Theory, condensed counterion

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). 

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

Theories of colloid stability based on electrostatics go way back beyond the DLVO theory, to the Gouy-Chapman theory of the electrical double layer proposed in the early 1910s and the Stem theory of counterion condensation proposed in 1924. There was much weighty speculation about the counterion distribution around colloidal particles throughout the 20th century, but nobody succeeded in measuring it until our work in 1997. This work is described in detail in Chapter 8. 

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. 

This property of P, derived within the framework of PB theory, can in turn be used to define the condensed fraction [4], It provides a suitable way to quantify counterion condensation beyond the scope of PB theory, and it is exact in the salt-free PB limit. From here on this method will be referred to as the inflection point criterion. This criterion has the advantages of (1) not fixing by definition the amount of condensed counterions (fe and RM can be determined independently of each other), (2) reproducing the salt-free PB limit, namely P(RM) = 1 l/ , and (3) quantifying the breakdown... 

Deserno M, Holm C, May S. The fraction of condensed counterions around a charged rod comparison of Poisson—Boltzmann theory and computer simulations. Macromolecules 2000 33 199. 

There are two cases. When < 1, dw/dO > 0 if 6 > 0. The equilibrium state of minimum free energy is therefore at 0 = 0, and there are no condensed counterions and no requirement for renormalization of polyion charge, and limiting Debye-Hiickel theory goes through unmodified. How-... 

Manning GS. A condensed counterion theory for polarization of polyelectrolyte solutions in high fields. J Chem Phys 1993 99 477-486. 

Two kinds of counterions, condensed and those constituting a diffuse ion atmosphere that may be treated in the Debye-Htickel approximation, are clearly recognized in their distinct spatial distributions, and introduction of the partial polarizability tensor enables us to distinguish between contributions to the polarizability from these two kinds of ions. The contribution from condensed counterions to the radial components of the polarizability tensor is very small, as has hitherto often been postulated in various theories. That from the diffuse ion atmosphere is very large and cannot be neglected in the calculation of the anisotropy. 

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... 

Muthukumar, M., 2004. Theory of counterion condensation on flexible polyelectrolytes Adsorption mechanism, J. Chem. Phys., 120, 9343-9350. 

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