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The Strong Influence of Internal Stresses on the Nucleation of a Nanosized, Deeply Undercooled Melt at a Solid–Solid Phase Interface


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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 S1S2 to 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 S1S2 phase transformations via IM take place, including electronic, geological, pharmaceutical, ferroelectric, colloidal, and superhard materials.
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The Strong Inuence of Internal Stresses on the Nucleation of a
Nanosized, Deeply Undercooled Melt at a SolidSolid Phase
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 eect of elastic energy on nucleation and
disappearance of a nanometer size intermediate melt (IM)
region at a solidsolid (S1S2) phase interface at temperatures
120 K below the melting temperature is studied using a phase-
eld approach. Results are obtained for broad range of the
ratios of S1S2to solidmelt 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 eects 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 eld approach, solidmeltsolid interface, nucleation, internal stresses
In this study, we investigate the appearance of phases at a
solidsolid (S1S2) boundary, detailing the inuence 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. Specically, 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 βδphase
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
crystalcrystal and crystalamorphous 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, solidsolid PT via
IM and surface-induced IM in PbTiO3nanobers 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
lms 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 crystalcrystal 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, 22982303
crystal transformation via intermediate (virtual) melting have
been discussed for a decade, indeed signicantly 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 simplied continuum thermodynamics.
Recently, however, we introduced a phase-eld approach for
the S1S2phase transformation via IM and the formation of
disordered interfacial phases both without
and with
mechanical eects. This approach yielded a more detailed
picture of the interface, including the appearance of a partial
IM and the substantial inuence of the parameter kδ,aneect
necessarily not present in sharp-interface theories. In refs 11
and 12, the eect of relaxation of internal stresses was briey
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 βδ
PTs in HMX crystals.
In this Letter, we employ our phase eld approach to study
eect of mechanics, that is, internal stresses (for dierent 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-eld 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: MS1, corresponding to variation in Υ
between 0 and 1 at ϑ=0;MS2, corresponding to variation
in Υbetween 0 and 1 at ϑ= 1; and S1S2, corresponding to
variation in ϑbetween 0 and 1 at Υ= 1, is described by single
order parameter with the other xed, 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 nite element package
Material parameters have been chosen for organic
HMX energetic crystal (Tables 1 and 2 in Supporting
Information). Problems have been solved for dierent 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 resolidies into βphase during β
δPT. Here the values of kEand kδare explored to determine
their inuence, partly as they are unknown; but also we expect
that these parameters will be sensitive in experiments to
impurities and other alloyingeects 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, xed 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-existingmeltconned 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 dened in the Supporting
Information (see eq 20 therein). Plane strain conditions in the
out-of-plane direction are assumed. The domain is meshed with
ve elements per S1S2interface width, using quadratic Lagrange
Figure 1. Eect 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/resolidication for small kδ= 0.3, and (b) jumplike IM
and resolidication 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, 22982303
elements. An implicit time-stepping integrator with variable-
step-size backward dierentiation is used with initial time step
of 1 ps and a relative tolerance of 104. The numerical model is
veried by solving the time-dependent GinzburgLandau
equations (phase elds) for the PT between two phases at
dierent temperatures without mechanics and comparing the
results with analytical solutions for the interface energy, width,
velocity, and prole,
which indicate perfect match.
Barrierless Nucleation. Here, the eect of thermal
uctuations 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/resolidication with increasing/decreasing kEand
presence of only a single solution independent of initial
conditions. Allowing for internal stresses generated by mist
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 dierent
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 dierent xed 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 eect 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 eect 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 rst 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 aected.
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 minmaxof 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 GinzburgLandau and mechanics equations
using an ane invariant form of the damped Newton method
Figure 2. Mechanics and scale eects 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.
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with initial conditions close to the nal conguration 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 matchedlayers of 10 nm in length
at the top and bottom that are used in the COMSOL code
mimic an innite sample length. Here, we focused on the eect
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 xed 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 prole for ϑand two back-to-back
interface proles 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 minmax 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). Prole of the order parameter Υ(r) along the horizontal line z= 30 nm is plotted in the top insets. Vertical
insets show the prole 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). Prole of order parameter Υ(r) along the horizontal line z= 30 nm is plotted in the top insets. Vertical
insets show the prole 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, 22982303
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 GinzburgLandau 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 dene the energy E1
CN of the CN1and the
energy E2
CN of the CN2. The dierence 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
CN Ess.
