The Mean-Spherical Equal Arm Stretching Approximation General Formalism

1 The Mean-Spherical Equal Arm Stretching Approximation General Formalism 

To analyze the effects of ionic strength and pH of the solution on the conformations of PE stars, we switch from the canonical cell model (where the number of ions was fixed) to the partially open ensemble. In the latter model, (a) one central star polymer occupies a spherical volume within radius R, and (b) the chemical potentials of all mobile ions are set equal to those in the bulk of the solution (infinite reservoir). 

The concentrations Cbj (or, equivalently, the chemical potentials) of all the mobile ions are assumed to be constant in bulk solution. Therefore, the relevant free energy of the corona is the Gibbs free energy. 

If all the star branches are assumed to be equally stretched (i.e., all the ends of the arms are localized at the edge of the star), the free energy of a star polymer can be presented as  

the first term accounts for the conformational entropy losses in non-uniformly extended branches that exhibit the Gaussian elasticity [122], whereas the second term is the interaction contribution to the free energy. The local chain extension (dr/dn) at distance r from the center is related to the local concentration of monomers, Cp(r), as  

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