Poisson-Boltzmann distribution

The Poisson-Boltzmann distribution for a spherical interface takes the form... 

For the case of two spherical particles of radii a and a2, Stern potentials, iftdi and i//d2, and a shortest distance, H, between their Stern layers, Healy and co-workers195 have derived the following expressions for constant-potential, V, and constant-charge, Fr, double-layer interactions. The low-potential form of the Poisson-Boltzmann distribution (equation 7.12) is assumed to hold and Kax and xa2 are assumed to be large compared with unity ... 

After application of a no slip boundary condition aty = 0 and AvJAy 0 asy oo. As can be seen from Eq. 19, varies exponentially from zero at the shear plane eventually reaching its steady state value of Veo at the edge of the double layer. In situations where the linear approximation is not accurate (typically when the -potential is very high), Eq. 19 can be further generalized by incorporating a nonlinear Poisson-Boltzmann distribution. This, however, typically involves the use of a numerical technique to find a solution. 

The free counterions form an electrical double layer in which the counterion concentrations around each micelle decrease in a Poisson-Boltzmann distribution into the aqueous phase. Figure 6 illustrates the double layer and the radial distributions of counterions at different salt concentrations obtained by solving the Poisson-Boltzmann equation. Note that the thickness of the double layer depends on the ion salt concentration. The graph also illustrates a two-site model for ion distribution used in the pseudophase models to describe measured ion distributions in solutions of ionic association colloids, that is, counterions are either bound or free (see below). However, explanations based only on coulombic interactions between headgroups and counterions fail to account for commonly observed trends in ion-specific effects, for example, the Hofmeister... 

Despite the vastly different nature of the media, the basic situation differs little in principle from that in aqueous solutions. There is a diffuse space charge layer, defined by Poisson-Boltzmann distributions of the positive (interstitial) and negative (vacancy) carriers in the electric field, which are truncated at x d, the distance of closest approach of the mobile species to the electrode surface. This truncation has the effect of placing a parallel-plate "inner layer" capacitor of plate separation d in series with the diffuse layer capacitance, giving the usual... 

In the DLVO theory, the distribution of counterions in the immediate vicinity of the charged colloidal particle is assumed to obey the Poisson-Boltzmann distribution. 

Jinnouchi and Anderson, as well as Goddard and coworkers, have instead adopted a Poisson Boltzmann distribution of countercharge (Model 2b.4). These methods couple this distribution with an implicit continuum solvation model for the solvent (water). The continuum model extends the double layer consideration to the diffusion layer region. Jinnouchi and Anderson highlight that strongly bound water molecules must still be included... 

There are a number of ways to model the interfacial electric field in SCMs, involving various combinations of a Poisson-Boltzmann distribution and/or one or more parallel-plate capacitors [21]. Originally, modelers treated essentially all the building blocks of an SCM - electrostatic model, site types, site densities, and equilibrium constants - as fitting parameters. It was soon discovered, however, that one could usually fit potentiometric titration data with multiple SCMs that posit widely diverging molecular-scale pictures of the interface [22],... 

If there are ions in the solution, they will try to change their location according to the electrostatic potential in the system. Their distribution can be described according to Boltzmarm. Including these effects and applying some mathematics leads to the final linearized Poisson-Boltzmann equation (Eq. (43)). 

The polyelectrolyte chain is often assumed to be a rigid cylinder (at least locally) with a uniform surface charge distribution [33-36], On the basis of this assumption the non-linearized Poisson-Boltzmann (PB) equation can be used to calculate how the electrostatic potential

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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... 

The EDL charge distribution can be modeled by a Poisson-Boltzmann equation (see, e.g., [46]). In many practical cases values for the layer thickness between 1 and 100 nm are obtained [47]. 

The ideal conductor model does not account for diffuseness of the ionic distribution in the electrolyte and the corresponding spreading of the electric field with a potential drop outside the membrane. To account approximately for these effects we apply Poisson-Boltzmann theory. The results for the modes energies can be summarized as follows [89] ... 

The second term in equation (9) is the usual electrostatic term. Here vA is the valency of the unit and e is the elementary charge, and ip(z) is the electrostatic potential. This second term is the well-known contribution accounted for in the classical Poisson-Boltzmann (Gouy -Chapman) equation that describes the electric double layer. The electrostatic potential can be computed from the charge distribution, as explained below. 

In the same way as described in Sec. 5.2 for a diifiise layer in aqueous solution, the differential electric capacity, Csc, of a space charge layer of semiconductors can be derived from the Poisson s equation and the Fermi distribution function (or approximated by the Boltzmann distribution) to obtain Eqn. 5-69 for intrinsic semiconductor electrodes [(Serischer, 1961 Myamlin-Pleskov, 1967 Memming, 1983] ... 

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. 

A more detailed view of the dynamies of a ehromatin chain was achieved in a recent Brownian dynamics simulation by Beard and Schlick [65]. Like in previous work, the DNA is treated as a segmented elastic chain however, the nueleosomes are modeled as flat cylinders with the DNA attached to the cylinder surface at the positions known from the crystallographic structure of the nucleosome. Moreover, the electrostatic interactions are treated in a very detailed manner the charge distribution on the nucleosome core particle is obtained from a solution to the non-linear Poisson-Boltzmann equation in the surrounding solvent, and the total electrostatic energy is computed through the Debye-Hiickel approximation over all charges on the nucleosome and the linker DNA. 

Equation (2.18) represents a linearized Boltzmann distribution. It contains two unknown variables, p(r) and >) (r). It is possible to reduce the problem of two unknown variables to a problem with one unknown variable by introducing a second equation expressing the relationship between the variables p(r) and t) (r). This second equation is known as the Poisson equation. The Poisson equation for spherically symmetrical charge distribution is given as... 

The third point implies that it is possible to develop a physical theory for ionic interactions that is relatively simple and still useful. The most frequently used is the Poisson-Boltzmann (P-B) equation, which combines the Poisson equation from classical electrostatics with the Boltzmann distribution from statistical mechanics. This is a second-order nonlinear differential equation and its solution depends on the geometry and the boundary conditions. 

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