The Mie-Grüneisen equation of state for metal nanoparticles

The Mie-Grüneisen equation is adapted for the description of the thermodynamic properties of nano-objects. It is shown that the Grüneisen parameter $\Gamma $ depends on the ratio of the number of surface atoms to the number of interior atoms of the nano-objects. This ratio is determined by the characteristic size of nano-object and its form. It increases with decrease of the characteristic size. Reduction of the characteristic size of nano-objects that have a free surface, leads to a change in the Grüneisen parameter. If the nano-object is embedded in a matrix and has with it the coherent boundary, at the decrease of the nano-object size occurs either an increase or decrease of $\Gamma $. Compression or stretching of nano-objects may bring about a change of the ratio between numbers of the interior and surface atoms caused by changing the form of nano-objects. The change of the Grüneisen parameter at the characteristic size decreasing of a nano-object is treated as the change of a degree of the anharmonicity of lattice thermal vibrations. It is established that an appreciable change of Γ occurs if the characteristic radius of a nano-object is less than 5 nm