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Coupled Concentration Polarization and Electroosmotic Circulation near Micro/Nanointerfaces: TaylorAris Model of Hydrodynamic Dispersion and Limits of Its Applicability Andriy Yaroshchuk,*,†,‡ Emiliy Zholkovskiy,§ Sergey Pogodin,|| and Vladimir Baulin†,|| †
Institucio Catalana de Recerca i Estudis Avanc-ats (ICREA), Barcelona, Spain Polytechnic University of Catalonia, Barcelona, Spain § Institute of Bio-Colloid Chemistry, National Academy of Sciences of Ukraine, Kiev, Ukraine University Rovira i Virgili, Tarragona, Spain
)
‡
ABSTRACT: Mismatches in electrokinetic properties between micro- and nanochannels give rise to superposition of electroosmotic and pressure-driven flows in the microchannels. Parabolic or similar flow profiles are known to cause the so-called hydrodynamic dispersion, which under certain conditions can be formally assimilated to an increase in the solute diffusivity (TaylorAris model). It is demonstrated theoretically that taking into account these phenomena modifies considerably the pattern of current-induced concentration polarization of micro/nanointerfaces as compared to the classical model of unstirred boundary layer. In particular, the hydrodynamic dispersion leads to disappearance of limiting current. At essentially “over-limiting” current densities, the time-dependent profiles of salt concentration in microchannels behave like sharp concentration “fronts” moving away from the interface until they reach the reservoir end of the microchannel. Under galvanostatic conditions postulated in this study, these “fronts” move with practically constant speed directly proportional to the current density. The sharp transition from a low-concentration to a high-concentration zone can be useful for the analyte preconcentration via stacking. The pattern of moving sharp concentration “fronts” has been predicted for the first time for relatively broad microchannels with negligible surface conductance. The TaylorAris approach to the description of hydrodynamic dispersion is quantitatively applicable only to the analysis of sufficiently “slow” processes (as compared to the characteristic time of diffusion relaxation in the transversal direction). A posteriori estimates reveal that the condition of “slow” processes is typically not satisfied close to current-polarized micro/ nanointerfaces. Accordingly, to make the description quantitative, one needs to go beyond the TaylorAris approximation, which will be attempted in future studies. It is argued that doing so would make even stronger the dampening impact of hydrodynamic dispersion on the current-induced concentration polarization of micro/nanointerfaces.
’ INTRODUCTION The phenomenon of current-induced concentration polarization close to electrodes and/or ion-exchange membranes is fundamental for and well known in electrochemistry.1,2 For example, it is a physicochemical basis of the technological process of electrodialysis widely used for water desalination.2 The physical cause of this phenomenon is the difference in the transport numbers of ions between two adjacent media, for example, an ion-exchange membrane and an electrolyte solution. Due to this difference, the electromigration flow of an ion is discontinuous at the interface. To ensure ion conservation, a gradient of concentration of this (and other) ion(s) arises in the system. The physical picture of this phenomenon will be considered in more detail below. At this point, let us just note that theoretically any difference in the ion transport numbers between the adjacent media can give rise to a current-induced concentration polarization. Recently, it has been demonstrated on several occasions that in charged nanochannels the concentrations of ions deviate from r 2011 American Chemical Society
their values in equilibrium solutions.36 The so-called counterions are pulled into the nanochannels, whereas the so-called coions are excluded from them to some extent. By using the definition of ion transport numbers, it is easy to demonstrate that these deviations automatically give rise to differences in the ion transport numbers between the nanochannels and the equilibrium electrolyte solutions, in particular, the neighboring parts of microchannels. Therefore, the phenomenon of concentration polarization can be expected to be very common close to nano/microinterfaces. Since even relatively small differences in the ion transport numbers can give rise to concentration polarization (at sufficiently large current densities), the overlap of diffuse parts of electric double layers inside the nanochannels is not a prerequisite of concentration polarization.
Received: April 13, 2011 Revised: August 3, 2011 Published: August 03, 2011 11710
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Langmuir This has already been pointed out in the literature (see, for example, refs 79). In micro/nanofluidic systems, the phenomenon of concentration polarization has already been observed experimentally in several distinct ways. Thus, for instance, considerable currentinduced enrichment/depletion in the concentration of charged fluorescent markers was directly observed close to micro/ nanointerfaces.7,8,11,13 Strong decreases in the local electric conductivity within such zones have been detected, too.14,15 Nonlinear and time-dependent currentvoltage characteristics have also been observed, which can be interpreted in terms of the concentration-polarization phenomenon.1620 This phenomenon has also been considered theoretically in a number of recent publications.9,12,13,2126 Typically, the theoretical models have been based on the coupled NernstPlanck and Poisson equations (the latter sometimes replaced by the condition of local electric neutrality,13,23,25). Attempts have also been made to take into account electroosmosis.9,12,13,2426 However, as discussed below, in most cases only the net electroosmotic flow was accounted for whereas the electroosmotic circulation (and the corresponding hydrodynamic dispersion, see below) was disregarded (with the possible exception of “fully numerical” models (see, for example, refs 21 and 26) where the circulation could be accounted for implicitly). Finally, in refs 13 and 25 the EO circulation has been included but the corresponding hydrodynamic dispersion was disregarded due to the use of the assumption of a local thermodynamic equilibrium in the microchannel. As compared to the “classical” configurations of the kind “stirred solution/unstirred layer/ion-exchange membrane”, the engineered micronanofluidic systems are characterized by the apparent lack of stirring as well as conditions for development of natural convection in microchannels. Besides, the length of microchannels (millimeters to centimeters) is usually much larger than the typical thicknesses of unstirred layers (tens to hundreds of micrometers) in membrane/electrode systems. In principle, this could lead to very pronounced limiting-current phenomena and very low limiting-current densities as discussed in ref 27. On the other hand, compared to the membrane or electrode configurations, at current densities typical for the electrically driven microfluidics, microchannels usually exhibit strong electroosmotic flows. In the previous theoretical studies, these flows have often been neglected on the grounds that the electroosmotic permeability of serial connections of nano- and microfluidic systems is controlled by their nanofluidic components (due to the much higher hydraulic resistance of the latter). Here, the electroosmotic permeability is known to be reduced due to the overlap of diffuse parts of electric double layers2830 and the local electric-field strength is also reduced owing to the surface conductance. This reasoning is basically correct but relates only to the electroosmotic flows averaged over the cross-section of microchannels. Actually, the situation is somewhat more complicated. In a microchannel “plugged” by a component with much higher hydraulic resistance and lower electroosmotic permeability (for example, a battery of parallel nanochannels), there is a gradient of hydrostatic pressure that drives the fluid in the direction opposite to the electroosmosis so that the net volume flow turns out equal to the reduced flow through the “nanoblock”. However, the velocity profiles of electroosmotic and pressure-driven flows are quite different. In sufficiently broad microchannels the electroosmotic profile is “plug-like” (i.e., the velocity is practically constant outside very thin diffuse parts of electric double layers in 1D flows) whereas the profile of pressure-driven flow is parabolic. As a result
