Superheating and Melting Within Aluminum Core–Oxide Shell Nanoparticles for a Broad Cite This: Phys

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Superheating and melting within aluminum core–oxide shell nanoparticles for a broad Cite this: Phys. Chem. Chem. Phys., 2016, 18, 28835 range of heating rates: multiphysics phase field 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 comparison 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 GPa1. For the heating rates Q r 109 Ks1, melting temperatures (surface and bulk start and finish melting temperatures, and maximum superheating temperature) are independent of Q. For Q Z 1012 Ks1, increasing Q generally increases melting temperatures and temperature for the shell fracture. Unconventional effects start for Q Z 1012 Ks1 due to kinetic Received 5th June 2016, superheating combined with heterogeneous melting and geometry. The obtained results are applied to Accepted 23rd September 2016 shed light on the initial stage of the melt-dispersion-mechanism of the reaction of Al nanoparticles. DOI: 10.1039/c6cp03897b 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- www.rsc.org/pccp sponding schemes.

Melting temperature of materials and melting mechanisms (including the effect of mechanics) for low6,7 and high10,11 depend on various parameters: size, shape, condition at the heating rates. Similar studies were performed for spherical surface, pressure (or, more generally, stress tensor), and heating particles without mechanics5,6 and with mechanics in quasi- rate, as well as on their interaction. Melting temperature depres- static formulation.6,7 Strong effects of the width of the external sion with reduction of the particle radius is well-known from surface and thermally activated nucleation were revealed within experiments,1,2 thermodynamic treatments,1,2 molecular dynamics the phase field approach in ref. 12. If the external surface of the simulations,3,4 and phase field studies without mechanics5,6 and material under study represents an interface with another solid, with mechanics (but without inertia eﬀects).6,7 surface melting depends on the type of interface. The low-energy Reduction in surface energy during melting leads to pre- coherent interfaces increase energy during melting and, con- melting below melting temperature followed by surface melting sequently, suppress surface nucleation and promote super- and solid–melt interface propagation through the entire sample heating.13,14 On the other hand, an incoherent interface, whose with increasing temperature. This was studied using the phase energy reduces during melting, promotes surface melting.2 field approach for the plane surface analytically8,9 and numerically Hydrostatic pressure inside a shell which can be created for materials with volume expansion during melting suppresses p melting and increases equilibrium melting temperature T eq a Department of Aerospace Engineering, Iowa State University, Ames, Iowa 50011, according to the Clausius–Clapeyron relationship. The effect of USA. E-mail: [email protected] b Departments of Aerospace Engineering, Mechanical Engineering, and Material pressure appears automatically within the phase field approach 6,15 Science and Engineering, Iowa State University, Ames, Iowa 50011, USA. if proper thermodynamic potential is implemented. Under E-mail: [email protected]; Fax: +1 801 788 0026; Tel: +1 515 294 9691 non-hydrostatic internal stresses that relax during melting,

This journal is © the Owner Societies 2016 Phys. Chem. Chem. Phys., 2016, 18, 28835--28853 | 28835 Paper PCCP e.g. under biaxial stresses due to constraint, melting tempera- leads to a wide spectrum of behavior from the reduction of the ture reduces. See thermodynamic11,16 and phase field11 treat- melting temperature due to size effect to a minor superheating ments for a layer. Melting temperature drastically decreases of up to 15 K32–35 due to sufficient time for stress relaxation. during very high strain-rate uniaxial compression in a strong Stress measurement in Al nanoparticles was performed in shock wave, as it was predicted thermodynamically and con- ref. 32 and 34–36. In contrast, fast heating with the rate higher firmed by molecular dynamics simulations.17 than 106–108 Ks1 can lead to the estimated superheating Metal can be kinetically superheated above its equilibrium by several hundred K due to the pressure increase37 because melting temperature when it is subjected to an extremely fast there is not sufficient time for phase transformations and other heating rate, for example, during irradiation by an ultra-fast laser stress relaxation mechanisms in the shell. For the higher with high energy, such as picosecond (ps) and femtosecond (fs) heating rates of 1011–1014 Ks1 used in experiments,38–40 both lasers. It has been observed in experiments18–20 and phase field the pressure-induced increase in melting temperature and kinetic simulations10,11 that an aluminum layer can be superheated up superheating are expected. to at least 1400 K,20 which is far above its equilibrium tempera- Thus, for understanding and quantifying melting of metallic ture, Teq = 933.67 K. The major reason for kinetic superheating, nanoparticles in a broad range of heating rates, one has to when heterogeneous surface melting initiates the process, is include and study the effect of an oxide shell and major physical the slower kinetics of solid–melt interface propagation than processes involved in melting, in particular, the effect of the heating.10,21 For very high heating rates Q Z 1012 Ks1, elastic generated pressure, kinetic superheating, heterogeneity of tem- wave propagation can affect the temperature of the material perature and stress fields, dynamics of elastic wave propagation, through thermoelastic coupling and melting temperature surface and interface energies and stresses, and coupling of the through the effect of stresses.22 Melting also influences the above processes. This is a basic outstanding multiphysics pro- temperature of materials through thermo-phase transforma- blem to be solved. Most of these processes strongly depend on tion coupling, mostly due to latent heat. the core radius Ri and the oxide shell thickness d; thus, their Thus, an analysis of kinetic superheating of materials effect should be studied in detail. should take several physical processes and their couplings into The understanding of the melting of Al nanoparticles at a account. Recently, there has been some research and suggested high heating rate is also very important for understanding and models23–27 to describe ultra-fast heating and melting with or controlling the mechanisms of their oxidation and combus- without mechanics, including thermoelastic coupling or thermo- tion.31,37,41 According to the melt-dispersion mechanism of the phase transformation coupling. However, those models neither reaction of Al particles,31,37,41 high pressure in the melt and describe a complete set of participated physical phenomena nor hoop stresses in the shell, caused by the volume increase include correct coupling terms rigorously derived from the during melting, break and spall the alumina shell. Then, the thermodynamic laws. Recently, we have developed a novel phase pressure at the bare Al surface drops to (almost) zero, while field model, which includes all of the above physical phenomena pressure within the Al core is not initially altered. An unloading and couplings in a single framework.10,11,22 spherical wave propagating to the center of the Al core generates a However, all the above modeling results have been obtained tensile pressure of up to 3 GPa at the center, which reaches 8 GPa for melting bare metallic nanostructures. In reality, metallic in the reflected wave. The magnitude of tensile pressure signifi- (e.g., Al, Fe, Cu, and others) particles and layers have a strong cantly exceeds the cavitation strength of liquid Al and disperses passivation oxide layer at the external surface. Thus, nano- the Al molten core into small bare drops. Consequently, the particles form a core–shell structure. The aluminum oxide or melt-dispersion mechanism breaks a single Al particle covered alumina passivation layer can be formed even at room tem- by an oxide shell into multiple smaller bare drops, which is perature28 by transporting Al cations driven by the non- not limited by diffusion through the initial shell. This mecha- equilibrium electrostatic field, the so-called Cabrera–Mott nism was extended for micron-scale particles42,43 and utilized mechanism.29,30 Aluminum oxide has a lower thermal expan- for increasing the reactivity of Al nano- and micron-scale sion coefficient than the aluminum core, so the compressive particles by their prestressing.44,45 However, there has been pressure in the core and the tensile hoop stress in the oxide no research for melting and kinetic superheating of Al nano- shell are generated due to volumetric expansion during heating particles within an oxide shell at high heating rates to the best before melting.31 Since melting of Al is accompanied by a of our knowledge. volumetric strain of 6%, pressure of several GPa can be In this paper, we study superheating and melting of Al nano- obtained in the melt and hoop stress in the alumina shell is particles covered by an alumina shell and the corresponding on the order of magnitude of 10 GPa. High pressure in the core physical processes under high heating rates. We utilize our results in an increase in the melting temperature according to recent model10,11,22 that includes the phase field model for the Clausius–Clapeyron relationship. The generated pressure melting developed in ref. 6 and 7, a dynamic equation of depends on the ultimate strength of the shell and relaxation motion, a mechanical model for stress and strain simulations, processes in it, including phase transformations from amor- and the thermal conduction model with thermo-elastic and phous alumina to crystalline g and d phases. Thus, slow heating thermo-phase transformation coupling, as well as with a dissi- of Al nanoparticles with an oxide layer at 20 K min1, depend- pation rate due to melting.10,11 The effects of geometric parameters ing on the Al core radius (Ri) and the oxide shell thickness (d), (which determine the stress-state and temperature evolution)