Similar, the activation energy of the S1S2CN1within S1MS2
interface is Qss
CN Esms and for CN2is Qsm
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 (4080)kBθ(ref
20), where kBis the Boltzmann constant. This is equal to 0.24
0.48 ×1018Jat 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
resolidication is much larger than the magnitude of thermal
uctuations 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
signicantly reduced energy of IM critical nucleus at the
surface (by 62 ×1018Jor by a factor of 15) and the energy of
the SS critical nucleus with solid at the center (by 51 ×
1018J). 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) signicantly changes their small
dierence Qsms
1. To ensure that our conclusions are physical
rather than due to numerical errors, we used dierent
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 resolidication for CN1.
Without and with mechanics, activation energies for both the
IM critical nucleus and the CN of a solidsolid interface are
much smaller (by 60 ×1018J) 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 2124. A
tensorial transformation strain for melting
and the eect of an
external load can be easily included as well. All these factors
may lead to new results and phenomena.
Previous results
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-eld
approach and applied it to study the eect of mechanics on
barrierless and thermally activated nucleation and disappear-
ance of nanoscale IM within an S1S2interface during S1S2PTs
120Kbelow the melting temperature. For dierent ratios kEand
kδ, various types of behavior, mechanics, and scale eects are
obtained. Barrierless intermediate melting/resolidication 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 dierence
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 (×1018J).
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DOI: 10.1021/nl504380c
Nano Lett. 2015, 15, 22982303
(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 resolidication 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 eects are expected to occur for other material
systems where solidsolid 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.
SSupporting Information
Details of the mathematical model and material properties. This
material is available free of charge via the Internet at http://
Corresponding Author
The authors declare no competing nancial interest.
This work was supported by ONR, NSF, ARO, DARPA, and
NIST. Certain commercial software and materials are identied
in this report in order to specify the procedures adequately.
Such identication 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 identied are necessarily the best available for the
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Nano Letters Letter
DOI: 10.1021/nl504380c
Nano Lett. 2015, 15, 22982303
Supporting Information
Kasra Momeni, Valery I. Levitas,and James A. Warren
Supporting Information
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
εtparts are:
ε= (u)sym;ε
εel +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
where σ
σis the stress tensor. The elasticity rule is
σ=∂ ψ
ε=∂ ψ e
=K(ϒ,ϑ)ε0el I
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 ψparts, and the term ˘
ψθdescribing double-well barriers between
ψθ+ψθ+ψ;ψθ=Gθ(θ,ϑ)q(ϒ,0); (4)
0el +2µ(ϒ,ϑ)|e
eel|2; (5)
ψ=0.5hβs0(ϑ)|∇ϒ|2+β21φ(ϒ,aφ,a0)|ϑ|2i; (6)
ψθ=As0(θ,ϑ)ϒ2(1ϒ)2+A21(θ)ϑ2(1ϑ)2q(ϒ,aA); (7)
εt= [ε
εt1+ (ε
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) = ax22(a2)x3+ (a3)x4, which varies between 0 and 1 when xvaries between 0 and
1 and has zero xderivative at x=0 and x=1, smoothly interpolates properties of three phases;
ais a parameter in the range 0 a6; if unknown, a=3 is accepted (see ref 2); the function
φϒ,aφ,a0=aφϒ22(aφ2(1a0))ϒ3+ (aφ3(1a0))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
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) = βs1m+βs2mβs1mq(ϑ,ams),(11)
the bulk moduli
K(ϒ,ϑ) = K0+ (Ks(ϑ)K0)q(ϒ,aK),(12)
Ks(ϑ) = Ks1+ (Ks2Ks1)q(ϑ,aks),(13)
and the shear moduli
µ(ϒ,ϑ) = µ0+ (µs(ϑ)µ0)q(ϒ,aµ),(14)
µs(ϑ) = µs1+ (µs2µs1)q(ϑ,aµs).(15)
The difference between the thermal energy of Ssand Mis
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
∂ φ (ϒ,aφ,a0)
∇ϒ; (17)
ϑ=Lϑ∂ ψ
∂ 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+ep(yvi jt)/δi j i;δi j =prβi j/h2Ai j (θ)3Gθ
i j(θ)i;
vi j =6Li jδi j Gθ
i j(θ)/p;Ei j =r2βi j Ai j (θ)3Gθ
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=3si j. Then
we have
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=3si j
(such a choice corresponds to the temperature-independent interface energies and widths4); θi j
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
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.
