pubs.acs.org/Langmuir © 2009 American Chemical Society
Electrohydrodynamic Deformation of a Miscible Fluid Stream by a Transverse Electric Field Paul A. Safier and James C. Baygents* Department of Chemical and Environmental Engineering, The University of Arizona, Tucson, Arizona 85721 Received July 7, 2008 . Revised Manuscript Received December 19, 2008 Electrohydrodynamic deformation of a cylindrical fluid stream is analyzed with a quasi-electroneutral model. The stream is miscible with the surrounding liquid, though of different electrical conductivity and permittivity, and is subject to an electric field that acts transverse to the axis of the cylinder. The formulation allows for natural gradients of electrical conductivity and dielectric constant in the transition region between the stream and the surrounding liquid; these property variations are fully coupled to the fluid motion and are assumed to stem from concentration gradients of charge-carrying solutes. Dielectric and Coulombic body forces attendant to the time-dependent, spatial nonuniformities are accounted for. The strength of the electrically driven flows is such that transport of solutes is dominated by advection. As a consequence, the initial conductivity and dielectric constant differences, between the interior of the stream and the surrounding liquid, persist through significant deformation of the stream and characterize the rate at which the stream (continuously) deforms. Calculations for aqueous systems dominated by conductivity effects agree with measurements of stream deformation made by Rhodes et al. [J. Colloid Interface Sci. 1989, 129, 78]. Calculations for systems controlled by dielectric effects show that relative permittivity differences must be at least O(1) if noticeable deformations are to occur in a matter of seconds, which may explain why Trau et al. [Langmuir 1995, 11, 4665] discerned no deformations controlled by dielectric effects in low permittivity, low conductivity systems. An implication of these latter predictions is that experiments to isolate the role of dielectric constant mismatch may not be practicable.
Introduction The use of electromechanical forces to manipulate nonhomogeneous fluids offers many possibilities for materials processing, including electrically driven separations schemes already in practice,1 and the developing areas of electrophoretic deposition2 and related microfluidic methods for particle assembly.3 Design and improvement of these technologies requires a sound understanding of the operative fluid physics. Electrohydrodynamic (EHD) deformation of immiscible liquid drops has been studied extensively from the perspective of theory, computation, and experiment.4-7 A review by Saville8 indicates that the elements of the theory are substantially correct: the sense and extent of the drop deformations are due to the action of interfacial electrical stresses, and moreover, these stresses are directly related to electrical conduction processes in and about the drop. The agreement between theory and experiment is far less satisfying when the deforming body is miscible with the surrounding liquid in which it is immersed. It is generally established that a motive force will arise if the electrical conductivity of the *Author to whom correspondence should be addressed. E-mail: jcb@ maxwell.che.arizona.edu. (1) Mosher, R. A.; Saville, D. A.; Thormann, W. The Dynamics of Electrophoresis; VCH: Weinheim, Germany, 1992. (2) Besra, L.; Liu, M. Prog. Mater. Sci. 2007, 52, 1. (3) Velev, O.; Bhatt, K. Soft Matter 2006, 2, 738. (4) Taylor, G. I. Proc. R. Soc. London A 1966, 291, 159. (5) Torza, S.; Cox, R. G.; Mason, S. G. Philos. Trans. R. Soc. London 1971, 269, 259. (6) Vizika, O.; Saville, D. A. J. Fluid Mech. 1992, 239, 1. (7) Feng, J. Q.; Scott, T. C. J. Fluid Mech. 1996, 311, 289. (8) Saville, D. A. Annu. Rev. Fluid Mech. 1997, 29, 27. (9) Rhodes, P. H.; Snyder, R. S.; Roberts, G. O. J. Colloid Interface Sci. 1989, 129, 78. (10) Saville, D. A. Phys. Rev. Lett. 1993, 71, 2907. (11) Trau, M.; Sankaran, S.; Saville, D. A.; Aksay, I. A. Langmuir 1995, 11, 4665.
