The 6-vertex model of hydrogen-bonded crystals with bond defects

Abstract

It is shown that the percolation model of hydrogen-bonded crystals, which is a 6-vertex model with bond defects, is completely equivalent with an 8-vertex model in an external electric field. Using this equivalence we solve exactly a particular 6-vertex model with bond defects. The general solution for the Bethe-like lattice is also analyzed.

Hydrophobic Hydration in an Orientational Lattice Model

“…A technical point which arises from our studies is the difference in the symmetry of the temperature−concentration phase diagram as derived from Bethe or Monte Carlo calculations (compare Figures , left, and 6, right). This result confirms an interesting theoretical question: the representation of the model on the Cayley tree leads to a loss of orientational information, as already pointed out by Izmailian at al . and confirmed in a previous work …”

“…Pure square water is a particular case of the general six-vertex model with bond defects discussed by Izmailian et al The authors show that the general six-vertex model with bond defects may be mapped onto an eight-vertex model in the presence of an electric field and that an exact solution is available only for a very particular situation: in the case where two specific states, among the six possible states, have zero Boltzmann weight (states 1 and 2 or 3 and 4 of Figure a). However, for the water model, this condition is not fulfilled and an exact solution is not available, so one has to resort to a Bethe lattice (quasichemical approximation) in order to obtain an analytical solution.…”

“…The neutrality condition represented by the Kronecker delta yields a very simple form for the partition function (eq 1) Defining R l = g l (−)/ g l (+), the recursion relations on the tree are given by where t = k B T /ε. For L → ∞, the fixed point solution converges to R = 1, and using the Gujrati prescription, , it is possible to write the free energy per particle for pure square water as: For the specific heat, we have: which is free of singularities. Note that an independent bond approximation for square water on a Bravais lattice, with Z N = (1 + e 1/ t ) 2 N for two states per bond, discussed by Nadler and Krausch, leads to the same expression for the specific heat.…”

“…

2 Square water on the Bethe lattice (quasichemical approximation). Each site of an usual Bethe lattice is placed by a square with four Ising spins on the vertexes . Square water states are identified by Ising spins σ = ±1 (σ = 1 denotes the presence of the hydrogen atom, while σ = −1 indicates its absence), and solute particles are associated with four spins σ = 0.

…”

“…In spite of its simplicity, square water has no analytical solution. As proven by Izmailian and collaborators, square water may be mapped on a nonsymmetric eight-vertex model that lacks an analytical solution . Monte Carlo simulations 24,26 show that the square water model does not exhibit a phase transition related to hydrogen bond ordering, and an independent bond approximation 24 gives a good description of its thermal behavior, a feature which disappears in the presence of solute 26 or under an electric field …”