  Counterion adsorption on a flexible polyelectrolyte chain in a spherical
cavity is considered by taking a "permuted" charge distribution on the chain so
that the "adsorbed" counterions are allowed to move along the backbone. We
compute the degree of ionization by using self-consistent field theory (SCFT)
and compare with the previously developed variational theory. Analysis of
various contributions to the free energy in both theories reveals that the
equilibrium degree of ionization is attained mainly as an interplay of the
adsorption energy of counterions on the backbone, the translational entropy of
the small ions, and their correlated density fluctuations. Degree of ionization
computed from SCFT is significantly lower than that from the variational
formalism. The difference is entirely due to the density fluctuations of the
small ions in the system, which are accounted for in the variational procedure.
When these fluctuations are deliberately suppressed in the truncated
variational procedure, there emerges a remarkable quantitative agreement in the
various contributing factors to the equilibrium degree of ionization, in spite
of the fundamental differences in the approximations and computational
procedures used in these two schemes. Nevertheless, since the significant
effects from density fluctuations of small ions are not captured by the SCFT,
and due to the close agreement between SCFT and the other contributing factors
in the more transparent variational procedure, the latter is a better
computational tool for obtaining the degree of ionization.
