Effects of surfactant solubility on the hydrodynamics of a viscous drop in a dc electric field
Herve Nganguia ,1Wei-Fan Hu,2Ming-Chih Lai,3and Y.-N. Young 4,*
1Department of Mathematics, Towson University, Towson, Maryland 21286, USA
2Department of Mathematics, National Central University, 300, Taoyuan 32001, Taiwan
3Department of Applied Mathematics, National Yang Ming Chiao Tung University, 1001, Ta Hsueh Road, Hsinchu 30050, Taiwan
4Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102, USA
(Received 30 April 2020; accepted 10 June 2021; published 29 June 2021) 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.
DOI:10.1103/PhysRevFluids.6.064004
I. INTRODUCTION
Electric field is widely utilized to deform a viscous drop in microfluidics and many petroleum engineering applications. Electrohydrodynamics (EHD), generally referred to as the motion of fluid induced by an electric field, is highly relevant to transport and manipulation of small liquid drops in microfluidic devices. Since the early 2000s, dielectrophoresis, electro-osmosis, and induced-charge electro-osmosis in EHD have deeply influenced the field of microfluidics. Moreover, the integration of EHD into microfluidic-based platforms has led to the development of technological platforms for manipulation of particles, colloids, droplets, and biological molecules across different length scales [1–6]. EHD has been used in a wide range of applications, such as spray atomization, fluid motion of bubble drop, electrostatic spinning, and printing [1,7–12]. In material and bioengineer- ing, EHD was utilized to manufacture nanostructured materials [13,14] and manipulate charged macromolecules [15].
For a leaky dielectric drop freely suspended in another leaky dielectric fluid, the bulk charge neutralizes on a fast timescale while “free” charges accumulate on (and move along) the drop surface. In this physical regime, the full electrokinetic transport model in a viscous solvent can be described by a charge-diffusion model that can be further reduced to derive the Taylor-Melcher (TM) leaky dielectric model [16]. In many physics and engineering applications with moderately
*Corresponding author: yyoung@njit.edu
TABLE I. Summary of published modeling work on the electrohydrodynamics (EHD) of a surfactant-laden viscous drop. SD denotes the small deformation (spherical harmonics) analysis, and LD refers to the large deformation (spheroidal harmonics) analysis. The abbreviations LS-RegM, BIM, SU-RM, and IIM stand for level-set regularized method, boundary integral method, successive under-relaxation method, and immersed interface method, respectively. Inertia-driven now (Navier-Stokes) is shortened using N.-S.
Fluids Electric field Surfactants Method References
Stokes dc, uniform Insoluble Analytical (SD) [42]
N.-S. dc, uniform Insoluble Numerical (LS-RegM) [43]
Stokes dc, uniform Insoluble (Semi-) analytical (LD) [44,45]
Stokes dc, uniform Insoluble Numerical (BIM) [46,47]
Stokes dc, uniform Insoluble Analytical (SD) & numerical (SU-RM) [48–50]
Stokes dc, nonuniform Insoluble Analytical and numerical [51]
Stokes dc, uniform Soluble Numerical (IIM) Present work
dissolvable electrolytes, the TM leaky dielectric model can capture the deformation of a viscous drop in both dielectric medium [17,18] and a conducting medium [19,20]. The TM model has been extended in recent years to include the effects of charge relaxation [21], charge convection [22–25], and the investigation of nonspherical drop shapes [26–29] and drop instabilities using direct numerical methods [30–36].
In the absence of surface-active agent (surfactant), the balance between the electric stresses and the hydrodynamic stress on the drop surface gives rise to a drop shape and a flow field that can be parametrized by the conductivity ratio and the permittivity ratio [37]. Under a small electric field, a steady equilibrium drop shape exists due to the balance between the electric and hydrodynamic stresses [34,38,39]. For a sufficiently large electric field, instabilities arise and the drop keeps deforming until it eventually breaks up into smaller drops [40,41].
Nonionic surfactant has been extensively used for stability control in experiments on elec- trodeformation of a viscous drop [42,46,52–54]. By reducing the surface tension and inducing a significant Marangoni stress due to the surfactant transport on the interface, surfactant could lead to drastically different EHD of a surfactant-laden viscous drop. Table Isummarizes the existing theoretical and numerical investigations in the literature. In most of these studies [42–51,55], surfactants are assumed to be insoluble and the surface tension is described using either a linear relationship, or more realistically the Langmuir equation of state
γ () = γ0+ RT ∞ln
1−
∞
, (1)
where R and T denote the gas constant and absolute temperature, respectively.γ0 is the surface tension of an otherwise clean drop, and∞is the maximum surface packing limit. A spheroidal model has been developed to predict the large electrodeformation of a viscous drop covered with insoluble surfactant [44]. Finite surfactant surface diffusivity has also been incorporated in such spheroidal model [45] with excellent agreement with full numerical simulations [47].
Studies have shown that sorption kinetics and interactions between surfactants molecules can be effectively used to alter the concentration of surfactants at the drop interface [56–59], and have profound effects on the drop shape and dynamics [60–65]. Electric field can in turn affect the rate of sorption kinetics [55]. These results naturally lead to the following inquiries: What effects does adsorption and/or desorption have on EHD and how do they affect the interplay between all the various stresses? To our knowledge these questions have yet to be addressed in the literature.
In this work we aim to fill the gap by numerically solving the coupled equations for the leaky- dielectric model and surfactant transport equations. While our method is general enough to handle interaction between surfactants molecules, here we assume the relation provided by the Langmuir
FIG. 1. Sketch of the problem: A leaky dielectric viscous drop (−) immersed in another dielectric fluid (+), with an external electric field 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 parameterθ = 0 and θ = π/2, respectively.
equation of state Eq. (1) to focus on the effects of surfactants solubility. In the present study, we investigate such dynamics in hopes of elucidating the physics governing the EHD of drops in the presence of soluble surfactants.
