Mathematical Physics in Diffusion Problems

Abstract

Using the divergence theorem and the coordinate transformation theory for the general Fickian second law, fundamental diffusion problems are investigated. As a result, the new findings are obtained as follows. The unified diffusion theory is reasonably established, including a self-diffusion theory and an N (N ≥ 2) elements system interdiffusion one. The Fickian first law is incomplete without a constant diffusion flux corresponding to the Brown motion in the localized space. The cause of Kirkendall effect and the nonexistence of intrinsic diffusion concept are theoretically revealed. In the parabolic space, an elegant analytical method of the diffusion equation is mathematically established, including a nonlinear diffusion equation. From the Schrödinger equation and the diffusion equation, the universal expression of diffusivity proportional to the Planck constant is reasonably obtained. The material wave equation proposed by de Broglie is also derived in relation to the Brown motion. The fundamental diffusion theories discussed here will be highly useful as a standard theory for the basic study of actual interdiffusion problems such as an alloy, a compound semiconductor, a multilayer thin film, and a microstructure material.

Fundamental Concept of Interdiffusion Problems

“…Nevertheless, the problem of coordinate system of diffusion equation was not mathematically investigated in accordance with the divergence theorem in those days. The problems of the coordinate transformation relevant to the diffusion equation had not been thus discussed in the diffusion history for a long time before the previous work [4]. The new findings, which are extremely dominant in the diffusion study, were reasonably obtained through the coordinate transformation theory then.…”

“…As is known from the unified theory of diffusion problems, the basic concept of diffusion phenomena is in the interdiffusion problems [4]. In the following discussion, the above raft model is reasonably applied to a typical binary system interdiffusion problem in accordance with the mathematical physics.…”

“…This indicates that the original diffusion equation is a moving coordinate system for the experimentation system outside the diffusion region. A discussion on the relation between the coordinate systems of diffusion equation is indispensable for understanding diffusion phenomena [4].…”

New Analytical Method of Interdiffusion Problems

“…has been widely used for analyzing interdiffusion problems [6] [8] [9] [10] [11] [12], the equation itself is not only mathematically wrong in the derivation process but also entirely meaningless because of using the intrinsic diffusion coefficients nonexistent in the recent diffusion theory [7] [15]. In the present work, the physical meaning of K effect is reasonably discussed compared with experimental results, regardless of the Darken equation.…”

“…It is also physically considered that the diffusion region space interacts with the free space near the surface of specimen considered to be the sink and source [18]. This means that the coordinate transformation of the diffusion equation is necessary for analyzing interdiffusion problems [15] [16]. Using the solutions obtained here, the relation 3) Determination of initial and/or boundary values of diffusivities in a ternary system interdiffusion It is revealed that the so-called interdiffusion coefficient in a many elements system is meaningful only in the differential equation of diffusion [15] [18].…”