Finite element approximation of a phase field model for void electromigration

We consider a fully practical finite element approximation of the nonlinear degenerate parabolic syste

On the stable discretization of strongly anisotropic phase field models with applications to crystal growth

We introduce unconditionally stable finite element approximations for anisotropic Allen--Cahn and Cahn--Hilliard equations. These equations frequently feature in phase field models that appear in materials science. On introducing the novel fully practical finite element approximations we prove their stability and demonstrate their applicability with some numerical results. We dedicate this article to the memory of our colleague and friend Christof Eck (1968--2011) in recognition of his fundamental contributions to phase field models.Comment: 20 pages, 8 figure

A phase field model for the electromigration of intergranular voids

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The Electromigration Force in Metallic Bulk

The voltage induced driving force on a migrating atom in a metallic system is discussed in the perspective of the Hellmann-Feynman force concept, local screening concepts and the linear-response approach. Since the force operator is well defined in quantum mechanics it appears to be only confusing to refer to the Hellmann-Feynman theorem in the context of electromigration. Local screening concepts are shown to be mainly of historical value. The physics involved is completely represented in ab initio local density treatments of dilute alloys and the implementation does not require additional precautions about screening, being typical for jellium treatments. The linear-response approach is shown to be a reliable guide in deciding about the two contributions to the driving force, the direct force and the wind force. Results are given for the wind valence for electromigration in a number of FCC and BCC metals, calculated using an {\it ab initio} KKR-Green's function description of a dilute alloy.Comment: 14 pages, 1 Postscript figur

Simulations of diffusion driven phase evolution in heterogenous solids

With reduction in size, ever greater operational demands are placed on electronics components at all levels of the device, starting from the transistor level to the level of the package and the solder interconnects. Concurrently, there has been a move to more complicated materials systems in order to meet health and environmental guidelines. These trends of reducing size, increasing loads have increased the necessity to understand the mechanisms of the failure. ^ As the length scales are reduced, it becomes increasingly important to consider interfacial and micro-structural effects that can be safely ignored at larger length scales owing to the randomness. It has become important to model the effect of interfacial motion and micro-structural evolution due to diffusion on the reliability of micro-electronics components. Examples of interfacial motion phenomena in solids include crack propagation, grain boundary motion, diffusion driven void motion through sur- face and bulk diffusion. The presence and evolution of these over the life-cycle of electronics components such as metal lines and solder joints presents a significant reliability challenge. The mathematical models that describe the evolution of these interfaces are usually formulated as systems of non-linear equations and hence, numerical methods provide an important method to study and understand them. The primary challenge in the study of these moving boundary problems is the tracking of the moving boundary and the application of appropriate boundary conditions on the moving boundary. ^ The phase field method tracks through smooth approximations of the Heaviside step and Dirac δ functions, which are maintained through the solution of a system of nonlinear differential equations. In this work, phase field approaches are developed for the study of diffusion driven phase evolution problems. First a phase field model for the evolution of voids in solder joints owing to electromigration and stress-migration both at the interface due to the surface gradients of the electric potential, temperature, curvature and strain energy, as well as self diffusion in the bulk on account of the chemical potential gradients as well as the electromigration force. This is modeled using a vacancy diffusion mechanism, while the growth of the voids is assumed to be due to the absorption of voids at the interface of pre-existing voids. A formal asymptotic analysis is performed to show the equivalence of the diffuse interface model to its sharp interface equivalents. Several numerical examples are presented. ^ Finally, an n-phase system of Cahn-Hilliard equations is developed to allow for the simulation of void evolution and growth in a multi-phase system. This is derived through a micro-force balance in order to eliminate the use of Lagrange multipliers that are commonly seen in such methods. A limited formal asymptotic analysis is performed to show the equivalence of the model to the standard surface diffusion model in regions where only two phase are present. This is numerically implemented and various numerical examples of phase evolution under simple surface diffusion, as well as surface diffusion with electromigration are demonstrated

Numerical Simulation of Grain Boundary Grooving By Level Set Method

A numerical investigation of grain-boundary grooving by means of a Level Set method is carried out. An idealized polygranular interconnect which consists of grains separated by parallel grain boundaries aligned normal to the average orientation of the surface is considered. The surface diffusion is the only physical mechanism assumed. The surface diffusion is driven by surface curvature gradients, and a fixed surface slope and zero atomic flux are assumed at the groove root. The corresponding mathematical system is an initial boundary value problem for a two-dimensional Hamilton-Jacobi type equation. The results obtained are in good agreement with both Mullins' analytical "small slope" solution of the linearized problem (W.W. Mullins, 1957) (for the case of an isolated grain boundary) and with solution for the periodic array of grain boundaries (S.A. Hackney, 1988).Comment: Submitted to the Journal of Computational Physics (19 pages, 8 Postscript figures, 3 tables, 29 references

Phase-field approaches to structural topology optimization

The mean compliance minimization in structural topology optimization is solved with the help of a phase field approach. Two steepest descent approaches based on L2- and H-1 gradient flow dynamics are discussed. The resulting flows are given by Allen-Cahn and Cahn-Hilliard type dynamics coupled to a linear elasticity system. We finally compare numerical results obtained from the two different approaches

Conformal mapping methods for interfacial dynamics

The article provides a pedagogical review aimed at graduate students in materials science, physics, and applied mathematics, focusing on recent developments in the subject. Following a brief summary of concepts from complex analysis, the article begins with an overview of continuous conformal-map dynamics. This includes problems of interfacial motion driven by harmonic fields (such as viscous fingering and void electromigration), bi-harmonic fields (such as viscous sintering and elastic pore evolution), and non-harmonic, conformally invariant fields (such as growth by advection-diffusion and electro-deposition). The second part of the article is devoted to iterated conformal maps for analogous problems in stochastic interfacial dynamics (such as diffusion-limited aggregation, dielectric breakdown, brittle fracture, and advection-diffusion-limited aggregation). The third part notes that all of these models can be extended to curved surfaces by an auxilliary conformal mapping from the complex plane, such as stereographic projection to a sphere. The article concludes with an outlook for further research.Comment: 37 pages, 12 (mostly color) figure