Size-dependent Diffusion Activation Energy

There is another world between the macroscopic one and the microscopic one, the mesoscopic world of the nanosystem where many special phenomena occur. The research on such state changes of condensed matter as melting and freezing is very important. Takagi in 1954 demonstrated for the first time that ultrafine metallic particles melt below their corresponding bulk melting temperature. It is now known that the melting temperature of all kinds of low-dimensional crystals, including metals, semiconductors and organic crystals, depends on their sizes. The melting temperatures could be higher or lower than the corresponding bulk ones depending on the surface states of low-dimensional materials. The study on superheating and surface stress related to surface phenomena is significant to stability of nanosized device. To understand the above problems, the essential way lies in the transition from microscopic properties of the material or the macroscopic ones to the mesoscopic ones. Thermodynamics is a simple method to study the transition from macroscopic world to the mesoscopic one. The application of thermodynamics to nanomaterials reveals that a new branch of thermodynamics appears, i.e., nanothermodynamics. Although there are relatively extensive investigations on the size-dependent melting of nanocrystals, it will be developed by the necessary investigation of the size-dependent thermodynamics of nanocrystals. A clear cognizance of the size-dependent thermodynamic function will help us to know more about the size-dependent energy transition law in the mesoscopic world. Based on the Lindemann criterion on melting and the Mott's expression for the vibrational entropy at the melting temperature, the thermodynamic expression for the size-dependent melting temperature without free parameters is presented; some quantities in the equation are determined. And this model is applied to study diffusion problem of kinetics. With further research of nanomaterial, its diffusion problem gradually becomes one of interesting domains of scientists. Experience results show nanomaterial diffuses much faster than bulk, namely, diffusion of materials is size-dependent. For example, diffusion coefficient of a 2 nm nanopaticles at room temperature is 1×10-24cm2s-1, however, that of bulk is 1×10-32cm2s-1, assumed to be 111 face, activation enthalpy of nanoparticles amounts to 75% of that of bulk. The size and temperature dependent diffusion coefficient function is also an important parameter for any phase transition process through nucleation and growth where the size of the formed nucleus is several nanometers and inoculation time and growth time are certainly related to the kinetic properties of a material. The understanding of this kind of scientific problem becomes more urgent due to recent development on the nanotechnology where the full size of the materials is in nanometer size range. Diffusion is a determining feature of a number of application-oriented properties of nanocrystalline materials, such as enhanced ductility﹑diffusion-induced magnetic, enhanced ionic mass transport and catalytic activity. Moreover, diffusion in nanocrystalline materials is also relevant to the basic physics of interface, which provide good approaches for understanding properties of nanocrystalline. Combining the thermodynamic size-dependent melting temperature model that is based on the Lindemann criterion on melting and Arrhenius diffusion equation, one can obtain size-dependent diffusion equation. There are several main parts in this paper, listed as follows:Size-dependent diffusion coefficient of materials Whether diffusion coefficient of bulk and that of corresponding nanomaterial essentially is equal at melting temperature, namely, size-independent. According to the assumption, combining the thermodynamic size-dependent melting temperature model that is based on the Lindemann criterion on melting and Arrhenius diffusion equation, one can obtain size-dependent diffusion equation. In our model, we know diffusion coefficient of nan...