A New Continuum Scalar Model of Facets
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A New Continuum Scalar Model of Facets Tinghui Xin and Harris Wong* Mechanical Engineering Department, Louisiana State University Baton Rouge, LA 70803-6413, USA. *correspondence: [email protected] ABSTRACT Facets or planar surfaces appear often on crystalline solids, and need to be accurately modeled in studying surface evolution. Here we propose a model in which the radius of curvature of the equilibrium crystal surface is prescribed as a function of crystallographic orientation. In this approach, a facet is represented by the Dirac delta function with the weight of the delta function equal to the width of the facet plane. This model allows sharp corners on solid surfaces, but avoids the non-uniqueness of equilibrium surface profiles that plagues previous facet models. We demonstrate this approach by solving the equilibrium shape and surface energy of triangular crystals. INTRODUCTION In a crystalline solid, atoms are arranged on a lattice, and the surface free energy varies with the crystallographic orientation of the surface plane. As a crystal forms, the minimum energy orientations are preferentially exposed to form planar surfaces or facets. Thus, facets are commonly observed on crystalline solids at low temperatures. Facets appear in many physical processes, such as crystal growth, solidification, and grain growth during annealing. To understand the effect of faceting in these processes, it is necessary to model facets accurately. Current models typically choose a particular form of anisotropic surface energy γ = γ(θ), where θ is a measure of crystallographic orientation [1-6]. They then compute the evolving profile of the solid surface. This approach works if the surface energy anisotropy is weak. However, when the anisotropy increases to the point that the reduced surface energy γ + d2γ/dθ2 < 0, the evolution problem becomes ill-posed [6]. In this work, we propose a new approach. Instead of prescribing γ(θ), we prescribe γ + d2γ/dθ2, which is proportional to the radius of curvature of the equilibrium crystal surface. Thus, in this paper we focus on equilibrium crystals and take the radius of curvature as a function of crystallographic orientation. Once the radius function is given, we can calculate the equilibrium crystal shape and the anisotropic surface energy. This approach prevents the reduced surface energy from becoming negative even for arbitrarily strong anisotropy, and thus eliminates the singularity in evolution problems. FORMULATION The new model is illustrated using a rod-like crystalline solid, as shown in Fig. 1. For such a two-dimensional structure, the chemical potential is
Y3.3.1
y θ1 θ
(x 1 , y 1 ) s
x0 x Figure 1. Half of a two-dimensional crystal. Cartesian coordinates (x, y) are defined with the x-axis placed at the symmetry plane. Crystallographic orientation is measured by θ, which is the angle that the surface normal makes with the x-axis. Arc-length s along the crystal surface starts at θ = 0 where x = x0. An intermediate facet is assumed at θ = θ1 and with center position (x1, y1)
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