Application of the Poisson-Boltzmann Equation

A basic approach to simplify the resolution of the PB equation is to linearize the exponential terms under the assumption of low-potential absolute values  

Note that due to the electroneutrality of the bulk solution, the first summation on the rightmost side of Equation 3.10 cancels out, so finally we obtain 

The Debye-Hiickel theory, originally developed to obtain expressions for activity coefficients, considers the charge distribntion around a single ion of charge z and radius Tq and assumes low-potential values, so that Equation 3.11 is solved in spherical coordinates with the boundary condition given by Gauss s law with spherical symmetry (Feynman, Leighton, and Sands 1970 Purcell 1984)  

FIGU RE 3.2 The interfacial region as predicted by the Debye-Hiickel theory, (a) Electrostatic potential as a function of distance from the surface at different ionic strengths, for a positively charged surface (h) ion concentrations as a function of distance for anions (c /c 1) and cations (c+/c 1) for the same case as (a). 

As we will be deahng with activities in the remainder of this book, it is worth noting that the Debye-Hiickel theory leads to a limiting law for the activity coefficient y, given by (McQuarrie and Simon 1997 Berry, Rice, and Ross 2000 Atkins and Panla 2009) 

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An alternative theoretical approach is the application of the Poisson-Boltzmann equation on the so-called cell model, assuming a parallel and equally spaced packing of rod-like polyions [62, 63]. This allows one to calculate at finite concentration according to ... 

An alternative approach is to prepare a reactive-ion surfactant for which the counterion is itself the reactant and inert counterions and interionic competition are absent (21-23). In principle, this method simplifies estimation of the concentration of an ionic nucleophile, for example, in the micellar pseudophase. Both these treatments of ionic reactions involve assumptions and approximations that seem to be satisfactory, provided that ionic concentrations are low, e.g., <0.1 M. These assumptions and approximations fail when electrolyte concentrations are high (24-25). A more rigorous treatment is based on application of the Poisson-Boltzmann equation in spherical symmetry (26-28), and this treatment accounts for some of the failures of the simpler models (29, 30). 

Tanford, C., Kirkwood, J. G. Theory of protein titration curves. I. General equations for impenetrable spheres. J. Am. Chem. Soc. 79 (1957) 5333-5339. 6. Garrett, A. J. M., Poladian, L. Refined derivation, exact solutions, and singular limits of the Poisson-Boltzmann equation. Ann. Phys. 188 (1988) 386-435. Sharp, K. A., Honig, B. Electrostatic interactions in macromolecules. Theory and applications. Ann. Rev. Biophys. Chem. 19 (1990) 301-332. 

The solution of the Poisson-Boltzmann equation with. the application to thermal explosions) 5) D.A. Frank-Kamenetskii, "Diffusion and Heat Exchange in Chemical Kinetics, pp 202-66, Princeton Uni v-Press, Princeton, NJ (1955) (Quoted from MaSek s paper) 6) L.N. Khitrin, "Fizika Goreniya i Yzryva (Physics of Combustion and Explosion), IzdMGU, Moscow (1957)... 

The second question concerns one particular aspect of general applicability of the simple mean field equations outlined above as opposed to more sophisticated statistical mechanical descriptions. In particular, the equilibrium Poisson-Boltzmann equation (1.24) is often used in treatments of some very short-scale phenomena, e.g., in the theory of polyelectrolytes, with a typical length scale below a few tens of angstroms (1A = 10-8 cm). On the other hand, the Poisson-Boltzmann equation implicitly relies on the assumption of a pointlike ion. Thus a natural question to ask is whether (1.24) could be generalized in a simple manner so as to account for a finite ionic size. The answer to this question is positive, with several mean field modifications of the Poisson-Boltzmann equation to be found in [5], [6] and references therein. Another ultimately simple naive recipe is outlined below. 

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. 

The limitations imposed on DDL theory as a molecular model by these four basic assumptions have been discussed frequently and remain the subject of current research.In Secs. 1.4 and 3.4 it is shown that DDL theory provides a useful framework in which to interpret negative adsorption and electrokinetic experiments on soil clay particles. This fact suggests that the several differences between DDL theory and an exact statistical mechanical description of the behavior of ion swarms near soil particle surfaces must compensate one another in some way, at least in certain applications. Evidence supporting this conclusion is considered at the end of the present section, whose principal objective is to trace out the broad implications of Eq. 5.1 as a theory of the interfacial region. The approach taken serves to develop an appreciation of the limitations of DDL theory that emerge from the mathematical structure of the Poisson-Boltzmann equation and from the requirement that its solutions be self-consistent in their physical interpretation. TTie limitations of DDL theory presented in this way lead naturally to the concept of surface complexation. 

The discussion of continuum electrostatics in Section 11.2.1 was limited to solution of Poisson s equation, which can be achieved exactly (for classical solutes) or to a good approximation (for QM solutes) using PCMs. In biomolecular applications, however, the objective is usually solution of the Poisson-Boltzmann equation [4, 33]. For low concentrations of dissolved ions, the latter is often replaced by the linearized Poisson-Boltzmann equation (LPBE),... 

