Shapes of Nonsymmetric Capillary Bridges

Here we study the shapes of droplets captured between chemically distinct parallel plates. This work is a preliminary step toward characterizing the influence of second-phase bridging between biomolecular surfaces on their solution contacts, i.e., capillary attraction or repulsion. We obtain a simple, variable-separated quadrature formula for the bridge shape. The technical complication of double-ended boundary conditions on the shapes of nonsymmetric bridges is addressed by studying waists in the bridge shape, i.e., points where the bridge silhouette has zero derivative. Waists are generally expected with symmetric bridges, but waist points can serve to characterize shape segments in general cases. We study how waist possibilities depend on the physical input to these problems, noting that these formulas change with the sign of the inside–outside pressure difference of the bridge. These results permit a variety of different interesting shapes, and the development below is accompanied by several examples.


■ INTRODUCTION
Here we study the shapes of nonsymmetric capillary bridges between planar contacts (Figure 1), laying a basis for studying the forces that result from the bridging.
The recent measurements of Cremaldi et al., 1 provide a specific motivation for this work. A helpful monograph 2 sketches adhesion due to symmetric capillary bridges, albeit with aspect ratio (width/length ≈10 3 ) vastly different than is considered below. Additionally, that sketch 2 does not specifically consider nonsymmetric cases surveyed by Cremaldi et al. 1 A specific description applicable to nonsymmetric cases is apparently unavailable 3 and, thus, is warranted here.
A background aspect of our curiosity in these problems is the possibility of evaporative bridging between ideal hydrophobic surfaces, influencing the solution contacts between biomolecules. 4−10 Assessment of critical evaporative lengths in standard aqueous circumstances on the basis of explicit thermophysical properties 8 sets those lengths near 1 μm. Though we do not specifically discuss that topic further here, our analytical development does hinge on identification of the length γ = |Δ | p 2 / , with γ the fluid interfacial tension, and Δp the pressure difference between inside and outside of the bridge. The experiments that motivate this study considered spans ≲(6 μL) 1/3 ≈ 1.8 mm. 1 A full development of the essential basics of this problem might be dense in statistical-thermodyamics. We strive for concision in the presentation below but follow 2 a Grand Ensemble formulation of our problem. We then develop the optimization approach analogous to Hamilton's Principle of classical mechanics. 2,11 That approach avoids more subtle issues of differential geometry related to interfacial forces and, eventually, should clarify the thermodynamic forces for displacement of the confining plates. Along the way, we support the theoretical development by displaying typical solutions of our formulation.

