3.3.1 Effect of Vacancy Transport

When an atom moves in an ideal crystal lattice, under the influence of the electromigration force, it leaves behind a vacancy. Considering a small test volume V in the crystal lattice, the number of atoms n_a which leave the volume is replaced by the number of vacancies n_v entering it [27]. Since the volume of a vacancy Ω_v is smaller than the volume of the atom by about 20-40% [91], the new test volume V' is given by

\[\begin{equation} V{'} = V- n_\text{a}\Omega_\text{a} + n_\text{v}\Omega_\text{v}=V- n_\text{a}\Omega_\text{a} + n_\text{v}f\Omega_\text{a}, \end{equation}\] (3.46)

The relative volumetric change of the test volume ΔV, due to the vacancy exchange mechanism and associated to the change in vacancy concentration is given as follows

\[\begin{equation} \cfrac{\Delta V}{V} =\cfrac{V{'}-V}{V}= -(1-f)\Omega_\text{a}\cfrac{n_\text{v}}{V}=-(1-f)\Omega_\text{a}\Delta C_\text{v}. \end{equation}\] (3.47)

A local volumetric strain ε^m is induced by the local volumetric changes caused by vacancy transport and is given by

\[\begin{equation} \cfrac{\Delta V}{V} =\epsilon^\text{m}_\text{xx}+\epsilon^\text{m}_\text{yy}+\epsilon^\text{m}_\text{zz}=3\epsilon^\text{m}, \end{equation}\] (3.48)

when the material is considered linear and isotropic. By evaluating the time derivative of equations (3.47) and (3.48), it is possible to obtain the mechanical relationship between volume change and strain

\[\begin{equation} \frac{\partial{\epsilon^\text{m}}}{\partial t} =-\cfrac{1}{3}(1-f)\Omega_\text{a}\frac{\partial{C_\text{v}}}{\partial t}. \end{equation}\] (3.49)

As discussed in Section 3.2, since the vacancy accumulation/depletion mechanism due to transport is described by the flux divergence during the vacancy flow in the form

\[\begin{equation} \frac{\partial{C_\text{v}}}{\partial t} =-\nabla \cdot \vec{J_\text{v}}, \end{equation}\] (3.50)

the components of the transport strain rate in equation (3.49) are given by

\[\begin{equation} \frac{\partial{\epsilon^\text{m}_\text{ik}}}{\partial t} =\cfrac{1}{3}(1-f)\Omega_\text{a}\nabla \cdot \vec{J_\text{v}}\delta_\text{ik}, \end{equation}\] (3.51)

where δ^ik is the Kronecker delta function, which has the properties

\[\begin{align} \delta_\text{ik} = \begin{cases} 1 & \mbox{for } i=k, \\ 0 & \mbox{for } i\neq k.\mbox{} \end{cases} \end{align}\] (3.52)

This function permits to reduce equation (3.51) to a single kinetic equation in terms of the trace of the transport strain tensor Tr(ε^m), when the transport strain tensor is diagonal with equal diagonal entries.