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of superposition of these two flows, the fluid close to the microchannel walls flows in the direction of electroosmosis (and the local fluid velocity here corresponds to the normal rates of electroosmosis in microchannels) whereas close to the middle of the microchannel the flow occurs in the opposite direction. This phenomenon is referred to as electroosmotic circulation and is important, for example, in the context of optical measurements of microelectrophoresis.31,32 Close to the micro/nanointerface, this flow pattern takes the form of a system of two counter-rotating vortices “returning” the electroosmotic wall flow to the middle of the microchannel as a pressure-driven backflow. In the context of micro/nanofluidic systems, this flow pattern has been experimentally observed by Han and co-workers.1517 The rate of electroosmosis is proportional to the electric-field strength. At the micro/nanointerface where the buffer concentration goes down, the electric-field strength strongly increases.1417 This causes a proportional increase in the local rate of electroosmosis close to the channel walls. Due to the circulation flow pattern (arising because of continuity of volume flow) this also means an increase in the rate of the pressuredriven backflow. Proceeding from the physics of these phenomena, it can be expected that the rate of such flow will be a superlinear function of current density and may become ever larger with time (due to the build up of concentration polarization). Superlinear dependences of the fluid circulation rate close to nano/microinterfaces on the applied voltage have been observed experimentally by Han et al.1517 They (and the authors of ref 14) have also directly observed the decrease in the local electric conductivity. The observed nonlinear electroosmosis has been qualitatively interpreted1517,33 in terms of nonequilibrium space charge and the so-called electroosmosis of the second kind (or inducedcharge electroosmosis) predicted theoretically by Dukhin and Mishchuk34 and studied experimentally and theoretically by several authors.3538 Qualitatively similar phenomena have also been predicted by Rubinstein.39 Although the theoretical description of these phenomena is quite intricate, in general terms it can be noted that they occur due to deviations from local electric neutrality in electrolyte solutions. Below, we shall demonstrate that close to micro/nanointerfaces nonlinear electroosmosis can be predicted in the approximation of local electric neutrality. Nevertheless, a posteriori estimates will demonstrate that deviations from local electric neutrality are probable in sufficiently narrow and long microchannels close to the micro/nanointerface at advanced stages of concentration polarization. At the same time, in relatively broad and short microchannels the mechanism considered in this study (and based on the “normal” electroosmosis) makes the concentration polarization not sufficiently strong for the deviations from the local electric neutrality in the microchannel to occur. Finally, we shall see that for the systems of interest the convective salt transfer toward polarized interfaces can be even essentially stronger than that captured by the TaylorAris approximation. Accordingly, the scope of applicability of the approximation of local electric neutrality may be still broader than identified below within the scope of TaylorAris model. In chemical engineering, it is well known that parabolic fluid velocity profiles give rise to the so-called hydrodynamic dispersion, which under certain conditions can be formally assimilated to an increase in the solute diffusivity.4043 The hydrodynamic dispersion is one of the principal sources of band broadening in 11711
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liquid chromatography, and it can also noticeably contribute to the band broadening in capillary electrophoresis.43 A classical approach to the description of hydrodynamic dispersion has been developed in seminal works of Taylor and Aris.4042 This study is an attempt to use this approach for analysis of the dynamics of concentration-polarization phenomena close to nano/microinterfaces. It will be shown that taking into account the hydrodynamic dispersion can lead to dramatic changes in the system behavior, in particular, to the disappearance of limiting current phenomenon. The time-dependent profiles of buffer/salt concentration also become quite different from the classical ones and take the form of propagating concentration “fronts”. At the same time, an a posteriori analysis will reveal that for realistic parameter combinations the system behavior turns out beyond the scope of quantitative applicability of approximations of the TaylorAris model. This occurs because the dynamics of concentration polarization is too “rapid” (see below). From physical considerations, we shall conclude that a more rigorous analysis would give rise to an even stronger dampening impact of electroosmotic circulation on the concentration-polarization phenomena.
’ THEORY In this study we shall explicitly consider only processes on the “micro” side of the micro/nanointerface (that is, in the microchannel). The “nanoblock” is approximated by a hydraulically impermeable ion-exchange “plug” whose presence is accounted for via boundary conditions. The geometry and system of coordinates are shown in Figure 1. As the starting point, for single salts (binary electrolytes) we shall use these well-known expressions for ion fluxes F ! ! Z( c( ∇ j þ uBc( ð1Þ B J ( ¼ D( ∇ c ( þ RT where D+,D are the ion diffusion coefficients, c( are the ion concentrations, and B u is the velocity of liquid flow. Electric current density is defined in this way J þ þ Z B J Þ B I FðZþ B
ð2Þ
It is beneficial to use the electric current density as a variable because electrical capacities charge very rapidly (in a matter of microseconds or less at typical buffer concentrations). Therefore, at much longer characteristic times typical for the processes considered in this study the divergence of current density can be considered zero. Due to this fact, in the 1D model analyzed below, the electric current density can be considered constant over any microchannel cross-section even under nonstationary conditions (at times longer than microseconds). By introducing the current density, one can eliminate the gradient of electric potential from eq 1 ! B I RT tþ t 1 ! ! þ ∇c ð3Þ þ ∇j ¼ g F 3 Zþ Z 3 c 3 where g
F2 ðZþ νþ Þ 3 c 3 ðZþ Dþ Z D Þ RT
ð4Þ
is the electric conductivity of solution inside the microchannel, c c( =ν(
ð5Þ
Figure 1. Model and system of coordinates.
