State Equation of a Nanocrystal with Vacancies

Abstract

An expression for the Helmholtz free energy is established and the equation of state is derived for a nanocrystal containing vacancies in the lattice and delocalized (diffusing) atoms. The calculations are performed for bcc (body-centered cubic) iron under the isothermal compression of a nanocrystal along the 300-K and 1000-K isotherms. The changes in the specific surface energy (σ), the probability of vacancy formation (ϕ_v) and the probability of atom delocalization (x_d) are studied depending on the size (N) and shape of the nanocrystal at different temperatures (T) and pressures (P). The size dependences are characterized along two isobars: at atmospheric pressure (P = 1 bar) and at P = 100 kbar. As is shown, two P-points arise in the σ(P) isotherms at T ≤ 300 K, where the specific surface energy is independent of the nanocrystal size, which is σ(N) = σ(∞). As the temperature increases, the P points approach each other and at T ≥ 1000 K they vanish in the isotherms. At atmospheric pressure and T = 300 K the amount of vacancies per atom in a nanocrystal is much lower than that in a macrocrystal; however, at T = 1000 K the shredding of the latter leads to an increase in the probability of vacancy formation. Moreover, the smaller the nanocrystal size, the higher the probability of atom delocalization (as well as the self-diffusion coefficient) at any pressure and temperature. The ratio ϕ_v/x_d decreases with decreasing size of the nanocrystal, and less than a certain size, there is an no-vacansies self-diffusion, at which the number of delocalized atoms is greater than the amount of vacant cells in the nanocrystal lattice, i.e., ϕ_v < x_d. As the nanocrystal shape becomes different from the energetically optimal, the size dependences of the lattice properties of the nanocrystal are enhanced.