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[Electroosmosis]
[Potential Flow]
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[Debye Layer]
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9.4 Accuracy of the Boltzmann and DebyeHückel approximations [electrical double layer top]
The two key approximations used in this chapter have been the DebyeHückel approximation(wall potential small
as compared to the thermal voltage) and the Boltzmann approximation (point charge, mean field). The accuracy of
these approximations are discussed in some detail in this section.
9.4.1 DebyeHückel approximation
The applicabilityof the DebyeHückel approximation is subtle, owing to the nonlinearity of the system and the wide
variety of double layer parameters under consideration. Though it is not mathematically rigorous to do so,
the DebyeHückel approximation is often used in situations where zφ^{*} is not small as compared to
unity, and we must understand when this approximation is acceptable. Depending on the situation or
application, we might be concerned with relations between the wall potential φ_{0}, c(y), ρ_{E}, the double layer
capacitance, or the normal electric field at the wall. The validity of the DebyeHückel approximation is not
equivalent for these different parameters. Table 9.1 lists some parameters and qualitatively indicates the
magnitude of the errors caused by the DebyeHückel approximation. This table is strictly qualitative,
as the errors caused by the DebyeHückel approximation are a strong function of the magnitude of
φ_{0}^{*}. However, this table gives a rough idea of when the DebyeHückel approximation is defensible.


 Parameter  Given φ_{
0}  Given q′′_{0} 


 φ_{0}  correctly predicted  grossly overpredicted 
u_{bulk}  correctly predicted  grossly overpredicted 
u(y)  slightly overpredicted  grossly overpredicted 
φ(y)  slightly overpredicted  grossly overpredicted 
streaming potential magnitude  slightly underpredicted  moderately overpredicted 
ρ_{E}(y)  grossly underpredicted  depends on y 
double layer capacitance  grossly underpredicted  correctly predicted 
_{wall}  grossly underpredicted  correctly predicted 
electric field magnitude normal to wall  grossly underpredicted  correctly predicted 
Navier slip velocity magnitude at wall  grossly underpredicted  correctly predicted 
 grossly underpredicted  correctly predicted 


 
Table 9.1: Predictions resulting from applying the DebyeHückel approximation to a PoissonBoltzmann
system with spatiallyuniform permittivity and viscosity, depending on whether the interface potential φ_{0} or
interface charge density q′′_{0} is known. Note that integral properties are more closely tied to φ_{0}, while
differential properties are more closely tied to q′′_{0}, especially near the wall. The absolute magnitude of any
error is a strong function of how large φ_{0}^{*} or q_{0}^{′′*} is; this table only estimates which predictions are more
or less accurate in a relative sense.
The conclusions from this table are as follows: given a specified φ_{0}^{*}, the integrated effect of the double layer on
the bulk electroosmotic velocity is independent of the model we use for the variation of potential in the double
layer—in fact, the expression for the bulk velocity was derived in Chapter 6 before any double layer model was
presented. The variation of the velocity and/or the electrical potential are only weakly affected by the
DebyeHückel approximation (see Figure 9.4), nor is streaming potential (see Section 10.6.3), especially since the
errors in net charge density caused by the DebyeHückel approximation are at regions where the flow velocity is
low. However, the charge density, normal derivatives of the potential and the velocity at the wall, the related wall
velocity predicted by the Navier slip model (Section ??), wall charge density, and double layer capacitance
(Section 16.2.1) are predicted poorly by the DebyeHückel approximation at all–in fact, these last parameters can
be off by a factor of 100 or so for a glass surface at neutral pH in a dilute buffer. For example, the potential gradient
at the wall (which is closely related to the surface charge) is graphed as a function of the potential at the wall in
Figure 9.9, and the electric field normal to the wall is graphed as a function of space (for φ_{0}^{*}=3) in
Figure 9.10.
__________________________________________________________________________________________________________________________________________________________ EXAMPLE PROBLEM 9.3:
Calculate and plot the dependence of the potential distribution φ^{*} on the surface charge density for
two sets of surface charge densities: first, q′′_{wall} = .0125,.025,.05,.1,.2, which corresponds to the
low surface charge density limit; and q′′_{wall} = .5,1,2,4,8, which corresponds to high surface charge
densities.
Solution:
See Figures 9.7 and 9.8.
__________________________________________________________________________________________________________________________________________________________
Overall, then, given φ_{0}, the results of the DebyeHückel approximation are reasonably accurate if we are
considering integrated values (u_{bulk}) but can be quite inaccurate if we are considering differentiated values (_{wall}).
The opposite is true if q′′_{wall} is given—differentiated values are reasonably accurate, but integrated values are
not.
9.4.2 Limitations of the ideal solution approximation
GouyChapman modeling of thedouble layer is a meanfield, pointcharge formulation that fails to account for
solvent structure, ion size, or ionion correlations. We focus in this section on the limitations caused by
the meanfield, pointcharge formulation and how these limitations can be overcome. Given a known
potential at an interface, the GouyChapman model of the electrical double layer describes the equilibrium
potential distribution near that wall, as well as the distribution of idealized point electrolytes interacting
only with that bulk potential field. As such, the GouyChapman model is an equilibrium, meanfield,
pointcharge formulation that fails to account for solvent structure, ion size, ionion correlations, or ion
dynamics.
Here we address the meanfield, pointcharge aspects of the GouyChapman double layer and highlight both the
limits of its applicability and modifications that can be used to extend its applicability. As was the case for the
DebyeHückel approximation, the errors caused by the Boltzmann approximation are generally most dramatic for
differential values and most moderate for integrated values.
[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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