Constancy of the surface energy during the melting of a nanocrystal

The relation for the surface free energy $\sigma$ as a function of size and shape of a nanocrystal is derived. The nanocrystal has the form of a parallelepiped with a square base. The ratio of the lateral edge length to the base edge length $f$ defines the shape of a system. It is shown that the value of $\sigma$ decreases with degreasing number of atoms $N$ in a nanocrystal. The higher the deviation of the form parameter $f$ from unity, the stronger the dependence $\sigma(N)$. It is found that the surface free energy decreases with the temperature $T$ in so doing, the quantity $(\mathrm{d}\sigma/\mathrm{d}T)$ is the greater, the smaller the size of the nanocrystal or the greater the deviation of the nanocrystal shape from the most thermodynamically stable, cubic, shape. It is shown that the nanocrystal melts when its surface energy decreases to a certain value independent of size and shape. The conditions are discussed under which the crystal fragmentation and dendritic growth are realized.