Appendix B. Multiple voids

We wish to determine conditions for which, given a fixed gas volume, a multiple number of smaller voids may have lower free energy than a single void in at least one of the configurations shown in Fig. 3. Let there be N voids, all having the same configuration³ but arbitrary volumes, and let ν(R) be the fraction of the total volume V occupied by any one of these voids having liquid–gas interface radius R. Furthermore, let ρ(ν) be the number of voids having volumes between νV and (ν + dν)V , such that

The radius of the liquid–gas interface for any void having volume νV is

and the interfaces have areas following the form

where A_pq(ν) is the area of the interface between phases p and q for a void with volume νV and A_pq,1 is the analogous quantity for a single void of the same configuration having the same total volume as the N-void system. Therefore, the total area of each type of interface bounding the voids in the N-void system is

where v1−1/d is the arithmetic mean value of v1−1/d. This relation holds for the area of all the types of interface, so

If φ_W ≤ π/2 and φ_B ≤ π/2, then these results indicate that Γ of an N-void system having a single type of configuration exceeds that of a single void with the same configuration and total volume. Furthermore, for a wall, base, or free configuration, regardless of the values of φ_W and φ_B, the area of the liquid–gas interface at constant volume increases with number of voids. Therefore, the free energy for all of these three configurations is minimized for a single void. However, the situation is more complicated for the corner configuration if φ_W + φ_B > 3π/2, because all three types of interface are present and Γ decreases with increasing A_BG (because cosφ_B < 0) and A_WG (because cosφ_W < 0).

For simplicity in analyzing the corner configuration when φ_W + φ_B > 3π/2, we assume that all N voids have the same volume.⁴ Substituting the 2-D results from Table 1 into Eq. (5) and differentiating with respect to N gives

where the expression for R from Table 2 has been substituted into the final result. The negative root applies only if the center of curvature of the interface is outside the void (φ_B + φ_W > 3π/2). Therefore, Eq. (B.8) shows that, neglecting other geometric constraints, the free energy of the 2C configuration is actually less than that for the C configuration only if φB +φW > 3π/2. Note that if φB +φW = 3π/2, then the liquid–gas interface is flat, the collection of terms in parentheses in Eq. (B.8) sums to zero, and therefore the C and 2C configurations have the same free energy. Finally, it also can be shown, using the same principles as in preceding paragraphs, that the 2C configuration has minimum free energy when both voids have the same volume.

So far, we have not addressed the case for which, in an N-void system, the voids adopt different configurations. But any void in an N-void system, if not already in the stable configuration for its values of φ_W and φ_B, can only lower its free energy by isometrically transforming to the stable configuration. This means that a system of N voids in different configurations has higher free energy than the same N voids all in the stable configuration. Therefore, the configurations considered in this paper (see Fig. 3) are the only ones eligible to have the minimum free energy.

³ The span configuration is not considered because there can only be one void of that type.