Analysis of Diffusion in the A, B, and C Regimes

A Regime Since diffusion in this regime is macroscopically similar to diffusion in a homogeneous material possessing an effective bulk diffusivity (Eq. 9.4), it may be analyzed by the methods described in Chapters 4 and 5. 

Fisher has produced a relatively simple solution for a specimen geometry that is convenient for experimentalists and which has been widely used in the study of boundary self-diffusion by making several approximations which are justified over a range of conditions [9, 10]. The geometry is shown in Fig. 9.8 it is assumed that the specimen is semi-infinite in the y direction and that the boundary is stationary. The boundary condition at the surface corresponds to constant unit tracer concentration, and the initial condition specifies zero tracer concentration within the specimen. Rapid diffusion then occurs down the boundary slab along y, while tracer atoms simultaneously leak into the grains transversely along x by means of crystal diffusion. The diffusion equation in the boundary slab then has the form 

the rapid saturation found in the boundary slab produces a quasi-steady-state condition along the boundary dcB/dtx in Eq. 9.8 can then be set equal to zero so 

The solution to Eq. 9.10 must satisfy the boundary condition cB(0, tx) therefore 

Putting this result into Eq. 9.9 in order to find cXL(x, y, t ), 

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