# Interfaces and Free Boundaries

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**Volume 9, Issue 2, 2007, pp. 171–210**

**DOI: 10.4171/IFB/161**

A phase field model for the electromigration of intergranular voids

John W. Barrett^{[1]}, Harald Garcke

^{[2]}and Robert Nürnberg

^{[3]}(1) Department of Mathematics, Imperial College London, South Kensington Campus, SW7 2AZ, LONDON, UNITED KINGDOM

(2) Naturwissenschaftliche Fakultät I - Mathematik, Universität Regensburg, 93040, REGENSBURG, GERMANY

(3) Department of Mathematics, Imperial College London, South Kensington Campus, SW7 2AZ, LONDON, UNITED KINGDOM

We propose a degenerate Allen--Cahn/Cahn--Hilliard system coupled to a quasi-static diffusion equation to model the motion of intergranular voids. The system can be viewed as a phase field system with an interfacial parameter $\gamma$. In the limit $\gamma \to 0$, the phase field system models the evolution of voids by surface diffusion and electromigration in an electrically conducting solid with a grain boundary. We introduce a finite element approximation for the proposed system, show stability bounds, prove convergence, and hence existence of a weak solution to this nonlinear degenerate parabolic system in two space dimensions. An iterative scheme for solving the resulting nonlinear discrete system at each time level is introduced and analysed, and some numerical experiments are presented. In the Appendix we discuss the sharp interface limit of the above degenerate system as the interfacial parameter $\gamma$ tends to zero.

*Keywords: *void electromigration, phase field model, degenerate Allen–Cahn/Cahn–Hilliard equation, fourth order degenerate parabolic equation, finite elements, convergence analysis, matched asymptotic expansions

Barrett J, Garcke H, Nürnberg R. A phase field model for the electromigration of intergranular voids. *Interfaces Free Bound.* 9 (2007), 171-210. doi: 10.4171/IFB/161