Diffusion
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Derivation of grain boundary flux equation
Total number of atoms per second = net flux × total area
\[JA = {J_{\rm{b}}}{A_{\rm{b}}} + {J_{{\rm{gb}}}}{A_{{\rm{gb}}}}\]
Dividing all parts by the area of the bulk grain gives;
\[\frac{{JA}}{{{A_{\rm{b}}}}} = \frac{{{J_{\rm{b}}}{A_{\rm{b}}}}}{{{A_{\rm{b}}}}} + \frac{{{J_{{\rm{gb}}}}{A_{{\rm{gb}}}}}}{{{A_{\rm{b}}}}}\]
if Ab >> Agb then A ≈ Ab And so these can cancel in the equation to give;
\[J = {J_{\rm{b}}} + {J_{{\rm{gb}}}}\frac{{2\delta }}{d}\]