On the nanosecond proton dynamics in phosphoric acid-benzimidazole and phosphoric acid-water mixtures

The unique proton conduction mechanism of phosphoric acid is important for the functions of complex phosphate containing biological and technological systems (e.g. phospholipid membranes and polybenzimidazole phosphoric acid membranes for high-temperature PEM fuel cells). In neat phosphoric acid structural proton diffusion, i.e. proton hopping between phosphoric acid molecules, is superimposed onto hydrodynamic diffusion of the molecules in the viscous liquid. In this study we separate the two dynamic contributions on the nanosecond timescale for the model systems phosphoric acid-water and phosphoric acid-benzimidazole. We demonstrate that 1H NMR dipolar relaxation measurements are controlled by hydrodynamic diffusion for the investigated conditions, while 17O NMR quadrupolar relaxation measurements reflect local proton displacement as part of structural diffusion. Quasielastic neutron scattering (QENS) applying high resolution backscattering spectroscopy (nBSS) confirms structural proton diffusion measurements using PFG-NMR in phosphoric acid-benzimidazole mixtures at different concentrations. With increasing benzimidazole content proton diffusion coefficients on the nanosecond scale decrease, thus following the trend of reduced hydrogen bond network frustration. The momentum transfer (Q) dependence of the width of the QENS spectra indicates the jump diffusion mechanism and can be scaled to a master plot both for different temperatures and different benzimidazole contents. This indicates a fundamentally unchanged structural proton diffusion process, however, with a lower probability of occurrence for successful intermolecular proton transfer with increasing benzimidazole content. Results of this work enable a better separation of different diffusion processes on short timescales also in more complex phosphoric acid containing systems.

suspect that a slightly unequal Teflon content and/or foil stretching in the different sample holders might be the reason. Therefore we have tried to minimise this effect for each sample by judging from the highest temperature runs with the widest QENS and by trying out different subtraction weights by adding to the empty cans with Teflon liner some empty can runs without Teflon liner. This method has the advantage that strength of the Teflon Bragg contamination could be slightly varied without affecting the flat background arising from other sources. The same weight of subtraction was then used for all temperatures of each sample. We conclude that in nearly all cases this subtraction problem can be considered as minor and does not affect the results from data fitting.

Data Fitting:
Data were fitted using the LAMP module 'str_fit'. Macros written in IGOR Pro 7 were then used to extract and visualize the fitresults. In a first step we tried to keep the fit model in LAMP as simple as possible, in order to avoid correlation of fit parameters.

Model 'LorBg':
This is the most simple fit model used which gave quite reasonable results as described in the paper. A single Lorentzian convoluted with the experimental resolution function in addition to a flat background is used. The flat background is the signature of fast picosecond proton motions seen in the wider energy window of neutron TOF instruments and known from simulation to happen on the sub-ps time-scale.
Phosphoric Acid (PA), H3PO4: At temperatures measured below the melting temperature (185K, 248K, 278K, 290K) the fitted Lorentzian FWHM was much smaller than the resolution (at T=290K a fitted Lorentzian FWHM~ 0.1 -0.2 µeV). The first measured temperature for which PA did show a significant broadening (0.8 µeV < FWHM < 30 µeV) was T = 339 K (PA339). The most important parameters of the fit are summarized in Fig. SI

Other Simple Fit Models:
Other simple fit models were tried in order to see how stable and reliable the resulting fit parameters are: 'ElLorBg', with an additional elastic resolution signal, 'LorLor', which only two Lorentzians convoluted with the resolution function and 'LorLorBg', where an additional flat background was added.

Master plots of fitted FWHM:
As model independent check for Q-regions where the width follows a FWHM ~ Q 2 behaviour we plot log10(FWHM) versus log10(Q 2 ). This is shown in  The FWHM of PA show at all temperatures above Tm a region where FWHM~Q 2 (dashed line marked slope=1) and turn away from this Q-dependence before leveling off at high Q in a roughly Q-independent linewidth, as would be expected for jump models like (Singwi-Sjölander, Hall-Ross, Chudley-Elliott) for which the high Q region can be interpreted by the FWHM being proportional to the inverse residence time in between fast proton jumps.
Clearly this leveling off is observed even for temperatures for which the fitted FWHM at high Q is in a range considered as reliable. From such type of plots we have determined shift factors for each temperature, which superimpose the data in an optimum way to a mastercurve as is shown for PA in Fig. 7 in the manuscript. The nice superposition in all Q-range   The parameters and individual detector fits for 6PA1BI at higher temperatures are shown in        (Figures SI-QENS_18). We ascribe this to the presence of even stronger additional elastic contributions or a change in line shape, though again we cannot exclude from these fits that the fast component, described as a flat background, has a curvature in the energy window of the spectrometer.     QENS_23 -SI-QENS_26 ). We ascribe this to the presence of even stronger additional elastic contributions or a change in line shape, though again we can not exclude from these fits that the fast component, described as a flat background, has a curvature in the energy window of the spectrometer.  6 PA1 bisBI fit s wit h ' LorBg' 28 4 K 32 7 K 36 2 K 37 9 K 39 4 K 43 0 K

Activation energies:
In the following we try to get a realistic estimate for the errors of the activation energies extracted from 'LorBg'-fits. In the chapters before the activation energies were determined either from fitting the shift factors in the low Q-range and the standard deviation resulted from this fit or from ~DQ2 fits in the low Q range. The corresponding fit error bars seem somewhat low and depend on the Q-and temperature range chosen for scaling the FWHM data and they do not take into account systematic errors. These Q-dependent activation energies are shown in Fig.SI-QENS30 a)  sample with high PA content is closest in activation energy to PA, but the sample with high BI content, 3PA1BI is clearly off. We may add as a critical comment that the fit quality of the LorBg fits is not taken into account in this estimation.

Fits to HWHM versus Q data
In the main text a selection of fit models for the Q dependence of HWHM for the different samples is discussed. In the following the all DQ^2 fits are shown trough which diffusion coefficients in fig. 11 in the main text have been obtained. Additionally we show the HR and SS model fits for all samples and temperatures.
The diffusion length on the nanosecond scale calculated with the Smolukowski-Einstein relation from the diffusion coefficient D and time τ of the HR fit is shown in figure SI-QENS 35.
We speculate that this length scale does not correspond to an individual jump length, but rather to a diffusion length on the nanosecond scale, as individual transfer events occur already on the picosecond scale. We plan to investigate this question again when evaluating additional neutron data taken on a ps-time scale. With free fit parameters, i.e., not fixing D to values obtained through PFG-NMR as in previous works (see main text), no clear trends in temperature dependence can be observed. Scattering is rather large due to the uncertainty in fitting τ. We feel that by itself there is at this moment insufficient additional information in this length scale.