PCCP PAPER. Yong Seok Hwang a and Valery I. Levitas* b

Transcription

1 PAPER Cite this: Phys. Chem. Chem. Phys., 2016, 18, Superheating and melting within aluminum core oxide shell nanoparticles for a broad range of heating rates: multiphysics phase field modeling Yong Seok Hwang a and Valery I. Levitas* b Received 5th June 2016, Accepted 23rd September 2016 DOI: /c6cp03897b 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 GPa 1. For the heating rates Q r 10 9 Ks 1, melting temperatures (surface and bulk start and finish melting temperatures, and maximum superheating temperature) are independent of Q. For Q Z Ks 1, increasing Q generally increases melting temperatures and temperature for the shell fracture. Unconventional effects start for Q Z Ks 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 corresponding schemes. Melting temperature of materials and melting mechanisms depend on various parameters: size, shape, condition at the surface, pressure (or, more generally, stress tensor), and heating rate, as well as on their interaction. Melting temperature depression with reduction of the particle radius is well-known from experiments, 1,2 thermodynamic treatments, 1,2 molecular dynamics simulations, 3,4 and phase field studies without mechanics 5,6 and with mechanics (but without inertia effects). 6,7 Reduction in surface energy during melting leads to premelting below melting temperature followed by surface melting and solid melt interface propagation through the entire sample with increasing temperature. This was studied using the phase field approach for the plane surface analytically 8,9 and numerically a Department of Aerospace Engineering, Iowa State University, Ames, Iowa 50011, USA. yshwang@iastate.edu b Departments of Aerospace Engineering, Mechanical Engineering, and Material Science and Engineering, Iowa State University, Ames, Iowa 50011, USA. vlevitas@iastate.edu; Fax: ; Tel: (including the effect of mechanics) for low 6,7 and high 10,11 heating rates. Similar studies were performed for spherical particles without mechanics 5,6 and with mechanics in quasistatic formulation. 6,7 Strong effects of the width of the external surface and thermally activated nucleation were revealed within the phase field approach in ref. 12. If the external surface of the material under study represents an interface with another solid, surface melting depends on the type of interface. The low-energy coherent interfaces increase energy during melting and, consequently, suppress surface nucleation and promote superheating. 13,14 On the other hand, an incoherent interface, whose energy reduces during melting, promotes surface melting. 2 Hydrostatic pressure inside a shell which can be created for materials with volume expansion during melting suppresses melting and increases equilibrium melting temperature T p eq according to the Clausius Clapeyron relationship. The effect of pressure appears automatically within the phase field approach if proper thermodynamic potential is implemented. 6,15 Under non-hydrostatic internal stresses that relax during melting, This journal is the Owner Societies 2016 Phys. Chem. Chem. Phys., 2016, 18,

2 Paper e.g. under biaxial stresses due to constraint, melting temperature reduces. See thermodynamic 11,16 and phase field 11 treatments for a layer. Melting temperature drastically decreases during very high strain-rate uniaxial compression in a strong shock wave, as it was predicted thermodynamically and confirmed by molecular dynamics simulations. 17 Metal can be kinetically superheated above its equilibrium melting temperature when it is subjected to an extremely fast heating rate, for example, during irradiation by an ultra-fast laser with high energy, such as picosecond (ps) and femtosecond (fs) lasers. It has been observed in experiments and phase field simulations 10,11 that an aluminum layer can be superheated up to at least 1400 K, 20 which is far above its equilibrium temperature, T eq = K. The major reason for kinetic superheating, when heterogeneous surface melting initiates the process, is the slower kinetics of solid melt interface propagation than heating. 10,21 For very high heating rates Q Z Ks 1, elastic wave propagation can affect the temperature of the material through thermoelastic coupling and melting temperature through the effect of stresses. 22 Melting also influences the temperature of materials through thermo-phase transformation coupling, mostly due to latent heat. Thus, an analysis of kinetic superheating of materials should take several physical processes and their couplings into account. Recently, there has been some research and suggested models to describe ultra-fast heating and melting with or without mechanics, including thermoelastic coupling or thermophase transformation coupling. However, those models neither describe a complete set of participated physical phenomena nor include correct coupling terms rigorously derived from the thermodynamic laws. Recently, we have developed a novel phase field model, which includes all of the above physical phenomena and couplings in a single framework. 10,11,22 However, all the above modeling results have been obtained for melting bare metallic nanostructures. In reality, metallic (e.g., Al, Fe, Cu, and others) particles and layers have a strong passivation oxide layer at the external surface. Thus, nanoparticles form a core shell structure. The aluminum oxide or alumina passivation layer can be formed even at room temperature 28 by transporting Al cations driven by the nonequilibrium electrostatic field, the so-called Cabrera Mott mechanism. 29,30 Aluminum oxide has a lower thermal expansion coefficient than the aluminum core, so the compressive pressure in the core and the tensile hoop stress in the oxide shell are generated due to volumetric expansion during heating before melting. 31 Since melting of Al is accompanied by a volumetric strain of 6%, pressure of several GPa can be obtained in the melt and hoop stress in the alumina shell is on the order of magnitude of 10 GPa. High pressure in the core results in an increase in the melting temperature according to the Clausius Clapeyron relationship. The generated pressure depends on the ultimate strength of the shell and relaxation processes in it, including phase transformations from amorphous alumina to crystalline g and d phases. Thus, slow heating of Al nanoparticles with an oxide layer at 20 K min 1, depending on the Al core radius (R i ) and the oxide shell thickness (d), leads to a wide spectrum of behavior from the reduction of the melting temperature due to size effect to a minor superheating of up to 15 K due to sufficient time for stress relaxation. Stress measurement in Al nanoparticles was performed in ref. 32 and In contrast, fast heating with the rate higher than K s 1 can lead to the estimated superheating by several hundred K due to the pressure increase 37 because there is not sufficient time for phase transformations and other stress relaxation mechanisms in the shell. For the higher heating rates of Ks 1 used in experiments, both the pressure-induced increase in melting temperature and kinetic superheating are expected. Thus, for understanding and quantifying melting of metallic nanoparticles in a broad range of heating rates, one has to include and study the effect of an oxide shell and major physical processes involved in melting, in particular, the effect of the generated pressure, kinetic superheating, heterogeneity of temperature and stress fields, dynamics of elastic wave propagation, surface and interface energies and stresses, and coupling of the above processes. This is a basic outstanding multiphysics problem to be solved. Most of these processes strongly depend on the core radius R i and the oxide shell thickness d; thus, their effect should be studied in detail. The understanding of the melting of Al nanoparticles at a high heating rate is also very important for understanding and controlling the mechanisms of their oxidation and combustion. 31,37,41 According to the melt-dispersion mechanism of the reaction of Al particles, 31,37,41 high pressure in the melt and hoop stresses in the shell, caused by the volume increase during melting, break and spall the alumina shell. Then, the pressure at the bare Al surface drops to (almost) zero, while pressure within the Al core is not initially altered. An unloading spherical wave propagating to the center of the Al core generates a tensile pressure of up to 3 GPa at the center, which reaches 8 GPa in the reflected wave. The magnitude of tensile pressure significantly exceeds the cavitation strength of liquid Al and disperses the Al molten core into small bare drops. Consequently, the melt-dispersion mechanism breaks a single Al particle covered by an oxide shell into multiple smaller bare drops, which is not limited by diffusion through the initial shell. This mechanism was extended for micron-scale particles 42,43 and utilized for increasing the reactivity of Al nano- and micron-scale particles by their prestressing. 44,45 However, there has been no research for melting and kinetic superheating of Al nanoparticles within an oxide shell at high heating rates to the best of our knowledge. In this paper, we study superheating and melting of Al nanoparticles covered by an alumina shell and the corresponding physical processes under high heating rates. We utilize our recent model 10,11,22 that includes the phase field model for melting developed in ref. 6 and 7, a dynamic equation of motion, a mechanical model for stress and strain simulations, and the thermal conduction model with thermo-elastic and thermo-phase transformation coupling, as well as with a dissipation rate due to melting. 10,11 The effects of geometric parameters (which determine the stress-state and temperature evolution) Phys. Chem. Chem. Phys., 2016, 18, This journal is the Owner Societies 2016