is the salt concentration, ν( are the stoichiometric coefficients of ions, and tþ
Zþ D þ Zþ Dþ Z D
ð6Þ
is the transport number of cations in the microchannel (assumed to be equal to that in the bulk electrolyte solution). The transport number of anions is t 1 tþ
ð7Þ
While expressing the ion concentrations via the salt concentration in eq 5, we used the assumption of local electric neutrality in the microchannel. The scope of applicability of this approximation will be discussed at the end of this paper. Now, we can substitute eq 3 to eq 1 to obtain B J( B I t( ! ¼ Ds ∇ c þ uBc þ ν( FZ( ν(
ð8Þ
where Ds
ðZþ Z ÞDþ D Z þ Dþ Z D
ð9Þ
is the molecular diffusion coefficient of the salt. Now let us use the conservation equations for individual ions ∂c( ! J( ¼ 0 þ ∇3B ∂t
ð10Þ
By combining eqs 310 we arrive at this equation for the description of time-dependent spatial distributions of salt concentration within the microchannel ∂c ! ! þ ∇ 3 ð Ds ∇ c þ uBcÞ ¼ 0 ∂t
ð11Þ
A term proportional to the current density does not appear in eq 11 because the divergence of electric-current density is equal to zero in the relatively slow processes considered in this study. Assuming zero normal flux of each of the ions at the microchannel walls and making use of eq 5, one can obtain the following boundary condition for eq 11 ! n 3 ∇ c ¼ 0 at the walls ð12Þ Further, we shall consider processes occurring at some distances (several microchannel heights) from the “plug” surface. It is assumed that all the flows and gradients here can be considered 1D and occurring only along the microchannel axis. However, due to the Poiseuille backflow, the local velocity of volume flow varies in the transversal direction (perpendicular to the microchannel walls). This makes the local ion fluxes and salt concentration dependent on the transversal coordinate(s). Nevertheless, due to the 1D character of flows, a considerable problem simplification can be attempted via averaging eq 11 over the microchannel cross-section. 11712
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Notably, the problem of eq 11 subject to the boundary condition of eq 12 is identical to the problem of transport of a nonelectrolyte solute advected by a hydrodynamic flow through a channel. A remarkable approximate approach to solving such a problem was proposed by Taylor and Aris4042 to address the flows satisfying the following conditions: (i) the microchannel length, L, is much larger than a lengthscale parameter, h, characterizing the cross-section geometry, i.e., h/L , 1 (ii) the process rate is characterized by a time-scale parameter, τ0, which is much larger than the characteristic diffusion time, h2/Ds, attributed to the cross-sectional length scale parameter, h, i.e., τ0 . h2/Ds. Below, we shall see that the second condition is typically not met in the systems of interest. Nevertheless, to be able to come to this conclusion (and outline the possible directions of improvement of the theoretical approach) we need to carry out this analysis by using the TaylorAris model as a first approximation. When the above conditions are satisfied, the solution of convective-diffusion problem of eqs 8 and 9 can be reduced to the solution of a purely diffusion equation (for the cross-sectionaveraged salt concentration) with an apparent diffusion , whose value is controlled by the (dispersion) coefficient, D(eff) s distribution of flow velocity within the cross-section. For perfectly plug-like velocity profiles, the dispersion and molecular = Ds. For pressure-driven diffusion coefficients coincide, D(eff) s flows, the increase in the dispersion coefficient is proportional to Ds ≈ u2P. the squared cross-section-averaged velocity, D(eff) s As discussed above, we assume the cross-section-averaged volume flow to be completely “stopped” by the hydraulically impermeable ion-exchange “plug”. Thus, the local volume flow is the superposition of oppositely directed (and equal on average) electroosmotic (plus capillary-osmotic, see below) and pressuredriven flows. In sufficiently broad microchannels, the electroosmotic and capillary-osmotic flows are perfectly plug-like, that is, their velocity is constant throughout any cross-section of microchannel. Such flows do not give rise to hydrodynamic dispersion. The pressure-driven counterflow can be considered perfectly parabolic. Accordingly, one can use the well-known results to obtain the coefficients of hydrodynamic dispersion for channels of various cross sections.4042 The corresponding expression for the effective diffusion coefficient has the form Dsðef f Þ ¼ Ds ð1 þ kT Pe2 Þ
ð13Þ
where the dimensionless Peclet number is defined with the transversal length-scale parameter, h, the average velocity of pressure-driven flow component, up, and the molecular salt diffusion coefficient up h Pe ̅ Ds
ð14Þ
kT is the dimensionless Taylor constant. For example, in slit-like microchannels kT = 2/105 and in cylindrical ones kT = 1/48 (the length-scale parameter, h, is equal to the cannel half-height and channel radius, respectively). The Taylor constants for a number of microchannel cross-sectional shapes have been calculated in ref 43. In particular, it was shown that for rectangular shapes this constant can be noticeably larger than for the idealized shapes mentioned above. Thus, for instance, in a 2/5 rectangular channel kT ≈ 0.16.
The averaged velocity of pressure-driven counterflow can be found proceeding from complete compensation of the plug-like flow. The latter is the superposition of electroosmosis and capillary osmosis (the latter arising due to gradients of salt concentration). The velocity profiles of capillary-osmosis flows in broad microchannels are also plug-like.44 Therefore, they can be considered in the same way as electroosmotic ones. For the rate of electroosmosis, we will use the classical expression νeo ðcðx, tÞÞ ¼ ̅
RTεε0 ζ~ðcðx, tÞÞ I ̅ 3 ηF gðcðx, tÞÞ ̅
ð15Þ
c(x,t) is the salt concentration averaged over the microchannel cross-section and I is the cross-section-averaged current density. In the case of sufficiently broad microchannels (where the surface conductance can be neglected), the cross-section-averaged current density coincides with the local one. In the case of 1:1 salts, the rate of capillary osmosis is given by the relationship44 νco ðcðx, tÞÞ ¼ ̅
ðRTÞ2 εε0 ζ~ðcðx, tÞÞ ∂c̅ ̅ θ jζ ~ ð cðx, tÞÞj 4 ln cosh 3 ̅ 3 ∂x 4 ηF2 cðx, tÞ ̅
ð16Þ In eqs 15 and 16, ζ~(c(x,t)) is the dimensionless zeta potential of the microchannel surface defined as ζ~ðcðx, tÞÞ ̅
Fζðcðx, tÞÞ ̅ RT
ð17Þ
Finally u̅ P ¼ ðνeo þ νco Þ ð18Þ Due to the liquid incompressibility the condition of zero averaged volume flow along the microchannel is applicable at any microchannel cross-section at any moment of time. Therefore, the averaged velocity of pressure-driven flow (and the strength of hydrodynamic dispersion with it) is a function of the axial coordinate due to the variation of electric conductivity along the microchannel caused by the concentration polarization. The hydrodynamic dispersion, in turn, essentially modifies the pattern of concentration polarization (via “renormalization” of the salt diffusion coefficient). Therefore, here we deal with two simultaneous strongly coupled and strongly nonlinear phenomena. As discussed above, the condition of ion conservation (remember that the cross-section averaged volume flow is assumed to be zero) leads to ∂c̅ ∂ ∂c̅ fÞ ¼ Dðef ð cðx, tÞÞ ð19Þ ̅ 3 ∂x ∂t ∂x s The boundary conditions follow from 1 given salt concentration at the end of microchannel in contact with the reservoir (which is assumed to be perfectly stirred) ð20Þ
