unknown; but also we expect

that these parameters will be sensitive in experiments to

impurities and other ”alloying”eﬀects and thus can be

experimentally controlled to some degree.

For barrierless processes, a rectangular 40 nm ×300 nm with

the symmetry plane at its left vertical edge, ﬁxed lower left

corner, and a stress-free boundary on the right side are

considered. A vertical initial interface was placed in the middle

of the sample. Two types of initial conditions have been used:

(i) A stationary S1S2interface, which is obtained by placing an

analytical solution for a stationary stress-free interface as an

initial condition (see eq 19 in Supporting Information), and (ii)

apre-existingmeltconﬁned between two solid phases

(designated as S1MS2), which is obtained as a stationary

solution with initial data corresponding to S1MS2with a

complete IM that is broader than in stationary solution.

Parameters kδand kEare explicitly deﬁned in the Supporting

Information (see eq 20 therein). Plane strain conditions in the

out-of-plane direction are assumed. The domain is meshed with

ﬁve elements per S1S2interface width, using quadratic Lagrange

Figure 1. Eﬀect of internal elastic stresses on thermodynamically equilibrium solutions as a function of kE. Initial conditions are shown in boxes and

correspond to S1S2(designated as SS) and S1MS2(designated as SMS) interfaces. Value of Υmin is shown for problems without and with mechanics

at θ=θe= 432 K, which is 120 K below the melting temperature. (a) Continuous premelting/resolidiﬁcation for small kδ= 0.3, and (b) jumplike IM

and resolidiﬁcation for kδ= 0.7. Allowing for elastic energy that relaxes during intermediate melting promotes melting for all cases.

Nano Letters Letter

DOI: 10.1021/nl504380c

Nano Lett. 2015, 15, 2298−2303

2299

elements. An implicit time-stepping integrator with variable-

step-size backward diﬀerentiation is used with initial time step

of 1 ps and a relative tolerance of 10−4. The numerical model is

veriﬁed by solving the time-dependent Ginzburg−Landau

equations (phase ﬁelds) for the PT between two phases at

diﬀerent temperatures without mechanics and comparing the

results with analytical solutions for the interface energy, width,

velocity, and proﬁle,

11

which indicate perfect match.

Barrierless Nucleation. Here, the eﬀect of thermal

ﬂuctuations is neglected and barrierless PTs are studied. The

IM exhibits itself as deviation of the order parameter Υwithin

otherwise S1S2interface from 1. If the minimum value Υmin

reaches zero, then IM is complete; otherwise, it is incomplete

IM. In Figures 1 and 2, the minimum value Υmin is presented

for the steady state solution using two initial conditions

(states): stationary S1S2and S1MS2interfaces.

Results for small kδvalues in Figure 1a revealed continuous

premelting/resolidiﬁcation with increasing/decreasing kEand

presence of only a single solution independent of initial

conditions. Allowing for internal stresses generated by misﬁt

strain at the S1S2interface promotes melt formation, that is,

reduces Υmin. In other words, melting results in the partial or

complete relaxation in internal stresses, or stress results in an

additional thermodynamic driving force for melting. Mechanics

also shifts the minimum value of kEfor initiation of disordering,

even below kE< 2.0, which is energetically impossible without

mechanics (because energy of two SM stress-free interfaces is

larger than energy of SS interface). For larger kδ= 0.7 (Figure

1b) a range of kEvalues is found for which two diﬀerent

stationary solutions exist depending on the chosen initial

conditions. Solutions for Υmin experience jumps from 1 to small

values after reaching some critical kEand then change

continuously with increasing or decreasing kE. Starting with

IM state, decreasing kEleads to jump to Υmin = 1. Thus, in

contrast to Figure 1a, there is a clear hysteresis behavior.

Internal elastic stresses reduce Υmin and shift the critical kE

values for loss of stability of S1S2and S1MS2interfaces to lower

values of kE, as well as increase hysteresis region, thus

promoting IM.

A much richer picture is observed when Υmin is plotted

versus scale parameter kδfor diﬀerent ﬁxed kE(Figure 2).

Figure 2b for kE= 2.6 shows that for small kδvalues, S1S2

interface does not exist and the only continuous reversible

intermediate melting/ordering occurs with increasing/decreas-

ing kδ. Elastic stresses promote IM again by reducing Υmin.

