Counterions valency

The effects of ion valence and polyelectrolyte charge density showed that at very low ionic strength found that when the counterion valence of added salt changes from monovalent (NaCl) to divalent (MgS04), the reduced viscosity decreases by a factor of about 4.5. If La(N03)3 is used, the reduced viscosity will be further decreased although not drastically. As for polyelectrolyte charge density, the intrinsic viscosity was found to increase with it because of an enhanced intrachain electrostatic repulsion (Antonietti et al. 1997). 

Note that if j/0 is not assumed to be large, T0 depends on both z and J/0. In this case the CCC is found to show a less sensitive dependence on the counterion valence than predicted by Equation (10). Example 13.1 examines this point. 

In the binding isotherm model [55], a polyion labeled P has N fixed charges. The poly ion is divided into sections [55]. There is no section of the polyion that has more than one binding site. Let the counterion valence be zc. Let. 1 denote a specific configuration of the nc counterions. In this configuration, the counterions bind to the poly ion and form the complex Dj(nc), and are described by the equilibrium [55]... 

The counterion valence is denoted by Z, the parameter Rg is the radius of gyration of RNA, i is the average charge per nucleotide after counterion condensation, and m is the number of nucleotides. 

There is a range of parameters other than polyelectrolyte charge density that has an important influence on the generated surface interactions, for instance, counterion valency and ionic strength of solution [121-123], the order of addition of polyelectrolyte and salt [124], polyelectrolyte concentration [125], presence of surfactants [31, 119, 126], and finally, the chemical structure of the polyelectrolyte itself [127]. A rich literature is available on these topics (see Ref. [115] and references therein). 

Attempts to utilize traditional DLVO approaches to quantify the Schulze-Hardy Rule have found limited and qualified success [23,59]. Although a qualitative agreement of the predicted dependence of ccc on counterion valence can be demonstrated, non-DLVO forces are typically ignored and analytical solutions of the DLYO equations predict unrealistically large ccc values [23,59]. 

The influence of counterion valence on the double layer thickness is described by the valency rule of Schulze and Hardy. It basically predicts that if a monovalent counterion is changed to a divalent counterion, the thickness of the double layer decreases by half and if the divalent counterion is changed to a trivalent ion, the thickness of the double layer decreases by three-quarters (Fig. 9.2). The relative amounts of counterions required to induce flocculation are 100 for a monovalent, 2 for a divalent, and 0.04 for a trivalent ion. 

This mathematical condition defines a critical separation in terms of a critical double layer thickness, k whidi can then be used to relate the critical salt concentration to the valence of the counterion. The result is the relation between the salt concentrations responsible for the CCC s being proportional to the counterion valences to the —6 power. 

According to O Brien and White the mobility maximum depends on xa as given in flg. 4.28. With increasing counterion valency the maximum becomes lower and is observed at lower (not shown). 

Ray J, Manning GS. Effect of counterion valence and polymer charge density on the pair potential of two polyions. Macromolecules 1997 30 5739-5744. 

Up to now only monovalent ions have been investigated. For multivalent ions the prediction of the PB theory is that for the distribution function P(r) only the product of the Manning parameter and the counterion valence v matters. Therefore a system of monovalent ions at ln = 3a is claimed to have the same distribution function as a system of trivalent ions at B = lo It will now be shown that this statement is an artifact of the PB approximation. Figure 9 shows examples of systems that are complementary in the described sense. Not only is the condensation enhanced as compared to PB theory, but the enhancement is stronger for the case involving multivalent ions. Two different reasons may be suggested to explain this effect ... 

FIG. 13 Osmotic coefficient f> as a function of density n for different valences. Heavy dots mark the measurements, while the solid lines are fits that merely serve to guide the eye. The dotted lines are the prediction of PB theory. From top to bottom the counterion valence v varies as 1, 2, 3, which gives the same value of v as the curves in Figure 12. The error in the measurement is roughly as big as the dots themselves. 

Theoretical predictions for a quadratic increase of the induced dipole moment in the presence of divalent counterions in comparison to that in the presence of monovalent counterions have been made [71,84,97,99]. An increase of the polarizability with increasing counterion valence is also predicted by Monte Carlo simulation, applied in the investigation of rodlike polyelectrolytes in solution [100]. Experimentally, however, the degree of DNA orientation has been found only slightly higher (ca. 5%) in the presence of divalent counterions than in monovalent solutions [96]. This means, according to the authors of Ref. 96, that the increased repulsion between divalent counterions counterbalances the expected increase in polarizability because of the increased charge fluctuation. 

Figure 10. Influence of counterion valence (X , n = i, 2) on the time dependence offemtosecoridy photoinduced, electron-transfer trajectories in aqueous ionic solutions (X , nCl, X = Na , Mg ). The upper part represents the absorption signal rise time at 1.77 eV following the femtosecond UV excitation of aqueous Cl (2X4 eV). The difference in the signal rise times of Na and Mg is due to the balance between two electronic transitions e iR e hyd fCl e X ...

For an annealing star polyion, the degree of ionization, a, becomes a function of the counterion valence, Z, and of the Z-ion bulk concentration, Ci,z, due to a progressive substitution of the monovalent counterion (/T " for a polyacid) by Z-ions and the corresponding increase in local pH inside the star. Implementation of the Donnan rule (51), together with the mass action law (5) and the osmotic balance (52), provides a scaling dependence for the star size ... 

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