Although electromigration refers to the transport of material caused by the momentum transfer from conducting electrons to metal atoms , the total atomic migration is influenced by other physical mechanisms. Electromigration constitutes a diffusion-convection problem, where atomic transport along the interconnect line occurs due to a combination of several driving forces. Since the atomic migration occurs via a vacancy exchange mechanism, the material transport can be, in general, described in terms of vacancies as
In sites of flux divergence vacancies can accumulate or vanish depending on the sign of the divergence, and the material balance is given by the usual continuity equation
Equations (2.11) and (2.12) are the basic continuum equations which describe the total mass transport along an interconnect line due to electromigration and accompanying driving forces.
The model proposed by Shatzkes and Lloyd  was the first that rigorously derived the interconnect lifetime with an inverse square dependence on current density. Considering only the influence of diffusion and electromigration on the vacancy flux, the continuity equation (2.12) along the interconnect length direction can be written as
For a semi-infinite line under the boundary conditions
Assuming that the failure occurs, when the vacancy concentration reaches a given critical value significantly higher than the initial equilibrium value , and that , then (2.15) is approximated  by
Note that this equation is similar to the original Black's equation (1.5), except for the multiplying term , and it also predicts a mean time to failure with an inverse square current density dependence. This is a result of the assumption that the failure takes place, when the vacancy concentration reaches a given critical value, which corresponds, in fact, to a void nucleation condition.
A similar model to (2.13) had already been proposed by Rosenberg and Ohring  including a source term, so that
The aforementioned models, where material transport only due to gradients of concentration and due to electromigration itself is considered, have two main shortcomings: first, the time scale to reach the steady state vacancy supersaturation lies, at most, in the order of minutes, which is too short compared to the typical failure times. Second, the maximum vacancy supersaturation is very low, in such a way that the energy barrier for void formation would be extremely high. This hinders void formation by means of vacancy condensation and, therefore, a critical vacancy concentration cannot be used to determine the failure of the interconnect. As will be shown in the next sections, these shortcomings can only be resolved with the introduction of mechanical stress in the model equations.