The paper is organized as follows: In Sec.II, we present the physical problem and formulate the governing equations. In Sec.IIIwe investigate how solubility affects the stability of various drop shapes by systematically exploring a range of electric parameters across the (σr, εr) phase diagram.
Then, in Sec.IV, we investigate the effects of the transfer parameter J, a measure of the strength of surfactant exchange between drop surface and bulk. In Sec.V, we end our study with a summary of how surfactants solubility affects deformation for surfactant-covered drops in electric fields.
II. THEORETICAL MODELING
We consider a viscous drop immersed in a leaky dielectric fluid in the presence of surfactants and subject to an electric field, as shown in Fig.1. Each fluid is characterized by the fluid viscosity μ, dielectric permittivity ε, and conductivity σ with the superscript denoting interior (−) or exterior (+) fluid. In this work we denote the contrasts of those properties by μr= μ+/μ−,εr = ε−/ε+, and σr = σ+/σ−. Following Ref. [41], note the inverse convention for the permittivity ratio. Moreover, in this workμ+= μ−.
A. Formulation
The fluids are governed by the incompressible Stokes equations, neglecting inertia
−∇p + μ∇2u= 0, (2)
where p and u are the pressure and velocity field, respectively. The electric field E= −∇φ, where φ is the electric potential that satisfies the Laplace equation both inside and outside the drop in the extended leaky dielectric model,
∇2φ = 0. (3)
The surfactant transport on the drop surface and in the exterior bulk fluid are described by the following set of coupled equations
∂
∂t + ∇s· (vs)+ (∇s· n)us· n = Ds∇s2 + βCs(∞− ) − α, (4)
∂C
∂t + u · ∇C = D∇2C, (5)
where n is the normal vector, us is the surface velocity on the drop, and vs= (I − nn) · us is the velocity tangential component along the drop (I is the identity tensor). and C are the surfactant concentration on the drop surface and in the bulk outside the drop, respectively; Cs is the concentration of surfactant in the fluid immediately adjacent to the drop surface;α and β are the kinetic constants for desorption and adsorption, respectively; Dsand D are the diffusion constant on the drop surface and in the bulk correspondingly.
At the drop interface, boundary conditions are imposed for the electric potentialφ, the flow field u, and the bulk surfactant concentration C. First, the electric potential is continuous and the total current is conserved,
φ = 0, σ∇φ · n
Ohmic current
= d ˜q
dt
Charge relaxation
+ ∇ s· ( ˜qus)
Charge convection
, (6)
where ˜q= −ε∇φ · n represents the surface charge density, and · denotes the jump between outside and inside quantities. The effects of charge relaxation on the transient behavior of drop [21], and of convection on equilibrium deformation [22,25,66], have been investigated analytically and numerically in the context of drops electrohydrodynamics. In the present study, we neglect these effects to more easily isolate the surfactant effects. This reduces Eq. (6) to:
φ = 0, σ∇φ · n = 0. (7)
Second, the electric and fluid problems are coupled through the stress balance
−p + μ
∇uT + ∇u
Hydrodynamic stress
·n + ε
EE−1
2(E· E)I
Electric stress
·n = γ (∇ s· n)n
Surface tension
− ∇sγ
Marangoni stress
. (8)
Surfactants act to lower the surface tension, which now depends on the concentration of surfactants through the equation of state Eq. (1). As a result, the nonuniform surfactant distribution induced by the flow in and around the drop yields a surface tension gradient (the Marangoni stress).
Finally, to close the system we need a third boundary condition that describes the flux of surfactants between the surface of the drop and the bulk. The interfacial condition for the surfactant concentration,
Dn· ∇C = βCs(∞− ) − α, (9)
where n· ∇C = ∂C/∂n denotes the normal derivative of C: Positive values denote adsorption- dominated surfactant kinetics, whereas negative values denote desorption-dominated surfactant kinetics. We henceforth concentrate on axisymmetric solutions only.
B. Nondimensionalization
We use the drop size r0 to scale length, capillary pressureγ0/r0 to scale pressure, equilibrium surfactant concentrationeqto scale, initial surfactant concentration C0to scale the bulk surfactant concentration, and electrically driven flow Ud = ε+E02r0/μ to scale velocity, in which E0denotes the intensity of the external electric field. Note that in the presence of soluble surfactant,eqdenotes the equilibrium with the bulk surfactant C0 [67], whereas for the insoluble caseeq is the initial uniform surfactant concentration.
There are nine independent dimensionless physical parameters that characterize this system: (1) the electric capillary number CaE ≡ μUd/γeq= ε+E02r0/γeq(ratio of electric pressure to capillary pressure), (2) permittivity ratioεr= ε−/ε+, (3) conductivity ratioσr = σ+/σ−, (4) the elasticity constant E = RT ∞/γ0in the Langmuir equation of state, (5) the surfactant coverageχ = eq/∞, (6) the surface surfactant Péclet number Pes= r0Ud/Ds, (7) the bulk surfactant Péclet number Pe= r0Ud/D, (8) the transfer parameter J = C0D/eqUd, and (9) the Biot number Bi= ατEHD(ratio of EHD characteristic timescaleτEHD= r0/Ud to desorption timescale).
The elasticity number E measures the sensitivity of the surface tension to the surface surfactant concentration, whereas in the presence of surfactant exchange between the bulk and the drop interface, the surfactant coverage is related to the adsorption constant k= βC0/α in Eq. (10) [56,57],
χ = k
k+ 1. (10)
Note that for the insoluble case, the surfactant coverage χ = eq/∞ results from scaling the surfactant concentration in Eq. (1). The bulk and surface Péclet numbers denote the relative strength of convective transport versus diffusive transport. These two numbers also represent the ratio of two timescales: Pe= τD/τEHD, whereτD= r20/D is the surfactant diffusion timescale. The parameter J gives a measure of transfer of surfactant between its bulk and adsorbed forms relative to advection on the interface. It is important to note the ratio Bi/J distinguishes two types of transport regime [68,69]: diffusion-controlled transport (Bi/J > 1) and sorption-controlled transport (Bi/J 1).
