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Effect of Finite Electric Double Layer Conductivity on Streaming Current Proceedings of the 20th National and 9th International ISHMT-ASME Heat and Mass Transfer Conference January 4-6, 2010, Mumbai, India 10HMTC89 Effect of Finite Electric Double Layer Conductivity on Streaming Potential and Electroviscous Effect in Nanofluidic Transport Siddhartha Das Suman Chakraborty Department of Mechanical Engineering Department of Mechanical Engineering Indian Institute of Technology, Kharagpur-PIN- Indian Institute of Technology, Kharagpur-PIN- 721302 721302 Email:[email protected] Email:[email protected] applications in diverse areas such as biomedical, ABSTRACT biotechnological, chemical, forensic etc [1-8]. In this paper, the role of finite Electric Because of strong dominance of surface effects Double Layer (EDL) conductivity on the and interfacial phenomena over reduced length Streaming Potential and the resulting scales, flow manipulation through exploitation of Electroviscous effect in pressure-driven electrical fields turns out to be convenient in nanofluidic ionic transport is investigated. nano-scale fluidic devices of practical relevance. Important dimensionless parameters like the Many such devices essentially function by Dukhin number (based on the EDL surface exploiting the formation of electrical double electrical conductivity, the bulk electrical layer (EDL) [9]), which is essentially a charged conductivity and the EDL thickness), ratio of the layer formed in vicinity of the fluid-substrate Stern Layer and the Diffuse Layer conductivities interface because of electro-chemical and the ratio of the channel height to the EDL phenomena. Under the action of driving forces, thickness are identified as important ionic charges in the mobile part of the EDL can dimensionless parameters that dictate the migrate along a preferential direction, thereby contribution of the finite EDL conductivity. For giving rise to local ionic currents. The ionic most of the combinations of these dimensionless transport in the EDL may also drag fluid parameters, the finite EDL effect lowers the elements along with the same as a consequence strength of the streaming potential and hence the of viscous shear, which may bear significant electroviscous action. However, for certain consequences with regard to the hydrodynamic choices of these parameters, particularly for transport in the narrow confinement. Critical to moderate Dukhin number and substantially large the investigation of electrokinetic transport, thus, zeta potential, finite EDL conductivity effect can lies the analysis of hydrodynamics within the actually enhance these effects. EDL. Advection of mobile ions present within the EDL, in presence of a pressure-driven 1. INTRODUCTION background flow, develops a so called streaming In the present decade, investigations on potential [9] in micro and nanochannels. transport in nanofluidic confinements have Streaming potential is effectively the induced received a tremendous boost, paving the way for electric field created due to a preferential a large number of novel chemical and biological downstream accumulation of the EDL 1 counterions and is directed opposite to the differences in EDL and bulk conductivities on pressure-driven transport. Consequently, it sets nanochannel streaming field and electroviscous up a back electrokinetic transport which opposes effect calculations. Only very recently, Davidson the pressure-driven flow. The resulting reduction and Xuan [17] elaborated the effects of Stern in the volume flow rate makes the flow appear to layer conductance on electrokinetic energy have an enhanced viscosity, an artifact known as conversion in nanochannels. However, in their Electroviscous Effect. Researchers have utilized work they did not consider the effect of the the generation of streaming potential for a diffuse layer, neither they pinpointed the number of applications which include the possible variations of the streaming potential and conversion of hydrostatic pressure differences the effective viscosity (characterizing the into useful electrical energy, characterization of electroviscous effect) as a function of the interfacial charge of organic thin films, different parameters governing the finite measurement of wall charge inversion in the conductivity within the EDL. presence of multivalent ions in a nanochannel, In the present paper we pinpoint the analysis of ion transport through nanoporous possible effects of considering finite EDL membranes, designing efficient nano-fluidic electrical conductivities (different from the bulk batteries, quantifying hydrodynamic dispersion value) in altering the streaming potentials and the in nanochannels, etc. resulting electroviscous action. The EDL itself is Streaming potential is estimated by assumed