Anharmonic and surface effects on thermodynamic properties of Ag, Al, and Cu nanostructures.

We present a theoretical approach combining the statistical moment method (SMM) and a bonding-energy model to investigate the size-, shape-, and temperature-dependent thermodynamic properties of Ag, Al, and Cu nanostructures. Analytical expressions are derived for the mean square displacement (MSD), thermal expansion coefficient, isobaric heat capacity, and Debye temperature. The results demonstrate that as the characteristic size decreases, the MSD, thermal expansion coefficient, and isobaric heat capacity increase, whereas the Debye temperature decreases. As the size increases, these thermodynamic quantities asymptotically approach their corresponding bulk values. Notably, at high temperatures, we observe pronounced nonlinear variations in nanostructures smaller than 5 nm, highlighting the critical role of anharmonic effects in lattice vibrations. Furthermore, the thermodynamic properties exhibit a strong shape dependence governed by the dimensionless shape factor that quantifies the effect of nanoparticle geometry on the surface-atom fraction. Regular tetrahedral particles show the greatest size-dependent variations due to their high shape factor, while thin films exhibit the lowest. This confirms that the surface area fundamentally drives the thermodynamic behavior of these nanomaterials. This study provides valuable theoretical insights for the design and optimization of nanostructure architectures tailored for specific practical applications.