Moreover the bulk Péclet number and the transfer parameter are related by J = 1/hadPe, where hadis the adsorption depth that measures the extent of surface dilution due to surfactant adsorption.
In terms of the above dimensionless parameters, the clean drop cases correspond to E = 0 or χ = 0 [Eq. (15)]. The case of insoluble surfactants corresponds to Bi= 0 [Eq. (16c)]. The nondiffusive case corresponds to Pe, Pes 1.
We obtain the following dimensionless equations:
−∇p + Ca∇2u= 0, (11)
∇2φ = 0, (12)
∂
∂t + ∇s· (vs)+ (∇s· n)us· n = 1
Pes∇s2 + Jn · ∇C, (13)
∂C
∂t + v · ∇C = 1
Pe∇2C, (14)
γ = 1 + E ln(1 − χ), (15)
where Ca= [1 + E ln(1 − χ )]CaE. On the drop surface, the dimensionless boundary conditions are given by
φ = 0, σ ∇φ · n = 0, (16a)
−p + Ca(∇uT+ ∇u) · n + CaE
EE−1
2(E· E)I
· n = γ (∇s· n)n − ∇sγ , (16b)
Jn· ∇C = Bi[Cs(1+ k − k) − ]. (16c)
In Eq. (11) and Eq. (16b) the capillary number Ca= μUd/γ0is the ratio of electric stress to tension in the absence of surfactant.
The right-hand side of the stress balance Eq. (16b) shows that two surfactant-related mechanisms govern the deformation of drops. The first mechanism is driven by the capillary pressure, and acts in the normal direction. It is further broken down into two phenomena: tip-stretching and surface dilution [70]. In tip-stretching, a decrease in surface tension γ < 1 at the pole due to surfactant yields larger drop deformation compared with the surfactant-free case. The area-average surface tensionγavg gives a global measure of the dilution effect: compared with the clean case, smaller deformations are attained forγavg> 1. The second mechanism is driven by the Marangoni stress, which acts to suppress [70,71] or even reverse [71] surface convective fluxes. The Marangoni stress acts in the tangential directions, and consists of two principal components: the derivative of surface tension as a function of surfactant concentration (∂γ /∂) and the surfactant concentration gradient (∂/∂θ), where θ is the angle parameter.
These nontrivial and highly nonlinear mechanisms pose challenges in studying the EHD of a surfactant-laden viscous drop. Analytical solutions of the transport equation are only possible in very restricted limits [72], and often numerical simulations are necessary. Several computational methods have been developed to simulate surfactants effects on droplets [73–76]. In the context of EHD, we refer the readers to the results in Refs. [43,46,77].
In this work we implement a numerical code based on the immersed interface method (IIM) integrating numerical tools developed by our group [35,78,79]. A description of the numerical setup is provided in AppendixA, together with numerical validation in AppendixBand convergence study in AppendixC.
Moreover we fix the elasticity constant E= 0.2 and conduct simulations with various combinations of parameters to investigate the effects of surfactant solubility on the drop electrohy- drodynamics. Our simulations show that deformation and flow patterns appear to be invariant with increasing surfactant solubility when the surfactant coverageχ < 0.8. We therefore focus our anal- ysis on elevated surfactant coverage withχ = 0.9. This surfactant coverage is in the relevant range in many experimental setups [42,67,80], and the corresponding (dimensionless) surface tension γeq= 1 + E ln(1 − χ ) = 0.54 and adsorption number k = χ/(1 − χ ) = 9. The Péclet numbers Pe= PeS = 100 in Sec.III. It is then increased to Pe= PeS= 500 for the prolate shapes in Sec.IV.
The transfer parameter J = 10−2for the oblate cases, and J= 2 × 10−3for the prolate cases. These values of the bulk Péclet number and transfer parameter yield an adsorption depth had= 1, which is physically relevant for a millimeter size drop with∞= 2.5 × 10−6m2/mol and bulk surfactant concentration ten times the critical micelle concentration [67]. The limit J 1 corresponds to the diffusion-controlled surfactant transport that is relevant in many practical applications [69].
At T = 0, the drop is spherical, and the surfactant concentrations C = 1 and = 1 are imposed uniformly in the bulk and on the drop surface, respectively.
III. GENERAL SURFACTANT EFFECTS ON A VISCOUS DROP UNDER AN ELECTRIC FIELD The shape of a clean viscous drop under an electric field can be either prolate, oblate or spherical with either a clockwise or counterclockwise circulation. The circulation is counterclockwise when ε+/σ+> ε−/σ−, and clockwise otherwise [81]. The dashed line in Fig. 2(a)represents Taylor’s discriminating function and delimits between prolate and oblate shapes. The circulation pattern, counterclockwise (equator-to-pole) or clockwise (pole-to-equator), are determined according to the flow inside the first quadrant of a axisymmetric drop in our numerical setup [Fig.14(b); also see Ref. [41] and references therein]. A steady equilibrium drop shape exists as long as the electric capillary number CaE is below the critical value. Beyond this value the viscous drop continues to elongate and eventually breaks up with complicated flow patterns inside the drop [41].
A. Insoluble surfactant: Bi= 0
When covered with insoluble surfactants (negligible adsorption-desorption kinetics), we find that insoluble surfactants may give rise to multiple toroidal vortices in a spheroidal drop, which is not observed for a clean drop [41]. This is summarized in Fig.2, which shows that due to an insoluble
FIG. 2. (a) Phase diagram for the insoluble case (Biot number Bi= 0) with surfactant coverage χ =
eq/∞= 0.9 and PeS= 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 (σr, εr)= (0.1, 10) reaches a steady equilibrium (solid line), while the drop with (σr, εr)= (10, 0.03) continues to elongate (dashed line) past T = 10. In each panel, the vector fields are plotted on a relative scale.
surfactant a prolate drop can have a clockwise circulation [Fig.2(b)], multiple toroidal vortices in a quadrant [Fig.2(c)] and a counterclockwise circulation [Fig.2(d)]. Figure2(f)shows that the flow inside an oblate drop can also have multiple toroidal vortices in a quadrant. In addition to changing the flow pattern, we also find that surfactants can reduce the critical capillary number to below CaE = 0.3 as shown in panel G, where a drop with multiple toroidal vortices continues to elongate past T = 10 at CaE = 0.3. Consequently the deformation number, defined as D = LL−B+B (L is the length of the major axis and B is the length of the minor axis of the ellipsoid), continues to increase (dashed curve) in this case.
