PoissonNernstPlancksystem

PoissonNernstPlancksystem, commonly referred to as the Poisson–Nernst–Planck (PNP) system, is a continuum model that describes the spatiotemporal evolution of ionic concentrations and electric potential in electrolytes under the influence of an electric field. It treats ions as charged species undergoing diffusion and migration in response to electrostatic forces, and it couples the Poisson equation for the electric potential with Nernst-Planck equations for each ionic species.

For each species i, with concentration c_i(x,t), valence z_i, diffusion coefficient D_i, and elementary charge e, the flux J_i is given by J_i = -D_i ∇c_i - μ_i z_i e c_i ∇φ, where φ is the electric potential and μ_i is the ionic mobility related to D_i by the Einstein relation D_i = μ_i k_B T / e. The conservation law is ∂c_i/∂t = -∇·J_i. The electrostatic potential satisfies the Poisson equation ∇·(ε ∇φ) = -∑_i z_i e c_i, with ε the permittivity of the medium.

Boundary conditions may specify fixed concentrations or fluxes, and electrode surfaces may impose a specified potential or reaction kinetics. Practically, the PNP system is solved numerically, often with finite element or finite volume methods, and sometimes with implicit time stepping due to stiffness.

Applications include modeling ion transport in microfluidic channels, nanopores, and biological ion channels, as well as in electrochemical cells. It is a mean-field model that neglects ion–ion correlations, finite-size effects, and nonlinear dielectric responses; various extensions incorporate steric effects, crowding, or more complex boundary interactions to address these limitations.