δβ(1 2) -141.654 432 1298.3 -16616 a2.4845 a-0.08
mδ(0 1) -793.792 550 2596.5 f(kE,kδ)g(kE,kδ)-0.067
mβ(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 (zz0W/2)/δ201+1+exp (z+z0W/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) = 1ϒ1(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.
[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–
[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.
... Hyperspherical phase-field models for rapid solidification neglecting the surface energy inhomogeneities have recently been developed for diffusionless processes neglecting elasticity [12], with elasticity [13][14][15], and with elasticity and surface tension [16] that satisfy all stability conditions for a three-phase system. Multiphase-field models have been developed and utilized to study the microstructure of printed Inconel 718 alloy [17] and solute trapping behavior during rapid solidification [18]. ...
... In this study, we develop a phase-field potential for binary alloys that satisfies the stability conditions at all temperatures by capitalizing on our models for diffusionless melting/solidification [12,15,33] and materials growth [34,35]. We analytically solved the governing equations for dilute solution approximation and calculated interface velocity as a function of undercooling. ...
Full-text available
The integrity of the final printed components is mostly dictated by the adhesion between the particles and phases that form upon solidification, which is a major problem in printing metallic parts using available In-Space Manufacturing (ISM) technologies based on the Fused Deposition Modeling (FDM) methodology. Understanding the melting/solidification process helps increase particle adherence and allows to produce components with greater mechanical integrity. We developed a phase-field model of solidification for binary alloys. The phase-field approach is unique in capturing the microstructure with computationally tractable costs. The developed phase-field model of solidification of binary alloys satisfies the stability conditions at all temperatures. The suggested model is tuned for Ni-Cu alloy feedstocks. We derived the Ginzburg-Landau equations governing the phase transformation kinetics and solved them analytically for the dilute solution. We calculated the concentration profile as a function of interface velocity for a one-dimensional steady-state diffuse interface neglecting elasticity and obtained the partition coefficient, k, as a function of interface velocity. Numerical simulations for the diluted solution are used to study the interface velocity as a function of undercooling for the classic sharp interface model, partitionless solidification, and thin interface.
... and 16.2.5 for details. Also, papers [277,352,353] include coupling with elasticity and [354] include interfacial stresses. Internal elastic stresses promote the existence and persistence of the IM . ...
... Internal elastic stresses promote the existence and persistence of the IM . In particular, in [352], the internal stresses decreased the activation energy of IM critical nucleus (Fig. 13b) by a factor of 16 for the HMX, making thermally activated nucleation of the IM possible. ...
Full-text available
Review of selected fundamental topics on the interaction between phase transformations, fracture, and other structural changes in inelastic materials is presented. It mostly focuses on the concepts developed in the author's group over last three decades and numerous papers that affected us. It includes a general thermodynamic and kinetic theories with sharp interfaces and within phase field approach. Numerous analytical (even at large strains) and numerical solutions illustrate the main features of the developed theories and their application to the real phenomena. Coherent, semicoherent, and noncoherent interfaces, as well as interfaces with decohesion and with intermediate liquid (disordered) phase are discussed. Importance of the surface-and scale-induced phenomena on interaction between phase transformation with fracture and dislocations as well as inheritance of dislocations and plastic strains is demonstrated. Some nontrivial phenomena, like solid-solid phase transformations via intermediate (virtual) melt, virtual melting as a new mechanism of plastic deformation and stress relaxation under high strain rate loading, and phase transformations and chemical reactions induced by plastic shear under high pressure are discussed and modeled. * Extended version of paper: Levitas V.I. Phase transformations, fracture, and other structural changes in inelastic materials.
