The distribution of counterions

Oppositely charged ions are attracted to each other by electrostatic forces and so will not be distributed uniformly in solution. Around each ion or polyion there is a predominance of ions of the opposite charge, the counterions. This cloud of counterions is the ionic atmosphere of the polyion. In a dynamic situation, the distribution of counterions depends on competition between the electrostatic binding forces and the opposing, disruptive effects of thermal agitation. 

Ion binding is affected by the size and charge of the counterion, the charge and conformation of the polyion, and states of hydration. We will examine these effects in some detail. 

This ratio is related linearly to the degree of polymerization n. In the case of a poly(acrylic acid) where n = 1000 and / = 20 nm, this ratio works out at 35. Thus, many of the counterions must enter the region of the polyion. Even when 90 % of the counterions are within the polyion this ratio is still high with a value of 3-5. A similar calculation for the rod-like random coil gives an energy ratio of 26 and similar arguments apply. 

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 

If V is the mean effective volume occupied by a single polyion and N is the number of polyions then v = nrH and VJN = nR l-, thus r /R = Nv/V=9diTid 

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The distribution of counterions at the charged surface and in the bulk solution was studied combining the radiotracer technique with foam fractionation technique. The relative adsorbability of counterions was determined from the ratio of mole ratios in an adsorbed and in the solution phases. The relative adsorbability was CT CH3COO Br ClOf NOs = 1 0.7 1.5 3.0 4.1 in solution of dodecylammo-nium(0.011 mole/liter), Cl Br ClOf NOf SO/2 = 1 1.2 2.8 1.6 2.3 in solution of dodecyltrimethylammo-nium(0.015 mole/liter), and Na+ Ca2+ = 1 210 in solution of sodium dodecylsulfate(0.006 mole/liter). 

The distribution of counterions in the vicinity of a single charged surface, at a high surface potential (rp = 0.1, Figure 3 a) is represented as a function of the distance x... 

We set the electric potential j/ r) to zero at points where the volume charge density p(r) resulting from counterions equals its average value (—zen). We assume that the distribution of counterions obeys a Boltzmann distribution, namely. 

Here we have assumed that the relative permittivity is assumed to take the same value inside and outside the membrane. We also assume that the distribution of counterions n x) obeys a Boltzmann distribution, namely,... 

Here we have assumed that the relative permittivity is assumed to take the same value inside and outside the membrane. We also assume that the distribution of counterions n(x) obeys Eq. (18.4) and thus the charge density PeiW is given by Eq. (18.5).Thus, we obtain the following Poisson-Boltzmann equations for the scaled potential y x) = ze l/(x)/kT ... 

Currently no adequate quantitative theory of the discrete-ion potentials for adsorbed counterions at ionized monolayers exists although work on this problem is in progress. These potentials are more difficult to determine than those for the mercury/electrolyte interface because the non-aqueous phase is a dielectric medium and the distribution of counterions in the monolayer region is more complicated. However the physical nature of discrete-ion potentials for the adsorbed counterions can be described qualitatively. This paper investigates the experimental evidence for the discrete-ion effect at ionized monolayers by testing our model on the results of Mingins and Pethica (9, 10) for SODS. The simultaneous use of the Esin-Markov coefficient (Equation 3) and the surface potential AV as functions of A at the same electrolyte concentration c yields the specific adsorption potentials for both types of adsorbed Na+ ions—bound and mobile. Two parameters which need to be chosen are the density of sites available to the adsorbed mobile Na+ ions and the capacity per unit area of the monolayer region. The present work illustrates the value... 

