The generation/annihilation of vacancies inside the test volume is caused by the change in the concentration of lattice sites. The new test volume V', due to the addiction/removal of lattice sites inside the initial volume V, is given by
The relative volumetric change of the test volume caused by the generation/annihilation process is obtained by following the same procedure used to derive equation (3.47) to obtain
\[\begin{equation} \cfrac{\Delta V}{V} =\cfrac{V{'}-V}{V}= \pm(1-f)\Omega_\text{a}\Delta C_\text{v}. \end{equation}\] | (3.54) |
Taking the time derivative of equation (3.54), the rate of the strain εm caused by vacancy recombination is given by
\[\begin{equation} \frac{\partial{\epsilon^\text{m}}}{\partial t} =\pm\cfrac{1}{3}f\Omega_\text{a}\frac{\partial{C_\text{v}}}{\partial t}. \end{equation}\] | (3.55) |
Since the change in vacancy concentration due to the generation/annihilation process is related to the source/sink term G of equation (3.26)
the components of the recombination strain rate are given by
\[\begin{equation} \frac{\partial{\epsilon^\text{m}_\text{ik}}}{\partial t} =\pm\cfrac{1}{3}f\Omega_\text{a} G\delta_\text{ik}. \end{equation}\] | (3.57) |
The complete kinetic relation for the strain εv induced by electromigration is given by the sum of the vacancy transport strain rate (equation (3.51)) and the vacancy recombination strain rate (equation (3.57)) as follows
\[\begin{equation} \frac{\partial{\epsilon^\text{v}_\text{ik}}}{\partial t} =\cfrac{\Omega_\text{a}}{3}\left((1-f)\nabla \cdot \vec{J_\text{v}}\pm f G\right)\delta_\text{ik}. \end{equation}\] | (3.58) |
The strain induced by variations in vacancy concentration εv can be determined from equation (3.58).