A simple theoretical model is developed to study the size and shape dependence of vibrational and thermodynamic properties of nanomaterials. To show the real connection with the nanomaterials we have studied Debye temperature, Debye frequency, melting entropy, and enthalpy in different shapes, namely, spherical, nanowire, and nanofilm of -Fe, Sn, Ag, and In. The results obtained are compared with the experimental data. A good agreement between the model predictions and the experimental data supports the theory developed in the present paper. 1. Introduction The bulk properties of crystals depend on their structure, but at the nanoscale, in addition to the structure, their size and shape are an important factor, which influences their properties. The most significant characteristic of materials at the nanoscale is their high surface to volume ratio, which affects their thermodynamical properties. It is now well known that the melting temperature of nanoparticles depends on their size [1]. Melting temperature depression occurs for almost all free nanoparticles [2–5] and superheating has been reported for nanoparticles embedded in other host materials [6–8]. Guisbiers [9] used a top down approach and reported a universal relation, which is particularly helpful when experiments are difficult to lead on a specific material property. To validate the relation Guisbiers [9] compared the theoretical relation with experimental data of cohesive energy of Mo and W nanoparticles and of activation energy of diffusion for Fe and Cu nanoparticles. Guisbiers [9] also compared his results with different models predicting the size-dependent behavior of the vacancy formulation energy. The vacancy formulation energy versus the size of a spherical gold nanoparticle has been studied [9]. It has been concluded that the models reported by earlier workers support the formulation given by Guisbiers [9]. Thus, there are relatively extensive investigations on the size dependence of melting and cohesive energy of nanocrystals. However, it has not been accompanied by the necessary investigation of size dependence of vibrational properties and thermodynamics of nanocrystals. Such investigations should depend on our understanding of the size effect of melting and cohesive properties. In particular, a complete understanding of the melting transition in nanocrystals cannot be obtained without a clear understanding of enthalpy and entropy of melting, which are important properties of melting as already discussed in detail by Zhang et al. [10] as well as Safaei and Shandiz [11]. Therefore,
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