pubs.acs.org/Langmuir © 2010 American Chemical Society
Effect of Conductivity Variations within the Electric Double Layer on the Streaming Potential Estimation in Narrow Fluidic Confinements Siddhartha Das and Suman Chakraborty* Department of Mechanical Engineering, IIT Kharagpur-721302, India Received March 5, 2010. Revised Manuscript Received April 16, 2010 In this article, we investigate the implications of ionic conductivity variations within the electrical double layer (EDL) on the streaming potential estimation in pressure-driven fluidic transport through narrow confinements. Unlike the traditional considerations, we do not affix the ionic conductivities apriori by employing preset values of dimensionless parameters (such as the Dukhin number) to estimate the streaming potential. Rather, utilizing the Gouy-ChapmanGrahame model for estimating the electric potential and charge density distribution within the Stern layer, we first quantify the Stern layer electrical conductivity as a function of the zeta potential and other pertinent parameters quantifying the interaction of the ionic species with the charged surface. Next, by invoking the Boltzmann model for cationic and anionic distribution within the diffuse layer, we obtain the diffuse layer electrical conductivity. On the basis of these two different conductivities pertaining to the two different portions of the EDL as well as the bulk conductivity, we define two separate Dukhin numbers that turn out to be functions of the dimensionless zeta potential and the channel height to Debye length ratio. We derive analytical expressions for the streaming potential as a function of the fundamental governing parameters, considering the above. The results reveal interesting and significant deviations between the streaming potential predictions from the present considerations against the corresponding predictions from the classical considerations in which electrochemically consistent estimates of variable EDL conductivity are not traditionally accounted for. In particular, it is revealed that the variations of streaming potential with zeta potential are primarily determined by the competing effects of EDL electromigration and ionic advection. Over low and high zeta potential regimes, the Stern layer and diffuse layer conductivities predominantly dictate the streaming potential variations whereas ionic advection governs the streaming potential characteristics over intermediate zeta potential regimes. It is also inferred that traditional considerations may grossly overpredict the magnitude of streaming potential for narrow confinements in which significant conductivity gradients may prevail across the EDL.
1. Introduction Investigations of transport phenomena in narrow fluidic confinements have been receiving an ever-increasing amount of attention, paving the way for diverse applications in biomedical, biotechnological, energy, chemical, and forensic sciences.1-8 Because of the strong dominance of surface effects and interfacial phenomena over reduced length scales, flow manipulation by the exploitation of electrochemical effects in the interfacial region has been demonstrated to be convenient in many of these applications. Importantly, such electrochemical functionalities often exploit the formation of the electrical double layer (EDL),9 which is essentially a charged layer formed in the vicinity of the fluid-substrate interface. Under the action of driving forces, ionic charges in the mobile part of the EDL can migrate along a preferential direction, thereby giving rise to local ionic currents. The ionic migration in the EDL may also drag fluid elements as a consequence of viscous shear, which may have significant consequences with regard to the interconversion of hydraulic and electrical forms of energy. *Corresponding author. E-mail:
[email protected]. (1) Watson, A. Science 2000, 289, 850. (2) Griffiths, S. K.; Nilson, R. H. Anal. Chem. 2006, 78, 8134. (3) De Leebeeck, A.; Sinton, D. Electrophoresis 2006, 27, 4999. (4) Chakraborty, S. Anal. Chim. Acta 2007, 605, 175. (5) van der Heyden, F. H. J.; Stein, D.; Besteman, K.; Lemay, S. G.; Dekker, C. Phys. Rev. Lett. 2006, 96, 224502. (6) Alkafeef, S. F.; Alajmi, A. F. J. Colloid Interface Sci. 2007, 309, 253. (7) Daiguji, H.; Oka, Y.; Adachi, T.; Shirono, K. Electrochem. Commun. 2006, 8, 1796. (8) Xuan, X.; Sinton, D. Microfluid. Nanofluid. 2007, 3, 723. (9) Hunter, R. J. Zeta Potential in Colloid Science: Principles and Applications; Academic Press: New York, 1981.
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The transport of mobile ions present within the EDL, under the actuation of a driving pressure gradient, may induce an electrical potential difference across the channel ends, also known as a streaming potential.9 The development of a streaming potential stems from the fact that the downstream motion of the mobile counterions in the EDL results in an advective ionic current, called the streaming current, that flows in the same direction as dictated by the imposed pressure gradient on the liquid. The resulting accumulation of ions in the downstream end of the channel, however, sets up its own electric field, which drives a current in a direction opposite to the incipient flow direction. This electromigrative current, also called the conduction current, balances the streaming current at steady state to satisfy the overall electroneutrality constraints. From this consideration, one may estimate the induced potential difference between the two ends of the channel, which is also known as the streaming potential. Researchers have utilized the generation of the streaming potential for a number of applications that include the conversion of hydrostatic pressure differences into useful electrical energy, the characterization of the interfacial charge of organic thin films, the measurement of wall charge inversion in the presence of multivalent ions in a nanochannel, the analysis of ion transport through nanoporous membranes, the design of efficient nanofluidic batteries, and the quantification of hydrodynamic dispersion in nanochannels, to name a few.10-13 (10) van der Heyden, F. H. J.; Bonthius, D. J.; Stein, D.; Meyer, C.; Dekker, C. Nano Lett. 2007, 7, 1022. (11) Schweiss, R.; Welzel, P. B.; Werner, C.; Knoll, W. Colloids Surf., A 2001, 195, 97.