With further increases in kδfor the same S1MS2interface, the

degree of disordering increases (and reversibly decreases with

decreasing kδ), the eﬀect of mechanics diminishes and

disappears when Υmin reaches zero. However, for large kδan

alternative solution Υmin = 1 exists and if S1S2interface is the

initial state, it does not change. Below some critical kδ, a jump

from S1S2interface to S1MS2interface occurs with reducing kδ

and the elastic energy increases slightly this critical value. A

reverse jump is impossible, thus S1MS2interface does not

transform to S1S2interface barrierlessly.

For smaller kE= 2.0 and 2.3, the eﬀect of the scale parameter

kδis nonmonotonous and thus more complex (Figure 2a). For

kE= 2.0 without mechanics, the only solution is the S1S2

interface. Elastic energy changes result qualitatively. Thus, for

small kδthe only solution contains IM; however, the degree of

disordering reversibly reduces with increasing kδ(opposite to

the case with larger kEin Figure 2b) and eventually disappears.

For large kδ, there are both (almost) complete S1MS2and S1S2

solutions. While initial S1S2does not change in this range, IM

reduces degree of disordering with reducing kδ, until IM

discontinuously disappears. For intermediate kδ, the only

solution is the S1S2interface. This region between two other

regions where IM exists we called the IM-free gap. For kE= 2.3,

IM-free gap exists without mechanics but disappears with

mechanics. Now, with mechanics the behavior is qualitatively

similar to that for kE= 2.6 (Figure 2b). Without mechanics, for

small kδthe value Υmin ﬁrst decreases and then increases up to

Υmin = 1 (i.e., exhibits local minimum), followed by IM-free gap

and then by two solutions. Thus, mechanics qualitatively

changes types of barrierless behavior. However, quantitatively

values Υmin are not drastically aﬀected.

Thermally Activated Nucleation. The presence of two

stationary solutions in Figure 1b, corresponding to local

minima of the energy, indicates existence of the third, unstable,

solution equivalent to the “min−max”of energy functional

corresponding to a CN between them. Critical nuclei are

studied at θ=θe= 432 K for kE= 2.6 and kδ= 0.7, that is, in the

range of parameters where two solutions exist for both cases

without and with mechanics (Figure 1b). Because of

thermodynamic instability, CN solutions are highly sensitive

to the initial conditions of the system and can be obtained by

solving stationary Ginzburg−Landau and mechanics equations

using an aﬃne invariant form of the damped Newton method

Figure 2. Mechanics and scale eﬀects on thermodynamically equilibrium solutions Υmin at θ=θe= 432 K for three values of kE. Elastic energy

promotes formation of melt and changes qualitatively types of behavior for some parameters.

Nano Letters Letter

DOI: 10.1021/nl504380c

Nano Lett. 2015, 15, 2298−2303

2300

with initial conditions close to the ﬁnal conﬁguration of the

CN.

We consider a cylindrical sample of R= 20 nm in radius, 100

nm in length along the axis of symmetry (z-axis), and capped

by two so-called “perfectly matched”layers of 10 nm in length

at the top and bottom that are used in the COMSOL code

17

to

mimic an inﬁnite sample length. Here, we focused on the eﬀect

of internal stresses and assumed that all external surfaces are

stress-free. Boundary conditions for both order parameters are

imposed in the form of zero normal components of the

gradient of the order parameters, which will guarantee that the

outer surface energy remain ﬁxed during a PT. Two CN were

considered; in one, CN1, the IM is at the center of a sample,

and in the other, CN2, the IM is at the surface. Initial conditions

for the simulations are obtained from the analytical solution for

a two-phase interface proﬁle for ϑand two back-to-back

interface proﬁles for Υ(see these conditions in the Supporting

Information and ref 11).

In Figure 3, for solutions that are without (do not consider)

mechanics (Figure 3a,b) and those with (that do consider)

mechanics (Figure 3c,d), we plot the distributions of the order

parameters, Υand ϑ, revealing the structure of CN for the case

when IM is at the center of a sample. Similar results for CN2are

presented in Figure 4. The solutions were tested to make sure

that they correspond to the energy min−max of the system.