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and heating rate on the characteristic melting and super- bulk and shear elastic moduli, DK = Ks Km; b is the gradient heating temperatures and melting behavior, as well as on the energy coeﬃcient; H is the latent heat; Tc is the melt instability maximum temperature corresponding to fracture of the shell, temperature assumed to be 0.8Teq. Using thermodynamic are simulated and analyzed by a parametric study. Several procedures, the following equations for the stress tensor r is nontrivial and unconventional phenomena are revealed. The obtained: influence of the above parameters at the initial stage of the @c @c 31 r ¼ J1rZ ¼ r þ r ; (7) melt-dispersion-mechanism of reaction of Al nanoparticles is @e @rZ e st evaluated and discussed. r y re ¼ Ke0eI þ 2mee; sst ¼ c þ c I brZ rZ; (8) 1 Models which consists of elastic stress re and interface stresses rst. The 1.1 Governing equations same procedure leads to the Ginzburg–Landau equation: Due to high heating rates and short melting time, we will neglect 1 @Z 1@c 1 @c ¼J þr J crystallization of the amorphous alumina shell and any inelastic L @t @Z @rZ flow in it. The model consists of the phase field equation, the e equation of motion, equations of elasticity theory, and the heat 1 @f ¼ J e0tpe þ 3peDa T Teq conduction equation; all of them are coupled. The phase field @Z (9) equation is applied only to the aluminum core since melting in T @f 46 1 2 aluminum oxide starts at a much higher temperature (2324 K ). J 0:5DKe0e þ mee:ee þ H 1 Teq @Z We designate contractions of tensors A ={Aij}andB ={Bji} over 2 one and two indices as AB ={AijBjk}andA:B = AijBji, respectively; 4AZð1 ZÞð0:5 ZÞþbr Z;

I is the unit tensor, r and r0 are the gradient operators in the r deformedandundeformedstates,and# designates the dyadic where L is the kinetic coefficient, pe = e:I/3 is the mean elastic product of vectors. stress, and p = pe is the pressure. Equations for phase transformation and deformation. Total For the aluminum oxide shell, eqn (1), (2), (3) and (7) are replaced by eqn (10) and (11): strain tensor e =(r0u)s (where u is the displacement vector and the subscript s designates symmetrization) can be additively e = ee + ey; ey = aox(T T0)I, (10) decomposed into elastic ee, transformation et, and thermal ey r r e m strains: = e = Kox 0eI +2 oxee. (11) Equation of motion. The momentum balance equation is e = ee + et + ey; e = 1/3e0I + e; (1) accepted in a traditional form: e = e I = e + e ; e = 1/3e (1 f(Z))I; (2) in in t y t 0t @2u r ¼rr: (12) @t2 f = Z2(3 2Z) for 0 r Z r 1; If the time scale of an elastic wave is much smaller than that of ey = as(Teq T0)I +(am + Daf(Z))(T Teq)I. (3) melting, the equation is replaced by the static equilibrium Here, Z is the order parameter that varies from 1 in solid to 0 in equation: rr =0. Heat transfer equation. The energy balance equation can be melt, as and am are the linear thermal expansion coeﬃcients for presented in the form of the following temperature evolution solid and molten Al, respectively, Da = as am, T0 is the initial equation, in which thermal conduction is assumed to be the only temperature, e0 is the total volumetric strain, e0t is the volumetric transformation strain for complete melting, and e is the mechanism of heat transfer: , deviatoric strain. The definition of f is modified to ensure a @T @p @Z 2 C ¼rðkrTÞ3TðÞa þ Daf e þ L single minimum of free energy for Z o 0 and Z 4 1 in the case @t m @t @t of T 4 1.2Teq or T o 0.8Teq; see below. (13) The Helmholtz free energy per unit undeformed volume is H @f@Z 3peDa T ; formulated as in ref. 7 and 10: Teq @Z @t

c ¼ ce þ Jcy þ cy þ Jcr; cy ¼ AZ2ð1 ZÞ2; (4) where C is the specific heat and k is the thermal conductivity. The time delay between electron gas and phonon47 is ignored e 2 y c = 0.5Ke0e + mee:ee; c = H(T/Teq 1)f(Z); (5) due to the fact that time scale is much longer than a few picoseconds in these simulations. In eqn (13), the second term cr = 0.5b|rZ|2, A:= 3H(1 T /T ). (6) c eq on the right hand side describes thermoelastic coupling (e.g., Here, ce, cy, cy, and cr are the elastic, thermal, double-well, cooling in an expansion wave or heating in a compression wave), and gradient energies, respectively; r0 and r are the mass the third term is the dissipation rate due to melting, and the last densities in the undeformed and deformed states, respectively; term is the heat source due to melting. For an aluminum oxide

J = r0/r =1+e0; K(Z)=Km + DKf(Z), and m(Z)=msf(Z) are the shell, only a thermoelastic coupling term is used.

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1.2 Numerical model, boundary and initial conditions, and external surfaces, we assume that interface stress is equal to the material properties interface energy. The boundary condition for the order para- Geometry. A 1D model in spherical coordinates is used to meter (16) is related to the change in the Al–alumina interface simulate Al superheating and melting in the Al core–alumina energy gcs during melting, when it changes from gs for solid Al to gm for molten Al. shell structure. Fig. 1 shows a 1D geometry, where Ri is the radius of the aluminum core, d is the oxide shell thickness, and For a solid phase, Z = 1, the thermodynamic driving force X = 0 in the Ginzburg–Landau eqn (9) and dg/dZ = 0 in eqn (16). Rs = Ri + d. Computational methods. The finite element method code Thus, melting cannot start without some perturbations at the 6 COMSOL Multiphysics48 is used for numerical simulation. Two boundary. We introduce perturbation Z^ =10 and the condi- time discretization methods, Backward Diﬀerentiation Formula tion that if Z 4 1 Z^ at the boundaries, then Z =1 Z^. This (BDF) and Generalized Alpha, used in COMSOL led to the same condition prevents the disappearance of the initial perturba- results; therefore, the first order BDF is selected for the stability tion when heating occurs below the melting temperature. The of computation. The quadratic Lagrangian elements were suﬃ- perturbation can bring about numerical oscillation, which cient to resolve a solid–melt interface, and higher order elements can cause the order parameter to grow larger than unity when made no improvement in the accuracy of the solution. Both the T 4 1.2Teq once Z 4 1. This happens because the traditional 2 time step and the size of the finite element have been reduced definition of f, f = Z (3 2Z), while fully satisfactory for until solutions with different discretization coincide. The time 0 r Z r 1, creates an unphysical minimum of the local order y y step was inversely proportional to the heating rate and provided parameter-dependent part of the energy, c þ c for Z 4 1, as at most 2–3 K increments per time step. It has been found that at shown in Fig. 2(a). To prevent the unphysical minimum, func- 2 least 5 elements are necessary to resolve the solid–melt interface, tion f is modified for Z 4 1 and T 4 Teq to f =2 Z (3 2Z)in which is about 2 nm wide for Al. Increasing the number of order to eliminate the artificial minimum, as illustrated in elements inside the interface did not result in a noticeable Fig. 2(a). The reinforced function f also produces a larger improvement of the solution. driving force, leading to returning f to the range Z r 1 for Boundary and initial conditions. The following boundary Teq r T r 1.2Teq. Also, the traditional definition of f can make conditions at points C and S, as well as jump conditions at unphysical growth of Z below zero when T o 0.8Teq once Z o 0. 2 point I are applied. In this case, function f = Z (3 2Z) for Z o 0 can guarantee a single minimum and stability of the computation. In summary, @Z At r ¼ 0: ¼ 0; u ¼ 0; h ¼ 0: (14) 2 @r f =2 Z (3 2Z) for Z 4 1 and T 4 Teq; f Z2 Z Z 4 At r = Ri: u1 = u2; sr,1 sr,2 = 2gcs/Ri; T1 = T2; h1 = h2. = (3 2 ) for 1 and T r Teq. (18) (15) 2 2 f = Z (3 2Z) for Z o 0 and T 4 Teq; @c @Z dgcs J n ¼ brZ n ¼ b ¼ ; gcsðZÞ¼gm þ ðÞgs gm fðZÞ: 2 @rZ @r dZ f = Z (3 2Z) for Z o 0 and T r Teq. (19) (16) At the external surface, a jump in normal stress from the value

