Stability diagrams like those in Fig. 4 can predict the equilibrium morphology of voids in a given system, and they also can address the more practical question of how the constitutive variables must be changed to reduce the stability of a trapped void. A striking prediction from Fig. 4A is that very small voids are stable only when the liquid perfectly wets both surfaces. If the practical objective is to eliminate all voids, regardless of their size, then either work must be supplied or the liquid and/or solids must be modified to promote perfect wetting. In many practical systems involving combinations of organic and metallic surfaces, perfect wetting may be a difficult task because these solid/vapor surfaces often have low surface energy densities and therefore are difficult to wet.
When it is not feasible to eliminate the possibility of void formation, it is natural to ask (1) how much work must be performed to free a void, and (2) whether there are other methods of introducing the liquid, instead of rapid immersion, that are less likely to form a void in the first place. The first question will be addressed here; the latter question is outside the scope of the paper.
By examining results like those in Fig. 5, we may calculate the work required to liberate a void of a given configuration. As mentioned earlier, a free void cannot be thermodynamically favored under these conditions.However, as Fig. 5 shows, any reduction in the contact angle between the liquid and the walls (or, in fact, the base) will reduce the work required to free a void. Noting the difference in vertical scale between Figs. 5A and 5B, the plots also confirm the intuitive notion that more work is required to liberate a void as its volume increases.
Work added to the system allows it to sample higher free energy states and therefore increases the likelihood of forming a free void. Once formed, a free void is unstable toward rising to the top of the liquid under buoyancy forces, but this will occur only if a path is available for the void to rise. Otherwise, the free void is more likely to reform in a trapped configuration of lower free energy.
We close this section with a few comments on the applicability of the model to real systems. Obviously, during the formation of a void, the system will often be far from equilibrium and hydrodynamic factors will play an important role in determining the initial configuration. More importantly, the initial configuration may be metastable relative to the configuration indicated in the diagram for the appropriate values of φW and φB. As an example, Fig. 4D indicates that, for void volumes that are large relative to the cavity volume, a liberated void is the only stable configuration if both φW and φB are sufficiently small. However, it may be likely for hydrodynamic reasons that a span configuration would form initially, making a bubble that spans the opening of the cavity. The span configuration has higher free energy than a free void when both wetting angles are small. But in the span configuration, the meniscus capping the cavity could adopt a local equilibrium shape, and transporting liquid to the base surface to liberate the void would undoubtedly require a supply of work in some form. Therefore, as with the application of any equilibrium analysis, one must be wary of metastable states.
We also may use this model to infer the behavior of voids at other types of rough surfaces. First, it is more common in practical situations for the surface to be more chemically homogeneous than is assumed by this model. Simply setting φW = φB will make the analysis here applicable to systems such as an embossed metal or the inside of a syringe.
The present model also may be extended to surfaces for which the "base" and "wall" are not mutually orthogonal. For such systems, the free energies of the wall, base, and free configurations at equilibrium must be basically unchanged by departures from orthogonality. The geometric limits of stability will be different, but even these changes should be small when the deviation from orthogonality is not too severe. On the other hand, the corner configuration should become increasingly less stable as the interior angle, θ , between the wall and base becomes more oblique. In the limiting case where the θ → π, i.e., the wall and base are coplanar, it is not possible for the liquid-gas surface to meet both types of surfaces while retaining constant mean curvature unless φW = φB. Thus equilibrium is impossible and the void will spontaneously migrate to the surface for which the contact angle is greater. Based on these considerations, the stability of a void is expected to be influenced increasingly by the least wetting surface as θ becomes more oblique. For the same reasons, the corner configuration should become increasingly favorable as θ becomes more acute.