Effects of particle size, shape and crystal structure on the formation energy of Schottky vacancies in free-standing metal nanoparticles: A model study

A simplified model based on cohesive energy is proposed to estimate the formation energy of Schottky vacancies (VFE) in free-standing metal nanoparticles with BCC and FCC crystal structures. To study the effect of particle size and shape, the surface energy, elastic contraction and average coordination number of particles at the surface and core was considered. It is shown that the energy of vacancy formation in FCC nanoparticles increases with decreasing the size while the effect of particle shape (sphere, cubic and icosahedral) is marginal. In spite of this behavior, BCC nanoparticles exhibit a critical particle size at around 25 Å, at which a minimum VFE is attained. Additionally, the energy of vacancy formation is notably lower for BCC nanoparticles with cubic shape than spherical ones. The application of the developed model is shown for free-standing Fe and Cu nanoparticles.