The Poisson–Boltzmann Equation

Chapter 4 is a thorough discussion of the theoretical underpinnings of the widely-used Poisson–Boltzmann (PB) equation. The tutorial is divided into four parts, the first of which is a brief history of the PB equation and its derivation. In the second part the PB equation is applied to several model systems for which exact or approximate analytical solutions can be found. In the third part of the tutorial numerical methods commonly used in applying the PB equation to more complicated systems than one-dimensional representations are provided. The fourth and final part of the chapter introduces topics of more advanced nature. The chapter covers: * Introduction State of the FieldOverview of the ChapterA Brief HistoryThe Poisson–Boltzmann Equation * Analytical Solutions to the Poisson–Boltzmann Equation Planar Geometry: The Membrane ModelCurved Surfaces: Cylinders and SpheresCylindrical Geometry: The Polymer ModelSpherical Geometry: The Micelle ModelMixed Geometry Studies * Numerical Solutions to the Poisson–Boltzmann Equation One-dimensional GeometriesFinite Difference/Finite Element AlgorithmsAlternative General-purpose MethodsLarge-scale Applications * Beyond the Poisson–Boltzmann Equation Assumptions of the Poisson–Boltzmann EquationCommon Approximations to the Poisson–Boltzmann EquationAlternatives to the Poisson–Boltzmann Equation