superheating and melting within aluminum core–oxide shell ... · and solid–melt interface...

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2016, 18, 28835

Superheating and melting within aluminumcore–oxide shell nanoparticles for a broadrange of heating rates: multiphysics phasefield modeling

Yong Seok Hwanga and Valery I. Levitas*b

The external surface of metallic particles is usually covered by a thin and strong oxide shell, which

significantly affects superheating and melting of particles. The effects of geometric parameters and

heating rate on characteristic melting and superheating temperatures and melting behavior of aluminum

nanoparticles covered by an oxide shell were studied numerically. For this purpose, the multiphysics

model that includes the phase field model for surface melting, a dynamic equation of motion,

a mechanical model for stress and strain simulations, interface and surface stresses, and the thermal

conduction model including thermoelastic and thermo-phase transformation coupling as well as

transformation dissipation rate was formulated. Several nontrivial phenomena were revealed. In com-

parison with a bare particle, the pressure generated in a core due to different thermal expansions of the

core and shell and transformation volumetric expansion during melting, increases melting temperatures

with the Clausius–Clapeyron factor of 60 K GPa�1. For the heating rates Q r 109 K s�1, melting tem-

peratures (surface and bulk start and finish melting temperatures, and maximum superheating tempera-

ture) are independent of Q. For Q Z 1012 K s�1, increasing Q generally increases melting temperatures

and temperature for the shell fracture. Unconventional effects start for Q Z 1012 K s�1 due to kinetic

superheating combined with heterogeneous melting and geometry. The obtained results are applied to

shed light on the initial stage of the melt-dispersion-mechanism of the reaction of Al nanoparticles.

Various physical phenomena that promote or suppress melting and affect melting temperatures

and temperature of the shell fracture for different heating-rate ranges are summarized in the corre-

sponding schemes.

Melting temperature of materials and melting mechanismsdepend on various parameters: size, shape, condition at thesurface, pressure (or, more generally, stress tensor), and heatingrate, as well as on their interaction. Melting temperature depres-sion with reduction of the particle radius is well-known fromexperiments,1,2 thermodynamic treatments,1,2 molecular dynamicssimulations,3,4 and phase field studies without mechanics5,6 andwith mechanics (but without inertia effects).6,7

Reduction in surface energy during melting leads to pre-melting below melting temperature followed by surface meltingand solid–melt interface propagation through the entire samplewith increasing temperature. This was studied using the phasefield approach for the plane surface analytically8,9 and numerically

(including the effect of mechanics) for low6,7 and high10,11

heating rates. Similar studies were performed for sphericalparticles without mechanics5,6 and with mechanics in quasi-static formulation.6,7 Strong effects of the width of the externalsurface and thermally activated nucleation were revealed withinthe phase field approach in ref. 12. If the external surface of thematerial under study represents an interface with another solid,surface melting depends on the type of interface. The low-energycoherent interfaces increase energy during melting and, con-sequently, suppress surface nucleation and promote super-heating.13,14 On the other hand, an incoherent interface, whoseenergy reduces during melting, promotes surface melting.2

Hydrostatic pressure inside a shell which can be created formaterials with volume expansion during melting suppressesmelting and increases equilibrium melting temperature T p

eq

according to the Clausius–Clapeyron relationship. The effect ofpressure appears automatically within the phase field approachif proper thermodynamic potential is implemented.6,15 Undernon-hydrostatic internal stresses that relax during melting,

a Department of Aerospace Engineering, Iowa State University, Ames, Iowa 50011,

USA. E-mail: [email protected] Departments of Aerospace Engineering, Mechanical Engineering, and Material

Science and Engineering, Iowa State University, Ames, Iowa 50011, USA.

E-mail: [email protected]; Fax: +1 801 788 0026; Tel: +1 515 294 9691

Received 5th June 2016,Accepted 23rd September 2016

DOI: 10.1039/c6cp03897b

www.rsc.org/pccp

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PAPER

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e.g. under biaxial stresses due to constraint, melting tempera-ture reduces. See thermodynamic11,16 and phase field11 treat-ments for a layer. Melting temperature drastically decreasesduring very high strain-rate uniaxial compression in a strongshock wave, as it was predicted thermodynamically and con-firmed by molecular dynamics simulations.17

Metal can be kinetically superheated above its equilibriummelting temperature when it is subjected to an extremely fastheating rate, for example, during irradiation by an ultra-fast laserwith high energy, such as picosecond (ps) and femtosecond (fs)lasers. It has been observed in experiments18–20 and phase fieldsimulations10,11 that an aluminum layer can be superheated upto at least 1400 K,20 which is far above its equilibrium tempera-ture, Teq = 933.67 K. The major reason for kinetic superheating,when heterogeneous surface melting initiates the process, isthe slower kinetics of solid–melt interface propagation thanheating.10,21 For very high heating rates Q Z 1012 K s�1, elasticwave propagation can affect the temperature of the materialthrough thermoelastic coupling and melting temperaturethrough the effect of stresses.22 Melting also influences thetemperature of materials through thermo-phase transforma-tion coupling, mostly due to latent heat.

Thus, an analysis of kinetic superheating of materialsshould take several physical processes and their couplings intoaccount. Recently, there has been some research and suggestedmodels23–27 to describe ultra-fast heating and melting with orwithout mechanics, including thermoelastic coupling or thermo-phase transformation coupling. However, those models neitherdescribe a complete set of participated physical phenomena norinclude correct coupling terms rigorously derived from thethermodynamic laws. Recently, we have developed a novel phasefield model, which includes all of the above physical phenomenaand couplings in a single framework.10,11,22

However, all the above modeling results have been obtainedfor melting bare metallic nanostructures. In reality, metallic(e.g., Al, Fe, Cu, and others) particles and layers have a strongpassivation oxide layer at the external surface. Thus, nano-particles form a core–shell structure. The aluminum oxide oralumina passivation layer can be formed even at room tem-perature28 by transporting Al cations driven by the non-equilibrium electrostatic field, the so-called Cabrera–Mottmechanism.29,30 Aluminum oxide has a lower thermal expan-sion coefficient than the aluminum core, so the compressivepressure in the core and the tensile hoop stress in the oxideshell are generated due to volumetric expansion during heatingbefore melting.31 Since melting of Al is accompanied by avolumetric strain of 6%, pressure of several GPa can beobtained in the melt and hoop stress in the alumina shell ison the order of magnitude of 10 GPa. High pressure in the coreresults in an increase in the melting temperature according tothe Clausius–Clapeyron relationship. The generated pressuredepends on the ultimate strength of the shell and relaxationprocesses in it, including phase transformations from amor-phous alumina to crystalline g and d phases. Thus, slow heatingof Al nanoparticles with an oxide layer at 20 K min�1, depend-ing on the Al core radius (Ri) and the oxide shell thickness (d),

leads to a wide spectrum of behavior from the reduction of themelting temperature due to size effect to a minor superheatingof up to 15 K32–35 due to sufficient time for stress relaxation.Stress measurement in Al nanoparticles was performed inref. 32 and 34–36. In contrast, fast heating with the rate higherthan 106–108 K s�1 can lead to the estimated superheatingby several hundred K due to the pressure increase37 becausethere is not sufficient time for phase transformations and otherstress relaxation mechanisms in the shell. For the higherheating rates of 1011–1014 K s�1 used in experiments,38–40 boththe pressure-induced increase in melting temperature and kineticsuperheating are expected.

Thus, for understanding and quantifying melting of metallicnanoparticles in a broad range of heating rates, one has toinclude and study the effect of an oxide shell and major physicalprocesses involved in melting, in particular, the effect of thegenerated pressure, kinetic superheating, heterogeneity of tem-perature and stress fields, dynamics of elastic wave propagation,surface and interface energies and stresses, and coupling of theabove processes. This is a basic outstanding multiphysics pro-blem to be solved. Most of these processes strongly depend onthe core radius Ri and the oxide shell thickness d; thus, theireffect should be studied in detail.

The understanding of the melting of Al nanoparticles at ahigh heating rate is also very important for understanding andcontrolling the mechanisms of their oxidation and combus-tion.31,37,41 According to the melt-dispersion mechanism of thereaction of Al particles,31,37,41 high pressure in the melt andhoop stresses in the shell, caused by the volume increaseduring melting, break and spall the alumina shell. Then, thepressure at the bare Al surface drops to (almost) zero, whilepressure within the Al core is not initially altered. An unloadingspherical wave propagating to the center of the Al core generates atensile pressure of up to 3 GPa at the center, which reaches 8 GPain the reflected wave. The magnitude of tensile pressure signifi-cantly exceeds the cavitation strength of liquid Al and dispersesthe Al molten core into small bare drops. Consequently, themelt-dispersion mechanism breaks a single Al particle coveredby an oxide shell into multiple smaller bare drops, which isnot limited by diffusion through the initial shell. This mecha-nism was extended for micron-scale particles42,43 and utilizedfor increasing the reactivity of Al nano- and micron-scaleparticles by their prestressing.44,45 However, there has beenno research for melting and kinetic superheating of Al nano-particles within an oxide shell at high heating rates to the bestof our knowledge.

