Poisson Boltzmann relation

Many more-sophisticated models have been put forth to describe electrokinetic phenomena at surfaces. Considerations have included distance of closest approach of counterions, conduction behind the shear plane, specific adsorption of electrolyte ions, variability of permittivity and viscosity in the electrical double layer, discreteness of charge on the surface, surface roughness, surface porosity, and surface-bound water [7], Perhaps the most commonly used model has been the Gouy-Chapman-Stem-Grahame model 8]. This model separates the counterion region into a compact, surface-bound Stern" layer, wherein potential decays linearly, and a diffuse region that obeys the Poisson-Boltzmann relation. 

The electric field in solution is given by the Poisson-Boltzmann relation. 

The value of O at the distance r = 1 / c, where the ionic atmosphere is most densely populated, gives an estimate of the validity of the Debye-Huckel approximation. Neglecting factors of the order of unity, it turns out that q = [ zeYlsrkT) K < 1 hence at 25°C, / c) < 5.10 v/z. Thus the linearised Poisson-Boltzmann relation underestimates the electrostatic interactions in polyvalent electrolytes and even for (1-1) salts in solvents of low dielectric constant. 

Since ionic association is an electrostatic effect for equilibrium properties of electrolyte solutions, it may be included in the Debye-Hiickel type of treatment by explicitly retaining further terms in the expansion of the Poisson-Boltzmann relation eqn. 5.2.8. - A similar calculation was attempted for conductance by Fuoss and Onsager. The mathematical approach and the model employed are similar to those used in their previous calculation, but they keep explicitly the exp (—0 y) term in the new calculation. The equation derived for A is... 

H. Qian and J. A. Schellman,/. Phys. Chem. B, 104,11528 (2000). Transformed Poisson-Boltzmann Relations and Ionic Distributions. 

Another way of calculating the electrostatic component of solvation uses the Poisson-Boltzmann equations [22, 23]. This formalism, which is also frequently applied to biological macromolecules, treats the solvent as a high-dielectric continuum, whereas the solute is considered as an array of point charges in a constant, low-dielectric medium. Changes of the potential within a medium with the dielectric constant e can be related to the charge density p according to the Poisson equation (Eq. (41)). 

The final class of methods that we shall consider for calculating the electrostatic compone of the solvation free energy are based upon the Poisson or the Poisson-Boltzmann equatior Ihese methods have been particularly useful for investigating the electrostatic properties biological macromolecules such as proteins and DNA. The solute is treated as a body of co stant low dielectric (usually between 2 and 4), and the solvent is modelled as a continuum high dielectric. The Poisson equation relates the variation in the potential (f> within a mediu of uniform dielectric constant e to the charge density p ... 

Oosawa (1971) developed a simple mathematical model, using an approximate treatment, to describe the distribution of counterions. We shall use it here as it offers a clear qualitative description of the phenomenon, uncluttered by heavy mathematics associated with the Poisson-Boltzmann equation. Oosawa assumed that there were two phases, one occupied by the polyions, and the other external to them. He also assumed that each contained a uniform distribution of counterions. This is an approximation to the situation where distribution is governed by the Poisson distribution (Atkins, 1978). If the proportion of site-bound ions is negligible, the distribution of counterions between these phases is then given by the Boltzmann distribution, which relates the population ratio of two groups of atoms or ions to the energy difference between them. Thus, for monovalent counterions... 

Here, b is the distance between the nearest unit charges along the cylinder (b = 0.34nm for the ssDNA and b = 0.17nm for the dsDNA), (+) and (—) are related to cations and anions, respectively, and a = rss for the ssDNA and a rds for the dsDNA. The expressions (5) and (6) have been obtained using the equations for the electrostatic potential derived in [64, 65], where a linearization of the Poisson-Boltzmann equation near the Donnan potential in the hexagonal DNA cell was implemented. 

Two remaining problems relating to the treatment of solvation include the slowness of Poisson-Boltzmann calculations, when these are used to treat electrostatic effects, and the difficulty of keeping buried, explicit solvent in equilibrium with the external solvent when, e.g., there are changes in nearby solute groups in an alchemical simulation. Faster methods for solving the Poisson-Boltzmann equation by means of parallel finite element techniques are becoming available, however.22 24... 

