Motion of spirals by crystalline curvature

Imai, Hitoshi; Ishimura, Naoyuki; Ushijima, Takeo

doi:10.1051/m2an:1999164

Cited by 6 publications

(10 citation statements)

References 17 publications

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“…For this direction there is a work by Forcadel, Imbert and Monneau [FIM] but their setting is somewhat restrictive. Imai, Ishimura and Ushijima [IIU99] presented a formulation of an evolving spiral by crystalline curvature flow with no driving force and gave some numerical simulations as well as a proof for local well-posedness.…”

Section: Introductionmentioning

confidence: 99%

A Level Set Approach Reflecting Sheet Structure with Single Auxiliary Function for Evolving Spirals on Crystal Surfaces

2014

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Abstract. We introduce a new level set method to simulate motion of spirals in a crystal surface governed by an eikonal-curvature flow equation. Our formulation allows collision of several spirals and different strength (different modulus of Burgers vectors) of screw dislocation centers. We represent a set of spirals by a level set of a single auxiliary function u minus a pre-determined multi-valued sheet structure function θ, which reflects the strength of spirals (screw dislocation centers). The level set equation used in our method for u − θ is the same as that of the eikonal-curvature flow equation.The multi-valued nature of the sheet structure function is only invoked when preparing the initial auxiliary function, which is nontrivial, and in the final step when extracting information such as the height of the spiral steps. Our simulation enables us not only to reproduce all speculations on spirals in a classical paper by Burton, Cabrera and Frank (1951) but also to find several new phenomena.

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Section: Introductionmentioning

confidence: 99%

A Level Set Approach Reflecting Sheet Structure with Single Auxiliary Function for Evolving Spirals on Crystal Surfaces

2014

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“…Among the rare, but undoubtedly valuable, papers devoted to this topic, we cite the works by Taylor [20] and Imai et al [10,11]. The last two of the mentioned publications encouraged us to undertake some research relating to this field.…”

Section: Introductionmentioning

confidence: 99%

“…The system of equations (4), (5) in Imai et al [10], governing the motion of the crystalline curvature on each of the line segments of the spiral-shaped polygonal curve, is the starting point of our considerations. However, our assumption varies from the conditions imposed in Imai et al [10].…”

Section: Introductionmentioning

confidence: 99%

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The Crystalline Dynamics of Spiral-Shaped Curves

Dudziński

Górka

2015

J Stat Phys

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“…Nakamura, T. Ogiwara and J.-C. Tsai ( [9]) gave some classification of rotating spirals. H. Imai, N. Ishimura and T. Ushijima ( [10]) showed the existence of solutions for an ordinary differential equation which describes the motion of spirals by crystalline curvature, and showed numerical examples.…”

Section: Introductionmentioning

confidence: 99%

Uniqueness and existence of generalized motion for spiral crystal growth

Goto¹,

Nakagawa²,

Ohtsuka³

2008

Indiana Univ. Math. J.

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Abstract. The uniqueness and existence of generalized solutions of 'spiral curves' for the mean curvature flow with driving force is studied by an adapted level set formulation. It is shown that the curves which are given by the level set formulation are unique with respect to initial spiral curves. For given spiral curves the method of a construction of an initial datum of a level set equation is also obtained by constructing a branch of the arguments from the centers of spiral curves, which has discontinuity at the given spiral curves. IntroductionVarious motions of spiral curves are observed on the growing surface of a crystal. This phenomena is come from spiral growth that occurs with aid of a screw dislocation intersecting the surface. In 1949, F. C. Frank pointed out important role of a screw dislocation to growth of a crystal in [5]. The theory of a growth of a crystal with aid of a screw dislocation is proposed by [1]. Recently this is called spiral growth of a crystal. According to the theory in [1], the spiral curves are described by steps on the surface of a crystal, and they move by the evolution law of the formwhere V is the normal velocity, κ is the curvature of the curve, and C is a constant Figure 1. A crystal surface.which denotes the driving force. This is an interface model of spiral crystal growth. We have two types of mathematical models of this phenomena, phase-field models or interface models. A. Karma and M. Plapp ([12]) proposed a phase-field model 2000 Mathematics Subject Classification. 53C44, 35K65.

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Motion of spirals by crystalline curvature

Cited by 6 publications

References 17 publications

A Level Set Approach Reflecting Sheet Structure with Single Auxiliary Function for Evolving Spirals on Crystal Surfaces

A Level Set Approach Reflecting Sheet Structure with Single Auxiliary Function for Evolving Spirals on Crystal Surfaces

The Crystalline Dynamics of Spiral-Shaped Curves

Uniqueness and existence of generalized motion for spiral crystal growth

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