Thermodynamics of Mobile Ion in Ion Exchange Membranes: Water-Swollen-Membrane Reference State and Quasi-Regular Solution Model

Thermodynamics of mobile ions in swollen ion exchange
membranes
(IEM) are essential for understanding their permselectivity. The theoretical
modeling of the ion activity coefficient and the ion partitioning
of the IEM is challenging. Based on a water-swollen membrane reference
state and a quasi-regular solution model, we successfully correlated
the ion activity coefficient in various IEMs. For all the studied
59 systems, the correlation coefficient r and determination
coefficient R2 are 0.973 and 0.947, respectively.
As expected, the water activity in the IEM and the ion partitioning
between the IEM and the external salt solution are represented. In
the new theory framework, the standard chemical potential of the counterion
is different from that of the aqueous solution. Moreover, the ion
activity coefficient in the IEM reaches unity for the infinite dilution
external salt concentration for all systems. Our quasi-regular solution
model only considered the short-range interactions between the co-ions
and the “effective free counterion”, but it showed excellent
correlation ability with the activity coefficient data of extensive
systems. A power-law relation between the effective concentration
of the free counterion and the apparent concentration of the free
counterion was suggested. This indicates that the long-range electrostatic
interactions in the swollen IEM are negligible due to abundant counterion
condensation shielding the charge of the polymer chain.