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[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]
[InducedCharge Effects]
[DEP]
[Solution Chemistry]
15.1 Unidirectional transport in infinitely long nanochannels [nanofluidics top]
For channels whose dimension is not large as compared to λ_{D}, 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 PoissonBoltzmann description described in Chapter 9 still applies:
and, for uniform properties and a simple interface, the velocity is still given by the analysis from Chapter 6:
 (15.2) 
We use these solutions in the detailed descriptions in the following sections.
Doublelayer 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^{*} = φ_{0}^{*} at the wall and = 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 λ_{D} ≃ 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 crosssectional areaaveraged solution of
the transport equations, which motivates the use of the electrokinetic couplingmatrix. This analysis is
valid when the nondimensional quantity Pe 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 crosssectionalareaaveraged properties is enormous, and the resulting errors are
negligible for channels of uniform crosssection. 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∕λ_{D}).
Recall that we can write the electrokinetic coupling matrix as
 (15.3) 
such that the electrokinetic coupling equationbecomes
 (15.4) 
For vanishingly thin double layers and small interfacial potentials, the electrokinetic coupling matrix was given
by
 (15.5) 
where r_{h} = R for a circular channel of radius R and r_{h} = 2d for infinite parallel plates separated by 2d. For
vanishingly thin λ_{D}, we assume d^{*}→∞, the system is independent of λ_{D}, and the only size dependence is on d
through the hydraulic radius dependence on χ_{11}. For λ_{D} small as compared to d but not vanishingly small, we
showed (see Equation 9.34) that the convective surface conductivity in the DebyeHückel limit is given by the first
term in the perturbation expansion for small λ_{D}∕d:
 (15.6) 
Here, ε^{2}φ_{0}^{2}∕λ_{D}^{2} 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 λ_{D} → 0 at constant εφ_{0}∕λ_{D} recovers
Equation 15.7.
Equation 15.6 is derived using the assumption that the double layer is given by the semiinfinite solution and
can be integrated to infinity. As the channel size d is reduced in comparison to λ_{D} 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
 (15.7) 
In this form, the coefficients c_{11}, c_{12}, c_{21}, and c_{22} illustrate the ratio of system response in a thick double layer
system to that in a system with vanishingly small λ_{D}.
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 λ_{D}). The hydraulic coupling coefficient (or hydraulic conductivity)
χ_{11} = r_{h}^{2}∕8η retains its functional dependence on r_{h} as long as viscosity is assumed independent of
the local ion concentration or electric field, and thus χ_{11} is independent of λ_{D}. Since r_{h} is linearly
proportional to the channel size, χ_{11} decreases as channel size is decreased. The electrical conductivity
χ_{22} increases with increasing φ_{0}^{*} and decreasing d^{*}, owing both to increased convective surface current and
increased ohmic conductivity in the double layer. The effective electroosmotic mobility χ_{12} increases with
increasing φ_{0}, but it decreases with decreasing d^{*}, as double layer overlap leads to a reduction in the total
flow. The effective streaming current coefficient χ_{21} 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 χ_{12} and χ_{21} are unequal; thus this system no longer exhibits Onsager
reciprocity.
In the dilute solution limit, the electroosmotic correction coefficient c_{12} can be calculated by solving the
onedimensional nonlinear PoissonBoltzmann equation and the onedimensional NavierStokes equation and
integrating:
 (15.8) 
We require approximate algebraic relations for c_{12} for common systems, for example for an electrolyte between two
plates at y^{*} = ±d^{*}, with uniform fluid properties assumed. In the DebyeHückel limit, this can be determined
analytically:
For wall potentials that cannot be assumed small, no closedform exact solution exists, but we can solve the equation
numerically and extract phenomenological relations to approximate c_{12}. The c_{12} 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 λ_{D}. Because of this behavior, we can approximate
c_{12} phenomenologically for a symmetric monovalent electrolyte with a logistic curve, where the parameters depend
on φ_{0}:
where
 (15.11) 
and
 (15.12) 
The numerical values for the φ_{0}^{*}dependence of α and d_{0}^{*} come from a numerical fit to full numerical solutions,
and are accurate to within 10% for φ_{0}^{*} < 6. Results for φ_{0}^{*} = 0,3,6 are shown in Figure 15.1.
In the dilute solution limit, the streaming current correction coefficient c_{21} can also be calculated by solving the
onedimensional nonlinear PoissonBoltzmann equation and the onedimensional NavierStokes equation and
integrating:
 (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:
 (15.14) 
In the DebyeHückel limit, c_{21} can be determined analytically:
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 closedform exact solution exists, but we can solve the equation numerically and extract
phenomenological relations to approximate c_{21}.
