Dynamic response of a thin sessile drop of conductive liquid to an abruptly applied or removed electric field

We consider, both theoretically and experimentally, a thin sessile drop of conductive liquid that rests on the lower plate of a parallel-plate capacitor. We derive analytical expressions for both the initial deformation and the relaxation dynamics of the drop as the electric field is either abruptly applied or abruptly removed, as functions of the geometrical, electrical, and material parameters, and investigate the ranges of validity of these expressions by comparison with full numerical simulations. These expressions provide a reasonable description of the experimentally measured dynamic response of a drop of conductive ionic liquid 1-butyl-3-methyl imidazolium tetrafluoroborate.

Figure 1

The geometry of a sessile drop resting on the lower plate of a parallel-plate capacitor. (All labels are described in Sec. 2.) (a) Definition of the cylindrical polar coordinates $r$ and $z$ and the dependent variables used in the model. (b) Drop with no electric field applied. (c) Drop deformed by the application of a DC voltage across the capacitor.

Figure 2

Experimental images of the thin [Bmim][${\mathrm{BF}}_4$] drop with (a) no applied voltage and (b) an applied voltage of 865 V, also showing its reflection in the substrate on which it rests within the parallel-plate capacitor.

Figure 3

The nondimensional equilibrium drop apex height difference $\mathrm{\Delta }{H}_{\mathrm{s}}$ plotted as a function of the electric Bond number $B$, obtained experimentally (shown with stars) with error bars corresponding to a 95% confidence interval using one pixel as the standard error, using the approximate analytical solution valid for $B\ll 1$ and $G=0$ given by Eq. (49) (shown with a dotted line), obtained numerically from the model using the thin-film approximation (shown with a dashed line), and obtained numerically from the full model without using the thin-film approximation (shown with a solid line).

Figure 4

The experimentally measured nondimensional drop apex height difference $\mathrm{\Delta }H$ (shown with circles) plotted as a function of nondimensional time $t$ for voltage pulses with four different electric Bond numbers $B=0.86$, 1.08, 1.50, 2.73, but a constant nondimensional pulse duration of $T=1.82$, along with the corresponding numerical solution of the thin-film equation (37) (shown with solid lines). For clarity, the experimental data is plotted using average values over 500 experimental times steps, although all of the analysis of the experimental data uses the complete data set.

Figure 5

Plot of $\lambda$, the coefficient of $Bt$ in the short-time expansion of the nondimensional drop apex height difference $\mathrm{\Delta }H=\lambda Bt+O\left(t^2\right)$ during switch on from $B_0=0$ to $B_1=B$ at time $t=0$, as a function of the gravitational Bond number $G$. The solid line represents the solution (53) and the dashed and dotted lines represent the two-term and three-term truncations of Eq. (54), respectively. The experimental value of $G$ in the present work, $G_{\exp }=0.9472$, is indicated with the vertical dashed line.

Figure 6

Extraction of the switch-on timescale from the experimental data: (a) data for $\mathrm{\Delta }H$ for short times after switch on, with larger electric Bond numbers $B$ corresponding to larger values of $\mathrm{\Delta }H$, plotted as a function of time $t$. For clarity, the experimental data is plotted using average values over five experimental time steps and for only 11 voltages, corresponding to $B=2.73$, 3.22, 4.29, 5.39, 6.42, 7.57, 8.63, 10.79, 11.87, 12.92, 14.04, although all of the analysis of the experimental data uses the complete data set; (b) experimental values of $\lambda$ assuming that $\mathrm{\Delta }H\left(t\right)\approx \mathrm{\Delta }{H}_0+\lambda Bt$ plotted as a function of $B$, together with the average value of $\lambda$ using all of the experimental data, namely $\lambda =0.471$ (shown with a solid line), and the average value of $\lambda$ using only the experimental data for values of $B$ satisfying $B\ge 4.29$, namely $\lambda =0.586$ (shown with a dashed line). The error bars on the values of $\lambda$ correspond to 95% confidence intervals.

Figure 7

Plot of $\lambda$, the coefficient of $Bt$ in the short-time expansion of the nondimensional drop apex height difference $\mathrm{\Delta }H=\mathrm{\Delta }{H}_{\mathrm{s}}+\lambda B(t-T)+O({(t-T)}^2)$ during switch off from $B_0=B$ to $B_1=0$ at time $t=T$, as a function of the electric Bond number $B$, determined from the dynamic numerical solution of the thin-film equation (37), and for values of the gravitational Bond number $G$ from zero to the experimental value $G_{\exp }=0.9472$.

Figure 8

Extraction of the switch-off timescale from the experimental data: (a) data for $\mathrm{\Delta }H$ for short times after switch off, with larger electric Bond numbers $B$ corresponding to larger values of $\mathrm{\Delta }H$, plotted as a function of time $t$. For clarity, the experimental data is plotted using average values over 5 experimental time steps and for only 11 voltages, corresponding to $B=2.73$, 3.22, 4.29, 5.39, 6.42, 7.57, 8.63, 10.79, 11.87, 12.92, 14.04, although all of the analysis of the experimental data uses the complete data set; (b) experimental values of $\lambda$ assuming that $\mathrm{\Delta }H\approx \mathrm{\Delta }{H}_{\mathrm{s}}+\mathrm{\Delta }{H}_0+\lambda B(t-T)$ plotted as a function of $B$, together with the best linear fit for $\lambda$ as a function of $B$, namely $\lambda =-(0.378+0.090B)$ (shown with a solid line). The error bars on the values of $\lambda$ correspond to 95% confidence intervals.