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Analysis

We wish to analytically justify the assertion that the mean curvature of a sufficiently small surface element of a condensed phase α is linear in (within an additive correction term) that portion of the volume of a small sphere, centered on the surface element, that does not contain α. The meaning of the terms "sufficiently small surface element" and "small sphere" will become apparent during the analysis.

Consider a point P lying within a small element S of a surface that bounds phase α, with outward unit normal vector (pointing away from α), as in Fig. 1(A). Assume that S is specified by a function, g, of two spatial coordinates, and that g is continuous on the element to at least second order in those coordinates. The principal radii of curvature at P, R_min = κ_max⁻¹ and R_max = κ_min ⁻¹ (R_min < R_max), can in general have any sign. Next, consider a small imaginary "template" sphere of radius b (b << R_min) centered at P (shown in Fig. 1(B)). Denote by V that portion of the sphere volume that does not intersect α.

Figure 1:(A) Point P centered within a two-dimensional surface element S, characterized by the two principal radii of curvature R_min and R_max. (B) Template sphere of radius b centered on P. The Cartesian reference frame is oriented such that the z-axis is normal to S at P, and such that P is at the origin. V (see text) is the portion of the template volume that lies entirely to one side of S (in this figure, the upper side).

Without loss of generality, we may orient a Cartesian coordinate frame such that (i) the tangent plane to S, passing through P is the xy-plane with P at the origin, and (ii) the projection of the two principal lines of curvature at P onto the xy-plane are the x and y axes (Fig. 1(B)). With this choice of coordinates, the slope of the tangent plane at P is zero, and we assume that S can be represented as a convergent Taylor expansion of g(x,y) about the origin:

where

Because of the placement of the coordinate system, the principal curvatures κ_max and κ_min at P are equal to the negative of the first and second partial derivatives, respectively, appearing in Eq. 3. Furthermore, by comparing Eq. 3 with an analogous expansion using orthogonal curvilinear parameters that define lines of curvature, it can be shown that the third 2nd-order partial derivative in Eq. 3 is zero [20,21]. Finally, we rewrite Eq. 3 in terms of dimensionless variables that scale with the radius of the template sphere, b (X = x/b, K_max = b κ_max, etc.),

where

We assume throughout this analysis that the maximum magnitude of the four ε_j terms, ε, is small compared to unity. In other words, we restrict the analysis to surfaces for which the expansion Eq. 5 is a good approximation.

It will prove convenient to use cylindrical coordinates R (= r/b), θ, and z rather than Cartesian coordinates. Then Eq. 5 can be rewritten as

where q = 2( K_max cos² θ + K_min sin² θ)⁻¹. The scaled volume ν = Vb⁻³ is then given by the integral expression

where Z_t is the equation for the lower half of the template sphere,

and R^*(θ) is determined by Z_s = Z_t,and can easily be shown to be

where

to order ε, with K = K_max + K _min. Substituting Eqs. 5 and 9 into Eq. 7 gives

and substituting Eq. 10 gives

Using the definition of q produces

where

If β < 1, then we may expand β ^{− n} according to

to order ε. Substituting into Eq. 12 yields

where

The second term on the right side of Eq. 16 also indicates one major effect of the template radius, b: The maximum magnitude of curvature that can be measured using the linear approximation is inversely proportional to b. Smaller templates therefore increase the range of measurable curvatures in the continuum limit. We will see in the next section, though, that when applying the method to digitized representations of surfaces, smaller templates decrease the resolution of the curvature measurement.

A different form for R^*(θ), Eq. 10, will generally result for different surface shapes. For example, a surface element that is a portion of a sphere gives

From Eq. 17 it is easily shown that

In other words, the linear approximation is exact for spherical surface elements.

To summarize, the linear approximation between mean curvature and V, Eq. 16, provides estimates of the mean curvature, typically accurate to within ±10%, at any point on a surface that can locally be approximated by a 2nd-order expansion, Eq. 5. The error in Eq. 5 generally increases with increasing displacement from the origin, that is, Eq. 5 becomes less accurate for larger template radii. Finally, it should be mentioned that the analysis is not strictly appropriate at points at which there is a discontinuity in the first or second order spatial derivatives, such as at sharp corners and edges. At such points the magnitude of the true mean curvature is unbounded, while the volume calculations shown in this section establish an upper limit on the magnitude at any point that is dictated by the volume of the template. Application of this method at these points can still, however, give some idea of an "apparent" curvature because the computed magnitude of the curvature will depend on the difference in slopes on either side of the singularity (razor edges, for example, will yield a higher computed value of curvature than right edges).

We now proceed to describe a few simple extensions of the template concept that are potentially useful under certain circumstances.