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deforming body differs from that of the continuous phase9-15 but at least two significant issues remain in play: one, the role of differences in dielectric constant11 and two, the influence of the continuous transition in electrical properties at the diffuse boundary between the deforming body and the surrounding liquid.10,12 In particular, Trau et al.11 were unable to discern EHD deformation in nonaqueous systems with permittivity (but little electrical conductivity) mismatch, and such an observation stands in marked contrast with the analysis of Rhodes et al.,9 which is predicated on the standard leaky dielectric model of EHD.4 In this article we consider a problem germane to the foregoing discussion, viz., the EHD deformation of a cylindrical stream that initially differs from the surrounding liquid by an increment of dielectric constant Δε and electrical conductivity Δσ. The dynamics of interest are driven by an electric field of magnitude E0, acting orthogonal to the axis of the cylindrical stream. We numerically solve the relevant balance laws, including one for charge-carrying solutes, as a coupled system evolving in space and time. Thus, a transition region between the miscible fluids unfolds naturally as the solutes diffuse and migrate in the field. The effect of the electric field on deformation of the cylinder is investigated for various Δσ and Δε, with a view toward isolating and quantifying the influence of the latter. The computation yields the rate and extent, as well as the sense, of the cross-section’s deformation. For the first time, a range of dielectric constant space is examined to determine the sensitivity of deformation to the relative permittivity mismatch Δε. For systems with matched conductivities (Δσ = 0), our computations clearly show that the (12) (13) 1481. (14) 1055. (15)
Saville, D. A.; Glynn, J. R. Ind. Eng. Chem. Res. 2006, 45, 6981. de Balmann, H. R.; Burgaud, C.; Sanchez, V. Sep. Sci. Technol. 1991, 26, Clifton, M. J.; de Balmann, H. R.; Sanchez, V. Can. J. Chem. Eng. 1992, 70, Heller, C.; Limat, L.; Sergot, P.; Viovy, J. L. Electrophoresis 1993, 14, 1278.
Published on Web 4/20/2009
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Figure 1. Schematic diagram of a CFE cell. The cross-section of the sample stream, which is initially circular, deforms under the action of the electric field, becoming elliptical at the exit of the cell. Note, the sample stream is miscible with the surrounding buffer.
time scale for deformation is μ/ε0ΔεE20, where ε0 is the permittivity of free space and μ is a viscosity representative of the (miscible) fluid system. One important implication of these results is that deformation times are obviously protracted if Δε , 1, and we suggest that this may explain why Trau et al.11 were unable to discern deformation attendant to dielectric mismatch. To set the context for our calculations, we have organized the balance of our presentation as follows. We first discuss extant analyses of EHD deformation experiments on miscible liquid systems, and then summarize the quasi-electroneutral16 conservation laws that we use to account for the salient processes. We show that numerical solutions to the model compare favorably with the measurements of Rhodes et al.,9 which were taken on aqueous streams dominated by conductivity effects (Δε z 0). Next, we show predictions for systems dominated by permittivity mismatch (Δσ = 0). There is but one published investigation on weakly conducting liquids,11 wherein dielectric effects ought to be more important, and these observations and our calculations are not in agreement. We close by positing explanations for this disagreement and suggest that, in light of the emerging importance of electromechanical schemes to manipulate fluid/particle systems,2,3,16 additional experiments on dielectric mismatch would be of interest. Beginning with Rhodes et al.,9 a series of experiments13-15 have shown that EHD deformations occur with sample streams in continuous flow electrophoresis (CFE), a separations process whereby an influent stream of analytes is fractionated with a transverse electric field (Figure 1). Saville and Glynn12 have recently reported analogous deformations of disk-shaped boluses in Hele-Shaw cells. Each of these experimental studies was performed on aqueous systems, and the EHD flows were dominantly influenced by conductivity gradients. Rhodes et al. used Taylor’s leaky dielectric model 4,8 to derive a discriminating function De that indicates the sense of the ribbonlike deformations observed in their experiments, i.e., 2 σs σs εs þ De þ 1 -3 σb σb εb
1 f e ¼ - ε0 E 3 Erε þ Fe E 2
ð1Þ
(16) Pundik, T.; Rubinstein, I.; Zaltzman, B. Phys. Rev. E 2005, 72, 061502-1.
