Symmetric and Asymmetric Coalescence of Drops on a Substrate

The coalescence of viscous drops on a substrate is studied experimentally and theoretically. We consider cases where the drops can have different contact angles, leading to a very asymmetric coalescence process. Side view experiments reveal that the “bridge” connecting the drops evolves with self-similar dynamics, providing a new perspective on the coalescence of sessile drops. We show that the universal shape of the bridge is accurately described by similarity solutions of the one-dimensional lubrication equation. Our theory predicts that, once the drops are connected on a microscopic scale, the bridge grows linearly in time with a strong dependence on the contact angles. Without any adjustable parameters, we find quantitative agreement with all experiments.

Figure 1

(a) Schematic of two coalescing viscous drops on a substrate, viewed from the side. The minimum height $h_0(t)$ characterizes the bridge height. The left-right contact angles ${\theta }_L$ and ${\theta }_R$ can be different at the moment of contact. The horizontal displacement $x_0$ results from the asymmetry in the contact angles. (b), (c) Typical frames of the experiments are shown for asymmetric contact angles (b) and symmetric contact angles (c). (d) Schematic of two coalescing viscous drops, viewed from the top. The lubrication model assumes that the flow is parallel to the solid wall and predominantly in the $x$ direction (arrows) [11]. The width of the bridge is $r(t)$.

Figure 2

Symmetric coalescence. (a) Height of the bridge $h_0$ as a function of time after contact $t$, for drops with ${\theta }_L={\theta }_R=22°$ ($\eta =12.2\mathrm{Pa}·\mathrm{s}$). Experiments are shown in red (●), the solid line is the prediction by Eqs. (2, 3). The dashed line represents the lower limit for spatial resolution. (b) Rescaled experimental profiles at different times, $\mathcal{H}=h(x,t)/h_0(t)$ versus $\xi =x\theta /h_0(t)$. The collapse reveals self-similar dynamics at the early stage of coalescence, in agreement with the similarity solution (solid line).

Figure 3

Asymmetric coalescence. (a) Horizontal and vertical position of the meniscus bridge, $x_0(t)$ and $h_0(t)$, for asymmetric drops (${\theta }_L=46°$, ${\theta }_R=13°$, viscosity $\eta =12.2\mathrm{Pa}·\mathrm{s}$). Blue (●) and red (▪) markers are experimental data for $x_0$ and $h_0$, respectively. Solid and dashed lines are the predictions from the similarity solutions. (b) Rescaled experimental profiles at different times, $\mathcal{H}=h(x,t)/h_0(t)$ versus $\xi =x_0{\theta }_L/h_0(t)$. The collapse reveals self-similar dynamics at the early stage of coalescence. The solid line is the similarity solution predicted by our analysis.

Figure 4

Contact angle dependence of coalescence velocity. (a) Dimensionless vertical speed, $V=3v\eta /(\gamma {\theta }_L^4)$, as a function of ${\theta }_R/{\theta }_L$. (b) Dimensionless horizontal speed, $U=3u\eta /(\gamma {\theta }_L^3)$, as a function of ${\theta }_R/{\theta }_L$. The horizontal speed vanishes for the symmetric case ${\theta }_R/{\theta }_L=1$, and displays a maximum around ${\theta }_L/{\theta }_R\sim 0.5$. Closed symbols: 75 experiments on completely wetting substrate. Open symbols: drops on partially wetting substrate (${\theta }_{\mathrm{eq}}=55°$). Solid lines: similarity solutions.