Uncharged Polymer with Excluded Volume

Careful numerical simulations (Cacciuto and Luijten 2006) show that Equation 5.28 is more accurate than the Flory result (Equation 5.26), for monomer concentrations in the semidilute regime. However, as the chain length increases so that the monomer concentration is in the concentrated regime (Table 2.2), 

The same result of Equation 5.30 is obtained if we use Equation 5.28 instead of Equation 5.26 in the derivation. The number of segments in the recipient compartment for the metastable conformation is simply the ratio of the volume of the recipient compartment to the total volume of the donor and recipient compartments. This formula turns out to be a good approximation for partitioning of segments when a chain straddles between two cavities (Kong and Muthukumar 2004). 

Basically, the confinement free energy of a polyelectrolyte chain inside a cavity is due to the translational entropy of counterions, electrolyte ions, and solvent molecules. Simple scaling formulas based on the radius of gyration of the polyelectrolyte, analogous to Equations 5.28 and 5.26, are not applicable for the confinement free energy of a polyelectrolyte in spherical cavities, unlike the case of uncharged polymers. 

Analysis of the numerical data on the free energy barrier associated with the end of a chain, conflned in volume R, searching for the pore entrance shows that (Kong and Muthukumar 2004, Kumar and Muthukumar 2009) 

---

The theory of solutions of flexible uncharged polymers with excluded volume is at present well developed, but the properties of polyelectrolytes and especially polyampholytes have been considered much less from the theoretical point of view. It is well known that polyampholytes exhibit a change in phase from the extended random flight configuration to a condensed microphase. The polyampholyte theory of Edwards et al. [6] considers the isoelectric state of polyampholytes as a microelectrolyte satisfying a Debye-Huckel-type of structure. The criterion of transition from the collapsed conformation to the extended one is described as follows ... 

Analogous to the derivation of Equations 2.73 and 2.74 for an uncharged polymer, a crossover formula for the radius of gyration of a flexible polyelectrolyte in a solution with high enough salt is obtained as follows. Substituting Equation 2.45 for the free energy of chain connectivity and Equation 2.68 (with w replaced by Weg) for the excluded volume effect in Equation 2.67, we get... 

---