Corner regularizations for nanoscale crystal growth

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The investigator focuses on a mathematical issue in thedescription of nanocrystal growth: how to properly resolve theill-posedness inherent in dynamic models involving crystallinesurfaces with strong anisotropy. Strong anisotropy in the surfaceenergy manifests itself physically in the formation of corners ona crystal. Traditional mathematical models applied to theformation of corners are mathematically ill-posed and thusintractable. Because the formation of corners during crystalgrowth is ubiquitous, the ill-posedness of corner formation is aproblem inherent in simulations of both industrial and naturallyoccurring crystal growth. Moreover, it is of critical importanceto modeling crystal growth of nanoscale materials because of thedominant role of surface effects at small length scales. Theinvestigator characterizes and evaluates different methodsproposed to remove or regularize the ill-posedness. A widelyemployed regularization is a singular perturbation. A significantmathematical challenge is to characterize the behavior of thesingularly perturbed corner. The investigator studies separatelythe role of the regularization in three dimensions and its effectin the presence of elastic stress. Another important scientificissue is to determine which of many regularization procedures istrue to the atomic-scale behavior of different materials. Theinvestigator also considers this question by studying the dynamicbehavior of regularizations in relation to experimentalobservations and the relation of regularizations to atomic-scalemodels. Overall, the project has the potential for significantimpact on the understanding of a key mathematical issue regardingregularization of ill-posedness in a classic moving boundaryproblem, and the impact of the work in a broader scientificcontext is that it contributes to the understanding of how tomodel the growth of crystalline solids in materials science. In the growth of crystals for nanotechnology and othermaterials applications, the formation of structures with corners(as on a grain of salt) is a natural occurrence. The physicaleffects responsible for the existence of a corner are wellunderstood and a mathematical description of an existing cornercan be accomplished with a classical mathematical model. However,the classical model is incapable of describing the actual dynamicsof corner formation. This problem is present in all mathematicalsimulations of crystal growth in which corners form. Moreover, itis of magnified importance in the simulation of the growth ofnanoscale structures: when the crystal decreases in size, cornersbecome an increasingly dominant part of the overall structure. Thus, to correctly describe the growth of nanostructuredmaterials, it is essential to have a correct model for cornerformation. To obtain tractable models for corner formation,different "regularization" ideas have been proposed to make themathematical problem of corner formation solvable, but there aremany different approaches and no universally accepted procedure. One aspect of this project is a critical comparison of thedifferent regularization approaches and how they behave inrelation to actual material systems. A second aspect of the workrelates to the fact that some of these models are "singularperturbations," which means that the results obtained when theregularization effect approaches zero can be different than if theregularization is not present at all. This type of unexpectedbehavior can mean that a small regularization that is added toallow for corner formation might give a different corner shape insimulations than should be present from the accepted classicalmodel. Thus, understanding such singular perturbation behavior isan important part of validating such regularization methods toensure that they give the correct overall behavior when used inlarge-scale crystal growth simulations. Taken as a whole, theproject has the potential for significant impact as a buildingblock in our ability to simulate the fabrication of nanomaterials,and by extension could contribute to the creation ofpurpose-specific materials, especially those with nanoscalefeatures, in electronics and other applications. In addition, theproject involves the training of a graduate student and includestwo undergraduate students, for whom the experience may serve asstimulus to pursue graduate degrees in the mathematical sciences.