Ind. Eng. Chem. Res. 2006, 45, 6981-6984
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Electrohydrodynamic Flows in Nonhomogeneous Liquids D. A. Saville* and J. R. Glynn† Department of Chemical Engineering, Princeton UniVersity, Princeton, New Jersey 08544
Small-scale electrohydrodynamic flows were investigated in a configuration where the motive force derives from charge generated in an electrolyte composition gradient. Using a thin Hele-Shaw cell filled with aqueous electrolyte, we studied flow through a disc-shaped bolus whose conductivity differed from that of the surrounding solution. When an ac electric field was applied, a thin layer of free charge arose from differences in conductivity between the bolus and its surroundings. The action of the field on the charge produced bulk flow, and since material interfaces were absent, the bolus deformed continuously. Deformations were prolate or oblate, depending on the conductivity of the bolus relative to the surrounding electrolyte. Modeling the flow with the Hele-Shaw formalism showed how electrical forces produce motion. To indicate the sense of the deformationsprolate or oblate with respect to the applied fieldsa discriminating function was derived. The discriminating function reduces to the Taylor-Melcher leaky dielectric model under the appropriate limiting conditions. Experiment and theory are in good agreement. The flows described here occur in the absence of rigid interfaces and equilibrium charge layers. Accordingly, they offer a means of controlling microfluidic flow and mixing. 1. Introduction Electrohydrodynamic and electrokinetic effects arise in different ways.1,2 As employed in lab-on-a-chip technologies,3,4 electrokinetic pumping involves electroosmosis due to the action of an electric field on the equilibrium free charge adjacent to oppositely charged surfaces. However, free charge also arises in nonequilibrium configurations. For example, in an applied field, variations in bulk conductivity lead to space charge and electrohydrodynamic flow. Here, we show how this class of flows can be understood in terms of an electrokinetic model. The novelty of the current configuration lies in the nonequilibrium origin of the free charge that drives the flow. Since they can be controlled electrically, such flows have many applications. 2. Experimental Section
Figure 1. Plan view schematic of the Hele-Shaw cell showing a prolate bolus deformation.
To identify the salient features, we used a three-dimensional Hele-Shaw cell with an active volume of 12.7 cm × 12.7 cm × 500 µm; see Figure 1. An ac field was generated between brass electrodes connected to a Trek 20/20A amplifier and controlled by a Tektronix CFG253 3 MHz-function generator. Frequency and amplitude were measured with a Tektronix 2235 100 MHz oscilloscope and a Keithley 157A autoranging multimeter, connected in series to the powered electrode through a Tektronix P6015A 1000 V high-voltage probe. Nominal field strengths were a few V/cm at frequencies of several kHz. The ac field eliminated electrokinetic effects due to charge on the cell walls and ensured that the flows were driven by induced charge. The thin configuration improved heat transfer, and temperature changes within the cell were always small ( 1) and oblate (σ j /σ < 1) deformations. As the figure illustrates, this linearity persists over a wide range of field strengthssup to 30 V/cm in the experiments described here. The absence of deformation in experiments with σ j /σ ) 1 testifies to the absence of polarization forces due to differences between the dielectric constant of the suspension and the surrounding fluid. Further insight can be obtained from a theoretical model. 3. Theory
Figure 3. Streamline pattern for a prolate deformation computed from the Hele-Shaw model.
images was recorded. Then the bolus outline was fitted to an elliptical shape, and the major and minor axis lengths were computed. This furnished information on the deformation magnitude as a function of time for a given field strength. The images disclose two important features. First, as illustrated in both parts of Figure 2, deformations (oblate or prolate) are symmetric in the early stages. The deformation shapes are also consistent with stream lines (Figure 3) computed with the theoretical model to be described shortly. Other results (not shown here) demonstrated that the rate of deformation is constant during the initial stage of deformation. Linearity persists for deformations up to 20% of the initial size. All these characteristics are consistent with the theoretical model; evidently, the model is robust. Second, and somewhat unexpected,
To formulate the theoretical model, the deformation process is envisioned as follows. At the outset, a disk-like bolus exists inside a homogeneous electrolyte confined to a thin cell; the conductivity of the bolus differs from the surrounding electrolyte, and the upper and lower surfaces of the cell are nonconducting. Although the compositions begin to equalize by diffusion, this is slow. When an oscillatory electric field is imposed, a thin layer of free charge forms in the circular transition region between the two electrolyte concentrations. The length scales for the inner and outer parts of the charge layer are the Debye lengths, κj-1 and κ-1. Compositions in the Debye layers stay in concert with the field because the layer is thin. Since the induced charge is proportional to the real part of the applied field, E∞ exp(i$t), the body force stemming from the product of the free charge and the field consists of a steady component and an oscillation. The steady part of the electrical body force produces the flow of interest here, inasmuch as the flow cannot track the oscillatory part.