of the gas pressure pg to the radial stress in a shell sr,2 due At r = Rs: sr,2 = 2gox/Rs + pg; h2 = h*. (17) to surface tension is applied; we use pg = 0 in simulations. Here n is the unit normal to the interface, which coincides with External time-independent heat flux h* is prescribed and its the radial direction; subscript 1 is used for the Al core and magnitude is iteratively chosen in a way that it produces the subscript 2 for the alumina shell. Eqn (14) describes traditional desired heating rate at point C. conditions at the center of symmetry: zero radial displacement u, The initial temperature is T0 = 293.15 K, initial stresses are heat flux h, and gradient of Z. At the internal core–shell interface, zero, and the initial order parameter is Z = 0.999 for all cases. continuity of the displacement, temperature, and heat flux is Material properties. The specific heat of aluminum, C = Cm + imposed, as well as jump in the normal stress component to the (Cs Cm)f(Z); according to ref. 49, interface is caused by interface stress. For both internal and 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)

3 3 1 Cs = 3308.87 10 J(m K) 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, Fig. 1 Domain for 1D simulation, where C, I, S represent the center, the respectively. The specific heat of aluminum oxide is assumed to be 6 3 1 interface and the surface, respectively. temperature independent, Cox = 4.924 10 J(m K) .

28838 | Phys. Chem. Chem. Phys., 2016, 18, 28835--28853 This journal is © the Owner Societies 2016 PCCP Paper

Table 2 Properties of aluminum oxide50,51

3 1 2 rox (K m ) Kox (GPa) mox (GPa) aox (K ) gox (J m ) 3000.0 234.8 149.5 0.778 105 1.050

temperature (down to 0.5 nm).6 Unfortunately, there are no experimental data detailing the melting of Al nanoparticles covered by an oxide shell at high heating rates. While there are data at slow heating rates,32–35,50 melting involves additional processes such as stress relaxation due to creep and transformation of amorphous to crystalline phases of alumina, which do not have time to occur at the high heating rates of interest here. That is why in order to validate the model, we solved the problems for laser melting of a thin Al nanolayer,18–20 for which experimental data are available for the heating rate, melting time, and temperature in the ranges similar to those here. Very good correspondence between experiments and modeling was obtained.22

2 Some definitions

The influence of the heating rate, particle radius, and oxide thickness on superheating of the particle is investigated by the parametric study. Six heating rates, Q,of108,109,1011,1012, 0.5 1013,1013 Ks1 are selected to explore kinetic superheating.

As a base case, we consider the Al core radius Ri =40nmandoxide thickness d = 3 nm; the eﬀect of oxide thickness is explored with d = 0 (bare particle), 2, and 4 nm and the eﬀect of particle size with

Ri =20and60nm. Surface premelting and melting initiates barrierlessly from the Al–alumina interface I, driven by reduction in interface energy during melting, and followed by solid–melt interface propagation 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) toward the center (Fig. 3). Homogeneous melt nucleation away and reinforced definition of f in eqn (18) and (19) for (a) T 4 1.2Teq and from the solid–melt interface was not observed here even above (b) T o 0.8Teq (dashed lines). the solid instability temperature Tsi =1.2Teq = 1120 K, because bulk fluctuations were not introduced and interface propagation

1 completes melting before any homogeneous nucleation becomes The thermal conductivity of aluminum, ks = 208 W (m K) visible.Thesameistrueforabareparticle. for solid and k = 102 W (m K)1 for melt,49 is used and k = m ox T t 7.5312 W (m K)1 for amorphous aluminum oxide. Coeﬃcients, The reduced temperature, T^ ¼ , and time, t^¼ , are T t constants, and other properties used for simulation are eq eq defined with normalization using bulk equilibrium tempera- included in Tables 1 and 2. They correspond to the width, ture, Teq = 933.67 K, and the time required to reach this energy, and mobility of a plane solid–melt interface of dsm = 2 1 1 temperature, teq. The heating rate for the core–shell structure 2.02 nm, gsm = 0.14 J m , and lsm = 1.7 m s K . Teq T0 Material parameters for melting of Al nanoparticles were is defined either as Q ¼ at the center of the particle or teq taken from our papers,6,10,22 where they were justified from the Teq T0 i known experimental and molecular dynamic simulations data as Qi ¼ , where teq is the time to reach Teq at the ti from the literature. Validity of a phase field melting model was eq interface I, if there is a significant heterogeneity in the tem- justified by reproducing the experimental results on radius- perature distribution. For a bare particle, the heating rate is dependent melting temperature of Al nanoparticles (down to 900 T0 radius of 2 nm) and the width of the surface melt versus defined as Q ¼ , where t900 is the time to reach 900 K at t900

Table 1 Properties of aluminum11

3 3 1 1 2 2 2 1 1 r0 (kg m ) Teq (K) H (J m ) Km (GPa) Ks (GPa) m (GPa) e0t am (K ) as (K ) gs (J m ) gm (J m ) b (N) L (m N s ) 2700.0 933.67 933.57 106 41.3 71.1 27.3 0.06 4.268 105 3.032 105 1.050 0.921 3.21 1010 532

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is defined as the temperature at which the order parameter at the interface I becomes smaller than 0.01 for the first time. In addition, two more characteristic temperatures are defined: the

maximum superheating temperature, Tms,whichisthemaximum temperature of the center of the solid core during melting,

and the maximum attainable temperature, Tma, which is the maximum temperature of aluminum at the interface attained before the fracture of the oxide shell. In this research, the fracture of oxide is assumed to occur once the maximum tensile

hoop stress s2 reaches the theoretical ultimate strength of 31 alumina, sth = 11.33 GPa. Note that Tma is practically achiev- able maximum temperature of aluminum with the core–shell structure, above which it ceases to exist. Characteristic times corresponding to each characteristic temperature are designated Fig. 3 Propagation of the solid–melt interface during melting for an Al by the same subscripts. 11 1 nanoparticle with Ri =40nmandd =3nmatQ =10 Ks . A summary of the main simulation results is presented in Table 3. the center, since temperature does not reach Teq due to the

Gibbs–Thomson effect. The surface melting start mark, Tsm,in all the following figures represents the initiation of surface 3 Superheating and melting of bare premelting when the order parameter reaches 0.5 for the first Al nanoparticles time at the interface I. The melting finish temperature, Tmf,is defined at the time when the order parameter reaches 0.5 for the For comparison and interpretation, the phase equilibrium r first time in the center C. The bulk melting start temperature, Tbm, temperature, Teq, corresponding to the interface radius ri and

Table 3 Summary of simulation conditions and results

1 a 1 Ri (nm) d (nm) MQ(K s ) teq ˆtsm ˆtbm ˆtms ˆtma ˆtmf Tsm (K) Tbm (K) Tms (K) Tma (K) Tmf (K) e_oxm,2 (s ) 40 0 108 6.1 ms 0.938 1.121 1.122 1.526 870.8 930.4 930.4 916.5 109 606.3 ns 0.938 1.121 1.122 1.542 870.8 930.4 930.4 918.8 1011 6.1 ns 0.939 1.128 1.595 1.595 871.4 933.5 946.8 946.8 5 1012 121.8 ps 1.057 1.255 2.230 2.230 947.2 1014.8 1308.1 1308.1 1013 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.

