Study On Debye Temperature, Coefficient Of Volume Thermal Expansion And Specific Heat Capacity Of Nano-sized Crystals

In recent years, thermal properties of nano-sized materials have been extensively investigated in many cases because of their widely use in nanoelectronics and their importance on scientific researches. It is well known that the properties of materials significantly different from that of bulk since the size of the crystal decreases into nano size. To solve the size dependence problem, the essential way lies in the transition from the material microscopic properties or the macroscopic ones to the microscopic ones. Thermodynamics is a simple method to study the transition from macroscopic world to the microscopic one.Takagi in 1954 demonstrated for the first time that ultra fine metallic particles melt below their corresponding bulk melting temperature. It is now known that the melting temperature of all low dimensional crystals, including metal, semiconductor and organic crystals, depends on their sizes. Among many melting theories, Lindemann criterion which was put forward about a century is experiential and quite effective for the studying of melting behavior of maters. It has been widely used to study the melting and solidification process of crystals, nanocrystals and organic materials. Experiments have shown that the melting process of nanocrystals follows Lindemann criterion as bulk crystals, it is feasible to further discuss the melting theory of nanocrystals based on the criterion in theory. In fact, the studying of melting thermodynamics of nanocrystals is to study size effect on the melting thermodynamic properties of crystals compared with bulk crystals. F. G. Shi theoretically presented a model for size-dependent melting temperature based on Lindemann criterion. The model can perfectly interpret melting behavior of nanocrystals, it is applicable both for melting depression and for superheating. For free-standing nanocrystals, the fraction of surface atoms increases with size decreasing and the thermal vibration of surface atoms is larger than that of internal atoms, mean square displacement increases with size decreasing, thus the melting temperature decreases with size decreasing. But this model has drawback for it including a undefined parameter which is obtained through experiment.It is known that the melting entropy of metallic nanocrystals is dominated by vibrational entropy of atoms and decreases with size decreasing. Based on the models above, a function of melting temperature dependent on the size which is very simple and free any adjustable parameter is developed by deciding the parameter of previous model.Based on the size dependent melting temperature model above, we have established simple models for size and interface effects on the specific heat capacity of nano-sized crystals. Our model has the following characteristics (1) Free of adjustable parameters; (2) full size range (3) each parameter within the model having clear physical meaning.Based on the function of Debye temperature of bulk material, the expression of Debye temperature of nano-sized crystals is obtained,Thus, the coefficient of volume thermal expansion can be described as the following function, A model of the specific heat capacity of nano-sized crystal which is size dependent is proposed based on the Debye model and the size dependent melting temperature model.It is obvious from the function above is that the specific heat capacity at constant volume is not only temperature dependent but also has the size effect. When the temperature is a constant, the specific heat capacity of nano-sized crystal is much larger than it of the bulk material. Although the trend of the model agrees well with the experimental data, the values are lack of good agreement with the experimental data.Based on the model of the cohesive energy of nanocrystals E(D) and some hypothesis, the specific heat capacity of nanocrystals has been described as the following function,In this contribution, the CP,m (D) of nanocrystals Cu and Pd dependent on size range from 150 K to 300 K has been computed. Comparisons of CP,m(D) of nanocrystals Cu and Pd between model predictions in term of Eq.(4) and available experimental results have been shown in the figures. As expected, the specific heat capacity of nanocrystal is larger than the specific heat capacity of the same bulk crystal. The enhancement of specific heat capacity may come from the contribution of lattice defects or grain boundary which modifies the phonon density of states. For the bulk material, since the number of surface atoms to the total number of atoms is very small, the contribution of surface atoms to heat capacity is neglectable. As the diminishing of the size of particles, the ratio of the number of surface atoms to the total number of atoms increases significantly. Thus the contribution of surface atoms to the specific heat capacity must be considered. Free boundaries provide the"softening"of vibration spectrum thus giving larger contribution to the specific heat capacity.