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[Kinematics]
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[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]
In this section, we describe the phenomena that lead to flux of species. These fluxes, when applied to a control
volume, lead to the NernstPlanck equations.
11.2.1 Species fluxes and constitutive properties
In the absence of chemical reactions, the two mechanisms that lead to flow of species into or out of a control volume
are diffusion and convection.
Diffusion
In the dilute solution limit with negligible thermodiffusion effects (which is applicable for most ionic species in the
conditions used in microfluidics), Fick’s lawdefines a flux density of species proportional to the gradient of the
species concentration and the diffusivity of the species in the solvent:
where _{diff,i} [mol∕s m^{2}] is thediffusive species flux density (i.e., the amount of species i moving across a surface
per unit area due to diffusion), D_{i} is the diffusivity of species i in the solvent (usually water), and c_{i} is the
concentration of species i. Fick’s law is a macroscopic way of representing the summed effect of the random motion
of species owing to thermal fluctuations. Fick’s law is analogous to theFourier law for thermal energy flux caused by
a temperature gradient and the Newtonian model for momentum flux induced by a velocity gradient, and the
species diffusivity D_{i} is analogous to the thermal diffusivity α = k∕ρc_{p} and the momentum diffusivity
η∕ρ.
Convection
In addition to the random fluctuations of ions due to thermal motion, the deterministic motion of ions due to fluid
flow and electric fields (i.e., convection) also leads to a species flux density:
where _{conv,i} [mol∕s m^{2}]is the convectivespecies flux density (i.e., the amount of species i moving across a surface
per unit area due to convection) and _{i} is the velocity of species i. As described in Equation 11.4, the velocity of
species i is given by the vector sum of (the velocity of the fluid) and _{EP,i} (the electrophoretic velocity of the ion
with respect to the fluid:
 (11.9) 
Since we often use units of mol∕L for species concentration, the species concentrations must be converted to
mol∕m^{3} if species fluxes are to be in SI units.
11.2.2 NernstPlanck equations
In general, the transport of a species i in the absence of chemical reactions can be described using theNernstPlanck
equations:
Here, D_{i} is the diffusivity of species i and _{i} is the velocity of species i. _{i} is the vector sum of the
fluid velocity and theelectrophoretic velocity of the species μ_{EP,i}. The first term in the brackets is
thediffusive flux and the second is theconvective flux. Restated verbally, theNernstPlanck equations
in this form say that the change in the concentration of a species is given by the divergence of the
species flux density. This relation can be derived (Exercise 11.3) by drawing a control volume and
evaluating the species fluxes. Such a control volume is shown (for a Cartesian system) in Figure 11.1.
Comparing the NernstPlanck and NavierStokes equations
Equation 11.10 can be reorganized into a form similar to the one we have used for the NavierStokes equations. For
example, assuming the diffusivity is uniform and implementing the product rule ∇(_{i}c_{i}) = _{i}∇c_{i}+c_{i}∇⋅_{i}, we
obtain
 (11.15) 
As compared to the NavierStokes equations for momentum transport, the NernstPlanck equations (shown here
without chemical reaction) have no source term akin to a body force term, nor a pressure term. In place of
these, the NernstPlanck equations have a term (c_{i}∇⋅_{i}) proportional to the divergence of the species
velocity. For incompressible fluid flows, the divergence of the fluid velocity ∇⋅ is zero owing to
conservation of mass; however, we cannot in general say that the divergence of the species velocity ∇⋅_{i} is
zero, since systems with spatiallyvarying conductivity have finite divergence in the species velocity
field.
[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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