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Langmuir 2008, 24, 2884-2891
Ion Size and Image Effect on Electrokinetic Flows Y. Liu,† M. Liu,† W. M. Lau,‡ and J. Yang*,† Department of Mechanical and Materials Engineering and Surface Science Western, UniVersity of Western Ontario, London, Ontario, N6A 5B7, Canada ReceiVed July 10, 2007. In Final Form: NoVember 4, 2007 Electrokinetic phenomena play a major role in microfluidic systems, and such a role becomes even more significant in nanofluidic systems due to the increase of the surface-to-volume ratio. Description of the electric double layer (EDL) at a solid-liquid interface is the key to understand and utilize electrokinetic phenomena. However, the traditional Gouy-Chapman (GC) theory for the EDL, which has been successfully used in many microfluidic applications, does not include some important characteristics such as ion size and image effect. These characteristics are indeed important in nanofluidics. This paper explores the impacts of ion size and the image effect on micro- and nanoscale electrokinetic flows. An advanced theory, the modified Poisson-Boltzmann (MPB) theory proposed by Outhwaite and his coworkers,1,26 is adopted to describe the EDL. Electrokinetic flows in micro- and nanochannels are reinvestigated. The results show that ion size has significant effects on electrokinetic flows in nanosystems in terms of both the flow field and the streaming potential, while the image effect only significantly affects the streaming potential.
I. Introduction Electrokinetic phenomena play a major role in microfluidic and nanofluidic systems for various applications in biology, medicine, analytical chemistry, and even energy conversion. In these applications, electrokinetic phenomena have been extensively exploited to develop, for example, sensing technology,2-4 electrokineticbatteries,5-7 electrokineticpumps,8,9 cellmanipulation,10-12 and electrokinetic flow focusing and switching.13,14 Electrokinetic phenomena are due to the existence of an electric double layer (EDL), i.e., a charge redistribution, near a solidliquid interface. Thickness of EDL typically ranges from several nanometers to hundreds of nanometers. This thin EDL, normally ignored in macroscale fluid systems, plays a significant role in microscale regimes and becomes more critical at nanoscales. It is crucial to know the exact flow behaviors at the primary stage of design of microfluidic or nanofluidic systems. Therefore, theoretical modeling and simulation are beneficial to dispelling divergence between theoretical anticipation and the real performance in practice.15 An elaborate EDL theory is required for a thorough understanding of electrokinetic flows. The Gouy* To whom correspondence should be addressed: Tel. (519) 6612111 ext. 80158, Fax (519) 6613020. Email:
[email protected]. † Department of Mechanical and Materials Engineering. ‡ Surface Science Western. (1) Outhwaite, C. W.; Bhuiyan, L. B. J. Chem. Soc., Faraday Trans. 1983, 79, 707-718. (2) Chow, A. W. AIChE J. 2002, 1590-1595. (3) Reyes, D. R.; Iossifidis, D.; Auroux, P. A. Anal. Chem. 2002, 74, 26232636. (4) Reyes, D. R.; Iossifidis, D.; Auroux, P. A. Anal. Chem. 2002, 74, 26372652. (5) Yang, J.; Lu, F. Z.; Kostiuk, L. W. J. Micromech. Microeng. 2003, 13, 963-970. (6) Van der Heyden, F.; Jan Bonthuis, D.; Stein, D.; Meyer, C.; Dekker, C. Nano Lett. 2006, 6, 2232-2237. (7) Canning, J.; Buckley, E.; Lyytikainen, K. Elec. Lett. 2004, 40, 298-299. (8) Koch, M.; Schabmueller, C. G. J.; Evans, A. G. R. Sens. Actuators, A 1999, 74, 207-210. (9) Laser, D. J.; Santiago, J. G. J. Micromech. Microeng. 2004, 14, 35-64. (10) Sia, S. K.; Whitesides, G. M. Electrophoresis 2003, 24, 3563-3576. (11) Johnson, R. D.; Badr, I. H. A.; Barrett, G. Anal. Chem. 2001, 73, 39403946. (12) Li, P. C. H.; Harrison, D. J. Anal. Chem. 1997, 69, 1564-1568. (13) Choi, S.; Park, J. K. Lab Chip 2005, 5, 1161-1167. (14) Yang, R. J.; Chang, C. C.; Huang, S. B. J. Micromech. Microeng. 2005, 15, 2141-2148. (15) Li, B. M.; Kwok, D. Y. J. Chem. Phys. 2004, 120, 947-953.
