Condensation 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). 

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  

The increase of counterion binding with the charge on the polyion has 

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

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The effect is more than just a matter of pH. As shown in Fig. XV-14, phospholipid monolayers can be expanded at low pH values by the presence of phosphotungstate ions [123], which disrupt the stmctival order in the lipid film [124]. Uranyl ions, by contrast, contract the low-pH expanded phase presumably because of a type of counterion condensation [123]. These effects caution against using these ions as stains in electron microscopy. Clearly the nature of the counterion is very important. It is dramatically so with fatty acids that form an insoluble salt with the ion here quite low concentrations (10 M) of divalent ions lead to the formation of the metal salt unless the pH is quite low. Such films are much more condensed than the fatty-acid monolayers themselves [125-127]. 

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. 

In polyelectrolyte solutions, the counterion condensation on linear polyelectrolyte chains is known to occur when the charge density along the chain exceeds the critical value [40]. Our work indicates the existence of a critical value for the separation distance between chains, where the interchain interaction changes drastically, most likely due to the transition in the binding mode of the counterions (see Fig. 13). Many peculiar forms of behavior, which are often interpreted by the cluster formation or the interchain organization of polyelectrolytes, have been reported for high concentrations of aqueous polyelectrolytes... 

The above treatment is based on the assumption that 0 is small. However, as Figure 4.3 shows, P does not greatly change with concentration so that counterion condensation is probably insensitive to concentration. The delayed binding of counterions is of some importance to the onset of gelation. 

Repulsive coulombic forces exist between charged polyions. These are attenuated by the bound counterions conversely they are stronger for polyions having a higher concentration of free counterions. When the charge along the polyion, Q, is small the forces involved are purely coulombic repulsion forces. However, when Q exceeds a certain value, counterions condense on the polyions and reduce the repulsive forces. 

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

Satoh, M., Komiyama, J. lijima, T. (1984). Counterion condensation in polyelectrolyte solutions a theoretical prediction of the dependences on the ionic strength and degree of polymerization. Macromolecules, 18, 1195-2000. 

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

As discussed extensively in this chapter, most of the surprising properties of polyelectrolyte dynamics are due to the coupling of counterion dynamics with polymer dynamics. But, there is no adequate understanding of how much of the counterions are mobile and how much are effectively condensed on polymer chain backbone. Theoretical attempts [77, 78] on counterion condensation need to be extended to concentrated poly electrolyte solutions. 

Figure 5. Regions of stability of isotropic and anisotropic solutions of 150 bp DNA, calculated according to Stigter 22L) The light band corresponds to the coexistence region for fully charged DNA, the dark band to DNA with 76% of charge neutralized by counterion condensation. The salt/DNA concentration regions where the gelation and ordinary-extraordinary transitions were studied are indicated by brackets.

A more recent SANS study revealed a 4% increase in dendrimer volume with decreasing pH (Chen et al. 2007). Although these stmctural variations were significantly less than predicted by the MD studies, this study confirmed the occurrence of both counterion condensation and water penetration of the dendrimer interior. 

The specific electro-diffusion phenomena, the field and force saturation and counterion condensation, as well as the corresponding features of the solutions to the Dirichlet problem for (2.1.2) to be addressed in this chapter, are closely related to those observed by Keller [7], [8] for the solutions of (2.1.3a) with f tp) positive definite, satisfying a certain growth condition. Keller considered f( 0, satisfying the condition... 

Another related phenomenon to be discussed in 2.3 is known in the polymer literature as counterion condensation. This term refers to a phase transition-like switch of the type of singularity, induced by a line charge to solutions of (2.1.2), occurring at some critical value of the linear charge density. Counterion condensation as a limiting property of the solutions of the Poisson-Boltzmann equation was studied in detail in [11]—[19]. Presentation of 2.3 follows that of [17]. 

The situation is completely different with counterion condensation, considered in this section. A natural order parameter here would be... 

Study of counterion condensation as a limiting property of the solutions of the Poisson-Boltzmann equation for arbitrary, charged cylindrical manifolds in H3 (see 2.3). 

Counterion condensation on a point charge in three dimensions for the excluded volume equation (1.68). It is expected that in contrast to the previous case (Problem 4) the limiting equilibrium solution will exist for the appropriate version of (1.68) and the corresponding limiting singularity in the electric potential should be studied. 

I. Rubinstein, Counterion condensation as an exact limiting property of the Poisson-Boltzmann equation, SIAM J. Appl. Math., 46 (1986), pp. 1024-1038. 

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