The dependence of the temperature of crystal-liquid phase transition on the size and shape of simple nanocrystal

The dependence of melting temperature $T_m$ on the size and shape of an $n$ -dimensional nanocrystal of elementary single-component substance is studied. The nanocrystal has the form of an $n$-dimensional parallelepiped with a square base. The ratio of the length of side rib to the length of base rib (which is equal to $f$) defines the form of the system. It is demonstrated that, if the surface pressure is ignored, the value of $T_m$ decreases with isomorphic ($f = \text{const}$) decrease in the size of nanocrystal. In so doing, the more the value of form parameter $f$ deviates from unity, the more appreciable the size dependence of $T_m$ will be. However, if the surface pressure ("Laplace pressure") is taken into account in the case of decrease in size, the value of $T_m$ may vary significantly. In so doing, if the surface pressure compresses the nanocrystal, this leads to an increase in $T_m$ when its size decreases. In the case of stretching of nanocrystal by surface pressure, the drop of $T_m$ with isomorphic decrease in size increases. It is demonstrated that the surface pressure may both attenuate (at low values of $T_m$) and intensify (at high values of $T_m$) the dimensional oscillation of melting temperature. This variation of oscillation of dimensional dependence will be most pronounced for substances with high values of Grueneisen parameter. The variation of melting temperature with decreasing crystal dimension $n$ is studied. It is demonstrated that, in the case of isomorphic decrease in the size of nanocrystal, the coefficient of thermal expansion at melting temperature increases and reaches a maximum, and the specific energies of activation processes and the surface energy decrease and reach a minimum.