^{1,a)}

### Abstract

A kinetic Monte Carlo model on a lattice, based on a reaction-like mechanism, is used to investigate the microscopic properties of the homogeneous melting of a metastable crystal. The kinetic Monte Carlo model relies on nearest-neighbors interactions and a few relevant dynamical parameters. To examine the reliability of the model, careful comparison with molecular dynamics simulations of a hard sphere crystal is drawn. A criterion on the critical nature of a microscopic configuration is deduced from the bimodal character of the probability density function of melting time. For kinetic Monte Carlo simulations with dynamical parameter values which fit the molecular dynamics results, the number of liquid sites of the critical droplet is found to be smaller than 300 and the ability of the critical droplet to invade the entire system is shown to be independent of the droplet shape as long as this droplet remains compact. In kinetic Monte Carlo simulations, the size of the critical droplet is independent of the system size. Molecular dynamics evidences a more complex dependence of melting time on system size, which reveals non-trivial finite size effects.

I would like to thank Michael Schindler for the MD code and Anthony Maggs for fruitful discussion.

I. INTRODUCTION

II. KINETIC MONTE CARLO AND MOLECULAR DYNAMICS

III. BIMODAL DISTRIBUTION OF MELTING TIME

IV. PROPERTIES OF A CRITICAL DROPLET

A. Influence of subcritical droplets

B. Size of the critical droplet versus boundary value of the stoichiometric coefficients between slow and fast processes

C. Shape of the critical droplet

D. System size

V. CONCLUSION

### Key Topics

- Fluid drops
- 88.0
- Monte Carlo methods
- 72.0
- Molecular dynamics
- 44.0
- Liquid solid interfaces
- 15.0
- Melting
- 14.0

## Figures

Number *N* _{ L } of liquid particles versus time *t*. The solid line gives the results of a molecular dynamics (MD) simulation for a volume fraction ϕ = 0.506 and a total number of particles *N* = 13 824. The dashed line gives the results of a kinetic Monte Carlo (KMC) simulation for the same total number of particles, the time step Δ*t* = 10^{−2} and the dynamical parameters *n* _{ l } = 5, *n* _{ s } = 7 and *l* _{0} = 0.8, *l* _{12} = 22, *s* _{0} = 0.5, *s* _{12} = 15.

Number *N* _{ L } of liquid particles versus time *t*. The solid line gives the results of a molecular dynamics (MD) simulation for a volume fraction ϕ = 0.506 and a total number of particles *N* = 13 824. The dashed line gives the results of a kinetic Monte Carlo (KMC) simulation for the same total number of particles, the time step Δ*t* = 10^{−2} and the dynamical parameters *n* _{ l } = 5, *n* _{ s } = 7 and *l* _{0} = 0.8, *l* _{12} = 22, *s* _{0} = 0.5, *s* _{12} = 15.

(a) Cumulative density function (cdf) of the waiting time τ_{KMC} deduced from kinetic Monte Carlo simulations (circles) for an initial configuration with a given initial number of liquid sites *N* _{ L } = 158 and for different seeds of the random number generator. The total number of sites is *N* = 1728 and the parameters are *n* _{ l } = 7, *n* _{ s } = 8 and *l* _{0} = 1, *l* _{12} = 24, *s* _{0} = 0.6, *s* _{12} = 11. The solid line is the best Weibull-type fit. The inset gives the corresponding probability density function (pdf) for KMC (circles) and the derivative of the cdf fit (solid line). (b) Same caption as (a) for *N* _{ L } = 204. (c) Same caption as (a) for *N* _{ L } = 226. (d) Same caption as (a) for *N* _{ L } = 229.

(a) Cumulative density function (cdf) of the waiting time τ_{KMC} deduced from kinetic Monte Carlo simulations (circles) for an initial configuration with a given initial number of liquid sites *N* _{ L } = 158 and for different seeds of the random number generator. The total number of sites is *N* = 1728 and the parameters are *n* _{ l } = 7, *n* _{ s } = 8 and *l* _{0} = 1, *l* _{12} = 24, *s* _{0} = 0.6, *s* _{12} = 11. The solid line is the best Weibull-type fit. The inset gives the corresponding probability density function (pdf) for KMC (circles) and the derivative of the cdf fit (solid line). (b) Same caption as (a) for *N* _{ L } = 204. (c) Same caption as (a) for *N* _{ L } = 226. (d) Same caption as (a) for *N* _{ L } = 229.

(a) Cumulative density function (cdf) of the waiting time τ_{MD} deduced from molecular dynamics (circles) for initial configurations with identical particle positions corresponding to an initial number *N* _{ L } = 337 of liquid particles, and different velocities. The total number of particles is *N* = 1728 and the volume fraction is ϕ = 0.499. The solid line is the best Weibull-type fit. The inset gives the corresponding probability density functions (pdf) for MD (circles) and the derivative of the cdf fit (solid line). (b) Same caption as (a) for a different choice of initial particle positions with *N* _{ L } = 432. (c) Same caption as (b) with *N* _{ L } = 494.

(a) Cumulative density function (cdf) of the waiting time τ_{MD} deduced from molecular dynamics (circles) for initial configurations with identical particle positions corresponding to an initial number *N* _{ L } = 337 of liquid particles, and different velocities. The total number of particles is *N* = 1728 and the volume fraction is ϕ = 0.499. The solid line is the best Weibull-type fit. The inset gives the corresponding probability density functions (pdf) for MD (circles) and the derivative of the cdf fit (solid line). (b) Same caption as (a) for a different choice of initial particle positions with *N* _{ L } = 432. (c) Same caption as (b) with *N* _{ L } = 494.

