Topological modelling of the hydrogen bond network of water cluster and proton hopping in a hydrated polyelectrolyte membrane

Abstract

The water cluster is characteristically constructed through the hydrogen bond formation in various types of aqueous solutions, especially in the pore of nano-materials such as polyelectrolyte membranes. The proton conduction in such a water cluster depends on the topology of the hydrogen bond network, since the movement of the proton is mainly driven by hydrogen bond exchange in the water cluster. The first-principle electronic state calculation should be straightforwardly adopted in order to estimate the proton conductivity because the change in hydrogen bond formation is associated with the change in the electronic states of the cluster. Although the first-principle calculations of proton conductivity are basically available to estimate the rate of proton hopping in the water clusters, it would be extremely time-consuming to simulate the dynamic structure in an inhomogeneous morphology of nanometre length scale such as in the polyelectrolyte membrane based on the electronic structure theory calculations. Here, the characterisation of the hydrogen bond network via a dynamically directed graph was evaluated to calculate the dynamical properties in the global structure of nanometre length scale inhomogeneous morphologies. The static and dynamic properties of the network structure were analysed by following the treatment of the graph theory. The mean rate of Hamming displacement (MHD) was defined to quantitatively express the dynamics of the hydrogen bond network. The average of the differences between the Hamming distances at different time steps, as the definition of the MHD, showed that the relaxation of the hydrogen bond network in polyelectrolyte membranes was slower than in the pure water because of the stronger clustering in a local area due to the mesoscopic morphology or confinement effect in nano-pores. Furthermore, proton hopping was modelled as a random walk on the dynamical hydrogen bond network, and then the diffusion coefficients of proton in the Nafion membranes were estimated as comparable results with the previous literatures.