## Corner regularizations for nanoscale crystal growth

##### Abstract

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