In this paper, we study superheating and melting of Al nano-particles covered by an alumina shell and the correspondingphysical processes under high heating rates. We utilize ourrecent model10,11,22 that includes the phase field model formelting developed in ref. 6 and 7, a dynamic equation ofmotion, a mechanical model for stress and strain simulations,and the thermal conduction model with thermo-elastic andthermo-phase transformation coupling, as well as with a dissi-pation rate due to melting.10,11 The effects of geometric parameters(which determine the stress-state and temperature evolution)

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and heating rate on the characteristic melting and super-heating temperatures and melting behavior, as well as on themaximum temperature corresponding to fracture of the shell,are simulated and analyzed by a parametric study. Severalnontrivial and unconventional phenomena are revealed. Theinfluence of the above parameters at the initial stage of themelt-dispersion-mechanism of reaction of Al nanoparticles31 isevaluated and discussed.

1 Models1.1 Governing equations

Due to high heating rates and short melting time, we will neglectcrystallization of the amorphous alumina shell and any inelasticflow in it. The model consists of the phase field equation, theequation of motion, equations of elasticity theory, and the heatconduction equation; all of them are coupled. The phase fieldequation is applied only to the aluminum core since melting inaluminum oxide starts at a much higher temperature (2324 K46).

We designate contractions of tensors A = {Aij} and B = {Bji} overone and two indices as A�B = {AijBjk} and A:B = AijBji, respectively;I is the unit tensor, r and r0 are the gradient operators in thedeformed and undeformed states, and # designates the dyadicproduct of vectors.

Equations for phase transformation and deformation. Totalstrain tensor e = (r0u)s (where u is the displacement vector andthe subscript s designates symmetrization) can be additivelydecomposed into elastic ee, transformation et, and thermal ey

strains:

e = ee + et + ey; e = 1/3e0I + e; (1)

ein = einI = et + ey; et = 1/3e0t(1 � f(Z))I; (2)

f = Z2(3 � 2Z) for 0 r Z r 1;

ey = as(Teq � T0)I + (am + Daf(Z))(T � Teq)I. (3)

Here, Z is the order parameter that varies from 1 in solid to 0 inmelt, as and am are the linear thermal expansion coefficients forsolid and molten Al, respectively, Da = as � am, T0 is the initialtemperature, e0 is the total volumetric strain, e0t is the volu-metric transformation strain for complete melting, and e is thedeviatoric strain. The definition of f is modified to ensure asingle minimum of free energy for Z o 0 and Z 4 1 in the caseof T 4 1.2Teq or T o 0.8Teq; see below.

The Helmholtz free energy per unit undeformed volume isformulated as in ref. 7 and 10:

c ¼ ce þ J �cy þ cy þ Jcr; �cy ¼ AZ2ð1� ZÞ2; (4)

ce = 0.5Ke0e2 + mee:ee; cy = H(T/Teq � 1)f(Z); (5)

cr = 0.5b|rZ|2, A:= 3H(1 � Tc/Teq). (6)

Here, ce, cy, �cy, and cr are the elastic, thermal, double-well,and gradient energies, respectively; r0 and r are the massdensities in the undeformed and deformed states, respectively;J = r0/r = 1 + e0; K(Z) = Km + DKf(Z), and m(Z) = msf(Z) are the

bulk and shear elastic moduli, DK = Ks � Km; b is the gradientenergy coefficient; H is the latent heat; Tc is the melt instabilitytemperature assumed to be 0.8Teq. Using thermodynamicprocedures, the following equations for the stress tensor r isobtained:

r ¼ @c@e� J�1rZ� @c

@rZ ¼ re þ rst; (7)

re ¼ Ke0eI þ 2mee; sst ¼ cr þ �cy� �

I � brZ�rZ; (8)

which consists of elastic stress re and interface stresses rst. Thesame procedure leads to the Ginzburg–Landau equation:

1

L

@Z@t¼ �J�1@c

@Z

��e

þr � J�1@c@rZ

� �

¼ J�1 �e0tpe þ 3peDa T � Teq

� � @f@Z

� J�1 0:5DKe0e2 þ mee:ee þHT

Teq� 1

� �� @f@Z

� 4AZð1� ZÞð0:5� ZÞ þ br2Z;

(9)

where L is the kinetic coefficient, pe = re:I/3 is the mean elasticstress, and p = �pe is the pressure.

For the aluminum oxide shell, eqn (1), (2), (3) and (7) arereplaced by eqn (10) and (11):

e = ee + ey; ey = aox(T � T0)I, (10)

r = re = Koxe0eI + 2moxee. (11)

Equation of motion. The momentum balance equation isaccepted in a traditional form:

r@2u

@t2¼ r � r: (12)

If the time scale of an elastic wave is much smaller than that ofmelting, the equation is replaced by the static equilibriumequation: r�r = 0.

Heat transfer equation. The energy balance equation can bepresented in the form of the following temperature evolutionequation, in which thermal conduction is assumed to be the onlymechanism of heat transfer:

C@T

@t¼ r � ðkrTÞ � 3T am þ Dafð Þ@pe

@tþ @Z

@t

� �2,

L

� 3peDa�H

Teq

�T@f@Z@Z@t;

(13)

where C is the specific heat and k is the thermal conductivity.The time delay between electron gas and phonon47 is ignoreddue to the fact that time scale is much longer than a fewpicoseconds in these simulations. In eqn (13), the second termon the right hand side describes thermoelastic coupling (e.g.,cooling in an expansion wave or heating in a compression wave),the third term is the dissipation rate due to melting, and the lastterm is the heat source due to melting. For an aluminum oxideshell, only a thermoelastic coupling term is used.

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1.2 Numerical model, boundary and initial conditions, andmaterial properties

Geometry. A 1D model in spherical coordinates is used tosimulate Al superheating and melting in the Al core–aluminashell structure. Fig. 1 shows a 1D geometry, where Ri is theradius of the aluminum core, d is the oxide shell thickness, andRs = Ri + d.

Computational methods. The finite element method codeCOMSOL Multiphysics48 is used for numerical simulation. Twotime discretization methods, Backward Differentiation Formula(BDF) and Generalized Alpha, used in COMSOL led to the sameresults; therefore, the first order BDF is selected for the stabilityof computation. The quadratic Lagrangian elements were suffi-cient to resolve a solid–melt interface, and higher order elementsmade no improvement in the accuracy of the solution. Both thetime step and the size of the finite element have been reduceduntil solutions with different discretization coincide. The timestep was inversely proportional to the heating rate and providedat most 2–3 K increments per time step. It has been found that atleast 5 elements are necessary to resolve the solid–melt interface,which is about 2 nm wide for Al. Increasing the number ofelements inside the interface did not result in a noticeableimprovement of the solution.

Boundary and initial conditions. The following boundaryconditions at points C and S, as well as jump conditions atpoint I are applied.

At r ¼ 0:@Z@r¼ 0; u ¼ 0; h ¼ 0: (14)

At r = Ri: u1 = u2; sr,1 � sr,2 = � 2gcs/Ri; T1 = T2; h1 = h2.(15)

J@c@rZ � n ¼ brZ � n ¼ b

@Z@r¼ �dgcs

dZ; gcsðZÞ ¼ gm þ gs � gmð ÞfðZÞ:

(16)

At r = Rs: sr,2 = �2gox/Rs + pg; h2 = h*. (17)

Here n is the unit normal to the interface, which coincides withthe radial direction; subscript 1 is used for the Al core andsubscript 2 for the alumina shell. Eqn (14) describes traditionalconditions at the center of symmetry: zero radial displacement u,heat flux h, and gradient of Z. At the internal core–shell interface,continuity of the displacement, temperature, and heat flux isimposed, as well as jump in the normal stress component to theinterface is caused by interface stress. For both internal and

external surfaces, we assume that interface stress is equal to theinterface energy. The boundary condition for the order para-meter (16) is related to the change in the Al–alumina interfaceenergy gcs during melting, when it changes from gs for solid Al togm for molten Al.

