Statistical model for nanoparticles formation: Self-consistent field approximation

In this paper, we propose a statistical model for the formation of metal nanoparticles in a homogeneous infinite medium. The Ising-like Hamiltonian of a system is analysed using the self-consistent field approximation for the lattice-gas model. Exclusively direct atom–atom interactions are considered. A special role of surface atoms and their interaction with the external medium are discussed in detail. The grand thermodynamic potential of a system is calculated and other thermodynamic functions are obtained. A consistent set of equations is found for describing both the radii of nanoparticles and their density (occupation number). The essential role of the medium nonhomogeneity in the process of nanoparticles arising is proved. The temperature dependence for the radii of the formed gold nanoparticles is estimated and the variance of their sizes is calculated.