Sharp interface model for solid-state dewetting problems with weakly anisotropic surface energies

Computing equilibrium shapes of crystals (ESC) is a challenging problem in materials science that involves minimizing an orientation-dependent (i.e., anisotropic) surface energy functional subject to a prescribed mass constraint. The highly nonlinear and singular anisotropic terms in the problem make it very challenging from both the analytical and numerical aspects. Especially, when the strength of anisotropy is very strong (i.e., strongly anisotropic cases), the ESC will form some singular, sharp corners even if the surface energy function is smooth. Traditional numerical approaches, such as the H −1 gradient flow, are unable to produce true sharp corners due to the necessary addition of a high-order regularization term that penalizes sharp corners and rounds them off. In this paper, we propose a new numerical method based on the Davis-Yin splitting (DYS) optimization algorithm to predict the ESC instead of using gradient flow approaches. We discretize the infinite-dimensional phase-field energy functional in the absence of regularization terms and transform it into a finite-dimensional constraint minimization problem. The resulting optimization problem is solved using the DYS method which automatically guarantees the mass-conservation and bound-preserving properties. We also prove the global convergence of the proposed algorithm. These desired properties are numerically observed. In particular, the proposed method can produce real sharp corners with satisfactory accuracy. Finally, we present numerous numerical results to demonstrate that the ESC can be well simulated under different types of anisotropic surface energies, which also confirms the effectiveness and efficiency of the proposed method.

We determine the linear stability of a rod or wire subject to capillary forces arising from an anisotropic surface energy. The rod is assumed to be smooth with a uniform cross–section given by a two–dimensional equilibrium shape. The stability analysis is based on computing the sign of the second variation of the total energy, which is examined by solving an associated eigenvalue problem. The eigenproblem is a coupled pair of second–order ordinary differential equations with periodic coefficients that depend on the second derivatives of the surface energy with respect to orientation variables. We apply the analysis to examples with uniaxial or cubic anisotropy, which illustrate that anisotropic surface energy plays a significant role in establishing the stability of the rod. Both the magnitude and sign of the anisotropy determine whether the contribution stabilizes or destabilizes the system relative to the case of isotropic surface energy, which reproduces the classical Rayleigh instability.

A holonomy invariant anisotropic surface energy in a Riemannian manifold

<abstract><p>We study a variational problem for hypersurfaces in a wedge in the Euclidean space. Our wedge is bounded by a finitely many hyperplanes passing a common point. The total energy of each hypersurface is the sum of its anisotropic surface energy and the wetting energy of the planar domain bounded by the boundary of the considered hypersurface. An anisotropic surface energy is a generalization of the surface area which was introduced to model the surface tension of a small crystal. We show an existence and uniqueness result of local minimizers of the total energy among hypersurfaces enclosing the same volume. Our result is new even when the special case where the surface energy is the surface area.</p></abstract>

A discrete lattice plane analysis of the composition profile and surface energy of binary H.C.P. alloys with applications to γ AlAg

Anisotropic and heterogeneous nucleation during the gamma to alpha transformation in iron

Shape evolution by surface and interface diffusion with rigid body rotations

Shape of hollow dislocation cores: anisotropic surface energy and elastic effects

The anisotropic surface energy of solid surfaces is mainly calculated using electron-based analyses. Experimental measurements are both difficult and subject to numerous errors. In this communication, by quantifying the surface area increment accompanying a vacancy formation process on solid surfaces, the anisotropic solid surface energy is approximated. Further analyses on the dependence of solid surface energy on planar packing fraction show that, however, the electron-based data may seem unrealistic in some cases.

Surface- and strain-induced anisotropy in epitaxial [formula omitted] films

Hydrates in some crude oils have a smaller tendency to form plugs than in others, and lately this is becoming a focus of research. To study this and the action of hydrate antiagglomerants in general, hydrate surface properties must be known. To help in characterizing the surface properties by simulation, the capillary waves of clathrate hydrate surfaces in vacuum are examined in all unique crystal faces by Molecular Dynamics, and an attempt is made to estimate the surface energies in the respective crystal faces from the wave fluctuations [1]. We also attempt to estimate solid/liquid surface energies of hydrate/oil and hydrate/water for a specific face, for comparison. The forcefield OPLS_AA is used for the organic compounds, while TIP4P/ice is used for the water framework. The anisotropy of the surface energy is then estimated and the result compared to the initial growth rate of different crystal faces as found in experiment [2].

Phase-field analysis of fracture-induced twinning in single crystals

In this article, an ultralow-carbon steel grade alloyed with Mn and Al has been investigated during α–γ–α transformation annealing in vacuum. Typical texture and microstructure has evolved as a monolayer of grains on the outer surface of transformation-annealed sheets. This monolayer consists of <100>//ND and <110>//ND fibre, which is very different from the bulk texture components. The selective driving force is believed to reside in the anisotropy of surface energy at the metal–vapour interface. The grain morphology is very different from the bulk grains. Moreover, 30–40% of the grain boundary interfaces observed in the RD–TD surface sections are tilt incoherent <110> 70.5° boundaries, which are known to exhibit reduced interface energy. Hence, the conclusion can be drawn that the orientation selection of surface grains is strongly controlled by minimization of the interface energy; both metal/vapour and metal/metal interfaces play a roll in this.

In this paper we present a new formulation of a large class of phase-field models, which describe solidification of a pure material and allow for both surface energy and interface kinetic anisotropy, in terms of the Hoffman–Cahn ξ-vector. The ξ-vector has previously been used in the context of sharp interface models, where it provides an elegant tool for the representation and analysis of interfaces with anisotropic surface energy. We show that the usual gradient-energy formulations of anisotropic phase-field models are expressed in a natural way in terms of the ξ-vector when appropriately interpreted. We use this new formulation of the phase-field equations to provide a concise derivation of the Gibbs–Thomson–Herring equation in the sharp-interface limit in three dimensions.

We present a variational formulation for the evolution of surface clusters in ℝ^3 by mean curvature flow, surface diffusion and their anisotropic variants. We introduce the triple junction line conditions that are induced by the considered gradient flows, and present weak formulations of these flows. In addition, we consider the case where a subset of the boundaries of these clusters are constrained to lie on an external boundary. These formulations lead to unconditionally stable, fully discrete, parametric finite element approximations. The resulting schemes have very good properties with respect to the distribution of mesh points and, if applicable, volume conservation. This is demonstrated by several numerical experiments, including isotropic double, triple and quadruple bubbles, as well as clusters evolving under anisotropic mean curvature flow and anisotropic surface diffusion, including computations for regularized crystalline surface energy densities.