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Diffusion theory

Diffusion is a process by which molecules spread from areas of high concentration, to areas of low concentration.

Atomic diffusion

Diffusion profile

An empirical law to describe the process of diffusion is called Fick’s first law which states that the atomic flux is linearly proportional to the concentration gradient, namely,


where J is the flux of a given species, c is the number of atoms per unit volume (concentration), D is the diffusion coefficient, and x is the position. Therefore, the diffusion coefficient, D, has a dimension of m2/s.

In the case of two materials are in contact at a interface (x=0) and the initial concentration of a given species is c=c1 for x<0 and c=c0 for x>0, we can obtain the concentration profile c(x, t), which is the solution of equation (1), as,


where erf(y) is the error function defined by:



The concentration profile c(x, t) can be experimentally determined, and the diffuison coefficient, D, can be obtained by fitting c(x, t) to equation (2).


A program for fitting.