Phase field simulation of kinetic superheating and melting of aluminumnanolayer irradiated by pico- and femtosecond laser

Yong Seok Hwang1,a) and Valery I. Levitas2,b)

1Department of Aerospace Engineering, Iowa State University, Ames, Iowa 50011, USA2Departments of Aerospace Engineering, Mechanical Engineering, and Material Science and Engineering,Iowa State University, Ames, Iowa 50011, USA

(Received 27 November 2013; accepted 7 December 2013; published online 26 December 2013)

Two melting mechanisms are reproduced and quantified for superheating and melting of Al

nanolayer irradiated by pico- and femtosecond laser using the advanced phase-field approach coupled

with mechanics and a two-temperature model. At heating rates Q � 79:04 K/ps induced by

picosecond laser, two-sided barrierless surface melting forms two solid-melt interfaces, which meet

near the center of a sample. The temperature for surface melting is a linear function, and for

complete melting it is a cubic function, of logQ. At Q � 300 K/ps induced by femtosecond laser,

barrierless and homogeneous melting (without nucleation) at the sample center occurs faster than due

to interface propagation. Good agreement with experimental melting time was achieved in a range of

0:95 � Q � 1290 K=ps without fitting of material parameters. VC 2013 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4858395]

The limits of superheating of solids and melting mecha-

nisms at very high heating rates are fundamental problems

with numerous applications, which are under intense

study.1–8 Experimentally, melting induced by irradiation of

the ultra-fast laser has been researched for the last several

decades and has been widely used for industrial manufactur-

ing processes.9 Various catastrophes (isochoric, isenthalphic,

isentropic) and Lindemann and shear instabilities10–12 have

been suggested and explored to predict the superheating

limit. Many researchers have described the limit of kinetic

superheating with homogeneous nucleation theory,1–4,13 but

at higher heating rates the homogeneous nucleation theory is

not in good agreement with experiments (see Ref. 8 and

below). Also, since the temperature may exceed the lattice

instability temperature Ti, the density-functional and phase-

field approach (PFA) result in zero energy of the critical

nucleus,14–16 and homogeneous nucleation theory based on

sharp interface is conceptually unacceptable; thus, barrierless

melting should be considered. For this case, solid phase does

not possess a local energy minimum and represents a transi-

tional state that melts barrierlessly at the ps time scale. At the

same time, for nanoparticles, surface premelting and melting

are observed below the melting temperature at slow heating,

and the melting temperature is reduced with the particle

size.17–19 For fast heating of nanoparticles, premelting and

superheating compete,6,20–22 and consequently barrierless

surface melting may contribute to the mechanism of super-

heating.23 Recent molecular dynamics (MD) simulations24

have revealed two mechanisms, surface-induced melting and

propagation of solid-melt interfaces (for slower heating) and

homogeneous melting (for faster heating), which compete for

the intermediate heating rates. However, MD simulations

have well-known time and space limitations. It is known that

PFA to melting16,21,22 can be applied for larger size and time

scales, which makes it favorable for practical applications.

However, we are not aware of any applications of PFA to

laser-induced melting. Mechanical strains and stresses have

also been found to be important.5,21,22,25

The goal of this work is to develop a simple PFA to ki-

netic superheating and melting under irradiation by ps and fs

lasers and verify it by comparison with known experiments

and MD simulations. For this purpose, we used an advanced

PFA to melting coupled to mechanics21,22 and combined it

with the two-temperature model (TTM).26 Our PFA21,22

described well the surface-induced melting (namely, the

width of the surface molten layer vs. temperature) and the

melting temperature of nanoparticles vs. particle radius, both

for very slow heating of Al samples. Information about the

kinetics of melting is included in terms of a single kinetic pa-

rameter v in the Ginzburg-Landau equation, which is related

to interface mobility. In the current work, we used v ¼532 m2=Ns justified by MD simulation for Al for small over-

heating27 and did not change any terms or other parameters

in the PFA model.21,22 Still, we obtained a good agreement

with known experiments8,28 in terms of the time for com-

plete melting for the heating rates Q from 1 to 1290 K/ps.

