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### 2.2. Three-dimensional model system

For arbitrary values of φB and φW, a 3-D analog to the system depicted in Fig. 2 is difficult to analyze because of the more complex geometries that must be evaluated to determine the volume and areas of the different interfaces. However, the situation simplifies considerably if we assume that φW = π/2 (i.e., γWG = γWL). And because we are interested mainly in identifying qualitative differences between the more artificial 2-D system and the 3-D system, we will accept that restriction for the 3-D model. The void will be assumed to reside in a tetragonal cavity (i.e., all walls of equal length D but Dh). Also, because the walls and base are all mutually orthogonal, the condition that φW = π/2 makes the problem mathematically identical to that of an axisymmetric sessile drop (void) on a rigid substrate. And for the latter situation, one may demonstrate, using variational principles to minimize the free energy, that the liquid/gas interface must assume the shape of a portion of a sphere when gravity is neglected.

Using these assumptions, one may readily obtain expressions analogous to those in Tables 1 and 2 for the 3-D model system. However, two additional configurations are possible in 3-D besides those identified already for 2-D. These are (1) the void situated along an edge between the base and a wall (denoted EBW) and (2) the void situated along the edge between two walls (denoted EWW). The results are given in Tables 3 and 4. The geometric restrictions on the 3-D void in a tetragonal cavity are virtually identical to those for the 2-D system described earlier, and will not be enumerated here.

For the special case of a span configuration, the height hp of the triple
junctions above the base determines the value of AWG.

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