28840 | Phys. Chem. Chem. Phys., 2016, 18, 28835--28853 This journal is © the Owner Societies 2016 PCCP Paper defined from the thermodynamic equilibrium conditions for curves for Q =108 and 109 Ks1 are almost overlapped, which r the stress-free case, H(Teq/Teq 1) = 2gsm/ri, is introduced. means there is no effect of kinetic superheating except at the very r - Thus, Teq reduces from Teq for the plane interface (ri N)to end of melting (which will be discussed below). Temperature zero for ri =2gsm/H = 0.279 nm, and for smaller ri the interface decreases with time and slightly (by 1 K) exceeds the equilibrium rp cannot be equilibrium. We also introduce phase equilibrium curve Teq(r). Thus, temperature reduction is due to the thermo- rp r temperature under the Laplace pressure p, Teq = Teq + DTp, dynamic effect of the interface radius on the phase equilibrium where the Laplace pressure p =2gm/Ri = 0.046 GPa is produced temperature. A small deviation cannot be considered a none- by the melt-vapor spherical particle surface with Ri = 40 nm and quilibrium effect because it is independent of the heating rate.

DTp is the Clausius–Clapeyron increase in the equilibrium tempera- It can be explained by the diﬀerence in the sharp interface model 1 rp ture due to this pressure, DTp = 0.046 GPa 60 K GPa =2.76K. for Teq(r) and the finite-width solid–melt interface in simula- Fig. 4(a) shows the variation of these two phase equilibrium tions. Note that for the plane solid–melt interface within a temperatures versus the radius of the propagating solid–melt nanolayer, for such heating rates melting occurs at a constant interface. temperature equal to the equilibrium temperature under the Fig. 4(b) shows the evolution of temperature at the center of corresponding stress.22 8 9 1 rp the bare particle for Q =10 and 10 Ks and a comparison with The strong decrease of Teq as ri approaches zero leads to r rp equilibrium melting temperatures, Teq and Teq. The simulated dependence of temperature evolution on the heating rate at the very end of melting (see Fig. 4(b) and inset in Fig. 5(a)). Since curves Tˆ(ˆt) for Q =108 Ks1 and Q =109 Ks1 (i.e., for diﬀerent rates of heat supply) coincide and the rate of heat absorption is

Fig. 4 (a) The variation of phase equilibrium temperature for the solid– r melt interface under stress-free conditions Teq and under Laplace pressure rp Teq versus the interface radius 1/ri and (b) evolution of temperature at the center of the Al particle during heating with two heating rates Q and T r rp Fig. 5 Evolution of (a) normalized temperature, , at the center (solid melting in comparison with Teq and Teq for a 40 nm bare particle. The 900 K r rp 8 9 1 equilibrium melting temperatures, Teq and Teq, versus time were obtained line) and the surface (dashed line); the inset is for Q =10 and 10 Ks . by substitution of the simulated interface position at each time instant (b) Evolution of the temperature diﬀerence between the center and the r rp ri(t) in equations Teq(ri)andTeq(ri). surface of a bare Al nanoparticle with Ri = 40 nm.

This journal is © the Owner Societies 2016 Phys. Chem. Chem. Phys., 2016, 18, 28835--28853 | 28841 Paper PCCP determined by the interface velocity, the interface velocity for these heating rates is determined by equality of heat supply and absorption. Fig. 5(a) shows the evolution of temperature at the center and the surface during heating and melting for a 40 nm radius bare Al nanoparticle. The bulk melting temperature for Q =108 9 1 and 10 Ks , Tbm = 930.4 K (Tˆ = 0.996), is slightly below Teq due to radius-dependence of the melting temperature, i.e., the Gibbs–Thomson effect. For higher heating rates, Q Z 1011 Ks1, kinetic superheating is observable, i.e., the evolution of temperature starts to deviate from that of Q =108 Ks1 and Q =109 Ks1 cases. Characteristic melting temperatures, also, deviate from equilibrium melting temperatures and increase according to increasing heating rates as shown in Table 3. The temperature drop at the final moment of melting disappears due to pre- vailing kinetic superheating over a relatively small heat sink by accelerated melting in the small volume of the final core. For Q Z 1012 Ks1, the heterogeneity of temperature becomes noticeable from the beginning of heating by inspecting the difference of temperature between the center and the surface (Fig. 5(b)); it becomes about 37 K during heating with Q = 1013 Ks1. The difference decreases at the beginning of surface melting due to heat absorption at the surface; starting with bulk melting it grows since the interface travels to the center absorbing heat while the surface is heated. After the completion of melting and disappearance of the interface, the difference decreases again. Wavy temperature evolution in Fig. 5 is caused by thermo-elastic interaction and becomes significant for Q Z 1012 Ks1.

4 Superheating and melting of an Fig. 6 Evolution of temperature and characteristic melting temperatures for an Al nanoparticle with Ri =40nmandd = 3 nm. (a) Evolution of Al core with a radius of 40 nm confined temperature at the center of the Al core for diﬀerent heating rates. The 8 1 by an alumina shell with a thickness inset is for Q =10 Ks . (b) The surface melting start temperature, Tsm, the bulk melting start temperature, Tbm, the maximum attainable temperature, of 3 nm Tma, and the maximum superheating temperature, Tms, as functions of the heating rate. 4.1 Eﬀect of confinement pressure on the melting of an Al nanoparticle

The melting temperature of an Al nanoparticle with an and Tˆ = 0.981, which is larger than Tsm of a bare particle oxide shell is neither Teq nor constant, as shown in the inset because of the pressure. After the start of surface melting and of Fig. 6(a), if a shell can sustain high pressure inside a core. then bulk melting, the temperature increases (in contrast to Increasing pressure within a core due to a less thermally that for a bare particle) since an increasing fraction of melt in a expanded shell and due to a transformation volumetric expan- core increases pressure (Fig. 7(a)). The eﬀect of pressure will be sion of 0.06 in the melt leads to increasing melting tempe- further elaborated for other Ri and M. rature, rationalized by the Clausius–Clapeyron relationship, dT T 4.2 Eﬀect of the heating rate ¼ e eq ¼ 60 K GPa1. dp 0t H Our previous phase field studies10,11 have demonstrated that The evolution of the core pressure at the interface I is shown ultrafast heating over 1011 Ks1 of an aluminum nanolayer in Fig. 7(a). The compressive pressure at ˆt = 1.13 (corresponding can kinetically superheat the material above the melting tem- to Tbm) for cases without kinetic superheating is 1.03 GPa, perature. While two interfaces propagate from both surfaces of which should result in the increase of the bulk melting tem- the layer until they meet in the central region of the layer, perature by 61.8 K in comparison with Tbm = 930.4 K for a bare temperature increases due to fast heating. The aluminum core– particle, i.e., in 992.2. This shows good agreement with the shell structure is subjected to kinetic superheating due to a simulated Tbm =986.1K(insetofFig.6(a)).Notethatthesurface similar mechanism, if the heating rate is fast enough. Since an melting start temperature is 915.6 K (Fig. 6(a) and Table 3), elastic wave traveling within ps time scale can possibly affect

