Coalescence of Spreading Droplets on a Wettable Substrate

We investigate experimentally and theoretically the coalescence dynamics of two spreading droplets on a highly wettable substrate. Upon contact, surface tension drives a rapid motion perpendicular to the line of centers that joins the drops and lowers the total surface area. We find that the width of the growing meniscus bridge between the two droplets exhibits power-law behavior, growing at early times as $t^{1/2}$. Moreover, the growth rate is highly sensitive to both the radii and heights of the droplets at contact, scaling as $h_o^{3/2}/R_o$. This size dependence differs significantly from the behavior of freely suspended droplets, in which the coalescence growth rate depends only weakly on the droplet size. We demonstrate that the scaling behavior is consistent with a model in which the growth of the meniscus bridge is governed by the viscously hindered flux from the droplets.

Figure 1

Sketch of two thin droplets coalescing on a flat substrate. (a) Elevation view. (b) Plan view. Dashed lines indicate the control volume around the meniscus bridge.

Figure 2

Optical microscopy images of two silicone oil droplets ($\eta =1000\mathrm{cS}$) spreading and coalescing on a flat polystyrene substrate. Dark regions are oil, light regions are background. Note that the droplet centers are outside the field of view. In later images, the light “halo“ in the center of the oil is an artifact due to the decreased curvature with respect to the incident light. Scale bar in (a) is $500\mu \mathrm{m}$.

Figure 3

(a) Width of the meniscus bridge versus time for three different viscosities. Symbols: ○, 100; □, 350; ▵, 1000 cS. Data represent 19 separate experiments; $t_o$ is the time of initial contact. (b) Width of the meniscus bridge in scaled coordinates, using all of the data presented in (a). Inset: logarithmic scale, $\mathrm{\text{fit slope}}=0.53$.

Figure 4

Numerical calculations of the meniscus width vs dimensionless time, ${\tau }_{*}=\gamma h_{*}^3(t-t_o)/\mu R_{*}^3$, for two different precursor film thicknesses. Inset: numerical calculations of the depth-averaged velocities versus time. Solid line: the maximum $x$ velocity calculated on the $x$ axis (the line joining the drop centers). Dashed line: the maximum $y$ velocity on the axis at the point of contact ($x=R_o$).