This test was done by taking the calculated solutions for CN

Figure 3. Structure for the CN1with IM at the center of a sample. Simulations are performed at θe= 432 K, kδ= 0.7, and kE= 2.6 for the cases

without (a,b) and with mechanics (c,d). Proﬁle of the order parameter Υ(r) along the horizontal line z= 30 nm is plotted in the top insets. Vertical

insets show the proﬁle of Υ(z) (top plots) and ϑ(z) (bottom plots) at r= 0. Solid line in the Υplots corresponds to Υ= 0.9 and determines the

boundary of disordered CN of IM within the S1S2interface. Dotted line in the ϑplots indicates the level line of ϑ= 0.5 and corresponds to the sharp

S1S2interface.

Figure 4. Structure for the CN2with IM at the surface of the sample. Simulations are performed at θe= 432K,kδ= 0.7, and kE= 2.6 for the case

without (a,b) and with mechanics (c,d). Proﬁle of order parameter Υ(r) along the horizontal line z= 30 nm is plotted in the top insets. Vertical

insets show the proﬁle of Υ(z) (top plots) and ϑ(z) (bottom plots) at r= 20 nm.

Nano Letters Letter

DOI: 10.1021/nl504380c

Nano Lett. 2015, 15, 2298−2303

2301

and slightly perturbing the CN solutions toward S1S2and

S1MS2solutions, obtaining nominally super- and subcritical

nuclei. These are then used as the initial conditions for the

time-dependent Ginzburg−Landau and mechanics equations.

As required for the unstable CN, the solutions with sub- and

supercritical IM nuclei evolved to the two stable S1S2and S1MS2

interfaces, respectively.

For models both without and with mechanics, the CN1with

the IM at the center has an ellipsoidal shape with Υmin = 0.24

and 0.30, respectively. The larger Υmin value for the sample with

mechanics is due to additional driving force associated with the

relaxation of elastic energy during melting. Allowing for

mechanics led to the formation of curved (bent) S1S2interface,

which is due to monotonically increasing volumetric trans-

formation strain across the S1S2interface. This bending cannot

be realized within the usual sharp-interface approaches,

suggesting that sharp-interface models should be improved to

include this phenomenon, for example, in refs 18 and 19. The

same interface bending is observed for CN2(Figure 4). Both

CN change local interface structure in terms of narrowing S1S2

interface in ϑdistribution within CN.

By construction, the energy of both bulk solid phases is equal

at their equilibrium temperature (with and without mechanics)

and thus the excess interface energy is calculated with respect to

any of homogeneous solid phase by integration of total energy

distribution over the sample. In such a way, we determine the

energy Ess of the S1S2and the energy Esms of the S1MS2ground

states. Similarly, we deﬁne the energy E1

CN of the CN1and the

energy E2

CN of the CN2. The diﬀerence between the energy of

each CN1and CN2, and each ground state gives the activation

energies for the corresponding PTs. Thus, the activation energy

of the S1MS2CN1at the center within S1S2interface is Qsms

1=

E1

CN −Ess and for CN2at the surface is Qsms

2=E2

CN −Ess.

Similar, the activation energy of the S1S2CN1within S1MS2

interface is Qss

2=E1

CN −Esms and for CN2is Qsm

2=E2

CN −Esms.

Each of the above-mentioned energies, which we will designate

by Ψfor conciseness, is the sum of three contributions: thermal

energy Ψθ, gradient energy Ψ∇, and elastic energy Ψe. Our

calculations for the energies of ground states and critical nuclei

are listed in Table 1.

