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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]

5.2 Electrodynamics [electrodynamics top]

Electrodynamics describes the physics of moving charges, which is critical because when we apply fields to aqueous solutions, the charged ions in the solution move, carrying a current. Fortunately, we need consider only the simplest aspects of electrodynamics—we can ignore many electrodynamic effects in most microfluidic systems. As mentioned earlier, we assume all magnetic fields are constant since time-varying magnetic fields are rarely used in microfluidic systems. Because the current in aqueous solutions is carried by ions, which move slowly, we typically ignore the Lorentz force (microfluidics textbook nanofluidics textbook Brian Kirby Cornell = qmicrofluidics textbook nanofluidics textbook Brian Kirby Cornellion ×microfluidics textbook nanofluidics textbook Brian Kirby Cornell) on a moving charge as well. Because of this, the impact of the rather extensive field of electrodynamics is seen largely in the charge conservation equation.

5.2.1 Charge conservation equation

Thecharge conservation equation, written in integral form as:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.35)

and in differentialform (upon application of the divergence theorem) as:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.36)

describes the motion of charge, and is thus a statement of electrodynamics rather than electrostatics. microfluidics textbook nanofluidics textbook Brian Kirby Cornell  is the current density (or, equivalently, the charge flux density), which in an aqueous solution includes both ohmic current stemming from the electromigration of ions with respect to the fluid as well as net convective and diffusive charge flux. The ohmic current flux is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.37)

where σ is the conductivity ofthe medium. In most regions of microfluidic system, this ohmic flux dominates. However, the solvent can play a role because when the solvent moves, it carries any net charge density with it. Further, the thermal energy of the solvent causes ions to diffuse, which leads to a net charge flux in the presence of an ion concentration gradient.

5.2.2 Electrodynamic boundary conditions

The electrodynamic boundary conditions for our systems come from a combination of Gauss’s law and the charge conservation equation appliedto the interface. The electrostatic condition for the normal field still applies:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.38)

as does the electrostatic condition for the tangent field:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.39)

Now, we add to this the charge conservation equation applied across the interface:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

which, if the current flux is strictly ohmic (that is, if ion migration is the only source of current, and ion diffusion and chemical reactions can be ignored), implies
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.41)

Thus, Gauss’s equation relates the normal fields to the interfacial charge density, while the charge conservation equation relates the normal fields to the time derivative of the interfacial charge density.


microfluidics textbook nanofluidics textbook Brian Kirby Cornell


Current boundary conditions at electrodes The boundary conditions at insulating walls are unaffected by the presence of current, as are the boundary conditions for fluid inlets and outlets. However, our treatment of metal electrodes often must be more sophisticated.

A voltage applied to a conducting metal electrode induces a current to pass through the interface. The interface, though, is complicated by the fact that the current in metals is carried by electrons, while the current in an aqueous solution is carried by ions. Thus, for current to pass through an electrode, an electrochemical reaction must occur. This reaction is driven by a potential difference between the electrode and the solution in contact with it, as denoted in Figure 5.8.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 5.8: Current and potential drop at an electrode.


The current density is defined by theButler-Volmer equation:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.49)

where n is the number of electrons transferred in the chemical reaction, F is the Faraday constant, ϕ1 and ϕ2 are the electrical potentials on the solution and electrode sides of theinterface, R = kBNA isthe universal gas constant, T is temperature, i0 is a constant referred to as the exchange current density, and α is a parameterbetween 0 and 1 which denotes the sensitivity of chemical transition states to ϕ1 and ϕ2. Lacking detailed information about the transition state, α is often assumed equal to (12), indicating that the chemical transition state is equally sensitive to both potentials.

If the potential drop is small as compared to RT ∕nF (approximately 25 mV at room temperature for a single-electron reaction), then the first-order Taylor series expansion of the Butler-Volmer equation gives
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.50)

In this limit, the interface has an effective resistance of RT ∕i0nF . As the current increases and the exponential term grows, the effective resistance drops. These relations are shown in Figure 5.9.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 5.9: Left: normal current density magnitude into the solution as a function of potential difference between electrode and solution (α = 0.5). For high exchange current density, the needed current density is achieved at low potential drops and the curve appears linear. For low exchange current density, the current density can only be achieved with large overpotentials and the exponential dependence is apparent. Right: effective resistance as a function of normal current density magnitude (α = 0.5).


5.2.3 Field lines at substrate walls

Perfect insulators and perfect conductors in contact with weakly conducting liquids lead to well-defined geometries of field lines at the interface, as denoted in Figure 5.10. Because the tangent electric field inside a conductor is zero, the tangent electric field just outside the conductor is also zero, and thus electrostatics dictates that the electric field just outside a perfect conductor is normal to the surface. Similarly, since the conductivity through a perfect insulator is zero, the current normal to the surface must be zero, and electrodynamics dictates that the electric field just outside a perfect insulator is tangent to the surface.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 5.10: Boundary conditions for the electric field lines at perfect insulators and conductors in contact with weakly conducting liquids.


[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.


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