Counterion condensation Manning

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. 

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

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

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]. 

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. 

Manning derived the following expression for based on counterion condensation for infinite dilution... 

Consider a counterion of absolute valence Zq that is held fixed at a distance r from the polyion. The polyion is modeled within the framework of the Manning line model [31] discussed in Section Chapter II.A. The idea here is that the reduction of a polyion charge is dictated by how far the counterion is from that charge. Hence, the renormalized charge due to counterion condensation is written as qnet = (1 — %(r))q, where q is the bare charge [43]. 

The Manning counterion condensation line model provides insights into the mean-square number fluctuations of the condensed counterions, ((A0)2), where 0 is the number of condensed counterions per polyion charge [50], Denote by 0O the value of 0 for which the total polyelectrolyte free energy G per charge is a minimum, that is, (0G/00)e=e vanishes. Expanding G in powers of 0 0O to quadratic order leads to [50]... 

A connection between Manning counterion condensation [31] and Kosterlitz-Thouless transition [58a] was conjectured by Mohanty, and supported by phenomenological arguments [58b]. An explicit calculation using nonlinear PB confirmed the hypothesis [59]. 

G. S. Manning, Biophys. Chem., 7, 95 (1977). Limiting Laws and Counterion Condensation in Polyelectrolyte Solutions. 4. Approach to Limit and Extraordinary Stability of Charge Fraction. 

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