For a solid phase, Z = 1, the thermodynamic driving forceX = 0 in the Ginzburg–Landau eqn (9) and dg/dZ = 0 in eqn (16).Thus, melting cannot start without some perturbations at theboundary. We introduce perturbation Z = 10�6 and the condi-tion that if Z 4 1 � Z at the boundaries, then Z = 1 � Z. Thiscondition prevents the disappearance of the initial perturba-tion when heating occurs below the melting temperature. Theperturbation can bring about numerical oscillation, whichcan cause the order parameter to grow larger than unity whenT 4 1.2Teq once Z 4 1. This happens because the traditionaldefinition of f, f = Z2(3 � 2Z), while fully satisfactory for0 r Z r 1, creates an unphysical minimum of the local order

parameter-dependent part of the energy, �cy þ cy for Z 4 1, asshown in Fig. 2(a). To prevent the unphysical minimum, func-tion f is modified for Z4 1 and T 4 Teq to f = 2 � Z2(3� 2Z) inorder to eliminate the artificial minimum, as illustrated inFig. 2(a). The reinforced function f also produces a largerdriving force, leading to returning f to the range Z r 1 forTeq r T r 1.2Teq. Also, the traditional definition of f can makeunphysical growth of Z below zero when T o 0.8Teq once Zo 0.In this case, function f = �Z2(3 � 2Z) for Z o 0 can guarantee asingle minimum and stability of the computation. In summary,

f = 2 � Z2(3 � 2Z) for Z 4 1 and T 4 Teq;

f = Z2(3 � 2Z) for Z 4 1 and T r Teq. (18)

f = Z2(3 � 2Z)2 for Z o 0 and T 4 Teq;

f = �Z2(3 � 2Z) for Z o 0 and T r Teq. (19)

At the external surface, a jump in normal stress from the valueof the gas pressure pg to the radial stress in a shell sr,2 dueto surface tension is applied; we use pg = 0 in simulations.External time-independent heat flux h* is prescribed and itsmagnitude is iteratively chosen in a way that it produces thedesired heating rate at point C.

The initial temperature is T0 = 293.15 K, initial stresses arezero, and the initial order parameter is Z = 0.999 for all cases.

Material properties. The specific heat of aluminum, C = Cm +(Cs � Cm)f(Z); according to ref. 49,

Cs = (2434.86 + (3308.87 � 2434.86)/(900.0 � 300.0)

� (T � 300.0)) � 103 J (m3 K)�1 for T o 900.0 K;(20)

Cs = 3308.87 � 103 J (m3 K)�1 for T Z 900.0 K; (21)

Cm = (2789.1 + (2713.72 � 2789.1)/(1173.0 � 933.0)

� (T � 933.0)) � 103 J (m3 K)�1, (22)

where Cs and Cm are specific heats of solid and molten aluminum,respectively. The specific heat of aluminum oxide is assumed to betemperature independent, Cox = 4.924 � 106 J (m3 K)�1.

Fig. 1 Domain for 1D simulation, where C, I, S represent the center, theinterface and the surface, respectively.

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The thermal conductivity of aluminum, ks = 208 W (m K)�1

for solid and km = 102 W (m K)�1 for melt,49 is used and kox =7.5312 W (m K)�1 for amorphous aluminum oxide. Coefficients,constants, and other properties used for simulation areincluded in Tables 1 and 2. They correspond to the width,energy, and mobility of a plane solid–melt interface of dsm =2.02 nm, gsm = 0.14 J m�2, and lsm = 1.7 m s�1 K�1.

Material parameters for melting of Al nanoparticles weretaken from our papers,6,10,22 where they were justified from theknown experimental and molecular dynamic simulations datafrom the literature. Validity of a phase field melting model wasjustified by reproducing the experimental results on radius-dependent melting temperature of Al nanoparticles (down toradius of 2 nm) and the width of the surface melt versus

temperature (down to 0.5 nm).6 Unfortunately, there are noexperimental data detailing the melting of Al nanoparticlescovered by an oxide shell at high heating rates. While there aredata at slow heating rates,32–35,50 melting involves additionalprocesses such as stress relaxation due to creep and transfor-mation of amorphous to crystalline phases of alumina, whichdo not have time to occur at the high heating rates of interesthere. That is why in order to validate the model, we solved theproblems for laser melting of a thin Al nanolayer,18–20 for whichexperimental data are available for the heating rate, melting time,and temperature in the ranges similar to those here. Very goodcorrespondence between experiments and modeling was obtained.22

2 Some definitions

The influence of the heating rate, particle radius, and oxidethickness on superheating of the particle is investigated by theparametric study. Six heating rates, Q, of 108, 109, 1011, 1012,0.5 � 1013, 1013 K s�1 are selected to explore kinetic superheating.As a base case, we consider the Al core radius Ri = 40 nm and oxidethickness d = 3 nm; the effect of oxide thickness is explored withd = 0 (bare particle), 2, and 4 nm and the effect of particle size withRi = 20 and 60 nm.

Surface premelting and melting initiates barrierlessly from theAl–alumina interface I, driven by reduction in interface energyduring melting, and followed by solid–melt interface propagationtoward the center (Fig. 3). Homogeneous melt nucleation awayfrom the solid–melt interface was not observed here even abovethe solid instability temperature Tsi = 1.2Teq = 1120 K, becausebulk fluctuations were not introduced and interface propagationcompletes melting before any homogeneous nucleation becomesvisible. The same is true for a bare particle.

The reduced temperature, T ¼ T

Teq, and time, t ¼ t

teq, are

defined with normalization using bulk equilibrium tempera-ture, Teq = 933.67 K, and the time required to reach thistemperature, teq. The heating rate for the core–shell structure

is defined either as Q ¼ Teq � T0

teqat the center of the particle or

as Qi ¼Teq � T0

tieq, where ti

eq is the time to reach Teq at the

interface I, if there is a significant heterogeneity in the tem-perature distribution. For a bare particle, the heating rate is

defined as Q ¼ 900� T0

t900, where t900 is the time to reach 900 K at

Table 1 Properties of aluminum11

r0 (kg m�3) Teq (K) H (J m�3) Km (GPa) Ks (GPa) m (GPa) e0t am (K�1) as (K�1) gs (J m�2) gm (J m�2) b (N) L (m2 N�1 s�1)

2700.0 933.67 933.57 � 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

Table 2 Properties of aluminum oxide50,51

rox (K m�3) Kox (GPa) mox (GPa) aox (K�1) gox (J m�2)

3000.0 234.8 149.5 0.778 � 10�5 1.050

Fig. 2 Comparison of the thermal part of the free energy, �cy þ cy, land-scape corresponding to the traditional function f = Z2(3 � 2Z) (solid lines)and reinforced definition of f in eqn (18) and (19) for (a) T 4 1.2Teq and(b) T o 0.8Teq (dashed lines).

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the center, since temperature does not reach Teq due to theGibbs–Thomson effect. The surface melting start mark, Tsm, inall the following figures represents the initiation of surfacepremelting when the order parameter reaches 0.5 for the firsttime at the interface I. The melting finish temperature, Tmf, isdefined at the time when the order parameter reaches 0.5 for thefirst time in the center C. The bulk melting start temperature, Tbm,

is defined as the temperature at which the order parameter atthe interface I becomes smaller than 0.01 for the first time. Inaddition, two more characteristic temperatures are defined: themaximum superheating temperature, Tms, which is the maximumtemperature of the center of the solid core during melting,and the maximum attainable temperature, Tma, which is themaximum temperature of aluminum at the interface attainedbefore the fracture of the oxide shell. In this research, thefracture of oxide is assumed to occur once the maximum tensilehoop stress s2 reaches the theoretical ultimate strength ofalumina, sth = 11.33 GPa.31 Note that Tma is practically achiev-able maximum temperature of aluminum with the core–shellstructure, above which it ceases to exist. Characteristic timescorresponding to each characteristic temperature are designatedby the same subscripts.

A summary of the main simulation results is presented inTable 3.

3 Superheating and melting of bareAl nanoparticles

For comparison and interpretation, the phase equilibriumtemperature, Tr

eq, corresponding to the interface radius ri and

Fig. 3 Propagation of the solid–melt interface during melting for an Alnanoparticle with Ri = 40 nm and d = 3 nm at Q = 1011 K s�1.

