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Effects of Stern Layer Conductance on Electrokinetic Energy Conversion In Electrophoresis 2008, 29, 1125–1130 1125 Christian Davidson Research Article Xiangchun Xuan Department of Effects of Stern layer conductance on Mechanical Engineering, Clemson University, electrokinetic energy conversion Clemson, SC, USA in nanofluidic channels Received July 27, 2007 Revised September 7, 2007 A thermo-electro-hydro-dynamic model is developed to analytically account for the effects of Accepted September 8, 2007 Stern layer conductance on electrokinetic energy conversion in nanofluidic channels. The optimum electrokinetic devices performance is dependent on a figure of merit, in which the Stern layer conductance appears as a nondimensional Dukhin number. Such surface con- ductance is found to significantly reduce the figure of merit and thus the efficiency and power output. This finding may explain why the recently measured electrokinetic devices performances are far below the theoretical predictions where the effects of Stern layer con- ductance have been ignored. Keywords: Dukhin number / Electrokinetic / Energy conversion / Figure of merit / Stern layer conductance DOI 10.1002/elps.200700549 1 Introduction effects of Stern layer conductance on electrokinetic devices performance (e.g., efficiency and power output) are still lar- Newfound attention has been given to electrokinetic energy gely unexplored. This article addresses this issue using an conversion devices including electroosmotic pumps ([1–12], analytical thermo-electro-hydro-dynamic approach devel- see ref. [7] for a review of micropumps by Laser and Santiago) oped by Xuan and Li [21]. A similar approach was also used and electrokinetic generators [13–17] due to their potential recently to analyze electrokinetic generating [13] and pump- for integration into micro- and nanofluidic systems. Recent ing efficiencies [12] in the absence of Stern layer con- efforts have focused on increasing the efficiency of electro- ductance. kinetic energy conversion in nanochannels, which, however, still seem much lower than that predicted by current theory [3, 9–17]. One likely explanation of this discrepancy is the 2 Theoretical formulation neglect of Stern layer conductance in the electrokinetic the- ory [3, 9, 10, 14]. The Stern layer is the layer of counterions 2.1 Thermodynamic analysis that attach to a charged surface. It is the inner immobile layer of the well-known electric double layer [18]. The electric Electrokinetic energy conversion devices can be thermo- conductance of the Stern layer has been found to signifi- dynamically described by the Onsager reciprocal relations for cantly affect the electrokinetic transport of colloidal particles volumetric flow rate, Q, and electric current, I, through a [19, 20]. Recently, van der Hayden et al. [14] included the channel of arbitrary geometry [22, 23] contribution of this surface conductance in a chemical equi- librium model, which explained fairly well their measure- Q ¼ GðDpÞþMðDfÞ (1) ments on a nanofluidic power generator. However, the real I ¼ MðDpÞþSðDfÞ (2) Correspondence: Professor Xiangchun Xuan, Department of Mechanical Engineering, Clemson University, Clemson, SC 29634-0921, USA where Dp and Df are the pressure difference and electric E-mail: [email protected] potential difference across the channel, G indicates the Fax: 11-864-656-7299 hydrodynamic conductance, M characterizes the EOF/ © 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.electrophoresis-journal.com 1126 C. Davidson and X. Xuan Electrophoresis 2008, 29, 1125–1130 streaming current, and S denotes the electrical conductance. surface conductance of the diffuse layer has been considered With these two phenomenological equations, Xuan and Li in the conduction current density in terms of the cosh(x) [21] have shown that the efficiency of electrokinetic energy function [26, 27]. This surface conductance should be dis- conversion is independent of the working mode (i.e., gen- tinguished from the Stern layer conductance. erator or pump) at the condition of maximum efficiency or Integrating Eqs. (6) and (7) over the channel cross-sec- maximum output power. Here, we present only the effi- tion and then comparing with Eqs. (1) and (2) yield the ciency Z and the output power W of an electrokinetic gen- expressions of phenomenological coefficients (see Section 6 erator at the condition of maximum efficiency for the derivation) z 2bh3 Z ¼ (3) G ¼ (8) max Z 22ðÞÀ z 3ml ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi ez p p 2 2bh 1 À Zð1 À 1 À ZÞ M ¼ g1 (9) W ¼ G ðDpÞ2 (4) ml max Z Z 4be2z2K2 b b 2 ¼ þ À coshðÞþC M S Ã2 1 g3 0 Du (10) Z ¼ (5) mhlz 4 4 GS Zh C y where Z is the previously termed “figure of merit” [21, 24, g ¼ 1 À d (11) 1 zà h 25]. This nondimensional parameter, as shown below, is a 0 function of fluid and channel properties. Zh y g ¼ coshðÞC d (12) 3 h 2.2 Electro-hydro-dynamic analysis 0 Consider a slit nanochannel of height 2h, width b, and length Within these definitions, C = zvec/kBT is the normalized l where b .. 