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Copyright Brian J. Kirby. With questions, contact Prof. Kirby here. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.

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Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

15.1 Unidirectional transport in infinitely long nanochannels [nanofluidics top]

For channels whose dimension is not large as compared to λD, the electroneutral “bulk” is no longer clearly defined. This phenomenon is commonly referred to asdouble layer overlap. However, if we consider unidirectional transport in infinitely long channels, the electrical double layer is still described by equilibrium relations, because the gradients are normal to the flow direction of flow.

15.1.1 Fluid transport

In the unidirectional case, the Poisson-Boltzmann description described in Chapter 9 still applies:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

and, for uniform properties and a simple interface, the velocity is still given by the analysis from Chapter 6:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.2)

We use these solutions in the detailed descriptions in the following sections.

Double-layer overlap and the breakdown of electroneutrality

As discussed in Chapter 9, the electroosmotic flow for narrow wall spacings can be determined by applying the boundary conditions that φ* = φ0* at the wall and microfluidics textbook nanofluidics textbook Brian Kirby Cornell = 0 at the axis of symmetry. This leads to the flow distributions shown in Figure 9.6 and represented in Equation 9.30. As compared to the thin double layer limit, for which the flow is approximately uniform, electroosmotic flow when λD d is nonuniform and therefore dispersive—since the velocity is a function of y throughout the flowfield, a sample bolus will be dispersed by the flow. Further, the total flow rate is reduced, since the potential difference between the wall and the bulk fluid is reduced (see Exercise 15.1).

15.1.2 Electrokinetic coupling matrix for thick double layer transport

Unidirectional flow through long, narrow channels can be analyzed with a cross-sectional area-averaged solution of the transport equations, which motivates the use of the electrokinetic couplingmatrix. This analysis is valid when the nondimensional quantity Pemicrofluidics textbook nanofluidics textbook Brian Kirby Cornell is small for ions, where the velocity used in the Peclet number is the characteristic velocity of ion motion through the channel, d is a measure of the channel depth, and L is a measure of the channel length. This criterion assures that the ion distribution normal to the direction of flow is governed by electrochemical equilibrium. Since d∕L is routinely as small as 1×10-5 in nanochannels, this criterion is satisfied for essentially all ions in nanochannels. The simplification from the use of cross-sectional-area-averaged properties is enormous, and the resulting errors are negligible for channels of uniform cross-section. With this approach, we can compare electrokinetic coupling phenomena in small channels both in terms of the absolute channel size (i.e., in terms of d) and in terms of the relative thickness of the channel with respect to the double layer (i.e., in terms of d* = d∕λD).

Recall that we can write the electrokinetic coupling matrix as
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.3)

such that the electrokinetic coupling equationbecomes
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.4)

For vanishingly thin double layers and small interfacial potentials, the electrokinetic coupling matrix was given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.5)

where rh = R for a circular channel of radius R and rh = 2d for infinite parallel plates separated by 2d. For vanishingly thin λD, we assume d*→∞, the system is independent of λD, and the only size dependence is on d through the hydraulic radius dependence on χ11. For λD small as compared to d but not vanishingly small, we showed (see Equation 9.34) that the convective surface conductivity in the Debye-Hückel limit is given by the first term in the perturbation expansion for small λD∕d:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.6)

Here, ε2φ02∕λD2 is equal to the square of the surface charge density, and the surface conductance is thus proportional to the squared surface charge density normalized by the viscosity and the ratio of the hydraulic radius to the Debye length. Taking the limit as λD 0 at constant εφ0∕λD recovers Equation 15.7.

Equation 15.6 is derived using the assumption that the double layer is given by the semi-infinite solution and can be integrated to infinity. As the channel size d is reduced in comparison to λD and d* can no longer be assumed large, the integral over finite domain must be employed, and several of the elements of this matrix take on a different functional form. This matrix analysis highlights the effects of double layer overlap on fluid and ion transport. We can parameterize these changes in the electrokinetic coupling matrix by writing it as
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.7)

In this form, the coefficients c11, c12, c21, and c22 illustrate the ratio of system response in a thick double layer system to that in a system with vanishingly small λD.