Table 3 Summary of simulation conditions and results

Ri (nm) d (nm) M Q (K s�1) teqa tsm tbm tms tma tmf Tsm (K) Tbm (K) Tms (K) Tma (K) Tmf (K) _eoxm,2 (s�1)

40 0 108 6.1 ms 0.938 1.121 1.122 1.526 870.8 930.4 930.4 916.5109 606.3 ns 0.938 1.121 1.122 1.542 870.8 930.4 930.4 918.81011 6.1 ns 0.939 1.128 1.595 1.595 871.4 933.5 946.8 946.85 � 1012 121.8 ps 1.057 1.255 2.230 2.230 947.2 1014.8 1308.1 1308.11013 60.5 ps 1.221 1.388 2.566 2.566 1052.7 1105.6 1500.1 1500.1

40 2 20 108 6.4 ms 0.945 1.114 1.420 1.412 1.449 904.2 972.2 992.0 991.9 984.9 5.39 � 103

109 641.0 ns 0.945 1.114 1.420 1.412 1.452 904.2 972.3 992.1 992.1 985.4 5.39 � 104

1011 6.4 ns 0.945 1.120 1.521 1.422 1.521 904.8 976.0 1026.1 1004.1 1026.1 5.34 � 106

5 � 1012 128.4 ps 1.103 1.260 2.159 1.570 2.159 1007.9 1071.2 1433.1 1179.3 1433.1 3.02 � 108

1013 61.2 ps 1.233 1.388 2.504 1.616 2.504 1106.2 1167.9 1671.6 1264.0 1671.6 6.73 � 108

40 3 13.3 108 6.4 ms 0.967 1.133 1.422 1.537 1.446 915.6 986.1 1011.5 1063.5 1005.2 5.15 � 103

109 638.5 ns 0.967 1.134 1.422 1.537 1.449 915.7 986.1 1011.7 1063.5 1005.7 5.15 � 104

1011 6.4 ns 0.968 1.141 1.510 1.538 1.510 916.3 990.2 1043.9 1063.7 1043.9 5.08 � 106

5 � 1012 128.9 ps 1.070 1.260 2.155 1.672 2.155 991.1 1081.4 1476.0 1246.3 1476.0 2.82 � 108

1013 61.5 ps 1.238 1.395 2.507 1.728 2.507 1116.9 1186.5 1732.4 1345.6 1732.4 6.34 � 108

40 4 10 108 6.4 ms 0.985 1.149 1.418 1.708 1.438 925.3 997.4 1026.3 1203.1 1020.4 4.94 � 103

109 640.3 ns 0.985 1.149 1.418 1.708 1.442 925.3 997.4 1026.5 1203.1 1020.9 4.94 � 104

1011 6.4 ns 0.985 1.155 1.506 1.708 1.506 925.9 1001.6 1061.4 1203.3 1061.4 4.85 � 106

5 � 1012 128.7 ps 1.140 1.289 2.154 1.774 2.154 1037.6 1103.8 1513.2 1320.2 1513.2 2.68 � 108

1013 62.2 ps 1.254 1.401 2.374 1.782 2.396 1139.4 1211.9 1662.6 1408.3 1648.8 5.79 � 108

20 1.5 13.3 108 6.4 ms 0.960 1.177 1.369 1.557 1.408 913.2 989.4 1003.1 1088.1 991.6 5.02 � 103

109 642.9 ns 0.960 1.177 1.369 1.557 1.411 913.2 989.5 1003.1 1088.1 988.5 5.02 � 104

1011 5.9 ns 0.961 1.181 1.460 1.558 1.460 913.8 992.1 1019.6 1088.1 1019.6 5.41 � 106

5 � 1012 129.2 ps 1.146 1.327 1.904 1.664 1.904 1015.8 1073.6 1291.0 1177.0 1291.0 4.02 � 108

1013 63.9 ps 1.285 1.457 2.147 1.715 2.147 1103.1 1159.2 1453.6 1242.2 1453.6 7.82 � 108

60 4.5 13.3 108 6.4 ms 0.969 1.117 1.440 1.530 1.460 916.4 984.4 1014.9 1055.3 1010.6 5.16 � 103

109 642.0 ns 0.969 1.117 1.443 1.530 1.463 916.4 984.5 1015.3 1055.3 1011.8 5.16 � 104

1011 6.4 ns 0.969 1.126 1.548 1.534 1.548 917.0 989.9 1064.2 1060.7 1064.2 5.05 � 106

5 � 1012 128.0 ps 1.079 1.218 2.318 1.641 2.318 1028.4 1099.9 1625.9 1302.9 1625.9 2.74 � 108

1013 63.5 ps 1.127 1.261 2.288 1.580 2.324 1134.1 1207.2 1635.4 1407.4 1539.2 6.06 � 108

a teq for the bare particle is defined as the time for T = 900 K.

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defined from the thermodynamic equilibrium conditions forthe stress-free case, H(Tr

eq/Teq � 1) = �2gsm/ri, is introduced.Thus, Tr

eq reduces from Teq for the plane interface (ri - N) tozero for ri = 2gsm/H = 0.279 nm, and for smaller ri the interfacecannot be equilibrium. We also introduce phase equilibriumtemperature under the Laplace pressure p, Trp

eq = Treq + DTp,

where the Laplace pressure p = 2gm/Ri = 0.046 GPa is producedby the melt-vapor spherical particle surface with Ri = 40 nm andDTp is the Clausius–Clapeyron increase in the equilibrium tempera-ture due to this pressure, DTp = 0.046 GPa � 60 K GPa�1 = 2.76 K.Fig. 4(a) shows the variation of these two phase equilibriumtemperatures versus the radius of the propagating solid–meltinterface.

Fig. 4(b) shows the evolution of temperature at the center ofthe bare particle for Q = 108 and 109 K s�1 and a comparison withequilibrium melting temperatures, Tr

eq and Trpeq. The simulated

curves for Q = 108 and 109 K s�1 are almost overlapped, whichmeans there is no effect of kinetic superheating except at the veryend of melting (which will be discussed below). Temperaturedecreases with time and slightly (by 1 K) exceeds the equilibriumcurve Trp

eq(r). Thus, temperature reduction is due to the thermo-dynamic effect of the interface radius on the phase equilibriumtemperature. A small deviation cannot be considered a none-quilibrium effect because it is independent of the heating rate.It can be explained by the difference in the sharp interface modelfor Trp

eq(r) and the finite-width solid–melt interface in simula-tions. Note that for the plane solid–melt interface within ananolayer, for such heating rates melting occurs at a constanttemperature equal to the equilibrium temperature under thecorresponding stress.22

The strong decrease of Trpeq as ri approaches zero leads to

dependence of temperature evolution on the heating rate at thevery end of melting (see Fig. 4(b) and inset in Fig. 5(a)). Sincecurves T(t) for Q = 108 K s�1 and Q = 109 K s�1 (i.e., for differentrates of heat supply) coincide and the rate of heat absorption is

Fig. 4 (a) The variation of phase equilibrium temperature for the solid–melt interface under stress-free conditions Tr

eq and under Laplace pressureTrp

eq versus the interface radius 1/ri and (b) evolution of temperature at thecenter of the Al particle during heating with two heating rates Q andmelting in comparison with Tr

eq and Trpeq for a 40 nm bare particle. The

equilibrium melting temperatures, Treq and Trp

eq, versus time were obtainedby substitution of the simulated interface position at each time instantri(t) in equations Tr

eq(ri) and Trpeq(ri).

Fig. 5 Evolution of (a) normalized temperature,T

900 K, at the center (solid

line) and the surface (dashed line); the inset is for Q = 108 and 109 K s�1.(b) Evolution of the temperature difference between the center and thesurface of a bare Al nanoparticle with Ri = 40 nm.

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determined by the interface velocity, the interface velocity forthese heating rates is determined by equality of heat supply andabsorption.

Fig. 5(a) shows the evolution of temperature at the centerand the surface during heating and melting for a 40 nm radiusbare Al nanoparticle. The bulk melting temperature for Q = 108

and 109 K s�1, Tbm = 930.4 K (T = 0.996), is slightly below Teq

due to radius-dependence of the melting temperature, i.e., theGibbs–Thomson effect. For higher heating rates, Q Z 1011 K s�1,kinetic superheating is observable, i.e., the evolution of tempera-ture starts to deviate from that of Q = 108 K s�1 and Q = 109 K s�1

cases. Characteristic melting temperatures, also, deviate fromequilibrium melting temperatures and increase according toincreasing heating rates as shown in Table 3. The temperaturedrop at the final moment of melting disappears due to pre-vailing kinetic superheating over a relatively small heat sink byaccelerated melting in the small volume of the final core. ForQ Z 1012 K s�1, the heterogeneity of temperature becomesnoticeable from the beginning of heating by inspecting thedifference of temperature between the center and the surface(Fig. 5(b)); it becomes about 37 K during heating with Q =1013 K s�1. The difference decreases at the beginning of surfacemelting due to heat absorption at the surface; starting withbulk melting it grows since the interface travels to the centerabsorbing heat while the surface is heated. After the comple-tion of melting and disappearance of the interface, the differ-ence decreases again. Wavy temperature evolution in Fig. 5 iscaused by thermo-elastic interaction and becomes significantfor Q Z 1012 K s�1.