cð0, tÞ ¼ c0 ̅
2 continuity of cation flux at the “plug” surface (the condition of continuity of anion flux is satisfied automatically) ðmÞ I 3 ðtþ tþ Þ ∂c̅ ðeff Þ tÞÞ ¼ ð21Þ -Ds ðcðL, ̅ 3 ∂x FZþ νþ x¼L
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where t(m) is the cation transport number in the “plug” + membrane. The initial condition is cðx, 0Þ ¼ c0 ̅
ð22Þ
In this study, it is assumed that at t = 0 the current density is increased stepwise from zero to a constant value. Below, we will introduce the classical limiting-current density. For the system of interest, it will be shown to be equal to Ilim ¼
FZþ νþ Ds c0 ðmÞ
L 3 ðtþ tþ Þ
2Fc0 Dþ L
ð24Þ
It is convenient to scale the current density on the limiting current density ~I I=Ilim
I3L 2Fc0 Dþ
ð25Þ
The axial coordinate, x, is scaled on the microchannel length, L ξ x=L
ð26Þ
Time is scaled according to this τ t=tch tch
L2 Ds
ð27Þ ð28Þ
The salt concentration is scaled on the initial salt concentration (which coincides with the time- and space-independent concentration in the reservoir) sðξ, tÞ cðξ, τÞ=c0 ̅
ð29Þ
In dimensionless variables, eq 19, the boundary and initial conditions can be represented in this way ∂s ∂s ∂ ¼ 1 þ kT Pe2 ðsðξ, τÞÞ 3 ∂τ ∂ξ ∂ξ h ζ~ðsðξ, tÞÞ Peðsðξ, τÞÞ m 3 3 L sðξ, tÞ 1 3 2 0 ζ~ðsðξ, tÞÞ C 4 3 ln cosh 6 B 1 C ∂sðξ, tÞ7 7 6 ~ B 4 7 t I þ B tþ C 3 36 4þ3 @ 2 A ∂ξ 5 jζ~ðsðξ, tÞÞj
ð30Þ
ð31Þ
where m 3as =rB
Ds
ð32Þ
kT 6πηas
ð33Þ
The boundary conditions are sð0, τÞ ¼ 1 ∂s ∂ξ
ð23Þ
where L is the length of the microchannel (from the reservoir to the “plug”). It can be demonstrated that the assumption of zero diffusion permeability of the “plug” (and thus of zero diffusion (m) flux of cations through it) can only be valid when t(m) + = 1 or t+ = 0. = 1 (an ideally permFor definiteness, let us assume that t(m) + selective cation-exchange “plug”). Besides, in this study, we will concentrate our attention on the analysis of simple case of 1:1 electrolytes. In this particular case Ilim ¼
is the so-called Levine parameter, rB is the Bjerrum length, and as is the “hydrodynamic radius” of the salt “molecule” formally defined by using the StokesEinstein relationship
¼ ξ¼1
ð34Þ ~I 1 þ kT Pe2 ðsð1, τÞÞ
ð35Þ
where we have taken into account that it is now assumed that t(m) + = 1. The initial condition is sðξ, 0Þ ¼ 1
ð36Þ
Extensive numerical analysis has revealed that taking into account capillary osmosis does not change the qualitative pattern of the phenomena of interest. Moreover, even the quantitative contribution of capillary osmosis is usually moderate. At the same time, by neglecting the capillary osmosis (the second term in the brackets in the right-hand side of eq 31) and by assuming that the zeta potential of the microchannel walls is a constant independent of salt concentration, eq 30 can be easily solved analytically in the steady state (fully developed concentration polarization). For the stationary salt concentration at the nano/microinterface, we obtain qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 sis sð1, ∞Þ ¼ ½ ð~I 3 ð1 þ A~I Þ 1Þ2 þ 4A~I 2 2 ð~I 3 ð1 þ A~I Þ 1Þ ð37Þ where we denoted h 2 A kT m 3 ζ~3 tþ 3 L
ð38Þ
The stationary salt-concentration profile along the microchannel is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ss ðξÞ sðξ, ∞Þ ¼ ½ ð~I 3 ðξ þ A~I Þ 1Þ2 þ 4A~I 2 2 ~ ðI 3 ðξ þ A~I Þ 1Þ ð39Þ Equation 37 shows that at A 6¼ 0 the interface concentration always remains positive whatever the dimensionless current density. This is qualitatively different from the “classical” mode (no hydrodynamic dispersion) and means that there is no limiting current.
’ RESULTS AND DISCUSSION Classical Picture of Current-Induced Concentration Polarization. For the description of the diffusion boundary layer near a
membrane or electrode surface, one often uses the so-called Nernst model. It is assumed that the gradual transition from the boundary layer where the ion transport is controlled predominantly by electrodiffusion to the stirred reservoir where this transport is dominated by convection can be replaced by a sharp boundary dividing the system in an absolutely stagnant layer (no stirring at all) and a perfectly stirred reservoir where the ion concentrations are considered homogeneous. The unstirred (or Nernst) layer is characterized by the thickness and ion diffusion coefficients and charges. The perfectly stirred reservoir is characterized 11714
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Figure 2. Dynamics of classical concentration polarization: concentration profiles (dimensionless times and dimensionless current density are indicated on the plots).
by its volume and the ion concentrations. The ion transport through the membrane is usually assumed to be due to electromigration alone (no diffusion), and the volume transfer through the membrane is also neglected. Therefore, within the unstirred layer, one can use eq 8 with B u = 0. In the steady state, the cation fluxes through the stagnant layer and the membrane must be equal, which yields ðmÞ
Ds
dc Iðt tþ Þ ¼ þ dx FZþ νþ
ð40Þ
where t(m) + is the cation transport number in the membrane. Since the right-hand side of eq 40 is just a constant, the stationary saltconcentration profile is linear and integration of eq 40 yields ðmÞ
I δ ðt tþ Þ c0 cm ¼ 3 3 þ Fzþ νþ Ds
ð41Þ
where δ is the thickness of the unstirred layer and cm is the salt concentration at the membrane surface. For negatively charged t+ > 0, so for positive (cation-exchange) membranes, t(m) + currents (cations moving toward the membrane) c0 cm > 0, that is, the salt concentration at the membrane surface goes down. On the opposite membrane surface it goes up. Equation 41 shows that the surface concentration decreases linearly with the current density. However, concentrations cannot be negative. Therefore, there is a maximum stationary current density that can occur in the system and which corresponds to cm = 0. This current density is referred to as limiting and is equal to Ilim ¼
FZþ νþ Ds c0 ðmÞ
δ 3 ðtþ tþ Þ
ð42Þ
Within the scope of this model, stationary current densities exceeding the limiting one cannot occur because the salt concentration going down to zero at the membrane surface makes the electrical resistance of the system increase to infinity. The voltage related to the so-called membrane potential strongly increases, too. However, this picture is based on the assumption that the thickness of the unstirred layer is independent of the current density. Due to a number of reasons (for example, electroconvection39), this assumption may not be valid, especially at larger current densities.
