FLOW IN POLYCRYSTALLINE ICE

Part 2 - Background information

By Chris Wilson and Brett Marmo

2.4 Bernard-Fowler rule

The Bernal-Fowler rule allows the rotation of water molecules within the Ice Ih lattice by hydrogen atoms jumping sites. This movement requires that all hydrogen atoms move simultaneously or the presence of point defects. Bernal & Fowler (1993) suggested that a single hydrogen atom lies on a line between each oxygen atom. The angle between oxygen atoms within Ice Ih is 109°, thus only a small variation of the H-O-H angle in the gas phase is necessary to accommodate hydrogen in the Ice Ih lattice. The hydrogen atoms lie 1Å from their associated oxygen atoms and 1.76Å from the closest neighbouring oxygen atom. There are six possible configurations of hydrogen atoms around oxygen atoms that satisfy this arrangement.

A statistical model of the position of hydrogen atoms was produced by Pauling (1935) based on the six possible configurations of hydrogen atoms within Ice Ih. The statistical model is known as the Bernard-Fowler rule and is defined as ideal crystal based on the assumptions that:

1. Each oxygen atom is bonded to two hydrogen atoms at a distance of 0.95Å to form a water molecule;
2. Each molecule is orientated so that its two hydrogen atoms face two, of the four, neighbouring oxygen atoms that surround in tetrahedral coordination;
3. The orientation of adjacent molecules is such that only one hydrogen atom lies between each pair of oxygen atoms;
4. Ice Ih can exist in any of a large number of configurations, each corresponding to a certain distribution of hydrogen atoms with respect to oxygen atoms.

There is an electrostatic attraction between the positively charged hydrogen nucleus () of one molecule and the negatively charged lone pair () of a neighbouring molecule. Hydrogen atoms therefore are most stable when aligned along an axis parallel to a neighbour's lone pair. The electrostatic attraction increases the distance from the oxygen atom and its associated hydrogen atoms from 0.96Å (in the gas phase) to 1.011Å. The lattice energy, L, for Ice Ih is the difference between the energy of a motionless water molecule in the gas phase at 0°K and its energy in Ice Ih at 0°K. The lattice energy for Ice Ih is L=0.58 eV (Hobbs 1974). If the lattice energy is attributed to hydrogen bonding, then the energy of the hydrogen bond will be half the value of the lattice energy (as each molecule contains two hydrogen atoms). Thus the energy of the hydrogen bond is 0.29 eV.