This means that in the first approximation the information

about melting kinetics at high superheating is present in the

model for slow heating. Two major melting mechanisms

were reproduced, in an agreement with MD results.24 In all

cases, surface premelting and melting represent the initial

stage of the process. At Q � 79:04 K/ps, two-sided barrier-

less surface melting forms two solid-melt interfaces, which

propagate toward each other and meet near the center of a

sample; melting time and, consequently, superheating

increase with the sample width. The temperature for com-

plete melting has a cubic relation on logQ, while the surface

melting temperature is a linear function of logQ and for

given Q is independent of fluence, pulse duration, and sam-

ple size. At Q � 300 K/ps, barrierless homogeneous melting

(without nucleation) at sample center occurs faster than due

to interface propagation. Melting time and, consequently,

superheating, are independent of the sample width in this

a)Electronic mail: yshwang@iastate.edub)Electronic mail: vlevitas@iastate.edu

0003-6951/2013/103(26)/263107/4/$30.00 VC 2013 AIP Publishing LLC103, 263107-1

APPLIED PHYSICS LETTERS 103, 263107 (2013)

regime and depend on heating rate only. They represent the

upper bounds for melting time and superheating temperature.

Model: The complete system of equations for PFA to

melting coupled with mechanics21,22 and combined with

TTM,26 as well as boundary conditions and all material pa-

rameters, are presented in supplementary material.27 The

order parameter g varies from 1 for solid to zero for melt.

Interface stresses are taken into account. In addition to the

equilibrium bulk-melting temperature Teq¼ 933.67 K, the

lattice instability temperature Ti¼ 1.2Teq¼ 1120.4 K is an

important material parameter. An infinite layer of the width

w is subjected to penetrating laser fluence orthogonal to the

right surface, which is modeled as a distributed heat source

with thermal isolation at surfaces.

Boundary conditions for the order parameter take into

account reduction in the surface energy of the external sur-

face during melting. TTM assumes that the irradiated energy

of the laser is first absorbed by electron gas near the metal

surface, then at the fs scale energy is distributed among elec-

tron gas in the entire sample by collision of electrons, and

finally the atomic lattice is heated by electron-phonon

coupling.26

No fluctuations were included—i.e., barrierless melting

is considered. Plane boundaries were stress-free, and one of

the boundaries was fixed to prevent translation and rotation

of a sample. Because the radius of the irradiated spot was

much larger than w, mm vs. nm, the problem could be safely

considered as a 1-D problem with zero displacements or-

thogonal to the laser axis, with all parameters varying along

the laser axis only.

The initial temperature was T0¼ 293.15 K and g ¼0:999 for all cases. The finite-element code COMSOL

Multiphysics was used for the simulations.29

Picosecond laser irradiation: The melting of a 25-nm

thin Al nanolayer irradiated by an ultrafast laser was simu-

lated to replicate the melting time measured in the experi-

ment.28 Conditions and results are shown in Table I. In each

simulation, the melting of a sample starts from the surface

because of the reduction in surface energy during melting.

The surface melting temperature Tsm was defined as the tem-

perature at which the order parameter first reached 0.5.

Despite the promoting effect of the surface, for fast heating

the surface melting occurred above Teq. A solid-melt inter-

face propagated from each surface toward each other, and

met at the melting center. This mechanism represents hetero-

geneous melting.

The melting center is shifted to the left from the sample

center because of the one-sided heating and heterogeneous

temperature (Fig. 1(a)). When the temperature exceeds Ti,

barrierless melting starts at each point of the solid for any

initial deviation from g ¼ 1. It is called a homogeneous

melting because it does not require interfaces. It has nothing

to do with homogeneous nucleation, which does require

interface and thermal fluctuations. The heating rate at each

point is practically constant during laser irradiation and can

be defined, for example, as Q ¼ ðTsm � T0Þ=tsm, where tsm is

the time of surface melting. For Q � 79:04 K=ps, homogene-

ous melting is negligible before the two interfaces meet,

even for the highest fluence of ps laser irradiation (Fig. 1(b)).

The time for complete melting, tm, is defined as time from

the moment of laser irradiation to the instant when the two

TABLE I. Summary of simulation results.

Fluence (mJ/cm2) texpm ðpsÞa tm (ps) Q (K/ps) Tsm (K) Ts (K) hc

7 1000 789.8 0.95 942.9 1035.5 1.11

8 350 296.8 3.03 1055.5 1187.0 1.27

9 180 161.5 6.51 1158.5 1344.5 1.44

10 115 106.4 11.21 1234.4 1491.0 1.60

11 60 63.6 23.15 1397.5 1703.0 1.82

14.67b … 63.5 23.15 1397.5 1784.8 1.91

13 20 27.7 79.04 1573.6 1921.4 2.06

19.5b … 26.8 79.04 1573.6 2524.7 2.70

aPulse duration is same as texpm in the experiment.

bAdditional simulation cases to attain maximum superheating temperature at

the same heating rate; an 80 ps pulse for 14.67 mJ/cm2 and an 30 ps pulse

for 19.5 mJ/cm2 were used.