... This technique avoids applying boundary conditions at an interface that is mathematically difficult and computationally expensive. Instead, it uses additional internal variables, called order parameters, to model the interfaces and microstructure of the material (Ref [11][12][13][14][15][16][17][18]. The method captures intermediate phases and applies to particles with a size comparable to the solid-melt interface width. ...
Aluminum alloys are among the top candidate materials for in-space manufacturing (ISM) due to their lightweight and relatively low melting temperature. A fundamental problem in printing metallic parts using available ISM methods, based on the fused deposition modeling (FDM) technique, is that the integrity of the final printed components is determined mainly by the adhesion between the initial particles. Engineering the surface melt can pave the way to improve the adhesion between the particles and manufacture components with higher mechanical integrity. Here, we developed a phase-field model of surface melting, where the surface energy can directly be implemented from the experimental measurements. The proposed model is adjusted to Al 7075-T6 alloy feedstocks, where the surface energy of these alloys is measured using the sessile drop method. Effect of mechanics has been included using transformation and thermal strains. The effect of elastic energy is compared here with the corresponding cases without mechanics. Two different geometric samples (cylindrical and spherical) are studied, and it is found that cylindrical particles form a more disordered structure upon size reduction compared to the spherical samples.
Systematic understanding on the magnetic field intensity dependent microstructure evolution and recrystallization behavior in a Co-B eutectic alloy under a constant undercooling (∆T≈100 K) were carried out. Absent of the magnetic field, the comparable size of divorced FCC-Co and Co3B eutectic ellipsoidal grains coexist with a few regular lamellas. When the magnetic field is less than 15 T, the elongated primary FCC-Co dendrites parallel to the magnetic field with the dispersed FCC-Co nano-particles embedded within the Co3B matrix occupy the inter-dendrite regions. Once the magnetic field increases to 20 T, the FCC-Co/Co2B anomalous eutectic colonies dominate. The formation mechanism of Co2B phase is discussed from several aspects of the competitive nucleation, the chemical redistribution induced by the thermomagnetic-induced convection and magnetic dipole interaction, and the strain-induced transformation. Furthermore, the application of magnetic field is found to promote recrystallization, proved by the lower density of misorientation, the appearance of FCC-Co annealed twins and more Co3B sub-grains. This work could further enrich our knowledge about the magnetic-dependent microstructure evolution and recrystallization process in the undercooled Co-B system and provide guidance for controlling the microstructures and properties under extreme conditions.
Full-text available
Crystal–crystal (c–c) transitions are often induced by large shear deformation, but the microscopic kinetics associated with c–c transitions are difficult to experimentally observe and are poorly understood theoretically. Here, we drive shear-induced phase transitions from square ($\square$) lattices to triangular ($\triangle$) lattices and directly observe the accompanying kinetics with single-particle resolution inside the bulk crystal. For relatively small oscillatory shear strain amplitude, γ m < 0.1, the initial nucleation is martensitic, and the late-stage growth exhibits shear-coupled interface propagation. By contrast, for large shear strain, 0.1 < γ m < 0.4, liquid nuclei form first, and then $\triangle$-lattice nuclei crystallize within liquid nuclei; $\triangle$-lattice nuclei are surrounded by a liquid layer throughout their growth due to localised shear strain at the interface; large localized strain enables the Lindemann melting criterion to be reached locally and drives virtual melting. Such virtual melting and interface propagation induced shear phenomena have been predicted in theory and simulation, but have not been observed in experiment.
Multimetallic layered composites (MMLCs) have shown an excellent potential for application under extreme environments, e.g., accident-tolerant fuel cladding, because of their low oxidation tendency and high corrosion resistance. Interfacial phases or complexions in nanocrystalline materials accelerate the annihilation of defects and enhance the radiation resistance of materials, making MMLCs with engineered interlayer phases compelling to deploy in extreme conditions. However, implementation of MMLCs in full capacity remained a challenge due to a lack of fundamental understanding of the underlying mechanisms governing the characteristics of the interface between the metallic layers. The precise role of interlayer phases in MMLCs and their interaction with defects, specifically under extreme conditions, is still unexplored. Pursuing atomistic simulations for various Inconel-Ni MMLCs model materials, we revealed accelerated defect mobility in interlayers with larger crystalline misorientation and the inverse relationship between the interface sink strength to the misorientation angle. Furthermore, we found a linear relation between interlayer misorientation angle with the density of radiation-induced defects and radiation enhanced displacements. Finally, our results indicate that radiation-induced material degradation is accelerated by the higher defect formation tendency of MMLCs with a high-angle interlayer interface. Data availability All data that was obtained during this project is available from the authors.