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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 Ks1 and d = 3 nm. R (c) Eﬀect of M ¼ i for Q =1013 Ks1 and Q =108 Ks1. d

the temperature, and hence melting, of the particle by thermo- for melting, ˆtmf ˆtsm. For lower heating rates without kinetic elastic coupling, the dynamic equation of motion is incorpo- superheating, the interface propagation is completely governed rated into the model for Q Z 1012 Ks1. Fig. 6(a) displays the by the equality of supplied and latent heats since heating is evolution of temperature at the center for various heating rates. slower than propagation. Thus, heating becomes the limiting While curves for 108 Ks1 and 109 Ks1 are overlapped (i.e., process, the time period for melting is inversely proportional there is no kinetic superheating), the curve for Q =1011 Ks1 to Q, and the normalized time period for completing melting shows a small deviation from them. This deviation becomes becomes independent of Q. For heating rates with kinetic obvious as the heating rate increases. For Q 4 1012 Ks1, the superheating, solid–melt interface propagation turns out to be temperature is affected by elastic waves, so that small oscilla- slower than heating and the interface kinetics becomes the tions appear on the temperature evolution curve. The magni- limiting process, which leads to the increase in the normalized tude and normalized period of oscillations grow as the heating time period for melting (refer to Fig. 11(d)). rate increases. While for Q r 1012 Ks1 temperatures of Fig. 6(b) shows the change of four characteristic tempera- initiation and end of bulk melting are clearly detectable by tures versus the heating rate (the data are also summarized in inspecting the change in the slope in the temperature evolution Table 3). All four temperatures are practically independent of curves, it is not the case for higher heating rates. the heating rate for Q r 109 Ks1, i.e., kinetic superheating is 11 1 Kinetic superheating for Q Z 10 Ks retards the beginning absent in any sense. Surface melting start temperature, Tsm,is of melting in terms of ˆt and extends the normalized time period as low as 915.6 K due to surface premelting, which is lower than

This journal is © the Owner Societies 2016 Phys. Chem. Chem. Phys., 2016, 18, 28835--28853 | 28843 Paper PCCP T the equilibrium temperature T p ¼ T þ pe eq , 995.5 K, eq eq 0t H predicted by the Clausius–Clapeyron relationship. The equilibrium temperature is quite close to Tbm, 986.1 K, and the difference is mostly due to the size effect. The maximum superheating temperature, Tms, is 1011.6 K and is higher than the equilibrium temperature. The maximum attainable temperature before 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 Ks1 above those for lower heating rates indicates kinetic superheating. This threshold heating rate for kinetic superheating is similar to that for the Al nanolayer, see ref. 11. The difference, 11 1 12 1 Tbm Tsm, increases for Q =10 Ks and 10 Ks but then 12 1 reduces for Q 4 10 Ks . Tms is largely affected by kinetic superheating and increases drastically. Tms is rapidly increased for Q 4 1011 Ks1, but it may not be realized in experiments for the given geometric parameters because oxide shell fractures for Q 4 1011 Ks1 before completing melting. Including fracture in the model and studying melting during and after fracture will be pursued in the future work.

Temperature Tma is independent of the heating rate for Q r 1011 Ks1 because the oxide shell can withstand pressure for complete melting and stress is the same after complete melting for any heating rate in this range (Fig. 11(a)). For higher heating rates, the tensile stress in oxide, s2, reaches the ultimate stress during melting and Tma increases with increasing Q.

4.3 Effects of elastic wave and heterogeneous temperature distribution For Q r 1011 Ks1, temperature within the Al core is practically homogeneous (Fig. 8a). For Q =1011 Ks1, the first temperature Fig. 8 (a) Temperature distribution along the radial direction for different heterogeneity is observed in the oxide shell (because of lower heating rates at a moment slightly before the surface melting starts for an heat conductivity of aluminum oxide than aluminum), but the Al nanoparticle with Ri =40nmandd = 3 nm. (b) The effect of M = Rs/d on the evolution of the temperature difference DT across the oxide shell temperature difference is only 2 K. For Q =1012 Ks1, the between interface I and surface S for an Al nanoparticle with Ri =40nmat temperature heterogeneity in a core also becomes visible and it Q =1013 Ks1. reaches 17.2 K in a shell. For Q =1013 Ks1, the temperature difference in the core reaches 38.4 K and the total temperature difference between the center C and the external oxide surface S Elastic wave (inertia effect) and the temperature gradient is almost 225 K. in a shell may be considered as the possible causes for the An increasing heating rate and a consequent strain rate pressure reduction in the core. In order to clarify this issue, enable dynamic processes and elastic waves to influence the melting of the particle with artificially large thermal conducti- temperature. In geometries considered in this research, the vity in dynamic and quasi-static formulations has been simu- order of magnitude of an acoustic time for the elastic wave to lated for Q =1013 Ks1 and compared with the base case. travel a particle is 10 ps (40 nm/(4 nm ps1) = 10 ps, where Thermal conductivity of both liquid and solid Al was increased 1 1 3 5 Ri = 40 nm and 4 km s = 4 nm ps is an estimated acoustic by a factor of 10 , and of alumina by a factor of 10 . The same speed in aluminum). Thermoelastic coupling produces a visible heat flux provided for Q =1013 Ks1 for the base case was effect on the temperature evolution with a small oscillating applied at the surface S. Fig. 7(b) shows the evolution of pressure pattern in Fig. 6(a) for Q 4 1012 Ks1. This correlates with the for four cases, and it is clear that the large thermal conductivity appearance of similar trends in the pressure evolution in eliminates the pressure drop. For the actual thermal conductivity, Fig. 7(a): the initial reduction in pressure and pressure oscilla- both quasi-static and dynamic solutions exhibit a pressure drop tions become obvious for Q 4 1012 Ks1. While pressure by 0.085 GPa in a core, and the dynamic solution oscillates oscillations due to multiple wave propagations and reflections around the quasi-static one with a relatively small amplitude. are not surprising, pressure reduction in a core is counter- Therefore, the pressure drop is not a result of inertia but of intuitive and intriguing, and the reasons for pressure reduction relatively slow heat conduction. Slow heat conduction initially with increasing heating rates are not evident. delays heating of a core in comparison with the infinite

28844 | Phys. Chem. Chem. Phys., 2016, 18, 28835--28853 This journal is © the Owner Societies 2016 PCCP Paper conductivity case. The delayed thermal expansion of a core retards pressure growth in the core and the difference of pressures for two cases is constant due to the same heat flux. Thus, the initial temperature drop shown in Fig. 6(a) and the corresponding initial pressure drop shown in Fig. 7(a) and (c) originate from an initially colder core than for slower heating. The pressure drop due to heterogeneous temperature affects the superheating temperature so 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 superheating. Also, the heterogeneity of temperature slightly reduces 13 1 the heating rate for the same heat flux: Qi =1.10 10 Ks for 13 1 finite k and Qi =1.13 10 Ks for infinite k. Therefore, the drop in temperature Tsm isthecombinedresultofbothphysical phenomena: a 0.085 GPa pressure drop corresponds to 5.1 K according to the Clausius–Clapeyron relationship, and lowering the heating rate corresponds to the remaining 5.7 K. However, the Fig. 9 Evolution of temperature at the core–shell interface I and the effect is smaller than the melting temperature change due to R surface of the alumina shell for two values of M ¼ i for R = 40 nm and kinetic superheating. The coincidence of pressure and tempera- d i Q =1013 Ks1. ture curves at ˆt = 1.0 in Fig. 6(a) and 7(a), (c) for different Q but the same particles is not surprising: by definition, T = Teq at ˆt =1.0for all cases, and the pressure is also almost the same since pressure rate for all three M. This deviation obviously originates from depends on the core temperature. the heterogeneous temperature and becomes significant as M decreases due to greater temperature heterogeneity. It almost disappears at ˆt = 1.0 because the temperature of the core is the Ri 5 Effect of the parameter M ¼ same as Teq for all cases due to normalization. The volumetric d transformation strain due to melting of the core increases the Here, we keep the fixed Al core radius Ri = 40 nm while varying pressure so that the slope of pressure evolution becomes steeper the shell thickness d = 2, 3 and 4 nm. As it follows from the after surface melting start mark. The kinetic superheating delays static analytical solution for stresses in ref. 31, reduction in the increase in pressure.

