Oosawa treatment

We intend to present first a set of experimental data concerning the equilibrium properties (pK, activity of counterions) of polyelectrolytes in salt free solutions in a second part, these results are compared with theoretical values obtained when a rod-like model is adopted in the Lifson-Katchalsky, Manning and Oosawa treatments. The polyelectrolytes tested are respectively polysaccharides (alginic and pectic acids), carboxymethylated derivatives of cellulose (CMC), dextran (CMD), amylose (CMA), and sulphonic derivatives of polyvinyl alcohol. The essential difference between them is local rigidity. It is possible to vary by synthesis the degree of substitution and consequently the linear charge density our data are given for dilute aqueous solutions (2 X 10 N-5 X 10" N) at 25°C. 

Oosawa (1971) developed a simple mathematical model, using an approximate treatment, to describe the distribution of counterions. We shall use it here as it offers a clear qualitative description of the phenomenon, uncluttered by heavy mathematics associated with the Poisson-Boltzmann equation. Oosawa assumed that there were two phases, one occupied by the polyions, and the other external to them. He also assumed that each contained a uniform distribution of counterions. This is an approximation to the situation where distribution is governed by the Poisson distribution (Atkins, 1978). If the proportion of site-bound ions is negligible, the distribution of counterions between these phases is then given by the Boltzmann distribution, which relates the population ratio of two groups of atoms or ions to the energy difference between them. Thus, for monovalent counterions... 

NUCLEATION. The polymer self-assembly theory of Oosawa and KasaP treats nucleation as a highly cooperative and unfavorable event. Their kinetic theory for nucleation permits one to obtain information about the size of the polymerization nuclei, provided that two basic assumptions can be satisfied experimentally. First, the rate of nuclei formation is assumed to be proportional to the ioth power of the protomer concentration, with io representing the number of protomers required to create the nucleus. Second, the treatment deals only with that period during which the polymerization rate greatly exceeds the rate of protomer loss from the polymers (i.e., the initial stage of polymerization when the protomer concentration is the highest). 

The results are recorded in Figure 1 for monovalent and divalent counterions. The experimental results are compared to the free fractions of counterions calculated by the treatment of Oosawa (yo) [5] and to the activity of the counterions given by Manning (y ) [6] these parameters are given for infinite dilution by ... 

A) of counterions is condensed the fraction of bound counterions in the polyelectrolyte phase according to Oosawa is equal to(l—yo) the calculated values for the density of bound ions are given in Figure 2 for infinite dilution in this representation, Oosawa and Manning s treatments give the same results. 

Anticipating a more rigorous treatment in Chap. 2, we already give the standard expression often used for the depletion interaction [40, 54]. Consider two colloidal spheres each with diameter 2R, each surrounded by a depletion layer with thickness S. In that case the depletion potential can be calculated from the product of P = rib kT, the (ideal) osmotic pressure of depletants with bulk number density rib, times Vov, the overlap volume of the depletion layers. Hence the Asakura-Oosawa-Vrij (AOV) depletion potential equals ... 

An independent justification has been given by Oosawa [8] who has shown that additivity is a consequence of the properties of the integrated coulomb potential of the macroion. The approximate nature of the Rule is recognized in Oosawa s treatment, and it would seem, by analogy with the development of the present understanding of low molecular weight electrolyte mixtures, that departures from additivity when carefully assessed would be more truly revealing of the complex interactions between a polyelectrolyte and added salt. 

The points of Figure 26 were calculated from Equation (25) using a= 12 A and assimilating the potentiometric results to the free fraction y. Oosawa s treatment best fits the experimental data for 2>1. The ion-selectivity [20] (Figure 25) or the ultrasonic absorption [21] increases with the number of counterions whilst the effective ionisation approaches the y2 1 limit. 

The characteristic 2 = 1 is a particular value in different treatments (Oosawa, Lifson-Katchalsky at infinite dilution) and is explained here by a new approach. 

Activity coefficients for Na+ and Ca in salt free solution with various stoechiometric ratios [27] (Oosawa s treatment)... 

Fig. 27. Comparison between experimental data (y, y ) and theoretical ones (Oosawa s treatment) CMCc = 2xlO-3N ODS A 55 1.7 O 55 2.49.

By opposition, Oosawa s treatment gives very good agreement between experimental and calculated values when 2 > 1. 

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