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On the modified Fick law and its potential applications

Piekarski S.

Journal of Technical Physics

2003

Vol. 44, no 2

125-131

It can be observed that the standard Einstein-Smoluchowski equation for isothermal processes can be written in the two equivalent forms which, however, become distinct when the temperature becomes spatially varying. Therefore, we have two different equations which could potentially describe the flux of diffusing particles subjected to the potential external forces and variable temperature. The first is the standard Einstein-Smoluchowski equation, while the second is its modification containing an additional term proportional to the temperature gradient. In order to fix the terminology, the above mentioned modification of standard Einstein-Smoluchowski equation shall be called '^the modified Fick law". In this paper, we shall discuss the origin of this equation and we shall try to make a preliminary discussion of its solutions and potential applications to thermodiffusion. We propose two models: one exploiting Onsager's thermodynamics and the second one independent from it. Elementary solutions of both models are identical in the limit of small concentrations.

On diffusion and thermodiffusion in a gravity field

Piekarski S.

Journal of Technical Physics

2003

Vol. 44, no 3

329-337

The modified Fick law has been introduced in the paper: S. Piekarski, On the modified Fick law and its potential applications (J. Tech. Phys., 44, 2, 125-131, 2003). The first purpose of the present paper is to give a simpler "derivation" of the modified Fick law than that originally presented. Namely, it can be introduced as a particular case of the Wojnar equation (R. Wojnar, "Nonlinear heat equation and thermodiffusion", p. 296, Eq. (2.1), Rep. on Math. Phys., No 1/2, Vol. 46, 2000). The third equation discussed is the Einstein-Smoluchowski equation. All equations mentioned above, contain a potential for the force, acting on the diffusing matter. The important fact is that, for isothermal conditions, all the equations mentioned above are identical. In this article we try to describe the effects of the constant gravitational field on the diffusion processes. We propose an explicit form of the potential for which the equilibrium concentration profiles are consistent with the experimental results. We compare shortly our results with two models of the diffusion processes in a gravitational field described in Huang's book [8] as well as with other opinions concerning this subject. As an example of technological application of the presented theory, the diffusion of the atomic hydrogen in fluid iron is proposed.