Electrokinetic coupling coefficients for thick double layers

The elements of the electrokinetic coupling matrix vary as a function of the characteristic size of a channel (both absolutely and relatively in comparison to λD). The hydraulic coupling coefficient (or hydraulic conductivity) χ11 = rh28η retains its functional dependence on rh as long as viscosity is assumed independent of the local ion concentration or electric field, and thus χ11 is independent of λD. Since rh is linearly proportional to the channel size, χ11 decreases as channel size is decreased. The electrical conductivity χ22 increases with increasing φ0* and decreasing d*, owing both to increased convective surface current and increased ohmic conductivity in the double layer. The effective electroosmotic mobility χ12 increases with increasing φ0, but it decreases with decreasing d*, as double layer overlap leads to a reduction in the total flow. The effective streaming current coefficient χ21 increases as d* decreases, since more of the net charge density is in regions of relatively fast fluid flow. When the wall potential is large and double layers are thick, the values of χ12 and χ21 are unequal; thus this system no longer exhibits Onsager reciprocity.

In the dilute solution limit, the electroosmotic correction coefficient c12 can be calculated by solving the one-dimensional nonlinear Poisson-Boltzmann equation and the one-dimensional Navier-Stokes equation and integrating:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.8)

We require approximate algebraic relations for c12 for common systems, for example for an electrolyte between two plates at y* = ±d*, with uniform fluid properties assumed. In the Debye-Hückel limit, this can be determined analytically:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

For wall potentials that cannot be assumed small, no closed-form exact solution exists, but we can solve the equation numerically and extract phenomenological relations to approximate c12. The c12 expression in Equation 15.9 is reminiscent of a logistic curve when plotted on a logarithmic axis, and for large wall potentials this curve shifts to lower d* and spreads out. This is because, at high wall potentials, the characteristic decay of the potential distribution occurs over a length shorter than λD. Because of this behavior, we can approximate c12 phenomenologically for a symmetric monovalent electrolyte with a logistic curve, where the parameters depend on φ0:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.11)

and
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.12)

The numerical values for the φ0*-dependence of α and d0* come from a numerical fit to full numerical solutions, and are accurate to within 10% for φ0* < 6. Results for φ0* = 0,3,6 are shown in Figure 15.1.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 15.1: Correction factor c12 for nanoslit geometries for symmetric monovalent electrolytes.


In the dilute solution limit, the streaming current correction coefficient c21 can also be calculated by solving the one-dimensional nonlinear Poisson-Boltzmann equation and the one-dimensional Navier-Stokes equation and integrating:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.13)

Here, u is the velocity profile generated by a pressure gradient. For example, for flow between plates at y* = ±d* with uniform fluid properties:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.14)

In the Debye-Hückel limit, c21 can be determined analytically:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

which echoes the result in Equation 15.9—Onsager reciprocity holds for thick double layers if the wall potential is low, since the deviation of ion distributions from the bulk concentrations is minor. For wall potentials that cannot be assumed small, no closed-form exact solution exists, but we can solve the equation numerically and extract phenomenological relations to approximate c21.

The thick double layer modification for c22 has two components, stemming both from ohmic and convective contributions to the current. Recall that the convective surface conductivity has already been presented in the thin double layer, Debye-Hückel limit (Equation 9.34). The average current density i = I∕A in any one-dimensional system is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.16)

Since the velocity magnitude of a species ui can be separated into ui = u+μEP,iE, we can split this into a convective component and an electrophoretic component:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.17)

No diffusion term appears in the previous equation, because we have assumed that the system has no gradients in the x-direction. Simplifying the above relation, we get
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.18)

where the net charge density ρE = iciziF and the conductivity σ = iciμEP,iziF . For a system in which the double layer is vanishingly thin, the current density microfluidics textbook nanofluidics textbook Brian Kirby Cornell  is uniform and given by microfluidics textbook nanofluidics textbook Brian Kirby Cornell= σmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell. When the double layer is finite, the first term is nonzero owing to nonzero ρE in the electrical double layer. When the wall potential is large, the second term is modified since σ is a function of y and increases in the electrical double layer.