This has been put forward as one of the explanations of the existence of experimentally observed overlimiting currents in membrane systems.45,46 Under nonstationary conditions, overlimiting currents can occur until the moment in time when the salt concentration at the membrane surface drops to zero. Consequently, the voltage drop on the system tends to infinity, and the galvanostatic mode cannot be sustained anymore. This is illustrated by the time evolution of salt-concentration profiles shown in Figure 2. As discussed below, the hydrodynamic dispersion makes the behavior qualitatively different. Numerical Analysis. Figure 3 shows examples of the time evolution of salt-concentration profiles within the microchannel with the due account of TaylorAris dispersion obtained via numerical integration of eq 30 in Matlab environment. In these calculations we neglected the contribution of capillary osmosis. Besides, we assumed that the zeta potential of the microchannel surface has a constant value independent of salt concentration. Figure 3a shows that at a moderately overlimiting current density the salt-concentration profiles initially resemble “classical” ones (occurring at no hydrodynamic dispersion). This is not surprising because at the relatively weak dispersion assumed in this calculation and at moderate current densities the effective diffusion coefficient is practically not influenced by the hydrodynamic dispersion until the salt concentration drops to a quite low level. Therefore, initially the profiles have a conventional shape. However, at a certain moment in time there is a dramatic change in the behavior. From the nano/microinterface, a zone starts to propagate where the salt concentration is very low and, accordingly, the hydrodynamic dispersion becomes sufficiently strong despite the moderate current density (due to the local increase in the electric-field strength caused by the dramatic decrease in the conductivity). This zone becomes ever more extended in the way that the crossover point moves with an approximately constant velocity away from the interface for a certain period of intermediate times. Outside this zone, the concentration profile takes an ever more linear shape. Finally, a stationary state is reached where the microchannel is clearly divided into two zones. Adjacent to the nano/microinterface, the salt concentration is very low. Because of this, the concentration gradients here are also very small but the salt flux is as large as outside this zone due to the increase in the effective salt diffusivity caused by the hydrodynamic dispersion. Outside this “depleted” zone, the concentration profile is practically linear. By using eq 39, at A 3 ~I 2 , 1, the crossover can be shown to occur at ξcr ≈ 1=~I
ð43Þ
Figure 3b shows the case of a rather strongly overlimiting current. Here, too, at very short times after the current switch on there are conventional concave-up concentration profiles (not shown). However, very rapidly, the behavior changes, and the profiles take the shape of concentration “fronts” moving away from the interface. They move with a constant speed (and preserving the shape) until they reach the reservoir. At this point in time, the pattern changes and there is a very rapid crossover to the stationary state. The dimensionless propagation speed is approximately equal to the dimensionless current density in this mode. Interestingly, the features of the stationary concentration profile still correspond to the behavior described above in relation with Figure 3a, and the point of crossover is given by eq 43. 11715
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Figure 3. Time evolution of concentration profiles (numerical). First profile from the right corresponds to τ = 0.1/~I ; for each subsequent profile the time is incremented by 0.1/~I from right to left; the last profile from the left corresponds to the steady state. Dimensionless current densities are indicated on the plots.
Figure 3c shows the case of a very strongly overlimiting current. The basic features are similar to Figure 3b, but the moving zone of transition from the depleted region to the highconcentration one is still narrower than at ~I = 10. This figure also features the pattern of steady propagation of a concentration “front” with the speed approximately equal to the dimensionless current density. However, the stationary concentration profile cannot be that clearly divided into two distinct zones anymore, and eq 43 does not apply. This is because the condition A 3 ~I 2 , 1 is not satisfied in this case. Finally, from Figure 3d it is seen that at extremely large dimensionless current densities the transition zone becomes broader. This is because A 3 ~I 2 . 1 in this case. In Figure 3 within the depleted zone details of the spatial distribution of the salt concentration are not visible. Therefore, to improve visibility, we plotted the data as the reciprocal dimensionless concentration. In addition to the improved visibility, this quantity controls the local electric-field strength, and the time evolution of its distribution is directly related to the time evolution of voltage drop on the microchannel. Figure 4 shows the data from Figure 3b presented in these coordinates. The pattern is amazingly simple, the reciprocal concentration being a linear function of coordinate within the depleted zone. Outside this zone, the reciprocal concentration is much smaller. Accordingly, the high-concentration zone makes practically no contribution to the microchannel electrical resistance and the voltage drop. Further calculations have revealed that the simple linear pattern is quite persistent and
Figure 4. Concentration profiles plotted in terms of reciprocal concentration (numerical); the profile sequence is the same as in Figure 3b. Dimensionless current density is indicated on the plot.
occurs within a broad range of dimensionless current densities. The moving transition region between the two zones is the narrowest at intermediate dimensionless current densities. A quantitative criterion of the narrowness of this region will be formulated below by means of approximate scaling-form solution. 11716
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Approximate Scaling-Form solution. A scaling-form mode occurs when there is a certain combination of space and time variables such that the solution depends only on this combination but not on the space and time variables separately. For the concentration “fronts” traveling backward with a constant speed, such a variable should have the form
y ¼ ξ þ ντ
ð44Þ
where v is the propagation speed. By substituting eq 44 into eq 30 and by taking into account that the scaling-form solution must depend only on this new variable we obtain ! ! ~I 2 ds ds d 1 þ A2 ð45Þ ν ¼ dy dy s dy By integrating this equation from 0 to y, we get ! ! ~I 2 ds ~I 2 ds 1 þ A2 ν 3 ðs 1Þ ¼ 1 þ A 2 s dy s dy
ð46Þ y¼0
As is seen from Figure 3b and 3c, while the concentration “front” is still sufficiently far away from the reservoir (and this is one of the prerequisites of applicability of the scaling-form solution), the space derivative of concentration close to the reservoir is practically equal to zero. Therefore, the second term in the right-hand side of eq 46 can be neglected to obtain ! ~I 2 ds ð47Þ ν 3 ð1 sÞ ¼ 1 þ A 2 s dy By integrating this equation from y to 1 we get 1 si ν 3 ð1 yÞ ¼ ð1 þ A~I 2 Þ 3 ln 1 sðyÞ 1 1 sðyÞ 2 ~ þ ln þ AI 3 si sðyÞ si
ð48Þ
The propagation speed can be found by using the boundary condition at the nano/microinterface (eq 35) and eq 47 to obtain ν¼
~I 1 si
Figure 5. Applicability of scaling-form solution to the analysis of concentration profiles; values of dimensionless time (in 1/~I units) are indicated on the plots. Dimensionless current density is indicated on the plot.