Chapman (GC) theory and its Debye-Hu¨ckel approximation, the linearized Gouy-Chapman (LGC) theory, are the most popular ones in EDL applications.16,17 However, the latter of them (LGC) is only valid for low surface potential cases, and both of them (GC and LGC) do not include effects of finite sizes of ions, nonelectrostatic interactions between counter- and coions at the interface, and image force between ions and the solid surface.18 In general, the ion size19 and the ion-ion correlation1,18 are important characteristics, which should be considered in EDL theories. Furthermore, the image effect20 related to dielectric constants of both solid and liquid can also significantly influence the charge distribution in EDL. To overcome the limitations of the traditional GC theory, several theories have been proposed in recent years. A modification to the Gouy-Chapman (MGC) theory was offered by Stern21 to take into account an exclusion layer which ions cannot enter. The Monte Carlo (MC) method as a statistical approach has been successfully used to simulate EDL near a charged wall with considerations of exclusion volume, the image effect,20 and ionion collisions.22 In MC simulations, a random walk through the phase space of the model system is made. Consequently, a sequence of microscopic states is generated, and the microscopic states are either accepted or not accepted based on some criteria.18 Based on the statistical thermodynamics of liquid, several integral theories including the hypernetted chain theory (HNC),18 the mean spherical approximation (MSA),23,24 and the modified Poisson-Boltzmann theory (MPB)1,25,26 were proposed to seek improvements upon the mean-field theory by incorporating the interionic correlations and the ionic exclusion volume due to the (16) Masliyah, J. H. Aostra Technical Publication Series, 1994. (17) Hunter, J. Oxford UniVersity Press, 1989. (18) Lyklema, J. Volume 2, Academic Press, 1995. (19) Carnie, S. L.; Torrie, G. M.; Valleau, J. P. Mol. Phys. 1984, 53, 235-256. (20) Bratko, D.; Jonsson, B.; Wennerstrom, H. Chem. Phys. Lett. 1986, 128, 449-454. (21) Stern, O. Z. Electrochem. 1924, 30, 508-526. (22) Jamnik, B.; Vlachy, V. J. Am. Chem. Soc. 1993, 115, 660-666. (23) Plischke, M.; Henderson, D. J. Chem. Phys. 1989, 90, 5738-5741. (24) Olevares, W.; Sulbaran, B.; Lozada-Cassou, M. J. Phys. Chem. 1993, 97, 4780-4785. (25) Outhwaite, C. W.; Bhuiyan, L. B.; Levine, S. J. Chem. Soc., Faraday Trans. II 1980, 76, 1388-1408. (26) Bhuiyan, L. B.; Outhwaite, C. W. Phys. Chem. Chem. Phys. 2004, 6, 3467-3473.
10.1021/la702059v CCC: $40.75 © 2008 American Chemical Society Published on Web 02/01/2008
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finite size nature of the ions. Additionally, MPB further includes an accurate description of the image effect, which is related to the dielectric material property of the solid surface.25,26 In terms of the image effect, several other theoretical approaches are also applicable.27-31 Compared with MPB, these approaches are too complex to be implemented computationally. With many years of continuous development, the computer program for MPB has evolved from the earliest version of MPB125 to the latest version of MPB5.26 MPB5 shows a good agreement with density functional theory (DFT) and MC. In this paper, MPB5 is used and generally termed as “MPB”. An earlier application of MPB was implemented by Bratko32 to explore the potential of mean force acting on the ions and then to determine the coefficients of self-diffusion of counterions in micellar solutions. It was proven that MPB works better than PB in calculation of coefficients of self-diffusion of divalent counterions in micellar solutions.32 The results predicted by MPB agreed well with previous experimental data. In fact, comparisons among MPB, MC, HNC/MSA, and GC have also shown that MPB agrees very well with MC and HNC/MSA,1,25,26,32 and reveals more insights of the EDL than GC. Since MPB keeps a similar format to that of GC theory facilitated with the classical Poisson-Boltzmann equation, common GC users typically find it easily comprehensible and handy. In the next sections, we will briefly summarize various EDL theories, investigate the charge distribution and potential distribution in EDL using MPB, and apply MPB to study electrokinetic flows in micro- and nanochannels with considerations of ion size and the image effect.
