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MIT10_626S11_lec24

# MIT10_626S11_lec24 - V Electrostatics Lecture 24 Diffuse...

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V. Electrostatics Lecture 24: Diffuse Charge in Electrolytes MIT Student 1. Poisson-Nernst-Planck Equations The Nernst-Planck Equation is a conservation of mass equation that describes the influence of an ionic concentration gradient and that of an electric field on the flux of chemical species, specifically ions. We can start with the general conservation of mass equation for an incompressible fluid ( ): The background velocity in the convection term ( ) is relatively easy to define in a dilute solution, where it is just the (mass averaged) velocity of the solvent. However, this velocity becomes more difficult to define in concentrated solutions, since the distinction between the “flux” of an ion and relative to the “flow” of the solvent becomes blurred. More general prescriptions are available that treat all molecules (ions and solvent) on an equal footing, such as the Stefan-Maxwell equations for coupled fluxes or the de Groot-Mazur equations of nonequilibrium thermodynamics, but we neglect such complexities here, since most electrolytes are dilute enough to be well described. The flux density for the Nernst-Planck Equation can be generally expressed as Using the Einstein relation, , and the gradient of the chemical potential for a dilute solution, we can rewrite the flux as , , where the first term on the RHS is the flux due to diffusion and the second term on the RHS is the flux due to electromigration (the nonlinear term). We can now insert this expression for the flux into the conservation of mass equations and we will obtain the Nernst-Planck Equation for a dilute solution:

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Lecture 24: Diffuse charge in electrolytes 10.626 (2011) Bazant The Nernst-Planck Equation gives us i equations with i+1 unknowns. Hence, in order to solve the system of equations, we need to come up with one more equation. We can describe the electrostatic potential by using the Poisson Equation (a mean field approach), , where ρ is the free charge density and D is the is the electric displacement field vector. If we assume that we have a linear dielectric material, we can describe the electric displacement field vector as , where ε is the permittivity of the material (mainly the solvent), and E is the electric field generated by charges in the system. For electrostatics we also know that Hence we can rewrite the expression for the electric displacement field vector as We can now insert this expression into the Poisson Equation to arrive at our final form for this equation: Using a mean field approximation, we can get another equation for the free charge density, defined in terms of the mean (volume averaged) ion concentrations, This equation sums up all the charges of all the ions per unit volume. Combining this equation and the Poisson equation, we can get a new equation for the electrostatic potential to combine with the Nernst Planck Equations: Our full set of Poisson-Nernst-Planck (PNP) Equations is then: The system is now fully specified with matching numbers of equations and variables.
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MIT10_626S11_lec24 - V Electrostatics Lecture 24 Diffuse...

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