A relaxation method for the energy and morphology of grain boundaries and interfaces

Abstract

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 molecular dynamics 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 interfacial energy combined with a general microstructure construction based on convexification is able to replicate complex interfacial morphologies, including thermally-induced morphological transitions. minima, or cusps, for special orientations. This property causes the corresponding energy functional to lack lower-semicontinuity, i. e., to lack stability with respect to microscopic fluctuations, in turn resulting in non-attainment of the energy minimum and the formation of fine microstructure (cf., e. g., [17,18]).This interpretation connects the theory of faceted structures to a variety of other nonconvex minimization problems in mechanics [19] and has led to a number of criteria for energy-minimal facet patterns, including Cahn's vectorized energy formalism for formulating a facet-optimization problem [13], Taylor's formulation of the problem in terms of convexity [14], the use of varifolds to represent infinitely-finely corrugated surfaces with a macroscopic shape [20][21][22], a common-tangent construction [23] and standard convexification with the problem posed in terms of graphs [24], among others. A quantitative estimate of the distance of low-energy states from the ideal Wulff shape was recently obtained by Figalli and Maggi [25]. The corresponding timeevolution problem has been the object of extensive mathematical study, cf., e. g., [26] and references therein. Further to these mathematical developments, materials scientists have also suggested that the energy of faceted interfaces can be estimated by rotational interpolation between the facet planes or by means of a lever-type rule [27,28].In the past, the relaxation interpretation of faceting has been presented mainly as a conceptual explanation...

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