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To work out the relevant scales, we use characteristics of aqueous systems with an ionic strength,2 n∞ ≈ 10-3 mol/L. So, κ-1 ≈ 10 nm and D ≈ 10-9 m2/s. For a millimeter size bolus, the characteristic diffusion time, a2/D, is 103 s. Since the ion electromigration velocity scales as eωE∞ ) eDE∞/kBT, electrical transport on the bolus scale is also slow for field strengths of the order of 10 V/cm. In contrast, diffusion and electromigration are both rapid on the Debye layer scales since (κ-1)2/D ≈ 10-7 s and (κ-1)/eωE∞ ≈ 10-4 s. Accordingly, at frequencies of a few kHz, ions in the diffuse layer keep pace with the applied field, and free charge is created in the transition zone by the combined effects of diffusion and electromigration. Analyzing the relevant equations will make the relationships clear. From Gauss’s law and the ion-conservation equations for a dilute binary electrolyte, we have2 2
-o∇2ψ )
∑1 ezknk
(1)
and
∂ k n + u‚∇nk ) ∇‚{ezkωknk∇ψ + kBTωk∇nk}, k ) 1, 2 ∂t (2) The symbols have their usual meanings: ψ denotes the electrostatic potential, nk is the ion concentration of the kth species with valence zk, and u is the local fluid velocity. The dielectric constant of the liquid is , o is the permittivity of free space, e is the unit of charge, ωk is the ionic mobility, and kBT is the product of Boltzmann’s constant and the absolute temperature. For simplicity, we assume a univalent system, z1 ) -z2 ) 1, with equal ion mobilities related to the ion diffusivity as D ) ωkBT. In the experiments, KCl was used as the electrolyte and the mobilities of potassium and chloride ions are virtually the same. The important length scales are the radius of the bolus, a, and the Debye thicknesses in the bolus, κj-1 ) okBT/2e2nj∞, and in the surrounding fluid, κ-1 ) okBT/2e2n∞. Here, nj∞ denotes an initial ion concentration in the bolus; n∞ corresponds to the surrounding fluid. In terms of the diffusivity, the electrical (ohmic) conductivity, σ, is equal to oκ2D. Next, we linearize the model to accommodate small differences between the electrolyte concentration inside and outside the bolus by writing concentrations as nk ) n∞k + mk and scale the variables using a as the length scale, the reciprocal of the applied frequency, $-1, as a time scale, and aE∞ as the scale for the potential. Upon noting that the hydrodynamic velocity is proportional to the charge, the O(δ2) convection term can be omitted from the charge balance and the dimensionless forms of eqs 1 and 2 are
∇2ψ ) -(κa)2δ
(3)
a2 $ ∂ δ ) ∇2ψ + ∇2δ D ∂t
(4)
and
Here, the ion-conservation equations have been combined to describe the distribution of (dimensionless) free charge, δ, as e(m1 - m2) ≡ oκ2aE∞δ. Note that, although the charge density is large, O(κa)2, it yields an O(1) force since it is localized in a thin region. The equations describing things inside the bolus have the same form with “overbars” inserted to distinguish the Debye thickness, potential, and charge. Since κa ≈ 105 and
a2$/D ≈ 107, eqs 3 and 4 indicate that the fluids remain electrically neutral on the scale of the bolus. However, on the Debye scale κ-2$/D ≈ 10-3 so the charge is quasi-static and keeps pace with the applied field. Analysis of eq 3, the steady version of eq 4, and the boundary conditions shows that, inside the bolus, the dimensionless charge is described by the real part of
[1 - κj/κ] exp[-κja(1 - r)] δ h ≈ -2 exp[-i$t] cos θ, [1 - (κj/κ)2] κjaxr 0 < r < 1 (5) Here, θ represents the angle with respect to the imposed field. Since κja . 1, the layer is exceedingly thin. A similar expression describes charge outside the bolus. The electric potentials inside and outside are described by a superposition of solutions of Laplace’s equation with a contribution from the charge layer. For example, the thin layer inside the bolus contributes a term proportional to the real part of exp[-κja(1 - r)] exp[-i$t] cos θ/κjaxr. The body force has a steady component derived from the product of the fluctuating field and the charge so, on the scale of the bolus, this force appears as a force singularity in the equations of motion. Note also that assuming a symmetrical electrolyte with equal mobilities for the co- and counterions suppresses, among other things, diffusion potentials. Such phenomena may be taken up in subsequent work. To analyze the flow, we use the Hele-Shaw formalism described by Batchelor;7 here, the relevant Viscous stresses arise from interactions with the upper and lower (planar) boundaries of the cell. The electrical force driving motion appears as an O(oE∞2) jump in normal stress at the cylindrical boundary between the bolus and the surrounding fluid. In the Hele-Shaw formalism, the pressure and velocity, averaged across the thin dimension, h, are harmonic functions. Here, we ignore deformation of the bolus and focus on the initial stages of the flow. Solving the equations and balancing the pressure and hydrodynamic normal stress with the steady electrical stresses at the edge of the bolus gives the (steady) radial velocity as
