Effects of surfactant solubility on the hydrodynamics of a viscous drop in a dc electric field

Surfactants (amphiphilic surface active agents) are often used to control the dynamics of viscous drops and bubbles in microfluidics applications. Surfactant sorption kinetics has been shown to play a critical role in the deformation of drops in extensional and shear flows, yet to the best of our knowledge these kinetics effects on a viscous drop in an electric field have not been accounted for. In this paper we numerically investigate the effects of sorption kinetics on a surfactant-covered viscous drop in an electric field. Over a range of electric conductivity and permittivity ratios between the interior and exterior fluids, we focus on the dependence of deformation and flow on the transfer parameter $J$, and Biot number Bi that characterize the extent of surfactant exchange between the drop surface and the bulk. Our findings suggest solubility affects the electrohydrodynamics of a viscous drop in distinct ways as we identify parameter regions where (1) surfactant solubility alters both the drop deformation and circulation of fluid around a drop and (2) surfactant solubility affects mainly the flow and not the deformation.

Figure 1

Sketch of the problem: A leaky dielectric viscous drop (${\mathrm{\Omega }}^{-}$) immersed in another dielectric fluid (${\mathrm{\Omega }}^{+}$), with an external electric field $\vec{\mathbf{E}}$ in the $z$ direction. The bead-rod particles represent surfactants at the interface or in the bulk. The double arrows denote adsorption-desorption kinetics while the curved arrows represent the induced flow. The curved arrow across the vertical axis shows the symmetry around the $z$ axis. The equator and pole are represented by the angle parameter $\theta =0$ and $\theta =\pi /2$, respectively.

Figure 2

(a) Phase diagram for the insoluble case (Biot number $\text{Bi}=0$) with surfactant coverage $\chi ={\mathrm{\Gamma }}_{\mathrm{eq}}/{\mathrm{\Gamma }}_{\infty }=0.9$ and ${\text{Pe}}_S=100$. [(b)–(d)] prolate drops with three types of circulation. [(e) and (f)] oblate drops with two types of circulation. (g) Drop deformation versus time. One drop with $({\sigma }_r,{\varepsilon }_r)=(0.1,10)$ reaches a steady equilibrium (solid line), while the drop with $({\sigma }_r,{\varepsilon }_r)=(10,0.03)$ continues to elongate (dashed line) past $T=10$. In each panel, the vector fields are plotted on a relative scale.

Figure 3

(a) Surfactant distribution for the three prolate drops in Figs 2, 2 and 2. (b) Surfactant distribution for the two oblate drops in Figs. 2 and 2. The values $\theta =0$ and $\theta =\pi /2$ denote the equator and pole, respectively.

Figure 4

For the three prolate drops in Fig. 2: (a) tangential velocity $u_t$, (b) Marangoni stress ${\gamma }_s$ (the spatial derivative of surface tension $\gamma$), (c) normal Maxwell stress, and (d) tangential Maxwell stress. The values $\theta =0$ and $\theta =\pi /2$ denote the equator and pole, respectively.

Figure 5

For the two oblate drops in Fig. 2: (a) tangential velocity $u_t$, (b) Marangoni stress ${\gamma }_s$ (the spatial derivative of surface tension $\gamma$), (c) normal Maxwell stress, and (d) tangential Maxwell stress. The values $\theta =0$ and $\theta =\pi /2$ denote the equator and pole, respectively.

Figure 6

Phase diagram for the soluble case (Biot number $\text{Bi}=10$) with surfactant coverage $\chi ={\mathrm{\Gamma }}_{\mathrm{eq}}/{\mathrm{\Gamma }}_{\infty }=0.9, {\text{Pe}}_S=100$, and $J={10}^{-2}$. (a) The green-shaded region is for adsorption-dominance, and the gray-shaded region is for desorption-dominance. In the unshaded region, the relative change in the total amount of surface surfactant is small. (b) The clockwise (c), counterclockwise (cc) and multiple toroidal vortices (s) circulation for a spheroidal equilibrium shape. (c) The red-shaded region denotes region in which solubility destabilizes the drop, and the blue-shaded region represents region in which solubility stabilizes the drop. The circle size denotes the relative change between a clean drop and a surfactant-covered drop.

Figure 7

Insoluble (solid line) versus soluble (dashed line) surfactants for a prolate drop (top row, ${\text{Ca}}_E=0.3$) and for an oblate drop (bottom row, ${\text{Ca}}_E=0.25$) on the [(a) and (d)] surfactant distribution, [(b) and (e)] tangential velocity $u_t={\mathit{v}}_s·\mathit{t}$, and [(c) and (f)] Marangoni stress ${\gamma }_s$ (the spatial derivative of surface tension $\gamma$). The Péclet number $\text{Pe}=100$ and $J={10}^{-2}$, and the values $\theta =0$ and $\theta =\pi /2$ denote the equator and pole, respectively.

Figure 8

(a) Deformation of a prolate drop with $({\sigma }_r,{\varepsilon }_r)=(0.1,5)$ as a function of dimensionless time $T$. The electric capillary number ${\text{Ca}}_E=0.25$. The Biot numbers $\text{Bi}=0$ (insoluble case) and $\text{Bi}=1$. The inset shows the drop shapes at various times. [(b)–(d)] Circulation for the prolate drop with $\text{Bi}=0$ at times $T=0.125, T=0.625$, and $T=4$, respectively. [(e)–(g)] Circulation for the prolate drop with $\text{Bi}=1$ and $J=2\times {10}^{-3}$ at times $T=0.125, T=0.625$, and $T=4$, respectively. The Péclet number $\text{Pe}={\text{Pe}}_S=500$.

