PoissonNernstPlanckNavierStokes

Poisson-Nernst-Planck equations describe the coupled transport of charged species under diffusion and electrostatic forces in liquids, solids, and porous media. They are used to model ionic currents in electrolytes, biological systems such as ion channels, and semiconductor devices, capturing how ion concentrations and electric potential influence each other.

The model consists of three sets of relations. The Poisson equation relates the electric potential φ to the charge density: ∇·(ε ∇φ) = -ρ, where ρ = ∑_i z_i e c_i + ρ_f, with ε the permittivity, z_i the valence, e the elementary charge, c_i the concentration of species i, and ρ_f any fixed background charge. The Nernst-Planck flux for each species i combines diffusion and migration: J_i = -D_i ∇c_i - (D_i z_i e /(k_B T)) c_i ∇φ, where D_i is the diffusion coefficient, k_B is Boltzmann’s constant, and T is temperature. The continuity (mass balance) equation for each species is ∂c_i/∂t + ∇·J_i = 0. Together, these equations form a nonlinear, strongly coupled system for the set {c_i, φ}.

In steady state, ∂c_i/∂t = 0 and ∇·J_i = 0, with appropriate boundary conditions such as applied potentials, fixed surface concentrations, or insulating boundaries. Applications span electrochemical energy storage, microfluidics, and biological membranes, as well as semiconductor device modeling and corrosion analysis.

Limitations include the mean-field nature of the model, which neglects finite ion size and correlations. Extensions such as size-modified PNP and coupling to fluid flow (Poisson–Nernst–Planck–Navier–Stokes) address some of these issues. Numerical solutions commonly employ finite element or finite difference methods.