3 and heating rate on the characteristic melting and superheating temperatures and melting behavior, as well as on the maximum temperature corresponding to fracture of the shell, are simulated and analyzed by a parametric study. Several nontrivial and unconventional phenomena are revealed. The influence of the above parameters at the initial stage of the melt-dispersion-mechanism of reaction of Al nanoparticles 31 is evaluated and discussed. 1 Models 1.1 Governing equations Due to high heating rates and short melting time, we will neglect crystallization of the amorphous alumina shell and any inelastic flow in it. The model consists of the phase field equation, the equation of motion, equations of elasticity theory, and the heat conduction equation; all of them are coupled. The phase field equation is applied only to the aluminum core since melting in aluminum oxide starts at a much higher temperature (2324 K 46 ). We designate contractions of tensors A ={A ij }andb ={B ji } over one and two indices as AB ={A ij B jk }anda:b = A ij B ji, respectively; I is the unit tensor, r and r 0 are the gradient operators in the deformedandundeformedstates,and# designates the dyadic product of vectors. Equations for phase transformation and deformation. Total strain tensor e =(r 0 u) s (where u is the displacement vector and the subscript s designates symmetrization) can be additively decomposed into elastic e e, transformation e t, and thermal e y strains: e = e e + e t + e y ; e = 1/3e 0 I + e; (1) e in = e in I = e t + e y ; e t = 1/3e 0t (1 f(z))i; (2) f = Z 2 (3 2Z) for 0 r Z r 1; e y = a s (T eq T 0 )I +(a m + Daf(Z))(T T eq )I. (3) Here, Z is the order parameter that varies from 1 in solid to 0 in melt, a s and a m are the linear thermal expansion coefficients for solid and molten Al, respectively, Da = a s a m, T 0 is the initial temperature, e 0 is the total volumetric strain, e 0t is the volumetric transformation strain for complete melting, and e is the deviatoric strain. The definition of f is modified to ensure a single minimum of free energy for Z o 0 and Z 4 1 in the case of T 4 1.2T eq or T o 0.8T eq ; see below. The Helmholtz free energy per unit undeformed volume is formulated as in ref. 7 and 10: c ¼ c e þ J c y þ c y þ Jc r ; c y ¼ AZ 2 ð1 ZÞ 2 ; (4) c e = 0.5Ke 0e 2 + me e :e e ; c y = H(T/T eq 1)f(Z); (5) c r = 0.5b rz 2, A:= 3H(1 T c /T eq ). (6) Here, c e, c y, c y, and c r are the elastic, thermal, double-well, and gradient energies, respectively; r 0 and r are the mass densities in the undeformed and deformed states, respectively; J = r 0 /r =1+e 0 ; K(Z) =K m + DKf(Z), and m(z) =m s f(z) are the bulk and shear elastic moduli, DK = K s K m ; b is the gradient energy coefficient; H is the latent heat; T c is the melt instability temperature assumed to be 0.8T eq. Using thermodynamic procedures, the following equations for the stress tensor r is obtained: J ¼ r e þ r st ; (7) r e ¼ Ke 0e I þ 2me e ; s st ¼ c r þ c y I brz rz; (8) which consists of elastic stress r e and interface stresses r st. The same procedure leads to the Ginzburg Landau equation: ¼ J 1@c L þr J ¼ J e 0t p e þ 3p e Da T T J 1 0:5DKe 2 0e þ me e :e e þ H 1 T 4AZð1 ZÞð0:5 ZÞþbr 2 Z; where L is the kinetic coefficient, p e = r e :I/3 is the mean elastic stress, and p = p e is the pressure. For the aluminum oxide shell, eqn (1), (2), (3) and (7) are replaced by eqn (10) and (11): (9) e = e e + e y ; e y = a ox (T T 0 )I, (10) r = r e = K ox e 0e I +2m ox e e. (11) Equation of motion. The momentum balance equation is accepted in a traditional form: u ¼rr: If the time scale of an elastic wave is much smaller than that of melting, the equation is replaced by the static equilibrium equation: rr =0. Heat transfer equation. The energy balance equation can be presented in the form of the following temperature evolution equation, in which thermal conduction is assumed to be the only mechanism of heat ¼rðkrTÞ 3Tða m þ 2 3p e Da ; Paper (13) where C is the specific heat and k is the thermal conductivity. The time delay between electron gas and phonon 47 is ignored due to the fact that time scale is much longer than a few picoseconds in these simulations. In eqn (13), the second term on 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 last term is the heat source due to melting. For an aluminum oxide shell, only a thermoelastic coupling term is used. This journal is the Owner Societies 2016 Phys. Chem. Chem. Phys., 2016, 18,