Published on Web 05/17/2010
DOI: 10.1021/la1009237
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Figure 1. Gouy-Chapman-Grahame model near a charged surface. In the inset, we provide the flow geometry and the coordinate axes.
The precise determination of the streaming potential depends on an accurate estimation of the conduction current, which in turn depends on an adequate description of the ionic conductivity of the solution within the EDL. Anatomically, the EDL may be conceptualized primarily as a combination of two distinct layers (Figure 1) formed adjacent to the channel wall as a consequence of surface charging. In Figure 1, the wall is assumed to be negatively charged with a bare wall charge density of σ0 and a potential of ψ0. The first layer of ionic charges adjacent to the wall consists of immobilized counterions. This compact layer, also known as the Stern layer or the Helmholtz layer, has a thickness on the order of the ionic diameter. The charge and the potential distributions within this layer are strong functions of the geometrical characteristics of the ions and the short-range interaction forces among the ions, wall, and adjacent dipoles. As depicted in Figure 1, this layer can be further divided into two sections. The inner section (i.e., the section nearer to the wall) is separated from the outer section by a plane called the inner Helmholtz plane (IHP). The IHP is assumed to be located at a distance of RSt from the channel wall. The electrical potential and the charge density (for an ion of species i) at the IHP are considered to be ψ0(RSt) and σSt,i, respectively. The outer section of the Stern layer is separated from the second layer of the EDL (also called the diffuse layer or the Gouy-Chapman layer) by a plane called the outer Helmholtz plane (OHP). The OHP is considered to be located a distance of βSt from the channel wall. The electrical potential and the charge density at the OHP are taken to be ζ (the zeta potential) and σd (the diffuse layer charge density), respectively. The diffuse layer is essentially a layer of mobile ions beyond the Stern layer. Unlike the Stern layer, this layer consists of both counterions and co-ions and diffuses well away from the wall. The outer edge of the diffuse layer is assumed to be located a distance λ from the channel wall. The characteristic length scale λ, also known as the Debye length, characterizes the distance from the OHP over which the EDL potential reduces to 1/e times ζ. The net charge in these two layers must be equal to the bare charge on the substrate. (12) Daiguji, H.; Yang, P.; Szeri, A. J.; Majumdar, A. Nano Lett. 2004, 4, 2315. (13) Huang, K.-D.; Yang, R.-J. Nanotechnology 2007, 18, 115701.
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One traditional approach to estimating the streaming potential is to characterize the ionic conductivity of the solution by a unique value that is dictated predominantly by the bulk ionic concentration. Researchers have subsequently augmented this elementary consideration to incorporate the effects of Stern layer conductivity14-20 for the streaming potential estimation. However, the Stern layer conductivity is usually prescribed in the literature to be independent of other pertinent interfacial considerations, such as the zeta potential, in estimating the streaming potential.21,22 The fact that the diffuse layer conductivity may significantly deviate from that of the bulk is also not duly accounted for in many of these calculations. We believe that the simplistic standard approaches of streaming potential estimation, delineated as above, may turn out to be grossly inconsistent and erroneous, particularly for narrow fluidic confinements in which the Debye lengths approach the channel heights. Here we attempt to overcome the above deficiencies in the standard practices of streaming potential estimation by employing a modified formalism that includes two important considerations. First, we consider that the Stern layer conductivity cannot be independently prescribed but is rather dictated by the zeta potential as well as the channel height to Stern layer thickness ratio. Fundamentally, the monoionic thick Stern layer develops when the ions undergo an adsorption-desorption mechanism with the charged wall. The resulting ionic number distribution determines the potential and the charge density variations across the Stern layer as a parametric function of the zeta potential, which finally leads to an estimation of the Stern layer ionic conductivity. Second, we consider the diffuse layer conductivity also to be a strongly dependent function of the zeta potential as well as the channel height to Debye length ratio. The diffuse layer, in principle, is characterized by an imbalance in the distribution of cations and anions. Such imbalances implicate a position-dependent conductivity (parametrized with respect to the zeta potential) within the diffuse layer, which needs to be integrated across the diffuse layer to obtain an effective diffuse layer conductivity. Following the above considerations, we express the dimensionless estimates of the conductivities of these two layers, normalized with respect to the bulk ionic conductivity (expressed in terms of two separate Dukhin numbers, DuSt and DuDiff, for the Stern layer and the diffuse layer, respectively), as functions of the dimensionless zeta potential and the channel height to Debye length ratio. Although separate accounting of the diffuse layer conductivity is not uncommon in the electrokinetics literature,14,23 the corresponding effects have been grossly overlooked in the context of the streaming potential estimation. Through the present analysis, we demonstrate that large alterations in the Stern layer and diffuse layer conductivities with simultaneous enhancements in the magnitudes of the dimensionless zeta potentials and Debye length to channel height ratios implicate estimations of streaming potentials that may be substantially overpredicted as (14) Carrique, F.; Arroyo, F. J.; Shilov, V. N.; Cuqejo, J.; Jimenez, M. L.; Delgado, A. V. J. Chem. Phys. 2007, 126, 104903. (15) Carrique, F.; Arroyo, F. J.; Delgado, A. V. J. Colloid Interface Sci. 2000, 227, 212. (16) Mangelsdorf, C. S.; White, L. R. J. Chem. Soc., Faraday Trans. 1990, 86, 1859. (17) Carrique, F.; Arroyo, F. J.; Delgado, A. V. J. Colloid Interface Sci. 2001, 243, 351. (18) Carrique, F.; Arroyo, F. J.; Delgado, A. V. J. Colloid Interface Sci. 2002, 252, 126. (19) Rubio-Hernandez, F. J.; Ruiz-Reina, E.; Gomez-Merino, A. I. J. Colloid Interface Sci. 2000, 226, 180. (20) Carrique, F.; Arroyo, F. J.; Delgado, A. V. Colloids Surf., A 2001, 195, 157. (21) Davidson, C.; Xuan, X. Electrophoresis 2008, 29, 1125. (22) Choi, Y. S.; Kim, S. J. J. Colloid Interface Sci. 2009, 333, 672. (23) Hughes, M. P.; Green, N. V. J. Colloid Interface Sci. 2002, 250, 266.