In solid-state diffusional phase transformations involving nucleation and growth, the size of the critical nucleus for a homogeneous process (rhomo*=r∗) has been assumed to be time invariant (constant). The strain associated with the process leads to an increase in the value of r∗, with respect to that for nucleation from a liquid. The strain energy stored in the matrix increases with transformation and the nuclei forming at a later stage encounter a strained matrix. Using devitrification of a bulk metallic glass as a model system, we demonstrate that r* is not a cardinal time invariant constant for homogeneous nucleation and can increase or decrease depending on the strain energy penalty. We show that the assumption regarding the constancy of r∗ is true only in the early stages and establish that with progress of the transformation leads to an altered magnitude of r*, which is a function of the microstructural details, geometrical variables and physical parameters. Using, HRLFI and computations, we argue that, 'liquid-like' homogeneous nucleation can occur and that the conclusions are applicable to a broad set of solid-state diffusional transformations. The above effect 'opens up' a lower barrier transformation pathway arising purely from the internal variables of the system.
In this work, molecular dynamics simulations have been used to undertake a computational study of the equilibrium crystal-melt interface stresses in face-centered-cubic (FCC) Ni and body-centered-cubic (BCC) Fe, BCC Nb, and a model BCC soft-sphere elemental system, for three different interface orientations, i.e., (100), (110), and (111). The sign, magnitude, and anisotropy of the excess interface stresses and their relationships with the corresponding interfacial free energies have been examined. The universality of a few trends regarding the interfacial stresses observed in FCC crystal-melt interfaces has been assessed for the BCC crystal-melt interfaces. The role of the interatomic bonding that affects the shape of the interfacial stress profiles, thus modulating the magnitude or sign of the excess interface stress, has been discussed through inspecting a particular type of crystal-melt interface over different materials. Besides, for the first time, we have demonstrated that the Irving-Kirkwood fine-grained algorithm for depicting microscopic pressure components and stresses in the vicinity of the crystal-melt interface is superior to the previously used per-particle virial stress algorithm. The reported data and new knowledgqe could enrich the accumulation for theory breakthroughs in predicting interface stresses and motivate future studies on the interfacial stresses for more types of solid-liquid interfaces.
We developed a combined finite element and CALPHAD based model of the Laser Powder Bed Fusion (LPBF) process for AA7075 alloy that considers the effect of feedstock composition and print parameters. A single-pass of a laser on a layer of AA7075 alloy powder has been considered. Sensitivity of temperature evolution and melt pool geometry to variation in the stoichiometry of the feedstock powder and laser source characteristics have been studied. Our results indicate that deviation (up to 10%) of the feedstock composition from the AA7075 raises the maximum temperature and increases melt pool size. Excess Cu content shows the largest melt pool width and depth among all the cases. The peak temperature is higher than the standard feedstock composition in all cases, except when the Cu concentration is reduced. Increasing the scan power also results in a higher peak temperature and a larger melt pool size. Furthermore, the temperature's rise time increases by lowering the scan speed.
We developed a coupled CALPHAD and finite element-based computational model of the Laser Powder Bed Fusion (LPBF) process for HAYNES230, considering the feedstock composition and packing density. We further used this model to investigate the effect of variation in feedstock composition and print parameters on the quality of the final printed part. Sensitivity of the maximum reached temperature to variations in characteristics of the laser source is also studied considering a single-track laser scan on a layer of metal powder. We analyzed temperature evolution in the powder bed and melt pool geometry along the path of the laser. Our results indicate that the LPBF process of HAYNES230 alloy requires a powder layer thickness of ∼20μm and laser spot size ∼30μm radius compared to other alloys. It is essential to achieve sufficient melt pool depth necessary for cohesion with the substrate while avoiding large melt pool width that adversely affects the formation of cracks and residual stresses. We also revealed that reducing the laser power or increasing scan speed drastically reduces peak temperature while less susceptible to solute composition.
