Nernst–Planck equation

Equation used to calculate the electromigration of ions in a fluid

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 c ( t , x ) {\displaystyle c(t,{\bf {x}})} of a chemical species:

∂ c ∂ t + ∇ ⋅ J = 0 {\displaystyle {\partial c \over {\partial t}}+\nabla \cdot {\bf {J}}=0}

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

J = − D ∇ c ⏟ Diffusion + c v ⏟ Advection + D z e k B T c E ⏟ Electromigration {\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 D {\displaystyle D} is the diffusivity of the chemical species, z {\displaystyle z} is the valence of ionic species, e {\displaystyle e} is the elementary charge, k B {\displaystyle k_{\text{B}}} is the Boltzmann constant, and T {\displaystyle T} is the absolute temperature. The electric field may be further decomposed as:

E = − ∇ ϕ − ∂ A ∂ t {\displaystyle {\bf {E}}=-\nabla \phi -{\partial {\bf {A}} \over {\partial t}}}

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

∂ c ∂ t = ∇ ⋅ [ D ∇ c − c v + D z e k B T c ( ∇ ϕ + ∂ A ∂ t ) ] {\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 ( ∂ c / ∂ t = 0 ) {\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:

∇ ⋅ { D [ ∇ c + z e k B T c ( ∇ ϕ + ∂ A ∂ t ) ] } = 0 {\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:

∇ ⋅ [ D ( ∇ c + z e k B T c ∇ ϕ ) ] = 0 {\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/(m2·s) and the gas constant R {\displaystyle R} , one obtains the more familiar form:[3][4]

∇ ⋅ [ D ( ∇ c + z F R T c ∇ ϕ ) ] = 0 {\displaystyle \nabla \cdot \left[D\left(\nabla c+{zF \over {RT}}c\nabla \phi \right)\right]=0}

where F {\displaystyle F} is the Faraday constant equal to N A e {\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

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.