Poisson-Boltzmann equation analytical solution

For our work, expressions of Ohshlma et. al. (37) obtained from an approximate form of the Poisson-Boltzmann equation were used. These analytical expressions agree with the exact solution for xRp 2. (All of our calculations meet this criterion.) The relation between the surface potential and the surface charge density Is (37)... 

An analytical solution of the above system can be obtained in the linear approximation as follows. The Poisson—Boltzmann equations become... 

In the paragraphs below, we first examine the simple, analytical results that can be derived from the linear Poission-Boltzmann equation for a single particle interacting with a flat surface. Next, more complicated physical situations are considered, including interactions between many particles and a wall between a particle and a deformable interface between a protein and a wall and between a moving particle and a wall. In Sec. Ill, solutions to the nonlinear Poisson-Boltzmann equation are considered, and comparisons are made between the linear and nonlinear versions and also with more... 

The solution procedure for a particle covered by a membrane is similar to that for a rigid surface, except that the membrane phase needs additional treatments. In this section, we introduce three methods for recovering the solution of a Poisson-Boltzmann equation. As in the case of a rigid surface, obtaining the exact analytical solution for a Poisson-Boltzmann equation is almost impossible, in general, and only approximate results are available. Procedures for the estimation of the basic thermodynamic properties for the problem under consideration are also discussed. 

When the magnimde of the surface potential is arbitrary so that the Debye-Hiickel hnearization cannot be allowed, we have to solve the original nonlinear spherical Poisson-Boltzmann equation (1.68). This equation has not been solved but its approximate analytic solutions have been derived [5-8]. Consider a sphere of radius a with a... 

By using an approximation method similar to the above method and the method of White [6], one can derive an accurate analytic expression for the potential distribution around a spherical particle. Consider a sphere of radius u in a symmetrical electrolyte solution of valence z and bulk concentration n[7]. The spherical Poisson-Boltzmann equation (1.68) in this case becomes... 

So far we have treated uniformly charged planar, spherical, or cylindrical particles. For general cases other than the above examples, it is not easy to solve analytically the Poisson-Boltzmann equation (1.5). In the following, we give an example in which one can derive approximate solutions. 

The Poisson-Boltzmann equation for the potential distribution around a cylindrical particle without recourse to the above two assumptions for the limiting case of completely salt-free suspensions containing only particles and their counterions was solved analytically by Fuoss et al. [1] and Afrey et al. [2]. As for a spherical particle, although the exact analytic solution was not derived, Imai and Oosawa [3,4] smdied the analytic properties of the Poisson-Boltzmann equation for dilute particle suspensions. The Poisson-Boltzmann equation for a salt-free suspension has recently been numerically solved [5-8]. 

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. 

Surface potentials at the electrode-solution interface have been described by a number of formalisms. The most successful of these was offered originally by Gouy and Chapman with subsequent elaborations from Chapman, Stem, Bockris etc. (outlined in ref 1 21). McLaughlin (22) and others (outlined in 1) suggested that a combination of the Poisson and Boltzmann equations best describes the state of affairs in the space between the membrane surface and the bulk phase aqueous solution ie. the electrode-water interface. The Poisson-Boltzmann equation, with defined boundary conditions can be solved analytically (1,22) to yield an expression for the surface potential as follows ... 

A crucial parameter-free test of the theory is provided by its application to micelle formation from ionic surfactants in dilute solution [47]. There, if we accept that the Poisson-Boltzmann equation provides a sufficiently reasonable description of electrostatic interactions, the surface free energy of an aggregate of radius R and aggregation number N can be calculated horn the electrostatic free energy analytically. The whole surface free energy can be decomposed into two terms, one electrostatic, and another due to short-range molecular interactions that, from dimensional considerations, must be proportional to area per surfactant molecule, i.e. 

The system that we consider is an infinitely long cylindrical poly-ion enclosed in an outer concentric cylindrical container filled with solvent and counter-ions of valence z but with no added salt this is one case in which analytic, as opposed to numerical, solutions of the Poisson-Boltzmann equation are available. Numerical solutions for the case where added salt is present show much the same picture, however, so this limiting case with counter-ions only is still of general interest. The Poisson-Boltzmann equation for this system was solved long ago (9,10). 

