^{1,a)}

### Abstract

Molecular dynamics simulations of a hard sphere crystal are performed for volume fractions ranging from solidification point to melting point. A local bond order parameter is chosen to assign a nature, liquid or solid, to a particle. The probability for a liquid or solid particle to change state presents a typical sigmoid shape as the nature of its neighbors changes. Using this property, I propose a reaction-like mechanism and introduce a small number of rate constants. A mean-field approach to melting and a kinetic Monte Carlo algorithm on a lattice are derived from these chemical processes. The results of these models successfully compare with molecular dynamics simulations, proving that the main properties of melting can be captured by a small number of dynamical parameters.

I. INTRODUCTION

II. MOLECULAR DYNAMICS SIMULATIONS OF A HARD SPHERE CRYSTAL

A. Local bond order parameter and liquid nature of a particle

B. Global bond order parameter and waiting time before melting

C. Rate constants

III. MEAN-FIELDMODEL AND KINETIC MONTE CARLO ALGORITHM BASED ON A REACTION-LIKE MECHANISM

IV. CONCLUSION

### Key Topics

- Monte Carlo methods
- 33.0
- Molecular dynamics
- 32.0
- Liquid solid interfaces
- 30.0
- Reaction rate constants
- 17.0
- Solid liquid phase transitions
- 17.0

##### B01J19/06

## Figures

(a) Probability density function (pdf) of the interparticle distance (solid line). The vertical solid line gives the chosen cutoff radius *r* _{ c }. The dashed-dotted, dashed, and dotted lines correspond to the first-, second-, and third-neighbor peaks. (b) Pdf of *S* for a metastable solid at ϕ = 0.506 (solid line), a stable solid at ϕ_{ m } = 0.55 (dotted line), and a stable liquid at ϕ_{ s } = 0.49 (dashed line). (c) Pdf of the number *n* _{ cn } of connected neighbors per particle for the same parameters as in (b).

(a) Probability density function (pdf) of the interparticle distance (solid line). The vertical solid line gives the chosen cutoff radius *r* _{ c }. The dashed-dotted, dashed, and dotted lines correspond to the first-, second-, and third-neighbor peaks. (b) Pdf of *S* for a metastable solid at ϕ = 0.506 (solid line), a stable solid at ϕ_{ m } = 0.55 (dotted line), and a stable liquid at ϕ_{ s } = 0.49 (dashed line). (c) Pdf of the number *n* _{ cn } of connected neighbors per particle for the same parameters as in (b).

(a) Number *N* _{ L } of liquid particles versus time *t* for the three different descriptions. The solid line gives the results of one 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 one KMC simulation for the same total number of particles and the time step Δ*t* = 10^{−2}. The dotted line gives the two stable stationary states of the mean-field model. In the two last approaches, the dynamical parameters are *n* _{ l } = 5, *n* _{ s } = 7 and *l* _{0} = 0.8, *l* _{12} = 22, *s* _{0} = 0.5, *s* _{12} = 15. (b) Time evolution of the global order parameter *Q* _{6} in the same MD simulation as in (a).

(a) Number *N* _{ L } of liquid particles versus time *t* for the three different descriptions. The solid line gives the results of one 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 one KMC simulation for the same total number of particles and the time step Δ*t* = 10^{−2}. The dotted line gives the two stable stationary states of the mean-field model. In the two last approaches, the dynamical parameters are *n* _{ l } = 5, *n* _{ s } = 7 and *l* _{0} = 0.8, *l* _{12} = 22, *s* _{0} = 0.5, *s* _{12} = 15. (b) Time evolution of the global order parameter *Q* _{6} in the same MD simulation as in (a).

Variation of deduced from molecular dynamics with the volume fraction ϕ for different total number of particles *N* = 576 (circles), *N* = 1728 (crosses), and *N* = 13 824 (plus). In the inset, the probability density function of the waiting time τ_{MD} (circle) for different initial particle velocities and its best fit *dW*/*d*τ with λ = 25 000 and *k* = 1.1 (solid line) are shown for *N* = 576 and ϕ = 0.497.

Variation of deduced from molecular dynamics with the volume fraction ϕ for different total number of particles *N* = 576 (circles), *N* = 1728 (crosses), and *N* = 13 824 (plus). In the inset, the probability density function of the waiting time τ_{MD} (circle) for different initial particle velocities and its best fit *dW*/*d*τ with λ = 25 000 and *k* = 1.1 (solid line) are shown for *N* = 576 and ϕ = 0.497.

Variation of the mean waiting time before melting occurs ⟨τ_{MD}⟩ deduced from molecular dynamics with the total number *N* of particles for a volume fraction ϕ = 0.499.

Variation of the mean waiting time before melting occurs ⟨τ_{MD}⟩ deduced from molecular dynamics with the total number *N* of particles for a volume fraction ϕ = 0.499.

