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The Strong Inﬂuence of Internal Stresses on the Nucleation of a
Nanosized, Deeply Undercooled Melt at a Solid−Solid Phase
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
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
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
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.
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
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
© 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
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
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.
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
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
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.
Similar to sharp-interface study of phase transformations in
HMX, which is consistent with experiments,
we assume that
internal stresses cannot cause nucleation of dislocations. The
model was implemented in the ﬁnite element package
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
Nano Lett. 2015, 15, 2298−2303
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,
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
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
Nano Lett. 2015, 15, 2298−2303
with initial conditions close to the ﬁnal conﬁguration of the
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
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
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
Nano Lett. 2015, 15, 2298−2303
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
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
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
CN −Ess and for CN2at the surface is Qsms
Similar, the activation energy of the S1S2CN1within S1MS2
interface is Qss
CN −Esms and for CN2is Qsm
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
CN and Ess) signiﬁcantly changes their small
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
and the eﬀect of an
external load can be easily included as well. All these factors
may lead to new results and phenomena.
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.
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
with the IM at the Center of a Sample and E2
CN with the IM
at the Surface
without mechanics with mechanics
Ess 1256.64 1269.281 21.4274 1290.7084
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
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
1100.418 96.6513 8.908 105.56
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
Nano Lett. 2015, 15, 2298−2303
(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
and formation of
interfacial and intergranular crystalline or amorphous phases
in ceramic and metallic systems and
developing corresponding interfacial phase diagrams.
Details of the mathematical model and material properties. This
material is available free of charge via the Internet at http://
The authors declare no competing ﬁnancial interest.
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
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Nano Letters Letter
Nano Lett. 2015, 15, 2298−2303
Kasra Momeni, Valery I. Levitas,∗and James A. Warren
Mechanics equations — The relationship between the strain tensor ε
ε, displacement vector u
and the decomposition of strain into elastic ε
εel and transformational ε
where ε0and e
eare the volumetric and deviatoric contributions to strain tensor; I
Iis the unit tensor,
∇is the gradient operator, and subscript sym means symmetrization. The equilibrium equation is
σis the stress tensor. The elasticity rule is
ε=∂ ψ e
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
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
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
the barrier between solid and melt
As0(θ,ϑ) = A10(θ) + A20(θ)−A10(θ)q(ϑ,aϑ),(10)
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
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
∂ φ (ϒ,aφ,a0)
∂ ϑ +∇
where Lϒand Lϑare the kinetic coefﬁcients and derivatives of ψ, evaluated at ε
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
vi j =6Li jδi j ∆Gθ
i j(θ)/p;Ei j =r2βi j Ai j (θ)−3∆Gθ
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
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
(such a choice corresponds to the temperature-independent interface energies and widths4); 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.
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.
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),
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
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.
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