Hydrogen bond ordering in ice V and the transition to ice XIII

The proton ordered version of ice V, ice XIII, was recently identified using Raman spectroscopy and neutron diffraction techniques. The transformation, between 108 and 117 K, only occurred in the presence of a small amount of dopant, similar to the proton ordering transition of ice Ih/XI. In this work, we investigate the hydrogen bond fluctuations in ice V and XIII with statistical mechanical techniques that use results from periodic electronic density functional theory calculations as input. We find a number of low-lying hydrogen bond configurations, approximately 20 within 10 K/water above the ground state state configuration, the structure of which agrees with fully ordered ice XIII. Using an analytic theory, graph invariants, we developed effective spin-lattice Hamiltonians governing hydrogen bond fluctuations to perform statistical mechanical calculations for a large simulation cell containing 6048 water molecules. Two models were constructed, one more elaborate than the first, to gauge the robustness of our methods when the unit cell is very complex and a large number of configurations lie close in energy to the ground state. The predicted proton ordering transitions, 62 and 72 K for the two models, are in qualitative agreement with experiment. Occupation probabilities, obtained from our simulations, compare well with values from recent neutron diffraction experiments and help verify our effective Hamiltonians. In both models, we find that a second order phase transition intervenes about 10 K above the transition to ice XIII, but its effect is negligible on the behavior of thermodynamic functions near the transition to ice XIII.