Ion Association of Polyions with Counter-ions

The expression (2.76) for the osmotic coefficient was previously derived by Lifson and Kachalsky (1954) and by Manning (1969) but shown by Manning to be valid only when f 1. For polyelectrolyte solutions free from added salt with f 1 the limiting value of the osmotic coefficient is (p= 1—Vaf. Only when the fixed charges on the polyelectrolyte are densely spaced, that is, when the average distance bi between them is smaller than 2q so that 1, would the electrical potential cause the (univalent) counter-ions to bind to ( condense on (Manning 1969)) the fixed ions, thereby reducing the effective value of to the critical value of unity. 

The Donnan salt exclusion parameter f expresses a property of polyelectrolyte solutions. It describes the bias when a salt is present in the bulk of the solution against the presence of mobile ions in the vicinity of the polyelectrolyte. In the case of the two phase system of a cross-linked ion exchange gel and an outer solution the exclusion is directly measurable experimentally. If the concentration of the salt in the external solution is CEex and that in the vicinity of the polyelectrolyte is CEnear, then the limiting value of Tfor a concentration Cpoiy of the polyelectrolyte tending to zero is formally defined by  

Manning (1969) showed that when f 1 then F= V2 (1 — V2 ) whereas if 1 then r= l/4f. He compared these expectations with experimental data available to him for a variety of polyelectrolytes sodium polyvinyl sulfate and sodium polyacrylate (of various degrees of neutralization a) and potassium polyphosphate and DNA, having values ranging from 0.29 to 4.20, with good results. Also the expectations for the osmotic coefficients of the sodium polyacrylate and polymethacrylate were well vindicated. 

The course of the titration is described by the following operative expression according to Harris and Rice (1954)  

Here/ is the degree of binding of the sodium counter-ion, is the intrinsic dissociation constant for the acid and Kip is that of the ion pair (the reciprocal of its association constant). This treatment assumes random distribution of the three types of sites along the polyelectrolyte chain, but restrictions can be introduced, such as requiring an ionized site to be adjacent to the one where ion pairing takes place. Lif-son and Katchalsky (1948) took into account the mutual interaction of neighboring ionized sites, but the final result was the same as Eq. (2.78), i.e., as for the case where no such interaction was considered. 

---