Size dependence of the temperature coefficient of surface tension for a solid nanoparticle at the interface with vapor

New relationships have been obtained for the temperature coefficient of surface tension for a solid spherical nanoparticle at the interface with vapor $d\sigma/dT$ as a function of the radius $r$ of the surface of tension for two different cases of two-phase equilibrium (at a constant radius of the curvature $r=\operatorname{const}$ and at a constant pressure in the vapor phase $P^{(\beta)}=\operatorname{const}$), as well as for the case of three-phase (solid nanoparticlevapor-liquid) equilibrium at arbitrary values of $r$ and $P^{(\beta)}$. The dependences of $d\sigma/dT$, $dT/dr$, and $\sigma$ on $r$ have been self-consistently calculated for many metals.