Crystal Shapes and Phase Equilibria: A Common Mathematical Basis

Abstract

Geometrical constructions, such as the tangent construction on the molar free energy for determining whether a particular composition of a solution, is stable, are related to similar tangent constructions on the orientation-dependent interfacial energy for determining stable interface orientations and on the orientation dependence of the crystal growth rate which tests whether a particular orientation appears on a growing crystal. Subtle differences in the geometric constructions for the three fields arise from the choice of a metric (unit of measure). Using results from studies of extensive and convex functions we demonstrate that there is a common mathematical structure for these three disparate topics, and use this to find new uses for well-known graphical methods for all three topics. Thus the use of chemical potentials for solution thermodynamics is very similar to known vector formulations for surface thermodynamics, and the method of characteristics which tracks the interfaces of growing crystals; the Gibbs-Duhem equation is analogous to the Cahn-Hoffman equation. The Wulff construction

Surface Structure, Morphology, and Stability of Li(Ni1/3Mn1/3Co1/3)O2 Cathode Material

“…In the present work, we use density functional theory (DFT) to identify the lowest energy surfaces of low-index facets of NMC111. The Wulff construction 23 is used to predict the thermodynamic equilibrium particle shape as a function of O and Li surface coverage. The results are compared with experimentally observed particle shapes of single-crystal NMCs, synthesized under various conditions and, as shown, may be extended, in general, to other NMC compositions.…”

3D electronic and photonic structures as active biological interfaces

“…Obviously, metals or intermediate phases as catalyst are necessary to grow semiconductors for VLS method. Relatively, as an alternative catalyst‐free path, the proposed VS method could eliminate the catalysts to grow semiconductors directly through the effect of imbalance in crystal growth velocities . Semiconductors like gallium nitride (GaN) and indium arsenide (InAs) are typically prepared via VS method.…”

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

“…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].…”

“…where σ 2 = 2k B T /w 0 and k B is Boltzmann's constant, T is the absolute temperature, and w 0 is an adjustable parameter. Physically, each term in (19) can be understood as measuring the difference between the atomic positions as they are and the atomic positions as they would be in a perfect lattice. Precisely, the term…”

“…If the two densities would coincide, the two lattices would be identical and there would be no interface, hence no interfacial energy. The interfacial energy in (19) measures the distance between the two densities, in an L 2 sense. We stress that we are only comparing the densities, in particular there is no reference configuration and the model has the full symmetry of the point groups of the lattices.…”