Equation 15.18 can be evaluated using numerical integration for general species with species-dependent μEP; however, we find it useful to approximate the μEP of all species as equal, resulting in a general relation for the enhancement of the effective conductivity of the solution as a function of the species valences and wall potential alone. Assuming that μEP is the same for all species, we can normalize the current density by σbulkE, resulting in
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.19)

where μEP is the electrophoretic velocity of the ions and μEO = -εφ0∕η is the electroosmotic mobility of the surface in the thin-double-layer limit. From the form of the first term, we can see that the relative contribution of the electroosmotic fluid flow to the current is large if the surface electroosmotic mobility is large as compared to the ion electrophoretic mobility. The relative contribution of the electroosmotic fluid flow to the current is also large when there is considerable overlap between the electroosmotic fluid velocity and the charge density, which occurs when the double layers are thick. The second term accounts for the surface conductance, i.e., the increase in conductivity from the increased ion concentration in electrical double layers.

From the relations above, c22 can be written as

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

The ohmic contribution is highest for large φ0 and small d*. The convective contribution is highest for large φ0 and d* near 1—for large d*, there is little overlap between the charge density and the velocity, and for small d*, the velocity magnitude is low.

Observed system dependence on d and d* when forcing functions are controlled

We can consider the pressure and electric field to be the forcing functions of the electrokinetic coupling equation, and consider flow and current to be the outcomes. If we control the forcing functions and observe the outcomes, the results are each a function of only one element of the electrokinetic coupling matrix. If we apply an electric field but fix the pressure gradient at zero, our measured current density I∕A is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.21)

and the cross-section-averaged electroosmotic velocity Q∕A is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.22)

This and the correction factors described earlier indicate that thick double layer systems (typically found in nanochannels) exhibit increased conductivity but decreased electroosmosis as compared to thin double layer systems (typically found in microchannels).

If we apply a pressure gradient -microfluidics textbook nanofluidics textbook Brian Kirby Cornell and fix the electric field at zero, we find that the mean flow rate Q∕A is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.23)

and the streaming current I∕A is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.24)

This magnitude increases as d* decreases (since χ21 increases). Thus, thick double layer systems have increased streaming current and small-diameter systems have reduced pressure-driven flow.

Observed system dependence on d and d* when one forcing function and one outcome are controlled

If we consider control of one forcing function (i.e., one of the elements on the right-hand side of Equation 15.7) and one outcome (i.e., one of the elements on the left-hand side of Equation 15.7), we find that the results are a function of two or four of the electrokinetic coupling coefficients.

If we apply an electric field in a mechanically closed system, E is specified and Q = 0. In this case, we can solve the matrix equation to find
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.25)

and
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.26)

Equation 15.25 shows that the pressure gradient generated by an electrokinetic pump is increased as d decreases owing to the d-dependence of χ11 until double layers overlap and performance is degraded by the decrease in χ12. Equation 15.26 shows that the current required to drive an electrokinetic pump is affected by double layer overlap in three ways: the conductivity is increased because the ohmic contribution goes up and the convective ion transport goes up, peaking at d* near 1, but the net current is reduced by the adverse streaming current caused by the adverse pressure-driven flow (which is, in turn, caused by the electroosmosis). The ohmic current is typically the dominant term, but the adverse convective current becomes important when d is small because of the d-dependence of χ11. The current required to drive an electrokinetic pump increases as d* decreases, but this effect is attenuated if d becomes so small that adverse streaming current plays a prominent role.

