Multiphase mean curvature flows with high mobility contrasts: A phase-field approach, with applications to nanowires

Abstract

The structure of many multiphase systems is governed by an energy that penalizes the area of interfaces between phases weighted by surface tension coefficients. However, interface evolution laws depend also on interface mobility coefficients. Having in mind some applications where highly contrasted or even degenerate mobilities are involved, for which classical phase field models are inapplicable, we propose a new effective phase field approach to approximate multiphase mean curvature flows with mobilities. The key aspect of our model is to incorporate the mobilities not in the phase field energy (which is conventionally the case) but in the metric which determines the gradient flow. We show the consistency of such approach by a formal analysis of the sharp interface limit. We also propose an efficient numerical scheme which allows us to illustrate the advantages of the model on various examples, as the wetting of droplets on solid surfaces or the simulation of nanowires growth generated by the so-called vapor-liquid-solid method.

Learning phase field mean curvature flows with neural networks

“…We also plot on the (15) the numerical approximation of the mean curvature flow obtained in the case of non-orientable initial sets. In each of these experiments, we observe the evolution of points with triple junction which seem to satisfy the Herring condition [45]. This result is all the more surprising since our learning base contained only circle evolutions and no interface with the triple point.…”

“…to approximate the mean curvature flow, evolutions starting from different initial sets are shown in Section 3. We observe in particular that S NN θ,2 is able to handle correctly non-orientable surfaces, even with singularities, and the Herring's condition [45] seems to be respected at points with triple junction. This is somewhat surprising because only evolving smooth sets are used to train our networks (e.g., circles in 2D, spheres in 3D).…”

“…where the Lagrange multiplier λ is associated with the partition constraint L k=1 u k = 1. As explained in [68,45], this PDE can be for instance computed in two steps:…”

Convergence of Landau-Lifshitz equation to multi-phase Brakke's mean curvature flow

“…Note that several equations similar to (1.4) have been studied as the phase field model of the multi-phase mean curvature flow [6,7,8,10,11]. As an analogy of those results, it is expected that when ε → 0, Ω is divided into Ω 1 t , Ω 2 t ,.…”

“…We rermark that it can be said that {Ω i t } N i=1 and its boundaries correspond to magnetic domains and domain walls, respectively. Recently, in [6,7], they studied the following system of the Allen-Cahn equations:…”