Figure2(a)is the phase diagram of a surfactant-laden viscous drop withχ = 0.9 and Bi = 0 (no adsorption-desorption kinetics). For each point we simulate the electrohydrodynamics up to T = 10 for several values of the electric capillary number up to CaE = 0.3. A blue circle denotes that a steady equilibrium is reached before T = 10 and the critical capillary number is larger than 0.3. A red star denotes that a steady spheroidal drop shape is not stable and the critical capillary number is less than 0.3. The dashed curve in Fig.2(a)separates an oblate drop (above the curve) from a prolate drop (below the curve). The shaded regions delimit the various circulation patterns: clockwise (c), counterclockwise (cc), and multiple toroidal vortices (s). Our numerical findings show that these circulation patterns are qualitatively similar for the range of electric capillary numbers we investigated (CaE 0.3).
The stability of spheroidal drop shape summarized in Fig.2(a)is consistent with prediction from our previous semianalytical model [45], which shows that equilibrium shapes exist below Taylor’s discriminating function (prolate) withσr 1: In these cases, the capillary pressure is greater than the normal component of the Maxwell stress (or electric pressure). The stability of a spheroidal drop is lost and equilibrium shapes cease to exist for drops below Taylor’s discriminating function with σr > 1, where the electric pressure is greater compared to the capillary pressure.
Figure3(a)shows the equilibrium surfactant distribution for the three prolate drops in Figs2, and Fig.3(b) is the corresponding surfactant distribution for the two oblate drops in Fig.2. We observe that regardless of the different flow patterns inside a drop, the surfactant concentration is always maximum at the pole atθ = π/2 for a prolate drop, while for an oblate drop reaches maximum at the equator atθ = 0. This is because at large Péclet number (low surfactant diffusivity) the dynamic surfactant redistribution from the initial homogeneous coverage is dominated by the surface dilation or contraction as the drop deforms. When the drop shape reaches an equilibrium,
(a) (b)
Γ
θ θ
Γ
0 0.5 1 1.5
0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02
r=1, r=0.03 (CC) r=0.1, r=10 (C) r=0.3, r=0.03 (S)
0 0.5 1 1.5
0.9 0.92 0.94 0.96 0.98 1 1.02 1.04
r=3, r=1 (S) r=0.3, r=10 (C)
FIG. 3. (a) Surfactant distribution for the three prolate drops in Figs2(b),2(c)and 2(d). (b) Surfactant distribution for the two oblate drops in Figs.2(e)and2(f). The valuesθ = 0 and θ = π/2 denote the equator and pole, respectively.
the surfactants also quickly approach a distribution that is close to the equilibrium distribution [70]:
For a prolate (oblate) drop, the interface’s contraction took place at the pole (equator) and hence the maximum surfactant atθ = π/2 (θ = 0).
For the prolate drops in Fig.3(a), Fig.4shows the corresponding tangential velocity ut = vs· t [Fig.4(a)], Marangoni stressγs= −Eχs/(1 − χ) [Fig.4(b)], normal Maxwell stress [Fig.4(c)], and tangential Maxwell stress [Fig. 4(d)]. For the oblate drops in Fig. 3(b), Fig. 5 shows the corresponding tangential velocity ut [Fig.5(a)], Marangoni stressγs [Fig.5(b)], normal Maxwell stress [Fig.5(c)], and tangential Maxwell stress [Fig.5(d)].
Combining the simulation results (such as those summarized in Figs.3,4, and5) for all the cases in Fig.2, we can draw the following conclusion: (1) The spatial distribution of the normal Maxwell stress correlates to the drop shape: Normal Maxwell stress is the largest at the pole for a prolate drop, while for an oblate drop the normal Maxwell stress reaches maximum at the equator. (2) Similarly to a clean drop, the flow magnitude in a clockwise circulation is larger than in a counterclockwise circulation. The tangential Maxwell stress on a prolate drop with a clockwise circulation is nearly zero as in Fig.4(c). This implies that, for a prolate drop with a clockwise circulation, the dominant balance in traction is between the viscous stress and the Marangoni stress. (3) For a prolate drop
FIG. 4. For the three prolate drops in Fig.2: (a) tangential velocity ut, (b) Marangoni stressγs(the spatial derivative of surface tensionγ ), (c) normal Maxwell stress, and (d) tangential Maxwell stress. The values θ = 0 andθ = π/2 denote the equator and pole, respectively.
FIG. 5. For the two oblate drops in Fig.2: (a) tangential velocity ut, (b) Marangoni stressγs (the spatial derivative of surface tensionγ ), (c) normal Maxwell stress, and (d) tangential Maxwell stress. The values θ = 0 andθ = π/2 denote the equator and pole, respectively.
with a counterclockwise circulation, the viscous stress is small and dominant balance in the traction is between the Maxwell stress and the Marangoni stress.
B. Soluble surfactant: Bi= 10 and J = 10−2
In this section we investigate the effects of surfactant exchange on drop dynamics. To quantify these effects, we consider cases discussed in the previous section to clearly highlight how increasing the Biot number affects the dynamics. To draw direct comparison with the phase diagram in Fig.2 for the insoluble cases, we summarize the results for the soluble cases in Fig.6: Filled circles denote parameters where a steady spheroidal drop is found at equilibrium up to Ca= 0.3, while the stars are for parameters where the drop does not reach a steady spheroidal equilibrium for Ca= 0.3.
