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In this chapter we will review the recent developments in simulating and modelling proton transport. We will put a special emphasis on studies employing classical and quantum molecular-dynamics simulations, but also include basic studies that have focussed on model systems using accurate quantum-chemical methods. Proton-transport and diffusion phenomena in liquids – such as water, inorganic acids, or organic liquids – will be discussed as well as in biomolecules, solid-state materials, and at the solid-liquid interface. Many of these materials are used in proton-transporting fuel-cell membranes, so that membrane materials will be the focus of the last section.

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