  We use molecular dynamics simulations to probe hydration, ion spacing, and
cation-anion interaction in two sulfonated polysulfones with different ion
distributions along the polymer backbone. At room temperature, these polymers
remain in a glassy state even with water contents more than 10%. At the
equilibrium water uptake, the ions exhibit a similar level of hydration as they
would in their saturated aqueous solution. The framework of Manning's limiting
law for counterion condensation is used to examine ionic interactions in the
simulated polysulfones. The dielectric constant that the ions experience can be
well approximated by a volume-weighted average of the dielectric constants of
the polymer backbone and water. Our results show that a reasonable estimate of
the average inter-ion distance, b, is obtained by using the distance where the
sulfonate-sulfonate coordination number reaches 1. The spacing of the sulfonate
ions along the polysulfone backbone plays a role in determining their spatial
distribution inside the hydrated polymer. As a result, the value of b is
slightly larger for polymers where the sulfonate ions are more evenly spaced
along the backbone, which is consistent with experimental evidence. The
simulations reveal that the sulfonate ions and sodium counterions form
fibrillar aggregates at water contents below the equilibrium water uptake. Such
extensive ionic aggregates are expected to facilitate ion transport in
sulfonated polysulfone membranes, without the need for long-range chain motion
as in the case of traditional rubbery ionic polymers. Our estimates for the
dielectric constant and b are used in conjunction with Manning's theory to
estimate the fraction of counterions condensed to the fixed ions. The
prediction of Manning's theory agrees well with the simulation result.