approximately equal to 1/~I . Moreover, the crossover from the scaling-form solution to the stationary one is very rapid. Thus, the scaling-form solution can be quite useful for analysis of the shape of concentration “fronts”. This solution is not applicable during an initial “induction” period of time. It also fails when the concentration “front” reaches the reservoir end of microcapillary. However, when the transition between the zones of low and initial salt concentration is sufficiently narrow, the scaling-form solution is applicable for a major part of the transient period. A quantitative measure of the sharpness of the transition zone is the space derivative of dimensionless salt concentration in the middle of it, for example, at s = 1/2. For the scaling-form solution to be useful, this derivative must be much larger than one. Equations 47 and 49 show that the derivative at s = 1/2 is equal to (remember that si is usually quite small) ~I ~I ds ≈ ¼ 2 dy 2ð1 þ 4A~I Þð1 si Þ 2ð1 þ 4A~I 2 Þ s ¼ 1=2
ð49Þ
Since the interface concentration is usually very small, eq 49 confirms our previous observation that the salt-concentration “fronts” are moving with a practically constant dimensionless speed equal to the dimensionless current density. By combining eqs 48 and 49 we get 2
0
0
ð51Þ The slope first increases, passes through a maximum, and then decreases with the dimensionless current density. The maximum occurs at 1 ~I ¼ p ffiffiffi 2 A
113
1 61 B1 B si 1 CC7 1 si 1 6 B B CC7 ~ þ ln@ 1 þ AI 3 @ y ¼ 1 ð1 si Þ4 3 ln AA5 ~I si sðyÞ 1 sðyÞ 1 sðyÞ
ð50Þ This solution can be used to model analytically the salt-concentration profiles as a function of interface salt concentration. Figure 5 shows the results of a comparison of concentration profiles obtained numerically and by means of eq 50 as well as the stationary solution of eq 39. The agreement is excellent (in fact, the numerical and approximate plots are hardly discernible) starting from 1/20 and until 9/10 of characteristic relaxation time
ð52Þ
At current densities around the value given by eq 52, the pattern is the most “scaling form”. Equation 51, also shows that the scaling-form behavior becomes ever more pronounced at smaller values of parameter A (as long as the dimensionless current densities are not too far away from the value given by eq 52). At the same time, this mode cannot occur at small to moderate dimensionless current densities, which is confirmed by the data shown in Figure 3a). Thus, the scaling-form mode is most pronounced at small A and large dimensionless current densities, ~I . The feature of a very narrow transition from the low-concentration (high-electric-field) to the high-concentration (low-electric-field) zone can 11717
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density can be obtained by combining the condition of current continuity and the Gauss law.51 In this way, for our 1D case, we obtain F ¼ εε0 3
I ∂g g 2 3 ∂x
ð53Þ
where we have taken into account that the local electric-field strength is equal to (I/g). By transforming to the dimensionless variables, we obtain 4tþ~I 1 ∂s 4tþ~I 1 ∂ 1 f ¼ ð54Þ ðk0 LÞ2 3 s3 3 ∂ξ ðk0 LÞ2 3 s 3 ∂ξ s where k0 is the reciprocal Debye screening length at the initial salt concentration and we denoted F ð55Þ f ¼ Fc0 s The criterion of applicability of approximation of local electric neutrality is f ,1
ð56Þ
The identical representation in the right-hand side of eq 54 shows that the probability of deviations from local electric neutrality gradually increases with time (until the steady state) and on the approach of the interface. Indeed, eq 54 shows that f is proportional to the product of reciprocal concentration and its spatial derivative. From Figure 4 it is seen that within the depleted zone (and just here the deviations of interest are most probable to occur) the derivative of reciprocal concentration is practically constant while the reciprocal concentration itself increases linearly both with time and on approach of the interface. Thus, deviations from local electric neutrality are most probable in the steady state and close to the nano/microinterface. Therefore, for overestimates of f we can use the analytical results of eqs 3739 obtained in the steady state. By using eq 35 besides that, we obtain f ¼
~I 2 4tþ 1 23 3 2 ðk0 LÞ sis sis þ A~I 2
ð57Þ
where sis is the stationary interface salt concentration given by eq 37. It can be shown that this concentration has a minimum at ~I ≈ 2 and sis ≈ 4A at the point of this minimum. It can also be shown that f is largest under the same conditions (at sufficiently small A). Therefore, the following overestimate can be obtained for f (at ~I = 2) f j
tþ ðk0 LÞ2 3 A2
ð58Þ
Accordingly, the criterion of applicability of approximation of local electric neutrality close to the interface (and, thus, all the more elsewhere in the microchannel) is ðk0 LÞ 3 A > 1
ð59Þ
where we replaced the “much larger” sign by the just “larger” one because of the square in eq 58. A typical value of (k0L) can be around 106 (for a 1 mM solution and a 1 cm long microchannel). Accordingly, the approximation of local electric neutrality is applicable when A > 106. By using the definition of parameter A of eq 38, the criterion of eq 59 can be rewritten 11718
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in this form
h >1 kT 3 ðmtþ 3 ζ~Þ2 3 ðk0 hÞ 3 L
ð60Þ
λ h=H
Thus, deviations from local electric neutrality can primarily be expected in narrow and long microchannels with weak hydrodynamic dispersion (for example, due to small zeta potentials and/or Taylor constants). It has to be stressed that the criterion of eq 60 has been obtained for the value of dimensionless current density (~I = 2) where the deviations from local electric neutrality are most probable. At smaller and larger current densities f is smaller and the approximation of local electric neutrality is more applicable. Besides, as shown in the next section, the TaylorAris approximation to the description of hydrodynamic dispersion is not quantitatively applicable to our system. Taking into account the stronger convective salt “delivery” to the polarized interface (see below) would make the electric fields there weaker and the deviations from local electric neutrality less probable. Applicability of Approximation of “Slow” Processes. Above we stated that the TaylorAris approach to the description of hydrodynamic dispersion is quantitatively applicable provided that the relevant processes are sufficiently “slow” in the sense that the diffusion relaxation in the transversal direction has enough time to occur. In this section, we shall use the results obtained above to verify a posteriori if this condition is satisfied. To do so, we shall briefly recall the principal steps leading to the Taylor expression for the effective diffusion coefficient of eq 13. The starting point is the convection-diffusion equation describing the nonstationary distribution of salt concentration within the microchannel ∂c ∂2 c ∂c ! ¼D þ ∇ 2 cÞ u 2 ∂t ∂x ∂x
where H is the length-scale parameter characterizing the longitudinal solute distribution, and the small parameter, λ, is given by
ð61Þ
where rB t Bi y(∂/∂y) + Bi z(∂/∂z) and subject to the boundary condition of zero normal salt flux at the channel walls ! n 3 ∇ c ¼ 0 at the walls ð62Þ B
Strictly speaking, H is not a fixed parameter dictated by the system geometry, but it can change during the concentrationprofile evolution. Moreover, in the case of complex concentration distributions (like those considered in this study), this parameter can have different values in various parts of the microchannel. Remarkably, under conditions of validity of the Taylor approach, the result obtained by employing the normalization of eqs 6567 turns out independent of H. However, the limits of applicability of the Taylor approach can depend on this parameter. Below, we discuss this possibility in more detail. By using eq 64, it can be shown that the concentration deviation, δc, can be represented as hu̅ ∂c̅ hu̅ ð69Þ δc ¼ λ 3 3 3 ωð~y, ~zÞ þ O λ 3 D ∂~x D where ω(~y,~z) is the solution of the following 2D boundary-value problem ~ 2 ω ¼ δu~ ∇
ð63Þ
where δc t c c and δu t u u are the local deviations from the cross-section-averaged values, c and u, of concentration and velocity, respectively. By combining eqs 61 and 63 we obtain the following equation λ2
∂δc ∂2 δc 2 ∂τ ∂ ~x
!
