Diffusion in a Chemical Potential Gradient

Diffusion of atoms under a pressure gradient in a grain (a) vacancy concentration below the surfaces where stresses are applied (b) diffusion paths of atoms indicated by curved arrows and (c) grain that has undergone creep. 

Diffusion of an atom to the nearest vacancy (a) The initial, intermediate, and final positions (b) variation of free energy versus distance. 

Let us say that the atom is diffusing under a chemical potential gradient of d 4./dx per mole. This gradient per atom (f) is given by Equahon 15.19. 

In this equation, H is the free energy difference between two sites X is the interatomic distance. Using Equations 15.19 and 15.20, we get an equation for the energy difference between two adjacent atom positions as given by Equation 15.21. 

A is the site fraction of vacancies, a is a geometric constant that depends on the crystal structure, is the coordination number of the vacancy, Vq is the frequency of vibration of the atoms, and AG m is the free energy change required for the diffusion to take place. The geometric constant takes into account that only a fraction of the total jumps, Vq, are jumps in one direction. In the case of cubic lattices, is 6, hence, only one-sixth of the total jumps are in one direction. Hence, the value of a is 1/6. It is also possible for the atoms to diffuse backward, provided they acquire sufficient energy. The frequency for the backward jump is given by  

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The tracer or self-diffusion coefficient represents only a random walk diffusion process, i.e. in the absence of chemical potential gradients. The true chemical diffusion coefficient refers to diffusion in a chemical-potential gradient and its expression is more complicated . ... 

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