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well as qualitatively inaccurate, for cases in which these effects are not adequately taken into consideration.
2. Mathematical Modeling 2.1. Traditional Considerations for the Streaming Potential Estimation. We consider the pressure-driven transport of a z/z ionic solution in a slit-like fluidic confinement of height 2H and length L. The necessary coordinate axes and the origin, along with the flow geometry, are provided in the inset of Figure 1. Velocities of the positive and negative ions (uþ and u-, respectively) may be described as the combined consequences of the local fluid velocities (u) and the electromigrative velocities so that one may write uþ ¼ u þ
zeEx zeEx , u- ¼ u - fþ f
ð1a, bÞ
where f þ and f - are the friction factors of the positive and negative ions, e is the charge of a proton, and Ex is the axial electric field. (For purely pressure-driven transport, it is equal to the streaming field Es.) Here both uþ and u- are along the axial directions (x). Furthermore, uþ = uþ(y) and u- = u-(y). The net ionic current through the channel section (per unit width) may be described as Z
2H
i ¼
ðzenþ uþ - zen - u - Þ dy
ð2Þ
0
where nþ and n- are the ionic number densities (in 1/m3) of the cations (or counterions in the present case) and anions (or co-ions in the present case). Considering the ionic friction factors to be identical for cations and anions ( f þ=f -=f ), eq 2, with the aid of eqs 1a and 1b, may be rewritten as Z z2 e2 Ex 2H þ ðn - n Þu dy þ ðn þ n -Þ dy i ¼ ze f 0 0 streaming current, istr conduction current, icond Z
2H
þ
-
ð3Þ
Expressing the ionic friction factor in terms of the bulk ionic conductivity (ΓB) (f=2n0z2e2/ΓB, where n0 is the number density of ions in the bulk solution, i.e., outside the EDL), eq 3 may be simplified as Z i ¼ ze
2H
ðnþ - n - Þu dy þ
0 streaming current, istr
ΓB Ex 2n0
Z
2H
ðnþ þ n -Þ dy
ð4Þ
0 conduction current, icond
(i.e., streaming-potential-driven) components so that one may write 1 dp ε0 εr ζEx ψ ð7Þ u ¼ up þ uE ¼ ð2Hy - y2 Þ 12μ dx ζ μ where ε0 is the permittivity of the vacuum, εr is the dielectric constant of the medium, - dp/dx is the applied pressure gradient, ψ is the EDL potential distribution, and μ is the fluid viscosity. The EDL potential field (ψ) may be described in terms of the net charge density (Fe) within the EDL by invoking the Poisson equation as d2 ψ F ¼ - e dy2 ε0 εr
Assuming that the ionic concentrations obey the Boltzmann distribution (i.e., n( = n0 exp[-(ezψmod/kBT)]), eq 8 may be analytically solved with the following boundary conditions: dψ ¼0 ψjy ¼ 0, 2H ¼ ζ and dy y ¼H
The resulting potential distributions of the two confining boundaries may be linearly superposed (provided that their respective EDLs do not overlap) so as to yield the resulting potential distribution9 8 " # 4kB T < ezζ y -1 exp ψ ¼ tanh tanh ze : 4kB T λ 9 " #= ezζ 2H y ð9Þ exp þ tanh - 1 tanh ; 4kB T λ where λ is the Debye length. Equation 9 may be applied even for noninfinitesimally thin EDLs (typically for H/λ > 4) and hence may be employed to probe EDL interactions in narrow fluidic confinements with significant EDL protrusion into the bulk but without incurring any appreciable EDL overlap.24 The streaming potential field (ES), considering the flow to be solely actuated by a pressure gradient of - dp/dx, may be estimated by noting that at steady state ð10Þ istr þ icond ¼ 0 so that one gets (considering the streaming potential field, ES = Ex), from the traditional considerations that disregard ionic concentration variations within the EDL for estimating the conduction current,
In traditional theory, icond is evaluated by disregarding the charge density variations within the EDL and hence by referencing to the bulk solution (i.e., nþ=n-=n0) so that one may write icond ¼ 2HΓB Ex
ΓB ¼ en0 ðΛ þ þ Λ -Þ
ð6Þ
where Λ( represents the mobilities of the cations (anions). To evaluate istr, one needs to obtain the ionic charge density distribution (Fe = ze(nþ - n-)) across the channel section as well as the velocity field (u) in the flow domain. The later is a combined consequence of the pressure-driven and electric-field-driven Langmuir 2010, 26(13), 11589–11596
ES ¼ Eref
ð5Þ
where ΓB is the bulk ionic conductivity, expressed in units of S/m. Furthermore, one writes
ð8Þ
R2
0 ½ð2y - y
Rþ
R2 0
ÞsinhðψÞ dy ! ψ sinhðψÞ dy 1ζ 2
ð11Þ
where ψ ¼
ezψ ezζ y ΓB μ ,ζ ¼ ,y ¼ ,R ¼ , Eref kB T kB T H n0 ezε0 εr ζ
¼ -
H 2 dp 2ε0 εr ζ dx
(24) Chakraborty, S.; Das, S. Phys. Rev. E 2008, 77, 037303.