4 Superheating and melting of anAl core with a radius of 40 nm confinedby an alumina shell with a thicknessof 3 nm4.1 Effect of confinement pressure on the melting of anAl nanoparticle

The melting temperature of an Al nanoparticle with anoxide shell is neither Teq nor constant, as shown in the insetof Fig. 6(a), if a shell can sustain high pressure inside a core.Increasing pressure within a core due to a less thermallyexpanded shell and due to a transformation volumetric expan-sion of 0.06 in the melt leads to increasing melting tempe-rature, rationalized by the Clausius–Clapeyron relationship,dT

dp¼ e0t

Teq

H¼ 60 K GPa�1.

The evolution of the core pressure at the interface I is shownin Fig. 7(a). The compressive pressure at t = 1.13 (correspondingto Tbm) for cases without kinetic superheating is 1.03 GPa,which should result in the increase of the bulk melting tem-perature by 61.8 K in comparison with Tbm = 930.4 K for a bareparticle, i.e., in 992.2. This shows good agreement with thesimulated Tbm = 986.1 K (inset of Fig. 6(a)). Note that the surfacemelting start temperature is 915.6 K (Fig. 6(a) and Table 3),

and T = 0.981, which is larger than Tsm of a bare particlebecause of the pressure. After the start of surface melting andthen bulk melting, the temperature increases (in contrast tothat for a bare particle) since an increasing fraction of melt in acore increases pressure (Fig. 7(a)). The effect of pressure will befurther elaborated for other Ri and M.

4.2 Effect of the heating rate

Our previous phase field studies10,11 have demonstrated thatultrafast heating over 1011 K s�1 of an aluminum nanolayercan kinetically superheat the material above the melting tem-perature. While two interfaces propagate from both surfaces ofthe layer until they meet in the central region of the layer,temperature increases due to fast heating. The aluminum core–shell structure is subjected to kinetic superheating due to asimilar mechanism, if the heating rate is fast enough. Since anelastic wave traveling within ps time scale can possibly affect

Fig. 6 Evolution of temperature and characteristic melting temperaturesfor an Al nanoparticle with Ri = 40 nm and d = 3 nm. (a) Evolution oftemperature at the center of the Al core for different heating rates. Theinset is for Q = 108 K s�1. (b) The surface melting start temperature, Tsm, thebulk melting start temperature, Tbm, the maximum attainable temperature,Tma, and the maximum superheating temperature, Tms, as functions of theheating rate.

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the temperature, and hence melting, of the particle by thermo-elastic coupling, the dynamic equation of motion is incorpo-rated into the model for Q Z 1012 K s�1. Fig. 6(a) displays theevolution of temperature at the center for various heating rates.While curves for 108 K s�1 and 109 K s�1 are overlapped (i.e.,there is no kinetic superheating), the curve for Q = 1011 K s�1

shows a small deviation from them. This deviation becomesobvious as the heating rate increases. For Q 4 1012 K s�1, thetemperature is affected by elastic waves, so that small oscilla-tions appear on the temperature evolution curve. The magni-tude and normalized period of oscillations grow as the heatingrate increases. While for Q r 1012 K s�1 temperatures ofinitiation and end of bulk melting are clearly detectable byinspecting the change in the slope in the temperature evolutioncurves, it is not the case for higher heating rates.

Kinetic superheating for Q Z 1011 K s�1 retards the beginningof melting in terms of t and extends the normalized time period

for melting, tmf � tsm. For lower heating rates without kineticsuperheating, the interface propagation is completely governedby the equality of supplied and latent heats since heating isslower than propagation. Thus, heating becomes the limitingprocess, the time period for melting is inversely proportionalto Q, and the normalized time period for completing meltingbecomes independent of Q. For heating rates with kineticsuperheating, solid–melt interface propagation turns out to beslower than heating and the interface kinetics becomes thelimiting process, which leads to the increase in the normalizedtime period for melting (refer to Fig. 11(d)).

Fig. 6(b) shows the change of four characteristic tempera-tures versus the heating rate (the data are also summarized inTable 3). All four temperatures are practically independent ofthe heating rate for Q r 109 K s�1, i.e., kinetic superheating isabsent in any sense. Surface melting start temperature, Tsm, isas low as 915.6 K due to surface premelting, which is lower than

Fig. 7 Evolution of pressure in the Al core at the interface I for an Al nanoparticle with Ri = 40 nm. (a) For various heating rates and shell widths d = 3 nm.(b) Comparison for static and dynamic formulations, as well as for dynamic formulation with infinite thermal conductivity for Q = 1013 K s�1 and d = 3 nm.

(c) Effect of M ¼ Ri

dfor Q = 1013 K s�1 and Q = 108 K s�1.

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the equilibrium temperature Tpeq ¼ Teq þ pe0t

Teq

H

� �, 995.5 K,

predicted by the Clausius–Clapeyron relationship. The equili-brium temperature is quite close to Tbm, 986.1 K, and thedifference is mostly due to the size effect. The maximum super-heating temperature, Tms, is 1011.6 K and is higher than theequilibrium temperature. The maximum attainable temperaturebefore shell fracture, Tma, is 1063.5 K and higher than Tms, i.e.,the oxide shell can withstand complete aluminum melting.Elevation of the characteristic temperatures for Q = 1011 K s�1

above those for lower heating rates indicates kinetic super-heating. This threshold heating rate for kinetic superheating issimilar to that for the Al nanolayer, see ref. 11. The difference,Tbm � Tsm, increases for Q = 1011 K s�1 and 1012 K s�1 but thenreduces for Q 4 1012 K s�1. Tms is largely affected by kineticsuperheating and increases drastically. Tms is rapidly increasedfor Q 4 1011 K s�1, but it may not be realized in experiments forthe given geometric parameters because oxide shell fracturesfor Q 4 1011 K s�1 before completing melting. Includingfracture in the model and studying melting during and afterfracture will be pursued in the future work.

Temperature Tma is independent of the heating rate forQ r 1011 K s�1 because the oxide shell can withstand pressurefor complete melting and stress is the same after completemelting for any heating rate in this range (Fig. 11(a)). For higherheating rates, the tensile stress in oxide, s2, reaches the ultimatestress during melting and Tma increases with increasing Q.

4.3 Effects of elastic wave and heterogeneous temperaturedistribution

For Q r 1011 K s�1, temperature within the Al core is practicallyhomogeneous (Fig. 8a). For Q = 1011 K s�1, the first temperatureheterogeneity is observed in the oxide shell (because of lowerheat conductivity of aluminum oxide than aluminum), but thetemperature difference is only 2 K. For Q = 1012 K s�1, thetemperature heterogeneity in a core also becomes visible and itreaches 17.2 K in a shell. For Q = 1013 K s�1, the temperaturedifference in the core reaches 38.4 K and the total temperaturedifference between the center C and the external oxide surface Sis almost 225 K.

An increasing heating rate and a consequent strain rateenable dynamic processes and elastic waves to influence thetemperature. In geometries considered in this research, theorder of magnitude of an acoustic time for the elastic wave totravel a particle is 10 ps (40 nm/(4 nm ps�1) = 10 ps, whereRi = 40 nm and 4 km s�1 = 4 nm ps�1 is an estimated acousticspeed in aluminum). Thermoelastic coupling produces a visibleeffect on the temperature evolution with a small oscillatingpattern in Fig. 6(a) for Q 4 1012 K s�1. This correlates with theappearance of similar trends in the pressure evolution inFig. 7(a): the initial reduction in pressure and pressure oscilla-tions become obvious for Q 4 1012 K s�1. While pressureoscillations due to multiple wave propagations and reflectionsare not surprising, pressure reduction in a core is counter-intuitive and intriguing, and the reasons for pressure reductionwith increasing heating rates are not evident.

Elastic wave (inertia effect) and the temperature gradientin a shell may be considered as the possible causes for thepressure reduction in the core. In order to clarify this issue,melting of the particle with artificially large thermal conducti-vity in dynamic and quasi-static formulations has been simu-lated for Q = 1013 K s�1 and compared with the base case.Thermal conductivity of both liquid and solid Al was increasedby a factor of 103, and of alumina by a factor of 105. The sameheat flux provided for Q = 1013 K s�1 for the base case wasapplied at the surface S. Fig. 7(b) shows the evolution of pressurefor four cases, and it is clear that the large thermal conductivityeliminates the pressure drop. For the actual thermal conductivity,both quasi-static and dynamic solutions exhibit a pressure dropby 0.085 GPa in a core, and the dynamic solution oscillatesaround the quasi-static one with a relatively small amplitude.Therefore, the pressure drop is not a result of inertia but ofrelatively slow heat conduction. Slow heat conduction initiallydelays heating of a core in comparison with the infinite

Fig. 8 (a) Temperature distribution along the radial direction for differentheating rates at a moment slightly before the surface melting starts for anAl nanoparticle with Ri = 40 nm and d = 3 nm. (b) The effect of M = Rs/d onthe evolution of the temperature difference DT across the oxide shellbetween interface I and surface S for an Al nanoparticle with Ri = 40 nm atQ = 1013 K s�1.

