Nernst–Planck equation

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

The Nernst–Planck equation is a continuity equation for the time-dependent concentration ${\displaystyle c(t,{\bf {x}})}$ of a chemical species:

${\displaystyle {\partial c \over {\partial t}}+\nabla \cdot {\bf {J}}=0}$

where ${\displaystyle {\bf {J}}}$ 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 ${\displaystyle \nabla c}$, flow velocity ${\displaystyle {\bf {v}}}$, and an electric field ${\displaystyle {\bf {E}}}$:

${\displaystyle {\bf {J}}=-\underbrace {D\nabla c} _{\text{Diffusion}}+\underbrace {c{\bf {v}}} _{\text{Advection}}+\underbrace {{Dze \over {k_{\text{B}}T}}c{\bf {E}}} _{\text{Electromigration}}}$

where ${\displaystyle D}$ is the diffusivity of the chemical species, ${\displaystyle z}$ is the valence of ionic species, ${\displaystyle e}$ is the elementary charge, ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant, and ${\displaystyle T}$ is the absolute temperature. The electric field may be further decomposed as:

${\displaystyle {\bf {E}}=-\nabla \phi -{\partial {\bf {A}} \over {\partial t}}}$

where ${\displaystyle \phi }$ is the electric potential and ${\displaystyle {\bf {A}}}$ is the magnetic vector potential. Therefore, the Nernst–Planck equation is given by:

${\displaystyle {\frac {\partial c}{\partial t}}=\nabla \cdot \left[D\nabla c-c\mathbf {v} +{\frac {Dze}{k_{\text{B}}T}}c\left(\nabla \phi +{\partial {\bf {A}} \over {\partial t}}\right)\right]}$

Simplifications

Assuming that the concentration is at equilibrium ${\displaystyle (\partial c/\partial t=0)}$ and the flow velocity is zero, meaning that only the ion species moves, the Nernst–Planck equation takes the form:

${\displaystyle \nabla \cdot \left\{D\left[\nabla c+{ze \over {k_{\text{B}}T}}c\left(\nabla \phi +{\partial {\bf {A}} \over {\partial t}}\right)\right]\right\}=0}$

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:

${\displaystyle \nabla \cdot \left[D\left(\nabla c+{ze \over {k_{\text{B}}T}}c\nabla \phi \right)\right]=0}$

Finally, in units of mol/(m^2·s) and the gas constant ${\displaystyle R}$, one obtains the more familiar form:[3][4]

${\displaystyle \nabla \cdot \left[D\left(\nabla c+{zF \over {RT}}c\nabla \phi \right)\right]=0}$

where ${\displaystyle F}$ is the Faraday constant equal to ${\displaystyle N_{\text{A}}e}$; the product of Avogadro constant and the elementary charge.

Applications

The 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

• Goldman–Hodgkin–Katz equation
• Bioelectrochemistry

References

1. Kirby, B. J. (2010). Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices: Chapter 11: Species and Charge Transport.
2. Probstein, R. (1994). Physicochemical Hydrodynamics.
3. Hille, B. (1992). Ionic Channels of Excitable Membranes (2nd ed.). Sunderland, MA: Sinauer. p. 267. ISBN 9780878933235.
4. Hille, B. (1992). Ionic Channels of Excitable Membranes (3rd ed.). Sunderland, MA: Sinauer. p. 318. ISBN 9780878933235.
5. Sparks, D. L. (1988). Kinetics of Soil Chemical Processes. Academic Press, New York. pp. 101ff.
6. 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.