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Copyright Brian J. Kirby. With questions, contact Prof. Kirby here.
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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.
[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]
Beyond the native bulk conductivity of the electrolyte solution, the perturbed ion distribution in the double layer
leads to two additional sources of electrical current when an extrinsic electric field is applied normal to the surface.
The excess current beyond that which would exist for an uncharged surface is called surface current, and the
property of the surface itself is written as a surface conductivity or surface conductance. Fluid flow combined with
net charge density leads to a net convective current, and the change in the ion concentrations changes the ohmic
conductivity of the fluid in the double layer. In this section, we concentrate on the former; the latter is
presented in Chapter 11. The convective surface current is present in the DebyeHückel limit, while the
conductive surface current is zero in the DebyeHückel limit. Both play a role when the surface potential is
large.
When an extrinsic electric field is applied tangent to a charged surface, the electroosmotic flow, combined with
the presence of a net charge density in the electrical double layer, causes a net electrical current. In
general, convective current (per length parallel to the wall and normal to the electric field) is given
by
 (9.31) 
where I′ denotes current per length. For a channel with perimeter P , the convective surface current is given by
P I′. In the DebyeHückel limit and assuming thin double layers, we can show (see Exercise 9.19)
by evaluating the integral in Equation 9.31 that the convective surface current per length I′ is given
by
The relative change in current owing to this phenomenon can be quantified by defining the surface conductivity σ_{s},
which is the ratio of the convective surface current (normalized by the crosssectional area of the channel) to the
applied electric field:
 (9.33) 
recalling that the hydraulic radius r_{h} = 2A∕P , the convective surface conductivity in the thindoublelayer,
DebyeHückel limit is given by
 (9.34) 
From this, we can see that the convective surface density scales with the surface potential squared, and is
inversely proportional to the hydraulic radius and the Debye length. Thus we expect convective surface
conductivity to be significant when the surface potential is large, the double layer is thin, and the channel is
small.
Recalling the electrokinetic coupling matrix, first presented in Chapter 3:
 (9.35) 
we see that convective surface conductivity is accounted for in the thindoublelayer, DebyeHückel limit by setting
χ_{22} = σ+. The first term is the contribution of the bulk ohmic conductivity, while the second is the
contribution of the convective surface conductivity.
[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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