Dependences of the Vacancy Concentration and the Self-Diffusion Coefficient on the Size and Shape of a Nanocrystal

Abstract

Based on the generalized Rectangular Parallelepiped (RP) model of a nanocrystal, an expression for the Helmholtz free energy is derived and the equation of state for the nanocrystal, which contains both lattice vacancies and delocalized (diffusing) atoms, is calculated. The dependences of the probability of formation vacancies (φ v ) and the probability of delocalization of an atom (φ v ) on the size and shape of a nanocrystal at different temperatures (T) and pressures (P) are studied. The calculations are performed for the BCC lattice of iron under the condition of isothermal compression of a nanocrystal along the isotherms at 300 and 1000 K. The size dependences are studied at the atmospheric pressure (P = 1 bar) and at P = 100 kbar. It is shown that the isothermal–isobaric growth of a nanocrystal at the atmospheric pressure and T = 300 K gives a lower number of vacancies per atom in a nanocrystal in comparison with a macroscopic crystal, but dispersing the latter at T = 1000 K increases the probability of formation of vacancies. Upon reducing the size of a nanocrystal, the probability of delocalization of an atom (the same as the self-diffusion coefficient) increases at any pressure and temperature. The φ v /φ v ratio decreases with a decrease in the size of a nanocrystal and the out-of-vacancy self-diffusion is observed at the size at which the number of delocalized atoms is larger than the number of vacant sites in the nanocrystal lattice. When the shape of a nanocrystal deviates from the most optimal shape (from the cubic shape in the case of the RP model), the size dependences of the nanocrystal lattice properties become stronger.