M (thicker shell) leads to higher pressure in the core and lower Temperature Tms (Fig. 10(b)) demonstrates a similar behavior tensile hoop stress in the shell for the same temperature, which with respect to the heating rate as Tsm in Fig. 10(a). The thicker should lead to higher melting temperatures and higher maxi- shell (smaller M) results in higher Tms because of higher pressure mum attainable temperature, Tma, before oxide fracture. Results inside the core. in Fig. 7(c), 10(a) and (b), and Table 3 confirm this qualitative A thicker shell (smaller M) results not only in the increase of prediction for all melting temperatures and heating rates. For all Tma due to reduced tensile hoop stress within oxide, but also in

M kinetic superheating becomes observable when Q reaches a slower increase of Tma due to growth in the heating rate in 12 1 s 10 Ks , and all melting temperatures strongly grow for Fig. 10(b). Stress 2 shown in Fig. 11(b) reaches the fracture larger heating rates. sth The different thicknesses of oxide produce different levels of stress (horizontal dashed line) around the melting finish mark 12 1 heterogeneity of temperature across the oxide shell as shown in for Q =10 Ks for a nanoparticle with M = 10, while for a nanoparticle with M = 13.3 this happens around the melting Fig. 8(b). However, its effect on Tsm appears to be quite limited 11 1 for the same heating rate in Fig. 10(a) since the difference in finish mark for Q =10 Ks in Fig. 11(a). That is why the 12 effect of the heating rate on the fracture starts at Q =1012 Ks1 Tsm among cases for Q Z 10 remains almost the same as for 13 1 Q o 1012. Note that we prescribed the heating rate at the center in Fig. 11(a) for M = 13.3 and at Q = 0.5 10 Ks in Fig. 11(b) of the core by adjusting the heat flux. A particle with M = 10 has for M = 10, which is in agreement with Fig. 10(b). As a result, the 12 1 a higher heat flux than with M = 20 (5.7 1011 Jm2 s1 versus difference of Tma between the cases in Fig. 10(b) for Q r 10 Ks 12 1 4.85 1011 Jm2 s1) in order to have the same heating rate at is larger than that for Q 4 10 Ks . the core. Such an adjustment of the heating rate diminishes by lowering of the heating rate due to heterogeneous temperature as described in the previous section, and this is one of the 6 Effect of the radius of an aluminum reasons of the weak effect of M on Tsm. Also, note that the shell core of the particle with M = 10 not only has a larger temperature difference, but also a higher average temperature (see Fig. 9). Particles with three Al core radii, Ri = 20 nm, 40 nm, and 60 nm Fig. 7(c) shows the pressure evolution in the Al core at the with M =13.3(i.e., with the thickness of the oxide shell d =1.5nm, interface I for Q =108 Ks1 and Q =1013 Ks1. There is an 3 nm, and 4.5 nm, respectively) are studied. Fig. 10(c) demon- initial deviation between the slowest and the fastest heating strates the effect of the Al core radius Ri on the heating-rate

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R Fig. 10 Effect of geometric parameters of a nanoparticle on characteristic melting temperatures versus heating rate. (a) Effect of M ¼ i on surface d melting start temperature, Tsm, (b) maximum attainable temperature, Tma, and maximum superheating temperature, Tms,forRi = 40 nm. (c) Effect of the radius of the core on surface melting start temperature, Tsm, (d) maximum attainable temperature, Tma, and maximum superheating temperature, Tms, for M = 13.3.

dependence of the surface melting temperature, Tsm. For dimension provides more time for interface propagation and 11 1 relatively small heating rate Q r 10 Ks , Tsm for Ri =20nm energy for superheating during melting. Unlike the minor effect is 2.4 K and 3.2 K lower than those for Ri = 40 nm and 60 nm, of M on kinetic superheating in Fig. 10(b), Ri has a significant respectively. This is typical size-dependence of the melting tem- influence on Tms. The increase in melting time shown in 6,7 perature, which is observed without an oxide shell. Kinetic Fig. 11(d) also has a significant effect on Tma, which will be superheating does not change the size-dependence, thus the explored in the following section. smaller particle has smaller Tsm.LargerRi results in a larger temperature difference along the radius and a larger heating rate 13 1 for the interface Qi for the same Q at the center: for Q =10 Ks , 7 Tensile hoop stress in and fracture of 13 1 we obtained Qi = 1.19 10 Ks for Ri = 60 nm and Qi = 13 1 the oxide shell 1.01 10 Ks for Ri = 20 nm. Thus, the difference in Tsm 13 1 with increasing Ri for Q =10 Ks increases mostly due to The maximum hoop stress in the shell is located at the inter- increased Qi. face I so that s2 at the interface and sth determine Tma. Elastic A larger particle shows a larger maximum superheating waves produce some contributions to change the maximum temperature, Tms, and kinetic superheating is observed at tensile hoop stress in an oxide shell for a high heating rate smaller Q (Fig. 10(d)). Thus, Tms for Ri = 40 and 60 nm starts (Fig. 11(a)–(c)). However, the major contribution to the maxi- 11 1 to increase below Q =10 Ks , while for Ri = 20 nm it remains mum tensile hoop stress in an oxide shell for high Q comes almost same until Q =1012 Ks1. This size effect on kinetic from kinetic superheating and increase in melting time due to superheating is governed by heterogeneous melting: a larger superheating, as will be shown below.

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 diﬀerent heating 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, ˆtmf–ˆtsm, versus heating rate for different core radii and M = 13.3.

The heating-rate dependence of Tma and the maximum For all heating rates, Tma and Tms increase with decreasing M superheating temperature at the center of the particle before becauseofincreasingpressureinthe core and reducing tensile 11 1 melting, Tms, for the three values of M are shown in Fig. 10(b). stresses due to a thicker shell. For Q o 10 Ks , the heating-rate

