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.