A schematic representation of the inner region of the double layer model is shown in Fig. 1. Figure lb describes the distribution of counterions and the potential profile /(a ) from a positively charged surface. The potential decay is caused by the presence of counterions in the solution side (mobile phase) of the double layer. The inner Helmholtz plane (IHP) or the inner Stem plane (ISP) is the plane through the centers of ions that are chemically adsorbed (if any) on the solid surface. The outer Helmholtz plane (OHP) or the outer Stem plane (OSP) is the plane of closest approach of hydrated ions (which do not adsorb chemically) in the diffuse layer. Therefore, the plane that corresponds to x = 0 in Eq. (4) coincides with the OHP in the GCSG model. The doublelayer charge and potential are defined in such a way that ao and /o, op and Tp, and <5d and /rf are the charge densities and mean potentials of the surface plane, the Stem layer (IHP), and the diffuse layer, respectively (Fig. 1). 

The electrostatic charges of surfactants seriously affect the localization of host molecules in the water pool. Monte Carlo simulation in which ionic reversed micelles are treated as spherical entities showed the presence of the electrical double layer in the interface of the water pool, and the distribution of counterions followed the Poisson-Boltzmann approximation [51]. Mancini and Schiavo [52] assumed recently, by the yield of halogenation, that the specific interactions between bromide or chloride ions and an ammonium head-group in cationic reversed micelles keep the ions in a defined position on the interface. 

Quadrupole relaxation reflects both the distribution of counterions and the dynamics of counterions associated with an aggregate. A full interpretation is typically difficult as information is not available on the quadrupole interaction of the associated state and is difficult to obtain independently because the relaxation dispersion is typically not observed. 

Chemical shifts give a direct picture of the distribution of counterions between different environments. 

In Fig. 7, the radial density profiles of charged monomers and of counterions (normalized for one branch) are shown for stars with different numbers of arms, p (in a salt-free system). For small p, the distribution of counterions is fairly uniform, whereas that for the stars with a large number of arms both distributions almost coincide. 

The distribution of counterions is essentially deter mined by their concentration and the geometry of the water core. Thus, in the case of large droplets, the assumption can be... 

In AOT microemulsions, where the aqueous core of the droplets also contains counterions, a considerable part of the dielectric response to the applied fields originates from the redistribution of the counterions. As mentioned in Sec. II, the counterions near th charged surface can be distributed between the Stem layer and the Gouy-Chapman diffuse double layer (28-31). The distribution of counterions is essentially determined by their concentration and the geometry of the water core. Thus, for very large droplets the diffuse double layer peters out and the polarization can be described by the Schwarz model (32). However, as already mentioned, this approach is more relevant to the dielectric behavior of emulsions than to that of micro emulsions. 

Consider a positively charged polyelectrolyte chain (see Figure 9). Let us place the center of the coordinate system at the chain s center of mass with the z-axis pointing along the direaion of the chain elongation. Counterions with valence cj = -l are distributed around a polyion with average local concentration qon(f). The distribution of counterions aeates distribution of the electrostatic potential P(r), which satisfies the Poisson equation ... 

Debye and Hiickel regarded all the nonideality as arising from the electrostatics, and none from short-ranged interactions. The nonideaiity reflected in the activity coefficient is modelled as arising from the nonuniformity in the distribution of counterions and co-ions that the central ion creates in its neighborhood. That is, an excess of counterions and a depletion of co-ions surround the central ion, in a spherically symmetrical way. 

Figure 6.9 Inhomogeneous charge distribution on the surface of DNA. Phosphates are positively charged, counterions are negatively charged. The strength of attraction depends on the distribution of counterions and on the ratio of the axial shift Az to the helical pitch H. [Reprinted with permission from A. A. Komyshev and S. Leikin, Phys. Rev. Lett., 82, Art. No. 4138. Copyright (1999) the American Physical Society]...

With isotopic labelling experiments, not only the polyion-polyion structure factor, but also other partial structure factors such as counterion-counterion structure factors, have been determined [137, 138]. First experiments provide evidence that the counterion-counterion structure factor also exhibits a peak at nearly the same position as the peak in the polyion-polyion structure factor. This is interpreted to originate from condensation of counterions onto the polyions. A similar observation was also made using SAXS by studying poly-potassiumstyrenesulphonate (KPSS) and NaPSS [139], where the scattering curve is mostly determined by the distribution of counterions because of their high contrast factor. 

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. 

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