Poisson-Boltzmann linearized, method

Davis, M. E., McCammon, J. A. Solving the finite difference linearized Poisson-Boltzmann equation A comparison of relaxation and conjugate gradients methods.. J. Comp. Chem. 10 (1989) 386-394. 

For studying the stability of colloidal particles in suspension (Chapter 13) or for determining the potential at the surface of particles (Chapter 12), one often needs expressions for potential distributions around small particles that have curved surfaces. Solving the Poisson-Boltzmann equation for curved geometries is not a simple matter, and one often needs elaborate numerical methods. The linearized Poisson-Boltzmann equation (i.e., the Poisson-Boltzmann equation in the Debye-Hiickel approximation) can, however, be solved for spherical electrical double layers relatively easily (see Section 12.3a), and one obtains, in place of Equation (37),... 

The solution of the linearized Poisson-Boltzmann equation around cylinders also requires numerical methods, although when cylindrical symmetry and the Debye-Hiickel approximation are assumed the equation can be solved. The solution, however, requires advanced mathematical techniques and we will not discuss it here. It is nevertheless useful to note the form of the solution. The potential for symmetrical electrolytes has been given by Dube (1943) and is written in terms of the charge density a as... 

A new theory of electrolyte solutions is described. This theory is based on a Debye-Hiickel model and modified to allow for the mutual polarization of ions. From a general solution of the linearized Poisson-Boltzmann equation, an expression is derived for the activity coefficient of a central polarized ion in an ionic atmosphere of non-spherical symmetry that reduces to the Debye-Hiickel limiting laws at infinite dilution. A method for the simultaneous charging of an ion and its ionic cloud is developed to allow for ionic polarization. Comparison of the calculated activity coefficients with experimental values shows that the characteristic shapes of the log y vs. concentration curves are well represented by the theory up to moderately high concentrations. Some consequences in relation to the structure of electrolyte solutions are discussed. 

ALTERNATIVE METHOD OF LINEARIZATION OF THE POISSON-BOLTZMANN EQUATION... 

In this section, we present a novel linearization method for simplifying the nonlinear Poisson-Boltzmann equation to derive an accurate analytic expression for the interaction energy between two parallel similar plates in a symmetrical electrolyte solution [13, 14]. This method is different from the usual linearization method (i.e., the Debye-Hiickel linearization approximation) in that the Poisson-Boltzmann equation in this method is linearized with respect to the deviation of the electric potential from the surface potential so that this approximation is good for small particle separations, while in the usual method, linearization is made with respect to the potential itself so that this approximation is good for low potentials. 

This chapter deals with a method for obtaining the exact solution to the linearized Poisson-Boltzmann equation on the basis of Schwartz s method [1] without recourse to Derjaguin s approximation [2]. Then we apply this method to derive series expansion representations for the double-layer interaction between spheres [3-13] and those between two parallel cylinders [14, 15]. 

We start with the simplest problem of the plate-plate interaction. Consider two parallel plates 1 and 2 in an electrolyte solution, having constant surface potentials i/ oi and J/o2, separated at a distance H between their surfaces (Fig. 14.1). We take an x-axis perpendicular to the plates with its origin 0 at the surface of one plate so that the region 0solution phase. We derive the potential distribution for the region between the plates (0linearized Poisson-Boltzmann equation in the one-dimensional case is... 

Potential energy descriptors proposed as an indicator of hydrophobicity [Oprea and Waller, 1997]. Originally, they were calculated using the finite difference approximation method the linearized Poisson-Boltzmann equation was solved numerically to compute the electrostatic contribution to solvation at each grid point. Desolvation energy field values were calculated as the difference between solvated (grid dielectric = 80) and in vacuo (grid dielectric = 1). 

To rectify this problem, efforts are underway to develop methods that predict binding free energies, at least qualitatively, for a medium-sized compound library (100-200 compounds) over a relatively short time period. The most promising methods to date include Linear Interaction Energy (LIE) analysis. Molecular Mechanics Poisson Boltzmann Surface Area (MM-... 

Implicit solvation models developed for condensed phases represent the solvent by a continuous electric field, and are based on the Poisson equation, which is valid when a surrounding dielectric medium responds linearly to the charge distribution of the solute. The Poisson equation is actually a special case of the Poisson-Boltzmann (PB) equation PB electrostatics applies when electrolytes are present in solution while the Poisson equation applies when no ions are present. Solving the Poisson equation for an arbitrary equation requires numerical methods, and many researchers have developed an alternative way to approximate the Poisson equation that can be solved analytically, known as the... 

M. E. Davis and J. A. McCammon, ]. Comput. Chem., 10, 386 (1989). Solving the Finite Difference Linearized Poisson-Boltzmann Equation A Comparison of Relaxation and Conjugate Gradient Methods. 

Those using analytical methods (often via perturbation approaches, which necessitate some level of approximation such as linearization of the Poisson-Boltzmann equation) to incorporate increasing complexity into the model of interfacial structure. 

Numerically, it is now a common practice to calculate within the dielectric continuum formulation but employing cavities of realistic molecular shape determined by the van der Waals surface of the solute. The method is based upon finite-difference solution of the Poisson-Boltzmann equation for the electrostatic potential with the appropriate boundary conditions [214, 238, 239]. An important outcome of such studies is that even in complex systems there exists a strong linear correlation between the calculated outer-sphere reorganization energy and the inverse donor-acceptor distance, as anticipated by the Marcus formulation (see Fig. 9.6). More... 

Particular attention should be paid to the DESMO method [43], as this model appears to be suitable for use with solvents containing dissolved ions that are described by the linearized Poisson-Boltzmann equation. DESMO shows promising accuracy with respect to benchmark LPBE calculations. This includes an anal3 ically solvable model problem consisting of multiple solute cavities, as would be encountered in a study of protein-protein interactions in implicit solvent. 

Implicit solvation models developed for condensed phases represent the solvent by a continuous electric field, and are based on the Poisson equation, which is valid when a surrounding dielectric medium responds linearly to the charge distribution of the solute. The Poisson equation is actually a special case of the Poisson-Boltzmann (PB) equation PB electrostatics applies when electrolytes are present in solution, while the Poisson equation applies when no ions are present. Solving the Poisson equation for an arbitrary equation requires numerical methods, and many researchers have developed an alternative way to approximate the Poisson equation that can be solved analytically, known as the Generalized Born (GB) approach. The most common implicit models used for small molecules are the Conductor-like Screening Model (COSMO) [96,97], the Dielectric Polarized Continuum Model (DPCM) [98], the Conductor-like modification to the Polarized Continuum Model (CPCM) [99], the Integral Equation Formalism implementation of PCM (lEF-PCM) [100] PB models and the GB SMx models of Cramer and Truhlar [52,57,101,102]. The newest Miimesota solvation models are the SMD (universal Solvation Model based on solute electron Density [57]) and the SMLVE method, which combines the surface and volume polarization for electrostatic interactions model (SVPE) [103-105] with semiempirical terms that account for local electrostatics [106]. Further details on these methods can be found in Chapter 11 of reference 52. 

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