4 Paper 1.2 Numerical model, boundary and initial conditions, and material properties Geometry. A 1D model in spherical coordinates is used to simulate Al superheating and melting in the Al core alumina shell structure. Fig. 1 shows a 1D geometry, where R i is the radius of the aluminum core, d is the oxide shell thickness, and R s = R i + d. Computational methods. The finite element method code COMSOL Multiphysics 48 is used for numerical simulation. Two time discretization methods, Backward Differentiation Formula (BDF) and Generalized Alpha, used in COMSOL led to the same results; therefore, the first order BDF is selected for the stability of computation. The quadratic Lagrangian elements were sufficient to resolve a solid melt interface, and higher order elements made no improvement in the accuracy of the solution. Both the time step and the size of the finite element have been reduced until solutions with different discretization coincide. The time step was inversely proportional to the heating rate and provided at most 2 3 K increments per time step. It has been found that at least 5 elements are necessary to resolve the solid melt interface, which is about 2 nm wide for Al. Increasing the number of elements inside the interface did not result in a noticeable improvement of the solution. Boundary and initial conditions. The following boundary conditions at points C and S, as well as jump conditions at point I are applied. At r ¼ ¼ 0; u ¼ 0; h ¼ 0: At r = R i : u 1 = u 2 ; s r,1 s r,2 = 2g cs /R i ; T 1 = T 2 ; h 1 = h 2. (15) n ¼ brz n ¼ dg cs dz ; g csðzþ¼g m þ ðg s g m ÞfðZÞ: (16) At r = R s : s r,2 = 2g ox /R s + p g ; h 2 = h*. (17) Here n is the unit normal to the interface, which coincides with the radial direction; subscript 1 is used for the Al core and subscript 2 for the alumina shell. Eqn (14) describes traditional conditions 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 is imposed, as well as jump in the normal stress component to the interface is caused by interface stress. For both internal and external surfaces, we assume that interface stress is equal to the interface energy. The boundary condition for the order parameter (16) is related to the change in the Al alumina interface energy g cs during melting, when it changes from g s for solid Al to g m for molten Al. 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). Thus, melting cannot start without some perturbations at the boundary. We introduce perturbation ^Z =10 6 and the condition that if Z 4 1 ^Z at the boundaries, then Z =1 ^Z. This condition prevents the disappearance of the initial perturbation when heating occurs below the melting temperature. The perturbation can bring about numerical oscillation, which can cause the order parameter to grow larger than unity when T 4 1.2T eq once Z 4 1. This happens because the traditional definition of f, f = Z 2 (3 2Z), while fully satisfactory for 0 r Z r 1, creates an unphysical minimum of the local order parameter-dependent part of the energy, c y þ c y for Z 4 1, as shown in Fig. 2(a). To prevent the unphysical minimum, function f is modified for Z 4 1 and T 4 T eq to f =2 Z 2 (3 2Z)in order to eliminate the artificial minimum, as illustrated in Fig. 2(a). The reinforced function f also produces a larger driving force, leading to returning f to the range Z r 1 for T eq r T r 1.2T eq. Also, the traditional definition of f can make unphysical growth of Z below zero when T o 0.8T eq once Z o 0. In this case, function f = Z 2 (3 2Z) for Z o 0 can guarantee a single minimum and stability of the computation. In summary, f =2 Z 2 (3 2Z) for Z 4 1 and T 4 T eq ; f = Z 2 (3 2Z) for Z 4 1 and T r T eq. (18) f = Z 2 (3 2Z) 2 for Z o 0 and T 4 T eq ; f = Z 2 (3 2Z) for Z o 0 and T r T eq. (19) At the external surface, a jump in normal stress from the value of the gas pressure p g to the radial stress in a shell s r,2 due to surface tension is applied; we use p g = 0 in simulations. External time-independent heat flux h* is prescribed and its magnitude is iteratively chosen in a way that it produces the desired heating rate at point C. The initial temperature is T 0 = K, initial stresses are zero, and the initial order parameter is Z = for all cases. Material properties. The specific heat of aluminum, C = C m + (C s C m )f(z); according to ref. 49, C s = ( ( )/( ) (T 300.0)) 10 3 J(m 3 K) 1 for T o K; (20) C s = J(m 3 K) 1 for T Z K; (21) C m = ( ( )/( ) (T 933.0)) 10 3 J(m 3 K) 1, (22) Fig. 1 Domain for 1D simulation, where C, I, S represent the center, the interface and the surface, respectively. where C s and C m are specific heats of solid and molten aluminum, respectively. The specific heat of aluminum oxide is assumed to be temperature independent, C ox = J(m 3 K) Phys. Chem. Chem. Phys., 2016, 18, This journal is the Owner Societies 2016