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conductivity case. The delayed thermal expansion of a core retardspressure growth in the core and the difference of pressures for twocases is constant due to the same heat flux. Thus, the initialtemperature drop shown in Fig. 6(a) and the corresponding initialpressure drop shown in Fig. 7(a) and (c) originate from an initiallycolder core than for slower heating. The pressure drop due toheterogeneous temperature affects the superheating temperatureso that Tsm = 1116.9 K for finite k and Tsm = 1127.7 K for infinite k,both with dynamics; a lower core pressure results in lower super-heating. Also, the heterogeneity of temperature slightly reducesthe heating rate for the same heat flux: Qi = 1.10 � 1013 K s�1 forfinite k and Qi = 1.13 � 1013 K s�1 for infinite k. Therefore, thedrop in temperature Tsm is the combined result of both physicalphenomena: a 0.085 GPa pressure drop corresponds to 5.1 Kaccording to the Clausius–Clapeyron relationship, and loweringthe heating rate corresponds to the remaining 5.7 K. However, theeffect is smaller than the melting temperature change due tokinetic superheating. The coincidence of pressure and tempera-ture curves at t = 1.0 in Fig. 6(a) and 7(a), (c) for different Q but thesame particles is not surprising: by definition, T = Teq at t = 1.0 forall cases, and the pressure is also almost the same since pressuredepends on the core temperature.

5 Effect of the parameter M ¼ Ri

dHere, we keep the fixed Al core radius Ri = 40 nm while varyingthe shell thickness d = 2, 3 and 4 nm. As it follows from thestatic analytical solution for stresses in ref. 31, reduction inM (thicker shell) leads to higher pressure in the core and lowertensile hoop stress in the shell for the same temperature, whichshould lead to higher melting temperatures and higher maxi-mum attainable temperature, Tma, before oxide fracture. Resultsin Fig. 7(c), 10(a) and (b), and Table 3 confirm this qualitativeprediction for all melting temperatures and heating rates. For allM kinetic superheating becomes observable when Q reaches1012 K s�1, and all melting temperatures strongly grow forlarger heating rates.

The different thicknesses of oxide produce different levels ofheterogeneity of temperature across the oxide shell as shown inFig. 8(b). However, its effect on Tsm appears to be quite limitedfor the same heating rate in Fig. 10(a) since the difference inTsm among cases for Q Z 1012 remains almost the same as forQ o 1012. Note that we prescribed the heating rate at the centerof the core by adjusting the heat flux. A particle with M = 10 hasa higher heat flux than with M = 20 (5.7 � 1011 J m�2 s�1 versus4.85 � 1011 J m�2 s�1) in order to have the same heating rate atthe core. Such an adjustment of the heating rate diminishes bylowering of the heating rate due to heterogeneous temperatureas described in the previous section, and this is one of thereasons of the weak effect of M on Tsm. Also, note that the shellof the particle with M = 10 not only has a larger temperaturedifference, but also a higher average temperature (see Fig. 9).

Fig. 7(c) shows the pressure evolution in the Al core at theinterface I for Q = 108 K s�1 and Q = 1013 K s�1. There is aninitial deviation between the slowest and the fastest heating

rate for all three M. This deviation obviously originates fromthe heterogeneous temperature and becomes significant as Mdecreases due to greater temperature heterogeneity. It almostdisappears at t = 1.0 because the temperature of the core is thesame as Teq for all cases due to normalization. The volumetrictransformation strain due to melting of the core increases thepressure so that the slope of pressure evolution becomes steeperafter surface melting start mark. The kinetic superheating delaysthe increase in pressure.

Temperature Tms (Fig. 10(b)) demonstrates a similar behaviorwith respect to the heating rate as Tsm in Fig. 10(a). The thickershell (smaller M) results in higher Tms because of higher pressureinside the core.

A thicker shell (smaller M) results not only in the increase ofTma due to reduced tensile hoop stress within oxide, but also ina slower increase of Tma due to growth in the heating rate in

Fig. 10(b). Stresss2sth

shown in Fig. 11(b) reaches the fracture

stress (horizontal dashed line) around the melting finish markfor Q = 1012 K s�1 for a nanoparticle with M = 10, while for ananoparticle with M = 13.3 this happens around the meltingfinish mark for Q = 1011 K s�1 in Fig. 11(a). That is why theeffect of the heating rate on the fracture starts at Q = 1012 K s�1

in Fig. 11(a) for M = 13.3 and at Q = 0.5 � 1013 K s�1 in Fig. 11(b)for M = 10, which is in agreement with Fig. 10(b). As a result, thedifference of Tma between the cases in Fig. 10(b) for Q r 1012 K s�1

is larger than that for Q 4 1012 K s�1.

6 Effect of the radius of an aluminumcore

Particles with three Al core radii, Ri = 20 nm, 40 nm, and 60 nmwith M = 13.3 (i.e., with the thickness of the oxide shell d = 1.5 nm,3 nm, and 4.5 nm, respectively) are studied. Fig. 10(c) demon-strates the effect of the Al core radius Ri on the heating-rate

Fig. 9 Evolution of temperature at the core–shell interface I and the

surface of the alumina shell for two values of M ¼ Ri

dfor Ri = 40 nm and

Q = 1013 K s�1.

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dependence of the surface melting temperature, Tsm. Forrelatively small heating rate Q r 1011 K s�1, Tsm for Ri = 20 nmis 2.4 K and 3.2 K lower than those for Ri = 40 nm and 60 nm,respectively. This is typical size-dependence of the melting tem-perature, which is observed without an oxide shell.6,7 Kineticsuperheating does not change the size-dependence, thus thesmaller particle has smaller Tsm. Larger Ri results in a largertemperature difference along the radius and a larger heating ratefor the interface Qi for the same Q at the center: for Q = 1013 K s�1,we obtained Qi = 1.19 � 1013 K s�1 for Ri = 60 nm and Qi =1.01 � 1013 K s�1 for Ri = 20 nm. Thus, the difference in Tsm

with increasing Ri for Q = 1013 K s�1 increases mostly due toincreased Qi.

A larger particle shows a larger maximum superheatingtemperature, Tms, and kinetic superheating is observed atsmaller Q (Fig. 10(d)). Thus, Tms for Ri = 40 and 60 nm startsto increase below Q = 1011 K s�1, while for Ri = 20 nm it remainsalmost same until Q = 1012 K s�1. This size effect on kineticsuperheating is governed by heterogeneous melting: a larger

dimension provides more time for interface propagation andenergy for superheating during melting. Unlike the minor effectof M on kinetic superheating in Fig. 10(b), Ri has a significantinfluence on Tms. The increase in melting time shown inFig. 11(d) also has a significant effect on Tma, which will beexplored in the following section.

7 Tensile hoop stress in and fracture ofthe oxide shell

The maximum hoop stress in the shell is located at the inter-face I so that s2 at the interface and sth determine Tma. Elasticwaves produce some contributions to change the maximumtensile hoop stress in an oxide shell for a high heating rate(Fig. 11(a)–(c)). However, the major contribution to the maxi-mum tensile hoop stress in an oxide shell for high Q comesfrom kinetic superheating and increase in melting time due tosuperheating, as will be shown below.

Fig. 10 Effect of geometric parameters of a nanoparticle on characteristic melting temperatures versus heating rate. (a) Effect of M ¼ Ri

don surface

melting start temperature, Tsm, (b) maximum attainable temperature, Tma, and maximum superheating temperature, Tms, for Ri = 40 nm. (c) Effect of theradius of the core on surface melting start temperature, Tsm, (d) maximum attainable temperature, Tma, and maximum superheating temperature, Tms,for M = 13.3.

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The heating-rate dependence of Tma and the maximumsuperheating temperature at the center of the particle beforemelting, Tms, for the three values of M are shown in Fig. 10(b).For M = 20, temperatures Tma and Tms almost coincide forQ r 109 K s�1, which means that the hoop stress in the shellreaches its ultimate strength almost at the end of the completemelting of a core. Temperature Tma is higher than Tms forQ r 1011 K s�1 of M = 13.3 and Q r 1011 K s�1 for M = 10, i.e.,fracture occurs after complete melting of the core. For all othercases, Tma o Tms and the fracture of the shell occurs beforecomplete melting of the core, followed by propagation of thepressure reduction wave, which can result in high tensilepressure.31 This process strongly depends on the fracture timeand will be studied elsewhere. Also, since it is highly probablethat the ultimate strength of the few nm thick alumina shell hassignificant scatter, the part of curves for Tms that are above thecurves for Tma still may have physical sense for higher ultimatestrength. If a shell is strong enough to contain complete melt forall heating rates or weak enough to break down before meltingstarts, Tma will be independent of the heating rate.

