Coalescence of sessile microdroplets subject to a wettability gradient on a solid surface

While there are intensive studies on the coalescence of sessile macroscale droplets, there is little study on the coalescence of sessile microdroplets. In this paper, the coalescence process of two sessile microdroplets is studied by using a many-body dissipative particle dynamics numerical method. A comprehensive parametric study is conducted to investigate the effects on the coalescence process from the wettability gradient, hydrophilicity of the solid surface, and symmetric or asymmetric configurations. A water bridge is formed after two microdroplets contact. The temporal evolution of the coalescence process is characterized by the water bridge's radii parallel to the solid surface $\left(W_m\right)$ and perpendicular to the solid surface $\left(H_m\right)$. It is found that the changes of both $H_m$ and $W_m$ with time follow a power law; i.e., $H_m={\beta }_1{\tau }^{\beta }$ and $W_m={\alpha }_1{\tau }^{\alpha }$. The growth of $H_m$ and $W_m$ depends on the hydrophilicity of the substrate. $W_m$ grows faster than $H_m$ on a hydrophilic surface, and $H_m$ grows faster than $W_m$ on a hydrophobic surface. This is due to the strong competition between capillary forces induced by the water-bridge curvature and the solid substrate hydrophobicity. Also, flow structure analysis shows that regardless of the coalescence type once the liquid bridge is formed the liquid flow direction inside the capillary bridge is to expand the bridge radius. Finally, we do not observe oscillation of the merged droplet during the coalescence process, possibly due to the significant effects of the viscous forces.

Figure 1

Schematic of the initial configuration for 3D computations.

Figure 2

Schematic of top and side views of two coalesced sessile drops. $W_m$ and $H_m$ are shown in top and side views, respectively.

Figure 3

Snapshots of the interfacial development and evolution of the coalescence process between a pair of identical sessile microdroplets. Note that time sequence is from (a) to (f).

Figure 4

Trimmed 3D flow streamlines inside the merged droplet at different arbitrary times during the symmetric coalescence process. Streamlines are released from arbitrarily selected $x$ planes.

Figure 5

Flow velocity vectors on the three different $x,y$, and $z$ planes inside the merged droplet with an elongated ellipsoid shape (beginning of the late-stage coalescence).

Figure 6

Variation of $W_m$ and $H_m$ with time in the early stage of coalescence for different sets of $\left({\theta }_0,{\theta }_f\right)$. T-100-92-Hm and T-100-92-Wm are for $\left({100}^{\circ },{92}^{\circ }\right)$, T-90-82-Hm and T-90-82-Wm are for $\left({90}^{\circ },{82}^{\circ }\right)$, and T-70-62-Hm and T-70-62-Wm are for $\left({70}^{\circ },{62}^{\circ }\right)$. Note that both the $x$ and the $y$ axes are in ${\text{log}}_{10}$ scale.

Figure 7

Snapshots of the interfacial development and evolution of the coalescence process between a pair of sessile microdroplets of different sizes. Note that time sequence is from (a) to (f).

Figure 8

Flow velocity vectors on the $x,y$, and $z$ planes for Fig. 7.

Figure 9

Variation of $W_m$ and $H_m$ with time for different values of $R_{r2}$ in the early stage of the coalescence process. Predicted lines for $W_m$ and $H_m$ by Eggers's model [Eq. (13)] also have been plotted. Note that both the $x$ and the $y$ axes are in ${\text{log}}_{10}$ scale.

Figure 10

Snapshots of the interfacial development and evolution of the coalescence process between a pair of sessile microdroplets. The left microdroplet is moving and the right microdroplet is stationary. Note that time sequence is from (a) to (f).

Figure 11

Trimmed 3D flow streamlines inside the merged droplet at different arbitrary times during the asymmetric coalescence process. Streamlines are released from arbitrarily selected $x$ planes.

Figure 12

Flow velocity vectors on $x,y$, and $z$ planes when the merged droplet reaches to an elongated ellipsoid shape in an asymmetric coalescence process.

Figure 13

Variation of $W_m$ and $H_m$ with time when one a sessile droplet moves towards a stationary droplet. A constant wetting gradient has been applied on the solid surfaces. However, the surface has different philicity properties. Note that the results are for the early stage of the coalescence process. Note that both the $x$ and the $y$ axes are in ${\text{log}}_{10}$ scale.

Figure 14

Comparison of $W_m$ and $H_m$ trends in time between symmetric and asymmetric coalescence. T-90-82-Wm, T-90-82-Hm, T-100-82-Wm, and T-100-82-Hm are for the case that both droplets move on the solid surface (symmetric coalescence). S-90-82-Wm, S-90-82-Hm, S-100-82-Wm, and S-100-82-Hm are for the cases that only one of the droplets moves while the other one is stationary (asymmetric coalescence). Note that the results are for the early stage of the coalescence process, and both the $x$ and the $y$ axes are in ${\text{log}}_{10}$ scale.