If we apply a pressure gradient to a system whose reservoirs are electrically isolated, -microfluidics textbook nanofluidics textbook Brian Kirby Cornell is specified and I = 0. In this case, we can solve the matrix equation to find
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.27)

and
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.28)

Equation 15.27 shows that streaming potential is dependent only on the relative size of the channel as compared to λD—the streaming potential is reduced when double layers overlap, because χ22 increases more than χ12 when the ion distribution of finite double layers is accounted for. Equation 15.28 shows that the flow through an electrically isolated channel is primarily a function of size, owing to the dependence of χ11 on d, but this flow is furthermore reduced at low d* (electroviscosity) because the streaming current induced by the pressure also creates an adverse electroosmotic flow. Since the first term is proportional to d2 and the second term is proportional to λD2, the relative importance of the two terms in the brackets in Equation 15.28 is dictated by d*2.

Streaming potential in thick double layer systems

Just as electroosmosis is affected by the relative size of λD as compared to d, the streaming potential is also changed. As compared to the approximate result for thin double layers with spatially-uniform fluid properties in the Debye-Hückel limit:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.29)

the thick double layer result is changed because the net ohmic and convective currents increase owing to the double layer overlap. Using the matrix formulation above, recall that the induced electric field is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.30)

which can be rewritten as
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.31)

Using the vanishingly thin double layer electrokinetic coupling matrix components gives Equation 15.29, and for thick double layer systems the streaming potential is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.32)

Electroviscosity

Electroviscosity is the term commonly used for the decrease in flowrate owing to adverse streaming potential-induced electroosmosis. Figure 15.2 illustrates the adverse electroosmosis induced by streaming potential in a system where a pressure is applied and the net current is zero. The term electroviscosity has the potential to be misleading, since no physical change in viscosity is implied—rather, the flow is attenuated owing to electrokinetic effects. For nanochannels with dimensions on the same order as the double layer, the flow reduction can be significant—approximately a factor of two in an extreme case. If electrokinetic effects are ignored when interpreting flow rates, this makes the fluid appear to be more viscous or the channel to be of smaller diameter than it really is. Electroviscosity is unrelated to the viscoelectric effect, described by Equation 10.31 and different in that the viscoelectric effect is a postulated change in the viscosity of water owing to the local electric field.

As discussed earlier, the flow induced by pressure for a system whose reservoirs are otherwise isolated electrically is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.33)

While an “effective viscosity” is commonly defined for this system by writing
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.34)

or
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.35)

we avoid this construction in this text in favor of a direct measure of the attenuation of the flow:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.36)

We evaluate the flow attenuation in the vanishingly thin double layer limit—the result in this limit overestimates electroviscous effects if applied to a thick double layer system, but it nonetheless provides an upper estimate of the magnitude of this flow attenuation and identifies situations when electroviscosity can be ignored. Substituting the vanishingly thin double layer relations for the components of χ, and noting that this approach sets an upper bound for the flow attenuation, we find:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.37)

This relation is reminiscent of the thin double layer, Debye-Hückel relation for convective surface conductivity, though it is a second-order perturbation in rh∕λD and is less often significant. Though equation 15.37 overestimates the flow attenuation significantly, its results are useful for determining those cases where electroviscosity can be ignored. For example, the correction term 8microfluidics textbook nanofluidics textbook Brian Kirby Cornell is less than 8% if the hydraulic radius is larger than
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(15.38)

If the hydraulic radius is smaller than this value, a full calculation to estimate the electroviscosity is warranted.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 15.2: Adverse electroosmosis induced by streaming potential—the cause of apparent electroviscosity.


