Counterion condensation site-bound counterions

More recently, Manning has extended his theory to include those counterions that are "territorially bound" or trapped in the domain of the polyelectrolyte, but are somewhat free to move along the polyion.- Counterions that are neither condensed site-bound nor territorially-bound are in the ion atmosphere, along with the coions, if simple salt is added. The idea of counterion condensation has recently received substantiation from theoretical investigations of the Poisson-Boltzmann equation for polyelectrolyte solutions.-... 

However the apparent charge of the counterions at low added salt is not as negative as expected, since it should be close to the charge of the polyion (about -100). Therefore some degrees of freedom remain for the cations which are allowed to move rapidly along the polyion. Indeed n.m.r. experiments have shown, at least for cobalt, that less than 40 percent of the condensed ions are site-bound (13) ... 

The theoretical prediction on condensation of Manning following which for 2 > 1 the density of bound sites should increase linearly with the charge density finds some confirmation in our work in all experiments the selectivity appears with monovalent counterions for a critical value of X (1

The condensed counterions are assumed to have negligible mobility along the length of the polyion. There is no implication here that the condensed ions are bound for long times to discrete sites or otherwise rigidly fixed to the polyion, only that their mobility is small compared to that of the uncondensed counterions. Justification for this assumption is its plausibility [47] and the close agreement with measurement of subsequently predicted transport properties. 

As for ion pair formation [15], PC, PC2 and PC3 represent three states of hydration of the counterion-polyion complex with the volume changes AV, and A V,2 associated respectively to reaction PCi PC2 and PC2- PC3. However, in this model, the formation of PCj from P and C is not considered as an equilibrium obeying the mass action law because Manning [16] has shown that even at infinite dilution a certain number of charged sites are neutralized by condensed counter-ions. It will be assumed that only those condensed counter-ions are active as far as ultrasonic absorption is concerned, while the free counter-ions do not take part to the relaxation processes. Free and bound counter-ions may however exchange very rapidly. As is shown ri Section 6 the model represented by Equation (5) is characterized by two concentration independent relaxation frequencies and two relaxation amplitudes proportional to c. 

Equation (6) has been verified to a high degree of accuracy for simple electrolytes. It assumes a complete ionization of the salt at infinite dilution. An examination of Table II reveals that for each of the polyelectrolytes studied in this work the values of (Fp )c depend on the counter-ion. This result means that the additivity law does not hold for polyelectrolyte solutions. A simple explanation for this behaviour is provided by Manning s theory which states that even at infinite dilution part of the counter-ions remain condensed on the polyions. On the other hand, dilatometry [2], refractometry [3] and ultrasonic absorption [9, 10] have shown that part of the condensed counterions and of the sites on which they are bound are dehydrated. Therefore, the value ()c of the apparent molal volume of the polyion P, as obtained from Equation (6) >vill include the volume change dVc associated with the binding of C. This is why the subscript C was placed on the apparent molal volume of the polyion P in Equation (6). If we call (Fp )true> the true apparent molal volume of the polyion P it may be assumed that (Fp )true (Fp)tma since there are numerous evidences [9,10, 18, 4c] that a negligible volume change is associated with the condensation of TMA" ion (this ion... 

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