Criteria for antibubble formation from drop pairs impinging on a free surface

Antibubbles are fluid entities with the inverse phase of regular bubbles. While the structure and stability of antibubbles have been studied, a fundamental understanding of antibubble formation remains limited. We report a theoretical and experimental study of antibubble formation. In the experiment, pairs of surfactant-laden water drops impinged successively on the surface of the same liquid reservoir to create antibubbles. We propose four criteria for antibubble formation from a scaling analysis. Two dimensionless groups prescribe the likelihood of antibubble formation, the summative Weber number and the ratio of timescales between the capillarity driven pinch-off and the viscous drainage of air.

Figure 1

Optical image of an antibubble separated from the bulk liquid by an air gap. Food coloring was added in the antibubble to visually distinguish it from the bulk liquid.

Figure 2

Schematic of the experimental setup. Surfactant-laden water drops impinge on the bulk liquid by a syringe pump to generate antibubbles. Antibubble formation process is captured by a high-speed camera.

Figure 3

Antibubble formation process. (a) Schematics and (b) corresponding experimental images in four states; two drops impinge successively onto the surface of reservoir (state 1), initial surface deformation by the first drop (state 2), further surface deformation by the second drop and pinch-off (state 3), leading to antibubble formation (state 4).

Figure 4

Crater evolution by the first drop impingement. (a) Schematic of the cavity and notations of moving reference frames. (b) Penetration depth from Eqs. (2) and (3) as a function of time for the impinging drop with a diameter of 2.48 mm and velocity of 1.24 m/s with corresponding experimental data points (open circles). (c) Time-lapse images of crater evolution from high-speed camera. (d) Dimensionless optimal time delay as a function of Weber number for different Froude numbers. (e) Optimal time delay as a function of impinging velocity for different diameters.

Figure 5

A representative failure case of antibubble formation due to insufficient total kinetic energy of two impinging drops at ${\mathrm{We}}_{\mathrm{sum}}<1$. An initial surface deformation was created by the first impinging drop at 0 ms. The second drop deformed the surface further at 5 ms. However, due to insufficient kinetic energy of these two drops, the second drop could not penetrate the surface to form an air column, and therefore, the deformed bulk liquid surface was recovered at 10 ms.

Figure 6

Experimental images and analysis of a failure case due to early drainage of air resulting in the collapse of the air gap. (a) Time-lapse experimental images of early drainage of air. A craterlike initial surface deformation was created by the first impinging drop at 0 ms. The second drop penetrated the deformed surface to form an air column at 3.5 ms. Collapse of the air gap occurred at 5 ms due to the early drainage of air before pinch-off. The second drop coalesced with the bulk liquid at 5.5 ms and recovered finally at 7 ms. (b) Schematics of an antibubble at the pinch-off step with a magnified view of the air flow within the gap, which is in the $\varphi$ direction in spherical coordinates.

Figure 7

Experimental results of antibubble formation plotted with the summative Weber number and the ratio of two timescales. Red circles, blue squares, and green triangles represent successful antibubble formation, insufficient deformation of the bulk liquid surface, and failures due to early drainage of air in the gap, respectively. (a) The boundary for summative Weber number is found to be ∼3. (b) Magnified view of the dashed box in (a) shows the boundary of the ratio of two timescales is ∼7, which shows good agreement with our theoretical analysis. (c), (d) The critical role of time delay between two-drop impingement. Antibubbles only form when the normalized time-delay difference is within a 30% range even when criteria for the summative Weber number and the ratio of two timescales are satisfied (shaded in red). The experimental data for time-delay effects are in Table S1 in the Supplemental Material, Sec. VII [20]. The error bars account for the uncertainties of drop diameter and impinging velocity, which are discussed in the Supplemental Material, Sec. VIII, in detail.