Snapshot of the initial MD spatial configuration used to obtain Fig. 3(b) . The white particles are solid and the red particles are liquid.

Snapshot of the initial MD spatial configuration used to obtain Fig. 3(b) . The white particles are solid and the red particles are liquid.

Cumulative density functions (cdf) of the waiting time τ_{KMC} deduced from kinetic Monte Carlo simulations from two configurations containing a single initial spherical drop of *N* _{ L } = 38 liquid sites (dashed line), *N* _{ L } = 68 liquid sites (solid line) and for different seeds of the random number generator. The other parameters are given in the caption of Fig. 2(a) . For the sake of efficiency, the simulations are stopped at time 2000.

Cumulative density functions (cdf) of the waiting time τ_{KMC} deduced from kinetic Monte Carlo simulations from two configurations containing a single initial spherical drop of *N* _{ L } = 38 liquid sites (dashed line), *N* _{ L } = 68 liquid sites (solid line) and for different seeds of the random number generator. The other parameters are given in the caption of Fig. 2(a) . For the sake of efficiency, the simulations are stopped at time 2000.

Plateau height *h* of the cumulative density function of melting time deduced from kinetic Monte Carlo simulations versus number *N* _{ L } of liquid sites in the initial configuration with a single spherical droplet for *n* _{ l } = 7, *n* _{ s } = 8 (circles, solid line), *n* _{ l } = 6, *n* _{ s } = 6 (squares, dashed line) and *l* _{0} = 1, *l* _{12} = 24, *s* _{0} = 0.6, *s* _{12} = 11.

Plateau height *h* of the cumulative density function of melting time deduced from kinetic Monte Carlo simulations versus number *N* _{ L } of liquid sites in the initial configuration with a single spherical droplet for *n* _{ l } = 7, *n* _{ s } = 8 (circles, solid line), *n* _{ l } = 6, *n* _{ s } = 6 (squares, dashed line) and *l* _{0} = 1, *l* _{12} = 24, *s* _{0} = 0.6, *s* _{12} = 11.

Plateau height of the cdf of the waiting time τ_{KMC} deduced from kinetic Monte Carlo simulations versus droplet anisotropy λ_{ max }/λ_{ min } for *N* = 1728, *l* _{0} = 1, *l* _{12} = 24, *s* _{0} = 0.6, *s* _{12} = 11, and an initial droplet of *N* _{ L } = 72 liquid sites. The squares correspond to *n* _{ l } = 6, *n* _{ s } = 7 and the circles to *n* _{ l } = 7, *n* _{ s } = 8. The lines give the plateau height of the cdf corresponding to the spherical droplet for *n* _{ l } = 6 and *n* _{ s } = 7 (solid line) and for *n* _{ l } = 7 and *n* _{ s } = 8 (dashed line).

Plateau height of the cdf of the waiting time τ_{KMC} deduced from kinetic Monte Carlo simulations versus droplet anisotropy λ_{ max }/λ_{ min } for *N* = 1728, *l* _{0} = 1, *l* _{12} = 24, *s* _{0} = 0.6, *s* _{12} = 11, and an initial droplet of *N* _{ L } = 72 liquid sites. The squares correspond to *n* _{ l } = 6, *n* _{ s } = 7 and the circles to *n* _{ l } = 7, *n* _{ s } = 8. The lines give the plateau height of the cdf corresponding to the spherical droplet for *n* _{ l } = 6 and *n* _{ s } = 7 (solid line) and for *n* _{ l } = 7 and *n* _{ s } = 8 (dashed line).

Variation of the waiting time before melting with the total number *N* of sites or particles. All the sites or particles are initially solid. The circles give the results of kinetic Monte Carlo simulations which scale as 1/*N* (solid line) for the same parameter values as in Fig. 2(a) . The pluses give the results of molecular dynamics simulations for ϕ = 0.499.

Variation of the waiting time before melting with the total number *N* of sites or particles. All the sites or particles are initially solid. The circles give the results of kinetic Monte Carlo simulations which scale as 1/*N* (solid line) for the same parameter values as in Fig. 2(a) . The pluses give the results of molecular dynamics simulations for ϕ = 0.499.

## Tables

Values of plateau height *h* of the cumulative density functions given in Figs. 3(a)–3(c) for the three different spatial configurations (a)–(c) and parameters λ_{ l }, *k* _{ l }, λ_{ r }, and *k* _{ r } of the corresponding Weibull-type fits. The values are given with an uncertainty of 1 on the last significant figure.

Values of plateau height *h* of the cumulative density functions given in Figs. 3(a)–3(c) for the three different spatial configurations (a)–(c) and parameters λ_{ l }, *k* _{ l }, λ_{ r }, and *k* _{ r } of the corresponding Weibull-type fits. The values are given with an uncertainty of 1 on the last significant figure.

Values of the plateau height *h* of the cumulative density function of melting time for different values of the parameters *n* _{ l } and *n* _{ s }, different droplet sizes *N* _{ L }, and two system sizes *N*. The value of the parameters *l* _{0}, *l* _{12}, *s* _{0}, *s* _{12} is the same as in Fig. 2(a) .

Values of the plateau height *h* of the cumulative density function of melting time for different values of the parameters *n* _{ l } and *n* _{ s }, different droplet sizes *N* _{ L }, and two system sizes *N*. The value of the parameters *l* _{0}, *l* _{12}, *s* _{0}, *s* _{12} is the same as in Fig. 2(a) .

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