5 Paper Table 2 Properties of aluminum oxide 50,51 r ox (K m 3 ) K ox (GPa) m ox (GPa) a ox (K 1 ) g ox (J m 2 ) 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, 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 Some definitions Fig. 2 Comparison of the thermal part of the free energy, c y þ c y, landscape corresponding to the traditional function f = Z 2 (3 2Z) (solid lines) and reinforced definition of f in eqn (18) and (19) for (a) T 4 1.2T eq and (b) T o 0.8T eq (dashed lines). The thermal conductivity of aluminum, k s = 208 W (m K) 1 for solid and k m = 102 W (m K) 1 for melt, 49 is used and k ox = W (m K) 1 for amorphous aluminum oxide. Coefficients, constants, and other properties used for simulation are included in Tables 1 and 2. They correspond to the width, energy, and mobility of a plane solid melt interface of d sm = 2.02 nm, g sm = 0.14 J m 2, and l sm = 1.7 m s 1 K 1. Material parameters for melting of Al nanoparticles were taken from our papers, 6,10,22 where they were justified from the known experimental and molecular dynamic simulations data from the literature. Validity of a phase field melting model was justified by reproducing the experimental results on radiusdependent melting temperature of Al nanoparticles (down to radius of 2 nm) and the width of the surface melt versus 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, of10 8,10 9,10 11,10 12, ,10 13 Ks 1 are selected to explore kinetic superheating. As a base case, we consider the Al core radius R i =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 R i =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 toward the center (Fig. 3). Homogeneous melt nucleation away from the solid melt interface was not observed here even above the solid instability temperature T si =1.2T eq = 1120 K, because bulk fluctuations were not introduced and interface propagation completes melting before any homogeneous nucleation becomes visible.thesameistrueforabareparticle. The reduced temperature, ^T ¼ T, and time, ^t ¼ t, are T eq t eq defined with normalization using bulk equilibrium temperature, T eq = K, and the time required to reach this temperature, t eq. The heating rate for the core shell structure is defined either as Q ¼ T eq T 0 at the center of the particle or t eq as Q i ¼ T eq T 0 t i, where t i eq is the time to reach T eq at the eq interface I, if there is a significant heterogeneity in the temperature distribution. For a bare particle, the heating rate is defined as Q ¼ 900 T 0, where t 900 is the time to reach 900 K at t 900 Table 1 Properties of aluminum 11 r 0 (kg m 3 ) T eq (K) H (J m 3 ) K m (GPa) K s (GPa) m (GPa) e 0t a m (K 1 ) a s (K 1 ) g s (J m 2 ) g m (J m 2 ) b (N) L (m 2 N 1 s 1 ) This journal is the Owner Societies 2016 Phys. Chem. Chem. Phys., 2016, 18,

6 Paper Fig. 3 Propagation of the solid melt interface during melting for an Al nanoparticle with R i =40nmandd =3nmatQ =10 11 Ks 1. the center, since temperature does not reach T eq due to the Gibbs Thomson effect. The surface melting start mark, T sm,in all the following figures represents the initiation of surface premelting when the order parameter reaches 0.5 for the first time at the interface I. The melting finish temperature, T mf,is defined at the time when the order parameter reaches 0.5 for the first time in the center C. The bulk melting start temperature, T bm, 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, T ms,whichisthemaximum temperature of the center of the solid core during melting, and the maximum attainable temperature, T ma, 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 s 2 reaches the theoretical ultimate strength of alumina, s th = GPa. 31 Note that T ma is practically achievable maximum temperature of aluminum with the core shell structure, above which it ceases to exist. Characteristic times corresponding to each characteristic temperature are designated by the same subscripts. A summary of the main simulation results is presented in Table 3. 3 Superheating and melting of bare Al nanoparticles For comparison and interpretation, the phase equilibrium temperature, T r eq, corresponding to the interface radius r i and Table 3 Summary of simulation conditions and results R i (nm) d (nm) M Q (K s 1 ) a t eq tˆsm tˆbm tˆms tˆma tˆmf T sm (K) T bm (K) T ms (K) T ma (K) T mf (K) _e oxm,2 (s 1 ) ms ns ns ps ps ms ns ns ps ps ms ns ns ps ps ms ns ns ps ps ms ns ns ps ps ms ns ns ps ps a t eq for the bare particle is defined as the time for T = 900 K Phys. Chem. Chem. Phys., 2016, 18, This journal is the Owner Societies 2016

7 defined from the thermodynamic equilibrium conditions for the stress-free case, H(Treq/Teq 1) = 2gsm/ri, is introduced. Thus, Treq reduces from Teq for the plane interface (ri - N) to zero for ri = 2gsm/H = nm, and for smaller ri the interface cannot be equilibrium. We also introduce phase equilibrium r temperature under the Laplace pressure p, Trp eq = Teq + DTp, where the Laplace pressure p = 2gm/Ri = GPa is produced by the melt-vapor spherical particle surface with Ri = 40 nm and DTp is the Clausius Clapeyron increase in the equilibrium temperature due to this pressure, DTp = GPa 60 K GPa 1 = 2.76 K. Fig. 4(a) shows the variation of these two phase equilibrium temperatures versus the radius of the propagating solid melt interface. Fig. 4(b) shows the evolution of temperature at the center of the bare particle for Q = 108 and 109 K s 1 and a comparison with equilibrium melting temperatures, Treq and Trp eq. The simulated Fig. 4 (a) The variation of phase equilibrium temperature for the solid melt interface under stress-free conditions Treq and under Laplace pressure Trp eq 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 melting in comparison with Treq and Trp eq for a 40 nm bare particle. The equilibrium melting temperatures, Treq and Trp eq, versus time were obtained by substitution of the simulated interface position at each time instant ri(t) in equations Treq(ri) and Trp eq(ri). This journal is the Owner Societies 2016 Paper curves for Q = 108 and 109 K s 1 are almost overlapped, which means there is no effect of kinetic superheating except at the very end of melting (which will be discussed below). Temperature decreases with time and slightly (by 1 K) exceeds the equilibrium curve Trp eq(r). Thus, temperature reduction is due to the thermodynamic effect of the interface radius on the phase equilibrium temperature. A small deviation cannot be considered a nonequilibrium effect because it is independent of the heating rate. It can be explained by the difference in the sharp interface model for Trp eq(r) and the finite-width solid melt interface in simulations. Note that for the plane solid melt interface within a nanolayer, for such heating rates melting occurs at a constant temperature equal to the equilibrium temperature under the corresponding stress.22 The strong decrease of Trp eq as ri approaches zero leads to 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 T (t ) for Q = 108 K s 1 and Q = 109 K s 1 (i.e., for different rates of heat supply) coincide and the rate of heat absorption is T, at the center (solid 900 K 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 the surface of a bare Al nanoparticle with Ri = 40 nm. Fig. 5 Evolution of (a) normalized temperature, Phys. Chem. Chem. Phys., 2016, 18,

