Parametric approximation of surface clusters driven by isotropic and 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 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.

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.

We present a new isogeometric analysis (IGA) approach based on extended Loop subdivision scheme for solving various geometric flows defined on subdivision surfaces. The studied flows include the second‐order, fourth‐order, and sixth‐order geometric flows, such as averaged mean curvature flow, constant mean curvature flow, and minimal mean‐curvature‐variation flow, which are generally derived by minimizing the associate energy functionals with ‐gradient flow respectively. The geometric flows are discretized by means of subdivision based IGA, where the finite element space is formulated by the limit form of the extended Loop subdivision for different initial control meshes. The basis functions, consisting of quartic box‐splines corresponding to each subdivided control mesh, are utilized to represent the geometry exactly. For the cases of the evolution of open surfaces with any shape boundary, high‐order continuous boundary conditions derived from the mixed variational forms of the geometric flows should be implemented to be consistent with the isogeometric concept. For time discretization, we adopt an adaptive semi‐implicit Euler scheme. By several numerical experiments, we study the convergence behaviors of the proposed approach for solving the geometric flows with high‐order boundary conditions. Moreover, the numerical results also show the accuracy and efficiency of the proposed method.

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.

In 2004 Chambolle proposed an algorithm for mean curvature flow based on a variational problem. Since then, the convergence, extensions and applications of his algorithm have been studied by many people. In this paper we give a proof of the convergence of an anisotropic version of Chambolle’s algorithm by use of the signed distance function. An application of our scheme to an approximation of the nonlocal curvature flow such as crystalline one is also discussed.

Shape evolution by surface and interface diffusion with rigid body rotations

Externally applied electric fields are widely employed during thin-film deposition to improve film uniformity, texture and densification. Despite extensive experimental evidence, the physical mechanisms by which such fields influence nucleation, surface diffusion, island coalescence and interface stability remain theoretically fragmented. Classical thin-film growth models assume a field-free energetic landscape and therefore provide limited predictive guidance for field-assisted manufacturing strategies. In this work, we introduce the Field-Driven Growth Model (FDGM), a unified theoretical framework that incorporates field-matter interactions directly into the free-energy functional governing thin-film growth. By explicitly accounting for effective dipolar coupling arising from field-induced polarization of surface species, predominantly quadratic in the field amplitude and consistent with linear-response polarization, the model consistently modifies the fundamental processes of nucleation, surface diffusion and coalescence. At the continuum scale, the FDGM predicts a field-induced stabilization mechanism that suppresses long-wavelength roughening modes and defines a field-controlled morphological crossover wavelength (field-controlled cutoff). The FDGM demonstrates that field-assisted nucleation bias, anisotropic surface diffusion, field-biased coalescence pathways and morphological stabilization are not independent phenomena, but multiscale manifestations of a single energy-minimization principle acting on a field-modified energy landscape. By providing analytical stability criteria and explicit links between external field parameters and morphological outcomes, the model establishes a predictive foundation for the manufacturing of thin films with improved uniformity in advanced thin-film-based devices. The framework is broadly applicable to deposition techniques such as sputtering, pulsed-laser deposition, chemical vapor deposition and atomic layer deposition.

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

<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>

In this paper, we prove short time existence, uniqueness, and regularity for a surface diffusion evolution equation with curvature regularization in the context of epitaxially strained two-dimensional films. This is achieved by using the H −1-gradient flow structure of the evolution law, via De Giorgi’s minimizing movements. This seems to be the first short time existence result for a surface diffusion type geometric evolution equation in the presence of elasticity.

Anisotropic motion laws play a key role in many applications ranging from materials science, biophysics to image processing. All these highly diversified disciplines have made it necessary to develop common math- ematical foundations and framworks to deal with anisotropy in geometric motion. The workshop brings together leading experts from various fields to address well-posedness, accuracy, and computational efficiency of the math- ematical models and algorithms.

The grain-boundary grooving of electropolished surfaces of polycrystalline tungsten annealed at 1350°C has been studied. Atomic force microscopy images of grooves were taken in the same locations after different annealing times. The profiles of the grooves developed between unfaceted grains were in qualitative agreement with the predictions of Mullins' theory of grooving by surface diffusion mass transport. In particular, the predicted secondary maxima next to the main groove maxima were often observed. Surface faceting strongly affected the grooving kinetics and groove shapes. Grooves developed between faceted and unfaceted grains were often asymmetric with unusual growth kinetics. Our observations suggest that certain faceted grains exhibited a negligible surface diffusion coefficient and that the surface fluxes at the associated groove root were non-zero. Numerical simulations assuming anisotropy of the surface diffusion coefficient (i.e. high diffusion coefficient on the unfaceted side, and negligible diffusion coefficient on the faceted side of the groove) were performed. The qualitative agreement between the simulated and observed groove shapes shows that the groove asymmetry can be explained by surface diffusion anisotropy.

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

Geometric evolution equations, such as mean curvature flow and surface diffusion, play an important role in mathematical modeling in various fields, ranging from materials to life science. Controlling the surface or interface evolution would be desirable for many of these applications. We attack this problem by considering the bulk contribution, which defines a driving force for the geometric evolution equation, as a distributed control. In order to solve the control problem we use a phase-field approximation and demonstrate the applicability of the approach on various examples. In the first example the effect of an electric field on the evolution of nanostructures on crystalline surfaces is considered. The mathematical problem corresponds to surface diffusion or a Cahn-Hilliard model. In the second example we consider mean curvature flow or a Allen-Cahn modelKeywordsGeometric evolutionsurface diffusionelectromigrationoptimal controlphase field approximation