Figure 9

Surfactant distribution (a), tangential velocity $u_t={\mathit{v}}_s·\mathit{t}$ (b), and Marangoni stress ${\gamma }_s$ (the spatial derivative of surface tension $\gamma$) (c) for the prolate drop in Fig. 8 at $T=19$. The solid lines are for $\text{Bi}=0$ (the insoluble surfactant case). The dashed and dash-dotted lines are for $\text{Bi}=1$ with $J=2\times {10}^{-3}$ and $J=10$, respectively. The Péclet number $\text{Pe}={\text{Pe}}_S=500$, and the values $\theta =0$ and $\theta =\pi /2$ denote the equator and pole, respectively.

Figure 10

A prolate drop with $({\sigma }_r,{\varepsilon }_r)=(1/3,1)$ and ${\text{Ca}}_E=0.3$. [(a) and (d)] Surfactant distribution, [(b) and (e)] tangential velocity $u_t={\mathit{v}}_s·\mathit{t}$, and [(c) and (f)] Marangoni stress ${\gamma }_s$ (the spatial derivative of surface tension $\gamma$) as a function of $\theta$. Solid lines are for $\text{Bi}=0$ (the insoluble surfactant case), dashed lines are for $\text{Bi}={10}^{-2}$, and dash-dotted lines are for $\text{Bi}=10$. The transfer parameter $J=2\times {10}^{-3}$ [(a)–(c)], and $J=10$ [(d)–(f)]. The Péclet number $\text{Pe}={\text{Pe}}_S=500$, and the values $\theta =0$ and $\theta =\pi /2$ denote the equator and pole, respectively.

Figure 11

Flow field for a prolate drop with $({\sigma }_r,{\varepsilon }_r)=(1/3,1)$ and ${\text{Ca}}_E=0.3$, corresponding to velocities in Figs. 10 and 10. (a) $\text{Bi}=0$, (b) $\text{Bi}={10}^{-2}$, and $J=2\times {10}^{-3}$; (c) $\text{Bi}=10$ and $J=2\times {10}^{-3}$; (d) $\text{Bi}=0$; (e) $\text{Bi}={10}^{-2}$ and $J=10$; (f) $\text{Bi}=10$ and $J=10$. The Péclet number $\text{Pe}={\text{Pe}}_S=500$.

Figure 12

Change in total amount $\mathcal{A}$ of surface surfactants for the prolate drop in Sec. 4b with $({\sigma }_r,{\varepsilon }_r)=(1/3,1)$ and ${\text{Ca}}_E=0.3$. The Biot number $\text{Bi}=1$ and the Péclet number $\text{Pe}={\text{Pe}}_S=500$.

Figure 13

Deformation as a function of dimensionless time $T$ for (a) a prolate drop with $({\sigma }_r,{\varepsilon }_r)=(0.97,0.75)$, and (b) the oblate drop with $({\sigma }_r,{\varepsilon }_r)=(4/3,2)$. ${\text{Ca}}_E=0.25$ for both cases. The solid and dotted lines are for low ($J={10}^{-3}$), and high ($J=10$) transfer parameter with $\text{Bi}={10}^{-1}$, respectively. The dashed and dash-dotted lines represent higher Biot number ($\text{Bi}=10$) with $J={10}^{-3}$ and $J=10$, respectively.

Figure 14

(a) The numerical algorithm for the second-order immersed interface method code: At $t_n$ the drop shape $\mathit{x}$, flow field $\mathit{u}$, and interface velocity, $\mathit{U}$ are computed using the electrohydrodynamic solver in Refs. [35, 78]. The information is then used as input to the surfactant transport solver [79], in order to determine the bulk ($\varphi$) and interface surfactant profile ($\mathrm{\Gamma }$). Given $\mathrm{\Gamma }$, we determine the change in surface tension $\gamma$, as well as the updated drop shape, flow field, and interface velocity at time $t_{n+1}$. This process is repeated either until a steady state is reached, or up to the onset of drop break-up. Flow circulation and direction are represented by the blue arrows. (b) Computational domain on the $(r,z)$ plane. On the walls BC 1, 2, 3, and 4 denote the boundary conditions (see text).

Figure 15

Comparison between published simulation results for the clean drop case [(a) and (c)] in [41] and the surfactant-covered drop cases [(b) and (d)] in Refs. [43, 47]. The solid lines represent simulations from boundary integral for the clean case and from level-set for the surfactant-covered drop case. The black triangles represent simulations from boundary integral, while the green circles represent simulations using the proposed IIM implementation. For the clean drop cases: (a) ${\sigma }_r=0.1,{\varepsilon }_r=0.1$; (c) ${\sigma }_r=0.5,{\varepsilon }_r=20$. For the surfactant-covered drop cases we set $E=0.2,\chi =0.3,{\text{Pe}}_s=10,\text{Bi}=0$: (b) ${\sigma }_r=0.3,{\varepsilon }_r=1$; (d) ${\sigma }_r=1,{\varepsilon }_r=2$. Volume and total surfactant are conserved to within 5% in all cases.