A thermally activated process can be experimentally observed

if the activation energy of CN is smaller than (40−80)kBθ(ref

20), where kBis the Boltzmann constant. This is equal to 0.24−

0.48 ×10−18Jat θe= 432 K. The results indicate that the only

possible thermally activated process is the formation of CN1of

IM within the S1S2interface at the center of a sample when

mechanics is included. Because the activation energy for

resolidiﬁcation is much larger than the magnitude of thermal

ﬂuctuations for both CN, the IM persists. Perhaps the most

surprising result is that including the energy of elastic stresses

reduced the activation energy for IM critical nucleus at the

center of a sample, by a factor of 16, making nucleation possible

despite large undercoolings. Similarly, internal stresses

signiﬁcantly reduced energy of IM critical nucleus at the

surface (by ∼62 ×10−18Jor by a factor of 15) and the energy of

the SS critical nucleus with solid at the center (by ∼51 ×

10−18J). Although elastic energy makes a positive contribution

to the energy of ground states and CN, it increases the energy

of ground states more than it increases the energy of CN. The

proximate cause of this phenomenon is the slight change in the

structure of the CN and alteration of the interface geometry

during appearance of the CN. Thus, a small change in two large

numbers (E1

CN and Ess) signiﬁcantly changes their small

diﬀerence Qsms

1. To ensure that our conclusions are physical

rather than due to numerical errors, we used diﬀerent

integration volumes enclosing IM critical nucleus. The

calculations are insensitive to the integration volume as long

as boundaries of this volume are far (>5 nm) from the

boundaries of CN. We note that mechanics surprisingly

increases activation energy for resolidiﬁcation for CN1.

Without and with mechanics, activation energies for both the

IM critical nucleus and the CN of a solid−solid interface are

much smaller (by ∼60 ×10−18J) for the CN1at the center in

comparison with the CN2at the surface. While results for CN1

are independent of the sample size and boundary conditions for

the order parameter (because CN1is much smaller than the

sample), this is not the case for the CN2at the surface.

Reducing the sample size reduces volume of the CN2and its

activation energy, and for some critical size nucleation of the

IM at the surface may be kinetically possible. Also, if surface

energy of the melt is smaller than the surface energy of the

solid, it promotes thermally activated nucleation of the IM at

the surface and may also lead to barrierless nucleation. This can

be studied using methods similar to those in refs 21−24. A

tensorial transformation strain for melting

25

and the eﬀect of an

external load can be easily included as well. All these factors

may lead to new results and phenomena.

Previous results

11

without mechanics showed very high

activation energy and the practical impossibility of thermally

activated intermediate melting, which contradicted the

experimentally observed thermally activated interface kinetics

and the overall kinetics for HMX.

1,2

The inclusion of elastic

stresses in the model results in a drastic reduction of activation

energy, resolving this discrepancy.

Concluding Remarks. We have developed a phase-ﬁeld

approach and applied it to study the eﬀect of mechanics on

barrierless and thermally activated nucleation and disappear-

ance of nanoscale IM within an S1S2interface during S1S2PTs

120Kbelow the melting temperature. For diﬀerent ratios kEand

kδ, various types of behavior, mechanics, and scale eﬀects are

obtained. Barrierless intermediate melting/resolidiﬁcation can

be continuous (reversible), jumplike in one direction and

continuous in another, and jumplike in both directions

Table 1. Total Energy, Ψ=Ψθ+Ψ∇+Ψe, and Its Individual

Contributing Terms, Thermal ΨθPlus Gradient Ψ∇

Energies, and Elastic ΨeEnergy, Calculated for Ground

States, Ess and Esms, as Well as for Interfaces with CN, E1

CN

with the IM at the Center of a Sample and E2

CN with the IM

at the Surface

a

without mechanics with mechanics

ΨΨ

θ+Ψ∇ΨeΨ

Ess 1256.64 1269.281 21.4274 1290.7084

E1

CN 1262.684 1269.444 21.6346 1291.0786

Qsms16.05 0.163 0.2072 0.37

Esms 1162.2663 1172.7927 12.7266 1185.5193

E2

CN 1323.0063 1277.2457 17.9696 1295.2153

Qss2160.74 104.453 5.243 109.7

Qsms266.3663 7.965 −3.4578 4.507

Qss

1100.418 96.6513 8.908 105.56

a

Activation energies Qfor appearance of the CN are the diﬀerence

between energies of interfaces with CN and ground states. Simulations

are performed for the cases without and with mechanics at θe= 432 K

for kδ= 0.7 and kE= 2.6. All the energies are expressed in (×10−18J).

