History
The ice rule was introduced by Linus Pauling in 1935 to account for the residual entropy of ice that had been measured by William F. Giauque and J. W. Stout. The residual entropy, of ice is given by the formula
where is Boltzmann's constant, is the number of oxygen atoms in the piece of ice, which is always taken to be large (the thermodynamic limit) and is the number of configurations of the hydrogen atoms according to Pauling's ice rule. Without the ice rule we would have since the number of hydrogen atoms is and each hydrogen has two possible locations. Pauling estimated that the ice rule reduces this to, a number that would agree extremely well with the Giauque-Stout measurement of . It can be said that Pauling's calculation of for ice is one of the simplest, yet most accurate applications of statistical mechanics to real substances ever made. The question that remained was whether, given the model, Pauling's calculation of, which was very approximate, would be sustained by a rigorous calculation. This became a significant problem in combinatorics.
Both the three-dimensional and two-dimensional models were computed numerically by John F. Nagle in 1966 who found that in three-dimensions and in two-dimensions. Both are amazingly close to Pauling's rough calculation, 1.5.
In 1967, Lieb found the exact solution of three two-dimensional ice-type models: the ice model, the Rys model, and the KDP model. The solution for the ice model gave the exact value of in two-dimensions as
which is known as Lieb's square ice constant.
Later in 1967, Bill Sutherland generalised Lieb's solution of the three specific ice-type models to a general exact solution for square-lattice ice-type models satisfying the zero field assumption.
Still later in 1967, C. P. Yang generalised Sutherland's solution to an exact solution for square-lattice ice-type models in a horizontal electric field.
In 1969, John Nagle derived the exact solution for a three-dimensional version of the KDP model, for a specific range of temperatures. For such temperatures, the model is "frozen" in the sense that (in the thermodynamic limit) the energy per vertex and entropy per vertex are both zero. This is the only known exact solution for a three-dimensional ice-type model.
Read more about this topic: Ice-type Model
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