Electrostatic interactions counterion distribution

In order to resolve these challenges, it is essential to account for chain connectivity, hydrodynamic interactions, electrostatic interactions, and distribution of counterions and their dynamics. It is possible to identify three distinct scenarios (a) polyelectrolyte solutions with high concentrations of added salt, (b) dilute polyelectrolyte solutions without added salt, and (c) polyelectrolyte solutions above overlap concentration and without added salt. If the salt concentration is high and if there is no macrophase separation, the polyelectrolyte solution behaves as a solution of neutral polymers in a good solvent, due to the screening of electrostatic interaction. Therefore for scenario... 

Since the ion association in micellar systems is due to the long-range electrostatic interactions it is preferable to describe the ion distribution around the charged micelle explicitly. This is customarily made in a model where the water is approximated by a dielectric continuum in which the counterions are distributed. Within such a model, there is a multitude of different degrees of refinements. We choose here to describe the, in our view, most straightforward scheme and postpone a discussion of the possible modifications to a later stage. 

All these factors combined are of great importance when considering PE adsorption to charged surfaces. We distinguish between physical adsorption, where chain monomers are attracted to surfaces via electrostatic or nonelectrostatic interactions, and chemical adsorption, where a part of the PE (usually the chain end) is chemically bound (grafted) to the surface. In all cases, the long-ranged repulsion of the dense layer of adsorbed PEs and the entropy associated with the counterion distribution are important factors in the theoretical description. 

We have already learnt that polyelectrolytes are much more soluble than the corresponding uncharged polymer, which we attribute to the entropy of the counterion distribution confining polymer molecules to part of the system costs little entropy due to the low number of entities. On the other hand, there is a large entropy loss on confining the (much more numerous) counterions. In mixed polymer systems, we see many consequences of the electrostatic interactions due to net charges. One is the low tendency to phase separation in a mixed solution of one nonionic and one ionic polymer in the presence of added electrolyte, this inhibition of phase separation is largely eliminated and typical polymer incompatibility is observed. 

The use of molecular dynamics to study the electric double-layer structure started a little over a decade ago, with the hope of determining more accurate structures because the classical description of an electric double layer based on the Poisson-Boltzmann equation is accurate only for low surface potential and dilute electrolytes. The Poisson-Boltzmann equation only considers the electrostatic interactions between the charged surface and ions in the solution, but not the ion-ion interactions in the solution and the finite molecule size, which can be taken into account in molecular dynamics simulations. It was shown [6, 7] that the ion distribution in the near-wall region could be significantly different from the prediction of classical theory. Typical molecular dynamics simulation results of counterion and co-ion concentrations in a nanochannel are shown in Fig. 2a. The ion distribution obtained... 

Equation [11] shows that the distribution constant for A depends on the ratio of activity coefficients and the concentration of the counterion B in the resin and external phases, respectively. When Cb,e increases, the main effect is a decrease in the electrostatic interaction to the fixed charges in the resin phase, which in turn affects the individual activity coefficients. It is therefore interesting to evaluate the ratio of activity coefficients in eqn [11]. Since eqn [6] applies individually to all ions in the electrolyte solution that are in equilibrium with the resin phase, a relation between Ka b, and Da can be found. 

It is obvious that both intra-molecular and inter-polymer phenomena in polyelectrolyte solutions are dominated by coulomb forces. Repulsive interactions would be expected to diminish aggregation. At the same time, extended chain dimensions and the long-range character of electrostatic effects can promote forms of ordering unique to polyelectrolytes. Hydrodynamic, spectroscopic and thermodynamic methods have all been brought to bear on the coupled problems of conformation, counterion distribution and inter-polymer ordering in polyion solutions. These approaches are well represented in Part III by the works on Paoletti, Berry and Jamieson. 

Subdivision of counterions into condensed and uncondensed populations according to the Manning/Oosawa depiction is not unique. For example, a hypothetical thermodynamically bound counterion population can be defined to account for the deviations of polyelectrolyte solutions from thermodynamic ideality (35) this population is not equal to the condensed population. One can also use different definitions of condensed. The inflection point in the radial ion distribution, the Bjerrum length, and the radial distance over which the electrostatic interaction energy decays to kT have all been employed as alternative criteria for defining a condensed fraction (50). 

Consider a polyelectrolyte chain with the degree of polymerization N, bond length b, and fraction / of the charged groups on the polymer backbone in a medium with the dielectric permittivity e. At very low polymer concentrations, almost all counterions are distributed outside the volume occupied by a chain. At these polymer concentrations, the electrostatic interactions between charged groups and chain s elasticity control chain dimensions. 

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