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Residual entropy of ice

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Click for interactive model:

Description

I'm interested in exploring ice-type models and their residual entropy.

There are many various phases of ice. Currently, I have a visualization of (the oxygen atoms of) Ice Ih, which is ordinary ice, Ice Ic and square ice.

Notes

The oxygen atoms of each of the ice type have a crystal structure. The "ice rule" applies to each ice type because the underlying graph of neighboring oxygen atoms is 4-valent (each oxygen atom has 4 neighboring oxygen atoms).

Since the 3D crystal structure does not involve the hydrogen, equivalent crystal structures are more often discussed in the context of different atoms in a 4-valent arrangement. For example, the oxygen atoms in cubic ice are arranged in a diamond cubic structure.

Each of the underlying graphs happen to be bipartite (neighboring oxygen atoms can be given alternating colors).

The structure of the underlying graphs can be distinguished by the respective coordination sequences, whose n-th element is the number of vertices which are a distance of n steps from a chosen vertex. These are:

  • Square: 1, 4, 8, 12, 16, 20, 24, ... (A008574)
  • Cubic (diamond): 1, 4, 12, 24, 42, 64, 92, ...(A008253)
  • Hexagonal (lonsdaleite): 1, 4, 12, 25, 44, 67, 96, ... (A008264)

(Each coordination sequence begins with 1, 4, since the graphs are all 4-valent.)

It is also interesting to consider the theta functions, which are the generating functions for the number of points of a given Euclidean squared-distance from a chosen point. These are:

  • Square: = 1 + 4 q + 4 q^2 + 4 q^4 + 8 q^5 + 4 q^8 + 4 q^9 + 8 q^10 + ... = θ₃(q)² (Jacobi theta)
  • Cubic (diamond): 1 + 4 q^9 + 12 q^24 + 12 q^33 + 6 q^48 + 12 q^57 + 24 q^72 + ...
  • Hexagonal (lonsdaleite): 1 + 4 q^9 + 12 q^24 + q^25 + 9 q^33 + 6 q^48 + 6 q^49 + 9 q^57 + 2 q^64 + 18 q^72 + + 9 q^81 + 12 q^88 + 3 q^89 + 6 q^96 + 6 q^97 + 18 q^105 + 3 q^113 + 12 q^120 + 7 q^121 + 3 q^129 + 12 q^136 + 6 q^137 + 6 q^144 + 6 q^145 + 6 q^152 + 12 q^153 + 12 q^160 + 24 q^168 + q^169 ...

Formulas for the theta functions for diamond and lonsdaleite structures were worked out by Sloane. Diamond is ½(θ₂(q¹²)³ + θ₃(q¹²)³ + θ₄(q¹²)³, while lonsdaleite has a rather messy formula, Eq. (20).