In the early 1980s Volino and Dianoux et al. used QENS to characterize the water dynamics in a hydrated Nafion® film [49, 50]. The Nafion® sample was 1200 EW and the water content was kept at approximately 15 % by mass. Experiments were carried out at 25 °C on the multichopper time-of-flight spectrometer (IN5) at the Institut Laue Langevin using three different neutron wavelengths of approximately 10, 11, and 13 A, corresponding to an energy resolutions (Лю) of 18.5, 14, and 9 peV, respectively. At each energy resolution, data was collected over the Q-range

0. 4—1.1 A-1. Along with the hydrated films, dry films were measured as a control. In early work, the quasi-elastic broadening was characterized using a Lorentzian to describe the long-range self-diffusion of the water molecules in the hydrated membrane. The dynamic scattering-function, S(Q, m), for diffusion of an incoherent — scattering particle inside an impermeable sphere of radius, a, was presented in the original work. Considering the broadening to be due to only long-range diffusion,

1. e. a ^ to, the scattering function reduces to:

1 dq2

p (DQ2)2+x2

where a Lorentzian function is used with full-width at half-maximum of Лю = 2DQ2 and D is the long-range diffusion coefficient.

It was determined that this model did not sufficiently describe the QENS spectra, and therefore inadequately described the water motions in Nafion®. Given the fact that a single Lorentzian function, as used in Eq. (10.1), could not completely describe the quasi-elastic broadening, a second model was proposed that considered the rotational motions of the water molecules within the membrane. This particular model assumes that the water molecules diffuse on a sphere of radius, p, with a rotational-diffusion coefficient, Dr. Using the distance between the centre-of-mass and the protons (0.95 A) as p, the researchers were unable to fit the data using any reasonable value for Dr. With p as adjustable parameter, they found reasonable fits with p ^ 3 A, but because this value did not correspond to any reasonable length — scale in the water molecule the analysis approach was not pursued further. Therefore, it was determined that the measured QENS spectra could not be suffi­ciently described by the rotational diffusion of water alone.

With the failure of the two simple models described above, a third model com­bining translational self-diffusion with rotational diffusion was proposed. Through various initial attempts at determining Dr, p, and D, it was found that the general form of the model could sufficiently describe the QENS data, particularly with p ^ 3 A. Again, given that this value did not match any reasonable length-scale for water rotation, it was determined that although the mathematical description was sufficient, the physical interpretation was lacking. Therefore, it was decided that the mathe­matical form of the model would be preserved and would be one in which the water molecules (viz. the protons) are restricted to local diffusion, Db inside a sphere of radius, a, combined with long-range translational diffusion, Dt, between the spheres. The best-fit curves were calculated using D[ = 1.8 x 10-5 cm2s-1, a = 4.25 A, and Dt = 1.6 x 10 6 cm2s 1. Although this model proved to provide a reasonable explanation, and fit, to the QENS data in the Q-range 0.4—1.1 A 1, it was suggested that more sophisticated models could be proposed and were being pursued. How­ever, this was not thought to change the overall conclusions of the work. This work showed that on a local size-scale (ca. 10 A) the water molecules diffuse with a self­diffusion coefficient that is similar to bulk water (Dwater = 2.2 x 10 5 cm2s :), but that the long-range self-diffusion is restricted due to the morphology of material. Essentially, water is relatively free to move within a water domain, or ionic aggregate, but motion between domains over large length-scales suffers. Subsequent measurements on oriented membranes use more sophisticated models describing diffusion within a cylinder instead of a sphere.

Pivovar and Pivovar [51] also used QENS to investigate the water dynamics in Nafion® over a range of hydration levels from X = 2 to 16 in order to determine how the local dynamical behaviour of water is correlated with proton conductivity, particularly in the low-X regime. The QENS data were collected at a time-of-flight instrument at NIST. The data were collected at 295 K over a Q-range from 0.4 to 2.0 A 1 with neutron wavelength of 6 A. The energy resolution was between 55 and 70 peV. Assuming that the QENS data were the result of both elastically and quasi­elastically scattered neutrons arising from immobile and mobile protons, respec­tively, the experimental results were successfully modelled using the following function:

