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Appendix: Relationship between curvature and pixel counting

In this appendix we show why the algorithm of counting the "air" pixels in a square which is centered on a point of a "solid" surface gives a measure of the curvature at that point. For purposes of discussion, the pixels on and within the surface are called "solid pixels" and those outside, "air pixels," but it is not necessary for either type of pixel to be solid or vapor phase. The algorithm simply counts the pixels of the opposite phase of that of which the surface is composed. The linear relationship that we present below holds for the limiting case that the pixel is small compared to the counting box (many pixels per box) and, in turn, the counting box is small compared to the local radius of curvature. These smallness conditions will not always be satisfied in numerical simulations and in those cases the counting will still increase with curvature, but not necessarily linearly. The relationship between curvature and counting can always be resolved numerically as in Section 2 of this paper. However, it is edifying to show analytically why the curvature-counting relationship works, in the continuum version of the digital algorithm.

A b x b counting box is centered on the surface of a circle of radius R at an angle , as in Fig. 14. It is clear from the inset of Fig. 14 that the area outside the circle, but inside the tangent to the circle (light gray in Fig. 14) increases with the curvature of the circle. This is the heuristic basis for the algorithm. The area outside the tangent, which is b²/2, is counted as well, but this is a constant which we neglect since we are only interested in curvature differences.

Figure 14: Showing a b x b square box centered at a surface point (R, ) of a circle of radius R. The inset shows a blow-up of the actual region whose area is being calculated.

The results can be calculated in the sector 0 < < /4, since the square has four-fold rotation plus mirror symmetry; all other sectors can be calculated by quadrature. One method is to write the equation of the circle in Cartesian coordinates with the origin placed at the bottom right of the box; integration takes place with the dependent coordinate running up the right-hand side of the box. After normalizing the integral by R and integrating the constant term:

The b²/2 term is the area outside the tangent and the R² term is the contribution of the region outside the circle but inside the tangent. If the circle does not intersect the top and bottom of the box, but instead intersects the bottom and side, the upper limit of the intergral is slightly different. However, this case can be neglected in the limit of b/R 0, since it only occurs in a region of width b/R around = /4. Expanding the integral in powers of b/R, we get (with the first two terms vanishing):

The last term in this equation is the area inside the tangent but outside the circle, which is proportional to the curvature.

The last term in the above equation is also proportional to cos^-3(), which is the angular dependence of the anisotropy in the square box counting scheme. With the smallness conditions discussed above, the equilibrium shape for this counting scheme is any surface that satisfies R cos^-3() = constant. This is the so-called Wullf shape [1]. For the cases illustrated in this paper, the effect of finite b/R and digitally rough surfaces seem to make the Wullf shape more isotropic. For example, the equilibrium shape for the sintered square in Fig. 5c was nearly a digitized circle.

If the counting is averaged over the entire circle, we get:

which demonstrates the b³ scaling of the slopes of the fitted straight lines plotted in Fig. 2. A similar calculation can be done for a counting cube centered at a surface point of a sphere, with results as in 2D, but with b⁴ appearing in the 1/R term instead of b³.

Figure 15 shows the cube root of the slopes of the fitted lines in Fig. 2 and the fourth root of the slopes of the fitted lines in Fig. 3, plotted against the box size b. The solid lines in Fig. 15 are fitted to the renormalized simulation slopes, clearly demonstrating the b³ dependence in 2D and the b⁴ dependence in 3D of the curvature slopes arising from the box counting algorithm. Uisng a circle counting method in 2D and a sphere counting method in 3D would give the same b dependence, where b would be the diameter of the counting circle or sphere [24].

Figure 15: Showing the cube root of the 2D curvature slopes from Fig. 2 and the 4th root of the 3D curvature slopes from Fig. 3, plotted against b. The solid lines are fit to the data points.

Consideration of the construction in Fig. 14 reveals that if a counting circle were employed instead of a counting box, then an isotropic count would be obtained. This method will be employed in future simulations [24]. A similar method was implied in an earlier paper [29], although no derivation explicitly relating area counting to curvature was given.

Using a counting circle also results in a method to compute the surface normal. If A is the region inside the counting circle, but outside the surface of interest, then the unit normal is given by: