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Refinements

We now describe briefly two refinements to the curvature computation when it is applied to digitized interfaces. The first represents a potential improvement in computational accuracy, while the second allows collection of information concerning the orientation of the interface.

At regions of an interface where sharp fluctuations in curvature occur over small distances, or at regions where two or more separate interfaces are closely spaced, the template method for computing curvature may generate errors not discussed in previous sections. We illustrate the source of this error schematically in Fig. 7. Figure 7(A) shows the region near the point of contact between two spheres. The true mean curvature at point P is simply the curvature of the right-hand sphere, namely B⁻¹. But since the template sphere may overlap a portion of the left-hand sphere, the curvature computation may generate too low a value for the curvature at point P, since it effectively takes the total number of template pixels and subtracts all solid pixels within the template. The curvature computation is especially prone to this error along crack surfaces or in partially-sintered powder compacts. One could, in principle, modestly decrease the error in these regions by decreasing the template radius, but an algorithm that tests for such regions and automatically adjusts the template radius in response to some criterion would likely be computationally intensive.

Figure 7: (A) Position of a template circle used to compute the curvature at point A in the vicinity of the point of contact between two larger circles. The intersection of the template with the left-hand circle will cause an erroneously low computed curvature. (B) Use of a burning algorithm to test for the connectivity of the solid pixels within the template circle. Cross-hatched (burned) pixels are those that are connected to point A. All unburned pixels within the template are counted in the curvature computation.

A simpler way to reduce errors caused by template overlap, without adjusting the template radius, is to incorporate a variant of the "burning" algorithm, frequently used in percolation models to assess the connectivity of a phase in a microstructure. For example, assume that the spheres in Fig. 7 are solid and the region surrounding them is a pore phase composed of some inert vapor. Then applying the burning algorithm at point P, one "burns" all the solid pixels within the template that can be reached from point P without encountering a pore pixel (shown in Fig. 7(B) with darker shading). The difference between the total number of template pixels and the burned solid pixels can then be used to obtain a more accurate estimate of the mean curvature at point P, since the region of overlap with the left-hand sphere does not burn. The burning algorithm can therefore substantially increase the accuracy of the curvature computation near these kinds of microstructural features. For example, in the relatively extreme 2D case of a flat interface of a crack that is one pixel wide, the unmodified method yields a curvature value of −0.442 ⁻¹ , while including the burning algorithm results in a curvature value of −0.003 ⁻¹, which is much closer to the true value of zero for the crack interface. Of course, there are limits to the effectiveness of the burning algorithm. If the template in Fig. 7 were moved several surface pixels closer to the point of contact (point Q), the burning algorithm would not help since all the solid pixels within the template would burn. Template diameter ultimately limits the resolution of the method in these instances.

The burning algorithm can also be applied if all pixels composing a given particle are assigned a unique label. For example, suppose the pixels composing the left particle in Figure 7 are assigned a different label than those composing the right particle, indicating it is a separate grain. By burning only pixels with like labels, the burning algorithm applied at Q could again produce a more accurate curvature value than that produced by the unmodified method. Assigning different labels to pixels in different particles is particularly useful in sintering and grain growth simulations where grain boundaries can be identified as the interface between two solid regions with different pixel labels [24,25]. The curvature along the grain boundaries can then also be computed using the burning algorithm.

Finally, the orientation of an interface can be both an important thermodynamic and kinetic factor when the specific interfacial free energy, γ, is anisotropic. In such cases theory predicts [26], and experiments confirm [27,28], that interface orientations with lower values of γ are favored over those of high γ. It is straightforward to compute the orientation, relative to some fixed coordinate frame, of the surface normal vector, , erected at any point P along a digitized surface. To determine the orientation of , one can simply keep track of the x, y, and z positions, relative to P, of each pixel in the template volume as it is counted. Using P as the origin, the direction of is given by the 3-tuple (X | |⁻¹,Y | |⁻¹, Z | |⁻¹), where | | = (X ² + Y² + Z²)^½ and, for example,

and m is the number of pixels counted in the curvature computation at P. The orientation of is then specified by, for instance, the two angles

used in a spherical polar coordinate frame (see Fig. 8(A)). Figure 8(B) shows the computed orientation φ of along the perimeter of a circle in 2D. To construct this figure, the diameter of the circle was chosen as 121 , and the template circle diameter was 13 . The computed values of φ are correct to within 3%. Figure 8(C) shows the computed value of φ for two different values of θ, and again the computed values of φ are quite accurate. Because it uses much of the same data and is accomplished at the same time as the curvature computation, determination of the surface normal orientation requires relatively little additional computational effort.