Manning condensation theory

The results reviewed above clearly indicate the features by which site-binding is distinguished from general electrostatic interactions, namely its specificity for one or more chemical groups on the polyion and the critical role played by the solvent. It remains to be considered how the demonstration of the existence of short-range interactions affects the validity of current theoretical treatment of polyelectrolyte solutions. We shall limit ourselves to two of these treatments, both of which apply to the cylindrical rod model, one involving the Poisson-Boltzmann equation, the other the Manning 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. 

According to Manning s theory, Na+ ions condense onto the polyanion until the fraction (1 — ) of the line charge is neutralized when , > 1 i.e. the effective... 

According to the counter ion condensation theory (Oosawa, 1971 Manning, 1978, 1980, 1985 Dautzenberg et al., 1994) 76% of the DNA phosphate charge is neutralized by territorially bound Na+ ions, in typical aqueous NaCl solutions (less than 1 M). Thus, it was expected that additional neutralization up to 90% is the necessary condition to cause DNA condensation , and that, with divalent cations, the maximum value is about 88%, being insufficient for condensation . As an exception to this theory, Mn2+ induced condensation of super-coiled plasmid DNA was discussed as the facilitating effect in the supercoiled state (Ma and Bloomfield, 1994). Condensation of DNA on a sohd surface inthe presence of divalent cations was also reported (Koltover et al., 2000). 

Within PB theory [2] and on the level of a cell model the cylindrical geometry can be treated exactly in the salt-free case [3, 4]. The Poisson-Boltzmann (PB) solution for the cell model is reviewed in the chapter in this volume on the osmotic coefficient. The PB approach can provide for instance new insights into the phenomenon of Manning condensation [5-7]. For example, the distance up to which counterions can be called condensed can be conveniently found via the inflection point in the log plot of the integrated radial distribution function P(r) of counterions [8, 9], defined as... 

Theoretical considerations of the coulombic interactions of dissolved biopolymers have produced a complete picture of the distributions of counter and coions under the influence of the electrostatic charge on the macroion(56,57). The counterion condensation theory of Manning(56) has stimulated a great deal of activity in the study of dissolved macroions, especially because it provides a group of limiting laws describing the contribution of electrostatic effects to the thermodynamic and transport properties of polyelectrolyte solutions. Data... 

Counterion condensation theory, however, does not provide a detailed picture of the distribution of the condensed Ions. Recent research using the Poisson-Boltzmann approach has shown that for cylindrical macroions exceeding the critical linear charge density the fraction of the counterions described by Manning theory to be condensed remain within a finite radius of the macroion even at infinite polyion dilution, whereas the remaining counterions will be infinitely dispersed in the same limit. This approach also shows that the concentration of counterions near the surface of the macroion is remarkably high, one molar or more, even at infinite dilution of the macromolecule. In this concentrated ionic milieu specific chemical effects related to the chemical identities of the counterions and the charged sites of the macroion may occur. 

Manning s condensation theory ( 1 ) describes the behavior of small ions around a long, highly charged polyelectrolyte by postulating that there are two populations of counter-ions, one normal, and one "condensed". If we have counter-ions with valence z, the fraction, Fm, of these ions in the condensed population is... 

The influence of salt on the distribution functions has been discussed within the PB theory in detail in Ref. 4. The general finding was that a low salt content leaves the picture of Manning condensation qualitatively unchanged, while at increasing salt concentration a crossover between Manning condensation and simple salt screening occurs. 

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

Manning GS. Counterion condensation theory constructed from different models. Physica A 1996 231 236-253. 

Ray J, Manning GS. Counterion and coion distribution functions in the counterion condensation theory of polyelectrolytes. Macromolecules 1999 32 4588-4595. 

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. 

We have also shown previously (12l by measuring D + when CaCl2 is added to ChSO.-Na that the small fraction (15 percent) of the sodium ions initially condensed are expelled from the condensation layer by the newly condensing Ca " " ions, which can be expected from Manning s theory. 

Mean field theories that implement a spherically symmetric distributions of counterions assume that the (effective) charge density on the star branches is below the Manning condensation threshold. In this limit, one can neglect the angular correlations between the positions of the arms of the star and those of its counterions. 

Manning proposed a linear counterion condensation theory to account for the low activity of counterions in polyelectrolyte solutions 14). The basic idea of the theo is that there is a critical charge density on a polymer chain beyond which some counterions will condense to the polymer chain to lower the charge density, otherwise the ener of the system would approach infinite. The concept of this theory has been widely accepted. The shortcoming of the linear countmon condensation is that it predicts that counterion condensation is independent of ionic strength in the solution, which is not in agreement with experimental observations. Counterion condensation can be obtained duectly by solving the nonlinear Poisson-Boltzmann equation. 

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)...

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 statistical thermodynamic counterion-condensation theory of Manning... 

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 ... 

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 GS (1996) Counterion condensation theory constructed from different models. Physica A 231(l-3) 236-253. doi 10.1016/0378-4371(95)00452-l... 

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