FIG. 1. Distribution of the order parameter for some time instants for

absorbed fluence of 11 mJ/cm2 (a), 13 mJ/cm2 of ps laser (b), and for irradi-

ated fluence of 70 mJ/cm2 of fs laser (c).

263107-2 Y. S. Hwang and V. I. Levitas Appl. Phys. Lett. 103, 263107 (2013)

interfaces meet each other, and the interface position is

defined by g ¼ 0:5. Table I shows good agreement between

calculated and experimentally determined tm within the

reported range of experimental error.

The maximum superheating temperature, Ts, is defined

as the temperature reached for the given heating rate during

the time for complete melting tm; i.e., Ts � Qtm þ T0.

Because irradiation for the two strongest fluences, 11 and 13

mJ/cm2, was ended before the melting was completed, Ts

was not reached in these simulations. To find Ts, we added

two more cases with fluences of 14.67 mJ/cm2 during 80 ps

and 19.5 mJ/cm2 during 30 ps, which have a longer duration

of irradiation but same heating rates, 23.15 and 79.04 K/ps.

As shown in Table I, much higher superheating temperature,

1784.8 and 2524.7 K, respectively, reached, while Tsm did

not change. The maximum superheating temperatures for

heterogeneous melting, Ts, is well described by a cubic

polynomial (R2¼ 0.9999), TsðKÞ ¼ 1041:54þ 252:29logQþ42:08ðlogQÞ2 þ 124:83ðlogQÞ3.

Our results for the melting time, while in agreement

with the experiments,28 differ significantly from the previous

models based on homogeneous nucleation theory. For

example, in Refs. 2 and 3, the following h� b� Q model is

suggested b ¼ ðA0 � b log10QÞhcðhc � 1Þ2, where b is a con-

stant that depends on the material, A0 and b are constants,

and hc ¼ Ts=Teq. In our study of heterogeneous melting,

maximum superheating for any heating rate depends not

only on the material but also on w because the interface ve-

locity and w determine the melting time, which controls the

maximum superheating. The homogeneous nucleation

model1 predicted tm¼ 1 ps for hc ¼ 1:31, but our prediction

was 296.8 ps for the similar superheating hc ¼ 1:27. The ho-

mogeneous nucleation model4 predicted tm¼ 10 ps for hc ¼1:936 in contrast to 63.5 ps for hc ¼ 1:91 in our simulations;

also tm¼ 10 ns for hc ¼ 1:33 vs. 296.8 ps for hc ¼ 1:27 in

our model. These contradictions confirm the inapplicability

of the homogeneous nucleation model above lattice instabil-

ity temperature for very high heating rates. Even MD simula-

tions,3 which did not consider surface melting, showed

hc ¼ 1:19 superheating for 5 K/ps, which should be between

hc ¼ 1:33 and hc ¼ 1:54 for this heating rate in our simula-

tions and experiment.

The surface melting temperature, Tsm, depends linearly

on logQ ðTsm ¼ 912:08þ 338:50 logQÞ and for given Q it is

practically independent of fluence, the duration of the laser

pulse (Table I), and sample size. The maximum compressive

pressure up to 4.5 GPa was observed for the strongest laser

fluence. It increases maximum superheating temperature by

305 K (Fig. 2), instead of the increase in equilibrium melting

temperature by 270 K.

Femtosecond laser heating: A 20-nm thin Al layer8,30

was irradiated with the fluence 70 mJ/cm2 during 120-fs

pulse duration. The simulated heating rate was not constant,

with 300 K/ps in average, and maximum value of

1360 K/ps. Surface melting started at 4.1 ps, but the homo-

geneous melting became the dominant mechanism after 5.7

ps until homogeneous melting in the central part completed

before the two interfaces meet each other (Fig. 1(c)). For

79:04 K=ps � Q � 300:0 K=ps, the mechanism changes

from heterogeneous to homogeneous melting as shown in

Fig. 1. A non-uniform distribution of temperature makes

homogeneous melting asymmetric, being faster at the right

side. The time and superheating for homogeneous melting

for the prescribed heating rates are independent of the sam-

ple size.

