The first such model was introduced by Linus Pauling in 1935 to account for the residual entropy of water ice.
[1] Variants have been proposed as models of certain ferroelectric[2] and antiferroelectric[3] crystals.
[4] The exact solution in three dimensions is only known for a special "frozen" state.
That is, each vertex of the lattice is connected by an edge to four "nearest neighbours".
At any vertex, there are six configurations of the arrows which satisfy the ice rule (justifying the name "six-vertex model").
are related by Several real crystals with hydrogen bonds satisfy the ice model, including ice[1] and potassium dihydrogen phosphate KH2PO4[2] (KDP).
Indeed, such crystals motivated the study of ice-type models.
In ice, each oxygen atom is connected by a bond to four hydrogens, and each bond contains one hydrogen atom between the terminal oxygens.
The hydrogen occupies one of two symmetrically located positions, neither of which is in the middle of the bond.
Pauling argued[1] that the allowed configuration of hydrogen atoms is such that there are always exactly two hydrogens close to each oxygen, thus making the local environment imitate that of a water molecule, H2O.
Thus, if we take the oxygen atoms as the lattice vertices and the hydrogen bonds as the lattice edges, and if we draw an arrow on a bond which points to the side of the bond on which the hydrogen atom sits, then ice satisfies the ice model.
Similar reasoning applies to show that KDP also satisfies the ice model.
Recently such analogies have been extended to explore the circumstances under which spin-ice systems may be accurately described by the Rys F-model.
associated with vertex configurations 1-6 determine the relative probabilities of states, and thus can influence the macroscopic behaviour of the system.
Slater[2] argued that KDP could be represented by an ice-type model with energies For this model (called the KDP model), the most likely state (the least-energy state) has all horizontal arrows pointing in the same direction, and likewise for all vertical arrows.
If there is no ambient electric field, then the total energy of a state should remain unchanged under a charge reversal, i.e. under flipping all arrows.
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.
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
Both the three-dimensional and two-dimensional models were computed numerically by John F. Nagle in 1966[15] who found that
Both are amazingly close to Pauling's rough calculation, 1.5.
[17] The solution for the ice model gave the exact value of
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.
In 1969, John Nagle derived the exact solution for a three-dimensional version of the KDP model, for a specific range of temperatures.
[19] This ice model provide an important 'counterexample' in statistical mechanics: the bulk free energy in the thermodynamic limit depends on boundary conditions.
[20] The model was analytically solved for periodic boundary conditions, anti-periodic, ferromagnetic and domain wall boundary conditions.
The six vertex model with domain wall boundary conditions on a square lattice has specific significance in combinatorics, it helps to enumerate alternating sign matrices.
occurs, in the thermodynamic limit, for periodic boundary conditions,[21] as used originally to derive
The number of states of an ice type model on the internal edges of a finite simply connected union of squares of a lattice is equal to one third of the number of ways to 3-color the squares, with no two adjacent squares having the same color.