Counterion condensation theory

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... 

Several chapters of this book discuss applications and extensions of the theory of polyelectrolyte solutions. Counterion condensation theory postulates that for a cylindrical macroion, if the linear charge density exceeds a well-defined critical value, a sufficient fraction of the counterions will "condense" into the immediate domain of the macroion so as to reduce the net charge density due to the macroion and Its condensed counterions to the critical value. No condensation is predicted for macroions with less than the critical charge density. 

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. 

A recent attempt to extend the scope of counterion condensation theory to the calculation of counterion-polyion, coion-polyion, and polyion-polyion pair potentials retains structural idealization but may nonetheless be capable of generating useful new information about the real molecular structure of polyelectrolyte solutions [57-59], The validity of the first and second stages of the theory, as discussed above, has been heavily documented, including the physical reality of the condensed layer and the onset of condensation at a critical charge density [55,56,60,61], In contrast, the inverted forces predicted by our extended theory have yet to be confirmed by experiment or simulation. We will argue, however, that their presence is at least suggested by current experimental knowledge. 

The organization of the chapter is straightforward. Section II reviews counterion condensation theory for an isolated polyion in the framework of the second stage as indicated above. Then in Sec. Ill the radial distribution function for the counterions and coions is discussed with emphasis on an inverted region at intermediate distances from the polyion. The pair potential for two identical polyions is also discussed in this section, and an inverted attraction is highlighted (an inverted repulsion is found for two polyions identical but for opposite charge). Finally, we review our work on polyion clustering in Sec. IV. 

Dais P, Peng QJ, Perlin AS. Binding of magnesium and lanthanum ions to heparin in the presence of sodium ions. A relationship between 13C chemical shift displacements and counterion condensation theory. Can J Chem 1987 65 1739-1745. 

Rivas BL, Moreno-Villoslada I. Binding of Cd2+ and Na+ ions by poly(sodium 4-styrenesulfonate) analyzed by ultrafiltration and its relation with the counterion condensation theory. J Phys Chem B 1998 102 6994-6999. 

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. 

Nucleic Acids are Polyelectrolytes Counterion Condensation Theory... 

A formal theoretical model for the association of cations with polyelectrolytes is provided by counterion condensation theory. In its ideal form this theory assumes a linear polyanionic chain consisting of regularly spaced charges and of infinite length, in a medium of constant dielectric constant. 

A better understanding of the physical mechanisms involved in the field-effect detection of DNA is fundamental in the development of reliable DNA microarrays based on FETs. Several aspects play a role in the detection mechanism. Counterion condensation theory can be used to evaluate the effective charge density of the DNA layer in contact with an electrol3d e, which partly screens its charge, its dependence on the ionic strength of the electrolyte, and the reduction of the charge fraction observed upon hybridization. Mathematical models have been used to describe the observed shifts in the I-V curves of the field-effect transistors. 

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. 

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