An exact solution of the nonlinear Poisson-Boltzmann equation in parallel-plate geometry

The Poisson-Boltzmann (PB) equation is a fundamental theoretical tool in understanding electric double layers (EDLs) at solid-liquid interfaces. Because of the intrinsic nonlinearity, finding exact analytical solutions of this equation is very difficult, and hitherto only very few exact analytical solutions are known. In this work, a new explicit exact solution for the nonlinear PB equation in parallel-plate geometry is derived in terms of Jacobi elliptic functions. A comparison of the sought solution with the finite element numerical simulation ensures correctness of the solution. We further found that the new solution is numerically consistent with the two existing solutions derived by Behrens & Borkovec (Phys. Rev. E 60:7040, [25]) and Johannessen (J. Math. Chem. 52:504, [36]) in spite of different expressions of the three solutions. This suggests equivalence of the three solutions. In addition, based upon the new solution, we suggest a method of determining the electrostatic potential profile inside the EDL with the experimental data of disjoining pressure.