Electro-osmotic drag phenomena are closely related to the distribution and mobility of protons in pores. The molecular contribution can be obtained by direct molecular d5mamics simulations of protons and water in single iono-mer pores, as reviewed in Section 6.7.2. The hydrod5mamic contribution to n can be studied, at least qualitatively, using continuum approaches. Solution of the Poisson-Boltzmann (PB) equation. 

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)... 

We can then relate the charge density, p, to the electrostatic potential nsing the one-dimensional Poisson-Boltzmann eqnation,... 

What are the assumptions that are needed to obtain the linearized Poisson-Boltzmann (LPB) equation from the Poisson-Boltzmann equation, and under what conditions would you expect the LPB equation to be sufficiently accurate What is the relation between the Debye-Huckel approximation and the LPB equation ... 

Another related phenomenon to be discussed in 2.3 is known in the polymer literature as counterion condensation. This term refers to a phase transition-like switch of the type of singularity, induced by a line charge to solutions of (2.1.2), occurring at some critical value of the linear charge density. Counterion condensation as a limiting property of the solutions of the Poisson-Boltzmann equation was studied in detail in [11]—[19]. Presentation of 2.3 follows that of [17]. 

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. 

Simultaneous measurements of d and osmotic pressure provide a relation between the separation of bilayers and their mutual repulsive pressure. Measurement of the electrostatic repulsion is, in fact, a determination of the electrostatic potential midway between bilayers relative to the zero of potential in the dextran reservoir. The full nonlinear Poisson-Boltzmann differential equation governing this potential has been integrated (I) from the midpoint to the bilayer surface to let us infer the surface potential. The slope of this potential at the surface gives a measure of the charge bound. 

Another limitation of the Poisson-Boltzmann approach is that the interaction between two surfaces immersed in water might not be exclusively due to the electrolyte ions. For instance, water has a different structure in the vicinity of the surface than in the bulk and the overlapping of such structures generates a repulsion even in the absence of electrolyte [20]. In this traditional picture, the hydration repulsion is not related to ion hydration actually it is not related at all to electrolyte ions. However, as recently suggested [21], this hydration interaction can still be accounted for within the Poisson-Boltzmann framework, assuming that the polarization is not proportional to the macroscopic electric field, but depends also on the field generated by the neighboring water dipoles and by the surface dipoles. 

Whereas the corrections to the traditional Poisson— Boltzmann approach could explain many experimental results, there are systems, such as the vesicles formed by neutral lipid bilayers in water, for which an additional force is required to explain their stability.4 This force was related to the organization of water in the vicinity of hydrophilic surfaces therefore it was called hydration force .5... 

A legitimate first question is related to the magnitude of errors generated by the discretization of the Poisson— Boltzmann equation (the replacement of eq 6 by eqs 7a—... 

In a typical macroscopic assumption of proportionality between polarization and applied electric field, P = e0(c — 1 )E, where e is the dielectric constant, and eq3 reduces to the traditional Poisson—Boltzmann equation (the concentrations cH and c0h being in general much smaller than ce). However, if the correlations between neighboring dipoles are taken into account, the following constitutive equation relating the polarization to the macroscopic electric field is obtained7... 

The polarization model is extended to account for the ion-ion and ion-surface interactions, not included in the mean field electrical potential. The role of the disorder on the dipole correlation length A, is modeled through an empirical relation, and it is shown that the polarization model reduces to the traditional Poisson Boltzmann formalism (modified to account for additional interactions) when X, becomes sufficiently small. 

Eqs. (5b) (5c) (5d) (5e). The constitutive equation of the polarization model contains only one unknown parameter, namely A, which expresses the correlation between neighboring dipoles. However, in order to solve the system of equations, both the surface charge density a (or the surface potential ip(z —d)) as well as the polarization of water near the surface have to be known. The latter can be related, in a microscopic model that will be examined in the next section, to the surface dipoles. In the limit A,—>0, the polarization model reduces to the Poisson— Boltzmann approach, and the two boundary conditions become dependent on each other, because in this case... 

Because in region I, the boundary of the atmosphere around each polymer chain was approximated with a cylinder, the electrical potential is related to the electrolyte concentration via a Poisson-Boltzmann equation in cylindrical... 

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