II. Theories of the Electric Double Layer We consider a symmetric (z0:z0) electrolyte solution contacting a flat solid wall, where z0 is the valence of ion species, y direction is assumed to be normal to the planar plate, and its origin is located at the wall. When an aqueous solution contacts the wall, the EDL is formed at the solid-liquid interface.16,17 According to the electrostatic theory, the relationship between the electrical potential ψ and the net charge density per unit volume, Fe, at a position y is depicted by the Poisson equation
Fe ∂2ψ )2 0 ∂y
(1)
where is the relative permittivity of the liquid and 0 is the permittivity of the vacuum, 8.854 × 10-12 C V-1 m-1. The net charge density is given by 2
Fe )
Boltzmann equation as a solution of the GC theory for the symmetric (z0:z0) electrolyte
( )
z0eψ ∂2ψ 2n0z0e ) sinh 2 0 kBT ∂y
(4)
with boundary conditions ψ(0) ) ψS and ψ(∞) ) 0 or ∂ψ(∞)/∂y ) 0. Under the Debye-Hu¨ckel approximation, when ψS < 25 mV, sinh(z0eψ/kBT) ≈ z0eψ/kBT. With the definition of the Debye length, κ-1, as
κ-1 )
( ) 0kBT 2
1/2
(5)
2
2e z0 n0
eq 4 reduces to its linear form
∂2Ψ ) κ2Ψ 2 ∂y
(6)
where Ψ is the dimensionless electrical potential defined as Ψ ) ez0ψ/kBT. Equation 6 is normally referred as the LGC theory. Since point ions are assumed in GC and LGC, GC and LGC do not distinguish different solute ions by size. In order to reflect the ion size effect, the modified Gouy-Chapman theory (MGC) was proposed by Stern,20,33 in which a compact layer of ions is assumed to be at the closest approach to the solid surface. Thus, the whole EDL are divided into linear and nonlinear regions
Ψ)
dΨ y + ΨS 0 e y e a/2 dy
∂2Ψ ) κ2 sinh(Ψ) y g a/2 ∂y2
(7)
where ΨS is the dimensionless surface potential and a is the ionic diameter. Although MGC considers the ion size effect, the basic assumption ignores details. A comprehensive EDL theory is needed to take more details of ion size into account in order to explain and predict more aspects of EDL. In this context, MPB1,26 includes more effects due to the image force, the fluctuation term, and the exclusive volume
2
Fi ) ∑ nizie ∑ i)1 i)1
(2)
where e is the elemental charge, 1.602 × 10-19 C, ni is the ionic number concentration per unit volume, and zi is the valence for the i type ion in the electrolyte. The Boltzmann distribution is
( )
ni ) n0 exp -
zieψ kBT
(3)
where kB is the Boltzmann constant 1.381 × 10-23 J/K, T is the absolute temperature, and n0 is the ionic number concentration in the bulk solution. Combining eqs 1-3 results in the Poisson(27) Carnie, S. L.; Chan, D. Y. C. Mol. Phys. 1984, 51, 1047-1047. (28) Vertenstein, M.; Ronis, D. J. Chem. Phys. 1987, 87, 4132-4146. (29) Kiellander, R.; Marcelja, S. J. Chem. Phys. 1985, 82, 2122-2135. (30) Kiellander, R.; Marcelja, S. J. Chem. Phys. 1988, 88, 7138-7146. (31) Kiellander, R. J. Chem. Phys. 1988, 88, 7129-7137. (32) Bratko, D. Bioelectrochem. Bioenerg. 1984, 13, 459-471.
ψ) ∂2ψ
∂ψ y + ψS 0 e y e a/2 ∂y
)-
∂y2
1 0
∑i zien0g0i(y)
y g a/2
(8)
where g0i(y) is the singlet distribution function for ith ion at a normal distance y from the wall into the electrolyte. g0i(y) can be expressed as
{
g0i(y) ) ξi exp -
[
zie zie (F - F0) + Fψ(y + a) + 2kBT 4aπ Fψ(y - a) -
F-1 a
∫yy-+aa ψ(X) dX
]}
(9)
(33) Torrie, G. M.; Valleau, J. P. Chem. Phys. Lett. 1979, 65, 343-346.
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Table 1. Comparison of the Dimensionless Diffuse Layer Potential Ψ(1/2) between Simulation Here with Previously Published MPB Results Ψ(0) c ) 0.1 M
Ψ(1/2)
3.98 5.83 2.17 3.79 1.71 5.62
c ) 1.0 M c ) 2.0 M
authors
ref 26
3.14475 4.14770 1.32646 2.10945 0.86634 2.26118
3.14 4.15 1.33 2.11 0.87 2.26
where ξi is the exclusive volume term. ξi can be expressed as
( ) {
ξi(y) ) H y -
a
2
exp -2π
x+a × ∫∞y ∑ n0i ∫max(a/2,y-a)
[
] }
i
1
(X - x)g0i(X) exp eiΦ(x, X) dX dx kBT
(10)
Figure 1. Dimensionless potential distribution predicted by different EDL theories for a (1:1) electrolyte at c ) 0.1976 M, ΨS ) -5.0, and f ) 0 for MPB.
