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. 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:
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}
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