3.3.2 Effect of Vacancy Generation/Annihilation

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

\[\begin{equation} V{'} = V\pm n_\text{v}f\Omega_\text{a}. \end{equation}\] (3.53)

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 ε^m 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)

\[\begin{equation} \frac{\partial{C_\text{v}}}{\partial t}=G, \end{equation}\] (3.56)

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 ε^v 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 ε^v can be determined from equation (3.58).