Effect of Dopants-Solute Drag

Dopant is also called solute, which is dissolved as solid solution in polycrystalline solids. If there is an interaction potential for the solute to be attracted to or repelled from the grain boundary, the solute atoms or ions will have a nonuniform distribution at the grain boundaries. The interaction could be due to lattice strain energy caused by size mismatch between the solute and host ions and/or electrostatic potential energy for aliovalent solutes. 

The chemical potential of the solute atoms in the near grain-boundary region is given by 

According to diffusion equation, the flux of the solute atoms is given by 

The concentration profile of the solute can now he calculated from the continuity, which is given by 

The solute atom applies a force —(df//dx) to the grain boundary, so that the net force applied by all the solute atoms is given by 

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