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2.2. Modified Considerations. The mathematical treatment presented in section 2.1 does not take into account the effect of local ionic concentration variations within the EDL with respect to assessing the effective EDL conductivity. Such approximations may turn out to be adequate only when the EDL spans over a negligible proportion of the channel height. However, for cases in which the EDL thicknesses turn out to be comparable to the channel height (without necessarily incurring EDL overlap), such considerations may indeed yield grossly erroneous predictions of the streaming potential so that modified considerations may be necessary, as detailed subsequently. Conductivity gradients across the EDL may be accounted for by considering the resultant conductivity to be a combined consequence of the Stern layer conductivity (ΓSt in units of S), the diffuse layer conductivity (ΓDiff in units of S), and the bulk conductivity (ΓB). Considering appropriate length scales of the Stern layer and diffuse layer to be βSt and λ, respectively, the net ionic conduction current (per unit width) may be expressed as Z icond, mod ¼ 2ΓSt ES þ 2ΓDiff ES þ ΓB ES
βSt
Z ¼e
λ
λ
ð12Þ
ðΛ þ n þ þ Λ - n -Þ dy
2H - βSt 2H - λ
ðΛ þ n þ þ Λ - n -Þ dy
ð13Þ
where n( = n0 exp(-exψmod/kBT) and ψmod is a modified potential distribution in the diffuse layer after accounting for the Stern layer effect. Some practical considerations for the diffuse layer conductivity estimation are outlined in Appendix B. It is also important to mention that with an adequate accounting of the finite thickness of the Stern layer the correct boundary conditions for the potential distribution become ψjy ¼ βSt , 2H - βSt
dψ ¼ ζ and dy
¼0 y¼H
This implies that the modified potential distribution outside the Stern layer will read 8 " # 4kB T < ezζ y - βSt -1 tanh tanh ψmod ¼ exp ze : 4kB T λ " þ tanh
-1
9 # ezζ 2H - y - βSt = exp tanh ; 4kB T λ
ð14Þ
Considering the modified expression for the conduction current (eq 12), a modified expression for the streaming potential (25) Schoch, R. B.; Renaud, P. Appl. Phys. Lett. 2005, 86, 253111.
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5.0 (1/nm2) 5.0 (1/nm2) 2.2 10-9 m2/s 2.2 10-9 m2/s 76.2 10-9 m2/(V s) 80 10-9 m2/(V s) 1 2
can be obtained as R2-γ
½2ðy - γÞ - ðy2 - γ2 Þsinhðψ mod Þ dy R2 2 0 2ðy - y ÞsinhðψÞ dy ! Z 1 2 ψ sinhðψÞ dy 1þ 1R 0 ζ ! Z 1 2-γ ψ mod DuSt þ DuDiff þ ð1 - λ=HÞ þ 1sinhðψ mod Þ dy R γ ζ ES, mod ¼ ES
γ
ð15Þ
dy
Details of the evaluation of ΓSt is outlined in Appendix A. Furthermore, ΓDiff may be expressed as25 Z
NKþ = Nþ NCl- = NDKþ = Dþ DCl- = DΛKþ = Λþ ΛCl- = ΛpKKþ = pKþ pKCl- = pK-
2H - λ
¼ 2ΓSt ES þ 2ΓDiff ES þ ΓB ES ð2H - 2λÞ
ΓDiff ¼ e
Table 1. Representative Parameter Values for KCl
where ψhmod =(ezψmod/kBT)γ=βSt/H, DuSt =ΓSt/ΓBH (Dukhin number denoting the ratio of the Stern layer conductivity and the bulk conductivity), and DuDiff = ΓDiff/ΓBH (Dukhin number denoting the ratio of the diffuse layer conductivity and the bulk conductivity). It is important to mention in this context that it may be rather inappropriate to prescribe DuDiff as an independent parameter because the same depends greatly on the ionic concentration distributions within the diffuse portion of the EDL, as elucidated through the results presented in the subsequent section.
