Manning’s counterion condensation theory

The derivation of the equations for the electrostatic part of the enthalpy function of a polyelectrolyte solution will be given hereafter within the frame of G. S. Manning s counterion condensation theory. It may be useful to recall the main assumptions (and limitations) of the theory,17 so that further assessment of the results is made easier ... 

Manning s counterion condensation theory, " developed for a line charge in a dilute electrolyte (kd 0), predicts... 

Figure 28 A plot of the ADH//NLDH fraction of surface charge neutralized by a bulk 1 1 electrolyte according to Eq. [265] for a charged cylinder as a function of the linear charge density zt,a ioi several values of the variable Kpu given at right. The prediction of Manning s counterion condensation theory (fu) for a line charge is shown by the dashed line.

The importance of the result is twofold. First, it opens up a novel path to a systematic extension, via liquid state theories [95], of Manning s counterion condensation formalism [96-101], Second, the methodology provides an alternative route to interpret and analyze the role of small ions in the binding characteristics of univalent and multivalent ions to nucleic acids and charged polymers [90,92,101],... 

The fact that the apparent linear charge density is lower than the actual charge density can be seen as confirmation that PB theory does predict some sort of counterion condensation. We mention that because the ADH method results from a (linear) DH interpretation of the (nonlinear) PB equation, it also has a (rather tenuous) mathematical connection to Manning s coimterion condensation theory, which also makes use of the DH solution. " From either Eq. [192] or Eq. [263], the fraction of linear charge density neutralized can be written as (ignoring a contribution from Q... 

Fenley, M. O. Manning, G. S. Olson, K. W. Electrostatic persistence length of a smoothly bending polyion computed by numerical counterion condensation theory. J. Phys. Chem. 1992 96 3963—3969. 

Figure 2. Conq)arison of experimental salt absorption with theoretical predictions based on ideal Donnan equUibrium (Cal-I), nonlinear counterion condensation theory (Cal-N), linear counterion condensation theory (Manning s theory) (Cal-L)...

Manning, G. S. 2011. Counterion condensation theory of attraction between like charges in the absence of multivalent counterions. Eur. Phvs. J. E.. 34(12), 1-18. 

G. S. Manning, Physica A, 231,236 (1996). Counterion Condensation Theory Constructed from Different Models. 

M. O. Fenley, G. S. Manning, and W. K. Olson, J. Phys. Chem., 96, 3963 (1992). Electrostatic Persistence Length of a Smoothly Bending Polyion Computed by Numerical Counterion Condensation Theory. 

G. S. Manning and J. Ray, Macromolecules, 32, 4588 (1999). Counterion and Coion Distribution Functions in the Counterion Condensation Theory of Polyelectrolytes. 

J. Ray and G. S. Manning, in Physical Chemistry of Polyelectrolytes, T. Radeva, Ed., Marcel Dekker, New York, 2001, pp. 111-133. Inverted Forces in Counterion Condensation Theory. 

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. 

Le Bret M, Zimm BH. Distribution of counterions around a cylindrical polyelectrolyte and Manning s condensation theory. Biopolymers 1984 23 287. 

According to Manning s theory for polyelectrolytes [72], a charged macromolecule in solution is submitted to counterion condensation when the so-called charge parameter, 4, is higher than unity, enabling this parameter to decrease. 

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... 

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