where Φ(x, X) is the fluctuation term for electric potential evaluated over the exclusive volume of the discharged ions.24 Other relevant parameters are26
F)
F)
(1 + fδ2) (1 + κ′a)(1 - fδ1) (4 + fδ3)
4 + κ′(a + 2y) + fδ4 δ1 )
a/2 e y e 3a/2
δ2[κ′a cosh(κ′a) - sinh(κ′a)] [(1 + κ′a) sinh(κ′a)]
δ2 ) δ3 )
y g 3a/2
exp{2κ′(a - y) sinh(κ′a)} 2κ′a
Figure 2. Dimensionless potential distribution of EDL for a (1:1) electrolyte with different ion sizes at ΨS ) -5.0 and f ) 0 for MPB.
1 1 a - 2y + (a2 + 2ay)1/2 + {1 y κ′
[
exp(κ′[(a2 + 2ay)1/2 - a - 2y])} a δ4 ) δ3 - (1 + exp{κ′[(a2 + 2ay)1/2 - a - 2y]}) y
[
κ′ )
e2 0kBT
∑i
zi2n0g0i(y) f )
]
- W + W
κ′0 ) lim κ′ F0 ) lim F ) (1 + κ′0a)-1 y′f∞
]
y′f∞
(11)
where f is the image factor, W is the dielectric constant of the solid planar plate, κ′ is the local Debye-Hu¨ckel parameter, and κ′0 is the traditional Debye-Hu¨ckel length. Definitions of other operators can be found in ref 26.
III. Discussions on Different EDL Theories As discussed above, MPB enriches the description of EDL in several aspects: the ion size effect, the exclusive volume, the fluctuation potential, and the image effect. The key of EDL theories is to determine ion distribution and potential distribution in the EDL. Next, we compare different EDL theories for potential distribution in terms of ion size, image effect, surface potential, and ionic concentration. MPB computation is carried out using the quasi-linearization technique introduced in ref 25. Computation accuracy has been verified by repeating the published results
in ref 26. A satisfactory agreement has been obtained as shown in Table 1. If not specified, these parameters for a (1:1) primitive electrolyte are used: the ionic diameter a ) 0.425 nm, the permittivity ) 78.5, the absolute temperature T ) 298 K, the ionic concentration c ) 0.1976 M and the dimensionless surface potential ΨS ) z0eψS/kBT ) -5.0. In Figure 1, we plot the dimensionless potential distribution Ψ using MPB, MGC, GC, and LGC. We find that these theories make distinct differences of potential distribution within the 3 nm vicinity of the solid wall. Out of this distance from the solid wall, there is no distinguishable difference among these theories. MPB is similar to MGC in the compact layer, but remarkably different from GC and LGC, which suggests that the ion size plays an important role. In addition, since ΨS ) -5.0, which corresponds to a dimensional surface potential ψS ≈ -130 mV, the assumption of sinh(z0eψ/kBT) ≈ z0eψ/kBT becomes invalid so that LPB deviates from the other three theories. A. Ion Size and Ionic Concentration. MPB includes not only the effect of ion size but also the ionic interactions. Thus, MPB also provides a more accurate description of EDL for an electrolyte solution of a high ionic concentration. In Figure 2, we plot the dimensionless potential distribution Ψ predicated by MPB for different ion sizes.16 These ion sizes correspond to some real positive ions: K+ is 0.138 nm in diameter; Na+ is 0.102 nm in diameter; Li+ is 0.076 nm in diameter; and H+ is 0.05 nm in diameter. Since a primitive electrolyte is assumed in MPB, the negative ions are assumed to have the same size as the positive ions. We find that ion size significantly affects the profile of potential distribution, whereas GC yields an identical
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Figure 3. Dimensionless potential distribution of EDL for a (1:1) KCl electrolyte with different ionic concentrations at ΨS ) -6.0 and f ) 0 for MPB.
Figure 6. Dimensionless potential distribution of EDL of a 2.0 M LiCl electrolyte with a ΨS ) -1.0 charged surface for different image factors.
Figure 4. Dimensionless potential distribution of EDL for an electrolyte with the ionic concentration c ) 0.1976M and the dimensionless surface potential ΨS ) -5.0 and different image factors.
Figure 7. Dimensionless potential distribution of EDL for a 0.01 M LiCl electrolyte at the dimensionless surface potential ΨS ) -10.0 and different image factors.