■ STATISTICAL THERMODYNAMIC FORMULATION
Consider two plates, not necessarily the same, oriented perpendicular to the z-axis and separated by a distance h ( Figure 1). A droplet captured between two parallel plates is assumed to be cylindrically symmetric about the z-axis. We want to determine the droplet shape ( Figure 2) in advance of analyses of the forces involved. We study a functional of the droplet radius r(z). Here ṙ= dr(z)/dz and r ± = r(z = ±h/2). γ is the tension between the droplet and the external solution. Δγ + is the inside−outside difference of the surface tensions of the fluids against the plate at z = +h/2 (and similarly for Δγ − with the fluids against the plate at z = −h/2); this differencing will be clarified below as we note how this leads to Young's Law. Δp is the traditional Laplace inside− outside pressure difference of the bridge. The usual Grand Ensemble potential for a single-phase uniform fluid solution being −Ω = pV, it is natural that ΔΩ[r] of eq 1 has Ω for the surrounding fluid solution subtracted away; i.e, the pressure− volume term of eq 1 evaluates the pressure inside times the bridge volume, minus the pressure outside times the same bridge volume. Formally with F the Helmholtz free energy. Therefore, the surface-area feature of eq 1 can be viewed as an addition of γA contribution to F, with A the surface area of contact of the bridge with the external fluid and γ is the tension of the fluid−fluid interface. We have not included a line tension associated with the top/ bottom contacts. Nevertheless, since eq 1 is firmly grounded in the basic physical description of our problem, line tension issues should be readily accommodated. An alternative perspective on ΔΩ (eq 1) is that it is a Lagrangian function for finding a minimum surface area of the bridge satisfying a given value of the bridge volume. Then Δp/ γ, which has dimensions of an inverse length, serves as a Lagrange multiplier. We then minimize ΔΩ with respect to variations of r(z), targeting a specific value of the droplet volume.
The first-order variation of ΔΩ is then The angle that the shape curve r(z) makes with the plane perpendicular to the z axis ( Figure 1 Depicted in Figure 1 is the choice of the bottom sign above, where 0 < ṙ< ∞. For θ + we change the choice so that the  , smaller than the radius of the upper contact, ≈0.08. The contact angle θ + = 60°together with R̃, eq 15 gives r̃+ ≈ 0.79, confirming the connection between branches above and below the waist. The dashed curve thus extends the solid curve. At each height, the red crosses mark the discrepancies of the Euler−Lagrange eq 9 from zero.
with the exterior angles contacting the upper and lower plates. The contact terms in eq 7 vanish if the contact angles obey the force balance of the traditional Young's Law. This reinforces the sign choice for eq 5. Equation 8 will provide boundary information for r(z). From eq 7, we require that the kernel with D a constant of integration. D + r 2 Δp/2 is non-negative according to eq 10. Recognizing that sign, then with 0 ≤ θ(z) ≤ π. The constant D can be eliminated in terms of boundary information, e.g., This helpfully correlates r(z) at other places too. For example, we will consider (Figure 2) intermediate positions where r(z) = 0 and sin θ(z) = 1. We call such a position a "waist". A waist is expected for symmetric cases that we build from here. Denoting the radius of a waist by R, then γ = + Δ R D R p/2 2 (13) from eq 10. This eliminates the integration constant D in favor of R, which may be more meaningful.
Considering Δp > 0 we can make these relations more transparent by nondimensionalizing them with the length γ = Δp 2 / . Then =r r and =R R , so Though this scaling with the length is algebraically convenient, Δp can take different signs in different settings; indeed, calculating from eq 9, at a waist Δp/γ = 1/R − r̈in the present setup, with r̈the curvature at that waist. Completing the square from eq 14 gives Equation 15 provides helpful perspective (Figure 3) for exploring different bridge sizes. Given θ − , this requires that (R̃− 1/2) 2 ≥ (cos θ − /2) 2 , as is evident there.
Interesting further consequences follow from considerations of the cases that the droplet is nearly tangent to the contact surfaces: θ ± = 0 or π. Consider first θ − → 0. The droplet approaches detachment from the lower surface. We expect r − → 0 then. Figure 3 shows that this can be achieved with R̃= 0 or 1. The R̃= 1 case produces a hemispherical lower portion on the bridge, with the hemisphere just touching the lower surface and When θ + → π for example, the droplet preferentially wets the upper surface. We expect r + to be relatively large then, and this force contribution describes interplate attraction, though not necessarily with a waist.
More Generally but Δp ≠ 0. Restoring in eqs 14 and 15 the dependence on is a signed length here. With these notations, . Thus, the solid black curves cover possible values of R̃for this θ − , and displacements upward from the green horizontal line, i.e., the arrows, show values of (r̃− − sin θ − /2) 2 . The θ − adopted for this drawing is π/6 as for the bottom branch shown in Figure 2, and the right-most dot locates the value of the waist radius there (Figure 2). Thus, the waist in that example is the slimmest waist in that range. Such considerations apply to both top and bottom contacts with their distinct contact angles. A contact angle near π/2 will correspond to a lower level for the horizontal line and thus be less restrictive of the possible values of a common waist radius R̃.
separates these variables for integration. We can still follow scaled lengths̃= | | r r/ and̃= | | R R/ . Then the analogue of eq 15 is when Δp < 0; see Figure 4. The analogue of eq 18 with this length scaling for Δp < 0 is To achieve Δp/γ = 1/R − r̈< 0 for a bridge with wiast radius R, clearly the curvature r̈at that waist should be substantially positive to ensure that the negative second contribution dominates. In addition, the radius at the waist should be fairly large, thereby reducing the contribution of the positive first term. These points combined suggest that to achieve adhesion the contact areas should be larger than the waist area, which itself should be substantial.
Waist R. Reaffirming the identification of R as the radius of a waist and specifically recalling that is a signed length: Factoring out the cot 2 θ(r̃2 = R̃2) = 0 feature gives

■ EXAMPLES
In the example Figure 2 (Δp > 0), R̃≈ 0.933 and (R̃− 1) 2 ≈ 0.067 2 , smaller than the radius of the upper cross-section, 0.08 2 , in that extended example. The slimmer second waist is not realized. Figure 5 shows a bridge shape for the slender waist identified for the contact angles specified in Figure 6 for Δp > 0.
In the example Figure 7, Δp < 0 and < 0 . Thus, ) , and the (−) of eq 23 is required to achieve a positive slope at the bottom plate. The aspect ratio of the bridge is vastly changed, as was true also in the discussion of capillary adhesion of ref 2; capillary adhesion would be expected for this shape.   In view of the variety of interesting shape possibilities, we reserve explicit study of the consequent interplate forces, and of the stability/metastability of these bridges, for a specific experimental context. Nevertheless, we outline here how such a practical study might be implemented. The setup above permits straightforward calculation of the thermodynamic potential Ω and