Coalescence-induced droplet jumping on superhydrophobic surfaces: Effects of droplet mismatch

On low-adhesion surfaces, coalescing droplets can spontaneously jump off, known as coalescence-induced droplet jumping. It is observed on a variety of synthetic and natural superhydrophobic surfaces, and gives rise to a range of applications, such as self-cleaning condensers, anti-icing coatings, and thermal diodes. Through three-dimensional simulations, this paper demonstrates the fluid dynamics of droplet jumping upon binary unequal-sized-droplet coalescence. Parametric studies show the influence of droplet mismatch, viscosity, and contact angle on jumping velocities, where liftoff regimes are defined on the basis of Ohnesorge number and droplet size ratio. Because of the strong asymmetric flow behavior, the well-known small conversion efficiency for equal-sized-droplet jumping, where around $6\%$ of the released surface energy is convertible into translational kinetic energy, is further reduced for unequal-sized-droplet jumping. The findings offer insights into their fluid dynamics and give a starting point for further modeling of dropwise condensation on superhydrophobic surfaces.

Figure 1

Mechanism of coalescence-induced droplet jumping. The coalescence of droplets leads to the release of surface energy, which is converted into translational motion. The static pressure field is scaled with the capillary pressure.

Figure 2

Surface and adhesion energy of three different static states.

Figure 3

Ratio between the excess energy of droplet merging and the adherence energy depending on droplet size ratio and contact angle (left). Regime diagram for the absolute jumping limit (right). Coinciding values are marked by the red crosses.

Figure 4

Shape of sessile drops at two different contact angles: present solver $(\cdots )$ and theory $(-)$. Errors are in the order of $\epsilon \approx {10}^{-2}$, measured by the location of the droplet centroid.

Figure 5

Nonlinear droplet oscillations: present solver $(\circ )$, interFoam $(\times )$, and numerical reference data from literature [35, 36] $(-)$. Dimensionless oscillation for high damping $(\mathrm{Re}=10$, left) and low damping $(\mathrm{Re}=100$, right). Due to the imprecise calculation of capillary pressures, interFoam fails to predict droplet oscillations accurately. The present solver agrees well with reference simulations.

Figure 6

Simulation case with boundary conditions. The smallest cell size after dynamic mesh refinement is $0.025/R_1$.

Figure 7

Time evolution of droplet shape for $\text{Oh}=0.01,k=0.67$, and $\theta ={170}^{\circ }$. The contours in the symmetry plane show the static pressure (adimensionalized with the capillary pressure). Two acceleration mechanisms are identified: impact of the capillary bridge and impingement of capillary waves. The droplet detaches from the solid surface at $t^{*}\approx 2.3$, initiating free motion in the gas phase. A movie is provided in the Supplemental Material [38].

Figure 8

Time evolution of droplet wall-normal (—left axis) and wall-parallel (- - - right axis) velocity for $\text{Oh}=0.01,k=0.67$, and $\theta ={170}^{\circ }$. Coalescence-induced droplet jumping can be divided into four stages: expansion of the capillary bridge (I), acceleration off the surface (II), deceleration (III), and free motion (IV). Stage II can be divided into substages based on dominating acceleration mechanisms: impact of capillary bridge (IIa), impingement of capillary wave on s.d.s. (IIb), no acceleration (IIc), and impingement of capillary wave on l.d.s. (IId).

Figure 9

Time evolution of droplet wall-normal velocity for $\text{Oh}=0.01$ and $\theta ={170}^{\circ }$, with droplet size ratios of $k=1,k=0.67$, and $k=0.5$. The scaling of the velocity axis is not uniform. As the travel times of the capillary bridge and the capillary waves depend on droplet geometries, the stages of coalescence-induced droplet jumping are shifted by droplet size ratio. Detachment and jumping velocity ${\overline{u}}_{\mathrm{j}}^{*}$ is defined by zero contact area $(\times )$.

Figure 10

Dimensionless jumping velocities as function of Ohnesorge number with contact angle $\theta ={170}^{\circ }$ and droplet size ratio $k=1$ from this work (o) in comparison with data from condensation experiments $(•)$ from Enright et al. for the “two-camera arrangement” [18], conducted on a superhydrophobic surface.

Figure 11

Dimensionless jumping velocities (left) and departure angles with respect to the solid surface (right) as a function of Ohnesorge number for different droplet size ratios with $k=1$ (o), $k=0.67$ $\left(◊\right)$, $k=0.5$ (x), and $k=0.33$ $(▹)$ for contact angle of $\theta ={155}^{\circ }$ and $\theta ={170}^{\circ }$.

Figure 12

Temporal evolution of the translational kinetic energy for a contact angle of $\theta ={170}^{\circ }$ (left) and a contact angle of $\theta ={155}^{\circ }$ (right) with a constant Ohnesorge number of $\text{Oh}=0.001$. The gray areas indicate the energy of adhesion needed to be overcome for droplet jumping.

Figure 13

Left: Ratio between translational and oscillatory energy at the point in time of maximal translational energy depending on droplet size, contact angle, and Ohesorge number. Right: Comparison between the adhesion energy of a static pending drop, the maximal translational energy, and the reduction in translational energy due to droplet detachment.

Figure 14

Regime diagram, differentiating between lift-off and no lift-off regimes on the basis of Ohnesorge number and droplet size ratio for two wetting conditions with contact angle $\theta ={170}^{\circ }$ and $\theta ={155}^{\circ }$, approximated with $\text{Oh}=0.001$.