| As the size of low-dimensional materials decreases to nanometer size range, electronic, magnetic, optic, catalytic and thermodynamic properties of the materials are significantly altered from those of either the bulk or a single molecule. Among the above special properties of nanocrystals, the size-dependent melting temperature of nanocrystals has received considerable attention since Takagi in 1954 experimentally demonstrated that ultrafine metallic nanocrystals melt below their corresponding bulk melting temperature. However, it has not been accompanied by the necessary investigation of other size-dependent thermodynamic properties of nanocrystals, such as the melting entropy, the melting enthalpy, and the cohesive energy. Such an investigation should deepen our understanding of the nature of the thermal stability of nanocrystals. Different from nanoparticles, nanovoids, being defined as a cluster of many vacancies in a given matrix, have negatively curved surfaces, which results in the unusual thermodynamic behavior. Note that many researches [13, 20-22] have been focused on the melting point of the crystals as a function of nanovoid size via computer simulations, and find that the melting point increases with void size decreasing. However, by far the mechanism of void melting is still lacking and the theoretical basis for its unusual melting behavior is still unknown. Since nanovoids with potential applications, exist in many real materials and lead to the formation of internal interfaces (surfaces), it is important to identify the void size effect on properties. Cohesive energy as one of the most important lattice parameter, should directly be related with the nature of the thermal stability of nanocrystals. The established size-dependent cohesive energy which is based on surface/volumeδ, is suitable for nanoparticles. As particle size decreases, the percent of surface atoms increases and thus results in higher energy, which is easily understood. However, this model is usually suitable for nanoparticles, and limited for other nanostructures. Recently, the cohesive energies of ZnO nanowires and ZnO single-walled nanotubes have obtained via first-principle simulation, and found that ZnO single-walled nanotubes is more stable only at small size. Thus, to estimate the cohesive energy of nanocrystals with different structure and the relative stability, it is urgent to establish a cohesive energy model which is related with both size and structure. It is clear that cohesive energy is a valid method to estimate structure stability which relies on size. For clusters with small number of atoms, all clusters have similar spherical shape with different structures, such as icosahedron and truncated decahedron. Due to the size dependence of structure, it is significant to construct a unite model to predict clusters cohesive energy, which will deepen our knowledge to this property. The concrete contents are listed as follows:1. The size-dependent melting temperature model for nanocrystals is summarized systematically. In terms of such model, the melting temperature of a free nanocrystal decreases as its size decreases while nanocrystals embedded in a matrix can melt below or above the melting point of corresponding bulk crystals. If the bulk melting temperature of nanocrystals is lower than that of the matrix, the atomic diameter of nanocrystals is larger than that of the matrix, and the interfaces between them are coherent or semi-coherent, an enhancement of the melting point is present. Otherwise, there is a depression of the melting point. Our model can predict melting temperuatures of low-dimensional crystals with different chemical bonds, different dimensions, and different surface or interface conditions and the surface melting temperature of nanocrystals. As an example, an agreement is found between model predictions and experimental results for In nanocrystals. Based on the above model and the size-dependent melting entropy model, we establisehed a model to predict the size dependence of the mleing enthalpy and further to predict the size effect on the cohesive energy of nanocrystals. It is found that our model predictions are consistent with available experimental results for the melting enthalpy of In nanocrystals and cohesive energies of Mo and W nanoparticles,and the validity of this model to predict cohesive energy of nanocrystals with surface/volume ratio smaller than 0.5 is apparent.2. With higher than the bulk melting point, the melting temperature of crystals from nanovoids is an inverse function of void size, which is different from the melting behavior of nanoparticles. According to the classical nucleation theory, the estimation of the equation for homogeneous nucleation temperature of liquid elements from nanovoids is developed by considering energetic contributions of surface (interface) energy and surface (interface) stress on total Gibbs free energy. The model predicts the superheating limit temperature, as a function of void size, which limits the crystal stability from kinetic point. The crystal can be significantly superheated with increase of superheating degree and this increase is enhanced as nanovoids size decreases, which can well explain the MD simulation results for melting temperatures of argon crystal from nanovoids. As void radius is larger than 2nm, this melting behavior is similar to that of bulk. In the solid-liquid transition, except the contribution of surface (interface) free energy change, taking the effect of elastic energy change into consideration is necessary to determine the superheating degree. Moreover, according to the established size-dependent surface (interface) energy functions for nanoparticles and taking the negative curvature for nanovoids, a similar function for negatively curved surface (interface) energy is obtained, which is consistent with other theoretical results. Similarly, the surface (interface) energy becomes slowly decrease as void radius is larger than 2nm.3. Although the synthesized ZnO single-walled nanotube has not obtained in experiments, ZnO single-walled nanotube can exist in principle since negative cohesive energy. Due to potential application, the stability of both ZnO single-walled nanotube and nanowires should be determined. Recently, ZnO single-walled nanotube is found to be stable only at small size, while ZnO nanowires is preferred at larger size. Based on the bond broken rule and introducing bond number which is directly related with cohesive energy, we develop a model of size-dependent cohesive energy without any adjustable parameters, which successfully predict the cohesive energy of ZnO nanowires. Moreover, the critical size at which the stability of ZnO single-walled nanotube and ZnO nanowires change is estimated. Good agreement between model predictions and simulation results is found.4. The structure and cohesive energy of clusters have been investigated by computer simulation. The structures of clusters depend on size and tend to bulk shape as size increasing. However, suitable models describing their cohesive energy and shape in a unified form are absent since the structure uncertainty is the key difficulty to realize the this object. For small size of clusters, the surface energy is the dominant factor to determine the structure, while the Wulff is preferred to be equilibrium shape for bulk crystals. Although Wulff construction does not account for the desire of clusters to minimize the total surface to volume ratio, Wulff construction has the same consideration to establish stable clusters?designing the substance with smallest surface energy. Owing to the densest consideration on packing, a size-dependent bond number of Wulff construction of Fcc structure is established. Substituting this bond number function into the cohesive energy model based on bond number, we find that the validity of Wulff construction to describe cohesive energy of cluters, since the good agreement between model predictions and simulation results for several metal clusters is found, without considering the detail of cluster structures.
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