Figure 5. Dimensionless potential distribution of EDL of a 2.0 M LiCl electrolyte with a ΨS ) -10.0 charged surface for different image factors.
curve for electrolytes of different ions. Although MPB has been very successful in studying ion size effect on EDL, it might lose some accuracy as the ion size goes to zero.19 The impact of the bulk ionic concentration is also examined using MPB. In general, when the bulk ionic concentration is higher, more counter-ions distribute in the region close to the charged solid surface. Consequently, EDL is reduced in thickness, or in other words, EDL becomes “compressed”. Figure 3 shows
the potential distribution of EDL for KCl solutions with different ionic concentrations: 0.01 M and 1 M, MPB predicts a thinner EDL of KCl solutions than that predicted by GC. B. Image Effect. In a real system, the dielectric discontinuities should be taken into account. Consequently, the image factor, defined as f ) ( - W)/( + W), is introduced in MPB to study such an effect. If W f ∞, f f -1; and if W f 1, f f 0.975. The former corresponds to a metallic surface and the latter corresponds to an insulating surface. Here, a glass surface of a dielectric constant W ) 5.0 is chosen, so that f ) 0.88. As a typical semiconductor material, a silicon surface of a dielectric constant W ) 11.5 is also chosen, whose corresponding image factor is f ) 0.7444. From Figure 4, where c ) 0.1976 M and the dimensionless surface potential ΨS ) -5.0, we can only find slight differences among potential distribution profiles for different image factors. For an electrolyte of a low or moderate concentration, the image factor has a weak effect on the potential distribution in EDL. 1. Image Effect for High Surface Potential and High Ionic Concentration. As shown in Figure 4, there is no remarkable effect of the image factor on EDL potential distribution when the ionic concentration is relatively low. But questions remain for solid-liquid interfaces of a highly concentrated electrolyte and/or a highly charged surface. In this simulation, we assume an electrolyte-surface system with 2.0 M Li+ and a high surface potential, ΨS ) -10.0 or ψS ≈ -257 mV. As shown in Figure 5, when the image factor increases from -1 to 0.975, the profile
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Figure 8. Schematics of the formed EDL and electrokinetic flow in a 2D parallel-plate channel.
of the EDL potential becomes sharper. We find that a critical value fC exists between -0.8 and -0.9 for this electrolytesurface system. If the image factor f is bigger than fC, one valley appears in the potential distribution profile, within which the sign of potential value is opposite to that of the surface potential ΨS. Although GC has never shown such a valley, MC and DFT34,35 both predict the existence of such a valley in EDL potential profiles for some electrolyte-surface systems. The reason for the appearance of this valley is that more bulk ions tend to be attracted into the EDL by the counter-ions. In addition, we also find that if the image factor is smaller than fC, the valley disappears and the profile of EDL potential distribution recovers to its classical shape, a monotonically decreasing curve.36 We further note that this critical value fC depends on the parameters of the specific electrolyte-surface system such as ion size, ion concentration, and surface charge density. A more general parameter, which governs the transition of the EDL potential distribution from a monotonically decreasing profile to a damped oscillating profile, is the dimensionless quantity κ′oa.25,26 2. Image Effect for High Ionic Concentration and Low Surface Potential. For ΨS < -1.0, a remarkable image effect is also found (see Figure 6). It is intriguing that the value of potential increases as the image factor increases in this case (Figure 6). However, in the case of high ionic concentration and high surface potential, the value of potential increases as the image factor decreases (Figure 5). For the case of high ionic concentration and low surface potential, the potential value in the entire EDL keeps the same sign as that of the surface potential (Figure 6). In other words, no valley appears. The reason is that the invasion capability of bulk ions into the EDL is not enough to exceed the resistance caused by counter-ions in the EDL region. 3. Image Effects for Low Ionic Concentration and High Surface Potential. The image effect on the EDL potential distribution for an electrolyte-surface system with a low ionic concentration and a high surface potential is studied here. We consider an electrolyte of 0.01 M Li+ and a charged surface with a dimensionless surface potential ΨS ) -10.0. From Figure 7, the image effect on EDL potential distribution is clear but not significant for the electrolyte-surface system of a low ionic concentration and a high surface potential. All of the results suggest that the image effect strongly depends on the ionic concentration and the surface potential. Based on above simulation and discussion, it is of great interest in studying the ion size and image effects on electrokinetic phenomena and related applications. In the next sections, MPB will be mainly (34) Torrie, G. M.; Valleau, J. P. J. Chem. Phys. 1980, 73, 5807-5816. (35) Gillespie, D.; Valisko´, M.; Boda, D. J. Phys.: Condens. Matter 2005, 17, 6609-6626. (36) Lamperski, S.; Outhwaite, C. W. Langmuir 2002, 18, 3423-3424.
utilized for analyzing electrokinetic flows in microchannels and nanochannels.