Comparing between the insoluble results in Fig.2and the soluble results in figures6we find that surfactant solubility changes the steady equilibrium of a spheroidal drop: For parameters below the dashed curve in Fig.2(a)and Fig.6, we observe that surfactant solubility destabilizes the spheroidal drop forσr < 1, while stabilizes the spheroidal drop for σr > 1.
In Fig. 6(a)the green region represents parameters where the surfactant solubility leads to a net adsorption as the total amount of surfactant on the drop increases. The gray region is for desorption-dominance, where the total amount of surfactant on the drop interfaces decreases. The
FIG. 6. Phase diagram for the soluble case (Biot number Bi= 10) with surfactant coverage χ = eq/∞= 0.9, PeS= 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.
FIG. 7. Insoluble (solid line) versus soluble (dashed line) surfactants for a prolate drop (top row, CaE= 0.3) and for an oblate drop (bottom row, CaE= 0.25) on the [(a) and (d)] surfactant distribution, [(b) and (e)]
tangential velocity ut = vs· t, and [(c) and (f)] Marangoni stress γs(the spatial derivative of surface tensionγ ).
The Péclet number Pe= 100 and J = 10−2, and the valuesθ = 0 and θ = π/2 denote the equator and pole, respectively.
sorption kinetics is minimal in the unshaded region inσr> 3, εr> 3, where the relative change in total surfactant concentration is less than 10−3. Figure6(b)shows the corresponding distribution of circulation patterns in the presence of soluble surfactants: The red region is for counterclockwise (cc) circulation (equator to pole), the green region is for clockwise circulation (pole to equator), and the gray region is for multiple toroidal vortices inside the first quadrant of the drop.
Focusing on the parameters where the surfactant solubility alters the stability of a spheroidal drop, we find that in the adsorption-dominant region the stability is lost when the surfactant solubility decreases the average surface tension, as shown in the red region in Fig. 6(c). On the other hand, in the desorption-dominant region, the stability of a spheroidal drop is retained when the surfactant desorption leads to sufficient increase in average surface tension, as shown in the blue region in Fig.6(c).
Next we consider how surfactant sorption and desorption kinetics affects the drop dynamics at various combination of (σr, εr): First we choose (σr, εr)= (0.3, 0.03) as a representative case where a prolate drop can reach a steady equilibrium when Bi= 0 (solid line in Fig. 7), and becomes unstable with Bi= 10 (dashed line). Figure7illustrates the effect of solubility on surface distribution of surfactant [Figs. 7(a) and 7(d)], the tangential velocity ut [Figs. 7(b) and7(d)], and Marangoni stressγs Figs. 7(c)and7(f)]. Adsorption onto the drop surface homogenizes the surfactant distribution [Fig. 7(a)], leading to smaller Marangoni stress [Fig. 7(c)] and stronger tangential flow [Fig. 7(b)]. The Maxwell stress also sees an increase due to solubility, albeit small in comparison to the change in Marangoni stress. As surfactant exchange takes place, the capillary pressure is lowered and is overtaken by the electric pressure leading to a loss in stability.
FIG. 8. (a) Deformation of a prolate drop with (σr, εr)= (0.1, 5) as a function of dimensionless time T . The electric capillary number CaE = 0.25. The Biot numbers Bi = 0 (insoluble case) and Bi = 1. The inset shows the drop shapes at various times. [(b)–(d)] Circulation for the prolate drop with Bi= 0 at times T = 0.125, T = 0.625, and T = 4, respectively. [(e)–(g)] Circulation for the prolate drop with Bi = 1 and J = 2× 10−3at times T= 0.125, T = 0.625, and T = 4, respectively. The Péclet number Pe = PeS= 500.
For an oblate drop, we choose (σr, εr)= (3, 1) as a representative case, where surfactant solu- bility stabilizes an oblate drop by piling up the drop surface with surfactant from the bulk. At this set of (σr, εr), an oblate drop lacks an equilibrium spheroidal shape in the presence of insoluble surfactants [Fig. 2(a)]. Increasing surfactant solubility suppresses the Marangoni stress almost entirely [Fig. 7(f)]. The diminished Marangoni stress yields a clockwise circulation [Fig.7(e)], identical to that observed for a clean spheroidal oblate drop. Put together, these results suggest solubility reestablishes the balance between viscous and Maxwell tractions leading to a steady spheroidal oblate drop.
For the remainder of our study, we investigate the effects of the transfer parameter J using values of the permittivity and conductivity ratios corresponding to existing data in the literature [38,41,43,46,51]. Comparison between Fig.2(a)and Fig.6(b)shows that the effects of solubility are much more pronounced for prolate drops. Thus in the following section we focus on the following three aspects of the solubility effects on a prolate drop: loss of stability, variability in the flow structure, and larger deformation with increasing surfactant exchange.
IV. EFFECTS OF SURFACTANT PHYSICOCHEMISTRY ON A PROLATE DROP A. Increasing Biot number destabilizes a prolate drop
Here we show that enhancing the surfactant solubility (by increasing Biot number) destabilizes a spheroidal prolate drop withσr < 1. We use the combination (σr, εr)= (0.1, 5), where a surfactant- free viscous prolate drop has a counterclockwise circulation under an electric field. For a clean drop the steady equilibrium exists at all values of CaE [41]. For a drop covered with insoluble surfactants (Bi= 0) and Péclet number PeS = 500, we find that an equilibrium drop shape exists for CaE 0.3 [Fig.2(a)]: At T = 0.125 the drop first elongates along the electric field direction with a tangential flow on the interface that moves the surfactant from equator to pole Fig. 8(b)]. As the surfactant accumulates and builds up the Marangoni stress, a circulation from pole to equator develops around T ∼ 0.6 [Fig.8(c)] and the drop reaches an equilibrium prolate shape with a clockwise circulation after T ∼ 4 [Figure8(d)]. This circulation at equilibrium is opposite to that of a clean prolate “A”
drop with a much weaker flow strength.
As we increase Bi (in the diffusion-controlled regime), we find that the steady spheroidal shape no longer exists and the drop keeps deforming until the end of simulations (up to T = 20), as illustrated by the dashed (J= 2 × 10−3) and dash-dotted (J = 10) lines in Fig.8(a).
