A Mathematical Primer on Water Ice

TL;DR

The paper surveys the rich geometry and topology of water ice polymorphs, recasting crystal structures as lattices and graphs to contrast Ih, Ic, II, XI, and other forms. It presents three key comparison tools—the radial distribution function, quotient graphs, and local hydrogen-bond topology—and demonstrates how seemingly similar Ih and Ic diverge in higher-order structure and energy. A central theme is the evolution from local tetrahedral motifs to complex stacking, proton ordering, and phase behavior, including the residual entropy of ice and its refined theoretical and experimental treatments. The work emphasizes mathematical formalisms that enable precise characterization of ice structures, with implications for modeling water in confined or biological environments and for understanding proton-ordering phenomena in astrophysical contexts.

Abstract

Water adopts many different crystal structures in its solid form. These provide insight into potential structures of water even in its liquid phase, and they can be used to calibrate pair potentials used for simulation of water. In crowded biological environments, water may behave more like ice than bulk water. The different ice structures have different dielectric properties. This brief primer is intended to facilitate further research.

Figures (22)

• Figure 1: The tetrahedral basis for the crystal structures of both forms of ice I. The triples of numbers are the Cartesian coordinates of the water positions in a cube of side two units. This local structure forms the basis for both the cubic structure of the diamond lattice for ice Ic and the hexagonal structure of ice Ih.
• Figure 2: Radial distances between oxygen centers in two different forms of ice one: ice Ih and ice Ic. The left two columns are the radial distances in mathematical and physical units, and the third and fourth columns are the numbers of water molecules at that distance: $N_h$ for ice Ih and $N_c$ for ice Ic.
• Figure 3: The radial distribution of ice Ic (a) and ice Ih (b). The horizontal axis is in mathematical units for ice, as in the first column in the table in Figure \ref{['tabl:radisticeo']} (multiply by 1.5935 to get Ångstroms or by $0.8433$ to get atomic units). The vertical axis represents the number of oxygen pairs at the given distance, that is, the third and fourth columns in the table in Figure \ref{['tabl:radisticeo']}.
• Figure 4: The fundamental finite graphs of ice Ih (a), ice Ic (b), and ice II (c).
• Figure 5: The local graph of hydrogen bonds in both forms of ice I.

Theorems & Definitions (3)

• Definition 1.1
• Definition 1.2
• Definition 1.3