Atomistic Approximation of Solid Surface Energy and Its Anisotropy

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.

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

A holonomy invariant anisotropic surface energy in a Riemannian manifold

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

At least five conclusions may be formulated after perusing this review.1. No method so far suggested for measuring the surface energy or surface tension of solids is satisfactory. 2. This failure may be caused, above all, by the fact that solids, contrary to liquids, cannot alter their shape without changing the strain energy in their volumes. The changes in strain energy are so much greater than those in surface energy that the latter remain unrecognized. 3. Solids possess an energy unknown in typical liquids. This cuticular energy exists because the surface region of innumerable solids has a chemical composition, a frequency of lattice defects, and so on, different from those in the bulk. 4. Small solid particles obtained by cooling of vapors, by grinding, or many other methods, usually have a less perfect lattice and more impurity than have bigger crystals of nominally identical composition. Hence, the cuticular energy of the former exceeds that of the latter. 5. Cuticular energy implies no tendency of the surface to contract, i.e., no surface tension. The theoretical calculations of the difference in energy between a broken and an unbroken crystal, if correct, afford a quantity which is related to cuticular energy and, like this, causes no contractile tendency.

<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>

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 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.

To a first approximation one may predict the orientation of the director of a thin film of nematic liquid crystal (L.C.) adjacent to a solid surface through the relative magnitude of the L.C. surface tension and solid surface energy. A deeper insight, taking into account the L.C. surface tension anisotropy leads to the conclusion that homeotropic orientation should be observed on high energy surfaces. Most experimental evidence indicates that L.C. orientation over inorganic surfaces is homogeneous. We will present the results of our measurements of inorganic solids surface tension which, in accordance with previously published results, show that inorganic surfaces, in contact with the atmosphere, have an average surface energy of about 40 mJm-2. This result has, in most cases, been explained through the absorption of a water layer. In this note we retain this explanation, as a first approximation. From the criterion for displacement of a liquid 1 by a liquid 2 and known values of the surface energy of inorganic solids, water and liquid crystals, it is shown that water will not be displaced by L.C. molecules. Thin L.C. films lie over a water layer of surface energy 40 ± 5mJm-2 and remain parallel to the solid surface.

: The concepts, experimental techniques, and theoretical analyses of the surface energy of solid ceramic materials are reviewed with the aim of condensing a large mass of unrelated data into a concise form for comparison and evaluation. It is shown that various experimental methods can be applied to the measurement of surface energy, but, that each of these has certain limitations which are often unstated. Furthermore, it is shown that theoretical analyses and empirical correlations, while sometimes rather imprecise, can be used to approximate surface energies, particularly as functions of temperature. While a few materials have been discussed in considerable detail (such as MgO, Al2O3 and some alkali halides), a review of the literature notes that there is a great paucity of information on the surface energies of many solids of interest. Improvements and extensions of experimental and empirical techniques are suggested that will help to fill the voids in the present understanding of ceramic solid surfaces, and specific analyses of experimental methods are forwarded. It is shown in the report that the proper use of thermodynamic techniques offers considerable potential for the measurement and interpretation of solid surface energies of a large number of materials that are poorly understood at present. In addition, the further development of thermodynamic methods presents an opportunity to investigate solid surface energies of nonbrittle materials, thus overcoming one of the basic limitations of the use of mechanical methods.

We propose a sharp interface model for simulating solid-state dewetting where the surface energy is (weakly) anisotropic. The morphology evolution of thin films is governed by surface diffusion and contact line migration. The mathematical model is based on an energy variational approach. Anisotropic surface energies lead to multiple solutions of the contact angle equation at contact points. Introduction of a finite contact point mobility is both physically based and leads to robust, unambiguous determination of the contact angles. We implement the mathematical model in an explicit finite difference scheme with cubic spline interpolation for evolving marker points. Following validation of the mathematical and numerical approaches, we simulate the evolution of thin film islands, semi-infinite films, and films with holes as a function of film dimensions, Young's angle $\theta_i$, anisotropy strength and crystal symmetry, and film crystal orientation relative to the substrate normal. We find that the contact point retraction rate can be well described by a power-law, $l \sim t^n$. Our results demonstrate that the exponent $n$ is not universal -- it is sensitive to the Young's angle $\theta_i$ (and insensitive to anisotropy). In addition to classical wetting (where holes in a film heal) and dewetting (where holes in a film grow), we observe cases where a hole through the film heals but leave a finite size hole/bubble between the continuous film and substrate or where the hole heals leaving a continuous film that is not bonded to the substrate. Surface energy anisotropy (i) increases the instability that leads to island break-up into multiple islands, (ii) enhances hole healing, and (iii) leads to finite island size even under some conditions where the Young's angle $\theta_i$ suggests that the film wets the substrate.

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

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

Hansen solubility parameters from surface measurements: A comparison of different methods

Modeling of a transition to diffusionless dendritic growth in rapid solidification of a binary alloy