FIG. 9. Surfactant distribution (a), tangential velocity ut= vs· t (b), and Marangoni stress γs(the spatial derivative of surface tensionγ ) (c) for the prolate drop in Fig.8at T = 19. The solid lines are for Bi = 0 (the insoluble surfactant case). The dashed and dash-dotted lines are for Bi= 1 with J = 2 × 10−3 and J= 10, respectively. The Péclet number Pe= PeS= 500, and the values θ = 0 and θ = π/2 denote the equator and pole, respectively.
For comparison, Figs.8(e)–8(g)shows the circulation with Bi= 1 and J = 2 × 10−3 at times T = 0.125, 0.625, 4.
Figure 9(c) shows that the Marangoni stress is reduced in magnitude because the surfactant on the drop surface is homogenized Fig.9(a)] by the adsorbed surfactants. Figure9(b)shows the corresponding tangential velocity on the drop interface.
For insoluble surfactants [solid curves in Figs. 9(a) and 9(c)], the surfactant has the most spatial inhomogeneity that corresponds to a large Marangoni stress. With soluble surfactant in the diffusion-controlled regime (Bi/J > 1, dashed curves) the surfactant sorption kinetics greatly reduces the Marangoni stress, giving rise to larger drop deformation. In the sorption-controlled regime (Bi/J = 0.1 < 1, dash-dotted curves) the surfactant concentration is nearly homogeneous and the Marangoni stress is quite small, corresponding to the largest and fastest deformation in Fig.8. Moreover, with surfactants accumulating on the drop surface in equilibrium with the bulk, the capillary pressure is reduced below the electric pressure, leading to the loss of stability of a spheroidal prolate drop.
B. Effects of Biot number on flow around a prolate drop
Here we investigate the effects of solubility on the flow field. Specifically we focus on the combination (σr, εr)= (1/3, 1) with CaE = 0.3, where equilibrium prolate drop is found for both insoluble (Fig.2) and soluble surfactants (Fig.6). In the diffusion-controlled regime, simulations show that the equilibrium drop deformation is minimally influenced by surfactant solubility for CaE ∈ [0, 0.3] because sorption kinetics induce little change in the total amount of surfactant [Fig.12(a)]. Consequently the average surface tension does not vary much with Bi, leading to little change in drop deformation with increased surfactant solubility.
The flow pattern, on the other hand, is highly dependent on the surfactant distribution and kinetics. Without surfactant a clean equilibrium prolate drop with (σr, εr)= (1/3, 1) has a coun- terclockwise flow under an electric field. For Bi= 0 the transport of an insoluble surfactant and the corresponding Marangoni stress gives rise to an interior flow dominated by a clockwise circulation with a small toroidal flow around the pole as shown in Fig.11(a). The corresponding tangential velocity is shown in Fig.10(b). As the Biot number is increased to Bi= 10−2the toroidal flow near the pole expands as shown in Fig.11(b), with the corresponding tangential velocity in Fig.10(b).
When we further increase the Biot number (Bi= 10), the flow transitions to a counterclockwise circulation with a small toroidal vorticity around the equator [Fig.11(c)]. The surfactant is more uniformly distributed [dash-dotted curve in Fig.10(a)and the Marangoni stress is of the smallest
FIG. 10. A prolate drop with (σr, εr)= (1/3, 1) and CaE = 0.3. [(a) and (d)] Surfactant distribution, [(b) and (e)] tangential velocity ut = vs· t, and [(c) and (f)] Marangoni stress γs (the spatial derivative of surface tensionγ ) as a function of θ. Solid lines are for Bi = 0 (the insoluble surfactant case), dashed lines are for Bi= 10−2, and dash-dotted lines are for Bi= 10. The transfer parameter J = 2 × 10−3[(a)–(c)], and J= 10 [(d)–(f)]. The Péclet number Pe = PeS= 500, and the values θ = 0 and θ = π/2 denote the equator and pole, respectively.
magnitude in Fig.10(c)]. This is because adsorption dominates in the surfactant kinetics, leading to a nearly uniform surfactant distribution on the drop surface and therefore smaller overall Marangoni stress at equilibrium. Results in Fig. 10 also show qualitative similarity in the effects of Bi between J= 2 × 10−3 [diffusion-controlled, Figs.10(a)–10(c)] and J= 10 [sorption-controlled, Figs.10(d)–10(f)].
Figure 11compares the flow pattern between various values of Bi and J. The surface of the drop is color-coded to indicate adsorption (blue) and desorption (red). The grayscale colorbar indicates the change in bulk surfactants. For this prolate drop, adsorption dominates the kinetics and we expect the total surface surfactants to increase. This leads to a region depleted of bulk surfactant C≈ 0.7. For J = 2 × 10−2 our simulations show that adsorption takes place around the pole [Fig.11(b)], and higher Biot number yields a wider surfactant-depleted region [Fig. 11(c)].
Whereas increasing the transfer parameter to J = 10 does not change the qualitative features of the flow, the enhanced surfactant kinetics homogenizes the bulk surfactant. The toroidal flow is still observed at intermediate Biot numbers [Fig.11(e)]; however, it is now concentrated midway between the pole and equator.
C. Effects of transfer parameter on total surface surfactant concentration and transient deformation dynamics
In this section we investigate how the total amount of surfactant on the drop surface depends on the transfer parameter J and Biot number Bi. We defineA as the difference in total amount of
FIG. 11. Flow field for a prolate drop with (σr, εr)= (1/3, 1) and CaE= 0.3, corresponding to velocities in Figs. 10(b) and 10(e). (a) Bi= 0, (b) Bi = 10−2, and J= 2 × 10−3; (c) Bi= 10 and J = 2 × 10−3; (d) Bi= 0; (e) Bi = 10−2and J= 10; (f) Bi = 10 and J = 10. The Péclet number Pe = PeS = 500.
surfactant on the drop surface between time T and initial time 0:
A ≡ 1
A0(AT− A0)≡ 1 (0)ds
(T )ds −
(0)ds
. (17)
Using this definition,A > 0 denotes adsorption, and A < 0 represents desorption. At T = 0, the surfactant concentrations are uniformly applied in the respective domains, andA = 0.
