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 = −∇φ −∂A/∂t.

{\frac {\partial c}{\partial t}}=-\nabla \cdot J

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

{\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, u is velocity of fluid, $\phi$ is the electric potential, $\mathbf {A}$ 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),

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

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]

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

where F is the Faraday constant equal to N_A e.

Notes

↑ Kirby, B. J. (2010). Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices: Chapter 11: Species and Charge Transport.
↑ 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.
↑ Hille, B. (1992). Ionic Channels of Excitable Membranes (3rd ed.). Sunderland, MA: Sinauer. p. 318.

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[Kirby-1] Kirby, B. J. (2010). Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices: Chapter 11: Species and Charge Transport.

[Probstein-2] Probstein, R. (1994). Physicochemical Hydrodynamics.

[Sparks1988-3] 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.

Nernst–Planck equation

Equation

See also

Notes