Effects of surfactant transport on electrodeformation of a viscous drop

In this work we quantify the effects of surfactant transport on the deformation of a viscous drop under a DC electric field. We study how convective and diffusive transport of surfactants at drop surfaces influence the equilibrium and dynamic deformation of a leaky dielectric drop and a conducting drop. Focusing on the prolate drop shape (elongates along the electric field), we show the differences in equilibrium deformation and flow circulation between a leaky dielectric drop and a conducting drop. We quantify the drop electrodeformation via its dependence on the interior flow circulation and the dominant surfactant transport regime (characterized by the surface Péclet number ${\text{Pe}}_s$). For a leaky dielectric drop with dominant surfactant diffusion (${\text{Pe}}_s\ll 1$), equator-to-pole (pole-to-equator) circulation yields smaller (larger) equilibrium deformation with increasing surfactant coverage, compared to a clean drop. However, when convection dominates (${\text{Pe}}_s\gg 1$), the equilibrium drop deformation increases (decreases) with larger surfactant coverage for equator-to-pole (pole-to-equator) circulation. Larger equilibrium drop deformation is found for a leaky dielectric drop than a conducting drop when the interior flow is from equator to pole. For an interior flow from pole to equator, we identify cases where larger deformation is found for a conducting interior fluid. Finally, we study the effect of the surfactant transport on the dynamic evolution of drop shape. We found the drop undergoes an overshoot in the early deformation phase, before settling to its equilibrium shape—similar to the overshoot observed for unsteady Stokes flow.

Figure 1

 A viscous drop, subject to a DC electric field $E_0{\mathit{e}}_z$ and covered with insoluble surfactant (bead-rod particles), separates two immiscible fluids, with physical parameters ${\varepsilon }_j,{\sigma }_j,{\mu }_j$, where $j=i$ denotes the interior of the drop, and $j=e$ its exterior. Application of the electric field induces flow inside and outside the drop, which deforms into a prolate-shaped spheroid at equilibrium. Deformation depends on the type of drops (leaky dielectric or conducting) and on the surfactant transport mechanism (diffusive transport for ${\text{Pe}}_s\ll 1$ and convective transport for ${\text{Pe}}_s\to \infty$). The drop shape parameter ${\xi }_0$ is a function of the aspect ratio between the major ($b$) and minor ($a$) axes.

Figure 2

Equilibrium deformation $D_{\text{eq}}$ as a function of electric capillary number for a clean conducting drop. The parameters ${\mu }_r=1/0.00126,{\sigma }_r={10}^{-8},{\varepsilon }_r=1/31.1$. The symbols denote experiments [6, 48]; the curves represent predictions using the boundary integral method (dashed) [48], Taylor's first-order theory (dotted), and Ajayi's second-order approximation (dash-dotted). The present model is shown in solid line.

Figure 3

Equilibrium deformation $D_{\text{eq}}$ as a function of electric capillary number for a surfactant-laden, leaky dielectric drop. The electric parameters are (a) for a prolate “A” (counterclockwise circulation) drop with ${\varepsilon }_r=1,{\sigma }_r=1/3$; and (b) for a prolate “B” (clockwise) drop with ${\varepsilon }_r=1/3.5,{\sigma }_r=1/3$. Symbols are from numerical simulations using a regularized level-set method [31] (green diamond), and a boundary integral equation method (blue circle) [35]; the current semianalytical approach is represented by the solid line.

Figure 4

Effects of Péclet number on the deformation number ($D_{\text{eq}}$) and flow strength ($A_3^3$) for a leaky dielectric prolate “A” drop with ${\varepsilon }_r=1,{\sigma }_r=1/3$. In (a) $D_{\text{eq}}$ shows dependence on both surfactant coverage and ${\text{Pe}}_s$. For ${\text{Pe}}_s⪅5, D_{\text{eq}}$ decreases with increasing $\chi$; above that threshold, the opposite holds: $D_{\text{eq}}$ increases with higher surfactant coverage. No such transition is observed in $A_3^3$. In (b), $A_3^3$ monotonically decreases with increasing ${\text{Pe}}_s$, and higher surfactant coverage yields stronger flow. In (c) the Marangoni stress (${\mathbf{\nabla }}_s\gamma$) and surface tension ($\gamma$, in inset) are plotted as a function of $\theta ={cos}^{-1}\left(\eta \right)$ for ${\text{Pe}}_s=0.1$. The electric capillary number ${\text{Ca}}_E=0.67$

