Phenomenological treatment of non-steady state diffusional processes in binary systems

Phenomenological treatment of non-steady state diffusional processes in binary systems 

(5-28) is not suitable for describing spatially and temporally variable diffusion processes. Because of the principle of the conservation of mass, however, the divergence of the flux can always be set equal to the time derivative of the local concentration. This leads to Pick s second law which may be written as follows for the case of binary systems with constant diffusion coefficients  

In the form of eq. (5-30), Pick s second law applies only to one-dimensional problems in an isotropic medium. The index i on the diffusion coefficient has been removed in order to make it clear that this is no longer the component diffusion coefficient D,-, but rather, it is the chemical interdiffusion coefficient. Normally, the chemical interdiffusion coefficient will be a function of the individual component diffusion coefficients Di because of the coupling of the fluxes in the lattice system. When local thermodynamic equilibrium prevails, the coefficients Di are, in turn, unique functions of the composition. From the thermodynamics of irreversible processes it can be shown [6] that in binary systems there is only one independent transport coefficient, and in general, in n-component systems there can only be (n - 1) /2 independent transport coefficients. 

If the chemical diffusion coefficient is concentration dependent, then, instead of eq. (5-30) we must write  

The diffusion problem as formulated in eq. (5-31) can be discussed in two ways 1. A solution of the second-order partial differential equation is sou t for a given set of initial and boundary conditions under the assumption that the chemical interdiflfusion coeffi- 

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