Nano Letters Letter

DOI: 10.1021/nl504380c

Nano Lett. 2015, 15, 2298−2303

2302

(hysteretic), partial and complete, with monotonous and

nonmonotonous dependence on kδand with IM-free gap

region between two IM regions along kδaxis. Internal elastic

stresses only slightly promote barrierless IM nucleation but

change type of system behavior, including appearance of IM for

kE< 2 (which is thermodynamically impossible without

mechanics) and elimination of IM-free gap region. To study

thermally activated nucleation, solutions for CN at the center

and surface of a sample are found and activation energies are

calculated and compared with the required values from a kinetic

nucleation criterion. We revealed an unanticipated, drastic (16

times for HMX energetic crystals) reduction in the activation

energy of IM critical nucleus when elastic energy is taken into

account. This reduction results in the system meeting the

kinetic nucleation criterion for the CN1at the center of a

sample, consistent with experiments for HMX. Because

thermally activated resolidiﬁcation is kinetically impossible,

IM persists during S1MS2interface propagation. For smaller

sample diameters and/or reduction of surface energy during

melting, mechanics can induce IM nucleation at the surface as

well. Similar eﬀects are expected to occur for other material

systems where solid−solid phase transformations via IM takes

place, including electronic (Si and Ge), geological (ice, quartz,

and coesite), pharmaceutical (avandia), ferroelectric (PbTiO3),

colloidal, and superhard (BN) materials. Similar approach can

be developed for grain-boundary melting

26

and formation of

interfacial and intergranular crystalline or amorphous phases

(complexions)

27−30

in ceramic and metallic systems and

developing corresponding interfacial phase diagrams.

■ASSOCIATED CONTENT

*

SSupporting Information

Details of the mathematical model and material properties. This

material is available free of charge via the Internet at http://

pubs.acs.org.

■AUTHOR INFORMATION

Corresponding Author

*E-mail: vlevitas@iastate.edu.

Notes

The authors declare no competing ﬁnancial interest.

■ACKNOWLEDGMENTS

This work was supported by ONR, NSF, ARO, DARPA, and

NIST. Certain commercial software and materials are identiﬁed

in this report in order to specify the procedures adequately.

Such identiﬁcation is not intended to imply recommendation or

endorsement by the National Institute of Standards and

Technology, nor is it intended to imply that the materials or

software identiﬁed are necessarily the best available for the

purpose.

■REFERENCES

(1) Levitas, V. I.; Henson, B. F.; Smilowitz, L. B.; Asay, B. W. Phys.

Rev. Lett. 2004,92, 235702.

(2) Levitas, V. I.; Henson, B. F.; Smilowitz, L. B.; Asay, B. W. J. Phys.

Chem. B 2006,110, 10105−10119.

(3) Henson, B. F.; Smilowitz, L.; Asay, B. W.; Dickson, P. M. J. Chem.

Phys. 2002,117, 3780−3788.

(4) Smilowitz, L.; Henson, B. F.; Asay, B. W.; Dickson, P. M. J. Chem.

Phys. 2002,117, 3789−3798.

(5) Levitas, V. I. Phys. Rev. Lett. 2005,95, 075701.

(6) Randzio, S. L.; Kutner, A. J. Phys. Chem. B 2008,112, 1435−

1444.

(7) Levitas, V. I.; Ren, Z.; Zeng, Y.; Zhang, Z.; Han, G. Phys. Rev. B

2012,85, 220104.

(8) Levitas, V. I.; Ravelo, R. Proc. Natl. Acad. Sci. U.S.A. 2012,109,

13204−13207.

(9) Ball, P. Nat. Mater. 2012,11, 747−747.

(10) Peng, Y.; Wang, F.; Wang, Z.; Alsayed, A. M.; Zhang, Z.; Yodh,

A. G.; Han, Y. Nat. Mater. 2015,14, 101−108.

(11) Momeni, K.; Levitas, V. I. Phys. Rev. B 2014,89, 184102.

(12) Levitas, V. I.; Momeni, K. Acta Mater. 2014,65, 125−132.

(13) Tiaden, J.; Nestler, B.; Diepers, H. J.; Steinbach, I. Physica D

1998,115,73−86.

(14) Folch, R.; Plapp, M. Phys. Rev. E 2005,72, 011602.

(15) Tóth, G. I.; Pusztai, T.; Tegze, G.; Tóth, G.; Gránásy, L. Phys.

Rev. Lett. 2011,107, 175702.

(16) Mishin, Y.; Boettinger, W. J.; Warren, J. A.; McFadden, G. B.

Acta Mater. 2009,57, 3771−3785.

