Phase field simulation of the void destabilization and splitting processes in interconnects under electromigration induced surface diffusion

Interconnect lines of integrated circuits inevitably exist micro-damage, such as voids, inclusions or cracks. Under the effect of different intrinsic physical mechanisms as well as external driving forces, the micro-damage will have different morphological evolution and even destabilize and split, which can affect the various properties of the interconnects. Based on the theory of diffusion interface of microstructure evolution in solid materials, a phase field model is established to simulate the morphological evolution of micro-damage in the interconnect line under electromigration-induced surface diffusion. Unlike the previously published work, the bulk free energy density and the degenerate mobility used in the model are both constructed by quartic double-well potential function. The applicability of the model for the morphological evolution of intracrystalline voids is proved by asymptotic analysis. The governing equation of the phase field method is solved by finite element method. And, the validity of the method is confirmed by the agreement of the numerical solutions with the theoretical solutions of a small circular void. The effects of the relative electric field intensity Χ, the linewidth h ∽ and the initial aspect ratio β on void evolution are discussed in detail. The results indicate that the intracrystalline voids drift in the direction of the electric field, and there is a destabilization critical value Χ cr . When Χ ≧Χ cr , there exist two splitting forms after destabilization for circular void, type I and type II, respectively. The value of Χ cr decreases as h ∽ decreases or β increases. The smaller h ∽ or the larger β is more prone to cause void destabilization. The effect of h ∽ or β on Χ cr is more significant as h ∽ or β is relatively small. In particular, when h ∽ or β is sufficiently large, there exists upper or lower limit for Χ cr , respectively.

Strong well-posedness and inverse identification problem of a non-local phase field tumour model with degenerate mobilities

We extend previous weak well-posedness results obtained in Frigeri et al. (2017, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Vol. 22, Springer, Cham, pp. 217–254) concerning a non-local variant of a diffuse interface tumour model proposed by Hawkins-Daarud et al. (2012, Int. J. Numer. Method Biomed. Engng.28, 3–24). The model consists of a non-local Cahn–Hilliard equation with degenerate mobility and singular potential for the phase field variable, coupled to a reaction–diffusion equation for the concentration of a nutrient. We prove the existence of strong solutions to the model and establish some high-order continuous dependence estimates, even in the presence of concentration-dependent mobilities for the nutrient variable in two spatial dimensions. Then, we apply the new regularity results to study an inverse problem identifying the initial tumour distribution from measurements at the terminal time. Formulating the Tikhonov regularised inverse problem as a constrained minimisation problem, we establish the existence of minimisers and derive first-order necessary optimality conditions.

Relaxation of the Cahn–Hilliard equation with singular single-well potential and degenerate mobility

The degenerate Cahn–Hilliard equation is a standard model to describe living tissues. It takes into account cell populations undergoing short-range attraction and long-range repulsion effects. In this framework, we consider the usual Cahn–Hilliard equation with a singular single-well potential and degenerate mobility. These degeneracy and singularity induce numerous difficulties, in particular for its numerical simulation. To overcome these issues, we propose a relaxation system formed of two second-order equations which can be solved with standard packages. This system is endowed with an energy and an entropy structure compatible with the limiting equation. Here, we study the theoretical properties of this system: global existence and convergence of the relaxed system to the degenerate Cahn–Hilliard equation. We also study the long-time asymptotics which interest relies on the numerous possible steady states with given mass.