Residual entropy of water ice

Water is known to form ~20 different solid phases. In many of them, the orientations of water molecules in the crystal lattice are disordered — the crystal itself is one of a macroscopic number of allowed configurations. This leads to residual entropy, seemingly violating the 3rd law of thermodynamics. Half a century ago, configurational entropy was calculated exactly for 2D ice model on a square lattice (Lieb, 1967). For other lattices, both 2D and 3D, not only the analytic values of the residual entropies are not known, but also their numerical estimates are surprisingly inaccurate. For 3D ice models on hexagonal (ice-Ih) and cubic (ice-Ic) lattices, their residual entropies have been estimated by Pauling (1932) and improved by Nagle and Onsager (1966) but since then there was no noticeable improvement neither in analytic results nor numerical calculations. This left some unresolved questions, which I will address during the talk.