IV. Electrokinetic Flow Analysis in Micro/ Nanochannels In numerous micro/nanosystems, micro/nanofluidics are driven by a pressure gradient, which results in the so-called streaming current and streaming potential between the two ends of a channel due to the movement of mobile ions. Meanwhile, a conduction current is induced by the streaming potential, which is opposite to the flow direction, to balance the streaming current in order to maintain a zero net current in the channel. The mechanism of such electrokinetic phenomena has been extensively discussed elsewhere.17,37 However, a comprehensive understanding including effects of ion size and the image factor on electrokinetic flows has not been addressed. Here, we consider a parallel-plate channel of height 2h as shown in Figure 8. The heights for microchannel and nanochannel in our simulation are 1 µm and 50 nm, respectively. The y′ direction is normal to the channel wall and the x′ direction is in the streamwise of the flow (Figure 8). At these sizes of channels, the Navier-Stokes solution of the Poiseuille flow in a parallel-plate channel is still reliable.38 The hydrodynamic Navier-Stokes equation and the corresponding boundary conditions for the flow motion are16,17
-
1 ∂p ∂2υ 1 1 ∂υ + + FE ) µ ∂x′ ∂y′2 µ e x′ ν ∂t υ(h, t) ) 0
∂υ(0, t) )0 ∂y′
(12)
where p is the pressure, υ is flow velocity, Ex′ is the electrical potential strength in the x′ direction, µ is the viscosity, ν is the kinematic viscosity of the liquid, and t is time. At nanoscales, the length of EDL may become comparable to the characteristic width of the nanochannel, resulting in an overlapped EDL. With regard to the overlapped EDL, the Poisson-Boltzmann equation can still be used as long as proper boundary conditions are treated.17,39 The boundary conditions traditionally accepted are17,39
ψ(y′ ) (h) ) ψs dψ )0 | dy′ y′)0
(13)
(37) Israelachvili, J. N. Academic Press Inc. Ltd. 1985. (38) Pennathur, S.; Santiago, J. G. Anal. Chem. 2005, 77, 6772-6781. (39) Verwey, E. J. W.; Overbeek, J. T. G. ElseVier, Amsterdam, 1948.
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The second equation of eq 13 is the symmetric boundary condition. For MPB, the potential distribution of EDL is solved using eqs 8-11 and the boundary conditions eq 13 as well. The definition of volumetric flow rate, q, per unit width of the parallel plate is
q)2
∫0h υ dy′
(14)
We can define the total electrical current as
i)2
∫0h Feυ dy′ +
2z0eDEx′ kBT
∫0h (F1 - F2) dy′
(15)
where D is the diffusion coefficient of the electrolyte. The first term of the right-hand side of eq 15 is the streaming current and the second term is the conduction current. Using eq 2 and eq 8 for a (z0:z0) electrolyte in MPB, we have F1 - F2 ) ez0n0[g01(y′) + g02(y′)]. At equilibrium, the net electric current is zero, which will be used to determine the strength of streaming potential Ex′ whose direction is opposite to the flow direction. Solving the Poisson equation (eq 1) and the Navier-Stokes equation (eq 12), solutions of velocity, current, and flow rate are obtained
υ(y′) )
i)-
{
q)
0 ∂p 1 2 (h - y'2) [ψ(y′) - ψS]Ex′ + 2µ ∂x′ µ
(
0 [2ψSh - 2 µ
-
Figure 9. The profiles for the singlet distribution function g0i(y) by MPB for 10-5 M KCl and 10-5 M LiCl in the 1 µm microchannel at a surface potential ψS ) -200 mV and an image factor f ) 0.74444 for MPB.
