Condensation and Screening of Charges

One might think at first that on dissolving a polyelectrol3de chain all counterions diffuse away and distribute themselves homogeneously in the solvent. As a consequence, the chain would get fully stretched in a reaction to the 

Consider a single rod-like chain with radius ro, which includes monovalent kations at regular distances lio, i.e., possesses a linear charge density e//jo. If it is brought in a solvent with a dielectric coefficient e it develops an electric field. The corresponding potential V follows from an application of basic laws 

In order to deal with this question, the cell structure shown in Fig. 3.12 is set up. A series of cylindrical cells i with inner radii ri and a constant thickness Ar surrounds the rod. We consider the probability of an occupation of cell i. The electrostatic energy of the anion in cell i is 

The entropy associated with the motion within the cell amounts to 

If there are many non-interacting anions, the number rii of anions in cell i at a temperature T follows from Boltzmann statistics as 

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