Electrowetting of a leaky dielectric droplet under a time-periodic electric field

The wetting and contact line dynamics of a leaky dielectric sessile droplet under an alternating (ac) electrostatic field applied in the vertical direction is investigated. A thin precursor film-based reduced-order model using the weighted residual integral boundary layer technique is developed. The limiting cases of perfect conducting and perfect dielectric droplets are also considered. It is shown that the droplet oscillates with a frequency twice that of the forcing potential due to the quadratic dependence of the Maxwell stress on the applied ac electric field. These oscillations take place about an equilibrium configuration, which can be achieved with a constant (dc) electric potential equivalent to the root-mean-square potential of the applied ac field. It is also shown that the contact line motion increases monotonically with the amplitude of the ac electric forcing. A significant increase in ac field leads to spiking and the interface ruptures at the top electrode. Depending on the static contact angle, the droplet deformation can become nonmonotonic as the applied frequency of the ac electric field increases. This behavior is attributed to the competition between the timescale of forcing and the timescale of the response as affected by the drop's wettability. The role of conductivity ratio, permittivity ratio, and different waveforms of ac forcing are also investigated.

Figure 1

(a) Schematic of a sessile droplet subjected to a periodic electrostatic forcing. The droplet is hydrodynamically active, while the ambient fluid is assumed to be hydrodynamically passive. (b) Different time-dependent waveforms of electric forcing $\left[\mathrm{\Psi }\right(t\left)\right]$ are considered in this study, namely sinusoidal (black solid line), triangular (red dashed line), and square (blue dash-dotted line).

Figure 2

The temporal variations of the dimensionless droplet height, $h_c$, subject to (a) dc electric forcing using different number of grid points with $E=9.8$ and (b) ac electric forcing using different number of grid points with $E=19.6$ and $f=6.25\times {10}^{-4}$. The rest of the dimensionless parameters are $S=26.8$, $G={10}^{-3}$, and $\mathrm{Re}=36$.

Figure 3

The temporal variation of the dimensionless droplet height, $h_c$, for different values of the electric potential, $E$, with $f=6.25\times {10}^{-4}$. The rest of the dimensionless parameters are $S=26.8$, $G={10}^{-3}$, and $\mathrm{Re}=36$. The droplet is a perfect conductor and ambient fluid is a perfect dielectric.

Figure 4

[(a) and (b)] The temporal variation of the dimensionless droplet height $h_c$ for different values of electrical field frequency: (a) the low-frequency range and (b) the high-frequency range. The variation of $h_c$ for the equivalent dc electric field $E/2$ is shown by the red dotted line in panels (a) and (b). The variation of (c) the droplet deformation, $D_h$, and (d) the right contact line motion, $D_{cR}$, with frequency. Here a perfect conducting sessile droplet in a perfect dielectric ambient medium is considered with $E=19.6$. The rest of the dimensionless parameters are $S=26.8$, $G={10}^{-3}$, and $\mathrm{Re}=36$.

Figure 5

The variations of (a) the deformation of the droplet, $D_h$ and (b) the right contact line motion, $D_{cR}$, with frequency $f$ for different values of the wetting parameter, $S$. (c) Panel (a) is replotted with the parameter, $f/S$, to depict the competition between forcing and wetting time scales in the problem. Here a perfect conducting sessile droplet in a perfect dielectric ambient medium is considered with $E=19.6$. The rest of the dimensionless parameters are $G={10}^{-3}$ and $\mathrm{Re}=36$.

Figure 6

The variations of the dimensionless droplet height, $h_c$ with dimensionless time for different values of the electric potential, $E$, with $f=6.25\times {10}^{-4}$, $\varepsilon =81$. Both the droplet and ambient fluid are perfect dielectric. The rest of the dimensionless parameters are $S=26.8$, $G={10}^{-3}$, and $\mathrm{Re}=36$.

Figure 7

[(a) and (b)] The temporal variations of the dimensionless droplet height, $h_c$, for different values of electrical field frequency: (a) the low-frequency range and (b) the high-frequency range. The $h_c$ for the equivalent dc electric field is shown by the red dotted line in panels (a) and (b). The variation of (c) the droplet deformation, $D_h$, and (d) the right contact line motion, $D_{cR}$, with $f$. Here, a perfect dielectric sessile droplet with $S=26.8$ in a perfect dielectric ambient medium is considered with $E=19.6$. The rest of the dimensionless parameters are $G={10}^{-3}$ and $\mathrm{Re}=36$.

Figure 8

The variations of (a) the droplet deformation, $D_h$, and (b) the right contact line motion, $D_{cR}$, with the permittivity ratio $\left(\varepsilon \right)$ for a perfect dielectric droplet. The rest of the parameters are $E=19.6$, $f=6.25\times {10}^{-4}$, $G={10}^{-3}$, $\mathrm{Re}=36$, and $S=26.8$.

Figure 9

Different regions in the electrical conductivity ratio $\left(\sigma \right)$ and permittivity ratio $\left(\varepsilon \right)$ space as shown in Ref. [37] for a planar interface between a pair of leaky dielectric fluids. The markers denote the parameter values used for nonlinear calculations.

Figure 10

Demonstration of the phase shift in the interfacial charge $\left(Q_{\max }\right)$ oscillations as one moves across the $\sigma =\varepsilon$ curve. The droplet and the ambient medium is leaky dielectric. The rest of the parameters are $E=4.9$, $f=2.5\times {10}^{-4}$, $G={10}^{-3}$, $\mathrm{Re}=36$, and $S=26.8$.

Figure 11

(a) Time sequence of the profile of the droplet, $h$, and (b) the interfacial charge, $Q$, of a leaky dielectric droplet under a sinusoidal electrical field. The rest of the parameters are $\varepsilon =9.45$, $\sigma =4$, $E=4.9$, $f=2.5\times {10}^{-4}$, $G={10}^{-3}$, $\mathrm{Re}=36$, and $S=26.8$. The image is not to scale and aspect ratio is chosen for visual clarity.

Figure 12

(a) Effect of permittivity ratio, $\varepsilon$, on the deformation of the droplet, $D_h$ ($\sigma =4$) and (b) effect of conductivity ratio, $\sigma$, on the deformation of the droplet, $D_h$ ($\varepsilon =1$) for $E=4.9$, $f=2.5\times {10}^{-4}$, $G={10}^{-3}$, $\mathrm{Re}=36$, and $S=26.8$. The droplet and the ambient medium is leaky dielectric.

Figure 13

Comparison of the droplet height oscillations, $h_c$, obtained using an ac electric forcing ($E=19.6$) with (a) $f=2.5\times {10}^{-4}$ and (b) $f=8.3$ with different waveforms. The droplet is perfect conductor. The sinusoidal waveform: black solid line, the triangular waveform: dashed black line, the square waveform: red dash-dotted line and the $E=19.6$ dc forcing: black solid line with circle symbols. The rest of the parameters are $G={10}^{-3}$, $\mathrm{Re}=36$, and $S=26.8$.