For M = 20, temperatures Tma and Tms almost coincide for dependence of both characteristic temperatures is weak. Tms 9 1 11 1 Q r 10 Ks , which means that the hoop stress in the shell increases with increasing heating rate for Q Z 10 Ks and Tma reaches its ultimate strength almost at the end of the complete increases with increasing heating rate for Q Z 1012 Ks1 for all M. melting of a core. Temperature Tma is higher than Tms for The wavy oscillation of s2, which originates from elastic Q r 1011 Ks1 of M =13.3andQ r 1011 Ks1 for M =10,i.e., waves traveling in the core, appears for Q 4 1012 Ks1. Note fracture occurs after complete melting of the core. For all other that the dynamic equation is used from Q =1012 Ks1 and no cases, Tma o Tms and the fracture of the shell occurs before oscillation is observed for this heating rate. The wave charac- complete melting of the core, followed by propagation of the teristics depend on the particle radius: oscillations of R =20nm pressure reduction wave, which can result in high tensile for Q =1013 Ks1 in Fig. 11(c) have the shortest period in terms pressure.31 This process strongly depends on the fracture time of T and it increases as R increases. However, the magnitude of and will be studied elsewhere. Also, since it is highly probable the wave is small for all tested cases, hence the eﬀect of elastic that the ultimate strength of the few nm thick alumina shell has waves on Tma is quite limited. significant scatter, the part of curves for Tms that are above the While the maximum superheating temperature is larger for curves for Tma still may have physical sense for higher ultimate larger particles and increases with the increase in the heating strength. If a shell is strong enough to contain complete melt for rate (Fig. 10(d)), the maximum attainable temperature, Tma, for all heating rates or weak enough to break down before melting Q r 1011 Ks1 has an opposite trend, i.e., it increases with the starts, Tma will be independent of the heating rate. reduction in the core radius (Fig. 10(d)). This happens due to

surfacetensionatthesurfaceSandtheinterfaceI,whichproduce Temperatures Tma and Tms (or times ˆtma and ˆtms) are compressive hoop stress in the oxide shell which increases with independent of Q for Q =108 Ks1 and Q =109 Ks1, and, 9 1 reduction in Ri. For example, the hoop stress at the interface I for consequently, at least for Q r 10 Ks . This is important for

Ri =20nmatT0 is 0.38 GPa in comparison with 0.13 GPa for the analysis of the melt-dispersion mechanism, because the 8 1 Ri = 60 nm, which delays the fracture of the oxide shell. This is to estimated heating rate at the reaction front was 10 Ks in some extent similar to pre-stressing of the Al core–shell structures ref. 31. For Q r 109 Ks1 and M = 20, the condition for com- by relaxing internal stresses by annealing at a higher temperature plete melting before oxide fracture is almost met, i.e. Tma B Tms, and quenching to ambient temperature in order to suppress which is consistent with ref. 31 and 53. However, for M = 20 and fracture and enhance the melt-dispersion mechanism.44,52 The Q Z 1011 Ks1, fracture occurs well before completing melting. trend of Tma in the size effect, however, has a crossover for higher Since all other cases in Table 3 have smaller M than 19, melting heating rates at 1011 Ks1 r Q r 1012 Ks1 in Fig. 10(d). The completes before fracture for Q r 1011 Ks1, excluding 11 1 reason for the crossover can be deduced from the evolution of the particles with Ri = 60 nm and M = 13.3 for Q =10 Ks . Note maximum hoop stress in the shell during heating in Fig. 11(c) just that Q =1011 Ks1 is the estimated heating rate in experiments by analyzing the position of the intersection point of the stress in ref. 39 and 40. For Q Z 1012 Ks1 (which is typical of the 9 1 s2/sth with the horizontal line s2/sth =1.0.ForQ =10 Ks , experiments in ref. 38), fracture occurs before completion of melting completes before fracture for all Ri, and the temperature at melting for all cases, which is far from optimal for melt dispersion. 11 12 1 the intersection (which is Tma) is higher for smaller Ri (as we Consequently, there is an upper bound of Q r 10 10 Ks for discussed, due to interface stresses). For Q =1012 Ks1,melting optimal melt dispersion, in contrast to the previous wisdom that almost completes at fracture points for Ri =20nmanddoesnot the larger Q is the better. 9 1 complete for Ri =40nmandRi = 60 nm; the temperature at the Thus, while predictions for Q r 10 Ks from the simpli- intersection point is lower as the radius is smaller, which results in fied theory in ref. 31 are confirmed by the current and more crossover. For Q 4 1012 Ks1, fracture occurs during melting, and again, temperature at the intersection point reduces with reducing radius Ri. The increase in the normalized melting time in Fig. 11(d) has an important role in the increase of Tma since volumetric transformation strain for the same temperature becomes smaller and so does hoop stress in a shell. That is why the slope of the curves in Fig. 11(c) increases with reduction in the core radius Ri.

8 Relationship to the melt-dispersion mechanism of the reaction of aluminum nanoparticles

The understanding of the melting of Al nanoparticles at high heating rates is very important for understanding and controlling the mechanisms of their combustion.37,41 According to the melt- dispersion mechanism of the reaction of aluminum nanoparticles describedinRef.31,37and41andIntroduction,highpressurein the melt and hoop stress in the shell, caused by the volume increase during thermal expansion and melting, break and spall the protective alumina shell, which traditionally suppresses the Al reaction with an oxidizer. Following dispersion of the molten core further promotes contact of Al with the oxidizer and drastically increases the reaction rate and flame speed. The main desirable condition in optimizing this mechanism is that the fracture of the shell occurs after complete melting of the Al core because only molten Al disperses and participates in the fast reaction. Thus, it is desired that Tma Z Tms (or tma Z tms)andthatTma (or tma)should not be sensitive to some scatter in geometric parameters of the core shell system and strength of the shell. In ref. 31, fracture of the shell occurred after complete melting for M r 19 and this result weakly depends on d and R separately since a simplified method of analysis for fracture of the shell in the previous research was @f @f Fig. 12 Evolution of (a) distribution and (b) volume integral of for an independent of the heating rate. Based on our much more precise @t @t 9 1 results in Fig. 10(b) and Table 3, we can conclude the following. Al nanoparticle with Ri =40nmandd = 3 nm for Q =10 Ks .

28848 | Phys. Chem. Chem. Phys., 2016, 18, 28835--28853 This journal is © the Owner Societies 2016 PCCP Paper precise simulations, results for Q Z 1011 Ks1 diﬀer quantita- with and without an oxide shell, and for all geometric para- tively and qualitatively. meters, as it follows from Table 3. The maximum temperature 9 1 Also, Table 3 contains data on the maximum rate of the hoop drop of Tmf Tms = 14.6 K is for Q =10 Ks , Ri = 20 nm, and 8 1 strain in the shell at the interface I, e_oxm,2.ForQ =10 Ks ,itisin d = 1.5 nm. This temperature drop comes from acceleration the range of 45 103 s1, which may be high enough for of melting when the interface reaches the center of a particle, r avoiding relaxation processes. It was roughly estimated in ref. 31 as T eq drastically reduces, and interfaces become incomplete 4 3.3 10 . The almost order of magnitude reduction in e_oxm,2 here (i.e., maximum Z reduces to smaller than 1). Fig. 12(a) shows is attributed to the more precise approach and insignificant @f a large negative magnitude of the local near the particle growth of temperature during melting with resultant reduction @t @f of eﬀective Q. Generally, e_oxm,2 is scaled proportionally to Q.Thus, center. Volume integration of the local presented in Fig. 12(b) for Q =107 Ks1 and even for Q =106 Ks1, for which melt- @t 2 1 takes into account that melting occurs within a smaller volume dispersion is still expected in ref. 31 and 41, e_oxm,2 is 45 10 s and 4050 s1, respectively. when the interface propagates toward the center. They show how melting at the center of a sample is drastically accelerated, which eventually results in the temperature drop of the particle 9 Temperature drop at the completion through eqn (13). For higher heating rates, the temperature of melting drop is absent and Tms = Tmf (Table 3). A similar but much larger temperature drop was observed at completion of melting An abrupt decrease in temperature by several degrees is observed of a plane nanolayer, when two solid–melt interfaces collided, for at the end of melting for Q =108 Ks1 and Q =109 Ks1, all heating rates (from 1.5 1010 Ks1 to 8.4 1013 Ks1)studied

Fig. 13 The map of physical phenomena aﬀecting melting temperatures (a) for Q r 109 Ks1 and (b) for Q Z 1012 Ks1.

