On the role of kinetic and interfacial anisotropy in the crystal growth theory

Abstract

A planar anisotropic curvature flow equation with constant driving force term is considered when the interfacial energy is crystalline. The driving force term is given so that a closed convex set grows if it is sufficiently large. If initial shape is convex, it is shown that a flat part called a facet (with admissible orientation) is instantaneously formed. Moreover, if the initial shape is convex and slightly bigger than the critical size, the shape becomes fully faceted in a finite time provided that the Frank diagram of interfacial energy density is a regular polygon centered at the origin. The proofs of these statements are based on approximation by crystalline algorithm whose foundation was established a decade ago. Our results indicate that the anisotropy of intefacial energy plays a key role when crystal is small in the theory of crystal growth. In particular, our theorems explain a reason why snow crystal forms a hexagonal prism when it is very small.

Oscillating facets

“…We will see, see Theorem 4.2, that jumps of the derivative may not be arbitrary. This fact is well-known for the crystalline motion, see [8] and the references therein. If at x 0 the interval with endpoints u +…”

On Large Time Behavior of Growth by Birth and Spread

“…Note that their proof based on characterization of lim t→∞ u(tx, t) works even when γ is crystalline, where u is in Theorem 1.1. For crystalline case evolution of a convex shape by (1.4) is analyzed in [15], where the role of anisotropy in M and γ is clarified. We expect that all results in Theorem 1.2 -Theorem 1.6 can be extended to crystalline γ if appropriate stability holds (See Subsection 5.2).…”

Effective Insensitiveness of Melamine Urea-Formaldehyde Resin via Interfacial Polymerization on Nitramine Explosives

“…On the other hand, since the solubility of the dispersant PVA in DMSO is small, when the MUF binders collide with each other, there is no good dispersing force, which causes them to stick to each other, eventually forming an irregular shape. In addition, the crystal growth theory [23] can also serve as a powerful support for explaining this irregular morphology. The drying bath method causes the explosive to undergo growth and development process of “the crystal embryo-nucleus-crystal”.…”