~ 2 δc ¼ 0 ∇
þ λ
~ 2 h2 ∇2 ∇
ð65Þ
~ δu=u δu ̅
ð66Þ
~x x=H
h2 u̅ ,1 HD
ð72Þ
ð73Þ
When this condition is satisfied the Taylor result is valid and insensitive to parameter H. In order to conduct a posteriori verification of whether a given concentration profile, c (x,t), satisfies the condition of eq 73, we will estimate the local time-dependent value of parameter H as Hðx, tÞ≈
cðx, tÞ ̅ j∂cðx, tÞ=∂xj ̅
ð74Þ
By using eq 74, we can rewrite the condition of eq 73 as the ratio of two time-scale parameters
ð64Þ
where the dimensionless 2D Laplace operator is scaled on the cross-sectional length scale parameter, h
ð71Þ
The Taylor result is obtained via substituting δc given by eq 69 into eq 63 while neglecting the last term in the brackets. Thus, the TaylorAris approach is valid when the terms of the order of O[(λ 3 (hu/D))2] can be neglected for being small. Consequently, by using eq 68, the following inequality can be obtained as a criterion of applicability of the Taylor approach
∂δc ∂ ~ ~ ∂c̅ þ δu ~ ∂δc hu̅ δuδc þ δu ∂~x ∂~x ∂~x ∂~x D
ð70Þ
! ~ ¼ 0 at the walls n 3∇ B ZZ ωð~y, ~zÞd~yd~z ¼ 0 over the channel cross-section
Averaging both sides of eq 61 over the channel cross-section and using eq 62 along with the 2D version of Gauss theorem yield ∂c̅ ∂2 c ∂c ∂ ¼ D 2̅ u̅ ̅ δuδc ∂t ∂x ∂x ∂x
ð68Þ
tD ,1 tu
ð75Þ
where tD h2 =D
ð76Þ
is the characteristic time of transverse diffusion and tu ðx, tÞ
ð67Þ 11719
cðx, tÞ ̅ uðx, tÞ 3 ∂cðx, tÞ=∂x ̅ ̅
ð77Þ
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is the characteristic time of longitudinal convective redistribution of solute. Thus, indeed, the Taylor approximation is applicable provided that the process of this convective solute redistribution is sufficiently slow everywhere in the microchannel as compared to the characteristic time of diffusion relaxation in the transversal direction. By using the definitions of eqs 4, 9, 15, 24, 25, 29, 32, and 33, the condition of eq 73 can be rewritten in this dimensionless form 2 ∂ 1 h ~ ~ ð78Þ m 3 tþ 3 ζ 3 I 3 ,1 ∂ξ s L As is seen from Figure 4, in the “propagation” mode within the depleted zone the reciprocal salt concentration is a linear function of coordinate. From the “scaling-form” solution (eq 50) it follows that within this zone ∂ð1=sÞ 1 = ∂ξ AI
ð79Þ
By using the definition of parameter A (eq 38) it is easy to show that the condition of eq 78 is satisfied provided that ζ.1 kT 3 m 3 tþ 3 ~
ð80Þ
Interestingly, this criterion does not depend, for example, on the microchannel dimensions and, thus, has a quite fundamental nature. In particular, it does not depend on the microchannel height despite the fact that the rate of diffusion relaxation in the transversal direction strongly increases with decreasing microchannel height. This occurs because in very narrow microchannels the hydrodynamic dispersion becomes noticeable only at extremely high current densities. The estimates leading to eq 80 have been performed for the mode of pronounced hydrodynamic dispersion. Due to the very large current densities needed to reach this mode in very narrow microchannels, the processes turn out to be not sufficiently “slow” in this case, too. Unfortunately, in view of the quite small typical values of the Taylor constant, kT, this condition is practically impossible to meet at realistic values of parameters. Therefore, the condition of “slow” processes is not satisfied in our system. Accordingly, the TaylorAris model of hydrodynamic dispersion cannot be used for a quantitative description of coupling between the electroosmotic-circulation and concentration-polarization phenomena at micro/nanointerfaces. To make such a description possible, a considerable modification of the TaylorAris approach has to be undertaken, which will be attempted in future studies. The physical cause of hydrodynamic dispersion is the additional salt transport along the microchannel due to convection. We demonstrated that concentration polarization becomes weaker because of this additional salt “delivery” to the polarized interface. The transversal salt diffusion makes this convective salt “delivery” weaker since part of the salt molecules diffuses in the transversal direction instead of advecting in the axial direction toward the polarized interface. Under conditions of incomplete salt redistribution in the transversal direction, the salt concentration is higher close to channel walls than in its center (assuming that electroosmosis is directed toward the ion-exchange “plug”). Due to a parallel connection of those two regions in electrical terms, the higher salt concentration close to the channel walls will bring about smaller (on average)
electric fields and weaker concentration polarization. Therefore, going beyond the approximation of “slow” processes may make the dampening impact of hydrodynamic dispersion on the concentration polarization even stronger than revealed by this analysis.