The thick double layer modification for c_{22} 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, DebyeHückel limit (Equation 9.34). The average current density i = I∕A in any onedimensional
system is given by
 (15.16) 
Since the velocity magnitude of a species u_{i} can be separated into u_{i} = u+μ_{EP,i}E, we can split this into a convective
component and an electrophoretic component:
 (15.17) 
No diffusion term appears in the previous equation, because we have assumed that the system has no gradients in the
xdirection. Simplifying the above relation, we get
 (15.18) 
where the net charge density ρ_{E} = ∑_{i}c_{i}z_{i}F and the conductivity σ = ∑_{i}c_{i}μ_{EP,i}z_{i}F . For a system in which the double
layer is vanishingly thin, the current density is uniform and given by = σ. When the double layer is finite,
the first term is nonzero owing to nonzero ρ_{E} 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 speciesdependent μ_{EP};
however, we find it useful to approximate the μ_{EP} 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 μ_{EP} is the same for all species, we can normalize the current density by σ_{bulk}E, resulting
in
 (15.19) 
where μ_{EP} is the electrophoretic velocity of the ions and μ_{EO} = εφ_{0}∕η is the electroosmotic mobility of the
surface in the thindoublelayer 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, c_{22} can be written as
The ohmic contribution is highest for large φ_{0} and small d^{*}. The convective contribution is highest for large φ_{0} 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
 (15.21) 
and the crosssectionaveraged electroosmotic velocity Q∕A is given by
 (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  and fix the electric field at zero, we find that the mean flow rate Q∕A is given
by
 (15.23) 
and the streaming current I∕A is given by
 (15.24) 
This magnitude increases as d^{*} decreases (since χ_{21} increases). Thus, thick double layer systems have increased
streaming current and smalldiameter systems have reduced pressuredriven 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 righthand side of Equation 15.7)
and one outcome (i.e., one of the elements on the lefthand 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
 (15.25) 
and
 (15.26) 
Equation 15.25 shows that the pressure gradient generated by an electrokinetic pump is increased as d decreases
owing to the ddependence of χ_{11} until double layers overlap and performance is degraded by the decrease in χ_{12}.
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
pressuredriven 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
ddependence of χ_{11}. 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,  is specified and I = 0.
In this case, we can solve the matrix equation to find
 (15.27) 
and
 (15.28) 
Equation 15.27 shows that streaming potential is dependent only on the relative size of the channel as compared to
λ_{D}—the streaming potential is reduced when double layers overlap, because χ_{22} increases more than χ_{12} 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 χ_{11} 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 d^{2} and the second term is
proportional to λ_{D}^{2}, 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 λ_{D} as compared to d, the streaming potential is also
changed. As compared to the approximate result for thin double layers with spatiallyuniform fluid properties in the
DebyeHückel limit:
 (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
 (15.30) 
which can be rewritten as
 (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
 (15.32) 
Electroviscosity
Electroviscosity is the term commonly used for the decrease in flowrate owing to adverse streaming
potentialinduced 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
 (15.33) 
While an “effective viscosity” is commonly defined for this system by writing
 (15.34) 
or
 (15.35) 
we avoid this construction in this text in favor of a direct measure of the attenuation of the flow:
 (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:
 (15.37) 
This relation is reminiscent of the thin double layer, DebyeHückel relation for convective surface conductivity,
though it is a secondorder perturbation in r_{h}∕λ_{D} 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 8 is less than 8% if the hydraulic radius is larger
than
 (15.38) 
If the hydraulic radius is smaller than this value, a full calculation to estimate the electroviscosity is warranted.
Valencedependent ion transport
Small ion transport in systems with d^{*} near unity exhibit valencedependent velocity even in channels of uniform
crosssection. 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
crosssectional 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.
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
HagenPoiseuille law Q = Δp∕R_{h}. 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 offdiagonal 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 crosssection is uniform
and for which gradients are normal to the flow, the only difference between thickdouble layer and
thindouble layer systems is in the electrokinetic coupling coefficients, as described earlier in this
section.
Equivalent circuit models are effective only when effects of changing crosssections 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 fluidmechanical 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 timedependent boundary conditions
(i.e., global nonequilibrium). From an ion transport standpoint, the use of an equilibrium distribution
(PoissonBoltzmann) 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 timedependent 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 φ_{0}^{*} 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 λ_{D}, 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 thickdouble layer channels have crosssection changes or at the location
where these channels intersect. At these intersections or for systems with variable crosssection, 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 crosssectional
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]
[InducedCharge 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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