ð2Þ
In eq 2, E is the local electric field and Fe = ε0r 3 (εE) is the free charge density. Spatial variations of ε and E result from composition differences between the sample and buffer fluids.18 The two terms on the right-hand side (RHS) of eq 2 account for dielectric and Coulombic forces, respectively. Incorporating fe into the equations of motion, Saville10 constructed an approximate solution that yielded oblate or prolate deformations, depending on Δσ = σs - σb and Δε = εs - εb, the respective differences in the conductivity and the relative permittivity of the sample and buffer fluids. Saville restricted his analysis to cases with no Coulombic body force, and, as a consequence, predicted no deformation when either Δσ or Δε was zero, though many studies 9,12-15 report observations of significant deformations in systems with Δε z 0. Subsequently, Trau et al.11 performed EHD experiments on spherical boluses of castor oil doped with barium titanate (BaTiO3) particles. Control of the sense of deformation (prolate vs oblate) was achieved by adjusting the conductivity mismatch between the bolus and exterior castor oil host. No EHD deforma6 0, but strong tion was observed19 with Δσ z 0 and Δε ¼ deformations were seen with Δσ ¼ 6 0 and Δε = 0. Trau et al.11 resolved the problem posed by Saville,10 allowing for Coulombic body forces. Their analysis showed that deformation would occur unless Δσ and Δε were both zero, which obviously raises the question as to why no deformations were uncovered in experimental systems with Δσ z 0 and Δε 6¼ 0. To address this specific question, we examine the dynamics of the deformation process, adopting a problem formulation similar to that of Alfonso and Clifton.18
Balance Laws: The Quasi-Electroneutral Model of EHD The velocity field is governed by the Navier-Stokes equations, including the electrical body force, viz., F
Here, σ denotes electrical conductivity and ε denotes relative permittivity; the subscripts s and b respectively refer to the sample and buffer fluids. When De is positive (negative), the stream cross-section is stretched parallel (orthogonal) to the applied field,
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and the deformations are said to be prolate (oblate); when De is nil, no EHD deformation occurs. Viovy and co-workers15,17 extended the leaky dielectric analysis to demonstrate that the oblate versus prolate deformations could be controlled/nullified by the use of crossed, oscillatory forcing fields. Saville10 criticized eq 1 on the basis that it is obtained by treating the sample and buffer fluids as distinct, uniform continua, differentiated by a mathematical boundary at which the electromechanical properties change. In fact, the sample and buffer fluids are miscible and, so, should be treated as a single, albeit nonuniform, continuum. The transition from sample to buffer occurs smoothly over a finite length scale characterized, for example, by electrodiffusional transport transverse to the main flow. The Maxwell stresses exerted on the fluid then give rise to an electrical body force fe,8,10,11 where
Du ¼ -rP þ μr2 u þ f e Dt
ð3Þ
r3u ¼ 0
ð4Þ
with
(17) Limat, L.; Stone, H. A.; Viovy, J. L. Phys. Fluids 1998, 10, 2439. (18) Alfonso, J.-L.; Clifton, M. J. Electrophoresis 1999, 20, 2801. (19) Saville, D. A. Private communication, 5 July 2004.
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Here u, F, and μ are, respectively, the fluid velocity, density, and viscosity, and P denotes the pressure. As we wish to isolate the effects of gradients of electrical conductivity and dielectric constant, we have taken F and μ to be constant. Owing to Maxwell’s equations, E = -rφ, and the (quasistatic) field is coupled to the distribution of ionic solutes. If we restrict the discussion to bipolar conductors, a balance on each charge-carrying species leads to20 DC þ u 3 rC ¼ Dr2 C Dt
ð5Þ
0zeðω þ -ω - ÞkB T r2 C þ r 3 ðσrφÞ
ð6Þ
Table 1. Comparison of Calculated Sample Stream Deformations with Measurements from Rhodes et al.,9 Which Were Taken in Separate Runs or Trialsa dfmeas E (V/cm) 10 10 20 20
σs/σb
trial 1
trial 2
1.63 0.51 0.24 0.61 -0.38 -0.26 1.63 0.97 0.59 0.61 -0.59 -0.63 a Conditions are as specified in Fig 2.
