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This brief book introduces the Poisson-Boltzmann equation in three chapters that build upon one another, offering a systematic entry to advanced students and researchers. Chapter one formulates the equation and develops the linearized version of Debye-Hückel theory as well as exact solutions to the nonlinear equation in simple geometries and generalizations to higher-order equations. Chapter two introduces the statistical physics approach to the Poisson-Boltzmann equation. It allows the treatment of fluctuation effects, treated in the loop expansion, and in a variational approach. First applications are treated in detail: the problem of the surface tension under the addition of salt, a classic problem discussed by Onsager and Samaras in the 1930s, which is developed in modern terms within the loop expansion, and the adsorption of a charged polymer on a like-charged surface within the variational approach. Chapter three finally discusses the extension of Poisson-Boltzmann theory to explicit solvent. This is done in two ways: on the phenomenological level of nonlocal electrostatics and with a statistical physics model that treats the solvent molecules as molecular dipoles. This model is then treated in the mean-field approximation and with the variational method introduced in Chapter two, rounding up the development of the mathematical approaches of Poisson-Boltzmann theory. After studying this book, a graduate student will be able to access the research literature on the Poisson-Boltzmann equation with a solid background. .

Statistical Physics. --- Electrochemistry. --- Differential equations. --- Surfaces (Technology). --- Thin films. --- Differential Equations. --- Surfaces, Interfaces and Thin Film. --- Films, Thin --- Solid film --- Solid state electronics --- Solids --- Surfaces (Technology) --- Coatings --- Thick films --- Materials --- Surface phenomena --- Friction --- Surfaces (Physics) --- Tribology --- 517.91 Differential equations --- Differential equations --- Chemistry, Physical and theoretical --- Physics --- Mathematical statistics --- Surfaces --- Statistical methods --- Equations. --- Poisson's equation. --- Differential equations, Elliptic --- Algebra --- Mathematics