2h and l .. 2h. One can then ignore the end double-layer potential where C0 the potential at the channel * effects and treat the channel as infinite parallel plates. In center; z = zvez/kBT the normalized zeta potential; K = kh the nondimensional channel height where such a channel, the axial fluid velocity u of a combined pres- qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 sure-driven and EOF is given by [26, 27] k ¼ 2zv e NAcb ekBT the inverse of Debye screening length, cb the ionic concentration of the bulk fluid, and NA is 2 h y2 Dp ez c Df the Avogadro’s number [18, 28]; b = Lm/eRT is the non- u ¼ 1 À À þ À 1 À (6) 2m h2 l m z l dimensional property of the working fluid where L is the molar conductivity and R is the universal gas constant (note: where m is the fluid viscosity, y the transverse coordinate in the reciprocal of b is previously termed the Levine number by the height direction that originates from the channel center, Griffiths and Nilson [10]); and Du = sStern/hsb = sStern/hLcb e the fluid permittivity, z the zeta potential on the channel is the Dukhin number. It is acknowledged that our Dukhin wall, and c is the electric double-layer potential. number is different from the traditional definition in which To account for the Stern layer conductance, we propose the numerator is the total surface conductance (i.e., the con- the introduction of a new term into the traditional current ductance through both the inner Stern layer and outer dif- density equation [26, 27], i.e., the last term in Eq. (7) fuse layer) instead of the current Stern layer conductance [29, 30]. Combining Eqs. (8)–(10) with Eq. (5), we arrive at the d2c z ec ÀDf s ÀDf i ¼e u þ s cosh v þ Stern (7) figure of merit, Z, in the presence of Stern layer conductance dy2 b k T l h l B effects Ã2 2 3z g1 where sb is the bulk conductivity of the fluid, zv the valence Z ¼ hi (13) 2 b b of ions, e the charge of proton, kB the Boltzmann’s constant, 2K 1 þ 4 g3 À coshðÞþC0 4 Du T the fluid temperature, and sStern is the Stern layer con- ductivity. The Stern layer conductance term in Eq. (7) is One can see that Z is a function of four nondimensional pa- * derived from (sSternC/l)(2Df)/A where, C =2(b 1 2h) and rameters, i.e., b, K, z , and Du, as defined above. Apparently, A =2bh are the perimeter and the area of the channel cross- the inclusion of Stern layer conductance decreases the figure section, respectively. The other two terms in Eq. (7) indicate of merit, Z, and hence reduces the efficiency of electrokinetic the streaming current density (first term) and the conduc- energy conversion, see Eq. (3). As will be seen shortly, Stern tion current density (second term), respectively. Note that the layer conductance also decreases the generation power in Eq. © 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.electrophoresis-journal.com Electrophoresis 2008, 29, 1125–1130 General 1127 (4) because the value of the figure of merit Z is reduced sig- nificantly. The double-layer potential C in the defined functions g1 and g3 are solved from the Poisson–Boltzmann equation [18] d2C ¼ k2 sinhðÞC (14) dy2 The analytical solution to C is expressed in terms of the Jacobian elliptical function [31] hi ky C ¼ C þ 2 ln JacCD eÀC0=2 e2C0 (15) 0 2 where C0 is the double-layer potential at the channel center first defined in Eq. (10) and can be determined from the known zeta potential z (more precisely, z*) on the channel wall. It is important to note that for a given fluid and channel combination, z will, in general, vary with the nondimen- sional channel height K = kh. One option to address this is to use a surface-charge based potential parameter for scaling instead of zeta potential [13, 16, 17, 32]. In this work and other studies [3, 4, 9–12, 15, 21, 26, 27, 33, 34], the zeta potential is used directly, as it may be readily determined through experiment and provides a direct measure of the electroosmotic mobility. 3 Results and discussion To examine quantitatively the effects of Stern layer con- ductance on electrokinetic devices performance, KCl aque- Figure 1. Contours of the figure of merit Z as a function of the normalized zeta potential z* and the nondimensional channel ous solution with the concentration of c =1025 M is used as b height K when Stern layer conductance is (a) considered and (b) the working fluid. Its properties include the visco- ignored. sity m = 0.961023 kg?m21?s21, dielectric constant e =7968.854610212 C?V21?m21, and molar conductivity L = 0.0144 S?m2?mol21 at temperature T = 298 K [14].
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