Bjerrum length, polyelectrolyte-counterion

In general, one of the characteristics of rod-like polyelectrolytes is the charge (Manning) parameter c which for monovalent counterions is defined through the ratio of the Bjerrum length Xp to the contour distance per unit charge b [22-24] ... 

Subdivision of counterions into condensed and uncondensed populations according to the Manning/Oosawa depiction is not unique. For example, a hypothetical thermodynamically bound counterion population can be defined to account for the deviations of polyelectrolyte solutions from thermodynamic ideality (35) this population is not equal to the condensed population. One can also use different definitions of condensed. The inflection point in the radial ion distribution, the Bjerrum length, and the radial distance over which the electrostatic interaction energy decays to kT have all been employed as alternative criteria for defining a condensed fraction (50). 

The conductivity data can be combined with the vb d ist to analyze CC effects during polymerization. CC in polyelectrolyte solutions [32,33] has been the topic of extensive analytical and numerical investigations [34, 35], but for which far fewer detailed experimental works exist [36, 37]. CC occurs when the electrostatic potential energy between charge sites on a polymer chain and counterions at the intermonomer backbone distance exceed kT. is defined in this context as the number of elementary charges per Bjerrum length /g. In the simplest theory, CC will prevent from surpassing unity. 

One other relevant theoretical set of work concerns the counterion distribution particularly in the dilute limit. Manning solved the Debye-HUckel equation for a single infinitely thin polyelectrolyte. He found that when a < Xb the counterions condense onto the hne polymer reducing the charge density until the charge separation becomes equal to the Bjerrum length. The details are altered when the Poisson-Boltzmann approximation is used for a cylindrical polyelectrolyte, " but the basic point of condensation occuring for A > 1 remains. In a similar vein, Oosawa proposed a two-phase model of bound and free counterions. These results are especially relevant, since many prototypical polyelectrolytes, such as DNA and NaPSS, have A 3,... 

This expression for differs by a factor of 2 from the length below which ion association is assumed to take place in electrolyte solutions, according to Bjerrum s theory, see (1.5.2.30a). The reason for this factor of 2 is that for counterion association on a polyelectrolyte only the former loses its kinetic energy, whereas for association of two small ions this occurs for both. At low polylon charge, is of course simply given by... 

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