3. Results and Discussions For the simulation results reported in the present study, we consider an aqueous KCl solution to be the fluid medium. The corresponding electrochemical parameters are enlisted in Table 1. In addition, we use the following simulation parameters: ε0 = 8.8 10-12 C/(V m), εr=79.8, ε1r=79.8, ε2r=79.8, RSt=βSt/2, T= 300 K, and kB =1.38 10-23 J/K. It is important to mention in this context that variations in the ratio ES,mod/ES are primarily dictated by four extrinsic dimensionless parameters, namely, ζh, R, H/λ, and γ = βSt/H (eq 15). However, the parameter R itself depends solely on the zeta potential for a given combination of the fluid viscosity, permittivity, bulk ionic concentration (which needs to be fixed to affix the H/λ ratio), and bulk ionic conductivity. Thus, in effect, one can reduce the numbers of important extrinsic dimensionless parameters to ζh, H/λ, and γ. It can also be noted in this context that although the Dukhin numbers are additional influencing parameters, they are intrinsic to the concentration variations within the EDL and hence cannot be independently prescribed. Figure 2 shows the variation of the Stern layer conductivity, ΓSt, with the dimensionless zeta potential (ζh) for different values of the dimensionless Stern layer thickness, γ. It is important to mention here that variations in H/λ result in imperceptible changes and are hence not considered while plotting this Figure. From the Figure, it is evident that it turns out to be a strong function of ζh, only if the magnitude of ζh remains below a critical limit (∼2). This may be qualitatively explained as follows. For lower magnitudes of the zeta potential, increments in the magnitudes of the zeta potential (which indicates an equivalent increase in the wall charge density) increase the concentration of counterions within the Stern layer (caused by a larger electrostatic attraction induced by the larger wall charge density). This, in Langmuir 2010, 26(13), 11589–11596
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Figure 2. Variation of the Stern layer conductivity with dimensionless zeta potential ζh = ezζ/kBT for different values of the Stern layer thickness (quantified by the dimensionless ratio γ = βSt/H). The Stern layer conductivity shows no dependence on the H/λ ratio.
Figure 3. Variation of the diffuse layer conductivity with dimensionless zeta potential ζh = ezζ/kBT for different values of the relative EDL thickness (quantified by the dimensionless ratio H/λ) for a γ = βSt/H = 0.004 diffuse layer conductivity showing a negligible dependence on γ.
effect, increases the Stern layer charge density, σSt, and hence the Stern layer conductivity. However, beyond a threshold magnitude of the zeta potential, the Stern layer becomes saturated with the counterions, forcing a subsequent invariance of σSt and ΓSt with further increments in the magnitude of ζh. For subsequent increments in the magnitude of ζh, the corresponding enhancement of the bare wall charge density σ0 is compensated for by the increase in the diffuse layer charge density σd. Figure 2 also depicts the dependence of ΓSt on γ. A thinner Stern layer (implicating a reduced value of γ) essentially signifies that a particular number of counterions are packed within a smaller volume, which increases the Stern layer conductivity. This consideration, however, cannot be indiscriminately extrapolated to the limit of γ f 0, for which the underlying physics forming the basis on which the present theoretical model is developed breaks down altogether. Figure 3 depicts the variation in the diffuse layer conductivity, ΓDiff, with the dimensionless zeta potential, ζh, for different values of the relative EDL thickness, H/λ, corresponding to γ=0.004. It is important to mention here that variations in γ result in imperceptible changes in ΓDiff and are hence not considered while plotting this Figure. From Figure 3, it is evident that increments in the magnitude of ζh result in consequent increments in the value of ΓDiff. This may be attributed to the increments in counterion concentration within the diffuse layer with enhancements in the Langmuir 2010, 26(13), 11589–11596
Figure 4. Variation of the ratio ES,mod/ES with the dimensionless zeta potential (ezζ/kBT) for different values of the dimensionless Stern layer thickness, γ, for (a) the case in which only the Stern layer conductivity (ΓSt) is considered (in the inset we magnify the corresponding variation for low magnitudes of the zeta potential), (b) the case in which only the diffuse layer conductivity (ΓDiff) is considered (as ΓDiff varies negligibly with γ, plots are identical for different values of γ), and (c) the case in which both ΓSt and ΓDiff are considered (in the inset we magnify the corresponding variations for small and large magnitudes of the zeta potential). In these plots, we use H/λ = 5.