8 Paper 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 =10 8 and 10 9 Ks 1, T bm = K (Tˆ = 0.996), is slightly below T eq due to radius-dependence of the melting temperature, i.e., the Gibbs Thomson effect. For higher heating rates, Q Z Ks 1, kinetic superheating is observable, i.e., the evolution of temperature starts to deviate from that of Q =10 8 Ks 1 and Q =10 9 Ks 1 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 prevailing kinetic superheating over a relatively small heat sink by accelerated melting in the small volume of the final core. For Q Z Ks 1, 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 = Ks 1. 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 Ks 1. 4 Superheating and melting of an Al core with a radius of 40 nm confined by an alumina shell with a thickness of 3 nm 4.1 Effect of confinement pressure on the melting of an Al nanoparticle The melting temperature of an Al nanoparticle with an oxide shell is neither T eq nor constant, as shown in the inset of Fig. 6(a), if a shell can sustain high pressure inside a core. Increasing pressure within a core due to a less thermally expanded shell and due to a transformation volumetric expansion of 0.06 in the melt leads to increasing melting temperature, rationalized by the Clausius Clapeyron relationship, dt dp ¼ e T eq 0t H ¼ 60 K GPa 1. The evolution of the core pressure at the interface I is shown in Fig. 7(a). The compressive pressure at tˆ = 1.13 (corresponding to T bm ) for cases without kinetic superheating is 1.03 GPa, which should result in the increase of the bulk melting temperature by 61.8 K in comparison with T bm = K for a bare particle, i.e., in This shows good agreement with the simulated T bm =986.1K(insetofFig.6(a)).Notethatthesurface melting start temperature is K (Fig. 6(a) and Table 3), Fig. 6 Evolution of temperature and characteristic melting temperatures for an Al nanoparticle with R i =40nmandd = 3 nm. (a) Evolution of temperature at the center of the Al core for different heating rates. The inset is for Q =10 8 Ks 1. (b) The surface melting start temperature, T sm, the bulk melting start temperature, T bm, the maximum attainable temperature, T ma, and the maximum superheating temperature, T ms, as functions of the heating rate. and Tˆ = 0.981, which is larger than T sm of a bare particle because of the pressure. After the start of surface melting and then bulk melting, the temperature increases (in contrast to that for a bare particle) since an increasing fraction of melt in a core increases pressure (Fig. 7(a)). The effect of pressure will be further elaborated for other R i and M. 4.2 Effect of the heating rate Our previous phase field studies 10,11 have demonstrated that ultrafast heating over Ks 1 of an aluminum nanolayer can kinetically superheat the material above the melting temperature. While two interfaces propagate from both surfaces of the 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 a similar mechanism, if the heating rate is fast enough. Since an elastic wave traveling within ps time scale can possibly affect Phys. Chem. Chem. Phys., 2016, 18, This journal is the Owner Societies 2016

9 Paper Fig. 7 Evolution of pressure in the Al core at the interface I for an Al nanoparticle with R i = 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 =10 13 Ks 1 and d = 3 nm. (c) Effect of M ¼ R i d for Q =1013 Ks 1 and Q =10 8 Ks 1. the temperature, and hence melting, of the particle by thermoelastic coupling, the dynamic equation of motion is incorporated into the model for Q Z Ks 1. Fig. 6(a) displays the evolution of temperature at the center for various heating rates. While curves for 10 8 Ks 1 and 10 9 Ks 1 are overlapped (i.e., there is no kinetic superheating), the curve for Q =10 11 Ks 1 shows a small deviation from them. This deviation becomes obvious as the heating rate increases. For Q Ks 1, the temperature is affected by elastic waves, so that small oscillations appear on the temperature evolution curve. The magnitude and normalized period of oscillations grow as the heating rate increases. While for Q r K s 1 temperatures of initiation and end of bulk melting are clearly detectable by inspecting the change in the slope in the temperature evolution curves, it is not the case for higher heating rates. Kinetic superheating for Q Z Ks 1 retards the beginning of melting in terms of tˆ and extends the normalized time period for melting, tˆmf tˆsm. For lower heating rates without kinetic superheating, the interface propagation is completely governed by the equality of supplied and latent heats since heating is slower than propagation. Thus, heating becomes the limiting process, the time period for melting is inversely proportional to Q, and the normalized time period for completing melting becomes independent of Q. For heating rates with kinetic superheating, solid melt interface propagation turns out to be slower than heating and the interface kinetics becomes the limiting process, which leads to the increase in the normalized time period for melting (refer to Fig. 11(d)). Fig. 6(b) shows the change of four characteristic temperatures versus the heating rate (the data are also summarized in Table 3). All four temperatures are practically independent of the heating rate for Q r 10 9 Ks 1, i.e., kinetic superheating is absent in any sense. Surface melting start temperature, T sm,is as low as K due to surface premelting, which is lower than This journal is the Owner Societies 2016 Phys. Chem. Chem. Phys., 2016, 18,

