Nernst–Planck equation

Equation

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

${\displaystyle \frac{\mathrm{\partial }c}{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \mathbf{J}=0}$

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

${\displaystyle \mathbf{J}=-\underset{\text{Diffusion}}{\underbrace{D\mathrm{\nabla }c}}+\underset{\text{Advection}}{\underbrace{c\mathbf{v}}}+\underset{\text{Electromigration}}{\underbrace{\frac{Dze}{k_{\text{B}}T}c\mathbf{E}}}}$

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:

${\displaystyle \mathbf{E}=-\mathrm{\nabla }\varphi -\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t}}$

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

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

Simplifications

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

${\displaystyle \mathrm{\nabla }\cdot \left\{D\left[\mathrm{\nabla }c+\frac{ze}{k_{\text{B}}T}c\left(\mathrm{\nabla }\varphi +\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t}\right)\right]\right\}=0}$

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:

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

Finally, in units of mol/(m²·s) and the gas constant ${\displaystyle R}$, one obtains the more familiar form:

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

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