Equation, Boltzmann, generalized Poisson

The electrostatic methods just discussed suitable for nonelectrolytic solvent. However, both the GB and Poisson approaches may be extended to salt solutions, the former by introducing a Debye-Huckel parameter67 and the latter by generalizing the Poisson equation to the Poisson-Boltzmann equation.68 The Debye-Huckel modification of the GB model is valid to much higher salt concentrations than the original Debye-Huckel theory because the model includes the finite size of the solute molecules. 

Finally, the extremum of the free energy with respect to /(z) yields the generalized Poisson-Boltzmann (GPB) equation for electrostatics... 

This is sometimes called the linearized Poisson-Boltzmann equation . The general solution of the linearized Poisson-Boltzmann equation is... 

Equation (6) is the linear Poisson-Boltzmann equation. Although generally considered to be less accurate than its nonlinear counterpart, it has the advantage of being considerably easier to solve. In addition, in several cases it has been shown to give results very close to Eq. (4), even when the surface potentials are as high as one to two times the thermal voltage kT/e (i.e., 25-5 mV). Hence, Eq. (6) can yield information relevant to real colloidal systems under certain conditions. 

Because this result has been obtained by solving a generalized Poisson-Boltzmann equation with the linearization approximation, it is necessary to compare it with the DLVO theory in the limit where the Debye approximation holds. In this case, Verwey and Overbeek [2], working in cgs (centimeter-gram-second) units, derived the following approximate equation for the repulsive potential ... 

Another theory for has been proposed by Adelman and Chen. Their approach differs from the theories described above in that they do not rely completely on statistical mechanical methods, but attempt to solve generalized Poisson-Boltzmann equations for ion-solvent mixtures. The approximation obtained predicts that increases with p+ for all con-... 

The governing equations for the DG-based solvation model with quantum mechanical charge distributions are determined by the calculus of variations. As before, variation of Eq. 12.24 with respect to the electrostatic potential gives the generalized Poisson-Boltzmann (GPB) equation [71, 74] ... 

Equation (55) is a generalized Poisson-Boltzmann equation including the free ions and the charged polymers. The first term represents the salt contrihu-tion and the second term is due to the charged monomers and their counterions. Equation (56) is a generalization of the self-consistent field equation of neutral polymers [9]. In the bulk, the above equations are satisfied by setting i/r — 0 and f (pb. 

Result for kR>1 including first correction to Gouy-Chapman solution to account for curvature (Evans and Ninham, 1983 Loeb et al., 1961). Note that no exact solution is available for the general Poisson-Boltzmann equation (V u = sinh ) in spherical coordinates ... 

We prefer to think of equation Cl) as the generalized Poisson-Boltzmann equation, because, unlike the situation in electrostatics with which Poisson was concerned, but like the problem in statistical mechanics with which Boltzmann was concerned, the charge density is assumed to be a known function of the potential. 

Consider a general Poisson-Boltzmann equation (cf. Eq. (6.101)) of the form... 

Figure 2 Illustration of a negatively charged biomolecular surface with charge density a in the presence of a mixed electrolyte. The surface may represent that of a colloidal or biophysical particle such as a membrane (plane), polynucleic acid (cylinder), or micelle (sphere) where the distance of closest approach of ions is designated x — a. n the solution of the Gouy-Chapman equation, and of the Poisson-Boltzmann equation in general, the charged surface is usually displaced from its actual position (relative to the solvent) to the plane of closest approach of nonadsorbed ions, also called the outer Helmholtz plane.

Y. Gur, I. Ravina, and A. J. Babchin, J. Colloid Interface Sci., 64, 326 (1978). On the Electrical Double Layer Theory. 1. A Numerical Method for Solving a Generalized Poisson-Boltzmann Equation. 

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 Poisson-Boltzmann equation is a modification of the Poisson equation. It has an additional term describing the solvent charge separation and can also be viewed mathematically as a generalization of Debye-Huckel theory. 

The continuum treatment of electrostatics can also model salt effects by generalizing the Poisson equation (12) to the Poisson-Boltzmann equation. The finite difference approach to solving Eq. (12) extends naturally to treating the Poisson-Boltzmann equation [21], and the boundary element method can be extended as well [19]. 

In addition to the nearest-neighbor interaction, each ion experiences the electrostatic potential generated by the other ions. In the literature this has generally been equated with the macroscopic potential 0 calculated from the Poisson-Boltzmann equation. This corresponds to a mean-field approximation (vide infra), in which correlations between the ions are neglected. This approximation should be the better the low the concentrations of the ions. 

The derivation of the Poisson equation implies that the potentials associated with various charges combine in an additive manner. The Boltzmann equation, on the other hand, involves an exponential relationship between the charges and the potential. In this way a fundamental inconsistency is introduced when Equations (26) and (28) are combined. Equation (29) does not have an explicit general solution anyhow and must be solved for certain limiting cases. These involve approximations that —at the same time —overcome the objection just stated. 

The Debye-Hiickel approximation to the diffuse double-layer problem produces a number of relatively simple equations that introduce a variety of double-layer topics as well as a number of qualitative generalizations. In order to extend the range of the quantitative relationships, however, it is necessary to return to the Poisson-Boltzmann equation and the unrestricted Gouy-Chapman theory, which we do in Section 11.6. 

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. 

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. 

Table I. Constants of Integration for the General Solution of the Linearized Poisson—Boltzmann Equation...

In many practical cases we can use the low-potential-assumption and it leads to realistic results. In addition, it is a simple equation and dependencies like the one on the salt concentration can easily be seen. In some cases, however, we have high potentials and we cannot linearize the Poisson-Boltzmann equation. Now we treat the general solution of the onedimensional Poisson-Boltzmann equation and drop the assumption of low potentials. It is convenient to solve the equation with the dimensionless potential y = ertp/kBT. Please do not mix this up with the spacial coordinate y In this section we use the symbol y for the... 

A number of methodologies have been developed and generalized in recent years to quantitatively describe the ion atmosphere around nucleic acids [11, 12, 17, 28, 29]. These include models based on Poisson-Boltzmann equation [11, 12], counterion condensation [17], and simulation methods, such as Monte Carlo, molecular dynamics, and Brownian dynamics [28, 29]. 

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