10 Paper the equilibrium temperature Teq p ¼ T T eq eq þ pe 0t, K, H predicted by the Clausius Clapeyron relationship. The equilibrium temperature is quite close to T bm, K, and the difference is mostly due to the size effect. The maximum superheating temperature, T ms, is K and is higher than the equilibrium temperature. The maximum attainable temperature before shell fracture, T ma, is K and higher than T ms, i.e., the oxide shell can withstand complete aluminum melting. Elevation of the characteristic temperatures for Q =10 11 Ks 1 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, T bm T sm, increases for Q =10 11 Ks 1 and Ks 1 but then reduces for Q Ks 1. T ms is largely affected by kinetic superheating and increases drastically. T ms is rapidly increased for Q Ks 1, but it may not be realized in experiments for the given geometric parameters because oxide shell fractures for Q K s 1 before completing melting. Including fracture in the model and studying melting during and after fracture will be pursued in the future work. Temperature T ma is independent of the heating rate for Q r Ks 1 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, s 2, reaches the ultimate stress during melting and T ma increases with increasing Q. 4.3 Effects of elastic wave and heterogeneous temperature distribution For Q r Ks 1, temperature within the Al core is practically homogeneous (Fig. 8a). For Q =10 11 Ks 1, the first temperature heterogeneity is observed in the oxide shell (because of lower heat conductivity of aluminum oxide than aluminum), but the temperature difference is only 2 K. For Q =10 12 Ks 1, the temperature heterogeneity in a core also becomes visible and it reaches 17.2 K in a shell. For Q =10 13 Ks 1, 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 is almost 225 K. An increasing heating rate and a consequent strain rate enable dynamic processes and elastic waves to influence the temperature. In geometries considered in this research, the order of magnitude of an acoustic time for the elastic wave to travel a particle is 10 ps (40 nm/(4 nm ps 1 ) = 10 ps, where R i = 40 nm and 4 km s 1 = 4 nm ps 1 is an estimated acoustic speed in aluminum). Thermoelastic coupling produces a visible effect on the temperature evolution with a small oscillating pattern in Fig. 6(a) for Q Ks 1. This correlates with the appearance of similar trends in the pressure evolution in Fig. 7(a): the initial reduction in pressure and pressure oscillations become obvious for Q K s 1. While pressure oscillations due to multiple wave propagations and reflections are not surprising, pressure reduction in a core is counterintuitive and intriguing, and the reasons for pressure reduction with increasing heating rates are not evident. Fig. 8 (a) Temperature distribution along the radial direction for different heating rates at a moment slightly before the surface melting starts for an Al nanoparticle with R i =40nmandd = 3 nm. (b) The effect of M = R s /d on the evolution of the temperature difference DT across the oxide shell between interface I and surface S for an Al nanoparticle with R i =40nmat Q =10 13 Ks 1. Elastic wave (inertia effect) and the temperature gradient in a shell may be considered as the possible causes for the pressure reduction in the core. In order to clarify this issue, melting of the particle with artificially large thermal conductivity in dynamic and quasi-static formulations has been simulated for Q =10 13 Ks 1 and compared with the base case. Thermal conductivity of both liquid and solid Al was increased by a factor of 10 3, and of alumina by a factor of The same heat flux provided for Q =10 13 Ks 1 for the base case was applied at the surface S. Fig. 7(b) shows the evolution of pressure for four cases, and it is clear that the large thermal conductivity eliminates the pressure drop. For the actual thermal conductivity, both quasi-static and dynamic solutions exhibit a pressure drop by GPa in a core, and the dynamic solution oscillates around the quasi-static one with a relatively small amplitude. Therefore, the pressure drop is not a result of inertia but of relatively slow heat conduction. Slow heat conduction initially delays heating of a core in comparison with the infinite Phys. Chem. Chem. Phys., 2016, 18, This journal is the Owner Societies 2016

11 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 T sm = K for finite k and T sm = K for infinite k, both with dynamics; a lower core pressure results in lower superheating. Also, the heterogeneity of temperature slightly reduces the heating rate for the same heat flux: Q i = Ks 1 for finite k and Q i = Ks 1 for infinite k. Therefore, the drop in temperature T sm isthecombinedresultofbothphysical phenomena: a 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 effect is smaller than the melting temperature change due to kinetic superheating. The coincidence of pressure and temperature curves at tˆ = 1.0 in Fig. 6(a) and 7(a), (c) for different Q but the same particles is not surprising: by definition, T = T eq at tˆ =1.0for all cases, and the pressure is also almost the same since pressure depends on the core temperature. 5 Effect of the parameter M ¼ R i d Here, we keep the fixed Al core radius R i = 40 nm while varying the shell thickness d = 2, 3 and 4 nm. As it follows from the static analytical solution for stresses in ref. 31, reduction in M (thicker shell) leads to higher pressure in the core and lower tensile hoop stress in the shell for the same temperature, which should lead to higher melting temperatures and higher maximum attainable temperature, T ma, before oxide fracture. Results in Fig. 7(c), 10(a) and (b), and Table 3 confirm this qualitative prediction for all melting temperatures and heating rates. For all M kinetic superheating becomes observable when Q reaches K s 1, and all melting temperatures strongly grow for larger heating rates. The different thicknesses of oxide produce different levels of heterogeneity of temperature across the oxide shell as shown in Fig. 8(b). However, its effect on T sm appears to be quite limited for the same heating rate in Fig. 10(a) since the difference in T sm among cases for Q Z remains almost the same as for Q o Note that we prescribed the heating rate at the center of the core by adjusting the heat flux. A particle with M = 10 has a higher heat flux than with M = 20 ( Jm 2 s 1 versus Jm 2 s 1 ) in order to have the same heating rate at 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 reasons of the weak effect of M on T sm. Also, note that the shell of the particle with M = 10 not only has a larger temperature difference, but also a higher average temperature (see Fig. 9). Fig. 7(c) shows the pressure evolution in the Al core at the interface I for Q =10 8 Ks 1 and Q =10 13 Ks 1. There is an initial deviation between the slowest and the fastest heating Fig. 9 Evolution of temperature at the core shell interface I and the surface of the alumina shell for two values of M ¼ R i d for R i = 40 nm and Q =10 13 Ks 1. rate for all three M. This deviation obviously originates from the heterogeneous temperature and becomes significant as M decreases due to greater temperature heterogeneity. It almost disappears at tˆ = 1.0 because the temperature of the core is the same as T eq for all cases due to normalization. The volumetric transformation strain due to melting of the core increases the pressure so that the slope of pressure evolution becomes steeper after surface melting start mark. The kinetic superheating delays the increase in pressure. Temperature T ms (Fig. 10(b)) demonstrates a similar behavior with respect to the heating rate as T sm in Fig. 10(a). The thicker shell (smaller M) results in higher T ms because of higher pressure inside the core. A thicker shell (smaller M) results not only in the increase of T ma due to reduced tensile hoop stress within oxide, but also in a slower increase of T ma due to growth in the heating rate in Fig. 10(b). Stress s 2 shown in Fig. 11(b) reaches the fracture s th stress (horizontal dashed line) around the melting finish mark for Q =10 12 Ks 1 for a nanoparticle with M = 10, while for a nanoparticle with M = 13.3 this happens around the melting finish mark for Q =10 11 Ks 1 in Fig. 11(a). That is why the effect of the heating rate on the fracture starts at Q =10 12 Ks 1 in Fig. 11(a) for M = 13.3 and at Q = Ks 1 in Fig. 11(b) for M = 10, which is in agreement with Fig. 10(b). As a result, the difference of T ma between the cases in Fig. 10(b) for Q r Ks 1 is larger than that for Q Ks 1. 6 Effect of the radius of an aluminum core Particles with three Al core radii, R i = 20 nm, 40 nm, and 60 nm with M =13.3(i.e., with the thickness of the oxide shell d =1.5nm, 3 nm, and 4.5 nm, respectively) are studied. Fig. 10(c) demonstrates the effect of the Al core radius R i on the heating-rate This journal is the Owner Societies 2016 Phys. Chem. Chem. Phys., 2016, 18,