For all heating rates, Tma and Tms increase with decreasing Mbecause of increasing pressure in the core and reducing tensilestresses due to a thicker shell. For Q o 1011 K s�1, the heating-ratedependence of both characteristic temperatures is weak. Tms

increases with increasing heating rate for Q Z 1011 K s�1 and Tma

increases with increasing heating rate for Q Z 1012 K s�1 for all M.The wavy oscillation of s2, which originates from elastic

waves traveling in the core, appears for Q 4 1012 K s�1. Notethat the dynamic equation is used from Q = 1012 K s�1 and nooscillation is observed for this heating rate. The wave charac-teristics depend on the particle radius: oscillations of R = 20 nmfor Q = 1013 K s�1 in Fig. 11(c) have the shortest period in termsof T and it increases as R increases. However, the magnitude ofthe wave is small for all tested cases, hence the effect of elasticwaves on Tma is quite limited.

While the maximum superheating temperature is larger forlarger particles and increases with the increase in the heatingrate (Fig. 10(d)), the maximum attainable temperature, Tma, forQ r 1011 K s�1 has an opposite trend, i.e., it increases with thereduction in the core radius (Fig. 10(d)). This happens due to

Fig. 11 Evolution of normalized maximum tensile hoop stress in the oxide shell at the interface I versus the temperature at the interface I, Ti, for differentheating rates and core radii: (a) for Ri = 40 nm, d = 3 nm, and M = 13.3, (b) for Ri = 40 nm, d = 4 nm, and M = 10, and (c) for three Ri with M = 13.3.(d) Normalized time for complete melting, tmf–tsm, versus heating rate for different core radii and M = 13.3.

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surface tension at the surface S and the interface I, which producecompressive hoop stress in the oxide shell which increases withreduction in Ri. For example, the hoop stress at the interface I forRi = 20 nm at T0 is �0.38 GPa in comparison with �0.13 GPa forRi = 60 nm, which delays the fracture of the oxide shell. This is tosome extent similar to pre-stressing of the Al core–shell structuresby relaxing internal stresses by annealing at a higher temperatureand quenching to ambient temperature in order to suppressfracture and enhance the melt-dispersion mechanism.44,52 Thetrend of Tma in the size effect, however, has a crossover for higherheating rates at 1011 K s�1 r Q r 1012 K s�1 in Fig. 10(d). Thereason for the crossover can be deduced from the evolution of themaximum hoop stress in the shell during heating in Fig. 11(c) justby analyzing the position of the intersection point of the stresss2/sth with the horizontal line s2/sth = 1.0. For Q = 109 K s�1,melting completes before fracture for all Ri, and the temperature atthe intersection (which is Tma) is higher for smaller Ri (as wediscussed, due to interface stresses). For Q = 1012 K s�1, meltingalmost completes at fracture points for Ri = 20 nm and does notcomplete for Ri = 40 nm and Ri = 60 nm; the temperature at theintersection point is lower as the radius is smaller, which results incrossover. For Q 4 1012 K s�1, fracture occurs during melting, andagain, temperature at the intersection point reduces with reducingradius Ri. The increase in the normalized melting time in Fig. 11(d)has an important role in the increase of Tma since volumetrictransformation strain for the same temperature becomes smallerand so does hoop stress in a shell. That is why the slope of thecurves in Fig. 11(c) increases with reduction in the core radius Ri.

8 Relationship to the melt-dispersionmechanism of the reaction ofaluminum nanoparticles

The understanding of the melting of Al nanoparticles at highheating rates is very important for understanding and controllingthe mechanisms of their combustion.37,41 According to the melt-dispersion mechanism of the reaction of aluminum nanoparticlesdescribed in Ref. 31, 37 and 41 and Introduction, high pressure inthe melt and hoop stress in the shell, caused by the volumeincrease during thermal expansion and melting, break and spallthe protective alumina shell, which traditionally suppresses the Alreaction with an oxidizer. Following dispersion of the molten corefurther promotes contact of Al with the oxidizer and drasticallyincreases the reaction rate and flame speed. The main desirablecondition in optimizing this mechanism is that the fracture of theshell occurs after complete melting of the Al core because onlymolten Al disperses and participates in the fast reaction. Thus, it isdesired that Tma Z Tms (or tma Z tms) and that Tma (or tma) shouldnot be sensitive to some scatter in geometric parameters of the coreshell system and strength of the shell. In ref. 31, fracture of theshell occurred after complete melting for M r 19 and this resultweakly depends on d and R separately since a simplified methodof analysis for fracture of the shell in the previous research wasindependent of the heating rate. Based on our much more preciseresults in Fig. 10(b) and Table 3, we can conclude the following.

Temperatures Tma and Tms (or times tma and tms) areindependent of Q for Q = 108 K s�1 and Q = 109 K s�1, and,consequently, at least for Q r 109 K s�1. This is important forthe analysis of the melt-dispersion mechanism, because theestimated heating rate at the reaction front was 108 K s�1 inref. 31. For Q r 109 K s�1 and M = 20, the condition for com-plete melting before oxide fracture is almost met, i.e. Tma B Tms,which is consistent with ref. 31 and 53. However, for M = 20 andQ Z 1011 K s�1, fracture occurs well before completing melting.Since all other cases in Table 3 have smaller M than 19, meltingcompletes before fracture for Q r 1011 K s�1, excludingparticles with Ri = 60 nm and M = 13.3 for Q = 1011 K s�1. Notethat Q = 1011 K s�1 is the estimated heating rate in experimentsin ref. 39 and 40. For Q Z 1012 K s�1 (which is typical of theexperiments in ref. 38), fracture occurs before completion ofmelting for all cases, which is far from optimal for melt dispersion.Consequently, there is an upper bound of Q r 1011�1012 K s�1 foroptimal melt dispersion, in contrast to the previous wisdom thatthe larger Q is the better.

Thus, while predictions for Q r 109 K s�1 from the simpli-fied theory in ref. 31 are confirmed by the current and more

Fig. 12 Evolution of (a)@f@t

distribution and (b) volume integral of@f@t

for an

Al nanoparticle with Ri = 40 nm and d = 3 nm for Q = 109 K s�1.

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precise simulations, results for Q Z 1011 K s�1 differ quantita-tively and qualitatively.

Also, Table 3 contains data on the maximum rate of the hoopstrain in the shell at the interface I, _eoxm,2. For Q = 108 K s�1, it is inthe range of 4�5 � 103 s�1, which may be high enough foravoiding relaxation processes. It was roughly estimated in ref. 31 as3.3 � 104. The almost order of magnitude reduction in _eoxm,2 hereis attributed to the more precise approach and insignificantgrowth of temperature during melting with resultant reductionof effective Q. Generally, _eoxm,2 is scaled proportionally to Q. Thus,for Q = 107 K s�1 and even for Q = 106 K s�1, for which melt-dispersion is still expected in ref. 31 and 41, _eoxm,2 is 4�5� 102 s�1

and 40�50 s�1, respectively.

9 Temperature drop at the completionof melting

An abrupt decrease in temperature by several degrees is observedat the end of melting for Q = 108 K s�1 and Q = 109 K s�1,

with and without an oxide shell, and for all geometric para-meters, as it follows from Table 3. The maximum temperaturedrop of Tmf � Tms = 14.6 K is for Q = 109 K s�1, Ri = 20 nm, andd = 1.5 nm. This temperature drop comes from accelerationof melting when the interface reaches the center of a particle,T r

eq drastically reduces, and interfaces become incomplete(i.e., maximum Z reduces to smaller than 1). Fig. 12(a) shows

a large negative magnitude of the local@f@t

near the particle

center. Volume integration of the local@f@t

presented in Fig. 12(b)

takes into account that melting occurs within a smaller volumewhen the interface propagates toward the center. They showhow melting at the center of a sample is drastically accelerated,which eventually results in the temperature drop of the particlethrough eqn (13). For higher heating rates, the temperaturedrop is absent and Tms = Tmf (Table 3). A similar but muchlarger temperature drop was observed at completion of meltingof a plane nanolayer, when two solid–melt interfaces collided, forall heating rates (from 1.5� 1010 K s�1 to 8.4� 1013 K s�1) studied

Fig. 13 The map of physical phenomena affecting melting temperatures (a) for Q r 109 K s�1 and (b) for Q Z 1012 K s�1.

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in ref. 22 since the volume of the colliding region for a planestructure is much larger than that for a spherical structure.

