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
editThe Nernst–Planck equation is a continuity equation for the time-dependent concentration of a chemical species:
where is the flux. It is assumed that the total flux is composed of three elements: diffusion, advection, and electromigration. This implies that the concentration is affected by an ionic concentration gradient , flow velocity , and an electric field :
where is the diffusivity of the chemical species, is the valence of ionic species, is the elementary charge, is the Boltzmann constant, and is the absolute temperature. The electric field may be further decomposed as:
where is the electric potential and is the magnetic vector potential. Therefore, the Nernst–Planck equation is given by:
![{\displaystyle {\frac {\partial c}{\partial t}}=\nabla \cdot \left[D\nabla c-c\mathbf {v} +{\frac {Dze}{k_{\text{B}}T}}c\left(\nabla \phi +{\partial {\bf {A}} \over {\partial t}}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e007862adf8a64468aef7a8124b61669ed0093d)
Simplifications
editAssuming that the concentration is at equilibrium and the flow velocity is zero, meaning that only the ion species moves, the Nernst–Planck equation takes the form:
![{\displaystyle \nabla \cdot \left\{D\left[\nabla c+{ze \over {k_{\text{B}}T}}c\left(\nabla \phi +{\partial {\bf {A}} \over {\partial t}}\right)\right]\right\}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1ac5644db3bf945bab1a5b600a378b1f09bbe2c)
Rather than a general electric field, if we assume that only the electrostatic component is significant, the equation is further simplified by removing the time derivative of the magnetic vector potential:
![{\displaystyle \nabla \cdot \left[D\left(\nabla c+{ze \over {k_{\text{B}}T}}c\nabla \phi \right)\right]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/735474d1f0364a55e8bf2a38a3d201ede909a4e8)
Finally, in units of mol/(m2·s) and the gas constant , one obtains the more familiar form:[3][4]
![{\displaystyle \nabla \cdot \left[D\left(\nabla c+{zF \over {RT}}c\nabla \phi \right)\right]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6637a7211576ff76cee40b792c958d328c253cf)
where is the Faraday constant equal to ; the product of Avogadro constant and the elementary charge.
Applications
editThe Nernst–Planck equation is applied in describing the ion-exchange kinetics in soils.[5] It has also been applied to membrane electrochemistry.[6]
See also
editReferences
edit- ^ Kirby, B. J. (2010). Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices: Chapter 11: Species and Charge Transport.
- ^ Probstein, R. (1994). Physicochemical Hydrodynamics.
- ^ Hille, B. (1992). Ionic Channels of Excitable Membranes (2nd ed.). Sunderland, MA: Sinauer. p. 267. ISBN 9780878933235.
- ^ Hille, B. (1992). Ionic Channels of Excitable Membranes (3rd ed.). Sunderland, MA: Sinauer. p. 318. ISBN 9780878933235.
- ^ Sparks, D. L. (1988). Kinetics of Soil Chemical Processes. Academic Press, New York. pp. 101ff.
- ^ Brumleve, Timothy R.; Buck, Richard P. (1978-06-01). "Numerical solution of the Nernst-Planck and poisson equation system with applications to membrane electrochemistry and solid state physics". Journal of Electroanalytical Chemistry and Interfacial Electrochemistry. 90 (1): 1–31. doi:10.1016/S0022-0728(78)80137-5. ISSN 0022-0728.