Counterion condensation threshold

The characteristic branching parameter (grafting density), n/m = specifies the onset of counterion localization inside the molecular brush. Note that in the osmotic regime, the spacers get fully extended, /t m. It is therefore not surprising, that the counterion localization in a cylindrical molecular brush coincides (in scaling terms) with the Manning condensation threshold [25] for a charged cylinder, qh = 1. 

Mean field theories that implement a spherically symmetric distributions of counterions assume that the (effective) charge density on the star branches is below the Manning condensation threshold. In this limit, one can neglect the angular correlations between the positions of the arms of the star and those of its counterions. 

A different effect occurs with the use of polycarboxy-lates in combination with zeolites. Small amounts of polycarboxylates or phosphonates can retard the precipitation of sparingly soluble calcium salts such as CaCOs (the threshold effect ). As they behave as anionic polyelectrolytes, they bind cations (counterion condensation), and multivalent cations are strongly preferred. Whereas the pure calcium salt of the polymer is almost insoluble in water, mixed Ca/Na salts are soluble, i.e. only overstoichiometric amounts of calcium ions can cause precipitation. Polycarboxylates are also able to disperse many solids in aqueous solutions. Both dispersion and the threshold effect result from the adsorption of the polymer on to the surfaces of soil and CaCOs particles, respectively. 

If the inequality [eq. (XI.20)] is violated, the counterions condense on the chain and decrease the charge density, bringing it back to the threshold value. Another limit to the assumption of negligible screening is given by the residual ionic content of the water itself. If we use pure water (completely salt free) as the solvent, we still have a certain number of and OH ions... 

A noteworthy conclusion is that for a polyion in aqueous solution, and in which kL is held fixed, the equilibrium state of the macromolecule is stable provided 0=1— c 1 [40]. Furthermore, the number of condensed counterions, as well as the threshold value for linear charge density, is unchanged for a polyion whose length is of the order of k 1 [40]. 

Experimentally, it is observed that divalent ions can trigger condensation in alcohol-water mixtures. In our model, both A and q also depend on the dielectric constant of the solvent, so that the attraction increases and the repulsion decreases with decreasing e this lowers the threshold value of Zmm. We note, however, that the effect of alcohol could be much more subtle and could depend on microscopic details such as the structure of water near the DNA and counterions that are neglected in our model. 

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