10 Physical phenomena involvedin melting and superheating ofAl nanoparticles

In this section, we summarize the effect of different parameterson the characteristic melting temperatures (Fig. 13) and maxi-mum attainable temperature, Tma, which is determined by thefracture of the oxide shell (Fig. 14). The maps are presented fortwo ranges of the heating rates: (a) for Q r 109 K s�1, when theeffect of Q is absent and melting is quasi-equilibrium, and (b) forQ Z 1012 K s�1, when effects of the heating rates are pronounced.For the intermediate heating rate, Q = 1011 K s�1, the effect ofQ appears but is weak.

In all maps:(a) Reduction of the radius of a core Ri and the solid–melt

interface radius ri leads to reduction in melting temperatures

according to the Gibbs–Thomson effect. The reduction of theradius of a core Ri in turn causes reduction of Tma due to shiftingvolumetric expansion and the corresponding stress increase in ashell to lower temperature if s2 reaches sth during melting.

(b) For small particles (Ri = 20 nm), surface tension at thecore–shell interface and the external shell surface producespressure in a core, which increases all melting temperatures and,consequently, Tma, if s2 reaches sth during melting.

(c) A decrease in M (an increase of thickness of oxide) causesan increase in pressure in a core and an increase of all meltingtemperatures and, consequently, Tma, if s2 reaches sth duringmelting.

For low heating rates of Q r 109 K s�1, in addition to theabove effects:

(a) Surface tensions at the core–shell interface and theexternal shell, and a decrease in M also decrease tensile stressesin a shell and, consequently, increases Tma. The same is true forhigh heating rates.

(b) There is a small reduction in temperature and, con-sequently, melt finish temperature Tmf at the end of melting at

Fig. 14 The map of physical phenomena affecting the maximum attainable temperature, Tma, (a) for Q r 109 K s�1 and (b) for Q Z 1012 K s�1.

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the core center. This reduction does not practically affect Tma;it is not visible for high heating rates because of kineticsuperheating.

For high heating rates, in addition to the above effects:(a) Kinetic superheating leads to an increase of all melting

temperatures and Tma if the oxide shell fractures during melting.Heterogeneity in temperature in a shell reduces pressure andheating rate in a core but this effect is small in comparison withkinetic superheating.

(b) An increase in the core radius significantly increaseskinetic superheating and the normalized time for completemelting, and hence it increases Tms and Tma if the oxide shellfractures during melting.

For bare particles, the effects of M, d, and surface tension atS, as well as temperature Tma are irrelevant. Elastic waves have amuch smaller magnitude than for particles with the shell andpractically do not affect melting temperatures. The main effectsare a reduction in melting temperatures due to reduction inRi (Gibbs–Thomson effect), kinetic superheating for high Q, anda small increase in melting temperatures for small particles dueto pressure caused by surface tension at Ri.

11 Conclusion

In the paper, we utilize our model10,11,22 that includes the phasefield model for surface and bulk melting, dynamic equation ofmotion, mechanical model for stress and strain simulations,interfacial and surface stresses, and the thermal conductionmodel with thermo-elastic, thermo-phase transformation couplingand transformation dissipation rate, to study the effects ofgeometric parameters and heating rate on characteristic meltingand superheating temperatures and melting behavior. Severalunconventional phenomena are revealed. The main results areenumerated below.

(1) In contrast to the plane interface, the spherical interfaceexhibits a strong decrease of equilibrium temperature at theinterface, Tr

eq, as ri approaches zero. This leads to a temperaturedrop for Q = 108 K s�1 and Q = 109 K s�1, which is slightlydifferent for Q = 108 K s�1 and Q = 109 K s�1 at the end ofmelting. Excluding the end of melting, curves T(t) for Q = 108 K s�1

and Q = 109 K s�1 coincide.(2) Increasing pressure within an Al core due to a less

thermally expanded oxide shell and, after initiation of melting,due to a transformation volumetric expansion of 0.06 leads toincreasing melting temperatures, rationalized by the Clausius–

Clapeyron relationship,dT

dp¼ e0t

Teq

H¼ 60 K GPa�1. Increasing

the fraction of melt during heating increases pressure and,consequently, the melting temperature of the particle evenwithout kinetic superheating. This is confirmed by coincidenceof the T(t) curves for Q = 108 K s�1 and Q = 109 K s�1. In contrast,for a bare particle, temperature slightly decreases for Q = 108 K s�1

and Q= 109 K s�1 due to reduction in ri. For small particles, thereis a small temperature increase due to pressure caused by surfacetension at the core external surface.

(3) Heating rates Q Z 1011 K s�1 trigger kinetic superheating(i.e., increase in melting temperatures Tsm, Tbm, Tmf, andTms with increasing Q), and the effect becomes obvious forQ Z 1012 K s�1.

(4) The heating rates Q Z 1012 K s�1 produce heterogeneityin the temperature distribution; thus, for Q = 1013 K s�1, thetemperature difference is 38.4 K across the 40 nm core and225 K across a 3 nm oxide layer. The heterogeneity in tempera-ture creates a colder core and corresponding to the pressuredrop in a core, which leads to lowering of superheating tem-peratures. It can change the heating rate across the core radiusso that Qi 4 Q. However, the effect of heterogeneity is minorrelatively to kinetic superheating.

(5) The heating rates Q Z 1012 K s�1 also result in wavepropagation within the core, which causes oscillation in pressureand temperature (due to thermoelastic coupling), but the effectof wave on melting temperatures and Tma is relatively small fortested heating rates.

(6) A reduction in M = Ri/d increases the pressure growth inthe core and leads to the increase of melting temperatures forall tested Q.

(7) For Q r 1011 K s�1, the oxide shell fractures aftercomplete melting and the heating-rate dependence of Tma isvery weak. Temperature Tma increases with reduction of M whichdecreases tensile hoop stress in the shell. Tma increases with thereduction in the core radius due to surface tension at the surfaceS and interface I which produce compressive hoop stress inthe oxide shell. If oxide fractures before completing melting(for Q 4 1011–1012 K s�1), the maximum attainable temperature,Tma, depends on the heating rate. It increases with an increase inthe heating rate if melting temperatures increase, which delaysthe stress growth due to transformation expansion. A thickershell increases the Q for which heat-rate dependence of Tma starts,and a larger Ri strengthens this dependence due to augmentedkinetic superheating and increased melting time.

(8) A parametric study of high-heating rate melting allows usto shed light on the melt-dispersion mechanism of combustionof an Al nanoparticle.31 It is desired for the promotion ofAl reactivity that oxide breaks after complete melting of theparticle, i.e., Tma Z Tms. Tma and Tms are independent of Q forQ r 109 K s�1, and for M = 20, the condition for completemelting before oxide fracture is met or almost met. However, forM = 20 and Q Z 1011 K s�1, fracture occurs well before completingmelting. For M r 13.3, melting completes before fracture forQ r 1011 K s�1, excluding particles with Ri = 60 nm and M = 13.3 forQ = 1011 K s�1. For Q Z 1012 K s�1, fracture occurs beforecompleting melting for all cases under study. An interaction withan elastic wave shows the oscillating evolution of s2 but its effect onTma is minor for all tested Q. Consequently, there is an upper boundof Q r 1011–1012 K s�1 for optimal melt dispersion, in contrast toprevious wisdom that the larger Q is better. While for Q r 109 K s�1

predictions of the simplified theory in ref. 31 are confirmed by thecurrent more precise simulations, for Q Z 1011 K s�1 results arequantitatively and qualitatively different. The maximum rate of thehoop strain in the shell, which characterizes the possibility ofavoiding stress relaxation before fracture, is determined.

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(9) Various physical phenomena that promote or suppressmelting and affect melting temperatures and Tma for differentheating rate ranges are summarized in the schemes inFig. 13 and 14.

The developed multiphysics phase field model is applicablefor other core and shell materials. As the next step, we plan toinclude the phase field simulation of the fracture of the shelland its effect on melting. A sharp drop in pressure causes largetensile pressure in the reflected wave and the correspondinglarge drop of melting temperature, which can lead to additionalsignificant overheating and ‘‘homogeneous’’ melting in theentire particle due to thermal fluctuations. It also can lead tocavitation which has been studied in ref. 54. A coherent core–shell interface, which excludes surface melting, will be treatedas well. Due to broad distributions of core and shell sizes andshell strengths in a manufactured powder sample, it will bedesirable to find connection of these distributions in a repre-sentative number of particles with a distribution of meltingtemperatures.

Acknowledgements

Support from ONR (N00014-16-1-2079), Agency for DefenseDevelopment, and Seyeon E&S corporation (all South Korea)is gratefully acknowledged.

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superheating and melting within aluminum core–oxide shell ... · and solid–melt interface...

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