Full-text available
The microscopic kinetics of ubiquitous solid-solid phase transitions remain poorly understood. Here, by using single-particle-resolution video microscopy of colloidal films of diameter-tunable microspheres, we show that transitions between square and triangular lattices occur via a two-step diffusive nucleation pathway involving liquid nuclei. The nucleation pathway is favoured over the direct one-step nucleation because the energy of the solid/liquid interface is lower than that between solid phases. We also observed that nucleation precursors are particle-swapping loops rather than newly generated structural defects, and that coherent and incoherent facets of the evolving nuclei exhibit different energies and growth rates that can markedly alter the nucleation kinetics. Our findings suggest that an intermediate liquid should exist in the nucleation processes of solid-solid transitions of most metals and alloys, and provide guidance for better control of the kinetics of the transition and for future refinements of solid-solid transition theory.
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Previously unknown phenomena, scale, and kinetic effects are revealed by introducing the finite width Δξ of the particle-exterior interface as the additional scale parameter and thermally activated melting in the phase field approach. In addition to traditional continuous barrierless premelting and melting for Δξ= 0, barrierless hysteretic jumplike premelting (melting) and thermally activated premelting (melting) via critical nucleus are revealed. A very rich temperature θ-Δξ transformation diagram is found, which includes various barrierless and thermally activated transformations between solid, melt, and surface melt, and complex hysteretic behavior under various temperature and Δξ trajectories. Bistable states (i.e., spontaneous thermally activated switching between two states) between solid and melt or surface melt are found for Al particles. Strong dependence of the melting temperature (which, in contrast to previous approaches, is defined for thermally activated premelting and melting) for nanoparticles of various radii on Δξ is found. Results are in good agreement with experiments for Al for Δξ=0.8-1.2nm. They open an unexplored direction of controlling surface melting and melting or solidification by controlling the width of the external surface and utilizing predicted phenomena. They also can be expanded for other phase transformations (e.g., amorphization, solid-solid diffusionless, diffusive, and electromagnetic transformations) and phenomena, imbedded particles, and mechanical effects.
Full-text available
Solid–solid (SS)(SS) phase transformations via nanometer-size intermediate melts (IMs)(IMs) within the SS interface, hundreds of degrees below melting temperature, were predicted thermodynamically and are consistent with experiments for various materials. A necessary condition for the appearance of IMs, using a sharp interface approach, was that the ratio of the energies of SS and solid–melt (SM)(SM) interfaces, kEkE, were >2. Here, an advanced phase-field approach coupled with mechanics is developed that reveals various new scale and interaction effects and phenomena. Various types of IM are found: (i) continuous and reversible premelting and melting; (ii) jump-like barrierless transformation to IMs, which can be kept at much lower temperature even for kE<2kE<2; (iii) unstable IMs, i.e. a critical nucleus between the SS interface and the IM. A surprising scale effect related to the ratio of widths of SS and SM interfaces is found: it suppresses barrierless IMs but allows IMs to be kept at much lower temperatures even for kE<2kE<2. Relaxation of elastic stresses strongly promotes IMs, which can appear even at kE<2kE<2 and be retained at kE=1kE=1. The theory developed here can be tailored for diffusive phase transformations, formation of intergranular and interfacial phases, and surface-induced phase transformations.
Full-text available
An advanced Ginzburg-Landau (GL) approach to melting and solidification coupled with mechanics is developed. It is based on the concept of a coherent solid-liquid interface with a transformation strain tensor, the deviatoric part of which is described by a thermodynamically consistent kinetic equation. Due to the relaxation of the elastic energy, a promoting contribution to the driving force for phase transformation in the GL equation appears, both for melting and solidification. Good agreement with known experiments is obtained for Al nanoparticles for the size-dependent melting temperature and temperature-dependent thickness of the surface molten layer. All types of interface stress distributions from known molecular dynamics simulations are obtained and interpreted. A similar approach can be applied for sublimation and condensation, amorphization and vitrification, diffusive transformations, and chemical reactions.
