Date of Award

Fall 2014

Degree Type


Degree Name

Doctor of Philosophy (PhD)


Mechanical Engineering

First Advisor

Ganesh Subbarayan

Committee Chair

Ganesh Subbarayan

Committee Member 1

R. Edwin Garcia

Committee Member 2

Marisol Koslowski

Committee Member 3

Anil K. Bajaj


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.