Water sub-diffusion in membranes for fuel cells

We investigate the dynamics of water confined in soft ionic nano-assemblies, an issue critical for a general understanding of the multi-scale structure-function interplay in advanced materials. We focus in particular on hydrated perfluoro-sulfonic acid compounds employed as electrolytes in fuel cells. These materials form phase-separated morphologies that show outstanding proton-conducting properties, directly related to the state and dynamics of the absorbed water. We have quantified water motion and ion transport by combining Quasi Elastic Neutron Scattering, Pulsed Field Gradient Nuclear Magnetic Resonance, and Molecular Dynamics computer simulation. Effective water and ion diffusion coefficients have been determined together with their variation upon hydration at the relevant atomic, nanoscopic and macroscopic scales, providing a complete picture of transport. We demonstrate that confinement at the nanoscale and direct interaction with the charged interfaces produce anomalous sub-diffusion, due to a heterogeneous space-dependent dynamics within the ionic nanochannels. This is irrespective of the details of the chemistry of the hydrophobic confining matrix, confirming the statistical significance of our conclusions. Our findings turn out to indicate interesting connections and possibilities of cross-fertilization with other domains, including biophysics. They also establish fruitful correspondences with advanced topics in statistical mechanics, resulting in new possibilities for the analysis of Neutron scattering data.


Sorption isotherms
. Sorption isotherms of Aquivion 790g/eq and Nafion 1100g/eq versus the relative humidity (RH). The water content is expressed as the water volume fraction, w (left), or the local hydration number , defined as the number of water molecules per sulfonic acid group (right). Figure S2. a) 1D SAXS spectra of Aquivion membranes as a function of the hydration number . b) Mean separation distance between hydrophobic aggregates, diono, obtained from the ionomer peak position Qiono, e.g. diono= 2/Qiono versus for Nafion (1100 g/eq) and Aquivion (790 and 850 g/eq). The mean size of aggregates is obtained from the extrapolated value in dried state, d0 = diono (=0). We find d0 = 29 Å in Nafion and 24 Å in Aquivion.

Quasi-Elastic Neutron Scattering (QENS): data analysis and parameters.
Models with variable degree of sophistication can be used to interpret QENS data, depending on the targeted information and the complexity of the expected intertwined mechanisms. For instance, analysis based on single-Lorentzian line shapes can be used to rapidly grasp the salient features of the dynamics, or compare first-order effects in distinct systems. We have followed this approach to compare the behavior of a large set of PFSA materials, establishing some common features. 1 On the other hand, a more complete description can require: i) the collection of QENS spectra on extended timescales; ii) the simultaneous analysis of multiple data-sets using a single model over the entire Q-range; and iii) an iterative data fitting process, without undefined parameters. This program can only be realized by combining experimental data taken at different spectrometers (ToF and BS) with different resolutions.
Multi-resolution QENS experiments were therefore conducted on hydrated Aquivion membranes. Data were analyzed using the method developed in Perrin et al., 2 based on the Gaussian model for translational diffusion. 3 Here we present a brief description of the methodology and modeling, all details can be found in the publications.

Multi-resolution QENS data analysis
Data taken on the time-of-flight and backscattering spectrometers were corrected using standard data reduction procedure as detailed by Perrin et al., 2   Inter-droplet diffusion is also accounted for and quantified by the nanoscale diffusion coefficient Dnano. The distance between two confinement domains, 2L, is defined as ). The slow localized motions are analysed using a back-and-forth hopping between equivalent sites, characterized by the residence time s and the jump distance s. The number of fast and slow protons are labeled as NF and NS.
The slow protons dynamical structure factor Ss(Q,) is the sum of an elastic term and a Lorentzian quasielastic component: where where Dloc, , Dnano, stand for the local diffusion coefficient inside a confinement domain (socalled droplet), the characteristic time of the local jump diffusion, the confinement domain (droplet) size, and the long-range (inter-droplet) diffusion coefficient, respectively.
The data analysis using the total dynamical structure factor of Eq. 1 is performed simultaneously on the multi-resolution sets of QENS data through a back-and-forth procedure, as described in our previous study on Nafion membranes. We find a very good agreement between the experimental spectra of each hydrated Aquivion membranes and the model using a single set of parameters (NF, NS, Dloc, ,, Dnano, S and s) (Fig. S4-S7). This is achieved at each instrumental resolution (1, 20, 30, 90 and 100 µeV) and over the whole Q-range, which strongly supports our interpretations and assesses the fitting procedure. Although a number of parameters have to be adjusted, they are unambiguously established given the number of constraints imposed by the extended set of data to be consistently reproduced simultaneously.

Parameters and their dependence upon hydration.
Values of the parameters: S, Dloc and Dnano were extracted from the fit of the extended set of QENS spectra. NF, NS, and S are extracted from the slow motions quasielastic structure factor analysis (see explanations below). All parameters are summarized in Tables S1 (slow dynamics) and S2 (fast dynamics), and plotted in Fig. S8-S10. Table S3 assembles the structural and dynamical parameters used in Fig. 4.  Table S1. Aquivion slow dynamics parameters obtained from the fits with the generalized Gaussian model.  Table S2. Aquivion fast dynamics parameters obtained from the fits with the generalized Gaussian model.    The values in Aquivion are compared to those found in Nafion and PFOS. 1 The behavior at large water volume fraction is found to be independent of the details of chemical architecture, as due to the driving "obstacle mechanism", e.g. water molecules diffuse within large water pools embedding hydrophobic aggregates.    Table S4. Models used in this work and previous publications.