to have two different conductivities; balancing the streaming current (current resulting one for the Stern Layer and another for the from flow induced downstream migration of Diffuse Layer. Based on the combination of mobile EDL ions) with the conduction current these conductivities with the bulk conductivity, (current resulting from the streaming potential along with the value of the EDL thickness, we induced reverse electro-migration of ions), under identify the relevant dimensionless parameters steady state condition. A precise determination that govern the extent of deviation of the of the conduction current, in turn, depends on an streaming potential for the cases with and accurate quantification of the conductivity of the without the considerations of varying electrical ionic solution. The standard practice is to conductivities over the channel section. characterize the conductivity by a unique value dictated by the bulk ionic concentration. This approach works fine for microchannels where 2. THEORY the EDL occupies only a negligible extent of the channel section. However, for nanochannels in which EDLs may protrude significantly into the 2.1 EDL potential and Streaming bulk, this approach may not turn out to be quite Field without Considering Finite EDL adequate. This inadequacy stems from the fact Conductivity that within the EDL the concentration of ions can We consider transport of an ionic liquid be much higher so that the corresponding value in a slit-like charged nanochannel of height 2H, of the conductivity is much larger than the bulk length L and width w. The EDL potential field conductivity. In fact, within the EDL itself, large (ψ) in the nanochannel is described in terms of differences in the ionic concentration between the net charge density ( ρe ) within the EDL as: the immobile Stern layer and the mobile diffuse layer may result in different conductivities at ∂∂22ψψ ρ different sections of the EDL. A correct e 22+=− (1) estimation of conduction current (and hence ∂∂xy ε 0ε r streaming potential and electroviscous effect estimation) in nanofluidic transport must take where ε 0 is the permittivity of free space and ε r into account these variable conductivities across is the relative permittivity of the solution. The different portions of the channel section. There charge density ρ is expressed in terms of the has been a large number of investigations e delineating the effects of finite EDL conductivity ionic number densities for symmetric electrolyte (or to be more specific, Stern Layer ( zz+−= −=z) as: conductivity) on different aspects of electrokinetic transport [10-16]. However, to the ρe =−ez( n+− n ) (2) best of our knowledge, there has been no investigation pinpointing the influences of the 2 Assuming that the ionic concentrations obey dp where is the pressure gradient, μ is the ⎛⎞ezψ dx Boltzmann distributions n= n exp ∓ ± 0 ⎜⎟ dynamic viscosity of the fluid and E is the ⎝⎠kTB S induced streaming potential field. where n0 is the bulk ionic concentration), eqs. The conduction current (Icond) results from the (1) and (2) are solved to obtain the solution of transport of the ions in response to the the potential field (under the condition that at established streaming potential, so that one may dψ write (assuming that the ions have identical ψζ= ()zeta potential and = 0 y,H=02 dy conduction velocities across the entire transverse yH= domain, i.e., the variation of the mobilities of the and the EDL thickness λ is comparable but ions depending on whether they are in the bulk always less than the channel half height) as [9]: or within the EDL is not accounted for) ⎧⎫⎡⎤⎛⎞ezζ ⎛⎞ y 2H ⎪⎪tanh-1 ⎢⎥ tanh exp−+ ⎜⎟⎜⎟ Icond =+enzu()++cond ,, + nzudy −−cond − (7) 4k T ⎪⎪⎣⎦⎢⎥⎝⎠4kB T ⎝⎠λ ∫ ψ = B ⎨⎬0 ze ⎪⎪-1 ⎡⎤⎛⎞ezζ ⎛⎞ 2H − y tanh⎢ tanh⎜⎟ exp − ⎥ ⎪⎜⎟⎪where u are the electromigration velocities ⎩⎣⎦⎢⎥⎝⎠4kB T ⎝⎠λ ⎭cond ,± (3) of the cations/anions in response to the streaming potential, and may be expressed as In presence of a pure pressure-driven transport downstream migration of the mobile EDL ions zeE u = ± S (8) give rise to what is known as the streaming cond ,± f± current. The resulting accumulation of the ions in the downstream establishes a field (called the Here f± are the friction coefficients of the ions, streaming field or streaming potential) in the which is assumed to have identical values for the upstream direction which drives a current called anions and the cations (i.e., the conduction current. The net ionic current is a f = ff==6πμ r, where r is the ionic resultant of these two effects and at steady state +− ion ion radius; here we assume, with some degree of is equal to zero. We thus have: approximation, that the cationic and anionic radii are approximately equal to around 1.5 A0), IIIionic=+ str cond =0 (4)
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