Electrokinetic transport phenomena in porous media have been studied much in the past decade, both theoretically and numerically however, most results were actually based on linearization approximations of the Poisson-Boltzmann equation [5] so that reliable applications were limited to cases where the EDL length was very thin or very thick compared to the channel size. To our... 

Most applications of the Poisson-Boltzmann (PB) equation can be placed into one of two areas investigations of the chemical physics of ionic or... 

L. Martinez, A. Hernandez, A. Gonzalez, and F. Tejerina,/. Colloid Interface Sci., 152, 325 (1992). Use of Variational Methods to Establish and Increase the Ranges of Application of Analytic Solutions of the Poisson-Boltzmann Equation for a Charged Microcapillary. 

J. Granot, Biopolymers, 22, 1831 (1983). Effect of Finite Ionic Size on the Solution of the Poisson-Boltzmann Equation Application to the Binding of Divalent Metal Ions to DNA. 

This theory of the diffuse layer is satisfactory up to a symmetrical electrolyte concentration of 0.1 mol dm-3, as the Poisson-Boltzmann equation is valid only for dilute solutions. Similarly to the theory of strong electrolytes, the Gouy-Chapman theory of the diffuse layer is more readily applicable to symmetrical rather than unsymmetrical electrolytes. 

The charge density x on any electrolyte lamina parallel to the electrode and a distance x from it can be obtained by the application of electrostatics (Poisson s equation) and the Boltzmann distribution. Similarly, one can write for the intrinsic semiconductor, Poisson s equation... 

We have presented the first application of the newly developed method for calculating the solvation free energy to protein, which is based on the extended scaled particle theory and the Poisson-Boltzmann equation. Although the results are still preliminary, it demonstrates a possibility of obtaining the quantity theoretically, which is difficult even for the modem... 

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. 

An alternative way of solving the Poisson Boltzmann equation is the finite element method, which uses nonuniform and not rectangular grids. For example, the grid may be made finer around an active site to accurately evaluate ligand binding, and coarser elsewhere. This achieves comparable accuracy with the finite difference methods, but with a smaller number of grid points. Unfortunately, the finite element method has not been used extensively in applications only implementations of the method have been reported to date [45 47],... 

Consider the interaction between polyions and small ions. For a polyelectrolyte chain, the charges on the polymer chains repel each other so that the polymer chain tends to assume a more extended configuration. Because the diameter of a polymer chain is very small compared with it length, in many applications, a polyelectrolyte chain can be treated as a charged cylinder. Tlie interaction between small ions and a charged cylinder (rf infinite length can be described by the Poisson-Boltzmann equation 16-18)... 

Due to their many applications, electrokinetic microfluidics and nanofluidics have been much studied in the past decades, both theoretically and numerically. Erom the macroscopic point of view, the EOEs are governed by the Poisson-Boltzmann equation for electric potential distributions and the Navier-Stokes equations for flows [1, 6]. Accurate and efficient solution of... 

These have formed the subject of a general review and of an issue of Berichte der Bunsengesellschaft fiir Physikalische Chemie specifically inorganic applications have also been briefly reviewed in a paper which attempts a correlation using the Poisson-Boltzmann equation applied to a parallel-rod cell model. 

The aim of this paragraph is to recall the basic principles of electrostatics and their application to electrolytes and interfaces. After presenting the quasi static electric fields, we discuss some dielectric properties. We emphasize then the Poisson-Boltzmann equation and its solutions for different symmetries, giving the Gouy-Chapman, the Debye-Hiickel and the Lifson-Katchalsky approximations. 

Attempts to improve the theory by solving the Poisson-Boltzmann equation present other difficulties first pointed out by Onsager (1933) one consequence of this is that the pair distribution functions g (r) and g (r) calculated for unsymmetrically charged electrolytes (e.g., LaCl or CaCl2) are not equal as they should be from their definitions. Recently Outhwaite (1975) and others have devised modifications to the Poisson-Boltzmann equation which make the equations self-consistent and more accurate, but the labor involved in solving them and their restriction to the primitive model electrolyte are drawbacks to the formulation of a comprehensive theory along these lines. The Poisson-Boltzmann equation, however, has found wide applicability in the theory of polyelectrolytes, colloids, and the electrical double-layer. Mou (1981) has derived a Debye-Huckel-like theory for a system of ions and point dipoles the results are similar but for the presence of a... 

Solutions to the Poisson—Boltzmann equation in which the exponential charge distribution around a solute ion is not linearized [15] have shown additional terms, some of which are positive in value, not present in the linear Poisson—Boltzmann equation [28, 29]. From the form of Eq. (62) one can see that whenever the work, q yfy - yfy), of creating the electrostatic screening potential around an ion becomes positive, values in excess of unity are possible for the activity coefficient. Other methods that have been developed to extend the applicable concentration range of the Debye—Hiickel theory include mathematical modifications of the Debye—Hiickel equation [15, 26, 28, 29] and treating solution complexities such as (1) ionic association as proposed by Bjerrum [15,25], and(2) quadrupole and second-order dipole effects estimated by Onsager and Samaras [30], etc. 

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