Boltzmann-Matano analysis of the concentration profile

In section 5.5.4, Pick s laws were solved for binary systems under the assumption that the chemical diffusion coefficient D is constant. This assumption is never strictly true, and is only approximately true in limiting cases. Thus, we must seek solutions for other cases as well. If we know the concentration dependence of the diffusion coefficient as well as the thermodynamics of the system, we can often reach conclusions regarding the type of disorder and the diffusion mechanism. Therefore, it is especially important in the study of point defect disorder and in the study of the elementary processes of solid state reactions that we know the concentration dependence of the diffusion coefficient. 

In the following discussion we shall once again be considering an isothermal, isotropic, one-dimensional diffusion problem. Local thermodynamic equilibrium of defects is assumed. Therefore, D will be a unique function of the composition at every coordinate x. 

Boltzmann and Matano showed [22, 23] how the concentration dependent chemical diffusion coefficient D (c) can be determined from the data obtained in a diffusion experiment with two semi-infinite regions and the initial conditions of Fig. 5-5 for the case where the molar volume Km binary system is independent of concentration. 

Pick s second law for the case of a concentration dependent chemical diffusion coefficient and the given initial and boundary conditions cannot be explicitly solved even by means of the substitution y = x/]/. However, if this substitution is made in eq. (5-31), then the equation 

The differential quotient (dx/dc) is taken at the point where the concentration is equal to c. Furthermore, it is evident that all material which has diffused out of the semi-infinite space X 0 must now be found in the semi-infinite space x 0. Therefore, the coordinate of the so-called Matano interface = 0 can be calculated by means of the equation  

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