Figure12showsA as a function of Biot number, for the prolate drop in Sec.IV Bwith an initially uniform surfactant distribution in the bulk. For J = 2 × 10−3 (solid curve), we see thatA exhibits a nonmonotonic behavior with a critical Biot number Bicr≈ 0.3, where the adsorbed surfactant concentration is maximized. Similarly for J= 10 (dashed curve) the difference in total surfactant reaches a maximum at a similar Biot number and then reaches a plateau.
We also observe that the transient dynamics of drop deformation depends on J. For a prolate drop with (σr, εr)= (0.97, 0.75) and a given value of Bi, Fig.13(a)shows that the drop deformation number D displays an overshoot en route to the equilibrium for small J (the diffusion-controlled regime). Such overshoot in the drop deformation is found for weakly diffusive insoluble surfactant [45]. However, as shown in Fig.13(see inset for close-up of the transient overshoot), the transient overshoot dynamics is suppressed at large J (the sorption-controlled regime). In this limit, the dynamics becomes closer to the uniform surface tension case [69] where the deformation mono- tonically reaches its equilibrium value. These observations also hold for oblate drops, as Fig.13(b) illustrates for (σr, εr)= (4/3, 2).
FIG. 12. Change in total amountA of surface surfactants for the prolate drop in Sec.IV Bwith (σr, εr)= (1/3, 1) and CaE = 0.3. The Biot number Bi = 1 and the Péclet number Pe = PeS= 500.
V. CONCLUSION
Experimental studies [55,82–84] show that the transport of surface active agents (surfactants) is nonlinearly coupled with drop shape, adsorption and desorption kinetics, and hydrodynamic flow. Consequently analytical investigation on electrohydrodynamics of a viscous drop with sur- factant sorption kinetics is challenging, and the numerical method in this study provides a tool to
FIG. 13. Deformation as a function of dimensionless time T for (a) a prolate drop with (σr, εr)= (0.97, 0.75), and (b) the oblate drop with (σr, εr)= (4/3, 2). CaE = 0.25 for both cases. The solid and dotted lines are for low (J= 10−3), and high (J= 10) transfer parameter with Bi = 10−1, respectively. The dashed and dash-dotted lines represent higher Biot number (Bi= 10) with J = 10−3and J= 10, respectively.
FIG. 14. (a) The numerical algorithm for the second-order immersed interface method code: At tn the drop shape x, flow field u, and interface velocity, 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 (φ) and interface surfactant profile (). Given , we determine the change in surface tension γ , as well as the updated drop shape, flow field, and interface velocity at time tn+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).
quantitatively investigate surfactant exchange between the bulk and the drop surface, and their effects on deformation. We show that surfactant solubility combines with the electric properties of the fluids in nontrivial ways to produce a wide range of electrohydrodynamics.
First, we find that solubility affects the stability of a spheroidal drop by changing the balance between the capillary and electric pressures. Specifically we find that the surfactant solubility makes a prolate spheroidal drop lose instability if the total amount of surfactant increases (forσr < 1), while the prolate drop retains its stability when the total amount of surfactant increases (forσr > 1).
Moreover, we show that solubility drastically changes the equilibrium flow: It suppresses multiple toroidal vortices that dominate in the absence of surfactant solubility. The resulting flow patterns resemble that observed for surfactant-free drops.
Second, we investigate the effects of the transfer parameter J. We find that increasing the transfer parameter J in the diffusion-controlled regime (Bi/J > 1) pushes the drop hydrodynamics toward that of a clean drop by homogenizing the surfactant concentration on the drop surface. Under the same conditions, the average surface tension is also increased. As a result, deformation is suppressed for a given CaE as in the diffusion-controlled regime for a surfactant-laden bubble [69].
Similarly increasing the surfactant solubility Bi 10 homogenizes the surfactant distribution even more and the Marangoni stress is almost completely suppressed. Under these conditions the drop behaves as a clean drop with a much lower average surface tension, and transient overshoots in the deformation dynamics are no longer observed. We tested this hypothesis, and our numerical results show deformations for the clean drop atγeq= 0.54 and surfactant-covered drop at large Bi and J are identical.
Further increase of J at a fixed Bi pushes the system into the sorption-controlled regime (Bi/J < 1). In this regime the total amount of surfactant on the drop interface increases little and the average surface tension is not affected much by surfactant sorption kinetics. However, the spatial
FIG. 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)σr = 0.1, εr= 0.1; (c) σr= 0.5, εr = 20.
For the surfactant-covered drop cases we set E = 0.2, χ = 0.3, Pes= 10, Bi = 0: (b) σr = 0.3, εr= 1; (d) σr = 1, εr = 2. Volume and total surfactant are conserved to within 5% in all cases.
variation in surface surfactants is sufficient to induce different flow patterns for the range of electric capillary number we used in the simulations.
Our results suggest that the critical CaE for an equilibrium drop shape also depends on surfactant solubility. We are now investigating the correlation between the critical CaE and other parameters (such as the surfactant coverageχ and Bi), and how the various flow patterns develop at a stronger electric field strength.
ACKNOWLEDGMENTS
H.N. acknowledges support from John J. and Char Kopchick College of Natural Sciences and Mathematics at Indiana University of Pennsylvania. W.F.H. acknowledges support from Ministry of Science and Technology of Taiwan under research Grant No. MOST-107-2115-M-005-004- MY2.
M.C.L. acknowledges support in part by Ministry of Science and Technology of Taiwan under research Grant No. MOST-107-2115-M-009-016-MY3 and the National Center for Theoretical Sciences. Y.N.Y. acknowledges support from NSF under Grant No. DMS-1614863, as well as support from Flatiron Institute, part of Simons Foundation.
