Entropic Effects on the Size Dependence of Cluster Structure

Abstract

We show that the vibrational entropy can play a crucial role in determining the equilibrium structure of clusters by constructing structural phase diagrams showing how the structure depends upon both size and temperature. These phase diagrams are obtained for example rare gas and metal clusters.PACS numbers: 61.46.+w,36.40.Mr,36.40.Ei Much of the interest in clusters or nanoparticles derives from the insights they can provide into how properties emerge and evolve on going between the atomic and molecular and bulk limits. Cluster structure provides a particular interesting example of this size evolution. At large enough sizes the clusters must display the bulk crystalline structure, but this limit may sometimes only be achieved at very large sizes (e.g. at least 20 000 atoms for sodium clusters [1]) and before that limit is reached unusual structural forms are often observed. For example, many clusters bound by van der Waals or metallic forces exhibit structures with five-fold axes of symmetry, a possibility that is forbidden in bulk crystalline materials. For these clusters the dominant structural motif typically changes from icosahedral ( Fig. 1(b)) to decahedral ( Fig. 1(c)) to face-centred-cubic (fcc) ( Fig. 1(a)) as the size increases.For many materials these structural changes occur at sizes that are too large for global optimization to be feasible. Therefore, the typical theoretical approach to systematically investigating the size evolution of cluster structure is to compare the energies of stable sequences of structures, such as the forms shown in Fig. 1. 'Crossover sizes' are then identified where the sequence with lowest energy changes. At this crossover the most common equilibrium structure is expected to change. This technique has been applied to rare gas [2,3,4], metal [5,6,7,8] and molecular clusters [9].The above approach is certainly valid at zero temperature, since the equilibrium structure then corresponds to the one with lowest energy. At other temperatures, however, the structure with lowest free energy needs to be found. However, perhaps through an expectation that entropic effects are unlikely to be important or are too complicated to take into account, size is usually the only variable that is considered both experimentally [10,11,12] and theoretically [2,3,4,5,6,7,8].In this paper we consider the role that entropy plays in the size evolution of cluster structure, and show that temperature can be a key variable in determining the equi- [14]. These clusters have the optimal shape for the three main types of regular packing seen in clusters: face-centred cubic, icosahedral and decahedral, respectively. The latter two structural types cannot be extended to bulk because of the five-fold axes of symmetry.librium structure of a cluster. A clue to this result can be garnered from the growing number of examples of solidsolid transitions in clusters where the structure changes from fcc or decahedral to icosahedral as the temperature increases [15,16,17,18,19]. The most well-investigated example...

Structural transition in(C60)nclusters

“…Since the icosahedral structures definitely do not represent the global energy minima ͑at Tϭ0 K) and are also considerably higher in potential energy than, for example, the tetrahedral structures, 28 they would only be preferred at high temperatures because of their larger vibrational entropy. 28,39 Consequently, this kind of equilibrium solid-solid transition, which was observed in some systems as a premelting effect, 34 -37 is most probably not responsible for the structural change in (C 60 ) n clusters in the investigated temperature range.…”

Melting of Lennard-Jones rare-gas clusters doped with a single impurity atom

“…To understand the solid-solid transitions that occur in a doped cluster between homotops of the same stoichiometry we have used the Harmonic Superposition Method (HSM) 22,23 . This method assumes that there is a number m of well defined states that make most of the contribution to the partition function in a certain range of temperatures.…”

Novel structures and energy spectra of hydroxylated (SiO2)8-based clusters: Searching for the magic (SiO2)8O2H3− cluster

“…[1][2][3] Recently, it has been increasingly realized that, for a particular known cluster size, the potential energy surface ͑PES͒ of cluster isomers, and their accessibility subject to the experimental conditions, can strongly influence the favored cluster structure. 4,5 Such considerations are particularly difficult to assess for clusters for which the PES exhibits many minima of similar energy with varying degrees of thermal accessibility. 6 From our, and other, investigations into pure [7][8][9][10][11][12] and hydroxylated [13][14][15][16][17] silica nanoclusters, it would appear from the evident rich structural diversity of low energy forms that SiO 2 -based nanostructures possess PES of this type.…”