(17) COMSOL Multiphysics, version 4.2; COMSOL, Inc.:

Burlington, MA, 2011,

(18) Javili, A.; Steinmann, P. Int. J. Solids. Struct. 2010,47, 3245−

3253.

(19) Levitas, V. I. J. Mech. Phys. Solids 2014,70, 154−189.

(20) Porter, D. Phase Transformation in Metals and Alloys; Van

Nostrand Reinhold: New York, 1981.

(21) Levitas, V. I.; Samani, K. Nat. Commun. 2011,2, 284−6.

(22) Levitas, V. I.; Javanbakht, M. Phys. Rev. Lett. 2011,107, 175701.

(23) Levitas, V., Javanbakht, M. Phys. Rev. Lett. 2010105.

(24) Levitas, V. I.; Samani, K. Phys. Rev. B 2014,89, 075427.

(25) Levitas, V. I.; Samani, K. Phys. Rev. B 2011,84, 140103(R).

(26) Lobkovsky, A. E.; Warren, J. A. Physica D 2002,164, 202−212.

(27) Luo, J. CRC Crit. Rev. Solid State 2007,32,67−109.

(28) Luo, J.; Chiang, Y.-M. Annu. Rev. Mater. Res. 2008,38, 227−249.

(29) Baram, M.; Chatain, D.; Kaplan, W. D. Science 2011,332, 206−

209.

(30) Frolov, T.; Olmsted, D. L.; Asta, M.; Mishin, Y. Nat. Commun.

2013,4, 1899.

Nano Letters Letter

DOI: 10.1021/nl504380c

Nano Lett. 2015, 15, 2298−2303

2303

Supporting Information

Kasra Momeni, Valery I. Levitas,∗and James A. Warren

E-mail: vlevitas@iastate.edu

Supporting Information

Model1

Mechanics equations — The relationship between the strain tensor ε

ε

ε, displacement vector u

u

u,

and the decomposition of strain into elastic ε

ε

εel and transformational ε

ε

εtparts are:

ε

ε

ε=1/3ε0I

I

I+e

e

e;ε

ε

ε= (∇u)sym;ε

ε

ε=ε

ε

εel +ε

ε

εt,(1)

where ε0and e

e

eare the volumetric and deviatoric contributions to strain tensor; I

I

Iis the unit tensor,

∇

∇

∇is the gradient operator, and subscript sym means symmetrization. The equilibrium equation is

∇

∇

∇·σ

σ

σ=0

0

0,(2)

where σ

σ

σis the stress tensor. The elasticity rule is

σ

σ

σ=∂ ψ

∂ε

ε

ε=∂ ψ e

∂ε

ε

εel

=K(ϒ,ϑ)ε0el I

I

I+2µ(ϒ,ϑ)e

e

eel,(3)

where Kis the bulk modulus and µis the shear modulus, which both are functions of polar order

parameters ϒand ϑ(see Eqs. (12)-(15)), ψand ψeare the total and elastic Helmholtz energies

that are calculated using Eqs. (4)1and (5), respectively.

∗To whom correspondence should be addressed

1

Thermodynamic functions — The Helmholtz energy per unit volume consists of elastic ψe,

thermal ψθ, and gradient ψ∇parts, and the term ˘

ψθdescribing double-well barriers between

phases:

ψ=ψe+˘

ψθ+ψθ+ψ∇;ψθ=∆Gθ(θ,ϑ)q(ϒ,0); (4)

ψe=0.5K(ϒ,ϑ)ε2

0el +2µ(ϒ,ϑ)|e

e

eel|2; (5)

ψ∇=0.5hβs0(ϑ)|∇ϒ|2+β21φ(ϒ,aφ,a0)|∇ϑ|2i; (6)

˘

ψθ=As0(θ,ϑ)ϒ2(1−ϒ)2+A21(θ)ϑ2(1−ϑ)2q(ϒ,aA); (7)

ε

ε

εt= [ε

ε

εt1+ (ε

ε

εt2−ε

ε

εt1)q(ϑ,atϑ)]q(ϒ,atϒ).(8)

Here, sub- and superscripts 0 are for melt Mand 1 or 2 for solids S1or S2;βs0and β21 are SM

and S1S2gradient energy coefﬁcients, respectively; ∆Gθand As0are the difference in thermal

energy and energy barrier between Mand Ss(s=1 or 2); A21 is the S1S2energy barrier; function

q(x,a) = ax2−2(a−2)x3+ (a−3)x4, which varies between 0 and 1 when xvaries between 0 and

