Dynamics of droplets under electrowetting effect with voltages exceeding the contact angle saturation threshold

To induce capillary waves on the surface of a droplet, the contact-line velocity has to overcome the inertial-capillary velocity $(\sigma/\rho r_{0})^{1/2}$, formulated using the stabilising velocity of a deformed droplet’s surface to minimise its curvature (Taylor, 1959). In normal wetting phenomena, the maximum velocity of the contact line that can be theoretically achieved is $\sigma/\lambda$. As a result, the condition enabling capillary waves becomes $\sigma/\lambda\geq(\sigma/\rho r_{0})^{1/2}$, or $\xi=\lambda/(\sigma\rho r_{0})^{1/2}\leq 1$. Here, the so-called damping coefficient $\xi$ measures the viscous effects in both the droplet and the outer oil against inertia for capillary flows. We note that $\xi$ is defined similarly to the Ohnesorge number ${\rm Oh}=\mu/(\sigma\rho r_{0})^{1/2}$, where $\mu$ is the liquid viscosity. In wetting phenomena driven by the electrowetting effect, $v_{\rm i}$ is determined by Eq. 2 and Eq. 3, resulting in another condition for capillary waves to occur: $v_{\rm i}\geq(\sigma/\rho r_{0})^{1/2}$. Using the experimental parameters in our experiment, the condition for the contact-line velocity to generate capillary waves is $v_{\rm i}\geq(\sigma/\rho r_{0})^{1/2}=0.27\,{\rm ms^{-1}}$, consistent with our experimental results shown in Fig. 2c, i.e., capillary waves are observable when $v_{\rm i}\geq 0.3\,{\rm ms^{-1}}$. We note that if we define a Weber-like number ${\rm We}=v_{\rm i}^{2}\rho r_{0}/\sigma$, the condition for capillary waves to occur then takes the form ${\rm We}\geq 1$.

To quantify the amplitude and phase of the capillary waves on a droplet having spherical-cap radius $r_{\rm s}$ and contact angle $\theta_{0}$, we use a polar coordinate system $(r,\phi)$ with origin $(r=0,\phi=0)$ at a distance $r_{s}$ below the droplet’s apex and the polar axis vertically upward (Fig. 3a, inset). For droplets in our experiment, $\theta_{0}\approx\pi$ and $r_{\rm s}\approx r_{0}$. In this coordinate system, the droplet-oil profile, which is also the capillary waveform, is $(r_{\rm d},\theta_{0}-\phi)$, where $\theta_{0}$ is the wave’s position of the contact line at $t=0$. The displacement of the capillary wave is therefore $a=r_{\rm d}-r_{s}$, measured by the deviation of the droplet-oil interface from the droplet’s initial shape ($r_{d}=r_{s}$). In Fig. 3a, we show $a$ vs. $\theta_{0}-\phi$ at several instances of time $t$ when $U=150\,$V is applied. When $t$ increases, the waveform moves from left to right indicating that the wave propagates further from the contact line and towards the apex of the droplet ($\phi=0$). Furthermore, we observe that while the wave’s propagating velocity reduces with time, its amplitude increases suggesting that the wave’s kinetic energy, originated from the fast initial motion of the contact line, is progressively converted to surface energy during its propagation.

In Fig. 3b, we show a plot of $a$ vs. $\theta_{0}-\phi$ at $t=\tau$ for different values of $U$ varying from $100$ V to $190$ V. Here, $\tau=(\rho r_{0}^{3}/\sigma)^{1/2}$ is the inertial-capillary time. We observe that the displacement at point M significantly increases with $U$, while the wave’s position of the point M does not depend on $U$. This observation is examined more quantitatively in Fig. 4a and b where we respectively show $a_{\rm(M)}$ and $\theta_{0}-\phi_{\rm(M)}$ vs. $t/\tau$ for all the values of $U$ ranging from 90 V to 190 V. Here, $a_{\rm(M)}$ and $\theta_{0}-\phi_{\rm(M)}$ denote the displacement and the position of the waves at point M, respectively. On the one hand, we observe that the slopes of $a_{\rm(M)}$ vs. $t/\tau$ increase with $U$ showing that the dependence of $a_{\rm(M)}$ on $t/\tau$ is more strongly at higher applied voltages. On the other hand, $\theta_{0}-\phi_{\rm(M)}$ is independent from $U$ at any values of $t/\tau$ indicating that the propagating velocity only depends on the hydrodynamical properties of the droplets, not the contact-line velocity.

The scaling laws for the displacement and the position of capillary waves on the surface of a spreading droplet were proposed by Ding et al. (2012) by adapting Keller & Miksis (1983)’s self-similarity theory for surface-tension driven flows:

