New mathematical solution for analyzing interdiffusion problems

Abstract

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. © 2011 The Japan Institute of Metals.