Sexp(Q, X = I(Q)(x • R(x) + (1 — x)R(rn) ® Цю)) + Bo (10.2)

where I(Q) is the scaled intensity at a given value of Q, R(oj) is the instrument resolution function, L(ffl) is a Lorentzian distribution, and Bo is a flat background. The quantity, x, is the elastic incoherent structure-factor (EISF), which is the ratio of the elastic scattering-intensity to the total scattering-intensity from both the quasi­elastic and elastic components. The EISF can yield information as the length-scale of the proton motions occurring within the sample. The fits to the experimental data based on Eq. 10.2 were interpreted using two different models. For Q < 0.7 A-1, the Lorentzian broadening was analysed according to a model where the incoherent scattering particles, i. e. protons, were thought to undergo continuous diffusion within a sphere as in the model proposed by Volino and Dianoux [55, 56]; above this Q-value a random unconstrained jump-diffusion model was used. From these two models the characteristic length-scales over which motion occurs, along with local (Dlocal) and jump (Djump) diffusion coefficients, were determined. In the context of the diffusion in a sphere model, the measured EISF was modelled with the following equation

x = EISF = fND + (1 — /nd)( jfl) (10.3)

where fND is the fraction of protons not diffusing in the timescale of the mea­surement, R is the characteristic radius of the sphere within which the protons are diffusion and j1 is the first-order Bessel function. The results of this analysis over the entire hydration range can be seen in Fig. 10.7.

The radius of the dynamic sphere is characterized by a linear increase from about 2 to 3.5 A in the X range from 1 to 7. Above a X of ca. 7 an asymptotic value of 3.68 A is reached for fully saturated membranes. Correspondingly, within the time­scale of the measurement, fND decreases with increasing water content. These results indicate that at low X values the hydrophilic domains are small and water molecules are likely to have strong interactions with the acid sites (i. e. the confining surface). These water molecules are effectively bound. As the domains swell, a smaller fraction of the protons feel the confining effects of these interactions. There is, however, inconsistency between the proposed size of these confined hydrophilic domains and the size as determined by SANS measurements. This can be accounted for by limitations in the time resolution of the instrument.

Following the work by Pivovar and Pivovar, further advancements in modelling of the QENS obtained from time-of-flight and backscattering spectrometers were made by Perrin and co-workers [52,53]. Their efforts aimed at developing a single model to account for bound-continuous diffusion, long-range diffusion, and atomic granularity (i. e. jump diffusion). This model used Gaussian statistics to describe the diffusion of the scattering particles within a restricted geometry with ill-defined boundaries. Essentially, the model accounted for two populations of diffusing protons: a popu­lation of fast protons and a second population of slower-moving protons.

Fig. 10.7 a The dynamic sphere radius, R, from a model fit to the EISF as a function of the water content, X, where the dashed lines demark the slope in the low-swelling regime of the asymptotic value at high hydration-level and b the fraction of non-diffusion hydrogen atoms, fND, as determined from the model fit as a function of water content, where the dashed line is the theoretically-calculated value offND assuming a single, non­diffusing hydronium ion for each sulfonic acid site. Reprinted with permission from (A. A. Pivovar,

B. S. Pivovar, J. Phys. Chem. B 109, 785 (2005)) [51]

© 2009 American Chemical Society

As described by Perrin, the slow protons are, for all intents and purposes, related to the hydronium ion formed when water abstracts a proton from the sulfonic acid site, but it is also likely that these are tightly bound or restricted, water molecules similar to those described by Pivovar and Pivovar [53]. The timescale of motion of this population becomes shorter with increasing hydration level and the characteristic length-scale of motions increases. No long-range diffusion was reported. With increased hydration, the “extra” water molecules comprise the fast protons whose motions are characterized over a similar length-scale as the slower protons along with long-range diffusion between adjacent confining domains. In general it was shown that for both populations of protons, the length-scale increases and the timescale of motions decrease with increasing hydration level. As with earlier studies, an asymptotic value was reached for these values at X of about 10.

Other studies have shown similar results. Paciaroni and co-workers [54, 55] studied the dynamics of the confined water in Nafion® films at low hydration levels (ca. X = 6) over a temperature range from 200 to 300 K. They characterized the motion according to random jumping inside a confining region, which they associate with the boundaries of the ionic aggregates, or clusters, of Nafion®. They also observed a transition in the water dynamics at approximately 260 K, which was reported to be related to an increase in the available degrees of freedom of the water created by a melting-like
process of the bound water. Their results concerning the local-diffusion coefficient of the water are similar to those previously reported [51]. Paciaroni et al. [55] also studied a composite membrane comprised of Nafion® with zirconium phosphate nano-filler. Again, the QENS data were modelled according to random jump-diffusion inside a confining spherical region. Ultimately, this work demonstrated that the local water dynamics were unaffected by the presence of the nano-filler.

