The basic 2-D model is shown in Fig. 2. It consists of a chemically homogeneous liquid (denoted as L) partially filling a reentrant surface feature. The remainder of the cavity is assumed to be filled with a gas (denoted G). The liquid is assumed to provide a thermal reservoir sufficient to keep the entire system isothermal. Furthermore, the liquid and external atmosphere with which it is in contact are assumed to provide a fixed hydrostatic pressure. As long as the pressure remains fixed, it is a good approximation to take the molar volume of the void to be constant also. The reentrant surface feature is a cavity having width D and depth h. Typical dimensions of D and h are often 10 to 1000 µm in engineering applications. For greater generality, and because of the close resemblance to features found in, for example, selective electrodeposition processes, the base of the cavity (denoted B) is assumed to have different wetting properties than the walls (denoted W). To avoid the complication of appreciable gas solubility or any chemical reactions, the liquid, solids, and gas are assumed to be mutually inert.
Fig. 2. A schematic of the 2-D void model.
The free energy of the system clearly should depend on the shape and location of the void within the cavity. The possible distinct configurations that a void of a specified volume 1 may adopt are shown in Fig. 3 along with a name and one-letter symbol by which each configuration will be denoted throughout this analysis. In principle, a volume of gas could be partitioned into multiple voids of different sizes, such as one void in each corner or one void at the wall and another at the base. However, as shown in Appendix B, the only circumstance for the system at hand in which this can happen is that for which the interface between the void and the liquid is convex when viewed from within the void. Such a condition is possible geometrically only for the corner configuration, and so we also consider a configuration, labeled as "2C" in Fig. 3, in which two voids having equal volumes form, one at each corner. By calculating the free energy of each configuration at constant void volume, subject to the geometric constraints, the most stable configuration may be determined.
Fig. 3. Different configurations of a void in a surface cavity.
Because the gas is assumed to have fixed molar volume and temperature, the free energy, F, written as a function of temperature and all other extensive variables, represents the thermodynamic potential governing equilibrium and stability. We will ignore the presence of body forces arising from gravitational,2 electrostatic, or centrifugal fields. For constant temperature, void volume and molar content, the bulk contributions to the free energy are invariant and may be subtracted from the true free energy without further loss of generality. We denote the four different types of materials by single capital letters as follows: B = base, W= wall, L = liquid, and G = gas. The area of the interface separating two phases i and j will then be denoted as Aij . With this notation, the free energy may be written as [2, 3]
where, for example, ABG is the total area of surface separating the base (B) and gas (G), and
Fig. 2 shows that ABL = D − ABG and AWL = 2h − AWG. Also, only the differences in free energies of the various con- figurations are relevant. We therefore subtract from Eq. (1) the constant terms γBLD and 2γWLh, integrate term by term, and normalize by γLG:
where gij ≡ γij / γLG .
It can be shown, by minimization of free energy [4-6],
that at equilibrium the liquid/gas interface must assume a
shape having convex mean curvature.
Δ P = γLGκ,
that at equilibrium the liquid/gas interface must assume a shape with
constant mean curvature. A wide variety of surface shapes with constant mean
curvature, so-called minimal surfaces, can be embedded in three dimensions. However, in 2-D the only
constant-curvature surface possible is an arc of a circle. Furthermore, provided
that both the walls and base of the cavity are essentially rigid solids, minimization of free
energy can be shown to require [7] that the geometry satisfy
Young’s equation [8],
The constitutive variables for the problem at hand are the void volume, V, and the thermodynamic contact angles φB and φW. Therefore we must express, for each configuration, the surface areas Aij in Eq. (5) as explicit functions of these variables. The mathematics are straightforward but tedious and will not be reproduced here. Briefly, the areas are written in Table 1 as functions of the interface radius of curvature, R and then R(V, φB, φW) is derived using trigonometry and catalogued in Table 2. An example derivation for the base configuration is given in Appendix A. Comparison of the free energies for various values of the constitutive variables will establish the global equilibrium configurations of the void as well as the relative work required to liberate the void to the bulk liquid. However, in addition to these free energy considerations, each void configuration can exist only within certain geometric constraints. These constraints are of two types: (1) upper bounds on the linear dimensions of the void, above which the void impinges on another boundary of the cavity; (2) for the C and 2C configurations, the requirement that the liquid–gas interface have constant curvature implies certain inequality relationships between φW and φB. The constraints are catalogued in Appendix C.
Evaluation of stable void configurations in the 2-D model is presented in the Discussion section. But first, we establish a restricted 3-Dmodel that will be useful for comparison to the 2-D results.
For the C and 2C configurations, R → ∞ as φW + φB →3π/2, so the void assumes the shape of a right triangle and the areas are functions of V . For the span configuration, hp is the height above the base of the triple junction formed at each wall.
1 Strictly speaking, "volume" is really an area in 2-D, but for purposed of clarity we will continue to refer to space in 2-D as having volume and to interfaces in 2-D as having area.
2 For large voids, buoyancy forces should be considered because they become
increasingly important, relative to capillary effects, as the surface/
volume ratio decreases. However, for small voids (≈100 µm effective diameter),
it is an excellent approximation to ignore gravity.
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