A charge-transfer resistance model and Arrhenius activation analysis for hydrogen ion transmission across single-layer graphene

Abstract

Transmission rates for protons and deuterons across single-layer graphene embedded in Nafion | graphene | Nafion sandwich structures are measured as a function of temperature in electrochemical hydrogen pump cells. Rates of ion transmission through graphene are obtained in the form of area-normalized ion-transfer resistances, and are interpreted in terms of ion-exchange current densities and standard heterogeneous ion-transfer rate constants. An encounter pre-equilibrium model for the ion-transfer step is then used to provide rate constants for the fundamental microscopic step of ion (proton or deuteron) transmission across graphene. Application of this rate model to interpret variable-temperature data on proton and deuteron transmission rates provides values for the activation energy and pre-exponential factor for the fundamental ion transmission step across graphene. Activation energies obtained from the Arrhenius plots for proton and deuteron transmission are as follows; for proton, Eact = 48 ± 2 kJ/mole (0.50 ± 0.02 eV) and for deuteron, Eact = 53 ± 5 kJ/mole (0.55 ± 0.05 eV). The difference between these two values of approximately 5 kJ/mole is in good agreement with the expected difference in vibrational zero-point energies for OH and O-D bonds, albeit with some uncertainty given the uncertainties in the activation energy values. Pre-exponential frequency factor values of 8.3 ± 0.4 × 1013 s−1 and is 4.7 ± 0.5 × 1013 s−1 were obtained for proton and deuteron transmission respectively across graphene. These pre-factor values are both quite large, on the order of the values predicted from the Eyring – Polanyi equation with a transmission coefficient near one. The ratio of 1.8 for the rate pre-factors (H/D) is in reasonable agreement with the value of 1.3 for the ratio of bond vibrational frequencies for OH and O-D stretching, respectively. Taken together, these data support a model in which proton and deuteron transmission across graphene are largely adiabatic processes for which the differences in transmission rate at room temperature are due largely to differences in activation energies.