Fig. 3 displays variation of the temperature and the func-

tion / gð Þ ¼ g2 3� 2gð Þ of the order parameter, which

describes change of any material property during melting27

in the middle plane of a sample. Superheating up to 1680 K

ðhc ¼ 1:80Þ was observed. In experiment,8,30 time for initia-

tion and completion of melting is determined by measuring

diffraction peaks intensity averaged over the sample thick-

ness. We will use for this purpose the averaged parameter�/ ¼

Ð/dx=w, defined in the reference configuration and

assume that melting starts at �/ ¼ 0:9 and completes at�/ ¼ 0:1. In the experiment,8 the time between detected dis-

ordering and complete melting was 2 ps. Calculated time for

complete melting is thm¼ 2.9 ps (for neglected mechanics, it

is 1.9 ps), which is in good agreement with experiment.

FIG. 2. Superheating temperature vs. heating rate at the melting center.

Rectangular solid symbols represent simulated data in the current research

for the experiment,28 triangles are the estimated maximum superheating

temperature Ts, and the red line is the fit to the maximum superheating tem-

perature Ts. The circular symbols are the superheating temperatures for the

stress-free case, and the rectangular hollow symbols represent the surface

melting temperature Tsm.

FIG. 3. Time evolution of the function / gð Þ ¼ g2 3� 2gð Þ of the order pa-

rameter and lattice temperature at a middle plane of sample.

263107-3 Y. S. Hwang and V. I. Levitas Appl. Phys. Lett. 103, 263107 (2013)

However, the time from the beginning of irradiation to initia-

tion of melting was 1.5 ps in Ref. 8 and 5.2 ps here (for

neglected mechanics, it is 4.6 ps). Note that simulation temper-

ature before melting at 1.5 ps is 1400 K, exactly the same as

temperature in the experiment.30 Therefore, one has to intro-

duce thermal fluctuations in g in order to describe shorter time

before initiation of melting. Our preliminary simulations con-

firmed this conclusion. We would like to keep the current

model as simple as possible and to include thermal fluctuations

along with thermomechanical coupling, heat of fusion, dynam-

ics and wave propagation,31 and stress relaxation via disloca-

tions32 in the future work. Even such a simplified model

describes experiments much better than the previous continuum

models. In particular, the melting time for the homogeneous

nucleation model1 is more than an order of magnitude shorter

than in experiment8 (which was mentioned in Ref. 8), with

lower superheating of hc ¼ 1:37 rather than hc ¼ 1:80 here.

The simulated compressive pressure reached 3.0 GPa. In

Fig. 3, the onset of melting in the stress-free simulations

starts 0.6 ps earlier than in the case with stresses, and

thm¼ 1.9 ps is shorter than for the stressed case. The temper-

ature for onset of homogeneous melting is practically the

same as the final superheating temperature, 1680 K, regard-

less of stressed or stress-free conditions.

In summary, a good correspondence between the simu-

lated and experimental melting time was obtained for laser

heating of the Al nanolayer for the heating rates from 0.95 to

1290 K/ps using PFA coupled with mechanics and TTM. This

did not require modification of the PFA in comparison with

the slow-heating regimes. We reproduced and quantified the

two main mechanisms found in MD simulation,24 namely (a)

heterogeneous melting initiated from surface melting at both

surfaces and propagation of two interfaces until they meet and

(b) homogeneous melting without interfaces above Ti. These

mechanisms substituted for the traditional homogeneous nucle-

ation mechanism, which is not applicable here because T > Ti.

Note that homogeneous melting under shock loading was also

obtained in MD simulations.33 The same approach can be

applied, e.g., for laser ignition of nano- and micron-scale Al

particle34,35 and nano structuring of thin metal film.36,37

Support from ONR, NSF, Agency for Defense

Development and Gyeongsang National University (both

South Korea), and ISU is gratefully acknowledged.

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263107-4 Y. S. Hwang and V. I. Levitas Appl. Phys. Lett. 103, 263107 (2013)

Phase field simulation of kinetic superheating and melting of

aluminum nanolayer irradiated by pico- and femtosecond laser

Yong Seok Hwang∗

Department of Aerospace Engineering,

Iowa State University, Ames, Iowa 50011, U.S.A.