’ CONCLUSIONS Electroosmotic circulation (arising due to the mismatch of electrokinetic permeabilities of micro- and nanochannels) gives rise to hydrodynamic dispersion close to micro/nanointerfaces. At the same time, these interfaces are concentration polarized by electric current because of the mismatch of electrochemical properties (ion transport numbers) of adjacent media. These two phenomena have been shown theoretically to be strongly coupled. Theoretical analysis was carried out within the scope of the TaylorAris model of hydrodynamic dispersion. By combining this model with the classical approximation of local electric neutrality in the microchannel, we reduced the mathematical problem to a single 1D equation of nonstationary diffusion with an effective diffusion coefficient being a function of cross-section averaged salt concentration and its spatial derivative. Analytical and numerical analyses of this equation have revealed several unusual features of current-induced concentration-polarization phenomena at micro/nanointerfaces as compared to the classical situations of polarization of macroscopic ion-exchange membranes or electrodes. The phenomenon of limiting current disappears. Besides, at sufficiently overlimiting current densities, the time-dependent salt-concentration profiles take the unusual shape of “concentration fronts” moving with practically constant speed away from the nano/ microinterface. These “fronts” clearly divide the microchannel into “depleted” and high-concentration zones. This pattern of moving “concentration fronts” has been predicted for the first time for microchannels with negligible surface conductance. The mechanism of a similar phenomenon previously predicted and observed in very narrow microchannels with noticeable surface conductance13,25 has been shown to be different from the one discovered in this study. A posteriori analysis of time-dependent concentration profiles and distribution of pressure-driven backflow has revealed that the condition of “slow” processes (implicit to the Taylor Aris model of hydrodynamic dispersion) is not satisfied in the systems considered in this study. Accordingly, the approach to the description of hydrodynamic dispersion in this context has to be considerably modified to include “rapid” processes. The latter may give rise to an even stronger dampening impact of electroosmotic circulation on the current-induced concentration polarization of micro/nanointerfaces than demonstrated in this study. The scope of applicability of the approximation of local electric neutrality in the microchannel (used in this study) appears to be broader than that of the approximation of “slow” processes. Moreover, the approach of local electric neutrality may become even more applicable due to the stronger convective “delivery” of salt to the polarized interface to be expected if the restriction of “slow” processes is relaxed in future studies. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. 11720
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’ ACKNOWLEDGMENT This study was partially supported by the Spanish Ministry of Science and Innovation (MICINN) within the scope of the project “Integraci on de procesos de extraccion reactiva y procesos de membranas en la eliminacion de compuestos indeseados en etapas de potabilizacion de aguas superficiales y de regeneracion de aguas tratadas (PERMEAR)”. E.Z. is grateful to the Agencia de Gestio dAjuts Universitaris i de Recerca (AGAUR) for financial support of a sabbatical stay in Barcelona. The valuable comments and suggestions of Reviewer 7 of the previous version of this publication are gratefully acknowledged. ’ REFERENCES (1) Kort€um, G., Bockris, J. O’M. Textbook of electrochemistry; Elsevier: New York, 1951. (2) Strathmann, H. Ion-Exchange Membrane Separation Processes; Elsevier Science: New York, 2004; 360 pp. (3) Karnik, R.; Fan, R.; Yue, M.; Li, D.; Yang, P.; Majumdar, A. Nano Lett. 2005, 5, 943–948. (4) Plecis, A.; Schoch, R. B.; Renaud, Ph. Nano Lett. 2005, 5, 1147–1155. (5) Oh, Y.-J.; Garcia, A. L.; Petsev, D. N.; Lopez, G. P.; Brueck, S. R. J.; Ivory, C. F.; Han, S. M. Lab Chip 2009, 9, 1601–1608. (6) Bottenus, D.; Oh, Y.-J.; Han, S. M.; Ivory, C. F. Lab Chip 2009, 9, 219–231. (7) Zhou, K.; Kovarik, M. L.; Jacobson, S. C. J. Am. Chem. Soc. 2008, 130, 8614–8616. (8) Kovarik, M. L.; Zhou, K.; Jacobson, S. C. J. Phys. Chem. B 2009, 113, 15960–15966. (9) Zangle, Th.A.; Mani, A.; Santiago, J. G. Chem. Soc. Rev. 2010, 39, 1014–1035. (10) Pu, Q.; Yun, J.; Temkin, H.; Liu, S. Nano Lett. 2004, 4, 1099–1103. (11) Datta, A.; Gangopadhyay, S.; Temkin, H.; Pu, Q.; Liu, S. Talanta 2006, 68, 659–665. (12) Dhopeshwarkar, R.; Crooks, R. M.; Hlushkou, D.; Tallarek, U. Anal. Chem. 2008, 80, 1039–1048. (13) Zangle, Th.A.; Mani, A.; Santiago, J. G. Langmuir 2009, 25, 3909–3916. (14) Huang, K.-D.; Yang, R.-J. Electrophoresis 2008, 29, 4862–4870. (15) Kim, S. J.; Li, L. D.; Han, J. Langmuir 2009, 25, 7759–7765. (16) Kim, S. J.; Wang, Y. C.; Lee, J. H.; Jang, H.; Han, J. Phys. Rev. Lett. 2007, 99, 044501. (17) Kim, S. J.; Song, Y.-A.; Han, J. Chem. Soc. Rev. 2010, 39, 912–922. (18) Kim, S. J.; Han, J. Anal. Chem. 2008, 80, 3507–3511. (19) Jung, J.-Y.; Joshi, P.; Petrossian, L.; Thornton, T. J.; Posner, J. D. Anal. Chem. 2009, 81, 3128–3133. (20) Kelly, K. C.; Miller, S. A.; Timperman, A. T. Anal. Chem. 2009, 81, 732–738. (21) Jin, X.; Joseph, S.; Gatimu, E. N.; Bohn, P. W.; Aluru, N. R. Langmuir 2007, 23, 13209–13222. (22) Hlushkou, D.; Dhopeshwarkar, R.; Crooks, R. M.; Tallarek, U. Lab Chip 2008, 8, 1153–1162. (23) Hughes, B. T.; Berg, J. M.; James, D. L.; Ibraguimov, A.; Liu, S.; Temkin, H. Microfluid Nanofluid 2008, 5, 761–774. (24) Plecis, A.; Nanteuil, C.; Haghiri-Gosnet, A.-M.; Chen, Y. Anal. Chem. 2008, 80, 9542–9550. (25) Mani, A.; Zangle, Th.A.; Santiago, J. G. Langmuir 2009, 25, 3898–3908. (26) Postler, T.; Slouka, Z.; Svoboda, M.; Pribyl, M.; Snita, D. J. Colloid Interface Sci. 2008, 320, 321–332. (27) Yaroshchuk, A. E.; Zhukova, O. V.; Ulbricht, M.; Ribitsch, V. Langmuir 2005, 21, 6872–6882. (28) Pennathur, S.; Santiago, J. G. Anal. Chem. 2005, 77, 6772–6781. (29) Petsev, D. N. J. Chem. Phys. 2005, 123, 244907.
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