Mechanisms of phase transitions in sodium clusters: From molecular to bulk behavior

Abstract

The thermodynamics of sodium clusters is investigated by means of a classical empirical potential and a simple quantal tight-binding model. Neutral and singly charged clusters of sizes ranging from 8 to 147 atoms are considered. A very particular attention is paid to the optimization and sampling problems. We determine the lowest-energy structures (global minima) with the “basin-hopping” technique, and the finite-temperature simulations are improved by using the “q-jumping” method and put together with the multiple histogram method. The clusters geometries may be very different on the model used, but also on the ionic charge, up to the size of about 40 atoms. The thermodynamical analysis is performed near the solid–liquid transition by calculating the complete calorific curves (heat capacities) as well as some microscopic parameters to probe the dynamics on the energy landscapes, including the spectra of isomers found by periodic quenching, isomerization indexes and the Lindemann parameter $\mathrm{\delta .}$ Up to the largest sizes, we find that the heat capacity generally displays several features within the two models, although structural differences in the lowest-energy isomers usually induce different calorific curves. These premelting phenomena are characteristic of isomerizations taking place in a limited part of the configuration space. The thermodynamics appears to be directly related to the lowest-energy structure, and melting by steps is favored by the presence of defects on its surface. We estimate the melting temperatures $T_{\mathrm{melt}}\mathrm{(n)}$ and latent heats of melting $\mathrm{L(n),}$ and we observe two very different behaviors of their variations with the size $\mathrm{n.}$ Below about 75 atoms, both $T_{\mathrm{melt}}$ and $L$ exhibit strong non-monotonic variations typical of geometric size effects. This “microscopic” behavior is caused by the dominating premelting effects, and is replaced by a more “macroscopic” behavior for sizes larger than about 93 atoms. The premelting phenomena become there less important, and the melting process is much like the bulk solid–liquid phase transition rounded by size effects. The continuous variations displayed by the melting temperature are the only remains of cluster size effects. The models used are discussed and criticized on the basis of the similarities and discrepancies between their predictions and the experimental data.