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

Abstract

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.

Growth Rate of Crystal Surfaces with Several Dislocation Centers

“…We obtained new estimates on the growth rates for several different configurations (co-rotating pairs of spirals, spirals whose centers are co-linear, and groups of spirals), and we showed that these new rates were in agreement with the numerical simulations computed by the level set method proposed in the previous paper by the authors. 6 We gave a new definition of the critical distance (of co-rotating pair) with the view point of effective growth rate. We also gave an improved estimate of the growth rate by a co-rotating pair with an estimate of the rotating single spiral by Ohara-Reid.…”

“…We choose the initial steps as The details of the initial data for these simulations, and the profiles of spiral steps at t = 0.5 are given in the previous paper. 6 Profiles are slightly different from each other. We now give a classification if these situations are a group of 4 co-rotating screw dislocations, or 2 pairs from a view point of the growth rates.…”

“…[7][8][9] In this paper, we study the growth rates of such crystals as described in the classical paper by Burton et al 10 using the method proposed by the authors. 6 In particular, we give a quantitative definition of the critical distance (of co-rotating screw dislocations) under which the effective growth resembles that of a single spiral. We conclude that the critical distance predicted by Burton et al 10 is too small compared with our definition.…”

“…Recently, Miura-Kobayashi 9 proposed a phase-filed formulation for spiral crystal growth, and they concluded that their numerical simulations agree with the prediction of Burton et al 10 One of the aims of this paper is to clarify some discrepancy between the growth rates computed by our method and those reported in previous papers. 9,10 Moreover, we study on the growth rate by a group including several rotational orientation, which is mentioned by Burton et al 10 but not treated by The numerical simulations reported in this paper were computed by an implementation of the algorithm proposed by the authors, 6 and it is reviewed in the next section.…”

Curvature flow with driving force on fixed boundary points

“…Although, in our paper, we only consider the problem under the condition a ≤ 1/A, the simulated results in [13] give the hit about this research. We are devoted to considering them in analytic way.…”