Vacancy diffusion and the hydrodynamics of crystals

TL;DR

This paper develops a hydrodynamic theory for crystals with vacancies, deriving transport coefficients and confirming that vacancy diffusion constitutes a key hydrodynamic mode through simulations.

Contribution

It introduces a thermodynamic framework incorporating vacancy chemical potential and establishes microscopic foundations for transport properties.

Findings

Derived Green-Kubo and Einstein-Helfand formulas for vacancy transport coefficients.

Established a relationship between vacancy conductivities and Fickian diffusion.

Confirmed vacancy diffusion as an essential hydrodynamic mode via numerical simulations.

Abstract

The hydrodynamics of crystals with vacancies is developed on the basis of local-equilibrium thermodynamics, where the chemical potential of vacancies plays a key role together with a constraint relating the concentration of vacancies to the density of mass and the strain tensor. The microscopic foundations are established, leading to Green-Kubo and Einstein-Helfand formulas for the transport coefficients, including the vacancy conductivities and the coefficients of vacancy thermodiffusion. As a consequence of having introduced the chemical potential of vacancies, a relationship is obtained between the conductivities and the Fickian diffusion coefficients for the vacancies. The macroscopic equations are linearized around equilibrium to deduce the dispersion relations of the eight hydrodynamic modes. The theoretical predictions are confirmed by numerical simulations of the hard-sphere crystal with vacancies. The study explicitly shows that the eighth hydrodynamic mode of nonperfect monatomic crystals is indeed a mode of vacancy diffusion.