Deformation of soap bubbles in uniform electric fields

The deformation of hemispherical bubbles under vertical static electric fields was studied using a plane capacitor. The deformation and the bubbles shape have been monitored according to the amplitude of the electric field and the initial volume of the bubbles. Two different substrates, a dry solid plate or a liquid pool, were used to inspect the influence of the pining of the contact line. The deformation of sessile (on dry solid plate) and floating (on a liquid pool) bubbles were compared. The deformation parameter has been rationalized using a simple model. More precisely, the number of interfaces has been found to be a relevant parameter that imposes the shape of the bubbles.

Figure 1

Pictures of a bubble floating on a soap pool under a uniform electric field. The picture (a) shows the bubble on a liquid pool when the intensity of the electric field $E_0=0$. (b) Deformation of a bubble induced by the uniform electric field. The bubble adopts a hemispheroidal shape when $E_0<E_T$ with $E_T$ the intensity needed to trigger the instability. (c) When $E_0>E_T$, the instability occurs, the bubble becomes pointy and starts to eject droplets to the top plate to discharge. This behavior is called the Taylor cone. The height $H$ and the radius $R$ of the bubble as well as the emerged volume $V_f$ and the apparent initial contact angle $\theta$ of the bubble are measured.

Figure 2

Sketches of the experimental setup used for the experiments performed on floating bubbles. (a) Plan of the elements composing the experimental setup. (b) Drawing of the geometrical measures of the experiment: the liquid depth $e$ between the immersed electrode and the liquid surface, the size of the capacitor $d$, the distance between electrodes $d_t$, the electric field ${\vec{E}}_0$, the penetration depth $h$, the height $H_b$ and the radius $R_b$ (measured at the top of the meniscus) of the bubble as well as its emerged and immersed volume $V_f$ and $V_i$. The scale of both sketches is the same and is indicated in the top right corner.

Figure 3

Height of the floating bubble $H$ as a function of time. The size of the capacitor was $d=20\mathrm{mm}$ and the volume of the bubble was $V_g=0.5\mathrm{ml}$. At time $t=0\mathrm{s}$ the DC supply was set on and the top plate reached the preset voltage. Here, only one step is needed to go from $0\mathrm{kV}$ [on picture (i)] to $9\mathrm{kV}$ [on picture (ii)]. The black line is the fitting function, described by Eq. (2), used to obtain the characteristic time ${\tau }_c$ and deformation $D$ as well as the steady height $H_b$ and radius $R_b$. The ${\tau }_c$ for this experiment was found equal to $4.7\mathrm{s}$.

Figure 4

Stationary height of bubbles $H_b$ at different intensity values of the applied fields $E_0$. Both curves correspond to bubbles of $V_g=0.25\mathrm{ml}$ and a capacitor of $d=25\mathrm{mm}$. The red and the blue points are respectively for a bubble on solid and liquid substrate. Pictures (i) and (ii) illustrate the shape of a bubble under $E_0=0.02\mathrm{kV}/\mathrm{mm}$ for a solid (i) and a liquid (ii) substrate. Pictures (iii) and (iv) show the deformation of a bubbles respectively resting on solid, under $E_0=0.46\mathrm{kV}/\mathrm{mm}$ and floating on a liquid under $E_0=0.52\mathrm{kV}/\mathrm{mm}$. A typical error bar, corresponding to the worst-case scenario, is indicated on each curve.

Figure 5

Aspect ratio of sessile bubbles ${\mathrm{Ar}}_s=H_b/R_b$ resting on a dry solid plate plotted as a function of the energy ratio ${\mathrm{Bo}}_{e,s}={\varepsilon }_0{E}_0^2{R}_b/4\gamma$. Three sizes of capacitors $d$ have been used: $d=15\mathrm{mm}$ (solid circles), $20\mathrm{mm}$ (solid triangles), and $25\mathrm{mm}$ (solid squares), as well as three internal volumes $V_g$: $V_g=0.25\mathrm{ml}$ (in mauve), $0.50\mathrm{ml}$ (in orange), and $1.00\mathrm{ml}$ (in green). The black bullets represent the data presented from Ref. [17] for a capacitor of $d=85.5\mathrm{mm}$ and an approximate internal volume of $V_g=35.14\mathrm{ml}$ (solid circles) or $V_g=30.05\mathrm{ml}$ (solid triangles). The black straight line correspond to the small deformations law adapted from Eq. (1). On this graphic and the following ones, error bars are smaller than the bullets.

Figure 6

Aspect ratio of floating bubbles ${\mathrm{Ar}}_s=H_b/R_b$ as a function of the energy ratio ${\mathrm{Bo}}_{e,s}={\varepsilon }_0{E}_0^2{R}_b/4\gamma$. The same three sizes of capacitors $d$ had been used: $d=15\mathrm{mm}$ (solid circles), $20\mathrm{mm}$ (solid triangles) and $25\mathrm{mm}$ (solid squares), as well as the same three internal volumes $V_g$: $V_g=0.25\mathrm{ml}$ (in mauve), $0.50\mathrm{ml}$ (in orange), and $1.00\mathrm{ml}$ (in green). The results for bubbles resting on a solid substrate (red solid circles) are drawn for the sake of the comparison.

Figure 7

Aspect ratio of floating bubbles $\mathrm{Ar}=H_b/R_c$ drawn as a function of the energy ratio ${\mathrm{Bo}}_e={\varepsilon }_0{E}_0^2{R}_c/2n\gamma$. The same three sizes of capacitors $d$ were used: $d=15\mathrm{mm}$ (solid circles), $20\mathrm{mm}$ (solid triangles) and $25\mathrm{mm}$ (solid squares), as well as the same three internal volumes $V_g$: $V_g=0.25\mathrm{ml}$ (in mauve), $0.50\mathrm{ml}$ (in orange), and $1.00\mathrm{ml}$ (in green). The straight lines correspond to Eq. (9) describing small deformations. The color of each line is linked to the corresponding $V_g$ and the only fitting parameter for each curve is the aspect ratio in absence of field. The results for bubble resting on solid substrate (red solid circles) as well as the corresponding law (black straight line) are drawn for the sake of comparison.

Figure 8

Variation of the aspect ratio $\mathrm{\Delta }\mathrm{A}r=H_b/R_c-\mathrm{Ar}({\mathrm{Bo}}_b,0)$ of sessile (in red) and floating (in green, orange and mauve) bubbles drawn as a function of the energy ratio $B{o}_e={\varepsilon }_0{E}_0^2{R}_c/2n\gamma$. The same three internal volumes $V_g$: $V_g=0.25\mathrm{ml}$ (in mauve), $0.50\mathrm{ml}$ (in orange), and $1.00\mathrm{ml}$ (in green) were used. The straight line corresponds to the law given by the Eq. (11) describing small deformations.

Figure 9

Aspect ratio of frustrated bubbles $\mathrm{Ar}=H_b/R_c$ drawn as a function of the energy ratio ${\mathrm{Bo}}_e={\varepsilon }_0{E}_0^2{R}_c/2n\gamma$ for different liquid depths $e$. The size of the capacitor is fixed to $d=25\mathrm{mm}$ and the internal volume to $V_g=0.25\mathrm{ml}$. The blue and red circles correspond respectively to liquid and solid results in the same situation. The different results are sorted in three categories: $e\ge 1.25\mathrm{mm}$ (in deep mauve), $e\approx 0.50\mathrm{mm}$ (in purple), and $e\le 0.20\mathrm{mm}$ (in pink).