Valence-dependent ion transport

Small ion transport in systems with d* near unity exhibit valence-dependent velocity even in channels of uniform cross-section. Because double layer overlap leads to a velocity distribution that is nonuniform over the entire channel depth, variations in species distributions normal to the channel surfaces lead to a difference in the net cross-sectional average of the species velocity. In systems with d* near 1, the distribution of small ions (i.e., ions that can be modeled as point charges) is affected by the valence of the ion. Thus, the convective velocity of monovalent ions is larger than the convective velocity of di- or trivalent ions.


microfluidics textbook nanofluidics textbook Brian Kirby Cornell


15.1.3 Circuit models for nanoscale channels

Recall from Chapter 3 that 1D fluid flow systems can be predicted using equivalent circuits and the Hagen-Poiseuille law Q = Δp∕Rh. This is used to create sets of algebraic equations that can be solved in matrix form for the pressure at nodes and the volumetric flowrate in channels. Further, in Chapter 5 we use an identical approach to solve electrical circuit equations in matrix form for the voltage at nodes and the current in channels. These two systems of equations are uncoupled if walls are uncharged, but they are coupled if walls are charged. For a single channel, this is illustrated in Chapter 10 by showing that the resulting 2×2 matrix has off-diagonal elements that are zero if the walls are uncharged (Equation 10.41) and are nonzero if the walls are charged (Equation 10.42). For a system with N nodes and M channels, the system for uncharged walls can be written as two separate N +M×N +M matrix systems, while the system for charged walls must be written as one coupled 2N +2M×2N +2M matrix system. For channels whose cross-section is uniform and for which gradients are normal to the flow, the only difference between thick-double layer and thin-double layer systems is in the electrokinetic coupling coefficients, as described earlier in this section.

Equivalent circuit models are effective only when effects of changing cross-sections and interfaces are negligible. Equivalent circuits can employ algebraic relations to describe flow in long, narrow channels because of several key approximations—for example, in thin double layer systems we pay little attention to the details of the geometry of the interface between several channels, presuming that the electrical and hydraulic resistance in a long, straight channel is much larger than that of the intersection itself. This approach inherently stems from a thermodynamic argument that local equilibrium can be applied to the system, even though global nonequilibrium holds. From a fluid-mechanical standpoint, the Stokes approximation implies local equilibrium of fluid momentum by neglecting the unsteady and convective terms. The effect on the flows is seen in Chapter 8 through the observation of instantaneity (i.e., local equilibrium)—Stokes flows can vary in time only through time-dependent boundary conditions (i.e., global nonequilibrium). From an ion transport standpoint, the use of an equilibrium distribution (Poisson-Boltzmann) for ion distributions normal to a surface implies equilibrium in ion distributions normal to surfaces (i.e., local equilibrium), which vary only if there is time-dependent surface potential (i.e., global nonequilibrium).

For systems with d* near unity, equivalent circuit models fail at interfaces and in regions of varying cross section, because there is no longer a clear separation between local phenomena that are assumed to be in equilibrium and global phenomena that are assumed to be out of equilibrium, and the length scale for double layer equilibration is similar to the characteristic scales of the global system. If φ0* cannot be assumed small (and thus the perturbation to ion distributions owing to the wall is significant), then significant global effects evince the absence of local equilibrium. This occurs also during electrophoresis of particles with a* near one (see Figure 13.5). In the case of particles with radius near λD, the flow distortion of the double layer leads to an adverse electrical field that attenuates the particle flow. The electric field and flow are at points in the same direction as ion gradients, and these gradients are large enough to lead to significant effects. In nanochannel systems, the effects of note occur when thick-double layer channels have cross-section changes or at the location where these channels intersect. At these intersections or for systems with variable cross-section, we can no longer employ local equilibrium to separate the ion distribution solution normal to the wall from the flow parallel to the wall. The net effect of this is concentration polarization, in which ion concentrations are increased at one end of a nanochannel and decreased at the other. Concentration polarization at the interfaces between channels can often dominate the transport properties of a nanofluidic network.

The following section considers transport in thick double layer systems with these interfaces or cross-sectional area changes.

[Return to Table of Contents]



Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

Copyright Brian J. Kirby. Please contact Prof. Kirby here with questions or corrections. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.


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