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#### 3.2 Electromigration in Bulk Metals

The mass transport due to electromigration is modeled as the flow of vacancies. First the equations for EM in bulk were written in this form in [116] and further developed in [34, 136]. The governing equation of the vacancy
concentration is given by the conservation law

with a flow term and a generation/annihilation term. The flow is the sum of the four flows driven by diffusion, by EM, a gradient of the pressure, and a gradient of the temperature distribution.

The first flow contribution is a diffusion induced flow expressed by

where is the diffusion
coefficient tensor. This coefficient is in general a tensor of second order as shown in Section 2.2.

The second component is the electromigration induced flow given by

where is the effective valence, is the elementary charge, the Boltzmann constant, the temperature, and the electrical field.

Due to a gradient in the hydraulic pressure a flux is driven which is modeled by the third term as

with being the relation between the volume of a vacancy
and the volume of an atom . Therefore, is in the range between zero and one. For the crystal
lattice of metals the ratio is in the range of [39, 41].

The fourth term is the flow driven by temperature gradient.

For vacancies there is an equilibrium concentration to which the concentration converges in the absence of any outer perturbation. This phenomenon is modeled by the so called Rosenberg-Ohring term [21, 110].

is the
equilibrium concentration and is the characteristic relaxation time of the vacancy concentration. The equilibrium
concentration depends on the temperature according to an Arrhenius law [102]

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