A relaxation method for the energy and morphology of grain boundaries and interfaces
The energy density of crystal interfaces exhibits a characteristic "cusp" structure that renders it non-convex. Furthermore, crystal interfaces are often observed to be faceted, i.e., to be composed of flat facets in alternating directions. In this work, we forge a connection between these two observations by positing that the faceted morphology of crystal interfaces results from energy minimization. Specifically, we posit that the lack of convexity of the interfacial energy density drives the development of finely faceted microstructures and accounts for their geometry and morphology. We formulate the problem as a generalized minimal surface problem couched in a geometric measure-theoretical framework. We then show that the effective, or relaxed, interfacial energy density, with all possible interfacial morphologies accounted for, corresponds to the convexification of the bare or unrelaxed interfacial energy density, and that the requisite convexification can be attained by means of a faceting construction. We validate the approach by means of comparisons with experiment and atomistic simulations including symmetric and asymmetric tilt boundaries in face-centered cubic (FCC) and body-centered cubic (BCC) crystals. By comparison with simulated and experimental data, we show that this simple model of interfacial energy combined with a general microstructure construction based on convexification is able to replicate complex interfacial morphologies, including thermally induced morphological transitions.
© 2015 Elsevier Ltd. Received 12 July 2015, Revised 24 October 2015, Accepted 21 November 2015, Available online 9 December 2015. Brandon Runnels and Michael Ortiz would like to thank the NNSA's High Energy Density Laboratory Plasmas program under award #DE-NA0001805. Brandon Runnels additionally thanks the Los Alamos National Laboratory Seaborg Institute for support during Summer 2014. Irene Beyerlein would like to acknowledge support by a Laboratory Directed Research and Development program award number 20140348ER. Sergio Conti would like to acknowledge support of the DFG under SFB 1060.