Cornell University - Visit www.cornell.edu Kirby Research Group at Cornell: Microfluidics and Nanofluidics : - Home College of Engineering - visit www.engr.cornell.edu Cornell University - Visit www.cornell.edu
Cornell University, College of Engineering Search Cornell
News Contact Info Login

Donations keep this resource free! Give here:

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.

[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]

9.3 Convective surface conductivity [electrical double layer top]

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 Debye-Hückel limit, while the conductive surface current is zero in the Debye-Hückel 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
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(9.31)

where Idenotes current per length. For a channel with perimeter P , the convective surface current is given by P I. In the Debye-Hückel 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 Iis given by

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

The relative change in current owing to this phenomenon can be quantified by defining the surface conductivity σs, which is the ratio of the convective surface current (normalized by the cross-sectional area of the channel) to the applied electric field:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(9.33)

recalling that the hydraulic radius rh = 2A∕P , the convective surface conductivity in the thin-double-layer, Debye-Hückel limit is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(9.35)

we see that convective surface conductivity is accounted for in the thin-double-layer, Debye-Hückel limit by setting χ22 = σ+microfluidics textbook nanofluidics textbook Brian Kirby Cornell. 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] [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.


Ad revenue from these pages is used to support student research. The presence of an advertisement on these pages does not constitute an endorsement by the Kirby Research Group or Cornell University.

Donations keep this resource free! Give here: