Poisson-Boltzmann equation

The Poisson-Boltzmann equation - named after Siméon Denis Poisson and Ludwig Boltzmann - describes the electrostatic interactions between molecules in liquids with ions dissolved in them . The Poisson-Boltzmann equation can be derived using a mean-field approximation .

It is particularly important in the fields of physical chemistry and biophysics . Here it is used to model the implicit solvation . With this method it is possible to approximate the effects of solvents on the structures and interactions of molecules in solutions of different ionic strength . Since the Poisson-Boltzmann equation cannot be solved analytically for complex systems, various computer programs were developed to solve it numerically. The Poisson-Boltzmann equation is used in particular for biologically relevant systems such as proteins , DNA or RNA .

Mathematical description

The equation can be written using SI units as follows:

${\displaystyle {\vec {\nabla }}\cdot \left[\varepsilon ({\vec {r}}){\vec {\nabla }}\Psi ({\vec {r}})\right]=-\left(\rho ^{f}({\vec {r}})+\sum _{i}c_{i}^{\infty }z_{i}\lambda ({\vec {r}})q\exp \left({\frac {-z_{i}q\Psi ({\vec {r}})}{kT}}\right)\right)}$

With

• ${\displaystyle \varepsilon ({\vec {r}})}$denotes the location-dependent dielectric conductivity
• ${\displaystyle \Psi ({\vec {r}})}$the electrostatic potential
• ${\displaystyle \rho ^{f}({\vec {r}})}$a fixed charge density
• ${\displaystyle c_{i}^{\infty }}$the concentration of the ion of the species at an infinite distance from the fixed charge density (in English: "bulk"). At an infinite distance one meets the convention ${\displaystyle i}$ ${\displaystyle \Psi ^{\text{bulk}}=0}$
• ${\displaystyle z_{i}}$ the valence of the ion
• ${\displaystyle q}$ the charge of a proton (elementary charge)
• ${\displaystyle k}$the Boltzmann constant
• ${\displaystyle T}$ the temperature
• ${\displaystyle \lambda ({\vec {r}})}$is a measure of the accessibility of the location to the ions of the solution. ${\displaystyle r}$

For small electrical potentials, the Poisson-Boltzmann equation can be linearized and then provides the Debye-Hückel approximation .

Extensions

If the ions are of a certain size, excluded volume effects can be described, for example, with the modified Poisson-Boltzmann equation.

restrictions

Due to its mean field character, the Poisson-Boltzmann theory is only valid in the case of weak electrostatic coupling and in the case of not too high ion concentrations. In the case of strong electrostatic coupling, the electrostatic Strong Coupling Theory can be used for the description .

There is a variation approach that is used to describe charged systems taking fluctuations into account. This theory of variation goes beyond the Poisson-Boltzmann approach.

Web links

• APBS PB solver
• Zap - A Poisson-Boltzmann electrostatics solver

Individual evidence

1. ↑ Federigo Fogolari, Alessandro Brigo, Henriette Molinari: The Poisson-Boltzmann equation for biomolecular electrostatics. A tool for structural biology. In: Journal of Molecular Recognition. Volume 15 (2002), Issue 6, pp. 377-392. PMID 12501158 doi : 10.1002 / jmr577 (currently not available)
2. ↑ Borukhov, Itamar, David Andelman, and Henri Orland. "Steric effects in electrolytes: A modified Poisson-Boltzmann equation." Physical review letters 79.3 (1997): 435.
3. ↑ Nir Gavish, Doron Elad and Arik Yochelis: From Solvent-Free to Dilute Electrolytes: Essential Components for a Continuum Theory . In: Journal of Physical Chemistry Letters . tape 9 , no. 1 , 2018, p. 36-42 , doi : 10.1021 / acs.jpclett.7b03048 (English).
4. ↑ Ralf Blossey, Sahin Buyukdagli: Beyond Poisson-Boltzmann: fluctuations and fluid structure in a self-consistent theory . January 4, 2016, doi : 10.1088 / 0953-8984 / 28/34/343001 .