MIT10_626S11_lec29

# MIT10_626S11_lec29 - VI Electrokinetics Lecture 29...

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VI. Electrokinetics Lecture 29: Electrokinetics MIT Student (and MZB) “Electrokinetics” refers to the study of electrically driven mechanical motion of charged particles or fluids. Sometimes it is used more narrowly for fluid or particle motion in electrolytes and ionic liquids (reserving the term “electrohydrodynamics” for weakly conducting dielectric liquids). Here we examine the basic electrokinetic equations for electrolytes, as well as several types of electrokinetic phenomena. 1. Basic Equations 1.1 Governing equations for flow, concentration, and electrical potential, and key assumptions A description of an electrokinetic system requires governing equations for the local bulk fluid velocity, local species concentrations {c i }, and (mean) electrical potential ( φ ). One can obtain concentration profiles from a conservation of mass and electrical potential from electrostatic considerations. These concepts are embodied in the Poisson-Nernst-Planck equations. Specifically, the conservation of mass combined with the Nernst-Planck expression for flux yields the mass conservation expression for an ionic species. The Poisson equation expresses the electrostatic phenomena that determine the potential. For a dilute solution, the Nernst-Planck equation takes the following form: The mean field approximation of the electrostatic potential is described by the Poisson equation, which relates the electrical potential to the charge density. The Navier-Stokes equation (NSE) is an expression of conservation of linear momentum for a Newtonian fluid with constant mass density. Now we allow for fluid motion due to electrostatic force by adding a term to the NSE to represent the body force density due to electrostatic force.

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Lecture 29: Electrokinetics 10.626 (2011) Bazant Where ρ m is the mass density of the fluid and is the electrostatic body force density. The left-hand side of the NSE represents the convective transfer of linear momentum. A source of linear momentum is the divergence of stress ( ). The divergence of the stress tensor captures the viscous stress effects. In addition, the continuity equation for an incompressible fluid reduces to One can express the electrostatic body force density as a function of the mean electrical potential ( φ ) using the Poisson equation. We now simplify the NSE by making the following assumptions: 1.) Neglect unsteady terms (no time derivatives). This is an appropriate assumption unless there is forcing at high frequency. It is justified by comparing the kinematic viscosity to the mass diffusivity. Given and D = mass diffusivity, one can assume that the fluid momentum diffuses quickly when ν >> D. For example, if the system is in water, the fluid momentum relaxes quickly, so one typically neglects the time derivative of the velocity. 2.) For low Reynold’s number flows,
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MIT10_626S11_lec29 - VI Electrokinetics Lecture 29...

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