An advanced three-phase phase-�eld approach (PFA) is suggested for a non-equilibrium phase interface which contains an intermediate phase, in particular, a solid-solid interface with a nanometersized intermediate melt (IM). Thermodynamic potential in the polar order parameters is developed, which satis�es all thermodynamic equilibrium and stability conditions. Special form of the gradient energy allowed us to include the interaction of two solid-melt interfaces via intermediate melt and obtain a well-posed problem and mesh-independent solutions. It is proved that for stationary 1D solutions to two Ginzburg-Landau equations for three phases, the local energy at each point is equal to the gradient energy. Simulations are performed for � $ � phase transformations (PTs) via IM in HMX energetic material. Obtained energy - IM width dependence is described by generalized force-balance models for short- and long-range interaction forces between interfaces but not far from the melting temperature. New force-balance model is developed, which describes phase �eld results even 100K below the melting temperature. The e�ects of the ratios of width and energies of solid-solid and solid-melt interfaces, temperature, and the parameter characterizing interaction of two solid-melt interfaces, on the structure, width, energy of the IM and interface velocity are determined by �nite element method. Depending on parameters, the IM may appear by continuous or discontinuous barrierless disordering or via critical nucleus due to thermal uctuations. The IM may appear during heating and persist during cooling at temperatures well below than it follows from sharp-interface approach. On the other hand, for some parameters when IM is expected, it does not form, producing an IM-free gap. The developed PFA represents a quite general three-phase model and can be extended to other physical phenomena, such as martensitic PTs, surface-induced premelting and PTs, premelting/disordering at grain boundaries, and developing corresponding interfacial phase diagrams.
The transformation kinetics of the β-γ solid state phase transition in the organic nitramine molecule octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX) was discussed using second harmonic generation. The quantitative measurement of the γ phase mole fraction in ensembles of free HMX crystals and crystals embedded in a visco-elastic polymer matrix was discussed. The analysis showed difference in nucleation kinetics between samples of free crystals and crystals embedded in a visco-elastic polymer matrix.
Recent observations of three classes of nanometer-thick, disordered, interfacial films in multicomponent inorganic materials are reviewed and critically assessed. The three classes of films are equilibrium-thickness intergranular films (IGFs) in ceramics, their free-surface counterparts, that is, surficial amorphous films (SAFs), and their metallic counterparts. Also briefly reviewed are several related wetting and adsorption phenomena in simpler systems, including premelting in unary systems, prewetting in binary liquids or vapor adsorption on inert walls, and frustrated-complete wetting. Analogous diffuse-interface and force-balance models are discussed with the goal of exploring a unifying thermodynamic framework. In general, the stability of these nanometer-thick interfacial films does not follow bulk phase diagrams. Stabilization of quasi-liquid interfacial films at subeutectic or undersaturation conditions in multicomponent materials can be understood from coupled interfacial premelting and prewetting transitions. More realistic models should include additional interfacial interactions, for example, dispersion and electrostatic forces, and consider the possibility for metastable equilibration. It is suggested that quasi-liquid grain boundary films in binary metallic systems can be used to validate a basic thermodynamic model. These nanoscale interfacial films are technologically important. For example, the short-circuit diffusion that occurs in interface-stabilized, subeutectic, quasi-liquid films explains the long-standing mystery of the solid-state activated sintering mechanism in ceramics, refractory metals, and ice.
This paper, in line with the previous works (Javili and Steinmann, 2009, 2010), is concerned with the thermomechanically consistent theory and formulation of boundary potential energies and the study of their impact on the deformations of solids. Thereby, the main thrust in this contribution is the extension to thermomechanical effects. Although boundary effects can play a dominant role in the material behavior, the common modelling in continuum mechanics takes exclusively the bulk into account, nevertheless, neglecting possible contributions from the boundary. In this approach the boundary is equipped with its own thermodynamic life, i.e. we assume separate boundary energy, entropy and the like. Afterwards, the derivations of generalized balance equations, including boundary potentials, completely based on a tensorial representation is carried out. The formulation is exemplified for the example of thermohyperelasticity.