Figure 5

Streamlines for the leaky dielectric prolate “A” drop with ${\varepsilon }_r=1,{\sigma }_r=1/3$, and $\chi =0.5$: (a) ${\text{Pe}}_s=0.01$; (b) ${\text{Pe}}_s=10$; and (c) ${\text{Pe}}_s\to \infty$ (the nondiffusive case from [32]). As the Péclet number ${\text{Pe}}_s$ increases from (a) to (c), we observe larger deformation. In the nondiffusive limit the flow is completely suppressed as shown in (c).

Figure 6

Effects of Péclet number on the deformation number ($D_{\text{eq}}$) and flow strength ($A_3^3$) for a leaky dielectric prolate “B” drop with ${\varepsilon }_r=1/3.5,{\sigma }_r=1/3$. The transition from diffusion- to convection-dominated dynamics occurs around ${\text{Pe}}_s\approx {10}^{-1}$. In the diffusion-dominated regime, higher surfactant coverage yields larger deformation (and stronger flow), though the difference between these is minimal. In the convection-dominated regime, we observe that $D_{\text{eq}}$ decreases with ${\text{Pe}}_s$ while the magnitude of flow strength ($|A_3^3|$) increases with ${\text{Pe}}_s$. The electric capillary number ${\text{Ca}}_E=1.4$.

Figure 7

Equilibrium deformation number as a function of Péclet number for a conducting drop. The effects of surfactant transport is shown for various values of surfactant coverage at ${\text{Ca}}_E=0.112$. The electric parameters are ${\sigma }_r={10}^{-3},{\varepsilon }_r=1$.

Figure 8

(a) Deformation number $D$ as a function of dimensionless time for a leaky dielectric drop with various values of ${\text{Pe}}_s$. The electric capillary number ${\text{Ca}}_E=0.5$ and surfactant coverage $\chi =0.3$; the electric parameters ${\varepsilon }_r=1,{\sigma }_r=1/3$, corresponding to the prolate “A” shape. The steady-state curve for the clean drop case (CD) is denoted by the dotted line. (b) Snapshot of the drop shape at various times for $D$ in (a) with ${\text{Pe}}_s=100$. The dashed shape denotes the maximum shape achieved at peak overshoot.

Figure 9

The surfactant profile (a) and Marangoni stress (b) corresponding to ${\text{Pe}}_s=100$ in Fig. 8 are plotted as a function of $\theta ={cos}^{-1}\left(\eta \right)$. Each curve denotes a fixed time, with the peak overshoot occurring at $T\approx 115$.

Figure 10

The semianalytical algorithm: At $t_n$ the drop shape ${\xi }_0$ and flow field $\mathit{u}$ is computed using the large deformation spheroidal analysis. This information is then used as input to the surfactant transport, to determine the surfactant profile $\mathrm{\Gamma }$ numerically with a collocation method. With the surfactant concentration at hand, we determine the change in surface tension, the drop shape and the flow field at time $t_{n+1}$. This process is repeated until a steady state is reached.

Figure 11

Effects of Péclet number on the deformation number ($D_{\text{eq}}$) for (a) a conducting prolate “A” drop with ${\varepsilon }_r=1,{\sigma }_r=1/3$ and ${\text{Ca}}_E=0.67$, and (b) a conducting prolate “B” drop with ${\varepsilon }_r=1/3.5,{\sigma }_r=1/3$ and ${\text{Ca}}_E=1.4$. In both cases the deformation remains relatively constant across ${\text{Pe}}_s⪅1$, then shows small increases as convection starts dominating. Moreover, higher surfactant coverage yields smaller deformation.