Optimal Shapes of Crystal Interfaces

The analysis of interfaces is crucial for the designing, processing, and application of modern materials. A variety of phenomena, ranging from mechanical to electronic and from magnetic to optical, are characterized by the presence of interfaces. The description and modelization of such interfaces is then critical to exploit the extraordinary properties of nanostructures for the development of new technologies, such as next-generation electronics, solar cells, and 3D printing. This project aims at providing advancements in the understanding of crystal interfaces by means of the analytical validation of different models introduced in Molecular, Statistical, and Continuum Mechanics. The main focus is on the justification at the microscopical level of the crystal Wulff shape and on the morphology of crystal interfaces, especially grain boundaries, material defects and cavities, and thin-film profiles. Special emphasis will be given to the fundamental multiscale aspect of the problem by looking for crystalline ground states at the microscale, showing the emergence of optimal shapes at the macroscale, and evaluating the fluctuations around the optimal shape at the mesoscale. Both discrete and elastic continuous models will be investigated. By moving within the framework of the Calculus of Variations and by combining different techniques from Combinatorics to Geometry Optimization, from Partial Differential Equations to Geometric Measure Theory, newly designed methods which have been already tested with promising preliminary results will be further developed. The project impact is not only related to the fundamental theoretical questions that inspire it, but also to technological applications. For example, the mathematical understanding of thin-film growth is essential to control, in the 3D printing, the energy balance between the laser-beam power used to melt the material and the energy required to form a desired shape. Campaigns of experimentshave been here planned to characterize optimum growth parameters for the synthesis and design of supported nanostructures with an emphasis on graphene nanoflakes and metal oxide materials. Moreover, an international research network among Vienna, Italy, and the USA will be strengthened and new partnerships enhanced.

The goal of the project is to advance the understanding of the mechanisms that induce both the morphology and the instabilities presented by the boundaries of crystals and by the interfaces between crystalline materials by means of the introduction and the analytical validation of variational models in the framework of both molecular and continuum mechanics. Characterizing energetically optimal crystalline morphologies is proven to be crucial in the design, processing, and employment of nanostructures for the development of new technologies, such as next-generation electronics, oxide batteries, and 3D printing, since a variety of phenomena, ranging from mechanical to electronic and from magnetic to optical, are strongly influenced by the presence of interfaces. By moving within the framework of the calculus of variations, the project allows to characterize crystalline interfaces as minima of configurational energies, optimal with respect to suitable discrete isoperimetric problems, and solutions of gradient flows. Furthermore, in the strive to account for the multi-scale aspect of crystallization, both discrete and continuum models are taken into account. At the microscopical level the main focus is on finding crystalline ground states, at the mesoscopic level on the microscopical justification of crystalline Wulff shapes and on the evaluation of the fluctuations around such optimal shapes, and finally at the macroscopical level on the charcaterization of the profiles and evolution of thin films and crystal cavities. The project impact is not only related to the fundamental theoretical questions that inspired it, but also to the provided advancements in the mathematical modeling in various settings, which is relevant in the long-run for technological applications. Such settings include supported nanostructures, epitaxially strained thin films, and magnetoelastic films. First, a mathematical framework able to guarantee the existence of minimizers has been provided in all these settings, then a unified model that allows to simoultaneously treat all of them has been introduced and finally such a unified model has been extended to the multiphase setting in the presence of both coherent and incoherent interfaces between the phases. Moreover, a microscopical justification of the variational models for solid-state wetting and dewetting, for the Winterbottom shapes, and for epitaxially strained thin films has been provided starting from properly introduced atomistic models taking into account the two-body atomistic interactions both between film atoms and between film and substrate atoms, by a formal discrete-to-continuum passage performed by Gamma-convergence.Finally, as a result of the project investigations the following classical laws have been analytically validated: the Young-Dupré law for crystalline films, the N^3/4 law for the fluctuation of atomistic ground states around the classical Wulff shape for 2- and 3-dimensional settings, and the Cauchy-Born rule for carbon nanotubes.