Ionic density, Boltzmann distribution

Modification of this model to get the potential function is obtained considering the Fermi-Dirac distribution function for the electron density and the Boltzmann distribution for the ionic density. This was done by Stewart and Pyatt [58] to get the energy levels and the spectroscopic properties of several atoms under various plasma conditions. Here the electron density was given by... 

The linear approximation of the Boltzmann distribution that allows one to calculate the time averaged density of opposite charge surrounding the central ion, is assumed to hold, and this is actually true only for very low ionic concentrations. 

Consider, then, a fluid containing ions that are nonuniformly distributed, producing a position-dependent electric field E. Since an electric field is conservative, it is given by the gradient of a potential, E = — VtJ/. Negative ions tend to collect at locations where this potential is positive relative to some datum, and positive ions collect where it is negative. At equilibrium, the number density n, of ionic species i at each location is given by the Boltzmann distribution ... 

By increasing pressure and/or decreasing temperature, ionic quantum effects can become relevant. Those effects are important for hydrogen at high pressure [7, 48]. Static properties of quantum systems at finite temperature can be obtained with the Path Integral Monte Carlo method (PIMC) [19]. We need to consider the ionic thermal density matrix rather than the classical Boltzmann distribution ... 

The problem is to get some device which would substitute the average distribution of the discrete ions in the ionic atmosphere around the centralj-ion, given by n, in the Maxwell-Boltzmann expression, by a continuous charge density which could be taken to be equivalent to pj in the Poisson equation. This would enable Poisson s equation to be combined with a Maxwell-Boltzmann distribution. 

For a symmetric electrolyte (i.e., both co-ions and counterions have the same charge valence, Icr+I = lcr l = cr), its ionic concentration distributions for both anions and cations are assumed to follow the Boltzmann distributions [9], and hence, the local charge density, pe, is given by... 

Counterion condensation theory, however, does not provide a detailed picture of the distribution of the condensed Ions. Recent research using the Poisson-Boltzmann approach has shown that for cylindrical macroions exceeding the critical linear charge density the fraction of the counterions described by Manning theory to be condensed remain within a finite radius of the macroion even at infinite polyion dilution, whereas the remaining counterions will be infinitely dispersed in the same limit. This approach also shows that the concentration of counterions near the surface of the macroion is remarkably high, one molar or more, even at infinite dilution of the macromolecule. In this concentrated ionic milieu specific chemical effects related to the chemical identities of the counterions and the charged sites of the macroion may occur. 

It should also be noted that the full complexity of the PB equation is not always necessary. In many cases mobile ion densities are small enough that the exponential factors in the mobile ion distribution can be approximated by keeping only terms linear in the potential, leading to the linearized Poisson— Boltzmann (LPB) equation. In this case the ionic terms can be added to the diagonal of the coefficient matrix A, and extremization along any component direction is analytic because the surface is parabolic. Another advantage of LPB is the additivity of terms in the electrostatic potential, which enables contributions made by any part of a molecule to be determined. 

Verwey and Niessen treated the interfacial ion distribution as two back-to-back double layers, each described using the Poisson-Boltzmann approach developed by Gouy and Chapman for electrode-electrolyte interfaces [13]. For a 1 1 electrolyte, the excess ionic charge density on the aqueous side of the interface, qw, is given by... 

The functional depends on three types of fields, namely, surface fluctuations (R), electrostatic potential < (z, R), and the ionic concentrations rtf (z, R). Minimizing it with respect to < i and nf at a given f(R), one obtains Poisson-Boltzmann equations that describe the distribution of the electrostatic potential and ionic concentrations. Substituting the results into the density functional, one can minimize that with respect to f (R). This... 

The term weak adsorption implies that the entropic free energy of a chain is comparable to its electrostatic attraction energy to the interface. The chain is assumed to be Gaussian and its conformations are only weakly perturbed by interactions with the surface. This is the most severe approximation of the current model. We also assume that the polyelectrolyte-density profile is built up near the adsorbing surface without disturbing the electrostatic potential and ionic distribution near the interface prescribed by the Poisson-Boltzmann theory. A more general... 

There is a report on the colombic force field of a polyelectrolyte gel based on the analysis of dielectric relaxation spectra. High electron density of a polymer ion forms an extremely strong coulombic field in its vicinity (see Fig. 5) [11]. This distribution diagram is obtained by the numerical calculation based on the Poisson-Boltzmann equation. An ionic polymer gel possesses a static potential well. The coimter ions that dissociated firom the polymer ions then gather arormd them and form a restricted phase. Unlike free ions, these restricted cormter ions show dielectricity. From dielectric relaxation spectra, the insight on the coulombic force field around the polymer ions and microscopic morphology of the gel can be obtained [12-15]. 

Finite space charge densities occur in two diffuse regions in contact with both sides of the membrane. Their presence results from the disturbance of ionic distribution due to the action of the specific pumpings of ions. The thickness of these regions is supposed to be small in comparison with the cell dimensions. The Poisson-Boltzmann equations are written after linearization as follows, the axes x being perpendicular to the membranes... 

Kotin and Nagasawa [6] defined the counter-ion binding in analogy to the definition of Bjerrum on ion-pair formation [36]. That is, it is assumed that a polyion is placed in an infinite volume of a neutral salt solution of uni-uni valent type and the polyion is a rod of infinite length having a charge density N/L, Moreover, it is assumed that the ionic distribution around the rod is determined from the Poisson-Boltzmann equation. Then, if one plots the distribution of counter-ions PcC ) against the distance from the axis of the polyion r,... 

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