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. 

Solvation energies for other multipoles inside a spherical cavity, including corrections due to salt effects, can be found, for example in Ref. 29. Analytical solutions of the Poisson equation for some other cavities, such as ellipse or cylinder, are also known [2] but are of little use in solvation calculations of biomolecules. For cavities of general shape only numerical solution of the Poisson and Poisson-Boltzmann equations is possible. There are two well-established approaches to the numerical solution of these equations the finite difference and the finite element methods. 

Solvation energies were computed at the double-c level using a self-consistent reaction field approach based on numerical solutions of the Poisson-Boltzmann equation 58-60). These were computed at the optimized gas-phase geometry utilizing an appropriate dielectric constant for comparison to the experimental conditions (e = 37.5 for acetonitrile e = 20.7 for acetone). The standard set of optimized radii in Jaguar were employed Mo (1.526 A), W (1.534 A), H (1.150 A), C (1.900 A), O (1.600 A). Vibrational analyses using analytical frequencies were computed at the double-q level, ensuring all stationary points to be minima. 

If the surfactant has an anionic or cationic head group, then ionic interactions occur at the micellar surface. The improved approximate analytical solution of the Poisson-Boltzmann equation was used by Nagarajan for the calculations... 

The need for computationally facile models for dynamical applications requires further trade-offs between accuracy and speed. Descending from the PB model down the approximations tree. Figure 7.1, one arrives at the generalized Born (GB) model that has been developed as a computationally efficient approximation to numerical solutions of the PB equation. The analytical GB method is an approximate, relative to the PB model, way to calculate the electrostatic part of the solvation free energy, AGei, see [18] for a review. The methodology has become particularly popular in MD applications [10,19-23], due to its relative simplicity and computational efficiency, compared to the more standard numerical solution of the Poisson-Boltzmann equation. 

Table 5.1. Analytical solutions of the Poisson-Boltzmann equation for single electrolytes ...

Gouy [29] and Chapman [30] independently proposed an alternative treatment. It is based on an analytical solution of the nonlinear Poisson-Boltzmann equation for the electric potential created by a system of mobile point charges near a charged wall imitating an electrode surface. However, in the original form of this theory, most of its predictions were at variance with experimental data. 

It can be noted that in practice, the Poisson-Boltzmann equation can be used to a good effect even in the presence of thick EDLs, provided that the Peclet number based on the EDL thickness (i.e., Pckd = Uj [k/D, where ref is the characteristic velocity scale along the axial direction and D is the diffusion coefficient of the solute) is small [5]. In such cases, a closed-form analytical expression for ij/ can be obtained from Eq. 27 as... 

Because of nonlinearity, the Poisson-Boltzmann equation (i.e., Eq. 10) can be solved numerically. Using the Debye-Hiickel approximation, an analytical solution of Eq. 10 for the EDL potential can be obtained as... 

Eigure 2 compares the LPM results for the nonlinear Poisson-Boltzmann equation (Eq. 13) and the analytical solutions of the linearized equation (Eq. 35), together with a numerical solution using the multigrid method. The parameters are the bulk ionic molar concentration c o = — 4 M, = c oNa where Na is Avogadro s number, z = 1 is the dielectric constant of the... 

For more complex geometries, it is sometimes possible to obtain approximate analytic expressions [24]. However, in general, the desired potential-to-charge relationship must be obtained via numerical solution of the Poisson-Boltzmann equation. In particular, for a long cylindrical cavity, the case most relevant to this study, the desired relation can be obtained by solving the system [25],... 

D yachkov, L. G. 2005a. Screening of macroions in colloidal plasmas Accurate analytical solution of the Poisson-Boltzmann equation. Physics Letters A 340, no. 5-6 440-448. 

Rice, R. E., and E. H. Home. 1981. Analytical solution of the linearized Poisson-Boltzmann equation in cylindrical coordinates. The Journal of Chemical Physics 75 5582. 

Van, H., and J. A. M. Smit. 1995. Approximative analytical solutions of the Poisson-Boltzmann equation for charged rods in the presence of salt An analysis of the cylindrical cell model. Journal of Colloid and Interface Science 170, no. 1 134—145. doi 10.1006/ jcis.1995.1081. 

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