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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.
Preface
Acknowledgements
Contents
Acronyms
The Poisson-Boltzmann Equation
Length Scales
The Poisson-Boltzmann Equation for (1:1) Salt
Solution of the Poisson-Boltzmann Equation for the Single-Plate Geometry
Slit Geometry
The Poisson-Boltzmann Equation in Cylindrical and Spherical Geometry
Generalized Poisson-Boltzmann Equations
A Higher-Order Poisson-Boltzmann Equation
Summary
Further Reading
References
Poisson-Boltzmann Theory and Statistical Physics
The Partition Function of a Coulomb Gas of Interacting Ions
Mean-Field Theory
One-Loop Correction
The Free Energy and the Surface Tension
The Variational Method
A Charged Polymer Interacting with a Like-Charged Membrane
Summary
Further Reading
References
Poisson-Boltzmann Theory with Solvent Structure
Nonlocal Electrostatics
The Dipolar Poisson-Boltzmann Equation: A Point-Dipole Theory
Finite-Size Dipoles
Fluctuation Effects in the Nonlocal Dipolar Poisson-Boltzmann Theory
The Dielectric Permittivity ε(k): Mean-Field Versus Variational Models
Dilute Solvents in Slit Nanopores
Concentrated Electrolytes in Slit Nanopores
The Green Function of the Dielectrically Anisotropic Laplace Equation
The Three Contributions to the Variational Grand Potential Ωv
The Anisotropic van der Waals Free Energy
The Correction Term Ωc
The Dipolar Contribution to the Grand Potential
Differential Capacitance
Going Simple Again: A Phenomenological Model for the Differential Capacitance
Summary
Further Reading
References
Conclusions
Index