Researchers have used QENS to look at novel PEM materials and to investigate the fundamental relationship between structure and transport. Peterson and co-workers used QENS to investigate the water dynamics in novel plasma-poly­merized PEMs [56, 57]. The membranes were synthesized using a pulsed-plasma enhanced chemical vapour-deposition system. The resulting membrane was a cross- linked polystyrene containing trifluoromethanesulfonic-like acid sites. The ion — exchange capacity of this novel membrane was shown to be somewhat less than that of Nafion® (0.7 mequiv/gm compared with 0.91 mequiv/gm) with a signifi­cantly lower water content. The quasi-elastic broadening was modelled using a Lorentzian. Plots of the half-width at half-maximum (HWHM) of the Lorentzian as a function of Q2 were used to calculate the diffusion coefficients. The water in the plasma produced PEM showed similar diffusion behaviour compared to Nafion®. However, they did observe an increase in HWHM for the plasma produced PEM at low Q values compared to that measured for Nafion®, but no explanation was given for this observation. A later, more detailed, QENS study from this group revealed interesting behaviour in the plasma-polymerized PEM (PP-PEM) [57]. They found that the PP-PEMs studied had a proton conductivity, as measured by impedance spectroscopy that was ^21 % higher than that of Nafion®. A careful QENS study, and subsequent fitting, of the proton motions revealed that the quasi-elastic broadening could be described by two components, or types of protons. One component was due to motions arising from a dispersed relaxation in the frequency domain, which is equivalent to a stretched exponential in the time domain, as proposed by Bergman [58]. These protons were labelled as type 2 and considered to diffuse much like the protons in Nafion® membranes. The other, faster component was described by a broad Lorentzian and accounted for 41(3) % of the protons. The Q2 dependence of the two types of protons was used to determine self-diffusion coefficients for the fast and slow diffusing protons. The fast protons, which accounted for 41(3) %, of the protons in the system, had a self-diffusion coefficient of 2.8(1) x 10-4 cm2s-1. This is an order of magnitude faster than that measured for the Nafion®-like protons (3.0(2) x 10 5 cm2s :). These results help to support and give an explanation as to the increased proton conductivity measured for the PP — PEMs, despite having a lower water uptake. It was proposed that the superfast proton diffusion is likely due to the arrangement of the sulfonates in PP-PEM. While detailed structural information was not presented, a simple model was pro­posed which involve the hydronium ion being passed between adjacent trifluoro — methanesulfonate groups. The local environments of these groups are thought to allow for rapid, relatively long-range proton transfer as compared to Nafion® [57]. Lyonnard et al. [59] have also used QENS to study the role of structure and confinement on water mobility. This study, however, was not carried out on PEMs, but on a perfluorinated surfactant (PFOS) bearing a similarity to the side-chain found in Nafion®. By varying the ratio of PFOS to water, they were able to create both lamellar and hexagonal phases. They showed that the water motions are spatially confined and, more importantly, that the geometry of the confinement affects the diffusion behaviour. In the hexagonal phase, the water dynamics were found to be almost bulk-like while for the lamellar phase there were serious restrictions in water mobility. This information could play a vital role in the rationale design of future, novel membrane materials.

While it is not in the scope of this chapter, it is worth mentioning that QENS has also been used to study the motions of hydrogen gas, the fuel in a fuel cell, on PEM fuel-cell catalyst supports. Such studies demonstrated that the interactions between H2 and the carbon support play a significant role in reactant transport in the PEM fuel cell [60, 61].

The aforementioned studies demonstrate the efficacy of QENS for understanding the water transport in PEM materials. Several key pieces of information concerning the fundamental nature of water mobility in PEMs have emerged thanks to detailed and clever analyses carried out by the researchers mentioned. In summary, one can say that the water motions in PEMs, especially Nafion®, have the following characteristics:

1. Water motions occur in confined domains, the nature of which is largely dictated by the nanoscale morphology of the material.

2. Within this confined geometry the water motions are influenced largely by the number of water molecules present. As the number of water molecules decreases, the dynamics are increasingly restricted.

3. With a sufficient level of hydration, the local dynamics are not unlike the motions that occur in bulk water, but the long-range motions are restricted due to the material morphology.

This body of information has contributed significantly to our current under­standing of these materials and will continue to illuminate the path toward the rational design of new PEM materials.