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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]

11.2 Conservation of species: Nernst-Planck equations [species/charge transport top]

In this section, we describe the phenomena that lead to flux of species. These fluxes, when applied to a control volume, lead to the Nernst-Planck 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:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where microfluidics textbook nanofluidics textbook Brian Kirby Cornell diff,i [mols m2] is thediffusive species flux density (i.e., the amount of species i moving across a surface per unit area due to diffusion), Di is the diffusivity of species i in the solvent (usually water), and ci 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 Di is analogous to the thermal diffusivity α = k∕ρcp 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:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where microfluidics textbook nanofluidics textbook Brian Kirby Cornell conv,i [mols m2]is the convectivespecies flux density (i.e., the amount of species i moving across a surface per unit area due to convection) and microfluidics textbook nanofluidics textbook Brian Kirby Cornelli is the velocity of species i. As described in Equation 11.4, the velocity of species i is given by the vector sum of microfluidics textbook nanofluidics textbook Brian Kirby Cornell (the velocity of the fluid) and microfluidics textbook nanofluidics textbook Brian Kirby CornellEP,i (the electrophoretic velocity of the ion with respect to the fluid:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(11.9)

Since we often use units of molL for species concentration, the species concentrations must be converted to molm3 if species fluxes are to be in SI units.

11.2.2 Nernst-Planck equations

In general, the transport of a species i in the absence of chemical reactions can be described using theNernst-Planck equations:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Here, Di is the diffusivity of species i and microfluidics textbook nanofluidics textbook Brian Kirby Cornelli is the velocity of species i. microfluidics textbook nanofluidics textbook Brian Kirby Cornelli is the vector sum of the fluid velocity microfluidics textbook nanofluidics textbook Brian Kirby Cornell and theelectrophoretic velocity of the species μEP,imicrofluidics textbook nanofluidics textbook Brian Kirby Cornell. The first term in the brackets is thediffusive flux and the second is theconvective flux. Restated verbally, theNernst-Planck 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.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 11.1: Species fluxes for a Cartesian control volume.



microfluidics textbook nanofluidics textbook Brian Kirby Cornell


Comparing the Nernst-Planck and Navier-Stokes equations

Equation 11.10 can be reorganized into a form similar to the one we have used for the Navier-Stokes equations. For example, assuming the diffusivity is uniform and implementing the product rule (microfluidics textbook nanofluidics textbook Brian Kirby Cornellici) = microfluidics textbook nanofluidics textbook Brian Kirby Cornellici+ci∇⋅microfluidics textbook nanofluidics textbook Brian Kirby Cornelli, we obtain
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(11.15)

As compared to the Navier-Stokes equations for momentum transport, the Nernst-Planck 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 Nernst-Planck equations have a term (ci∇⋅microfluidics textbook nanofluidics textbook Brian Kirby Cornelli) proportional to the divergence of the species velocity. For incompressible fluid flows, the divergence of the fluid velocity ∇⋅microfluidics textbook nanofluidics textbook Brian Kirby Cornell is zero owing to conservation of mass; however, we cannot in general say that the divergence of the species velocity ∇⋅microfluidics textbook nanofluidics textbook Brian Kirby Cornelli is zero, since systems with spatially-varying 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] [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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