Buoyancy and capillary effects on floating liquid lenses

We study the equilibrium shape of a liquid drop resting on top of a liquid surface, i.e., a floating lens. We consider the surface tension forces in nonwetting situations (negative spreading factor), as well as the effects of gravity. We obtain analytical expressions for the drop shape when gravity can be neglected. Perhaps surprisingly, when including gravity in the analysis, we find two different families of equilibrium solutions for the same set of physical parameters. These solutions correspond to drops whose center of mass is above or below the level of the external liquid surface. By means of energetic considerations, we determine the family that has the smallest energy, and therefore is the most probable to be found in nature. A detailed explanation of the geometrical differences between the two types of solutions is provided.

Figure 1

Dimensionless scheme of a liquid lens (fluid $A$) over a deep external liquid (fluid $B$), surrounded by air (fluid $C$). The triple contact line is at $r=R_d$ and $z=0$ ($R_d\ll R_{\text{wall}}$). The arrows over each curve indicate the increasing direction of the arclength, $s$. The length scale is ${\mathcal{R}}_0$ [Eq. (1)].

Figure 2

Contact angle definitions for a liquid lens (fluid $A$) over a deep liquid substrate (fluid $B$), surrounded by air (fluid $C$).

Figure 3

Schematic of drop shapes for a given $\eta$ as a function of $\zeta$, with ${\sigma }_1>{\sigma }_2$ (top panel) and ${\sigma }_1<{\sigma }_2$ (bottom panel). The central dashed area where $0<\zeta <1$ ($S^{*}<S<0$) corresponds to the partial wetting scenario studied in this work. For $\zeta <0$ ($S>0$), we have complete wetting, so that the drop spreads indefinitely. Instead, for $\zeta >1$ ($S<S^{*}$) a nonwetting case occurs, where the drop remains on top or above the free surface, depending on the relative values of ${\sigma }_1$ and ${\sigma }_2$.

Figure 4

Solution without gravity: Drop profiles for $\eta =1.75$ and $\zeta =0.1,0.3,0.5,0.9$ for cases A (${\sigma }_1<{\sigma }_2$) and B (${\sigma }_1>{\sigma }_2$). Note that the free surface of the liquid substrate (curve 3) is flat for all $\zeta$.

Figure 5

Solution without gravity: $R_d$ as function of $\zeta$ for $\eta =1.01$, 1.25, 1.50, 1.75, and 2.00 for cases A (${\sigma }_1<{\sigma }_2$) and B (${\sigma }_2<{\sigma }_1$). The arrows indicate the direction of increasing $\eta$.

Figure 6

Solution without gravity: Radius of curvature of each surface of the lens as a function of $\zeta$ for $\eta =1.01$, 1.25, 1.50, 1.75, and 2.00. (a) For curve 1 and (b) curve 2 in case A (${\sigma }_1<{\sigma }_2$), and (c) for curve 1 and (d) curve 2 in case B (${\sigma }_2<{\sigma }_1$). The arrows indicate the direction of increasing $\eta$.

Figure 7

Example of the two families of solutions with gravity for $\eta =1.75$ and $\zeta =0.70$. (a) Case A (${\sigma }_1<{\sigma }_2$) and (b) Case B (${\sigma }_2<{\sigma }_1$).

Figure 8

Case A: Two families of solutions with gravity for $\eta =1.75$ and several values of $\zeta$. (a) Solutions type 1. (b) Solutions type 2. The arrows indicate the direction of increasing $\zeta$.

Figure 9

Case B: Two families of solutions with gravity for $\eta =1.75$ and several values of $\zeta$. (a) Solutions type 1 ($\text{Bo}=0.49$). (b) Solutions type 2 ($\text{Bo}=0.49$). (c) Solutions type 1 ($\text{Bo}=4.32$). (d) Solutions type 2 ($\text{Bo}=4.32$). The arrows indicate the direction of increasing $\zeta$.

Figure 10

Case B: (a) Two families of solutions with gravity for $\eta =1.75$, $\zeta =0.85$, and $\text{Bo}=4.32$ (practically coincident) compared with the analytical solution without gravity. (b) Evolution of angle $\gamma$ as a function of $\eta$ for both Sol 1 and Sol 2 for two extreme values of $\zeta$ and $\text{Bo}=0.49$. As $\eta$ increases, $\gamma$ decreases toward zero. The convergence to a flat Curve 3 is faster for larger $\zeta$.

Figure 11

(a) Initial and (b) final states of the approach used to calculate the system energy: it corresponds to a finite volume configuration with fixed $V_f$ and variable $h_f$.

Figure 12

Case A: Total energy variation for $\text{Bo}=1.22$, Sol 1 (solid lines) and Sol 2 (dashed lines). (a) $\eta =1.01$, (b) $\eta =1.5$, (c) $\eta =2$. The curves in (a) do not go beyond $\zeta \approx 0.8$ because of precision issues in the numerical scheme appearing when both $\xi$ and $\eta$ are very close to unity (their respective maximum and minimum limiting values).

Figure 13

Case B: Total energy variation for $\text{Bo}=0.49$, Sol 1 (solid lines) and Sol 2 (dashed lines). (a) $\zeta =0.1$, (b) $\zeta =0.5$, (c) $\zeta =0.9$.

Figure 14

(a) Experiment: Silicon oil (PDMS) drop on water (case A with $\eta =1.79$ and $\zeta =0.04$). (b) Comparison of drop shape with solutions Sol 1 and Sol 2.

Figure 15

Air bubble trapped between two dense immiscible liquids with gravity: ${\rho }_A=0.0013{\mathrm{g}/\mathrm{cm}}^3$ (air), ${\rho }_B=1.00{\mathrm{g}/\mathrm{cm}}^3$ (water), and ${\rho }_C=0.97{\mathrm{g}/\mathrm{cm}}^3$ (oil). Here, Sol 1 is the black solid line, while Sol 2 is the blue dashed line.

Figure 16

Sol 1 (blue, solid) reported in Fig. 4(a) in [12], and Sol 2 (red, dashed) the other possible solution obtained for the same set of physical parameters (see the text).