Polarizable Poisson-Boltzmann equation

There are many possible improvements to the Poisson-Boltzmann equation and an extensive discussion of the refinements has been presented by Bell and Levine3 01). The relative permittivity is field dependent and the ions are polarizable. In Eq.(6.2) the correlation between the ions is neglected so are specific chemical effects in the... 

A computationally efficient analytical method has been developed for the crucial calculation of Born radii, which is required for each atom of the solute that carries a (partial) charge, and the Gpoi term has been parameterized to fit atomic polarization energies obtained by Poisson-Boltzmann equation [57]. The GB/SA model is thus fully analytical and affords first and second derivatives allowing for solvation effects to be included in energy minimizations, molecular dynamics, etc. The Gpoi term is most important for polar molecules and describes the polarization of the solvent by the solute. As force fields in general are not polarizable, it does not account for the polarization of the solute by the solvent. This is clearly an important limitation of this type of calculations. 

In the case of electrode-electrolyte solution interfaces, the Poisson-Boltzmann equation has been modified for integrating many effects as, for example, finite ion size, concentration dependence of the solvent, ion polarizability, and so on. More often, this modification consists in the introduction of one or several supplementary terms to the energetic contribution in the distribution, which leads to modified Poisson-Boltzmann (MPB) nonlinear differential equations [52],... 

Ion distribution around a spherical micelle can also be described with models that consider that polarizable and not very hydrophilic species (such as Br ) interact both coulombically and by a specific, noncoulombic, interaction [26]. This latter interaction allows the ion to intercalate at the micellar surface and to neutrahze an equivalent number of head groups. Ion distribution around a micelle is then calculated by solving the Poisson-Boltzmann equation (PBE) in the spherical symmetry with allowance for specific interactions via a Langmuir or Volmer isotherm [31]. The original kinetic treatment for a micelle of radius a, aggregation number iV in a cell of radius R yields [31] ... 

Abstract The electric fields and potential in a pore filled with water are calculated, without using the Poisson-Boltzmann equation. No assumption of macroscopic dielectric behavior is made for the interior of the pore. The field and potential at any position in the pore are calculated for a charge in any other position in the pore, or the dielectric boundary of the pore. The water, represented by the polarizable PSPC model, is then placed in the pore, using a Monte Carlo simulation to obtain an equilibrium distribution. The water, charges, and dielectric boundary, together determine the field and potential distribution in the channel. The effect on an ion in the channel is then dependent on both the field, and the position and orientation of the water. The channel can exist in two major configurations open or closed, in which the open channel allows ions to pass. In addition, there may be intermediate states. The channel has a water filled pore, and a wall... 

Equation [18] is valid when the polarizability of the dielectric is proportional to the electrostatic field strength. " The operator V in the Cartesian coordinate system has the form (didx, didy, d/dz). When one deals with a system composed of a macromolecule immersed in an aqueous medium containing a dissolved electrolyte, the partial charges of each atom of the macromolecule can be described as fixed charges charges of the dissolved electrolyte can be described as mobile charges with density determined by a Boltzmann s distribution, and Eq. [18] can be written in the following form, known as the Poisson-Boltzmann equation ... 

As an example, values of the one- and two-electron reduction potential of pheophytin-a were calculated in N-N-dimethylformamide by a combination of quantum mechanical, statistical mechanical, and polarizable continuum methods (Mehta and Datta 2008). Two different computational methods gave -0.92 and -1.03 V for the one-electron, while -1.34 and -1.30 V for the two-electron reduction potential values, respectively. The observed one-and two-electron potentials are -0.90 and -1.25 V, respectively. Solution of the Poisson-Boltzmann equation gave for the reduction potential of pheophytin-a within the thylakoid membrane -0.58 V, which is in good agreement with the reported value of -0.61 V. 

---