Physics:Nernst–Planck equation
The time dependent form of the Nernst–Planck equation is a conservation of mass equation used to describe the motion of a charged chemical species in a fluid medium. It extends Fick's law of diffusion for the case where the diffusing particles are also moved with respect to the fluid by electrostatic forces:[1][2] It is named after Walther Nernst and Max Planck.
Equation
It describes the flux of ions under the influence of both an ionic concentration gradient ∇c and an electric field E = −∇[math]\displaystyle{ \phi }[/math] −∂A/∂t.
- [math]\displaystyle{ \frac{\partial c}{\partial t} = -\nabla \cdot J \quad | \quad J = -\left[ D \nabla c - u c + \frac{Dze}{k_\mathrm{B} T}c\left(\nabla \phi+\frac{\partial \mathbf A}{\partial t}\right) \right] }[/math]
- [math]\displaystyle{ \iff\frac{\partial c}{\partial t} = \nabla \cdot \left[ D \nabla c - u c + \frac{Dze}{k_\mathrm{B} T}c\left(\nabla \phi+\frac{\partial \mathbf A}{\partial t}\right) \right] }[/math]
Where J is the diffusion flux, t is time, D is the diffusivity of the chemical species, c is the concentration of the species, z is the valence of ionic species, e is the elementary charge, kB is the Boltzmann constant, T is the temperature, [math]\displaystyle{ u }[/math] is velocity of fluid, [math]\displaystyle{ \phi }[/math] is the electric potential, [math]\displaystyle{ \mathbf A }[/math] is the magnetic vector potential.
If the diffusing particles are themselves charged they are influenced by the electric field. Hence the Nernst–Planck equation is applied in describing the ion-exchange kinetics in soils.[3]
Setting time derivatives to zero, and the fluid velocity to zero (only the ion species moves),
- [math]\displaystyle{ J = -\left[ D \nabla c + \frac{Dze}{k_\mathrm{B} T}c\left(\nabla \phi+\frac{\partial \mathbf A}{\partial t}\right) \right] }[/math]
In the static electromagnetic conditions, one obtains the steady state Nernst–Planck equation
- [math]\displaystyle{ J = -\left[ D \nabla c + \frac{Dze}{k_{\rm B} T}c(\nabla \phi) \right] }[/math]
Finally, in units of mol/(m2·s) and the gas constant R, one obtains the more familiar form:[4][5]
- [math]\displaystyle{ J = -D\left[ \nabla c + \frac{zF}{RT}c(\nabla \phi) \right] }[/math]
where F is the Faraday constant equal to NAe.
See also
Notes
- ↑ Kirby, B. J. (2010). Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices: Chapter 11: Species and Charge Transport. http://www.kirbyresearch.com/index.cfm/wrap/textbook/microfluidicsnanofluidicsch11.html.
- ↑ Probstein, R. (1994). Physicochemical Hydrodynamics.
- ↑ Sparks, D. L. (1988). Kinetics of Soil Chemical Processes. Academic Press, New York. pp. 101ff.
- ↑ Hille, B. (1992). Ionic Channels of Excitable Membranes (2nd ed.). Sunderland, MA: Sinauer. p. 267. ISBN 9780878933235. https://archive.org/details/ionicchannelsofe00hill.
- ↑ Hille, B. (1992). Ionic Channels of Excitable Membranes (3rd ed.). Sunderland, MA: Sinauer. p. 318. ISBN 9780878933235. https://archive.org/details/ionicchannelsofe00hill.
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