2(0)2 µ
( )
∂p ∫0h ψ(y′) dy′] - ∂x′
dy′] + [∫ (∂ψ ∂y′) 2
h
0
20 1 2 3 ∂p + h ( µ 3 ∂x′ µ
( )(
)
)
2z0eD kBT
+
∫0h (F1 - F2) dy′
∫0h ψ(y′) dy′ - hψS)Ex′
}
Ex′
(16)
A. Ion Size Effect on Electrokinetic Flows. We can imagine that difference in flow rate and streaming potential should occur when different electrolyte solutions flow through the same microor nanochannel. Explanation of such a discrepancy could be partially based on difference of ion mobility. But here, we will explore such a discrepancy mainly from the aspect of the ion size. We set the following parameters to investigate the effect of ion size on fully developed electrokinetic flows in a 1 µm microchannel and a 50 nm nanochannel: ψS ) -200 mV, µ ) 0.9 × 10-3 kg m-1 s-1, e ) 1.602 × 10-19 C, ) 78.5 × 8.854 × 10-12 C V-1 m-1, -∂p/∂x′ ) 1 × 106 Pa m-1, and the image factor f ) 0.74444 for a silicon-based microchannel. 10-5 M LiCl and 10-5 M KCl are used as sample electrolytes. Although we also performed the simulations for the cases of higher ion concentration, for example, 0.01 M, the ion size effect is not remarkable, so that we do not show the results for c ) 0.01 M in this section. Based on MPB, the singlet distribution function g0i(y) (i ) 1 and 2) in the microchannel is shown in Figure 9, which will be used to calculate the charge density of counterion F1(y) and coion F2(y). Thereafter, the comparison of the flow velocity profiles predicted by GC, MGC, and MPB for a 10-5 M LiCl electrolyte is presented in Figure 10. Apparently, all of these, EDL theories predict a resistant effect of EDL on the flow, or the so-called electroviscous effect. The prediction by GC (dot line) is very close to that of MGC (dashed line), but both of them in fact underestimate the effect of EDL on flow velocity compared with that predicted by MPB (solid line). To investigate the effect of ion size, we introduce a 10-5 M KCl electrolyte, and keep other
Figure 10. Flow velocity profiles predicted using MGC, GC, and MPB for 10-5 M LiCl in microchannel at a ψS ) -200 mV surface charged potential.
parameters the same as those of the 10-5 M LiCl electrolyte except the ion size “a”. The results are shown in Figure 11a and the enlarged velocity profiles at the center of the microchannel are presented in Figure 11b. The effect of ion size on electrokinetic flows is clear. A quantitative comparison of flow rate and streaming potential predicted by different EDL theories for different electrolytes is given in Table 2. For the 1 µm microchannel, GC and MGC predict about a 4.7% reduction in flow rate compared with the flow rate without consideration of EDL; MPB predicts an even larger decrease of flow rate, or a larger electroviscous effect. With the consideration of ion size difference between K+ and Li+, the flow rate is reduced by 5.5% and 7.5%, respectively, for KCl and LiCl electrolytes. In terms of the streaming potential, MPB predicts a significantly higher streaming potential than that predicted by GC. In terms of the ion size effect on the streaming potential in MPB, the streaming potential of the 10-5 M LiCl electrolyte is 11.9% higher than that of the 10-5 M KCl electrolyte. In a recent experimental work by Ren et al.,40 it was found that different salt solutions (e.g., KCl and LiCl) of the same ion concentration cause different flow rates in the same microchannel and under the same pressure. Specifically, a 10-4 M LiCl solution (40) Ren, L. Q.; Li, D. Q.; Qu, W. L. J. Colloid Interface Sci. 2001, 233, 12-22.
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Figure 12. Flow velocity profiles predicted by GC and MPB for 10-5 M KCl and 10-5 M LiCl in a 50 nm nanochannel at a surface charged potential ψS ) -200 mV and an image factor f ) 0.74444 for MPB.
Figure 11. Flow velocity profiles predicted by GC and MPB for 10-5 M KCl and 10 -5 M LiCl in the 1 µm microchannel at a surface potential ψS ) -200 mV and an image factor f ) 0.74444 for MPB: (a) Velocity profiles predicted using GC and MPB for electrolytes with different ion sizes. (b) An enlarged view of velocity profiles in the center of the microchannel. Table 2. Strength of Streaming Potential and Flow Rate Predicted by Different EDL Theories for 10-5 M LiCl and 10-5 M KCl Electrolytes 10-5 M LiCl Ex ′(V/m) without EDL GC MGC MPB without EDL GC MGC MPB
10-5 M LiCl Q (m3/s)
10-5 M KCl Ex′ (V/m)
10-5 M KCl Q (m3/s)
-0.3092 -0.3093 -0.4013
1 µm Microchannel 9.2593 × 10-11 8.8225 × 10-11 8.8224 × 10-11 8.5675 × 10-11 -0.3586
8.7475 × 10-11
-0.1109 -0.1027 -0.1205
50 nm Nanochannel 1.1574 × 10-14 1.0571 × 10-14 1.0660 × 10-14 1.0425 × 10-14 -0.0984
1.0774 × 10-14
has a smaller flow rate in a 14.1 µm microchannel than that of 10-4 M KCl solution.40 In other words, a LiCl solution appears more viscous than a KCl solution. Ren and co-workers40 concluded that such discrepancy is caused by two reasons: (i) ion size is different; and (ii) surface potential changes when the solid channel wall contacts different salt solutions. However, such a discrepancy due to the ion size cannot be interpreted by GC, since GC does not distinguish electrolytes by ion size. MPB is able to overcome the limit of the GC in this regard. Although