trial 3 -0.42
dfcalc 0.35 -0.34 0.80 -0.79
and
In eqs 5 and 6, D (z+ - z-)ω+ω-kBT/(z+ω+ - z-ω-) is an effective diffusivity, σ e2(z+ω+ - z-ω-)C is the local electrical conductivity, and C z+C+ = -z-C-, where z(, ω(, and C( are the (signed) valence, mobility, and number density of the ionic solutes, kB is Boltzmann’s constant, e is the charge on a proton, and T is the absolute temperature; the superscripts + and - respectively refer to the cationic and anionic solute. These coupled balance laws are a quasi-electroneutral model16 that we solve for u, C, and φ. Equations 3, 4, and 6 were discretized with a penalty finite element formulation.21 Petrov-Galerkin weighting was used on convective terms, and the time integration was done with a semi-implicit backward Euler method. Equation 5 is convectively dominated and was solved with a flux corrected transport algorithm.22 To increase resolution at minimal computational effort, bilinear elements were constructed in a nonuniform mesh. To refine the region around the cylinder, continuous mesh-stretching equations described elsewhere23,24 were used. Figure 1 shows a CFE apparatus with the computational grid at a fixed position along the direction of flow of the sample. It should be noted that the calculation is conducted in a Lagrangian sense, i.e., the mesh moves downward, following the stream, in the direction perpendicular to the x-y plane. In all calculations shown, a nonuniform mesh of 100 elements in each dimension was used. The initial condition consisted of a circular region (b = 0.5 mm) of solute with a sharp interface separating it from the buffer. This distribution was allowed to diffuse for 15 min prior to the application of the electric field. A no-slip condition was imposed on u at the boundaries of the computational domain (x,y = (5), although the radius of the cylinder is small enough to preclude influence from the domain boundaries. No-flux boundary conditions are applied for the solution of eq 5 and in eq 6. Dirichlet conditions specifying the electric potential at the electrodes (cf. Figure 1) are imposed. Numerical convergence was established by increasing grid resolution until negligible change in the solution was observed. To create the nonuniform mesh, eq 5-223 from Anderson24 was used with yc = 0.0, xc = 0.0, and τ = 5.0.
Results and Discussion To test our formulation, we compare selected calculations with the measurements of Rhodes et al.9 Their CFE experiments (20) Newman, J. S. Electrochemical Systems; Prentice Hall: Upper Saddle River, NJ, 1991. (21) Heinrich, J. C.; Pepper, D. W. Intermediate Finite Element Method; Taylor and Frances: Philadelphia, 1999. (22) Boris, J. P.; Book, D. L. J. Comput. Phys. 1973, 11, 38. (23) Roberts, G. O. Proc. Second Int. Conf. Num. Meth. Fluid Dyn. Lect. Notes Phys. 1971, 8, 171. (24) Anderson, D. A. Computational Fluid Mechanics and Heat Transfer; Hemisphere Publishing Co.: New York, 1984.
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Figure 2. Contours of constant solute concentration for conditions consistent with the experiments of Rhodes et al.9 Contours are for Θ = 0.4. Contour 0 shows the initial condition. Contours 1 and 2 are at t = 16 s with σs/σb = 0.61 and 1.63, respectively. Δε = 0, E0 = 20 V/cm, ε = 78, μ = 0.001 Pa 3 s, F = 1.0 g/mL, b = 0.5 mm.
