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 = −∇${\displaystyle \phi }$ −∂A/∂t.

${\displaystyle {\frac {\partial c}{\partial t}}=-\nabla \cdot J\quad |\quad J=-\left[D\nabla c-uc+{\frac {Dze}{k_{\mathrm {B} }T}}c\left(\nabla \phi +{\frac {\partial \mathbf {A} }{\partial t}}\right)\right]}$

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

Where J is the diffusion flux density, 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, k_B is the Boltzmann constant, T is the temperature, ${\displaystyle u}$ is velocity of fluid, ${\displaystyle \phi }$ is the electric potential, ${\displaystyle \mathbf {A} }$ 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),

${\displaystyle J=-\left[D\nabla c+{\frac {Dze}{k_{\mathrm {B} }T}}c\left(\nabla \phi +{\frac {\partial \mathbf {A} }{\partial t}}\right)\right]}$

In the static electromagnetic conditions, one obtains the steady state Nernst–Planck equation

${\displaystyle J=-\left[D\nabla c+{\frac {Dze}{k_{B}T}}c(\nabla \phi )\right]}$

Finally, in units of mol/(m²·s) and the gas constant R, one obtains the more familiar form:[4][5]

${\displaystyle J=-D\left[\nabla c+{\frac {zF}{RT}}c(\nabla \phi )\right]}$

where F is the Faraday constant equal to N_Ae.

See also

• Goldman–Hodgkin–Katz equation
• Bioelectrochemistry

Notes

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. Sparks, D. L. (1988). Kinetics of Soil Chemical Processes. Academic Press, New York. pp. 101ff.
4. Hille, B. (1992). Ionic Channels of Excitable Membranes (2nd ed.). Sunderland, MA: Sinauer. p. 267.
5. Hille, B. (1992). Ionic Channels of Excitable Membranes (3rd ed.). Sunderland, MA: Sinauer. p. 318.