magnitude of ζh, as dictated by the Boltzmann distribution of the ionic species. Such increments are substantially more prominent at larger magnitudes of ζh as a consequence of the fact that the corresponding proportionate enhancements in the bare wall surface charge density need to be compensated for solely by increments in the diffuse layer charge density, σd. From Figure 3, it is also evident that larger values of H/λ indicate higher values of ΓDiff. This may be explained from the fact that for a given ζh a thinner EDL for a given channel height (i.e., a greater H/λ ratio) indicates the packing of a given number of counterions in a smaller volume, thereby enhancing ΓDiff. DOI: 10.1021/la1009237
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Figure 4a-c depicts the individual as well as the cumulative effects of the EDL conductivity components (ΓSt and ΓDiff) on the zeta potential dependence of streaming potential ratio ES,mod/ES (ES,mod is the streaming potential with considerations of conductivity variations within the domain) for different values of γ. Figure 4a considers ΓSt as the sole contributor to the EDL conductivity variations. Because of significant increments in ΓSt with increases in the magnitude of ζh for low magnitudes of ζh (Figure 2), an equivalent decrement occurs in the ES,mod/ES ratio over that regime (inset of Figure 4a). With further increments in the magnitude of ζh, however, ionic advection currents strengthen substantially and tend to compete with the conductive effects. A simultaneous saturation of ΓSt with the corresponding enhancements in the magnitude of ζh tends to offset the disparities between ES,mod and ES so that ES,mod/ES increases with increments in the magnitudes of ζh over the regimes characterized by high magnitudes of ζh (Figure 4a). Interestingly, smaller values of γ result in higher values of ΓSt (Figure 2) and consequently lower values of the ES,mod/ES ratio. Figure 4b depicts a situation in which the EDL conductivity effects are accounted for only through the consideration of ΓDiff. ΓDiff is negligibly small for low magnitudes of ζh but monotonically increases with further increments in the magnitudes of ζh (Figure 3). Consequently, the ratio ES,mod/ES turns out to be close to unity for low magnitudes of ζh and monotonically decreases with increments in the magnitude of ζh to attain low (with significant deviation from unity) values at very high zeta potentials. The fact that ΓDiff has a negligible dependence on γ implies that ES,mod/ES is also virtually unaffected by the variation in γ. Figure 4c represents a situation in which both ΓSt and ΓDiff constitute the EDL conductivity variations. For low and intermediate magnitudes of ζh, ΓDiff is significantly small, which in turn implies that the variation of ES,mod/ES follows a trend analogous to that obtained with the consideration of ΓSt being the sole contributor to the EDL conductivity (cf. Figure 4a and its inset with Figure 4c and its inset 1). However, for greater magnitudes of ζh, ΓDiff in coordination with ΓSt becomes large enough to result in a relative dominance of the EDL conductive current enhancement effects as compared to the corresponding augmentations in the advective current so that the ratio ES,mod/ES again starts to decrease with increments in the magnitude of ζh (inset 2 in Figure 4c). It is also interesting that the variations in ES,mod/ES with alterations in γ are determined by the corresponding variations in ΓSt only. Furthermore, we observe that in terms of the qualitative zeta potential dependence of the ES,mod/ES ratio the EDL conductivity effects considered in this work turn out to be the most prominent only at very large and very small magnitudes of the zeta potential whereas the variations in the advective transport of ions dictate the behavior of the streaming potential in the intermediate zeta potential regime. Furthermore, from all of the cases studied in Figure 4a-c, we may infer that the consideration of conductivity differences between the EDL and the bulk may result in predictions of the magnitudes of the streaming potential that may be substantially lower than the corresponding predictions made by considering the bulk conductivity as the sole reference parameter. This suggests that by ignoring the distinctive contributions of different layers of the EDL with respect to determining the resultant conduction current one may grossly overpredict the magnitude of the streaming potential in practical calculations. Figure 5a-c depicts cases identical to those portrayed in Figure 4a-c, but for different H/λ ratios rather than for different γ values. It is to be noted in this respect that although changes in 11594 DOI: 10.1021/la1009237
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Figure 5. Variation of the ratio ES,mod/ES with the dimensionless zeta potential (ezζ/kBT) for different values of the relative EDL thickness, H/λ, for (a) the case in which only the Stern layer conductivity (ΓSt) is considered, (b) the case in which only the diffuse layer conductivity (ΓDiff) is considered (in the inset we magnify the corresponding variations for large magnitudes of the zeta potential), and (c) the case in which both ΓSt and ΓDiff are considered (in the inset we magnify the corresponding variations for large magnitudes of the zeta potential). In these plots, we use the dimensionless Stern layer thickness, γ = 0.004.
H/λ do not alter the values of ΓSt to any extent the corresponding implications with respect to ionic advection may not be trivially overruled. As such, increments in H/λ essentially signify enhancements in the ionic advection strength. This, in turn, tends to nullify the effects of variations in ΓSt toward determining the streaming potential and attempts to lower the differences between ES,mod and ES. Thus, for a given magnitude of ζh, the ratio ES,mod/ ES turns out to be higher for a larger value of H/λ, for the case in which the EDL conductivity effects are accounted for by sole considerations of ΓSt (Figure 5a). Langmuir 2010, 26(13), 11589–11596
Das and Chakraborty
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Figure 5b depicts the implications of variations in the magnitude of ζh with respect to the streaming potential predictions for different values of H/λ, with ΓDiff as the sole contributor to the EDL conductivity. Although ΓDiff increases with increments in H/λ, it remains significantly low for low and intermediate zeta potential regimes. This, in turn, implies that the H/λ dependence of the ES,mod/ES ratio is solely dictated by the corresponding variation of the ionic advection strengths over this regime. Accordingly, the ES,mod/ES ratio increases with progressive increments in the H/λ ratio (Figure 5b). However, for large magnitudes of ζh, ΓDiff is significantly enhanced so that its influence can successfully outweigh the effects due to alterations in the ionic advection strength. Consequently, the augmentation of ΓDiff with H/λ (Figure 3) shows that the ES,mod/ES ratio decreases with increments in H/λ (inset of Figure 5b). Figure 5c depicts the implications of variations in the magnitude of ζh with respect to the streaming potential predictions for different values of H/λ, with ΓSt and ΓDiff being the simultaneous contributors to the EDL conductivity. In this case, the ionic advection primarily dictates the back electromotive transport for lower and intermediate magnitudes of ζh, resulting in increments in the ES,mod/ES ratio with enhancements in H/λ (Figure 5c). However, for large magnitudes of ζh, the H/λ dependence of ΓDiff turns out to be critical (Figure 5b), resulting in decrements in the ES,mod/ ES ratio with increments in H/λ (inset of Figure 5c). Overall, like Figure 4, Figure 5 also shows that a consideration of the deviation of the EDL conductivity from the bulk may lead to significantly reduced magnitudes of the streaming potential. This may fundamentally be attributed to appreciably augmented strengths of the conduction current with such a consideration. However, it may be noted that qualitative variations in the streaming potential are not solely determined by the EDL conductivity effects. Rather, those evolve as a competing consequence of the ionic advection and the EDL conduction effects, as elucidated earlier.