where $A$ and $\Phi$ denote the dimensionless displacement and the dimensionless position of the waves, respectively. In the insets of Fig. 4a and b, we show the log-log plots of $a_{\rm(M)}$ vs. $t/\tau$ and $\theta_{0}-\phi_{\rm(M)}$ vs. $t/\tau$, respectively. We observe that $a_{\rm(M)}\sim(t/\tau)^{2/3}$ and $\theta_{0}-\phi_{\rm(M)}\sim(t/\tau)^{2/3}$, implying that Eq. 4 is also applicable in our system in which the capillary waves are generated by electrowetting actuation. To further test Eq. 4, in Fig. 4c, we show the dimensionless waveforms, i.e., $A=ar_{0}^{-1}(t/\tau)^{-2/3}$ vs. $\Phi=(\theta_{0}-\phi)(t/\tau)^{-2/3}$ when $t/\tau$ varies from $0.05$ to $1.4$. We observe that all the waveforms for $0.2\leq t/\tau\leq 1$ collapse onto a single one confirming the self-similar behaviour of the capillary waves in this time range. However, we notice that capillary waves are not self-similar for $t/\tau<0.2$ and $t/\tau>1$ (see the dashed curves in Fig. 4c). At $t/\tau<0.2$, i.e., a short time after wetting is initiated, the waves are local at the contact line region (Cox, 1986), whereas, at $t/\tau>1$ the capillary waves reach the apex of the droplet and start interacting with the waves coming from the other side of the droplet. As a result, at both extremes the capillary waves on the droplet’s surface are more complicated and cannot be described by a single self-similar scaling law.

With the self-similar behaviour of the capillary waves, it is possible to reduce the wave’s profile $a(\phi,t)$ at any time $t$ by a time-invariant dimensionless profile $A(\Phi)$. The first solution for $A(\Phi)$ was proposed by Keller & Miksis (1983) who modified the calculation of Jeffreys & Jeffreys (1999) for inviscid, self-similar capillary-driven flows in a fluid wedge. Later, Billingham (1999) theoretically incorporated viscosity effects using the capillary-viscosity velocity. However, both Keller & Miksis (1983)’s and Billingham (1999)’s solutions fail to fit our experimental data. While Keller & Miksis (1983)’s lacks of the viscous effects, the model of Billingham (1999) is only applicable at a very early time after the wave is generated, and in a very small region close to the wave’s source, e.g., the time and length scales respectively are $\approx 0.2\,$ns and $\approx 10\,$nm for water.

We seek an alternative expression of $A(\Phi)$ taking into account the viscosity effect in our system. The self-similar profiles $A(\Phi)$ (see the profiles of $0.2\leq t/\tau\leq 1$ in Fig. 4c) can be approximated by spatially decaying waves driven by interfacial tension and resisted by inertia and viscous forces. For such waves, we expect that $A(\Phi)$ is the solution of an one-dimensional time-invariant wave equation with a spatial damping coefficient $\xi=\lambda/(\sigma\rho r_{0})^{1/2}$:

where $k=2\pi/\lambda_{\rm\Phi}=2\pi$ is the wave number, $\lambda_{\rm\Phi}=1$ is the dimensionless wavelength. We note that the damping coefficient $\xi$ does not depend on the strength of the driving force at the contact line. Our discussion is also limited to the case that $\xi<1$ for the possibility of capillary wave generation as discussed in Section. 3.2.1.

If we denote $\Phi_{0}\geq 0$ the dimensionless position of the contact line, the profile $A$ in Eq. 5 is required to satisfy the following boundary conditions:

where $a_{\rm ct}=r_{\rm ct}(\sin\phi_{\rm ct})^{-1}-r_{\rm 0}$ is the contact line’s displacement, $\phi_{\rm ct}$ its wave’s position, $r_{\rm ct}$ the contact radius, $\dot{a}_{\rm ct}$ denotes the time derivatives of $a_{\rm ct}$ and $\phi_{\rm ct}$ the time derivatives of $\dot{\phi}_{\rm ct}$. As we consider the capillary waves generated short-time after the electrowetting effect is activated, we can approximate $a_{\rm ct}\approx 0$, $\phi_{\rm ct}\approx\theta_{0}$, $a_{\rm ct}t^{-1}\approx\dot{a}_{\rm ct}\approx v_{\rm i}\sin\theta_{0}$ and $-(\theta_{0}-\phi_{\rm ct})t^{-1}\approx\dot{\phi}_{\rm ct}=-k_{\phi}r_{0}^{-1}(\sigma/\rho r_{0})^{1/2}$, where $k_{\phi}$ is a positive constant. As a result, we obtain

The general solution of Eq. 5 is

Here, $k_{\rm d}=2\pi(1-\xi^{2})^{1/2}$ is the attenuated wave number due to the damping coefficient $\xi$. Applying Eq. 8 and Eq. 9 to Eq. 10, we obtain $\alpha=0$ and $\beta={\rm We}^{1/2}\sin\theta_{0}(k_{\rm d}k_{\phi})^{-1}$. Consequently, we obtain an explicit expression of $A(\Phi)$:

To verify Eq. 11, we use the least-mean-square method to fit the experimental data of $A$ vs. $\Phi$ to Eq. 11 with the fitting parameters $\Phi_{0}$ and $k_{\phi}$. Examples of the fittings at $t/\tau=0.25$ and for different applied voltages $U$ are shown in Fig. 5a. Similar results obtained for different values of $t/\tau$ varying from $0.2$ to $1$ indicate a good agreement between Eq. 11 and the experimental data. In Fig. 5b, we show the values of $k_{\phi}$ (left axis) and $\Phi_{0}$ (right axis) that both can be approximated as constants, i.e., $k_{\phi}=2.4\pm 0.7$ and $\Phi_{0}=1.32\pm 0.06$, for $U$ varying from 90 V to 190 V. All solutions for $90\leq U\leq 190$ V are plotted in Fig. 5c.