(Invited) An Activity Model for Swelling of Nafion® in Varied Solvents
Authors: Leddy Johna , Rathuwadu Nadeesha 
Published: February 2020

Nafion is used ubiquitously in aqueous electrochemical systems and commonly in nonaqueous systems. Nafion solvation varies with solvent and cation. Its concomitant swelling complicates Nafion application in various electrochemical technologies. For direct reformation PEM fuel cells that operated with solvents of alcohols and aqueous alcohols, crossover of alcohols through Nafion to the cathode degrades fuel cell performance. The cation in Nafion impacts methanol permeability where permeability ranges over three orders of magnitude for various cations, as found by Kunz and coworkers [1,2]. Aquated, Li⁺ exchanged Nafion placed in organic solvents changes level of solvation; across almost 30 solvents, swelling mapped as change in volume over aquated volume, ΔV/V ranged from 40 to 730 % [3].
Properties of Nafion have been variously reviewed, with a recent review focused on aqueous systems and change in properties with cation [4]. Recent modeling [5] of cation impact on properties of Nafion in water has developed from activity and electrochemical potential models. Here, extension of the model to nonaqueous solvents for cation exchanged Nafion is considered.
Classical theories of activity evolved from Debye Hückel and Extended Debye Hückel (EDH) models where the models are applicable for ionic strength I≲0.01 M and ≲0.1 M, respectively. Activity for species i is defined as ai = γi[i] where γi is the activity coefficient and [i] is the concentration. The ionic strength in the hydrated domains of Nafion is about 10 M [4]. In further development of activity in high ionic strength, Stokes and Robinson [6] divided the solvent molecules into free and bound waters. The free waters act as solvent but bound waters are removed from the bulk solvent that dissolves the solute. The Stokes Robinson model describes the activity coefficient γSR,i for a solute ion relative to the activity coefficient for the ion according to EDH, γEDH,i.
γSR,i/γEDH,i =aw
-n/v [1 - 0.018 (n-v) c*]-1

where n is the number of waters bound to ion i, v is the number of ions formed when i dissolves (e.g., for NaCl, v=2), and c* is concentration of i. Because the number of free waters in high ionic strength is reduces, the activity of solvent water, aw< 1. The value 0.018 embeds properties of molecular weight and density for water. This has been used to model various properties of Nafion in an aqueous environment [5].
The Stokes Robinson model for Nafion will be adapted to model swelling of Nafion in various solvents, based on the data set in reference [3]. Preliminary results for 29 solvents yield a monotonic fit of ΔV/V with γSR,Li/γEDH,Li and a linear fit of ΔV/V with [γSR,Li/γEDH,Li]-1. A effective activity model based on the Stokes Robinson model is anticipated for various solvents. With an activity model, properties of Nafion in different solvents can be tailored for specific applications.
--- References

1. H. Kabu, H.R. Kunz, J.M. Fenton, Ionic Conductivity and Methanol Permeability of Modified Nafion® Membranes, www.electrochem.org/dl/ma/201/pdfs/0096.pdf
2. J.C. Lin, M. Ouyang, J. M. Fenton, H. R. Kunz, J. T. Koberstein and M. B. Cutlip, J. of Appl. Polym. Sci., 700, 121 (1998).
3. G. Gebel, P. Aldebertt, M. Pineri, Polymer 1993, 34, 333-339.
4. J. Leddy, Chapter 6, in Nanomaterials for Sustainable Energy, ACS Symp. Ser. 1213, 99-133.
5. J. Leddy, unpublished work
6. R.H. Stokes and R.A. Robinson JACS (1948) 70, 1870-1878.