1 and has zero x−derivative at x=0 and x=1, smoothly interpolates properties of three phases;

ais a parameter in the range 0 ≤a≤6; if unknown, a=3 is accepted (see ref 2); the function

φϒ,aφ,a0=aφϒ2−2(aφ−2(1−a0))ϒ3+ (aφ−3(1−a0))ϒ4+a0differs from qin that it is

equal to a0(rather than 0) at ϒ=0. Below we present the difference between the thermal energy

of the solids and the melt

∆Gθ(ϑ) = ∆Gθ

10 + (∆Gθ

20 −∆Gθ

10)q(ϑ,aϑG),(9)

the barrier between solid and melt

As0(θ,ϑ) = A10(θ) + A20(θ)−A10(θ)q(ϑ,aϑ),(10)

2

the gradient energy coefﬁcient

βms(ϑ) = βs1−m+βs2−m−βs1−mq(ϑ,ams),(11)

the bulk moduli

K(ϒ,ϑ) = K0+ (Ks(ϑ)−K0)q(ϒ,aK),(12)

Ks(ϑ) = Ks1+ (Ks2−Ks1)q(ϑ,aks),(13)

and the shear moduli

µ(ϒ,ϑ) = µ0+ (µs(ϑ)−µ0)q(ϒ,aµ),(14)

µs(ϑ) = µs1+ (µs2−µs1)q(ϑ,aµs).(15)

The difference between the thermal energy of Ssand Mis

∆Gθ

s0=−∆ss0(θ−θs0

e);(s=1,2),(16)

where θs0

eis the equilibrium temperature between the solid phase Ssand M, and ∆ss0is the jump

in entropy between Ssand M.

Ginzburg-Landau equations — Applying the ﬁrst and second laws of thermodynamics to the

system with a non-local free energy, and assuming a linear relationship between thermodynamic

forces and ﬂuxes, we obtain the Ginzburg-Landau equations:1

˙

ϒ=Lϒ−∂ ψ

∂ϒ+β21

2

∂ φ (ϒ,aφ,a0)

∂ϒ|∇

∇

∇ϑ|2+∇

∇

∇·βs0∇

∇

∇ϒ; (17)

˙

ϑ=Lϑ−∂ ψ

∂ ϑ +∇

∇

∇·β21φ(ϒ,aφ,a0)∇

∇

∇ϑ,(18)

where Lϒand Lϑare the kinetic coefﬁcients and derivatives of ψ, evaluated at ε

ε

ε=const

const

const.

3

Analytical solutions. One of the advantages of the Eqs. (4)-(16) is that, in contrast to multi-

phase models in Refs. 2,3, each of three PTs is described by a single order parameter, without

additional constraints on the order parameters. An analytical solution for each interface between i

and jphases, propagating along y-direction is1

ηi j =1/h1+e−p(y−vi jt)/δi j i;δi j =prβi j/h2Ai j (θ)−3∆Gθ

i j(θ)i;

vi j =6Li jδi j ∆Gθ

i j(θ)/p;Ei j =r2βi j Ai j (θ)−3∆Gθ

i j(θ)/6,(19)

where p=2.415,2η10 =ϒat ϑ=0; η20 =ϒat ϑ=1, and η21 =ϑat ϒ=1; vi j is the interface

velocity. These equations allow us to calibrate the material parameters βi j,Ai j ,θi j

c, and Li j when

the temperature dependence of the interface energy, width, and velocity are speciﬁed.

Using the Eq. (19), we deﬁned two dimensionless parameters, kEand kδ, that characterize

the energy and width ratios of SS to SM interfaces. We also assumed that energy and width of

interfaces are temperature independent, which can be achieved by substituting Ai j

c=−3∆si j. Then

we have

kE=E21

Es0=sβ21

βs0

∆s21(θ21

c−θ21

e)

∆ss0(θs0

c−θs0

e);kδ=δ21

δs0=sβ21

βs0

∆ss0(θs0

c−θs0

e)

∆s21(θ21

c−θ21

e); (20)

where in the presented simulations, the energy and width of SS interface are considered to be ﬁxed,

E21 =1J/m2and δ21 =1nm. Energy and width of SM interface will be determined by changing

the kEand kδ.

