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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]
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 timevarying 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 ( = q_{ion} ×) 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:
 (5.35) 
and in differentialform (upon application of the divergence theorem) as:
 (5.36) 
describes the motion of charge, and is thus a statement of electrodynamics rather than electrostatics. 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
 (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:
 (5.38) 
as does the electrostatic condition for the tangent field:
 (5.39) 
Now, we add to this the charge conservation equation applied across the interface:
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
 (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.
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.
The current density is defined by theButlerVolmer equation:
 (5.49) 
where n is the number of electrons transferred in the chemical reaction, F is the Faraday constant, ϕ_{1} and ϕ_{2} are the
electrical potentials on the solution and electrode sides of theinterface, R = k_{B}N_{A} isthe universal gas constant, T is
temperature, i_{0} 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 ϕ_{1} and ϕ_{2}. Lacking detailed information about the transition
state, α is often assumed equal to (1∕2), 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
singleelectron reaction), then the firstorder Taylor series expansion of the ButlerVolmer equation
gives
 (5.50) 
In this limit, the interface has an effective resistance of RT ∕i_{0}nF . As the current increases and the exponential term
grows, the effective resistance drops. These relations are shown in Figure 5.9.
5.2.3 Field lines at substrate walls
Perfect insulators and perfect conductors in contact with weakly conducting liquids lead to welldefined 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.
[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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