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 δ. 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 Tmelt(n) and latent heats of melting L(n), and we observe two very different behaviors of their variations with the size n. Below about 75 atoms, both Tmelt 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.
Melting of neutral and singly charged sodium clusters Na n and Na 1 n is investigated for several sizes ͑8 # n # 139͒ using both Monte Carlo and molecular dynamics simulations with an empirical many-body potential. Up to n 75, we find that the solid-liquid-like phase change is a multistage process initiated by "premelting." The equilibrium structure is found to play an important role in the thermodynamics, in particular when such premelting features are induced by diffusion processes at the surface. However, larger sizes n $ 93 exhibit preferentially a single process similar to bulk melting.[S0031-9007(99)08675-5] PACS numbers: 36.40.Ei, 36.40.Sx, 82.20.Wt The thermodynamics of finite systems is an important part of the clusters' chemical physics. Understanding how phase changes occur far below the bulk limit has motivated a lot of theoretical work on various species [1-10] as well as experimental investigations [11][12][13]. Size affects the thermal properties of nanoscale materials in several ways. A first-order phase transition, such as the solidliquid transition, becomes "rounded," spread over a finite temperature range. As the size decreases from the bulk, the melting point and the latent heat of fusion also decrease, and the transition region itself widens. When the size gets low enough, it has been observed both in simulations [14] and in experiments [13] that the melting point could undergo strong, nonmonotonic variations with size, even if only a single atom is added or removed. In the case of argon clusters, these differences have been attributed to the underlying geometrical "magic numbers" governing the extra stability of icosahedral clusters [14]. In the case of metal clusters, electronic shell effects are also expected to play a significant role [13].The experimental approaches to phase changes in free clusters, as developed, e.g., in the group of Haberland [12], are macroscopic in that they try to access thermodynamical indicators such as the heat capacity or the melting temperature. Theoretically, the most relevant numerical approaches, although they can also provide the thermodynamic functions, generally attempt to relate the solidlike or liquidlike character of a cluster to its microscopic dynamics, especially to the details of its potential-energy surface (PES) [4,10,15,16]. Solid and liquid phases cannot, however, always be identified unambiguously with a single peak in C͑T ͒. Indeed, for small clusters, the concept of phases and phase changes is often related to isomerization phenomena, and many theoretical studies seem to show an intricate, multimodal behavior of C͑T ͒. Bumps in C͑T ͒ may be due to preliminary isomerization processes [16], partial melting [17,18], orientational disorder in molecular clusters [4], or to more complex features such as multiple funnels on a PES landscape [10]. In some cases, such as ionic [16,19] or metallic [5,20,21] clusters, the solidlikeliquidlike transition may involve intermediate phases fluctuating between several isomers. These phases cannot be fully ...
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