4. Conclusions In the present study, we demonstrate that inadequate considerations of conductivity variations within the EDL, as employed in many traditional models, can lead to gross inaccuracies in qualitative as well as quantitative predictions of the streaming potential. We show that a correct estimation of the streaming potential can be made only with an appropriate zeta-potential-modulated representation of the Stern layer and the diffuse layer conductivities, which are sensitively dependent on the relative lateral extent of the fluidic channel in reference to the Debye length and the relative thickness of the Stern layer with respect to the channel height. We also reveal contrasting natures of zeta potential dependences of the Stern layer and the diffuse layer conductivities, which, in conjunction with the effects of ionic advection, may lead to nontrivial implications in the streaming potential estimation. Such sensitive variations of conductivity values within the EDL need to be appropriately taken into account for physically consistent estimations of the streaming potentials, bearing immense consequences on an accurate quantification of energy transfer and conversion effects in narrow fluidic confinement.
Appendix A: Estimation of the Stern Layer Conductivity We consider a Gouy-Chapman-Grahame model16 of the EDL, with various layers as marked in Figure 1. The IHP and OHP are assumed to be located at distances of RSt and βSt respectively, from the solid wall. Let σ0 be the bare charge density at the solid-solution interface and let ψ0 be the corresponding value of the potential. Also, let Langmuir 2010, 26(13), 11589–11596
ψ0(RSt) be the potential at the IHP. Quantitities ψ0 and ψ0(RSt) are connected through σ0 by the equation ψ0 - ψ0 ðRSt Þ ¼ σ0
RSt ε0 ε1r
ðA1Þ
where ε1r is the relative permittivity of the region between the wall and the IHP. The potential at the OHP is the zeta potential (ζ) and is connected to ψ0(RSt) through the average diffuse layer charge density σd by the equation ψ0 ðRSt Þ - ζ ¼ - σ d
βSt - RSt ε0 ε2r
ðA2Þ
where ε2r is the relative permittivity of the region between the IHP and the OHP. Physically, eqs A1 and A2 point to the fundamental relationship that the potential difference across any section is equal to the ratio of the charge contained in the section to the capacitance of the section. For example, in the section of the Stern layer bounded by the channel wall and the IHP, the potential drop is ψ0 - ψ0(RSt), the charge contained is σ0A (A is the area), and the capacitance is ε0ε1rA/RSt; consequently, we arrive at eq A1. Similarly arguments can be forwarded to arrive at eq A2.The average charge density, σd, within the diffuse layer is obtained as Rλ σd ¼
R 2H - βSt Dψ Dψ dy 2H - λ ε0 εr Dy dy Dy ¼ λ - βSt λ - βSt
βSt ε0 εr
ðA3Þ
Equation A3 physically points to the fact that the net charge density at the plane separating the Stern layer and the diffuse layer is screened entirely within the diffuse layer (i.e., beyond the EDL, the net charge density is zero, or in other words, beyond the EDL, the effect of the charge on the wall is not felt). Considering eqs A1-A3 and eq 14, the transverse potential field across the entire channel section can be expressed as ψ0 - ψ0 ðRSt Þ ¼ σ 0
RSt for 0 < y < RSt ε0 ε1r
βSt - RSt for RSt < y < βSt ε0 ε2r ( " # 4kB T ezζ y - βSt -1 tanh exp ψ ¼ tanh ze 4kB T λ " #) ezζ 2H - y - βSt -1 exp tanh þ tanh 4kB T λ ψ0 ðRSt Þ - ζ ¼ - σ d
for βSt < y < 2H - βSt βSt - RSt for 2H - βSt < y < 2H - RSt ε0 ε2r RSt ψ0 - ψ0 ðRSt Þ ¼ σ 0 for 2H - RSt < y < 2H ðA4Þ ε0 ε1r
ψ0 ðRSt Þ - ζ ¼ - σ d
Furthermore, as a result of electroneutrality, the total charge density of the system must be zero so that σ 0 þ σ St, i þ σd ¼ 0
ðA5Þ
where σSt,i is the Stern layer charge density of ion type i. With a prior knowledge of the zeta potential (ζ), eqs A1, A2, A3, and A5 involve five unknowns (namely, ψ0, ψ0(RSt), σ0, σSt,i, DOI: 10.1021/la1009237
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Das and Chakraborty
and σd) that need to be mathematically closed with an additional constraint considering the ionic reaction equilibria. In particular, σSt,i may be obtained by an adsorption-desorption isotherm model16 that considers that ions can be adsorbed onto free available surface sites, with the possibility for all types of ions to be adsorbed onto any site, leading to a possible competition between the existing ionic species. Let us consider an adsorption-desorption reaction in which the ith ionic species Xi gets adsorbed onto an empty Stern layer site Si such as Si þ Xi TSXi