Material properties. For simplicity, we assume all transformation strains are purely volumetric.

Properties of the melt, δphase (S1) and βphase (S2) of energetic material HMX (C4H8N8O8)

are used (Table 1). It is assumed that for all subscripts a=3 except aA=0; Ai j

c=−3∆si j

(such a choice corresponds to the temperature-independent interface energies and widths4); θi j

c=

4

θi j

e+pEi j /(∆si j δi j )and βi j =6Ei j δi j /p;2E21 =1J/m2and δ21 =1nm. We have also assumed

δ10 =δ20 for simplicity.

Table 1: Elastic properties of HMX crystal.

Property Value

K0=K1=K215 (GPa)5

µ00(GPa)

µ1=µ27(GPa)5

aCalculated for β-HMX at θ=θ21

eand considered as a constant within small-strain approximations.

Table 2: Thermophysical properties of melt (phase 0), δ(phase 1), and β(phase 2) HMX.

∆s(kPa/K)6θe(K)7L(mm2/N·s)θc(K)β(nJ/m)ε0t8

δ−β(1 −2) -141.654 432 1298.3 -16616 a2.4845 a-0.08

m−δ(0 −1) -793.792 550 2596.5 f(kE,kδ)g(kE,kδ)-0.067

m−β(0 −2) -935.446 532.14 2596.5 f(kE,kδ)g(kE,kδ)-0.147

aThis value was calculated using Eq. (19), assuming E21 =1J/m2and δ21 =1nm.9

Finding the Critical Nucleus. To ﬁnd the structure of the critical nucleus (CN), the stationary

Ginzburg-Landau equations must be solved using proper initial conditions for a distribution of the

order parameters close to the ﬁnal conﬁguration of the CN. In this process, the order parameter

associated with the phase transformation between two solid phases, ϑ, is initialized using Eq.

(19)1. The other order parameter, describing the solid-melt phase transformation, ϒ, is initialized

for the CN1at the sample center as

ϒ1(r,z) = h1+exp −(z−z0−W/2)/δ20−1+1+exp (z+z0−W/2)/δ10−1iH(r0),

(21)

where z0determines the width of IM, and Wis the length of the simulation domain (excluding

perfectly matched layers), His the Heaviside function, and δ10 =δ20 are the widths of S1M

and S2Mrespectively. Different widths and radii of the initial CN conﬁguration is modeled by

substituting different z0and r0values in Eq. (21). Based on a trial process, we found reasonable

5

initial conditions using z0=0.5δ21 for modeling CN of IM within the S1S2interface. The initial

conditions for the CN2, for which the IM is located at the surface of S1S2interface within a pre-

existing interfacial melt, can be determined using ϒ2(z,r) = 1−ϒ1(z,r), and choosing a large z0

value (e.g., z0=8δ10 kδ). For the model with mechanics, a two-step process is pursued for ﬁnding

the conﬁguration of the CN. In the ﬁrst step, we found the CN for the sample without mechanics.

Then in the second step, we used the solution obtained for order parameters in previous step to

initialize the system of equations for the sample with mechanics.

References

[1] Levitas, V. I.; Momeni, K. Acta Mater. 2014,65, 125–132.

[2] Levitas, V.; Preston, D.; Lee, D.-W. Phys. Rev. B 2003,68, 134201.

[3] Tiaden, J.; Nestler, B.; Diepers, H. J.; Steinbach, I. Physica D 1998,115, 73–86.

[4] Levitas, V. I. Phys. Rev. B 2013,87, 054112.

[5] Sewell, T. D.; Menikoff, R.; Bedrov, D.; Smith, G. D. J. Chem. Phys. 2003,119, 7417–7426.

[6] Henson, B. F.; Smilowitz, L.; Asay, B. W.; Dickson, P. M. J. Chem. Phys. 2002,117, 3780–

3788.

[7] McCrone, W. C. Analytical Chemistry 1950,22, 1225–1226.

[8] Menikoff, R.; Sewell, T. D. Combust. Theor. Model. 2002,6, 103–125.

[9] Porter, D. Phase Transformation in Metals and Alloys; Van Nostrand Reinhold, 1981.

6