Effect of Multivalent Counterions

The average number of copolymer chains per aggregate, m, found in [171] is shown in Fig. 39 as a function of temperature for the systems containing counterions of different valence. It is seen that regardless of the valence of the counterion, the average aggregate size increases as the temperature is decreased. This is, of course, a quite expected result. One can conclude that for all the systems, the aggregation process becomes well pronounced in the 

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Hayakawa K, Kwak JCT. Study of surfactant—polyelectrolyte interactions. 2. Effect of multivalent counterions on the binding of dodecyltrimethylammonium ions by sodium dextran sulfate and sodium poly(styrenesulfonate) in aqueous solution. J Phys Chem 1983 87 506-509. 

The nature of the charged groups (sulfate or sulfonate) and the nature of the backbone of the polyelectrolyte have little effect on the phase diagram of polyelectrolyte/counterion mixtures. Therefore the electrostatic attractions can be considered as the principal driving force for phase separation of highly charged strong polyelectrolytes in the presence of multivalent counterions. A universal behavior is observed whatever the nature of polyelectrolyte or counterions. 

What are the limits of the approximated expression Eq. 4 Mainly those due to the mean-field nature of PB. For, say, 99 % of the studied systems, the ions are monovalent, ion-ion correlations in water can be safely ignored, and the standard expression is valid. This is no more the case in presence of multivalent counterions (or monovalent ions in solvent of low e). That opens to the fascinating concept of electrostatic attraction between hke-charged colloids, subject of numerous false analyses, debates, and controversies in the literature for 30 years. Figure 1 presents Monte Carlo (MC) simulations data for the force vs. separation law within the primitive model (two latex colloids and ions in continuous solvent) in presence of counterions of increasing valence. While the PB/DLVO prediction remains everywhere repulsive, the exact MC behavior deviates at intermediate separation and develops an attractive well deeper and deeper as the valence increases above 3. This non mean-field effect is due to the repulsions and correlations among the counterions localized in the intersticial region (discreteness of the condensed layer). The same type of colloidal attraction is responsible for a liquid-gas (concentrated solution-dilute solution) phase separation, observed... 

A macroscopic phase separation only occurs if the concentration of DNA is high enough. For dilute solutions another phenomenon takes place, namely the compaction or condensation of DNA. This can to some extent be considered as a phase separation for a single DNA molecule surfactant molecules bind to DNA not as individual ions but as self-assembly aggregates. These have a high charge and act effectively as multivalent counterions. As surfactant is progressively bound to DNA... 

The procedure used for testing the ideal Donnan theory is applicable to any model that decouples ionic effects from network elasticity and polymer/solvent interactions. Thus we require that nnet depend only on EWF and not C. While this assumption may seem natural, several models which include ionic effects do not make this assumption. For example, the state of ionization of a polymer chain in the gel and the ionic environment may affect the chain s persistence length, which in turn alters the network elasticity [26]. Similarly, a multivalent counterion can alter network elasticity by creating transient crosslinks. 

As y tends to decrease with z because multivalent ions compensate more charge In the Inner layer, and as this is a strong effect because of the fourth power dependence, the Schulze-Hardy rule remains explainable In fact, the precise dependence on z and the nature of the counterion now depends on the system, as Is experimentally found. This re-lnterpretatlon of classical knowledge followed from double layer studies with silver Iodide. 

Qualitatively, the phase diagram fits very well to the phase diagram known for single-chain polyelectrolytes, the phase boundaries are only slightly shifted. In principle, the parameter space for polyelectrol ffes has far more dimensions, such as the solvent quality parameter, the valency of monomers or counterions, and additional salt concentration in the system. Especially for multivalent counterions, one can expect an even more complex picture, since correlation effects are known to play an important role even for single chains. 

It can be shown that the addition of trace amounts of Z-ions to the solution leads to a rapid substitution of monovalent counterion in the star corona by Z-ions. This is due to their stronger attraction to the oppositely charged PE star polymer. Since a smaller number of Z-ions is needed to ensure the electroneutrality of the star interior, an increase in (i.e., in relative amount of Z-ions in the bulk of the solution) leads to a rapid decrease in the osmotic pressure inside the corona and, consequently, to a de-swelling of the PE star. This effect, of replacing monovalent counterions by multivalent ones is most pronounced at low salt concentrations (in the osmotic regime), where ... 

A polymer may modify this entropy contribution in a number of different ways. If it is ionic and has a similar charge, then we have a simple and relatively moderate electrolyte effect. If its charge is opposite, and it acts as a multivalent counterion, then the interaction becomes very strong since an association between polymer and micelle leads to a release of the counterions of both the micelles and the polymer molecules a very similar effect will be obtained in mixtures of two oppositely charged polymers. Indeed there is for such a case a lowering of the CMC by orders of magnitude. 

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