ðA6Þ
Considering Ki to be the equilibrium constant for the desorption process, one gets Ki ¼
½Si ½Xi ½SXi
ðA7Þ
where [Si] and [SXi] are expressed in 1/m2 whereas [Xi] is the ion volumetric concentration (in 1/m3) expressed as zi e ψ0 ðRSt Þ ðA8Þ ½Xi ¼ n0 exp kB T The Stern layer charge density of the ith ionic species can be expressed as σSt, i ¼ ezi ½SXi
ðA9Þ
By considering Ai to be the cross-sectional area of the ith ionic species, the total number of sites available per unit area to the ith ionic species, Ni, is given by Ni ¼ 1=Ai
ðA10Þ
Accordingly, one can infer that the concentration of empty Stern layer sites for the ith ionic species is given by 0 ½Si ¼ Ni @1 -
N X ½SXj j ¼1
Nj
1 A
ðA11Þ
Using eqs A7 and A11 and noting that [Sj]=Nj, we obtain 0 1 N X ½Si ½Xi Ni ½Xi @ ½SXj A ½SXi ¼ ¼ 1Ki Ki Nj j ¼1 ¼
Ni ½Xi N ½X P j Ki 1 j ¼1 Kj
!- 1
Ni ½Xi
! N ½X P j Ki 1 þ j ¼1 Kj
ðA12Þ
(because the dissociation constants Kj are very high (Table 1), making [Xj]/Kj , 1). Henceforth, using eq A12 in eqs A8 and A9, we get n0 zi e ψ0 ðRSt Þ exp k T Ki B σ St, i ¼ N n P zj e 0 ψ0 ðRSt Þ 1þ exp kB T j ¼1 Kj ezi Ni
ðA13Þ
Equation A13 closes the system of equations and unknowns through an explicit determination of the Stern layer charge 11596 DOI: 10.1021/la1009237
density (for ion type i). Considering the same, the Stern layer conductivity may be obtained as ΓSt ¼
e ðD þ σSt, þ þ D - σ St, -Þ kB T
ðA14Þ
where D( represents the diffusivities of the cations (anions).
Appendix B: Ionic Mobility Values for Diffuse Layer Conductivity Calculations From eq 13, it becomes evident that the estimation of the diffuse layer conductivity strongly depends on a correct assessment of the ionic mobility (Λ() values. In practice, values of Λ( need not be constants but may be rather dependent on local variations in the electrical potential. In cases in which the ionic species in the solution are contributed by a partial dissociation of weak acids and bases, the respective ionic mobilities may be expressed as Λ ( ¼ β ( , free Λ ( , ¥
ðB1Þ
where β (,free represents the fractions of free cations (anions) in the solution and Λ(,¥ represents the mobilities of the cations (anions) at infinite dilution. When the anionic species come from a weak monobasic acid, β-,free can be expressed as26 β - , free ¼
10 - pKa 10 þ 10 - pH - pKa
ðB2Þ
where pKa is the pKa of the acid that furnishes the analyte. Similarly, when the cationic species come from a weak monoacidic base, β (,free can be expressed as26 β þ , free ¼
10 -pH 10 þ 10 -pH -pKa
ðB3Þ
where pKa is the pKa of the conjugate acid of the base contributing the cations. When EDL effects are neglected in the conductivity calculations, the buffer pH value used in eqs B2 and B3 remains cross-sectionally invariant. However, when EDL effects are included in the ionic mobility calculations, the hydrogen ions in the solution also obey the Boltzmann distribution within the EDL so that one may write ! - eψ ðB4Þ pH ¼ pHðyÞ ¼ pH0 - log10 exp kB T where pH0 is a reference pH value corresponding to the bulk state (the state in the “well”). Because ψ is a function of the transverse (y) coordinate, a combinations of eqs B1-B4 indicates that the ionic mobilities in the diffuse portion of the EDL for weak acid/base solutions are themselves functions of the y coordinate. In the present study, however, we limit the presentation of results to the cases of salts of strong acids and bases only (for example, KCl). For such a case, one may write Λ( ≈ Λ(,¥. Nevertheless, it needs to be mentioned that we make such a choice of salts without losing any generality in the mathematical model outlined in this work and our model remains valid even for cases in which such approximations cease to be appropriate. The main purpose of considering a salt solution such as KCl is based on the relatively well-known parameter values (in particular, those that are needed for the Stern layer conductivity calculations) for Kþ and Cl- that are not easily available for many other salts. (26) Baldessari, F. J. Colloid Interface Sci. 2008, 325, 526.
Langmuir 2010, 26(13), 11589–11596
