Next: Analysis Up: Main Previous: Introduction

Background

The mean curvature of an infinitesimal element along a condensed-phase interface represents the quantity δA/δV , where δA is the incremental change in the element's area when it is normally displaced by local addition of material of volume δV [5]. Since a finite positive energy always accompanies formation of a unit area of an interface, mean curvature plays an important role in governing the thermodynamics of interfacial phenomena.

The effects of mean interfacial curvature are manifested in a wide range of both equilibrium and non-equilibrium phenomena. An example of the former is the dependence of the equilibrium vapor pressure of a one-component liquid drop on the drop size. If there are no surface tractions or body forces deforming the drop, its equilibrium shape is a sphere of radius R, and its vapor pressure, p, is related to its mean curvature ¹, H = R⁻¹, by the Kelvin equation,

where p_o is the equilibrium vapor pressure above a planar surface, Ω is the molecular volume of the species composing the drop, γ is the surface free energy density, k_B is the Boltzmann constant, and T is the absolute temperature. Throughout this paper we adopt the convention from differential geometry of denoting H at any point P as the arithmetic mean of the two principal curvatures, κ₁ and κ₂, where κ₁ and κ₂ are the reciprocals of the radii of two mutually orthogonal "osculating" circles constructed tangent to the surface at P. We also adopt the convention of sintering theory of taking a principal curvature to be positive if the surface normal vector points away from the center of curvature. Non-equilibrium phenomena in which mean curvature plays an important role include those processes acting to reduce the overall interfacial energies during microstructural development of polycrystalline solids: sintering [6,7], grain growth [8], and Ostwald ripening [9]. For example, the driving force for diffusive mass transport during these processes is the gradient in chemical potential, µ , between portions of the interfaces (assuming that no other driving forces, like those for phase transformations, are present). When the interfacial energy density, γ, is independent of crystallographic orientation, then the excess contribution to µ along the interface is determined by H. The dependence on mean curvature of _s µ along the surface can then be found from the Gibbs-Thomson equation [10] for an interface composed of one chemical species,

sµ = 2γ ΩH.

(2)

Eq. 2 indicates why theoretical calculations of curvature-driven processes are usually difficult. An analytic description of H, as a function of position along the interface, is required in order to calculate the instantaneous driving force for curvature-driven mass transport. When the position of a 2D surface can be described by an analytic function, an analytical calculation of H is tractable at any twice-differentiable point on the surface. If H is also defined at that point, the driving force for mass transport is accessible from Eq. 2, provided that the surface energy is isotropic. Unfortunately, the surfaces and interfaces in random, porous polycrystalline solids are tortuous and usually defy analytical description. Furthermore, curvature-driven transport generally causes the surface shape to evolve with time. Modeling such transport therefore demands tracking the evolving surface, a generally formidable task. For these reasons, theoretical studies of mass transport during sintering have been limited to simple geometries, like the two-sphere model of Kingery and Berg [11], and more recently the linear particle array models of Carter and Cannon [12] and Kellett and Lange [13]. These and other models have proven to be very important in elucidating the major features of curvature-driven processes. However, further progress in modeling the development of real microstructures requires a different approach for assessing and tracking mean curvature variations along geometrically complex interfaces.

One way of approximating the curvature of an arc or an interface of arbitrary shape is to first represent the interface as a polynomial fit to consecutive points along the interface, and to then compute the necessary spatial derivatives [14]. A related method consists of constructing two orthogonal osculating circles by fitting each circle to three consecutive points along the surface coplanar with that circle [15]. The curvature is then determined by the sum of the inverses of the radii of the osculating circles. These methods require, at any point P on the surface, selection of several closely-spaced points within a small neighborhood of P to which a curve is fit. The shape of that curve, and therefore the computed curvature at P, can therefore depend heavily on the choice of fitting points, especially at regions of high curvature.

A third procedure for obtaining relative estimates of curvature consists, at any point P on the interface, of computing the portion of the volume enclosed by a small template sphere, centered at P, that lies on one side of the interface. This computation is particularly straightforward when the surface can be adequately represented by a collection of discrete elements or pixels. A similar technique has been employed to obtain qualitative curvature distributions in simulations of diffusion-controlled growth of aggregates in 2D with finite values of γ [16,17,18] and for cellular automata simulations of curvature-driven sintering in 2D [4] and 3-d [3]. For those simulations, the investigators obtained a relative curvature measure by counting the number of pixels external to the surface but within either a square [4,16,17,18] or a spherical [3] template centered on a given surface pixel. Such template schemes represent a generalization of the method, used by Holm and coworkers for Potts-model-type Monte Carlo simulations of grain growth [19] on a discrete lattice, of computing the spin Hamiltonian by counting the number and type of dissimilar nearest neighbors at a given grain boundary site. Although phrased in terms of a lattice Hamiltonian, their model qualitatively predicts the same grain boundary motion as that predicted by assuming curvature-driven motion [19].

Previous use of the template method for computing curvature has been justified by the demonstration that the pixel count described in the preceding paragraph is roughly proportional to mean curvature [4,16,17,18] for a square template. For square templates in 2D and cubic templates in 3-d, the curvature computation is subject to variability along surfaces due to anisotropy in the template shape (see Ref. [4]). This effect is significantly reduced by using circular (2D) or spherical (3D) templates. In 3D, Bentz and coworkers [3] showed analytically that, for the special case of a spherical template centered on the surface of a much larger sphere, the enclosed volume is exactly proportional to the mean curvature of the larger sphere. Analytical calculations were not provided for non-spherical surfaces [3].

In this paper we present a more general justification of the validity of the template method for computing curvature than has been given in previous papers [3,4,16,17,18] by analytically deriving the relationship between mean curvature and the spherically-bounded volume described in the preceding paragraph, for an arbitrary curve (in 2D) or surface (in 3D). The derivation demonstrates that this bounded volume is approximately linear in the mean curvature, to within higher-order correction terms. Estimates of the magnitudes of the correction terms for typical values of the curvature show these terms to be negligible in most cases. We then describe and illustrate application of the procedure to computing the discrete analogue of mean curvature along interfaces whose shapes are represented as a collection of discrete pixel elements. Digitization of the surface introduces larger errors that can cause the result of an individual curvature computation to deviate substantially from the true mean curvature. Even so, the method produces a reliable measure of the spatial distribution of curvature along non-uniform interfaces and throughout complex microstructures. This latter quality of the method makes it a useful tool for sintering simulations.