12 Paper Fig. 10 Effect of geometric parameters of a nanoparticle on characteristic melting temperatures versus heating rate. (a) Effect of M ¼ R i on surface d melting start temperature, T sm, (b) maximum attainable temperature, T ma, and maximum superheating temperature, T ms,forr i = 40 nm. (c) Effect of the radius of the core on surface melting start temperature, T sm, (d) maximum attainable temperature, T ma, and maximum superheating temperature, T ms, for M = dependence of the surface melting temperature, T sm. For relatively small heating rate Q r Ks 1, T sm for R i =20nm is 2.4 K and 3.2 K lower than those for R i = 40 nm and 60 nm, respectively. This is typical size-dependence of the melting temperature, which is observed without an oxide shell. 6,7 Kinetic superheating does not change the size-dependence, thus the smaller particle has smaller T sm.largerr i results in a larger temperature difference along the radius and a larger heating rate for the interface Q i for the same Q at the center: for Q =10 13 Ks 1, we obtained Q i = Ks 1 for R i = 60 nm and Q i = Ks 1 for R i = 20 nm. Thus, the difference in T sm with increasing R i for Q =10 13 Ks 1 increases mostly due to increased Q i. A larger particle shows a larger maximum superheating temperature, T ms, and kinetic superheating is observed at smaller Q (Fig. 10(d)). Thus, T ms for R i = 40 and 60 nm starts to increase below Q =10 11 Ks 1, while for R i = 20 nm it remains almost same until Q =10 12 Ks 1. This size effect on kinetic superheating is governed by heterogeneous melting: a larger dimension provides more time for interface propagation and energy for superheating during melting. Unlike the minor effect of M on kinetic superheating in Fig. 10(b), R i has a significant influence on T ms. The increase in melting time shown in Fig. 11(d) also has a significant effect on T ma, which will be explored in the following section. 7 Tensile hoop stress in and fracture of the oxide shell The maximum hoop stress in the shell is located at the interface I so that s 2 at the interface and s th determine T ma. Elastic waves produce some contributions to change the maximum tensile hoop stress in an oxide shell for a high heating rate (Fig. 11(a) (c)). However, the major contribution to the maximum tensile hoop stress in an oxide shell for high Q comes from kinetic superheating and increase in melting time due to superheating, as will be shown below Phys. Chem. Chem. Phys., 2016, 18, This journal is the Owner Societies 2016

13 Paper Fig. 11 Evolution of normalized maximum tensile hoop stress in the oxide shell at the interface I versus the temperature at the interface I, T i, for different heating rates and core radii: (a) for R i = 40 nm, d = 3 nm, and M = 13.3, (b) for R i = 40 nm, d = 4 nm, and M = 10, and (c) for three R i with M = (d) Normalized time for complete melting, tˆmf tˆsm, versus heating rate for different core radii and M = The heating-rate dependence of T ma and the maximum superheating temperature at the center of the particle before melting, T ms, for the three values of M are shown in Fig. 10(b). For M = 20, temperatures T ma and T ms almost coincide for Q r 10 9 Ks 1, which means that the hoop stress in the shell reaches its ultimate strength almost at the end of the complete melting of a core. Temperature T ma is higher than T ms for Q r Ks 1 of M =13.3andQ r Ks 1 for M =10,i.e., fracture occurs after complete melting of the core. For all other cases, T ma o T ms and the fracture of the shell occurs before complete melting of the core, followed by propagation of the pressure reduction wave, which can result in high tensile pressure. 31 This process strongly depends on the fracture time and will be studied elsewhere. Also, since it is highly probable that the ultimate strength of the few nm thick alumina shell has significant scatter, the part of curves for T ms that are above the curves for T ma still may have physical sense for higher ultimate strength. If a shell is strong enough to contain complete melt for all heating rates or weak enough to break down before melting starts, T ma will be independent of the heating rate. For all heating rates, T ma and T ms increase with decreasing M because of increasing pressure in the core and reducing tensile stresses due to a thicker shell. For Q o Ks 1, the heating-rate dependence of both characteristic temperatures is weak. T ms increases with increasing heating rate for Q Z Ks 1 and T ma increases with increasing heating rate for Q Z Ks 1 for all M. The wavy oscillation of s 2, which originates from elastic waves traveling in the core, appears for Q Ks 1. Note that the dynamic equation is used from Q =10 12 Ks 1 and no oscillation is observed for this heating rate. The wave characteristics depend on the particle radius: oscillations of R = 20 nm for Q =10 13 Ks 1 in Fig. 11(c) have the shortest period in terms of T and it increases as R increases. However, the magnitude of the wave is small for all tested cases, hence the effect of elastic waves on T ma is quite limited. While the maximum superheating temperature is larger for larger particles and increases with the increase in the heating rate (Fig. 10(d)), the maximum attainable temperature, T ma, for Q r Ks 1 has an opposite trend, i.e., it increases with the reduction in the core radius (Fig. 10(d)). This happens due to This journal is the Owner Societies 2016 Phys. Chem. Chem. Phys., 2016, 18,

