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.
Author(s): Ralf Blossey
Series: SpringerBriefs in Physics
Publisher: Springer
Year: 2023
Language: English
Pages: 112
City: Cham
Preface
Acknowledgements
Contents
Acronyms
1 The Poisson-Boltzmann Equation
1.1 Length Scales
1.2 The Poisson-Boltzmann Equation for (1:1) Salt
1.3 Solution of the Poisson-Boltzmann Equation for the Single-Plate Geometry
1.4 Slit Geometry
1.5 The Poisson-Boltzmann Equation in Cylindrical and Spherical Geometry
1.6 Generalized Poisson-Boltzmann Equations
1.7 A Higher-Order Poisson-Boltzmann Equation
1.8 Summary
1.9 Further Reading
References
2 Poisson-Boltzmann Theory and Statistical Physics
2.1 The Partition Function of a Coulomb Gas of Interacting Ions
2.2 Mean-Field Theory
2.3 One-Loop Correction
2.4 The Free Energy and the Surface Tension
2.5 The Variational Method
2.6 A Charged Polymer Interacting with a Like-Charged Membrane
2.7 Summary
2.8 Further Reading
References
3 Poisson-Boltzmann Theory with Solvent Structure
3.1 Nonlocal Electrostatics
3.2 The Dipolar Poisson-Boltzmann Equation: A Point-Dipole Theory
3.3 Finite-Size Dipoles
3.4 Fluctuation Effects in the Nonlocal Dipolar Poisson-Boltzmann Theory
3.5 The Dielectric Permittivity ε(k): Mean-Field Versus Variational Models
3.6 Dilute Solvents in Slit Nanopores
3.7 Concentrated Electrolytes in Slit Nanopores
3.8 The Green Function of the Dielectrically Anisotropic Laplace Equation
3.9 The Three Contributions to the Variational Grand Potential Ωv
3.9.1 The Anisotropic van der Waals Free Energy
3.9.2 The Correction Term Ωc
3.9.3 The Dipolar Contribution to the Grand Potential
3.10 Differential Capacitance
3.11 Going Simple Again: A Phenomenological Model for the Differential Capacitance
3.12 Summary
3.13 Further Reading
References
4 Conclusions
Index
