Contact line friction of electrowetting actuated viscous droplets

We examine the contact line friction coefficient of viscous droplets spreading and retracting on solid surfaces immersed in ambient oil. By using the electrowetting effect, we generate a surface tension imbalance to drive the spreading and the retracting motion of the three-phase contact line (TCL). We show that neither the driving force intensity nor TCL direction significantly influences the friction coefficient. Instead, the friction coefficient depends equivalently on the viscosity of liquid droplets and the surrounding oil. We derive and experimentally verify a transient timescale that can be used to characterize both the spreading and retracting dynamics.

Figure 1

(a) When a potential difference between a droplet and a substrate is applied, the droplet spreads from an initial state with contact angle ${\theta }_0$ to a new equilibrium state with contact angle ${\theta }_{\text{e}}$. At any transient state with contact angle ${\theta }_{\mathrm{t}}$, the driving force per unit length at the contact line is calculated as $F=\sigma (cos{\theta }_{\mathrm{e}}-cos{\theta }_{\mathrm{t}})$. The positive direction of the contact line motion is indicated by the $+$ sign. (b) When the voltage is released, the droplet retracts freely and the contact angle changes from ${\theta }_{\mathrm{e}}$ to ${\theta }_0$, assuming sufficiently small hysteresis. The driving force is now $F=\sigma (cos{\theta }_0-cos{\theta }_{\mathrm{t}})$. (c) Typical plots of the contact radius $r$ versus time $t$ (left axis) and contact angle ${\theta }_{\mathrm{t}}$ versus time $t$ (right axis) for a droplet of initial radius $R=0.5\mathrm{mm}$ when a voltage $V=100\mathrm{V}$ is applied for 50 ms and then released. Inset: schematic of the experimental setup. The liquid used to generate droplets is water-glycerol solution having viscosity $\mu =17.6\mathrm{mPas}$. The outer phase is silicone oil having viscosity ${\mu }_{\mathrm{o}}=4.6\mathrm{mPas}$.

Figure 2

Plots of contact line velocity $u$ versus time $t$ (left axis) and contact line force, $F$ versus time $t$ (right axis) for the spreading stage (a) and the retracting stage (b). Insets: data in the main plots are replotted to show the relation between $F$ and $u$. The results were obtained by applying 80 V to a 1 mm diameter droplets ($\mu =8.2$ $\mathrm{mPa}\mathrm{s}$) immersed in silicone oil (${\mu }_{\mathrm{o}}=4.6$ $\mathrm{mPa}\mathrm{s}$). The best fits to the data in the insets give $\lambda =0.19\pm 0.01$ $\mathrm{Pa}\mathrm{s}$ for the spreading stage and $\lambda =0.17\pm 0.02$ $\mathrm{Pa}\mathrm{s}$ for the retracting stage.

Figure 3

Driving force $F$ at the contact line versus contact line velocity $u$ for different applied voltages ($20\mathrm{V}\le \mathrm{V}\le 100\mathrm{V}$). Inset: zoom in of data near the origin. The working fluid's viscosity is $\mu =17.6\mathrm{mPa}\mathrm{s}$, while the oil viscosity is ${\mu }_{\mathrm{o}}=4.6\mathrm{mPa}\mathrm{s}$. The radius of droplets is 0.5 mm. The friction coefficient $\lambda =0.28\pm 0.05\mathrm{Pa}\mathrm{s}$ is calculated from the slope of the solid line.

Figure 4

(a) Log-log plot of $\lambda$ versus viscosity $\mu$ when ${\mu }_{\mathrm{o}}$ is fixed at $4.6\mathrm{mPa}\mathrm{s}$. The solid line represents the scaling law $\lambda \sim {\mu }^{1/2}$. (b) Log-log plot of $\lambda$ versus ${\mu }_{\mathrm{o}}$ when $\mu$ is fixed at $17.6\mathrm{mPa}\mathrm{s}$. The solid line represents the scaling law $\lambda \sim {\mu }_{\mathrm{o}}^{1/2}$. (c) Log-log plot of $\lambda$ versus $\mu {\mu }_{\mathrm{o}}$ for our experimental data shown in (a) and (b). The solid line represents the scaling law $\lambda \sim {\left(\mu {\mu }_{\mathrm{o}}\right)}^{1/2}$. Inset: Linear plot of $\lambda$ versus ${\left(\mu {\mu }_{\mathrm{o}}\right)}^{1/2}$. The solid line represents the relation $\lambda =C{\left(\mu {\mu }_{\mathrm{o}}\right)}^{1/2}$, where $C=26.24\pm 4.6$ is the fitting parameter.

Figure 5

Linear plots of the collapsed data of experiments with different droplet's viscosity $\mu$ for spreading stage (a) and retracting stage (b). The raw data of each plot is shown by the corresponding inset.

Figure 6

Linear plots of the collapsed data of experiments with different oil's viscosity ${\mu }_{\text{o}}$ for spreading stage (a) and retracting stage (b). The raw data of each plot is shown by the corresponding inset.