The Poisson–Boltzmann equation in micro- and nanofluidics: A formulary

The Poisson–Boltzmann (PB) equation provides a mean-field theory of electrolyte solutions at interfaces and in confinement, describing how ions reorganize close to charged surfaces to form the so-called electrical double layer (EDL), with numerous applications ranging from colloid science to biology. This formulary focuses on situations of interest for micro- and nanofluidics, and gathers important formulas for the PB description of a Z:Z electrolyte solution inside slit and cylindrical channels. Different approximated solutions (thin EDLs, no co-ion, Debye–Hückel, and homogeneous/parabolic potential limits) and their range of validity are discussed, together with the full solution for the slit channel. Common boundary conditions are presented, the thermodynamics of the EDL is introduced, and an overview of the application of the PB framework to the description of electrokinetic effects is given. Finally, the limits of the PB framework are briefly discussed, and Python scripts to solve the PB equation numerically are provided.