In two dimensions, specification of the three constitutive variables (V , φW, and φB) and the cavity dimensions D and h is sufficient to determine the thermodynamically favored void configuration consistent with the geometric restrictions. The information can be catalogued conveniently in a "phase diagram" that maps the ranges of constitutive variables within which a given void configuration is stable. Fig. 4 shows four isometric (i.e., constant V/Vc) cuts of such a diagram, using cavity dimensions D = h = 100 µm, for which the total cavity volume Vc = 104 µm2. The stable configuration in any region is denoted by a single capital letter as shown in Fig. 3. Chatain et al. [1] collect their stability data using the same principle, although in their paper they employ a different set of constitutive variables.
If the dimensions of the cavity are large compared to the dimensions of the void, then none but the span configuration are geometrically constrained. Therefore, the stability diagram shown in Fig. 4A, where V/Vc → 0, is determined only by the minimization of the free energy at constant volume. The symmetry of Fig. 4A therefore is expected because, from the perspective of the void, the cavity is essentially a single corner with base and wall extending away from it indefinitely. When the cavity dimensions are more comparable to the void dimensions, the base, wall, or corner configurations can become geometrically impossible, especially at larger values of φW or φB for which the void becomes extremely anisometric. This is reflected in Figs. 4B-4D. The broken symmetry in Figs. 4B and 4C is also due partially to the fact that the cavity itself is asymmetric, being open at the top but bounded on the bottom and sides.
Fig. 4. Isometric cuts of a void stability diagram calculated by the 2-D model for D = h = 100 µm (Vc = 104 µm2). (A) V / Vc = 10−3, (B) V / V c = 0.02, (C) V / Vc = 0.2, and (D) V / Vc = 2.0.
To be more quantitative about the changes that occur at
increasing void volumes, notice in Fig. 4B that regions of
stability for the base and wall configurations appear in narrow
bands for which φW or φB exceed critical values
or
, respectively. Both
decrease from a maximum
value of π as the void volume increases. To see why
this is, take as an example the base configuration. A value of φB =
π means that a void in contact with the base will spread
indefinitely, but because the walls limit the extent of spreading,
the base configuration is geometrically impossible for φB = π at
any finite volume. For very small void volumes,
V/Vc < 0.01 the value of the critical angles are sufficiently
close to π that the new stability regions, although present,
are not visually discernible in Fig. 4A.
We can approximate these critical angles as a function of
volume. For example, by explicitly writing the constraint in
Eq. (C.13), with R taken from Table
2, the value of
will
be that for which
If we use α = π −
as a variable, then for α→0 we can
expand Eq. (6) to second order and solve to find
Another change in the stability diagrams at larger void volumes
is the appearance of a region near the point (φW = π/2,
φB = π) in which a span configuration is favored. Again,
when φB → π a base configuration is not possible because
the void would spread indefinitely, leading to a span configuration. However, the span configuration is only possible,
at small void volumes, if the meniscus is a flat line in 2-D
(a plane in 3-D). If the meniscus is convex or concave, then
it is geometrically impossible to enclose a sufficiently small
volume under such a meniscus that spans the cavity. But as
long as themeniscus is flat or nearly so (i.e., φW ≈ π/2) then
any void volume can be contained by adjusting the height of
the meniscus above the base surface. The range of φW over
which the span configuration is favored, when φB = π, can
be estimated by following the same procedure as that leading
to Eq. (7). The result is written compactly as
As shown in the upper right corner of Fig. 4B, the C and 2C configurations have equal free energies along the boundary defined by φW + φB = 3π/2. It is exactly this condition for which the liquid-gas interface has zero curvature. As shown in Appendix B, when the liquid-gas interface has zero curvature, the free energy of N voids, having total volume V, in corner configurations is independent of N and there is no driving force either for coarsening of the voids or for anticoarsening. This result agrees with previous results by Chatain et al. [1]. For the 3-D model, Fig. 5 shows plots of Γ − Γfree vs φB for two different void volumes. The new "edge" configurations available in 3-D are in no case the most stable. With the exception of these two configurations, we can directly compare the results of Fig. 5 to the 2-D model. To make that comparison as meaningful as possible, we first normalize the free energy of each configuration, in 2-D and 3-D, by the free energy of a free void. The result is a set of dimensionless functions αi2 and αi3 in two and three dimensions, respectively, for configuration i . These functions each have range [0, 1]. Furthermore, recalling that the 3-D model in its present form is valid only for φW = π/2, we will assume this value for the 2-D model as well. Fig. 6 plots these normalized functions for corner, wall, and base configurations. The additional edge configurations available in the 3-D model are omitted from the figure for clarity. For every configuration, the values for the 3-D model differ from those of the 2-D model by no more than 15%. Therefore, we may have confidence that the 2-D model is sufficient for giving qualitatively meaningful insights and predictions.
Fig. 5. Free energies calculated by the 3-D model of each configuration when D = h = 100 µm. (A) V/Vc = 10−3 and (B) V/Vc = 0.5. For both plots, φW = π/2, and the terminating points of some of the curves indicate the range over which a given configuration can satisfy its geometric constraints imposed by the cavity dimensions.
Fig. 6. Normalized free energies α = Γ/Γfree for various configurations predicted by the restricted 3-D model (solid curves) and by the 2-D model (dashed curves) for voids with V/Vc = 10−3.