Counterion correlation

The PRISM (Polymer-Reference-Interaction-Site model) theory is an extension of the Ornstein-Zernike equation to molecular systems [20-22]. It connects the total correlation function h(r)=g(r) 1, where g(r) is the pair correlation function, with the direct correlation function c(r) and intramolecular correlation functions (co r)). For a primitive model of a polyelectrolyte solution with polymer chains and counterions only, there are three different relevant correlation functions the monomer-monomer, the counterion-counterion, and the monomer-counterion correlation function [23, 24]. Neglecting chain end effects and considering all monomers as equivalent, we obtain the following three PRISM equations for a homogeneous and isotropic system in Fourier space ... 

In the next two figures we discuss the pair-correlation functions as obtained from the two-density theory and computer simulations. First, in Fig. 3 we compare the counterion-counterion pair-distribution function as obtained theoretically (lines) and from simulations (symbols). The numerical calculations apply to cp = 0.0001 and ce = 0.005 mol dm-3 the results show that the theory underestimates the counterion-counterion correlation. Next, in Fig. 4 the macroion-counterion pair-distribution is shown for the same set of parameters. Finally, in Fig. 5 the macroion-macroion pair-distribution functions are calculated by both theoretical approaches at cp = 0.0001 mol dm-3 solution and for zp = —10 and —30. 

Liao Q, Dobrynin AV, Rubinstein M (2006) Counterion-correlation-induced attraction and necklace formation in polyelectrolyte solutions theory and simulations. Macromolecules 39 (5) 1920-1938. doi 10.1021/ma052086s... 

Diehl A, Carmona HA, Levin Y (2001) Counterion correlations and attraction between like charged macromolecules. Phys Rev E 64 011804 1 011804 6... 

Figure 5 displays the counter- and coion correlation functions (equivalent to the concentration profile per bulk concentration) according to the Gouy-Chapman, Dehye-Hiickel, and apparent Dehye-Hiickel solutions (Eqs. [23], [32], and [93]) for a bulk symmetric electrolyte at 0.1 M near a planar surface with charge density ct = 0.01 ealK. Far from the surface, where the potential is small, the correlation functions approach unity and the amounts of counter- and coions are equal close to the surface the number of counterions increases and the number of coions decreases as a result of the electrostatic attraction to and repulsion from the surface, respectively. The apparent DH counterion correlation function is, as can be seen, a marked improvement over the standard DH result. 

Dunn C M, Robinson B FI and Leng F J 1990 Photon-correlation spectroscopy applied to the size characterization of water-in-oil microemulstion systems stabilized by aerosol-OT effect of change in the counterion Spectrochim. Acta. A 46 1017... 

Further evidence for an increased efficiency of complexation in the presence of micellar aggregates with bivalent metal counterions is presented in Table 5.4. The apparent rate constants of the reaction of 5.1c with 5.2 in the presence of micelles of Co(DS)2, Ni(DS)2, Cu(DS)2 and Zn(DS)2 are compared to the rate constants for the corresponding bivalent metal ion - dienophile complexes in the absence of micelles. The latter data are not dependent on the efficiency of the formation of the catalyst - dienophile complex whereas possible incomplete binding will certainly be reflected in the former. The good correlations between 1 and and the absence of a correlation between and... 

These examples indicate that the (Z)-syn,(E)-antt correlation should be considered to be a rule with many exceptions. Two explanations may be given in order to rationalize the manifold stereochemical results in aldol additions. Firstly, it seems plausible that the many different reaction conditions and starting materials (e.g., various types of enolates, counterions, etc.) may cause the aldol addition to follow different reaction mechanisms, so that different types of transition states are involved. Secondly, in a single type of transition state model, the reactants may have different orientations to each other, so that the formation of different stereoisomers may result even for one and the same transition state model. 

A fundament of the quantum chemical standpoint is that structure and reactivity are correlated. When using quantum chemical reactivity parameters for quantifying relationships between structure and reactivity one has the advantage of being able to describe the nature of the structural influences in a direct manner, without empirical assumptions. This is especially valid for the so-called Salem-Klopman equation. It allows the differentiation between the charge and the orbital controlled portions of the interaction between reactants. This was shown by the investigation of the interaction between the Lewis acid with complex counterions 18> (see part 4.4). 

The results of the simple DHH theory outlined here are shown compared with DH results and corresponding Monte Carlo results in Figs. 10-12. Clearly, the major error of the DH theory has been accounted for. The OCP model is greatly idealized but the same hole correction method can be applied to more realistic electrolyte models. In a series of articles the DHH theory has been applied to a one-component plasma composed of charged hard spheres [23], to local correlation correction of the screening of macroions by counterions [24], and to the generation of correlated free energy density functionals for electrolyte solutions [25,26]. The extensive results obtained bear out the hopeful view of the DHH approximation provided by the OCP results shown here. It is noteworthy that in... 

Structural analyses of the triiodide ion, in crystals of this ion with various counterions, show that the I3 unit is always linear, or nearly so, and that there is considerable variety in the I I distances, which range from 2.67 A (covalent bond length) to 4.30 A (nonbonded contact distance). Further, there is strong correlation between the two I I distances in the I I I ion. Very similar correlations were obtained for the S-S S grouping in the thia-thiophthenes 37, for the O-H O groupings in a number of hydrogen-bonded... 

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