2 The Diffuse Interface Model

The diffuse interface model equations for the void evolving in an unpassivated interconnect line are the balance equation for the order parameter $\phi$ [56,57,58,59],

$$\frac{\partial \phi} {\partial t} = \frac{2D_{s}} {\epsilon \pi} \nabla \cdot (\nabla \mu - \vert e\vert Z^{*} \nabla \varphi),$$ (260)

$$\mu = \frac{4\Omega \gamma_{s}}{\epsilon \pi}(f'(\phi) - \epsilon^{2} \Delta \phi),$$ (261)

and for the electrical field,

$$\nabla \cdot (\gamma_E(\phi) \nabla \varphi)= 0.$$ (262)

where $\mu$ is the chemical potential, $f(\phi)$ is the double obstacle potential as defined in [85,86], and $\epsilon$ is a parameter controlling the void-metal interface width. The electrical conductivity was taken to vary linearly from the metal ( $\gamma_E = \gamma_{E,metal}$) to the void area ( $\gamma_E = 0$) by setting $\gamma_E = \gamma_{E,metal} (1+\phi)/2$. The equations (4.50) and (4.52) are solved on the two-dimensional polygonal interconnect area $T$.
It has been proven [57,59] that in the asymptotic limit for $\epsilon\rightarrow 0$ the diffuse interface model defined by equations (4.50)-(4.52) describes the same voids-metal interface behavior like equations (4.43)-(4.48). The width of the void-metal diffuse interface is approximately $\epsilon\pi/2$, and in order to numerically handle sufficiently thin interfaces one needs a very fine locally placed grid around it.