Finite size effects on coexistence at equilibrium in molecular dynamics. (a) Time evolution of the global order parameter *Q* _{6} for a total number of particles *N* = 110 592 at the volume fraction ϕ = 0.504 (solid line) and for *N* = 13 824 at ϕ = 0.499 (dashed line). The last value of ϕ is chosen such that the mean waiting times compare in both cases. (b) Pdf of *S* for coexistent liquid and solid at equilibrium at ϕ = 0.504 for *N* = 110 592 (solid line), a stable solid at ϕ_{ m } = 0.55 for *N* = 110 592 (dotted line), and an equilibrated liquid at ϕ = 0.504 for *N* = 13 824 (dashed line).

Finite size effects on coexistence at equilibrium in molecular dynamics. (a) Time evolution of the global order parameter *Q* _{6} for a total number of particles *N* = 110 592 at the volume fraction ϕ = 0.504 (solid line) and for *N* = 13 824 at ϕ = 0.499 (dashed line). The last value of ϕ is chosen such that the mean waiting times compare in both cases. (b) Pdf of *S* for coexistent liquid and solid at equilibrium at ϕ = 0.504 for *N* = 110 592 (solid line), a stable solid at ϕ_{ m } = 0.55 for *N* = 110 592 (dotted line), and an equilibrated liquid at ϕ = 0.504 for *N* = 13 824 (dashed line).

Molecular dynamics results for a volume fraction ϕ = 0.506, a total number of particles *N* = 13 824, and the threshold values *S* _{ tr } = 0.4 and *n* _{ tr } = 4: Scaled probabilities *l* _{ n } = *P* _{ n }(*S* → *L*)/Δ*t* _{MD} for a solid particle surrounded by *n* liquid neighbors to become liquid during Δ*t* _{MD} versus *n* for two values of the time interval Δ*t* _{MD} = 10^{−3} (solid line) and Δ*t* _{MD} = 10^{−4} (dashed line). Same results for *s* _{ n } = *P* _{ n }(*L* → *S*)/Δ*t* _{MD} for a liquid particle surrounded by *n* solid neighbors to become solid, for Δ*t* _{MD} = 10^{−3} (dashed-dotted line) and Δ*t* _{MD} = 10^{−4} (dotted line).

Molecular dynamics results for a volume fraction ϕ = 0.506, a total number of particles *N* = 13 824, and the threshold values *S* _{ tr } = 0.4 and *n* _{ tr } = 4: Scaled probabilities *l* _{ n } = *P* _{ n }(*S* → *L*)/Δ*t* _{MD} for a solid particle surrounded by *n* liquid neighbors to become liquid during Δ*t* _{MD} versus *n* for two values of the time interval Δ*t* _{MD} = 10^{−3} (solid line) and Δ*t* _{MD} = 10^{−4} (dashed line). Same results for *s* _{ n } = *P* _{ n }(*L* → *S*)/Δ*t* _{MD} for a liquid particle surrounded by *n* solid neighbors to become solid, for Δ*t* _{MD} = 10^{−3} (dashed-dotted line) and Δ*t* _{MD} = 10^{−4} (dotted line).

Same caption as in Fig. 6 for *l* _{ n } (a), *s* _{ n } (b), Δ*t* _{MD} = 10^{−3}, and different threshold values: *S* _{ tr } = 0.2 for *n* _{ tr } = 7 (solid line) and *n* _{ tr } = 8 (dotted line), *S* _{ tr } = 0.3 for *n* _{ tr } = 6 (solid line) and *n* _{ tr } = 7 (dotted line), *S* _{ tr } = 0.4 for *n* _{ tr } = 4 (solid line) and *n* _{ tr } = 5 (dotted line). Lines become bolder as *S* _{ tr } increases.

Same caption as in Fig. 6 for *l* _{ n } (a), *s* _{ n } (b), Δ*t* _{MD} = 10^{−3}, and different threshold values: *S* _{ tr } = 0.2 for *n* _{ tr } = 7 (solid line) and *n* _{ tr } = 8 (dotted line), *S* _{ tr } = 0.3 for *n* _{ tr } = 6 (solid line) and *n* _{ tr } = 7 (dotted line), *S* _{ tr } = 0.4 for *n* _{ tr } = 4 (solid line) and *n* _{ tr } = 5 (dotted line). Lines become bolder as *S* _{ tr } increases.

## Tables

Variation of the mean waiting time before melting occurs ⟨τ_{KMC}⟩ deduced from the KMC simulations for different values of the boundaries *n* _{ s } and *n* _{ l } and the same other parameter values as in Fig. 2.

Variation of the mean waiting time before melting occurs ⟨τ_{KMC}⟩ deduced from the KMC simulations for different values of the boundaries *n* _{ s } and *n* _{ l } and the same other parameter values as in Fig. 2.

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