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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]

9.4 Accuracy of the Boltzmann and Debye-Hückel approximations [electrical double layer top]

The two key approximations used in this chapter have been the Debye-Hückel 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 Debye-Hückel approximation

The applicabilityof the Debye-Hückel 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 Debye-Hückel approximation is often used in situations where * 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 φ0, c(y), ρE, the double layer capacitance, or the normal electric field at the wall. The validity of the Debye-Hückel 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 Debye-Hückel approximation. This table is strictly qualitative, as the errors caused by the Debye-Hückel approximation are a strong function of the magnitude of φ0*. However, this table gives a rough idea of when the Debye-Hückel approximation is defensible.





Parameter

Given φ 0  Given q′′0 



|φ0|

correctly predicted grossly overpredicted

|ubulk|

correctly predicted grossly overpredicted

|u(y)|

slightly overpredicted grossly overpredicted

|φ(y)|

slightly overpredicted grossly overpredicted

streaming potential magnitude

slightly underpredicted moderately overpredicted

|ρE(y)|

grossly underpredicted depends on y

double layer capacitance

grossly underpredicted correctly predicted

microfluidics textbook nanofluidics textbook Brian Kirby Cornellwall

grossly underpredicted correctly predicted

electric field magnitude normal to wall

grossly underpredicted correctly predicted

Navier slip velocity magnitude at wall

grossly underpredicted correctly predicted

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

grossly underpredicted correctly predicted




Table 9.1: Predictions resulting from applying the Debye-Hückel approximation to a Poisson-Boltzmann system with spatially-uniform permittivity and viscosity, depending on whether the interface potential φ0 or interface charge density q′′0 is known. Note that integral properties are more closely tied to φ0, 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 φ0* or q0′′* 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 φ0*, 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 Debye-Hückel 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 Debye-Hückel 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 Debye-Hückel 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 φ0*=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.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 9.7: Dependence of the potential distribution on the surface charge density at low surface charge density. Note that the normalized wall potential and normalized wall surface charge density are the same in the low surface charge density limit.



microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 9.8: Dependence of the potential distribution on the surface charge density at high surface charge density. Note that increases in the wall charge density lead to less and less increase in the wall potential.


__________________________________________________________________________________________________________________________________________________________


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 9.9: Dependence of the potential derivative at the wall on the potential at the wall.



microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 9.10: Spatial dependence of the normal electric field for φ0 = 3.


Overall, then, given φ0, the results of the Debye-Hückel approximation are reasonably accurate if we are considering integrated values (ubulk) but can be quite inaccurate if we are considering differentiated values (microfluidics textbook nanofluidics textbook Brian Kirby Cornellwall). 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

Gouy-Chapman modeling of thedouble layer is a mean-field, point-charge formulation that fails to account for solvent structure, ion size, or ion-ion correlations. We focus in this section on the limitations caused by the mean-field, point-charge formulation and how these limitations can be overcome. Given a known potential at an interface, the Gouy-Chapman 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 Gouy-Chapman model is an equilibrium, mean-field, point-charge formulation that fails to account for solvent structure, ion size, ion-ion correlations, or ion dynamics.

Here we address the mean-field, point-charge aspects of the Gouy-Chapman 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 Debye-Hückel approximation, the errors caused by the Boltzmann approximation are generally most dramatic for differential values and most moderate for integrated values.

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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]

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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