(2) at any given time a proton may be part of a Grotthuss chain (GC), which we define as a chain of 3 or more OH groups connected by hydrogen bonds arranged such that a proton can move forward and backward along the chain without any OH rotations; (3) the length of a GC is finite and unlikely to include more than about 10 OH groups; (4) GCs can break and form new GCs through OH rotations; (5) the GCs are long-lived compared with the time required for proton to hop from one O to a neighboring O; multiple hops along the chain will therefore typically take place before the chain breaks and a new chain is formed.