This journal is © the Owner Societies 2016 Phys. Chem. Chem. Phys., 2016, 18, 28835--28853 | 28849 Paper PCCP in ref. 22 since the volume of the colliding region for a plane according to the Gibbs–Thomson effect. The reduction of the structure is much larger than that for a spherical structure. radius of a core Ri in turn causes reduction of Tma due to shifting volumetric expansion and the corresponding stress increase in a 10 Physical phenomena involved shell to lower temperature if s2 reaches sth during melting. (b) For small particles (Ri = 20 nm), surface tension at the in melting and superheating of core–shell interface and the external shell surface produces Al nanoparticles pressure in a core, which increases all melting temperatures and, consequently, Tma,ifs2 reaches sth during melting. In this section, we summarize the effect of different parameters (c) A decrease in M (an increase of thickness of oxide) causes on the characteristic melting temperatures (Fig. 13) and maxi- an increase in pressure in a core and an increase of all melting mum attainable temperature, Tma, which is determined by the temperatures and, consequently, Tma,ifs2 reaches sth during fracture of the oxide shell (Fig. 14). The maps are presented for melting. two ranges of the heating rates: (a) for Q r 109 Ks1, when the For low heating rates of Q r 109 Ks1, in addition to the effect of Q is absent and melting is quasi-equilibrium, and (b) for above effects: Q Z 1012 Ks1, when effects of the heating rates are pronounced. (a) Surface tensions at the core–shell interface and the For the intermediate heating rate, Q =1011 Ks1, the effect of external shell, and a decrease in M also decrease tensile stresses

Q appears but is weak. in a shell and, consequently, increases Tma. The same is true for In all maps: high heating rates.

(a) Reduction of the radius of a core Ri and the solid–melt (b) There is a small reduction in temperature and, con- interface radius ri leads to reduction in melting temperatures sequently, melt finish temperature Tmf at the end of melting at

9 1 12 1 Fig. 14 The map of physical phenomena aﬀecting the maximum attainable temperature, Tma, (a) for Q r 10 Ks and (b) for Q Z 10 Ks .

11 1 the core center. This reduction does not practically affect Tma; (3) Heating rates Q Z 10 Ks trigger kinetic superheating it is not visible for high heating rates because of kinetic (i.e., increase in melting temperatures Tsm, Tbm, Tmf, and superheating. Tms with increasing Q), and the effect becomes obvious for For high heating rates, in addition to the above effects: Q Z 1012 Ks1. (a) Kinetic superheating leads to an increase of all melting (4) The heating rates Q Z 1012 Ks1 produce heterogeneity 13 1 temperatures and Tma if the oxide shell fractures during melting. in the temperature distribution; thus, for Q =10 Ks , the Heterogeneity in temperature in a shell reduces pressure and temperature difference is 38.4 K across the 40 nm core and heating rate in a core but this effect is small in comparison with 225 K across a 3 nm oxide layer. The heterogeneity in tempera- kinetic superheating. ture creates a colder core and corresponding to the pressure (b) An increase in the core radius significantly increases drop in a core, which leads to lowering of superheating tem- kinetic superheating and the normalized time for complete peratures. It can change the heating rate across the core radius melting, and hence it increases Tms and Tma if the oxide shell so that Qi 4 Q. However, the effect of heterogeneity is minor fractures during melting. relatively to kinetic superheating. For bare particles, the effects of M, d, and surface tension at (5) The heating rates Q Z 1012 Ks1 also result in wave

S, as well as temperature Tma are irrelevant. Elastic waves have a propagation within the core, which causes oscillation in pressure much smaller magnitude than for particles with the shell and and temperature (due to thermoelastic coupling), but the effect practically do not affect melting temperatures. The main effects of wave on melting temperatures and Tma is relatively small for are a reduction in melting temperatures due to reduction in tested heating rates.

Ri (Gibbs–Thomson effect), kinetic superheating for high Q,and (6) A reduction in M = Ri/d increases the pressure growth in a small increase in melting temperatures for small particles due the core and leads to the increase of melting temperatures for to pressure caused by surface tension at Ri. all tested Q. (7) For Q r 1011 Ks1, the oxide shell fractures after

complete melting and the heating-rate dependence of Tma is 11 Conclusion very weak. Temperature Tma increases with reduction of M which decreases tensile hoop stress in the shell. Tma increases with the In the paper, we utilize our model10,11,22 that includes the phase reduction in the core radius due to surface tension at the surface field model for surface and bulk melting, dynamic equation of S and interface I which produce compressive hoop stress in motion, mechanical model for stress and strain simulations, the oxide shell. If oxide fractures before completing melting interfacial and surface stresses, and the thermal conduction (for Q 4 1011–1012 Ks1), the maximum attainable temperature, model with thermo-elastic, thermo-phase transformation coupling Tma, depends on the heating rate. It increases with an increase in and transformation dissipation rate, to study the eﬀects of the heating rate if melting temperatures increase, which delays geometric parameters and heating rate on characteristic melting the stress growth due to transformation expansion. A thicker and superheating temperatures and melting behavior. Several shell increases the Q for which heat-rate dependence of Tma starts, unconventional phenomena are revealed. The main results are and a larger Ri strengthens this dependence due to augmented enumerated below. kinetic superheating and increased melting time. (1) In contrast to the plane interface, the spherical interface (8) A parametric study of high-heating rate melting allows us exhibits a strong decrease of equilibrium temperature at the to shed light on the melt-dispersion mechanism of combustion r 31 interface, Teq,asri approaches zero. This leads to a temperature of an Al nanoparticle. It is desired for the promotion of drop for Q =108 Ks1 and Q =109 Ks1, which is slightly Al reactivity that oxide breaks after complete melting of the 8 1 9 1 diﬀerent for Q =10 Ks and Q =10 Ks at the end of particle, i.e., Tma Z Tms. Tma and Tms are independent of Q for melting. Excluding the end of melting, curves Tˆ(ˆt)forQ =108 Ks1 Q r 109 Ks1, and for M = 20, the condition for complete and Q =109 Ks1 coincide. melting before oxide fracture is met or almost met. However, for (2) Increasing pressure within an Al core due to a less M =20andQ Z 1011 Ks1, fracture occurs well before completing thermally expanded oxide shell and, after initiation of melting, melting. For M r 13.3, melting completes before fracture for 11 1 due to a transformation volumetric expansion of 0.06 leads to Q r 10 Ks , excluding particles with Ri =60nmandM = 13.3 for increasing melting temperatures, rationalized by the Clausius– Q =1011 Ks1.ForQ Z 1012 Ks1, fracture occurs before

dT Teq 1 completing melting for all cases under study. An interaction with Clapeyron relationship, ¼ e0t ¼ 60 K GPa . Increasing dp H an elastic wave shows the oscillating evolution of s2 but its eﬀect on the fraction of melt during heating increases pressure and, Tma is minor for all tested Q. Consequently, there is an upper bound consequently, the melting temperature of the particle even of Q r 1011–1012 Ks1 for optimal melt dispersion, in contrast to without kinetic superheating. This is confirmed by coincidence previous wisdom that the larger Q is better. While for Q r 109 Ks1 of the Tˆ(ˆt)curvesforQ =108 Ks1 and Q =109 Ks1.Incontrast, predictions of the simplified theory in ref. 31 are confirmed by the for a bare particle, temperature slightly decreases for Q =108 Ks1 current more precise simulations, for Q Z 1011 Ks1 results are 9 1 and Q=10 Ks due to reduction in ri. For small particles, there quantitatively and qualitatively different. The maximum rate of the is a small temperature increase due to pressure caused by surface hoop strain in the shell, which characterizes the possibility of tension at the core external surface. avoiding stress relaxation before fracture, is determined.

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