(Figure 1) were carried out in aqueous electrolyte with little to no permittivity mismatch (ε = 78; Δε z 0), and data were obtained for different conductivity mismatches and voltage gradients. Density and viscosity differences between the sample stream and the surrounding buffer were negligible. The deformation of sample streams at the CFE chamber exit are summarized in Table 1 as dfmeas = (d1 - d2)/(d1 + d2) - [(d1 - d2)/(d1 + d2)]o; residence time in the apparatus was approximately 16 s. d1 and d2 are the sample stream dimensions parallel and transverse to the applied electric field, respectively, as indicated on contour 1 of Figure 2. The subscript ‘o’ refers to (small) deformations, measured at zero applied electric field, that presumably stem from the hydrodynamics of the CFE apparatus. This portion of the deformation was subtracted from df for comparisons with our numerical calculations, which are summarized under the heading dfcalc in Table 1. Contours 1 and 2 of Figure 2 respectively show the computed cross-section of the sample stream at the chamber exit (t = 16 sec) for σs/σb of 0.61 and 1.63, Δε = 0, and E0 = 20 V/cm. In the figure, the contour Θ = 0.4, where Θ = (C - Cb)/ (Cs - Cb), is used to delineate the outer boundary of the samplestream cross-section; the subscripts s and b respectively denote the uniform properties of the influent sample and buffer fluids (t = 0). The solute transport is advectively dominated, so the use of alternative contours (e.g., Θ = 0.9) does not significantly change the picture. Note that, based on the definition Langmuir 2009, 25(10), 6000–6004
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be of O(1) for noticeable deformations to occur in a matter of seconds. BaTiO3 particles have a large dielectric constant (300-10 000), and an estimate of the dielectric constant mismatch between bolus and host fluid follows from the Maxwell-Garnett formula,25 viz., εeff ¼ εb þ 3f εb
Figure 3. Time tdef required for deformation df as a function of dielectric mismatch Δε. Conditions are for BaTiO3 particles in castor oil as in Trau et al.:11 εb = 4.43; μ = 1 Pa 3 s; F = 0.96 g/mL; E0 = 2000 V/cm; Δσ = 0.
of σ that follows eq 6, the conductivity varies locally with Θ, viz., σ = σb + ΔσΘ. The entries in Table 1 show that the model compares favorably with measurements on systems dominated by conductivity effects. Such agreement is not entirely unexpected since Clifton et al.14 used a similar model to explain EHD deformations they measured in a CFE apparatus. Our primary objective in this communication, however, is to consider the sensitivity of deformation to Δε. To do this, we performed a series of computations with Δσ set to zero and assumed that the dielectric constant varied linearly with Θ,11,18 i.e., ε = εb + ΔεΘ. For these circumstances, Θ (or C) is understood to account for an entity, such as a colloidal particle, that can influence the effective permittivity of the liquid in which it is dispersed; it is for illustrative purposes alone, that we assume Θ has no effect on the electrical conductivity (nor F and μ). Results of these calculations are summarized in Figure 3, where tdef, the time to deform a given extent, is shown as a function of Δε for selected values of df. The shape of the deformed crosssections are included on the figure, and tdef is scaled on μ/Δεε0E20, which is clearly the appropriate normalization. Cross-sections are initially circular, as depicted for contour 0 of Figure 2, and the time scale holds even for large deformations. The calculations demonstrate, then, that tdef varies inversely with Δε. The values prescribed for the various parameters used in the calculation ( μ, E0, etc.) were chosen to correspond with the experimental conditions of Trau et al.,11 who conducted experiments of EHD deformation of spherical boluses in castor oil. Fields of 2000 V/cm were used to deform boluses that were doped with BaTiO3 particles to create a dielectric constant mismatch. The influence of conductivity mismatches, between bolus and host fluid (clear castor oil), were also investigated by adding small amounts of tetrabutylammonium tetraphenylborate. With no conductivity mismatch, no EHD deformation was observed; however, in approximately one-second intervals, boluses were significantly deformed as a result of a conductivity mismatch. According to our results for Δε = 0.05, the times required for deformations of -0.05, -0.1 and -0.2 are, respectively, 8.2, 16.6, and 34.1 s, all of which are considerably longer than the times reported by Trau et al. for deformations resulting from conductivity mismatches. Moreover, μ/ε0E20 is 2.8 s when E0 = 2000 V/cm, so, based on our calculation, Δε would need to Langmuir 2009, 25(10), 6000–6004
εp -εb εp þ 2εb -f ðεp -εb Þ