TABLE II. Numerical convergence for the flow field variables u= (u, v) and the pressure p.
N ||uN− u2N||∞ Rate ||wN− w2N||∞ Rate ||pN− p2N||∞ Rate
32 1.769 × 10−1 — 1.777 × 10−1 — 7.303 —
64 2.724 × 10−2 2.7 1.086×10−1 0.711 5.408 × 10−2 7.08
128 2.436 × 10−2 0.161 4.21×10−2 1.37 1.402 × 10−2 1.95
256 2.498 × 10−3 3.29 9.36×10−3 2.17 2.025 × 10−3 2.79
APPENDIX A: NUMERICAL IMPLEMENTATION
We solve the governing equations in the axisymmetric cylindrical coordinates (r, z) [Fig.14(b)], considering only the r 0 half-plane. Once the solution is obtained, it is extended to the left half- plane by symmetry.
Figure14(a)illustrates the algorithm. The droplet shape and position x, flow field u and interface velocity U are computed using the IIM solver in Refs. [35,78]. The boundary conditions in the computational domain = [0, L] × [−L, L] in Fig.14(b)are given as follows: For the electric po- tential,φ+= ∓E0L/2 at z = ±L (the bottom BC3 and top BC4 of the computational domain), while a Neumann boundary condition∂φ/∂r = 0 is imposed on the sides (r = 0, L) of the computational domain. For the Stokes equations, the pressure and velocity∂ p/∂r = 0, ∂w/∂r = 0, u = 0, at r = 0 (BC1), while Dirichlet boundary conditions are imposed on the other three sides (BC2–BC4) [35].
For the bulk surfactant concentration C, Neumann (BC1) and no flux (zero Neumann) (BC2–BC4) boundary conditions are imposed [79]. Note that the soluble surfactant only exists outside of the drop.
For more detailed implementation steps and numerical methods, the reader is referred to Ref. [35]
for the electrohydrodynamic solver. The three-dimensional axisymmetric soluble surfactant solver is a straightforward extension of the two-dimensional scheme in Ref. [79]. The main difference is in the treatment of the correction term for the curvature at the irregular grid nodes.
APPENDIX B: VALIDATION
We validate our numerical codes by comparing against results in the literature where the equilibrium deformation number Deqis reported as a function of the electric capillary number CaE, for both a clean drop and and a drop laden with insoluble surfactant. L and B are the drop size along the major and minor axes, respectively. At moderate CaE, the equilibrium drop shape under a dc electric field could be either prolate or oblate. For an oblate drop, the circulation is always from TABLE III. Numerical convergence for the component of interface markers X= (X,Y ), the surface surfactant concentration, the surface tension γ , and surface tension gradient dγ .
M ||XM− X2M||∞ Rate ||M− 2M||∞ Rate
16 4.384×10−2 — 1.053×10−1 —
32 2.29×10−3 4.29 9.954×10−3 3.4
64 4.992×10−4 2.17 1.72×10−3 2.53
128 1.07×10−4 2.2 5.522×10−4 1.64
M ||γM− γ2M||∞ Rate ||dγM− dγ2M||∞ Rate
16 9.213×10−3 — 2.27×10−3 —
32 8.493×10−4 3.44 1.223×10−3 0.892
64 1.477×10−4 2.52 2.897×10−4 2.08
128 4.744×10−5 1.64 4.765×10−5 2.6
TABLE IV. Numerical convergence for the staggered variables: The electric potential (φ) and the bulk surfactant concentration (C).
N ||φN− φ2N||∞ Rate ||CN− C2N||∞ Rate
32 1.556 — 2.016 —
64 4.179 × 10−1 1.9 1.691 0.254
128 6.779 × 10−2 2.62 1.451 0.221
256 1.035 × 10−2 2.71 6.025 × 10−1 1.27
the pole to the equator, while the flow inside a prolate drop can be either from the equator to the pole (prolate “A”) or from the pole to the equator (prolate “B”). In our simulations the computational domain size is [0, 5] × [−5, 5]. The step size h = 5/N where N = 256, and the timestep t = h/10.
Figure15shows comparisons for a clean drop [Figs.15(a)and15(c)] and for a surfactant-covered drop [Figs.15(b)and15(d)]. We test our implementation against the boundary integral (BI) results from Figs. 5 and 19 in Ref. [41]. Figure15(a)shows the equilibrium deformation number Deqas a function of the capillary number CaE for a prolate drop withσr= 0.1, εr = 0.1, while the oblate drop is shown in Fig.15(c)withσr = 0.5, εr = 20. These comparisons show good agreement with the present IIM results.
For the surfactant-covered drop, we consider the work in Refs. [43,47] to validate the prolate and the oblate shapes. For these simulations, the electric parameters are set toσr= 0.3, εr = 1 for the prolate drop (case A in Ref. [43]), andσr= 1, εr= 2 for the oblate drop (case C in Ref. [43]). The elasticity constant E = 0.2 and the surfactant coverage χ = 0.3. Other surfactant-related parameters are as follows: the surface and bulk Peclet numbers PeS = Pe = 10, respectively, and the Biot number Bi= 0 (the insoluble surfactant limit). Figures15(b)and15(d)show excellent agreement between all three numerical methods: BI, IIM, and regularized level-set method.
APPENDIX C: MESH REFINEMENT STUDY
We perform a grid analysis (or mesh refinement) study. We consider a computational domain
= [0, 5] × [−5, 5], to compute the L∞error and determine the ratio Rate = ||AN− A2N||∞
||A2N − A4N||∞, (C1)
where N is the grid size. The number of Lagrangian markers for the interface M = N/2. We run simulations to a final time T = 0.5 with timestep t = 10−3. The electric parameters are CaE = 0.1, εr= 1, σr= 0.3, corresponding to the prolate “A” drop shape (case A in [43]). The surfactant parameters are E = 0.2, Pe = 10, Pes= 10, χ = 0.3, and the solubility parameter Bi= 0.01. TablesII,III, andIVshow the results of the analysis.
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