Autonomous transport and splitting of a droplet on an open surface

Pumpless transport of droplets on open surfaces has gained significant attention because of its applications starting from vapor condensation to Lab-on-a-Chip systems. Mixing two droplets on open surfaces can be carried out quickly by using wettability patterning. However, it is quite challenging to split a droplet in the absence of external stimuli because of the interfacial energy of the droplet. Here, we demonstrate a standalone power-free technique for transport and splitting of droplets on open surfaces using continuous wettability gradients. A droplet moves continuously from a low to a high wettability region on the wettability-gradient surface. A Y-shaped wettability-gradient track – laid on a superhydrophobic background – is used to investigate the dynamics of the splitting process. A three-dimensional phase-field Cahn-Hilliard model for interfaces and the Navier-Stokes equations for transport are employed and solved numerically using the finite element method. Numerical results are used to decipher the motion and splitting of droplet at the Y junction using the principle of energy conservation. It is observed that droplet splitting depends on the configuration of the Y junction; droplets split faster for the superhydrophobic wedge angle of ${90}^{\circ }$ and the splitting ratio (ratio of the sizes of daughter droplets) depends on the widths of the Y branches. A critical branch-width ratio ($\frac{w_2}{w_1}=0.79$) is identified below which the droplet does not split and moves towards the branch of higher width and settles there. The present study provides the required theoretical underpinnings to achieve autonomous transport and splitting of droplets on open surfaces, which has clear potential for applications in Lab-on-a-Chip devices.

Figure 1

A droplet of diameter $d_0=0.5\mathrm{mm}$ is placed on a Y-shaped wettability-gradient track laid on a superhydrophobic background. The superhydrophobic background of the Y track has a contact angle of ${170}^{\circ }$. The initial position of the droplet is at a distance $d_0$ from the left. The base track of the Y-shaped wettability-gradient track has width $w$ and is divided into two branches by a superhydrophobic wedge of wedge angle $\alpha$. The branches of the Y track have widths $w_1$ and $w_2$. The length of the branches are $L_2=1.5\mathrm{mm}$ and the length of the base track of Y is $L_1=2\mathrm{mm}$. The wettability gradient is linear and the variation of contact angle is from ${\theta }^H={165}^{\circ }$ to ${\theta }^M={25}^{\circ }$ for length $L_1$ and from ${\theta }^M={25}^{\circ }$ to ${\theta }^L=5^{\circ }$ for length $L_2$. At the bifurcation region, the contact angle is close to ${25}^{\circ }$, which is represented by yellow color at the base track and the light green at the branches. The transition from yellow to light green is just to distinguish the base track and the branches.

Figure 2

The variation of the nondimensional wetted area of the droplet $A^{*}$ with nondimensional time $t^{*}$ plotted for different element sizes $h_1$ to get the independent grid size. The dotted line represents the independent grid.

Figure 3

The variation of the capillary number of the droplet Ca while it transports on a wettability-gradient surface. The red square and green triangle correspond to the Ca of Ahmadlouydarab and Feng [43]. The blue and black lines represent the present simulated Ca.

Figure 4

Droplet splitting on a Y-shaped bifurcated wettability-gradient track with the branch width ratio $w_2/w_1=1$ and the superhydrophobic wedge angle $\alpha ={90}^{\circ }$.

Figure 5

(a) Variation of the nondimensional front and back position of the droplet ${x_f}^{*}$ and ${x_b}^{*}$, respectively, with $t^{*}$. The droplet shape evolution with closeup looks (side view, top view, and sectional profile) of droplet spreading: (b) position A (end of Stage I), (c) position B (initial phase of Stage II), and (d) position C (end of Stage II) for $w_2/w_1=1$ and the wedge angle $\alpha ={90}^{\circ }$.

Figure 6

Velocity contour with overlaying vector plot inside the droplet during transport on a Y-shaped bifurcated wettability-gradient track having a branch-width ratio $w_2/w_1=1$ and the superhydrophobic wedge angle $\alpha ={90}^{\circ }$. The reference point $(X,Y,Z)=(0,0,0)$ is the bifurcation point of the Y track. The contours and velocity vectors are plotted on three different cross sections at (a) $t^{*}=112$, (b) $t^{*}=150$, (c) $t^{*}=238$, and (d) $t^{*}=634$. The velocity vectors represent the direction of fluid velocity is shown by black arrows. The droplet interface is shown by the solid blue line.

Figure 7

(a) Variation of the nondimensional viscous and contact line dissipation rates of the droplet ${\dot{E_v}}^{*}$ and ${\dot{E_{cl}}}^{*}$, respectively, with $t^{*}$. (b) Variation of the nondimensional rate of change of surface energy of the droplet ${\dot{E_s}}^{*}$ and the kinetic energy of the droplet ${\dot{E_k}}^{*}$ with $t^{*}$. (c) Variation of $({\dot{E_s}}^{*}+{\dot{E_k}}^{*})$ and $({\dot{E_v}}^{*}+{\dot{E_{cl}}}^{*})$ with $t^{*}$.

Figure 8

(a) Variation of the nondimensional wetted area of the droplet $A^{*}$ and (b) variation of the nondimensional height of the droplet $h^{*}$ are plotted as a function of nondimensional time $t^{*}$ while the droplet transport and split on the Y track with superhydrophobic wedge angle $\alpha ={90}^{\circ }$ at the bifurcation point. (c) Nondimensional splitting time ${t_s}^{*}$ taken by the droplet to split into two parts is plotted for different superhydrophobic wedge angles $\alpha$.

Figure 9

The three distinct regimes during the droplet-splitting stage (Stage II). In Regime 1, ${p_1}^{*}=0.01t^{*}-0.92$ with $R^2=0.98$, whereas in Regime 3, ${p_1}^{*}=0.78{t^{*}}^{1/7}-1$ with $R^2=0.98$.

Figure 10

The splitting volume ratios of the droplet $v_2/v_1$ is plotted with the branch-width ratios of the Y track $w_2/w_1$ for the superhydrophobic wedge angle $\alpha ={90}^{\circ }$.

Figure 11

Droplet dynamics for width ratio $w_2/w_1=0.714$ and the superhydrophobic wedge angle $\alpha ={90}^{\circ }$.

Figure 12

Schematic of forces involved during the splitting of the droplet for $\frac{w_2}{w_1}=1$ and (a) $\alpha ={90}^{\circ }$, (b) $\alpha ={30}^{\circ }$. Red line represents the droplet contact line. The dotted line corresponds the attachment area between two parts of the droplet.

Figure 13

(a) Schematic of droplet initial and final shapes of droplet-splitting process for $\frac{w_2}{w_1}=1$ and $\alpha ={90}^{\circ }$. (b) Corresponding cross section 2-2 of the branch fluid. (c) Corresponding section 1-1 of the branch fluid.