Sharp Interface Limits of the Cahn-Hilliard Equation with Degenerate Mobility

In this work, the sharp interface limit of the degenerate Cahn-Hilliard equation (in two space dimensions) with a polynomial double well free energy and a quadratic mobility is derived via a matched asymptotic analysis involving exponentially large and small terms and multiple inner layers. In contrast to some results found in the literature, our analysis reveals that the interface motion is driven by a combination of surface diffusion flux proportional to the surface Laplacian of the interface curvature and an additional contribution from nonlinear, porous-medium type bulk diffusion, For higher degenerate mobilities, bulk diffusion is subdominant. The sharp interface models are corroborated by comparing relaxation rates of perturbations to a radially symmetric stationary state with those obtained by the phase field model.Comment: 27 pages, 2 figure

Simulation of coarsening in two-phase systems with dissimilar mobilities

In this work, we apply phase field simulations to examine the coarsening behavior of morphologically complex two-phase microstructures in which the phases have highly dissimilar mobilities, a condition approaching that found in experimental solid-liquid systems. Specifically, we consider a two-phase system at the critical composition ($50\%$ volume fraction) in which the mobilities of the two phases differ by a factor of 100. This system is simulated in two and three dimensions using the Cahn-Hilliard model with a concentration-dependent mobility, and results are compared to simulations with a constant mobility. A morphological transition occurs during coarsening of the two-dimensional system (corresponding to a thin film geometry) with dissimilar mobilities, resulting in a system of nearly-circular particles of high-mobility phase embedded in a low-mobility matrix. This morphological transition causes the coarsening rate constant to decrease over time, which explains why a previous study found lack of agreement with the theoretical $t^{1/3}$ power law. Three-dimensional systems with dissimilar mobilities resulted in bicontinuous microstructures that evolve self-similarly, as determined by quantitative analysis of the interfacial shape distribution. Coarsening kinetics in three dimensions agreed closely with the $t^{1/3}$ power law after the initial transient stage. A model is derived to explain a nearly-linear relationship between the coarsening rate constant and the variance of scaled mean curvature that is observed during this transient stage.Comment: 25 pages, 12 figure

Phase Field Simulations of the Coarsening of Complex Microstructures

Coarsening is a fundamental phenomenon that occurs in a wide range of engineering materials, from polymer blends to cast aluminum alloys to functional nanostructured materials. The physics of coarsening is well understood. Differences in interfacial curvatures provide a driving force for mass transport, and the resulting evolution reduces the overall interfacial energy of the system as the average length scale of microstructural features increases. For simple particulate systems, such as those consisting of spherical precipitates at low volume fractions, analytical descriptions for the evolution are available and provide powerful tools for engineers to predict the microstructure for a given material and processing conditions. However, it is more difficult to predict the evolution of complex, well-connected structures like those present in dendritic solid-liquid systems and nanoporous metals. In these cases, simulations are necessary to develop fundamental understanding of coarsening and to gain the ability to predict microstructures that undergo coarsening. This dissertation consists of a series of simulation studies of coarsening of microstructures with complex morphologies. The simulation results and theories obtained here represent a fundamental contribution to the understanding of coarsening in complex microstructures. Coarsening with phases that have dissimilar mobilities is a condition typical of experimental solid-liquid systems. In a two-dimensional simulation, coarsening with dissimilar mobilities resulted in a morphological transition, as the initially complex, labyrinthine microstructure transforms into a system of high-mobility particles in a low-mobility matrix. In contrast, coarsening in three dimensions with dissimilar mobilities resulted in a stable bicontinuous structure after an initial transient stage. In this transient stage, we observed a theoretically predicted relationship between the coarsening rate constant and the variance of scaled mean curvature. Another important class of coarsening systems is those evolving by surface diffusion, including nanoporous metals. Intermediate volume fractions (between 36% and 50% minority phase) resulted in bicontinuous structures that coarsened self-similarly; that is, their morphologies became time-invariant when scaled by an evolving length scale. Morphologies of structures coarsening via surface diffusion were quantitatively different from those coarsening via bulk diffusion, but the difference was smaller than that of volume fraction. Simulations at a lower volume fraction, 32%, found coexistence of independent particles with well-connected domains. The effect of regularization in a phase field model with strongly anisotropic interfacial energy was quantified to understand and mitigate the error. An asymptotic analysis was performed to derive the expression for the effective interfacial energy for a given input interfacial energy. Simulated equilibrium shapes confirmed the prediction. The result can be used to parameterize the input anisotropic interfacial energy to implement desired interfacial anisotropy. To examine the origin of the ubiquity of the coarsening power law (the length scale proportional to the cubed root of time) two-dimensional simulations were conducted with a bimodal particle distribution. Particles with small radii were found to dominate the overall evolution within these simulations. The small particles evolved self-similarly, leading to agreement with the theoretical power law despite a lack of self-similarity in the overall structure. This set of simulations verified the existence of inactive length scales in coarsening, which was hypothesized in experiments. Additionally, a model was developed to study the role of topological singularities in bicontinuous structures during coarsening. Preventing topological singularities was found to reduce the coarsening rate, but further analysis of the data is required to fully understand the role of topological singularities.PHDMaterials Science and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/149950/1/wband_1.pd