Modeling of Water Sorption and Swelling in Polymer Electrolyte Membranes
Authors: Safiollah Motahareh , Melchy Pierre-Éric Alix , Eikerling Michael 
Published: February 2020

Energy conversion in polymer electrolyte fuel cells depends on the ability of the polymer electrolyte membrane (PEM) to transport protons from anode to cathode. The PEM, in turn, requires sufficient hydration for efficient proton conduction. The water content is the main thermodynamic variable to characterize the state of a PEM. It determines structure formation, proton conductivity and mechanical response of the membrane. As a result, a consistent description of water sorption and swelling under relevant conditions of PEFC operation is essential for the understanding of transport properties, performance, and degradation phenomena in PEM.
We present a theoretical model of water sorption and swelling in PEMs [1]. The model employs conditions of thermal, electrochemical and mechanical equilibrium of water in the porous network.
The single pore model relates the charge density, σ0, at the pore walls to a microscopic swelling parameter, η. The single pore model is expanded to an ensemble model using a statistical distribution function of the surface charge density, n(σ0). The volumetric expansion of the PEM can be expressed as a convolution integral,
[equation attached as a picture, equation 1]
normalized to the dry volume of the PEM, V
0. Furthermore, we obtain microscopic statistical distributions of pore size and elastic pressure in pores.
We performed an extensive parametric study. Water sorption and swelling behaviour of the PEM are sensitive to variations in ion exchange capacity, IEC, and elastic modulus, G. They are rather insensitive to variation in external gas pressure.
Freger [2] proposed that the deformation of pore walls is isotropic in the plane perpendicular to the stress caused by the expanding water domains. We have evaluated different deformation scenarios corresponding to anisotropic expansion of pore walls or uniaxial stretching of ionomer bundles. The anisotropic deformation scenario shows the best agreement with experimental sorption data [3, 4], as shown in Figure 1. The best result is obtained with a microscopic elastic modulus of 380 MPa. The statistical distributions of pore size and elastic pressure at pore walls are shown in Figure 2a and b, respectively.
The model can be applied to the analysis of water sorption in a PEM that is undergoing degradation (Figure 3). Figure 3a shows water sorption isotherms after 0, 80 and 280 days. The data were extracted from a study of Gebel et al.[5]. The total water content decreases because of the decrease in IEC and increase in elastic modulus of the membrane. Surface charge density distributions corresponding to the water sorption isotherms are shown in Figure 3b. The model-based analysis suggests that the total number of pores decreases while the statistical distribution is shifting to pores with larger surface charge density.
References:
[1] M.Eikerling and P.Berg, Soft Matter, (2011)

[2] V. Freger, Polymer, (2002)
[3] T.A. Zawodzinski et al., J.Phys. Chem (1991)

[4] L. Maldonado et al., J. Membrane Science (2012)
[5] F.M.Collette, et al., Membrane Science (2013)