14 Paper surfacetensionatthesurfacesandtheinterfacei,whichproduce compressive hoop stress in the oxide shell which increases with reduction in R i. For example, the hoop stress at the interface I for R i =20nmatT 0 is 0.38 GPa in comparison with 0.13 GPa for R i = 60 nm, which delays the fracture of the oxide shell. This is to some extent similar to pre-stressing of the Al core shell structures by relaxing internal stresses by annealing at a higher temperature and quenching to ambient temperature in order to suppress fracture and enhance the melt-dispersion mechanism. 44,52 The trend of T ma in the size effect, however, has a crossover for higher heating rates at Ks 1 r Q r Ks 1 in Fig. 10(d). The reason for the crossover can be deduced from the evolution of the maximum hoop stress in the shell during heating in Fig. 11(c) just by analyzing the position of the intersection point of the stress s 2 /s th with the horizontal line s 2 /s th =1.0.ForQ =10 9 Ks 1, melting completes before fracture for all R i, and the temperature at the intersection (which is T ma ) is higher for smaller R i (as we discussed, due to interface stresses). For Q =10 12 Ks 1,melting almost completes at fracture points for R i =20nmanddoesnot complete for R i =40nmandR i = 60 nm; the temperature at the intersection point is lower as the radius is smaller, which results in crossover. For Q Ks 1, fracture occurs during melting, and again, temperature at the intersection point reduces with reducing radius R i. The increase in the normalized melting time in Fig. 11(d) has an important role in the increase of T ma 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 R i. Temperatures T ma and T ms (or times tˆma and tˆms ) are independent of Q for Q =10 8 Ks 1 and Q =10 9 Ks 1, and, consequently, at least for Q r 10 9 Ks 1. This is important for the analysis of the melt-dispersion mechanism, because the estimated heating rate at the reaction front was 10 8 Ks 1 in ref. 31. For Q r 10 9 Ks 1 and M = 20, the condition for complete melting before oxide fracture is almost met, i.e. T ma B T ms, which is consistent with ref. 31 and 53. However, for M = 20 and Q Z Ks 1, fracture occurs well before completing melting. Since all other cases in Table 3 have smaller M than 19, melting completes before fracture for Q r K s 1, excluding particles with R i = 60 nm and M = 13.3 for Q =10 11 Ks 1. Note that Q =10 11 Ks 1 is the estimated heating rate in experiments in ref. 39 and 40. For Q Z Ks 1 (which is typical of the experiments in ref. 38), fracture occurs before completion of melting for all cases, which is far from optimal for melt dispersion. Consequently, there is an upper bound of Q r Ks 1 for optimal melt dispersion, in contrast to the previous wisdom that the larger Q is the better. Thus, while predictions for Q r 10 9 Ks 1 from the simplified theory in ref. 31 are confirmed by the current and more 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 meltdispersion 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 T ma Z T ms (or t ma Z t ms )andthatt ma (or t ma )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 independent of the heating rate. Based on our much more precise results in Fig. 10(b) and Table 3, we can conclude the following. Fig. 12 Evolution distribution and (b) volume integral of Al nanoparticle with R i =40nmandd = 3 nm for Q =10 9 Ks Phys. Chem. Chem. Phys., 2016, 18, This journal is the Owner Societies 2016

15 precise simulations, results for Q Z Ks 1 differ quantitatively and qualitatively. Also, Table 3 contains data on the maximum rate of the hoop strain in the shell at the interface I, _e oxm,2.forq =10 8 Ks 1,itisin the range of s 1, which may be high enough for avoiding relaxation processes. It was roughly estimated in ref. 31 as The almost order of magnitude reduction in _e oxm,2 here is attributed to the more precise approach and insignificant growth of temperature during melting with resultant reduction of effective Q. Generally, _e oxm,2 is scaled proportionally to Q. Thus, for Q =10 7 Ks 1 and even for Q =10 6 Ks 1, for which meltdispersion is still expected in ref. 31 and 41, _e oxm,2 is s 1 and s 1, respectively. 9 Temperature drop at the completion of melting An abrupt decrease in temperature by several degrees is observed at the end of melting for Q =10 8 Ks 1 and Q =10 9 Ks 1, Paper with and without an oxide shell, and for all geometric parameters, as it follows from Table 3. The maximum temperature drop of T mf T ms = 14.6 K is for Q =10 9 Ks 1, R i = 20 nm, and d = 1.5 nm. This temperature drop comes from acceleration of 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 near the center. Volume integration of the presented in Fig. takes into account that melting occurs within a smaller volume 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 through eqn (13). For higher heating rates, the temperature drop is absent and T ms = T mf (Table 3). A similar but much larger temperature drop was observed at completion of melting of a plane nanolayer, when two solid melt interfaces collided, for all heating rates (from Ks 1 to Ks 1 )studied Fig. 13 The map of physical phenomena affecting melting temperatures (a) for Q r 10 9 Ks 1 and (b) for Q Z Ks 1. This journal is the Owner Societies 2016 Phys. Chem. Chem. Phys., 2016, 18,

16 Paper in ref. 22 since the volume of the colliding region for a plane structure is much larger than that for a spherical structure. 10 Physical phenomena involved in melting and superheating of Al nanoparticles In this section, we summarize the effect of different parameters on the characteristic melting temperatures (Fig. 13) and maximum attainable temperature, T ma, which is determined by the fracture of the oxide shell (Fig. 14). The maps are presented for two ranges of the heating rates: (a) for Q r 10 9 Ks 1, when the effect of Q is absent and melting is quasi-equilibrium, and (b) for Q Z Ks 1, when effects of the heating rates are pronounced. For the intermediate heating rate, Q =10 11 Ks 1, the effect of Q appears but is weak. In all maps: (a) Reduction of the radius of a core R i and the solid melt interface radius r i leads to reduction in melting temperatures according to the Gibbs Thomson effect. The reduction of the radius of a core R i in turn causes reduction of T ma due to shifting volumetric expansion and the corresponding stress increase in a shell to lower temperature if s 2 reaches s th during melting. (b) For small particles (R i = 20 nm), surface tension at the core shell interface and the external shell surface produces pressure in a core, which increases all melting temperatures and, consequently, T ma,ifs 2 reaches s th during melting. (c) A decrease in M (an increase of thickness of oxide) causes an increase in pressure in a core and an increase of all melting temperatures and, consequently, T ma,ifs 2 reaches s th during melting. For low heating rates of Q r 10 9 Ks 1, in addition to the above effects: (a) Surface tensions at the core shell interface and the external shell, and a decrease in M also decrease tensile stresses in a shell and, consequently, increases T ma. The same is true for high heating rates. (b) There is a small reduction in temperature and, consequently, melt finish temperature T mf at the end of melting at Fig. 14 The map of physical phenomena affecting the maximum attainable temperature, T ma, (a) for Q r 10 9 Ks 1 and (b) for Q Z Ks Phys. Chem. Chem. Phys., 2016, 18, This journal is the Owner Societies 2016