Phase-field theory of edges in an anisotropic crystal

Abstract

In the presence of sufficiently strong surface energy anisotropy the equilibrium shape of an isothermal crystal may include corners or edges. Models of edges have, to date, involved the regularisation of the corresponding free boundary problem resulting in equilibrium shapes with smoothed out edges. In this paper we take a new approach and consider how a phase-field model, which provides a diffuse description of an interface, can be extended to the consideration of edges by an appropriate regularisation of the underlying mathematical model. Using the method of matched asymptotic expansions we develop an approximate solution which corresponds to a smoothed out edge from which we are able to determine the associated edge energy.

Asymmetric shape transitions of epitaxial quantum dots

“…Since our model is for a 2D island, we can take into account corner energy, a 2D analogue for edge energy, to examine the effect of edge energy in our model. To estimate the magnitude of the edge/corner energy effect, we consider the corner energy regularization model discussed in [36][37][38][39][40] for the energy of a rounded corner between two facets. For a round corner region with size comparable to the lattice parameter a, we have…”

Concepts of modeling surface energy anisotropy in phase-field approaches

“…A similar regularization term of the classical model is found by Wheeler (2006) as well as by Ghosh et al (2015), but with an alternative evaluation of the mean curvature, given by H = ∇ · n. Note, furthermore, that the energetics of the interface, originally comprised of gradient energy and potential, is now modified by the incorporation of an additional term. Thus, the contribution of the curvature is interpreted as a higher order term in the power expansion of the surface tension (Torabi and Lowengrub 2012).…”

“…This interface behavior for strong anisotropy functions is well known (Taylor and Cahn 1998;Torabi et al 2008) and provides a rethinking of the original models (Wheeler 2006;Torabi et al 2008). Thus, an additional higher order term is incorporated into the objective functional, with the objective to smoothen the zigzags in the interface.…”

“…On the other hand, for a non-convex Frank plot, the equilibrium shape consists of the so-called "ears", due to the missing orientations with σ (θ) + σ ′′ (θ) < 0, which do not belong to the convex hull. Equivalently, the Cahn-Hoffman vector (Hoffman und Cahn 1972;Cahn und Hoffman 1974;Wheeler 2006), "which provides a graphical representation of the equilibrium shape in the presence of anisotropic interfaces and their propensity to break up into facets", is not unique- Abdeljawad et al (2016). In addition to the shape anisotropy, the kinetic anisotropy plays a crucial role for the crystal growth, as reported particularly by Rappel (1998), McFadden et al (1993) and Sekerka (2005).…”

Simulating Interface Growth and Defect Generation in CZT – Simulation State of the Art and Known Gaps

“…With the classical interfacial energy formulation, the phase-field evolution equation becomes ill-posed for the sharp interface case (i.e., the interfacial energy function has cusps and/or is non-convex). Various attempts have been made to extend phase-field models to strongly anisotropic (non-convex) interfacial energy [179][180][181][182]. Torabi el at.…”