Free energy and concentration of crystalline vacancies by molecular simulation

We present an approach for evaluating the concentration of vacancy defects in crystalline materials by molecular simulation. The proposed method circumvents the problem of equilibration of the number of ‘lattice sites’ M, which characterises the trade-off between more, smaller lattice cells (with some vacant), versus fewer, larger cells. Working in a grand-canonical framework, we instead fix M and solve for the chemical potential μ that ensures thermodynamic consistency of the ensemble-averaged pressure and the grand potential. Having determined μ this way for the given M, the equilibrium vacancy concentration and free energy are easily determined. Methods are demonstrated for the classical Lennard–Jones fcc crystal, examining all states where the crystal is stable. We find for this system that the effect of equilibrating M is negligible at all conditions. Also, although the vacancy fraction varies by many orders of magnitude with temperature and density, we find that the value at melting is nearly independent of density, equal to about (Formula presented.). Results further show that a lattice-energy approximation (ignoring entropic effects) underestimates the correct concentration by four orders of magnitude at almost all conditions; ignoring only anharmonic contributions underestimates the vacancy concentration at melting by nearly one order of magnitude.