Valery I. Levitas†

Departments of Aerospace Engineering,

Mechanical Engineering, and Material Science and Engineering,

Iowa State University, Ames, Iowa 50011, U.S.A.

(Dated: November 26, 2013)

Supplementary material

1 Governing Equations

1.1. Phase Field Equations

The complete system consists of the following equations [1]:

Kinematics

Relationship between the displacement vector u and the strain tensor ε with respect to

undeformed solid

ε = ∇0u, (1)

where ∇0 is the gradient operator in the undeformed state (reference configuration).

Additive decomposition of strain tensor into elastic εe, transformation εt, and thermal

εθ strains

ε = εe + εt + εθ; ε = 1/3ε0I + e; (2)

εt = 1/3ε0t (1− φ (η)) I; εθ = αs (Teq − T0) I + (αm + ∆αφ (η)) (T − Teq) I, (3)

∗ yshwang@iastate.edu† vlevitas@iastate.edu

2

where η is the order parameter that varies from 1 in solid to 0 in melt, αs and αm are the

linear thermal expansion coefficient for solid and melt respectively, ∆α is difference in the

thermal expansion coefficients of solid and melt, I is the unit tensor, T is the temperature,

T0 is the initial temperature, Teq is the solid-melt phase equilibrium temperature, ε0 is the

total volumetric strain, ε0t is the volumetric strain for complete melting, e is the deviatoric

strain, and φ (η) = η2 (3− 2η).

Helmholtz free energy per unit undeformed volume of solid

ψ = ψe + Jψθ + ψθ + Jψ∇; (4)

J = ρ0/ρ = 1 + ε0; (5)

ψe = 0.5{Km + ∆Kφ(η)}ε20e + µφ(η)ee : ee; (6)

ψθ = H(T/Teq − 1)φ(η), ψθ = Aη2(1− η)2; (7)

ψ∇ = 0.5β|∇η|2, A := 3H(1− Tc/Teq). (8)

Here, ψe is elastic energy, ψθ is the thermal energy, ψθ is the double-well energy, ψ∇ is the

gradient energy; ρ0 and ρ are the mass densities in the undeformed and deformed states,

respectively, K and µ are the bulk and shear moduli, β is the gradient energy coefficient,

H is the heat of fusion, ∇ is the gradient operator in the current (deformed) state, and Tc

is the melt instability temperature, which is assumed to be 0.8Teq.

Decomposition of the stress tensor σ into elastic stress σe and surface tension

at interfaces σst

σ =∂ψ

∂ε− J−1∇η ⊗ ∂ψ

∂∇η= σe + σst; (9)

σe = {Km + ∆Kφ(η)}ε0eI + 2µφ(η)ee; σst = (ψ∇ + ψθ)I − β∇η ⊗∇η, (10)

where ⊗ designates dyadic product of vectors.

3

Ginzburg-Landau equation for the order parameter

1

χη = −J−1∂ψ

∂η

∣∣∣ε

+ ∇ ·(J−1

∂ψ

∂∇η

)= J−1{−ε0tpe + 3pe∆α(T − Teq)}

∂φ

∂η(11)

− J−1{0.5∆Kε20e + µee : ee +H

(T

Teq− 1

)}∂φ∂η

− 4Aη(1− η)(0.5− η) + β∇2η,

where χ is kinetic coefficient and pe = σe : I/3 is the mean elastic stress.

1.2. Equilibrium equation

∇ · σ = 0. (12)

1.3. Boundary conditions for the order parameter η

J∂ψ

∂∇η· n = β∇η · n = −dγ

dη, γ(η) = γm + (γs − γm)φ(η), (13)

where γs and γm are the solid-vapor and melt-vapor surface energies, n is the unit normal

to the external surface.

