Corner Regularization for Nanoscale Crystal Growth

The investigator focuses on a mathematical issue in the description of nanocrystal growth: how to properly resolve the ill-posedness inherent in dynamic models involving crystalline surfaces with strong anisotropy. Strong anisotropy in the surface energy manifests itself physically in the formation of corners on a crystal. Traditional mathematical models applied to the formation of corners are mathematically ill-posed and thus intractable. Because the formation of corners during crystal growth is ubiquitous, the ill-posedness of corner formation is a problem inherent in simulations of both industrial and naturally occurring crystal growth. Moreover, it is of critical importance to modeling crystal growth of nanoscale materials because of the dominant role of surface effects at small length scales. The investigator characterizes and evaluates different methods proposed to remove or regularize the ill-posedness. A widely employed regularization is a singular perturbation. A significant mathematical challenge is to characterize the behavior of the singularly perturbed corner. The investigator studies separately the role of the regularization in three dimensions and its effect in the presence of elastic stress. Another important scientific issue is to determine which of many regularization procedures is true to the atomic-scale behavior of different materials. The investigator also considers this question by studying the dynamic behavior of regularizations in relation to experimental observations and the relation of regularizations to atomic-scale models. Overall, the project has the potential for significant impact on the understanding of a key mathematical issue regarding regularization of ill-posedness in a classic moving boundary problem, and the impact of the work in a broader scientific context is that it contributes to the understanding of how to model the growth of crystalline solids in materials science. In the growth of crystals for nanotechnology and other materials applications, the formation of structures with corners (as on a grain of salt) is a natural occurrence. The physical effects responsible for the existence of a corner are well understood and a mathematical description of an existing corner can be accomplished with a classical mathematical model. However, the classical model is incapable of describing the actual dynamics of corner formation. This problem is present in all mathematical simulations of crystal growth in which corners form. Moreover, it is of magnified importance in the simulation of the growth of nanoscale structures: when the crystal decreases in size, corners become an increasingly dominant part of the overall structure. Thus, to correctly describe the growth of nanostructured materials, it is essential to have a correct model for corner formation. To obtain tractable models for corner formation, different "regularization" ideas have been proposed to make the mathematical problem of corner formation solvable, but there are many different approaches and no universally accepted procedure. One aspect of this project is a critical comparison of the different regularization approaches and how they behave in relation to actual material systems. A second aspect of the work relates to the fact that some of these models are "singular perturbations," which means that the results obtained when the regularization effect approaches zero can be different than if the regularization is not present at all. This type of unexpected behavior can mean that a small regularization that is added to allow for corner formation might give a different corner shape in simulations than should be present from the accepted classical model. Thus, understanding such singular perturbation behavior is an important part of validating such regularization methods to ensure that they give the correct overall behavior when used in large-scale crystal growth simulations. Taken as a whole, the project has the potential for significant impact as a building block in our ability to simulate the fabrication of nanomaterials, and by extension could contribute to the creation of purpose-specific materials, especially those with nanoscale features, in electronics and other applications. In addition, the project involves the training of a graduate student and includes two undergraduate students, for whom the experience may serve as stimulus to pursue graduate degrees in the mathematical sciences.