Electrolytic drops in an electric field: A numerical study of drop deformation and breakup

The deformation and breakup of an axisymmetric, conducting drop suspended in a nonconducting medium and subjected to an external electric field is numerically investigated here using an electrokinetic model. This model uses a combined level set-volume of fluid formulation of the deformable surfaces, along with a multiphase implementation of the Nernst-Planck equation for transport of ions, that allows for varying conductivity inside the drop. A phase diagram, based on a parametric study, is used to characterize the stability conditions. Stable drops with lower ion concentration are characterized by longer drop shapes than those achieved at higher ion concentrations. For higher drop ion concentration, greater charge accumulation is observed at drop tips. Consequently, such drops break up by pinching off rather than tip streaming. The charge contained in droplets released from unstable drops is shown to increase with drop ion concentration. These dynamic drop behaviors depend on the strength of the electric field and the concentration of ions in the drop and result from the interplay between the electric forces arising from the permittivity jump at the drop interface and the ions in the bulk.

Figure 1

Schematic of an axisymmetric drop (the equations are solved on the unshaded region with symmetry boundary conditions applied on the vertical centerline) suspended in another immiscible liquid, acted on by an external electric field parallel to the $z$ direction. The dotted lines show the profile of the drop deformed by the electric field.

Figure 2

Comparison of axial ratio $\mathrm{\Gamma }(l/b)$ predictions for this model (shown as black dots) with the minimum energy model [8] for (a) $S=5$ and (b) $S=50$ for a range of ${\mathrm{Ca}}_{\mathit{E}}$. Results of Hua et al. [40] are shown as black snowflakes.

Figure 3

Composite equilibrium drop shapes for values of ${\mathrm{Ca}}_{\mathit{E}}$ from 1.0 to 9.0. Included results are Sherwood [34] (top, reproduced from Sherwood, J. D. (1988) Breakup of fluid droplets in electric and magnetic fields. Journal of Fluid Mechanics 188: 133-146).

Figure 4

Images of an unstable drop with ${\mathrm{Ca}}_{\mathit{E}}=0.24$ and $K_{\mathit{d}}=7$ for $t=70$, 75, 80, 85, and 90. Charge contours are embedded inside the drop and lines of constant potential are also shown.

Figure 5

Evolution of drop deformation with time for (a) ${\mathrm{Ca}}_{\mathit{E}}=0.26$ and $K_{\mathit{d}}=0$–10 (Only the $K_{\mathit{d}}=0$ case is stable) and (b)$K_{\mathit{d}}=3$ and ${\mathrm{Ca}}_{\mathit{E}}=0.22$–0.27 (only the ${\mathrm{Ca}}_{\mathit{E}}=0.22$ case is stable).

Figure 6

(a) Evolution of $z$-averaged mean ion location [Eq. (17)] with time and (b) total electric force comparison for ${\mathrm{Ca}}_{\mathit{E}}=0.26$ and $K_{\mathit{d}}=1$–10 (only the $K_{\mathit{d}}=0$ case is stable).

Figure 7

Phase diagram for overall drop deformation behavior. The stable and unstable boundaries have been drawn connecting the cases to direct the reader's eyes.

Figure 8

(a) Comparison of breakup modes [(i) tip streaming and (ii) pinch-off] for $K_{\mathit{d}}=1$ and $K_{\mathit{d}}=10$ for ${\mathrm{Ca}}_{\mathit{E}}=0.28$ [scale for charge contours shown in (b)] and (b) charge in first drop ejected (red circles) and permittivity force as a percentage of total electric force at breakup (green squares) vs $K_{\mathit{d}}$ for ${\mathrm{Ca}}_{\mathit{E}}=0.25$ (all unstable cases). Drop profiles just prior to breakup for selected $K_{\mathit{d}}$ with embedded charge contours are also included.

Figure 9

(a) Evolution of permittivity and charge force [Eq. (12)] as a ratio of total electric force for selected $K_{\mathit{d}}$ (all unstable cases) at ${\mathrm{Ca}}_{\mathit{E}}=0.25$ with time and (b) evolution of percentage of both ion species in the top half of drop with time for selected dimensionless ion concentration $K_{\mathit{d}}$ (all unstable cases) at ${\mathrm{Ca}}_{\mathit{E}}=0.25$.

Figure 10

(a) Evolution of electric field magnitude at center of drop with time for selected dimensionless ion concentration $K_{\mathit{d}}$ (all unstable cases) at ${\mathrm{Ca}}_{\mathit{E}}=0.25$ and (b) evolution of the electric field magnitude with time across the horizontal centerline in the domain for ${\mathrm{Ca}}_{\mathit{E}}=0.25$ (all unstable cases) and $K_{\mathit{d}}=9$. The black dotted line indicates the vertical line of symmetry.

Figure 11

(a) Evolution of drop deformation of ${\mathrm{Ca}}_{\mathit{E}}=0.26$ ($K_{\mathit{d}}=4$) with time (with embedded charge contours) and (b) charge in first drop ejected (red circles) and permittivity force [Eq. (12)] as a percentage of total electric force at breakup (green squares) vs ${\mathrm{Ca}}_{\mathit{E}}$ for $K_{\mathit{d}}=4$ (all unstable cases)

Figure 12

(a) Evolution of percentage of both ion species in the top half of drop with time for selected ${\mathrm{Ca}}_{\mathit{E}}$ (all unstable cases) at $K_{\mathit{d}}=4$. (b) Evolution of permittivity and charge force [Eq. (12)] as a ratio of total electric force with time for selected ${\mathrm{Ca}}_{\mathit{E}}$ (all unstable cases) at $K_{\mathit{d}}=4$.

Figure 13

(a) Evolution of $z$-averaged mean ion location [Eq. (17)] with time for selected ${\mathrm{Ca}}_{\mathit{E}}$ (all unstable cases) at $K_{\mathit{d}}=4$. (b) Evolution of electric field magnitude at center of drop with time for selected ${\mathrm{Ca}}_{\mathit{E}}$ (all unstable cases) at $K_{\mathit{d}}=4$.