The Fickian second law is widely applicable not only to the analysis of various diffusion problems in material science but also to that of phenomena for the Brownian motion in other science fields, such as the behavior of neurons in life science. It is thus one of the most dominant equations in science. In 1894, Boltzmann transformed it into an ordinary differential equation applicable to the analysis of the interdiffusion problems. Since then, however, the mathematical solutions have not yet been obtained. Here we derived two new equations superior in calculation to the ones of Fick and Boltzmann. Using the derived integro-differential equation, their solutions were obtained as analytical expressions in accordance with the results of the experimental analysis. Hereafter, the basic equations derived here will be exceedingly useful for the analysis of the nonlinear problems concerning the Brownian motion in various science fields.
Based on the divergence theorem, we reveal that the Fickian first law relevant to the diffusion flux |<i>J(t,x,y,z) </i>> in the time and space is incomplete without an integral constant |<i>J<sub>0</sub>(t) </i>> for the integral of Fickian second law. The new diffusion flux (NDF) taking it into account shows that we can systematically understand the problems of one-way diffusion, impurity diffusion and self-diffusion as a special case of the interdiffusion. Applying the NDF to the interdiffusion problem between metal plates, it is clarified that the Kirkenkall effect is caused by |<i>J<sub>0</sub>(t) </i>> and also that the interdiffusion coefficients in alloy can be easily obtained. The interdiffusion problems are reasonably solved regardless of the intrinsic diffusion conception. Thus the NDF to replace the Fickian first law is an essential equation in physics
In a series of diffusion studies, some new diffusion theories different from the existing ones have been reported. One of their theories revealed that the concept of intrinsic diffusion, which is conceived to understand the well-known Kirkendall effect relevant to the interdiffusion, is an illusion. Based on the theory, a new interdiffusion theory is established as a Brown motion in many elements system, regardless of the diffusion field (solvent) in a crystal material or an amorphous one. At the same time, the vacancy diffusion equation of interdiffusion problems in crystal is also presented. The present analytical system is useful for analyzing actual interdiffusion problems in the many elements system. By means of investigating vacancy behavior in the diffusion field composed of many elements, a more accurate analysis of interdiffusion problems will be possible.
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