Existence of self-similar evolution of crystals grown from supersaturated vapor

Giga, Yoshikazu; Rybka, Piotr

doi:10.4171/ifb/107

Cited by 13 publications

(46 citation statements)

References 18 publications

(5 reference statements)

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“…This is a continuation of our series of papers, [GR1], [GR2], [GR3], [GR4]. We studied there a one phase Stefan-like problem with Gibbs-Thomson and kinetic effect where the interfacial energy is so singular that its Wulff shape-equilibrium shape is a regular circular cylinder.…”

Section: Introductionmentioning

confidence: 99%

“…The set of equations we are going to study has been justified in [GR1], [GR3], [GR4]. We also presented there its origin.…”

Section: Introductionmentioning

confidence: 99%

“…This equation expresses the properly rescaled mass conservation, see [GR3]. The value of σ at the surface is coupled to the surface velocity and its curvature through the Gibbs-Thomson relation…”

mentioning

confidence: 99%

“…It evolves at the expense of vapor (or other matter) whose supersaturation (see [GR3] for explanation) is denoted by σ. We are going now to present the evolution equations.…”

mentioning

confidence: 99%

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Stability of facets of crystals growing from vapor

Giga¹,

Rybka

2006

Discrete &Amp; Continuous Dynamical Systems - A

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Consider a Stefan-like problem with Gibbs-Thomson and kinetic effects as a model of crystal growth from vapor. The equilibrium shape is assumed to be a regular circular cylinder. Our main concern is a problem whether or not a surface of cylindrical crystals (called a facet) is stable under evolution in the sense that its normal velocity is constant over the facet. If a facet is unstable, then it breaks or bends. A typical result we establish is that all facets are stable if the evolving crystal is near the equilibrium. The stability criterion we use is a variational principle for selecting the correct Cahn-Hoffman vector. The analysis of the phase plane of an evolving cylinder (identified with points in the plane) near the unique equilibrium provides a bound for ratio of velocities of top and lateral facets of the cylinders.

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Section: Introductionmentioning

confidence: 99%

“…The set of equations we are going to study has been justified in [GR1], [GR3], [GR4]. We also presented there its origin.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

“…It evolves at the expense of vapor (or other matter) whose supersaturation (see [GR3] for explanation) is denoted by σ. We are going now to present the evolution equations.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Stability of facets of crystals growing from vapor

Giga¹,

Rybka

2006

Discrete &Amp; Continuous Dynamical Systems - A

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“…Mathematical analysis is not well-developed for this system. However, it is known that the problem is locally solvable in a class of admissible evolving crystals [GR02]. In some cases there even exist self-similar solutions [GR05] at least for small crystal.…”

Section: Introductionmentioning

confidence: 99%

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

Giga¹,

Giga²

2013

Interfaces Free Bound.

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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.

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