Self-Regularization of Polymer Charge

Let us consider a single chain of N monomers in volume V. Each monomer is monovalently charged, and i is the distance between two successive monomers along the polymer. Due to the electroneutrality condition, there are N monovalent counterions. Let M be the number of counterions adsorbed on the polyelectrolyte so that M/N is the degree of counterion adsorption and a = 1 — M/N) is the degree of ionization of the polyelectrolyte. In addition. 

Entropy of adsorbed ions. Since (1 -a )A counterions are adsorbed to each chain on an average and there are A /[((l -a)A ) ][(aA ) ] ways of formation of ion pairs, the free energy associated with the entropy of adsorbed counterions, with the Stirling approximation, is 

All nonlinear effects resulting from possible cooperative features associated 

Entropy of unadsorbed ions. The free energy due to the translational entropy associated with (N - M + n+) unadsorbed counterions and n coions in volume V is the familiar entropy of mixing term 

Correlations among dissociated ions. Using the Debye-Hiickel result given by Equation 3.44, we get 

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