Analytical solution of the grain boundary diffusion problem

Analytical solution of the grain boundary diffusion problem 

The result of this percolation into the grain is that, after some time, the concentration distribution in the grain boundary can be approximated as a steady state distribution. This means that d Ci/d t in eq. (5-50) almost vanishes. If we let dcijdt be zero, then the solution of eq. (5-51) is  

Since the left-hand-side of eq. (5-54) can be determined from experiments, one can derive the grain boundary diffusion coefficient with the help of eq. (5-54), provided that the volume diffusion coefficient Dj is known, and that an assumption has been made concerning the thickness S of the grain boundary. A characteristic value for S in metals is 5 10 cm. 

The above calculations, which are a zeroth order approximation, and which follow essentially the ideas of Fisher [28], can be further refined. Higher order approximations can 

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