ð7Þ
if we neglect double layer effects. Here, εeff is the effective dielectric constant of a dilute suspension of dielectric spheres and f and εp are the particle volume fraction and dielectric constant, respectively. The result is an effective dielectric constant in the bolus that differs by less than 0.1% from that of the host fluid at f = 2.5 10-4, the particle volume fractions used by Trau et al. On the basis of eq 7, f ≈ 0.1 is required to achieve an O(1) dielectric constant mismatch, if εp ≈ 104. At such particle volume fractions, buoyancy effects would rival the EHD flows. One feature evident in Figure 3 is that the deformation time varies inversely with Δε for both small and large deformations, meaning the dielectric mismatch persists as the cross-section of the stream distorts. By a conservative estimate, ε0Δε(E0b)2/μD, the Peclet number for the solute/particle transport, is at least of order 104. The EHD therefore alters the shape of the cross-section of the miscible stream, but the flow does not truly disperse the solute and dissipate the dielectric mismatch within the time frames that were studied. The transition region between the stream and its surroundings conforms to the distorted cross-section of the stream and remains comparatively narrow. The Peclet numbers for the aqueous systems depicted in Figure 2 are on the order of 10-102, so diffusion also contributes little to dissipation of the conductivity mismatch. Inasmuch as the initial property mismatch between stream and surroundings is not appreciably altered by deformation, one can revisit the work of Rhodes et al.,9 and extract an analytical estimate of the deformation time scale τ. Doing so yields ! 6μb ð2 þ Δσ=σ b Þ2 Δμ τ ¼ j ð8Þ 1þ 2μb εb ε0 E0 2 ð3 þ Δσ=σ b ÞΔσ=σb -3Δε=εb where eq 8 allows for the possibility that the viscosity of the cylindrical stream may differ from μb, the viscosity of the surrounding liquid, by an increment Δμ. This time scale is set by the (bulk) viscous response of the miscible fluids to the electrical body forces exerted in the thin fluid layer over which the electromechnical properties vary (see eq 2). Equation 8 reduces to τ ¼
8μb 1 þ Δσ=σb 2 jΔσ=σ -Δε=ε , εb ε0 E0 b b
Δσ=σ b ,1, Δμ=μb ,1 ð9Þ
which, consistent with Figure 3, implies that τ ∼ μb/ε0ΔεE20 as Δσ/σb tends to zero. Recall that Rhodes et al.9 envision the stream and surrounding liquid as distinct, uniform continua, so their analysis presumably works because diffusive solute transport is subordinate and the transition from stream to surroundings occurs over a length that is in some sense small compared to b.
Conclusions In the work presented here, the EHD of a miscible liquid stream has been investigated with a quasi-electroneutral model.16 The (25) Sihvola, A. Electromagnetic Mixing Formulas and Applications; IEE Electromagnetic Waves Series 47; IEE: London, 1999.
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formulation allows for natural gradients of electrical conductivity and dielectric constant in the transition region between the stream and the surrounding liquid; such gradients are assumed to be associated with concentration gradients of charge-carrying solutes and/or small particles. Electrical body forces that arise from dielectric and Coulombic effects are accounted for. Owing to the strength of the electrically driven flows, transport of solutes is dominated by advection. As the solutes follow the fluid elements and the cross-section of the miscible stream deforms, the electromechanical properties of the liquid within the stream remain largely unaffected, and the region over which σ and ε vary conforms to the evolving shape of the stream. Consequently, Δσ and Δε, which characterize the initial differences between the interior of the stream and the surrounding
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liquid, set the time scale on which the stream deforms, even at appreciable extents of deformation. Calculations for aqueous systems dominated by conductivity effects are consistent with measurements of stream deformation made by Rhodes et al.9 Predictions for systems controlled by dielectric effects indicate that Δε must be on the order of unity before noticeable deformations occur. This may explain why Trau et al.11 discerned no deformations controlled by dielectric effects in low permittivity, low conductivity systems. An implication of these predictions is that experiments to isolate the role of dielectric constant could be difficult to do. Manipulating dielectric constant mismatches, without affecting the conductivity match, may not be practical in miscible systems, and examples of such experiments have not appeared in the open literature.
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