u)
oE∞2h2 2(x - 1) [1 + x + x2 + 3x3] cos 2θ (6) aµ 9(x2 + 1)2
at the edge of the bolus. Note that x ≡ κj/κ ) xnj/n ) xσ j /σ. Equation 6 is, in Taylor’s language,8 a discriminating function. When x > 1, i.e., when the conductivity of the bolus exceeds that of the surrounding fluid, the deformation is prolate; x < 1 signifies an oblate form. 4. Results and Discussion Figure 3 depicts stream lines computed from the model, and Figure 5 shows the measured (scaled) interface velocity and the discriminating function for several electrolyte concentrations. As expected from the linearization, the relationship holds best for conductivity ratios close to unity. It is also worth noting that the magnitude of the measured velocity is close to that predicted from the model, despite the many simplifications. The phenomena described here are electrohydrodynamic in the sense that they are driven by forces stemming from free charge induced by an applied field. However, the force distribution differs from that predicted by a straightforward application of the Taylor-Melcher approach, which omits diffusive transport. Rhodes et al.9 treated a situation similar to that investigated here and derived a discriminating function
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Acknowledgment D.A.S. has enjoyed a 30-year collaboration with Bill Russel; this has been one of the highlights of his time at Princeton. The research described here was supported by the NASA Microgravity Science and Applications Division. Literature Cited
Figure 5. Dimensionless interfacial velocity in the direction of the applied field correlated with the Hele-Shaw discriminating function. The sense and magnitude of the deformation are consistent with the model, i.e., prolate, oblate, and no deformation when the discriminating function is greater than, less than, and equal to zero, respectively.
proportional to (σ j /σ)2 + σ j /σ - 1 for an infinitesimally thin transition region. Allowing for a continuous variation of conductivity across an interface, but still neglecting diffusive transport,10 also fails to capture the features identified here. The absence of “square roots” in the expressions derived from the Taylor-Melcher approach testifies to the omission of diffusion. In addition to the study reported here, we also carried out similar experiments with nonaqueous systems, adjusting the conductivity with an organic electrolyte. Good agreement between theory and experiment was found. These results will be reported in due course.
(1) Saville, D. A. Electrohydrodynamics: The Taylor-Melcher Leaky Dielectric Model. Annu. ReV. Fluid Mech. 1997, 29, 27. (2) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, U.K., 1989; paperback edition, 1992. (3) Culbertson, C. T.; Jacobson, S. C.; Ramsey, J. M. Microchip Devices For High-Efficiency Separations. Anal. Chem. 2000, 72, 5814. (4) Bousse, L.; Cohen, C.; Nikiforov, T.; Chow, A.; Kopf-Sill, A. R.; Dubrow, R.; Parce, J. W. Electrokinetically Controlled Microfluidic Analysis Systems. Annu. ReV. Biophys. Biomol. Struct. 2000, 29, 155. (5) Sankaran, S.; Saville, D. A. Experiments On The Stability Of A Liquid Bridge In Axial Electric Field. Phys. Fluids, A 1993, 5, 1081. (6) Burcham, C. L.; Saville, D. A. The Electrohydrodynamic Stability Of A Liquid Bridge: Microgravity Experiments On A Bridge Suspended In A Dielectric Gas. J. Fluid Mech. 2000, 405, 37. (7) Batchelor, G. K. An Introduction to Fluid Dynamics; Cambridge University Press: London, 1967. (8) Taylor, G. I. Studies In Electrohydrodynamics. I. The Circulation Produced In A Drop By An Electric Field. Proc. R. Soc. London, Ser. A 1966, 291, 159. (9) Rhodes, P. H.; Snyder, R. S.; Roberts, G. O. Electrohydrodynamic Distortion Of Sample Streams In Continuous Flow Electrophoresis. J. Colloid Interface Sci. 1989, 129, 78. (10) Saville, D. A. Electrohydrodynamic Deformation Of A Particulate Stream By A Transverse Electric Field. Phys. ReV. Lett. 1993, 71, 2907.
ReceiVed for reView November 3, 2005 ReVised manuscript receiVed July 18, 2006 Accepted July 20, 2006 IE0512237
