Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

DoITPoMS Teaching & Learning Packages Diffusion Derivation of grain boundary flux equation
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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;

\[J = {J_{\rm{b}}} + {J_{{\rm{gb}}}}\frac{{2\delta }}{d}\]