2 Two Temperature Model (TTM)

Two temperature model assumes that irradiated energy of laser is first absorbed by

electron gas near the metal surface, then at the fs scale energy is distributed among electron

gas in the entire sample by collision of electrons, and finally the atomic lattice is heated

by electron-phonon coupling [2]. The heating process can be described by the following

equations

Cl∂T

∂t=

∂

∂x

(κl∂Te∂x

)+G (Te − T ) ; (14)

Ce∂Te∂t

=∂

∂x

(κe∂Te∂x

)+ I −G (Te − T ) , (15)

where T and Te are the temperature of lattice and electron gas, respectively, Cl is the heat

capacity of lattice,

Cl = Clm + (Cls − Clm)φ(η),

4

where Clm and Cls are heat capacity of melt and solid, respectively;

Cls = [2434.86 + (3308.87− 2434.86)/(900.0− 300.0)× (T − 300.0)] × 103 J/m3K for

T < 900.0K [3];

Cls = 3308.87× 103 J/m3K for T > 900.0K;

Clm = [2789.1 + (2713.72− 2789.1)/(1173.0− 933.0)× (T − 933.0)]× 103 J/m3K [3];

Ce = γTe is the heat capacity of electron gas, γ is the electron heat capacity constant,

91.2 J/m3K2 [4] , G is the electron-phonon coupling coefficient, κl and κe are the thermal

conductivity of lattice and electron gas respectively, and I is the laser power absorbed by

the electrons. For the electron-phonon coupling coefficient G, the following theoretically

calculated data, which include temperature variation and are supported by experiments,

have been presented in the form of curve in Ref. [4] and table in Ref.[? ]. They are

approximated by the following equation

G = (3.663 + (2.445− 3.663)/(1 + (Te × 10−4/0.221)2.294))× 1017 J/(m3K).

The electron thermal conductivity was approximated as

κe = κe,eqTe/T

to take into account the non-equilibrium effect [5, 6];

κe,eq = κem,eq + φ(η)(κes,eq − κem,eq),

where κes,eq = 208W/(mK) for solid and κem,eq = 102W/(mK) for melt[3];

The lattice thermal conductivity is κl = 0.01κe [7].

Attenuation of an irradiated laser was modeled using the Beer-Lambart law8:

I = I0exp(−ζ(w − x));

I0 = ζW/(1− exp(−ζw)); (16)

W = (1−R)F0/tp,

where ζ is the absorption coefficient, which is 1.21 × 108 m−1 for the 1064-nm laser in

the picosecond experiment and 1.4616 × 108 m−1 for the 700-nm laser in the femtosecond

experiment; w is thickness of a sample, which is 25 nm for the picosecond experiment and

20 nm for the femtosecond experiment; R is the reflectance, which is 0.87 for fs experiment9

and 0 for ps experiment since absorbed fluence was reported10; tp is the pulse duration; and

F0 is the fluence of laser.

5

FIG. 1: Schematics of computational sample distribution of irradiated energy.

3 Boundary and initial conditions and schematics of computation

Figure 1 shows the schematics of computational domain. The plane boundaries were

stress-free, and one of the boundaries was fixed to prevent translation and rotation of the

sample. Because the radius of the irradiated spot was much larger than w, mm vs. ns, the

problem could be safely considered to be a 1-D problem with zero displacements orthogonal

to the laser axis; all parameters vary along the laser axis only. The initial temperature was

T0 = 293.15K and initial η = 0.999. For the Ginzburg-Landau equation, the boundary

conditions at the left and right planes were given by Eq.(13). As for TTM, the laser was

irradiated at the right boundary only. Energy irradiated by the laser was included as a

volumetric heat source as described by Eq.(16), see Fig. 1. The heat flux was zero at both

plane boundaries.

4 Material parameters for simulation of aluminum

Coefficients, constants, and properties used for simulation are presented in Table I.

6

TABLE I: Properties of aluminum11

Teq (K) H(J/m3)

Km

(GPa)Ks

(GPa)µ (GPa) ε0t αm

(K−1)αs

(K−1)γs(J/m2)

γl(J/m2)

β(N) χ(m2/Ns)

933.67 933.57×106

41.3 71.1 27.3 0.06 4.268×10−5

3.032×10−5

1.050 0.921 3.21×10−10

532

To determine kinetic coefficient χ, we utilize an analytical solution for interface velocity,

c, in Ref. (it is better to use Levitas V.I., Lee D.-W. and Preston D.L. International J.

Plasticity, 2010, Vol. 26, No. 3, 395-422. ) 13

c =6χρ0∆G

T (T )

k; k =

√6ρ0H (Teq − Tc)

βTeq; ∆G(T ) =

H

Teq(T − Teq) . (17)

Then the interface mobility is

µ =dc

dT=

√6ρ0Hβ

Teq(Teq − Tc)χ (18)

Substituting µ = 1.7 m/(sK) obtained with MD simulation in12, we obtain χ = 532 m2/Ns.

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