Growth Rate of Crystal Surfaces with Several Dislocation Centers

Abstract

We study analytically and numerically the growth rate of a crystal surface growing by several screw dislocations. To describe several spiral steps we use the revised level set method for spirals by the authors (Journal of Scientific Computing 62, 2015).We carefully compare our simulation results on the growth rates with predictions in a classical paper by Burton et al. (Philos Trans R Soc Lond Ser A Math Phys Sci 243,299-358, 1951). Some discrepancy between the growth rate computed by our method and reported by the classical paper is observed. In this paper we propose improved estimates on the growth rate with several different configurations. In particular we give a quantitive definition of the critical distance of co-rotating screw dislocations under which the effective growth resembles that of a single spiral. The proposed estimates 1 are in agreement with our numerical simulations. The influence of distribution of screw dislocations in a group on a line to the growth rate, and the growth rate by a group including different rotational orientations of spirals are also studied in this paper.

On obstacle problem for mean curvature flow with driving force

“…Such a problem has been arisen when one discusses spiral growths. In [10,11], a spiral growth by V = −κ + 1 is discussed for the Neumann boundary condition by using a modified level-set method. It seems to be possible to discuss the Dirichlet problem by using this obstacle approach.…”

Numerical analysis comparing ODE approach and level set method for evolving spirals by crystalline eikonal-curvature flow

“…Because of the above differences, we have no conjectures of convergence between Γ L (t) and Γ D (t) now. Moreover, from the numerical results of the isotropic case in [10,12], not only tending the approximation parameters to zero but also letting ρ → 0 is required for numerical accuracy. Thus, we shall check the numerical results with fixed radius 0 < ρ 1 and reducing radius ρ = O(∆x).…”

“…For this problem, we introduced an approximation of level set equation for the crystalline curvature flow, which is established with the approximation of the characteristic function as in [3]. To measure the difference between the two curves obtained by the discrete model and the level set method, we introduced an area difference function defined by (12). It is consist of the L 1 difference of the height function as in [10] with the step-height h 0 = 1.…”

Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow

“…Because of the above differences, we have no conjectures for the convergence of Γ L (t) and Γ D (t) now. Moreover, the numerical results of the isotropic case in [17,21] shows that taking the approximation parameters to zero, as well as letting ρ → 0 is required for numerical accuracy. We therefore check the numerical results with fixed radius 0 < ρ 1 and reducing radius ρ = O(∆x).…”