Phase transition of an ice-proton system into a Bernal-Fowler state Ivan A. Ryzhkin 1 and Victor F. Petrenko 2 1 Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, Moscow Region 142432, Russia 2 Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire 03755 ~Received 3 May 2000! We present a microscopic model of an ice-proton system. A mean-field approximation was used to study the disorder-partial order transition in the proton subsystem of ice. This analysis revealed that ice rules arise as a result of the second-order phase transition. From the theoretical point of view, above the phase-transition temperature the protons should be distributed over all possible positions without any restrictions. However, in real ice under zero pressure the disordering of the proton lattice apparently leads to ice melting. We also derived the Landau-Ginzburg equation suitable for the model under consideration and showed that the theories based on ice rules must be modified when one is working in the region of the critical temperature. Ordinary water can solidify in 12 crystalline and several amorphous forms. 1 In this paper, however, we deal only with the most widespread form, hexagonal ice, known as ice Ih. This ordinary form of ice can be obtained either by freezing water at atmospheric pressure or directly from water vapor at not too low temperature. 1 The atomic lattice of the hexagonal ice is made up of oxygen ions and protons ~see Fig. 1!. The oxygen ions ~shown by open circles! are arranged in a regu- lar wurtzite lattice, which, if oxygen ions are substituted for those of groups II and VI, is identical to the lattices of II-VI compounds. The important property of the oxygen lattice is that each oxygen ion has four neighbors, at the corners of a regular tetrahedron. There are two possible proton sites on each hy- drogen bond and four of these sites adjacent to each oxygen. Thus for the crystal with N water molecules there are 4 N positions for 2 N protons. The proton distribution over these positions must satisfy two ice rules: ~i! there are two and only two protons adjacent to each oxygen; and ~ii! there is one and only one proton per bond. 2 Any proton configuration satisfying the ice rules is called a Bernal-Fowler configura- tion ~Bernal-Fowler state!. Pauling calculated the number of the proton configurations and assumed that each one has the same energy. 3 This led him to a conclusion that contradicted the third law of thermodynamics. Namely, he found that zero-temperature entropy in ice had nonzero value equal to k ln( 3 2 ) per molecule; his results were in a remarkable agree- ment with the experimental work of the time. 4 Since then, the ice rules have become a cornerstone of ice theory. A series of statistical models has also been named after them. 5 Why and when are the ice rules satisfied? What happens to the ice rules when temperature rises? Under what circum- stances might the ice rules be violated? Until now we have not had well-founded answers to these questions. The pur- pose of the present paper is to formulate a microscopic model capable of predicting the behavior of the proton sys- tem in ice. We will then study that model using a mean-field approximation. We will show that the ice rules arise as a result of the second-order phase transition as the temperature decreases. Above the critical temperature the ice rules are not satisfied and the proton subsystem becomes totally disor- dered. That disordered phase of ice may have very high elec- trical conductivity if it could be actualized. We begin our study with introduction of a model Hamil- tonian, which comprises the basic experimental facts about the structure of an ice proton system. Hereinafter we will use a Hamiltonian of the lattice gas model, usually applied in the theory of fast ionic transport: 6 H 5 1 2 ( i a, j b V ~ r i a 2r j b ! n i a n j b , ~1! where i a designates the proton lattice sites ~i designates an oxygen site and a designates four proton sites adjacent to it!. The sum is taken over all pairs of nearest neighbors, V ( r i a 2r j b ) is the energy of interaction between the protons lo- cated at sites i a and j b , and n i a are occupation numbers, equal to 0 or 1 for empty or occupied sites, respectively. FIG. 1. An elementary cell of hexagonal ice Ih. The light circles are oxygen ions ~labeled by i!; the dark circles are protons ~labeled i a !. The unit vector e i a is directed from the oxygen ion i along the hydrogen bond a. There are four proton sites adjacent to each oxy- gen ion. The oxygen-hydrogen distance r OH 50.1 nm, the hydrogen bond length r OO 50.276 nm, and the distance between proton sites on the same hydrogen bond r HH 50.076 nm. PHYSICAL REVIEW B 1 NOVEMBER 2000-I VOLUME 62, NUMBER 17 PRB 62 0163-1829/2000/62~17!/11280~4!/$15.00 11 280 ©2000 The American Physical Society