Residual Entropy of Square Ice

Abstract

At low temperatures, ice has a residual entropy, presumably caused by an indeterminacy in the positions of the hydrogen atoms. While the oxygen atoms are in a regular lattice, each O-H-O bond permits two possible positions for the hydrogen atom, subject to certain constraints called the "ice condition." The statement of the problem in two dimensions is to find the number of ways of drawing arrows on the bonds of a square planar net so that precisely two arrows point into each vertex. If $N$ is the number of molecules and (for large $N$) ${W}^{N}$ is the number of arrangements, then $S=Nk\mathrm{ln}W$. Our exact result is $W={(\frac{4}{3})}^{\frac{3}{2}}$.