^{1}, A. Mijailović

^{2}, C. V. Achim

^{1,2}, T. Ala-Nissila

^{1,3}, R. E. Rozas

^{2}, J. Horbach

^{2}and H. Löwen

^{2}

### Abstract

We determine the orientation-resolved interfacial free energy between a body-centered-cubic (bcc) crystal and the coexisting fluid for a many-particle system interacting via a Yukawa pair potential. For two different screening strengths, we compare results from molecular dynamics computer simulations, density functional theory, and a phase-field-crystal approach. Simulations predict an almost orientationally isotropic interfacial free energy of 0.12*k* _{ B } *T*/*a* ^{2} (with *k* _{ B } *T* denoting the thermal energy and *a* the mean interparticle spacing), which is independent of the screening strength. This value is in reasonable agreement with our Ramakrishnan-Yussouff density functional calculations, while a high-order fitted phase-field-crystal approach gives about 2−3 times higher interfacial free energies for the Yukawa system. Both field theory approaches also give a considerable anisotropy of the interfacial free energy. Our result implies that, in the Yukawa system, bcc crystal-fluid free energies are a factor of about 3 smaller than face-centered-cubic crystal-fluid free energies.

We thank Martin Oettel, Martin Grant, Ken Elder, and Mikko Karttunen for helpful discussions. This work was supported by the DFG within SPP 1296, and by the Academy of Finland through its Centres of Excellence Program (Project No. 251748).

I. INTRODUCTION

II. COMPUTER SIMULATIONS OF INTERFACE FREE ENERGY

III. DENSITY FUNCTIONAL THEORY OF FREEZING

A. Liquid state integral theory

IV. PHASE-FIELD CRYSTAL MODEL

V. RESULTS

A. Capillary wave analysis from MD simulations

B. Field theoretical approaches

VI. SUMMARY AND CONCLUSIONS

### Key Topics

- Free energy
- 51.0
- Density functional theory
- 33.0
- Correlation functions
- 14.0
- Anisotropy
- 13.0
- Liquid crystals
- 9.0

## Figures

The radial distribution function as a function of the inter-particle distance *r* obtained from Monte Carlo simulations and the ERY method (*a*′ = 3.89 and ɛ/*a*′*k* _{ B } *T* = 389.47).

The radial distribution function as a function of the inter-particle distance *r* obtained from Monte Carlo simulations and the ERY method (*a*′ = 3.89 and ɛ/*a*′*k* _{ B } *T* = 389.47).

*q*-dependent interfacial free energy for the (100) and (111) orientations as well as for the two directions of the (110) orientation. Here, the Yukawa system ^{ 23 } with *a*′ = 2.5 is considered. The solid lines are fits from which in the limit *q* → 0 is extracted (see text). Error bars are shown for the (100) orientation. For the other orientations, similar error bars are obtained (cf. Table II ).

*q*-dependent interfacial free energy for the (100) and (111) orientations as well as for the two directions of the (110) orientation. Here, the Yukawa system ^{ 23 } with *a*′ = 2.5 is considered. The solid lines are fits from which in the limit *q* → 0 is extracted (see text). Error bars are shown for the (100) orientation. For the other orientations, similar error bars are obtained (cf. Table II ).

Density profile in *z* direction (i.e., the direction perpendicular to the solid-liquid interface), as obtained in the framework of DFT. The density ρ is scaled by the liquid coexistence density while the distance *z* is in units of the linear dimension of a primitive bcc cell *a* _{bcc}. The inset shows the average density profile.

Density profile in *z* direction (i.e., the direction perpendicular to the solid-liquid interface), as obtained in the framework of DFT. The density ρ is scaled by the liquid coexistence density while the distance *z* is in units of the linear dimension of a primitive bcc cell *a* _{bcc}. The inset shows the average density profile.

## Tables

Box lengths , , and , pressure *P*′ and particle number *N* used in the MD simulations for *a*′ = 2.5 and *a*′ = 4.0.

Box lengths , , and , pressure *P*′ and particle number *N* used in the MD simulations for *a*′ = 2.5 and *a*′ = 4.0.

Dimensionless values for the stiffnesses ( ) from MD simulations.

Dimensionless values for the stiffnesses ( ) from MD simulations.

Melting temperature , coexistence densities for the fluid, , and the crystal, , and , as obtained from the MD simulation (the densities and are given in units of ρ_{s}/κ^{3}).

Melting temperature , coexistence densities for the fluid, , and the crystal, , and , as obtained from the MD simulation (the densities and are given in units of ρ_{s}/κ^{3}).

Parameters of the cubic harmonic expansion (see text for details).

Parameters of the cubic harmonic expansion (see text for details).

Dimensionless values for the interfacial energies (γρ^{−2/3}/*k* _{ B } *T* _{m}) from MD simulations.

Dimensionless values for the interfacial energies (γρ^{−2/3}/*k* _{ B } *T* _{m}) from MD simulations.

Melting temperature , coexistence densities for the fluid, , and the crystal, , and , as obtained from DFT and PFC (the densities and are given in units of ρ_{s}/κ^{3}).

Melting temperature , coexistence densities for the fluid, , and the crystal, , and , as obtained from DFT and PFC (the densities and are given in units of ρ_{s}/κ^{3}).

Dimensionless values for the interfacial energies (γρ^{−2/3}/*k* _{ B } *T* _{m}) from the DFT and PFC calculations.

Dimensionless values for the interfacial energies (γρ^{−2/3}/*k* _{ B } *T* _{m}) from the DFT and PFC calculations.

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