Counterion Manning

Another result concerns the charge stoichiometry within one complex. For excess DNA, Z /Z = 0.8 whereas for excess polycation Z /Z = 3. It should be noted that these numbers do not represent the effective charge but the chemical charge of the complexes, which may include stericaUy shielded charges in the interior of the cylindrical brushes and ignores possible reduction of charges by counterions (Manning condensation, ion pair formation). 

Although reaching the same predictions for the fraction of condensed counterions, Manning s derivations of counterion condensation are superior to the one just given (Oosawa s) in the sense that Manning s derivations allow in the region of free ions to be screened in the Debye-Hiickel approximation according to... 

The properties of polyelectrolyte solutions depend strongly on the interactions between the polymers and the surrounding counterions. Manning s theory of counterion condensation predicts that a certain quantity of counterions condenses onto a polymer, whose charge density exceeds a critical value [44]. This leads to an effective decrease in the polymer charge. The macroscopic properties of the polyelectrolyte are not determined by its bare charge but by an effective charge. In particular, the flexibility and hydrophobicity of the polyelectrolyte chain, the... 

Manning s theory does not take the local effective dielectric constant into consideration, but simply uses the a value of bulk water for the calculation of E,. However, since counterion condensation is supposed to take place on the surface of polyions. Manning s 2, should be modified to E, by replacing a with aeff. The modified parameters E, is compared with E, in Table 1, which leads to the conclusion that the linear charge density parameter calculated with the bulk dielectric constant considerably underestimates the correct one corresponding to the interfacial dielectric constant. 

The theory of counterion condensation is implicit in Oosawa (1957) but the term was coined later (Imai, 1961). The phenomenon was demonstrated by Ikegami (1964), using refractive index measurements of the interaction between sodium and polyacrylate ions. It has since been confirmed for many mono-, di- and trivalent counterions and polyionic species (Manning, 1979). 

Manning (1969) suggested that there is a critical charge density above which counterions condense on the surface of the polyion. This phenomenon is most clearly illustrated by the simple case of iiffinite dilution. As 0->O in equation (4.11), the graph of P against Q falls into two parts about the critical point 2=1 ... 

Theories of counterion condensation have been reviewed by Manning (1979, 1981) and Satoh, Komiyama lijima (1984) have extended the theory. 

From all of this discussion it is apparent that, as Manning (1979) said, the binding between counterion and polyion can range from atmospheric to covalent site binding. 

Manning, G. S. (1969). Limiting laws and counterion condensation in polyelectrolyte solutions. 1. Colligative properties. Journal of Chemical Physics, 51, 924-33. 

Manning, G. S. (1979). Counterion binding in polyelectrolyte theory. Accounts of Chemical Research, 12, 443-9. 

The Manning theory(16,17) predicts a critical value c above which counterions are condensated on polyion. In the case of monovalent ions, =1 and for > c, the chain expansion... 

G.S. Manning, Counterion condensation on a helical charge lattice. Macromolecules 34, 4650-4655... 

A polyelectrolyte solution contains the salt of a polyion, a polymer comprised of repeating ionized units. In dilute solutions, a substantial fraction of sodium ions are bound to polyacrylate at concentrations where sodium acetate exhibits only dissoci-atedions. Thus counterion binding plays a central role in polyelectrolyte solutions [1], Close approach of counterions to polyions results in mutual perturbation of the hydration layers and the description of the electrical potential around polyions is different to both the Debye-Huckel treatment for soluble ions and the Gouy-Chapman model for a surface charge distribution, with Manning condensation of ions around the polyelectrolyte. 

This paper reports the systematic study of the apparent anhydrous micellar weights of the three principal bile salts found in man and their glycine and taurine conjugates in respect to bile salt concentration, counterion concentration, temperature, pH, and urea concentration. On the basis of these studies, model structures of bile salt micelles are proposed. Sodium dehydrocholate, a triketo bile salt, was also studied but was not found to form micelles. 

Counterions are necessary to ensure electroneutrality in polyelectrolyte solutions. Therefore, it can be energetically advantageous if a fraction of counterions are situated in the vicinity, or at the surface, of the polyion in order to reduce the charge of the polyion. To answer the question under which conditions this occurs, the concept of the counterion condensation has been introduced by Fuoss, Katchalsky and Lifson [98],Alexandrowicz and Katchalsky [99] or Oosawa [100] and subsequently theoretically developed by Manning [101-108]. 

Fig. 14. Concentration dependence of the counterion activity coefficients, fa, of DADMAC, PDADMAC with different molar masses, and comparison with theoretical predictions (T=20 °C DADMAC PDADMAC with Mn 12,000 g mol"1 72,000 g mol"1 170,000 g mol-1 .Manning ------Gueron) (Data taken from [38])...

Ray, J. and Manning, G.S. (1994) An attractive force between two rodlike polyions mediated by the sharing of condensed counterions. Langmuir, 10,2450-2461. 

Recently, the stiff-chain polyelectrolytes termed PPP-1 (Schemel) and PPP-2 (Scheme2) have been the subject of a number of investigations that are reviewed in this chapter. The central question to be discussed here is the correlation of the counterions with the highly charged macroion. These correlations can be detected directly by experiments that probe the activity of the counterions and their spatial distribution around the macroion. Due to the cylindrical symmetry and the well-defined conformation these polyelectrolytes present the most simple system for which the correlation of the counterions to the macroion can be treated by analytical approaches. As a consequence, a comparison of theoretical predictions with experimental results obtained in solution will provide a stringent test of our current model of polyelectrolytes. Moreover, the results obtained on PPP-1 and PPP-2 allow a refined discussion of the concept of counterion condensation introduced more than thirty years ago by Manning and Oosawa [22, 23]. In particular, we can compare the predictions of the Poisson-Boltzmann mean-field theory applied to the cylindrical cell model and the results of Molecular dynamics (MD) simulations of the cell model obtained within the restricted primitive model (RPM) of electrolytes very accurately with experimental data. This allows an estimate when and in which frame this simple theory is applicable, and in which directions the theory needs to be improved. 

---