a full comparison between MPB simulation and experimental data40 cannot be done because the surface potential is actually an unknown parameter, MPB indeed predicts a correct effect of ion size on the electrokinetic flows in microchannels: the velocity of LiCl solution is smaller than that of KCl solution as shown in Figure 11 for a 1 µm microchannel. It is obvious that the ion size effect shown in Figure 11 should be more remarkable than that in the experimental situation,40 where a 14.1 µm microchannel was used and the ion concentration used was 10-4 M. In principle, the effect of ion size should play a more significant role in nanofluidics. Therefore, a 50 nm nanochannel is chosen and other parameters are kept the same. The velocity profiles for different EDL theories and different electrolytes are shown in Figure 12, and the corresponding results of flow rate and streaming potential are summarized in Table 2. As expected, the flow rate is further reduced due to the increasing electroviscous effect in nanochannels. For example, GC predicts an 8.7% reduction of the flow rate compared with the flow rate without EDL effects in this nanochannel. As for MPB, the flow rate is reduced by 6.6% and 9.9% for KCl and LiCl electrolytes, respectively. In nanochannels, the ion size effect on the streaming potential is more remarkable, the predicted streaming potential of the 10-5 M LiCl electrolyte is 19% higher than that of the 10-5 M KCl electrolyte (Table 2). Another interesting finding is that the streaming potential of the KCl electrolyte predicted by MPB is larger than that predicated by GC in the microchannel, while the streaming potential predicted by MPB is smaller than that predicated by GC in the nanochannel. This is mainly due to the overlap of EDL in nanochannels. In most of the microchannels, an overlapped EDL rarely exists. B. Image Effect on Electrokinetic Flows. Other than the surface charges, the dielectric property of the channel wall also influences the EDL potential distribution. For an electrolytesolid system with a high ionic concentration and a high surface potential, the image factor has a more significant impact on the EDL potential distribution. In this simulation, 2.0 M LiCl and a surface potential ψS ) -256 mV are chosen. Two image factors are used: f ) -1 (metal) and f ) 0.74444 (silicon). It is wellknown that the ionic concentration determines the effective range of EDL, which is however only several nanometers for the 2.0 M LiCl electrolyte. Therefore, along the entire cross section of the 1 µm microchannel and 50 nm nanochannel, one may not be able to notice any remarkable difference in velocity profiles calculated by MPB for the 2.0 M LiCl electrokinetic flow with
Electrokinetic Flows
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Table 3. Strength of Streaming Potential and Flow Rate Predicted by Different EDL Theories for a 2.0 M LiCl Electrolyte at Different Image Factors Ex′ (V/m) 1 µm Microchannel without EDL GC MPB f ) 0.7444 MPB f ) -1
-0.0054 -0.0052 -0.0043 50 nm Nanochannel
without EDL GC MPB f ) 0.7444 MPB f ) -1
-0.0012 -0.0011 -0.0005
Q (m3/s) 9.2595 × 10-11 9.2594 × 10-11 9.2594 × 10-11 9.2594 × 10-11 1.1580 × 10-14 1.1568 × 10-14 1.1569 × 10-14 1.1574 × 10-14
different imaging factors. However, as listed in Table 3, the streaming potential shows a strong dependence on the image factor. MPB predicts a 20.9% bigger streaming potential for f ) 0.74444 than that for f ) -1 in the microchannel, and the streaming potential for f ) 0.74444 is twice of that for f ) -1 in the nanochannel. Thus, we suggest that the image effect should be considered in nanofluidic systems, particularly for electrokinetic flows in nanochannels smaller than 100 nm.
V. Conclusions A comprehensive EDL theory is crucial for predicting electrokinetic transport phenomena in microfluidic and nanofluidic systems. We have investigated potential distribution in EDL and electrokinetic flows using MPB, and compared results predicted by MPB with those predicted by other EDL theories. Effects of ion size, ionic concentration, surface potential, and the image factor on potential distribution in EDL and electrokinetic flows have been systematically analyzed using MPB. Simulation results have shown that, in addition to traditional EDL effects, effects of ion size and the image factor also play a significant role in micro and nanosystems. This study based on MPB has advanced our understanding of electrokinetic flows in terms of velocity, flow rate, and streaming potential. This study shows great potential in design and development of next generation nanofluidic Lab-on-a-chip devices and other nanofluidic systems. Acknowledgment. The authors gratefully acknowledge financial support of this research from Natural Science and Engineering Research Council of Canada (NSERC), and fruitful discussion with Professor. C. W. Outhwaite. LA702059V
