A new phase-field model for strongly anisotropic systems

Abstract

We present a new phase-field model for strongly anisotropic crystal and epitaxial growth using regularized, anisotropic Cahn–Hilliard-type equations. Such problems arise during the growth and coarsening of thin films. When the anisotropic surface energy is sufficiently strong, sharp corners form and unregularized anisotropic Cahn–Hilliard equations become ill-posed. Our models contain a high-order Willmore regularization, where the square of the mean curvature is added to the energy, to remove the ill-posedness. The regularized equations are sixth order in space. A key feature of our approach is the development of a new formulation in which the interface thickness is independent of crystallographic orientation. Using the method of matched asymptotic expansions, we show the convergence of our phase-field model to the general sharp-interface model. We present two- and three-dimensional numerical results using an adaptive, nonlinear multigrid finite-difference method. We find excellent agreement between the dynamics of the new phase-field model and the sharp-interface model. The computed equilibrium shapes using the new model also match a recently developed analytical sharp-interface theory that describes the rounding of the sharp corners by the Willmore regularization.

Geometric evolution of bilayers under the functionalized Cahn–Hilliard equation

“…Higher-order energies, which resemble the FCH with η 2 < 0 and an untilted well W, have been proposed, see in particular equations (1.5) of Loreti & March [12] and (3.16) of Torabi et al [13]. Indeed, the De Giorgi conjecture, which concerns the Γ limit of the FCH energy for η 2 < 0 with an untilted well has been established [14].…”

Phase‐field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy

“…An expedite and homogenized approach to deal with strongly anisotropic surface energies is to convexify the surface energy [37], at the expense of missing interesting details of the geometric structure of the free-surface. Because strongly anisotropic surface energies are important in applications and the details about the freesurface matter, researchers have developed other remedies to this difficulty, for instance, regularizing the phase-field functional by adding the square of the Laplacian or a phase-field approximation to the Willmore curvature energy; see [38] and references therein. This method works well for models with phase-field variations across the interface approximating smoothly step functions, such as hyperbolic tangent profiles, but unfortunately cannot be used for phase-field model of fracture.…”

Phase-Field Modeling of Equilibrium Precipitate Shapes Under the Influence of Coherency Stresses

“…This will require regularization of the interfacial energy forms that have been performed in previous contributions from Eggleston et al [56] and Voigt and colleagues. [57] Apart from this, we also compare and contrast our model formulation with other diffuse-interface and sharp-interface models. Here, while we discuss that the sharp-interface model descriptions are retrieved in the different diffuse-interface models in the sharp-interface limit, we also highlight the conceptual differences between our model formulation and the other diffuse-interface methods.…”