Geometrical Frustration - Water Ice

Water Ice

Although most previous and current research on frustration focuses on spin systems, the phenomenon was first studied in ordinary ice. In 1936 Giauque and Stout published The Entropy of Water and the Third Law of Thermodynamics. Heat Capacity of Ice from 15K to 273K, reporting calorimeter measurements on water through the freezing and vaporization transitions up to the high temperature gas phase. The entropy was calculated by integrating the heat capacity and adding the latent heat contributions; the low temperature measurements were extrapolated to zero, using Debye’s then recently derived formula . The resulting entropy, S₁ = 44.28 cal/(K•mol) = 185.3 J/(mol•K) was compared to the theoretical result from statistical mechanics of an ideal gas, S₂ = 45.10 cal/(K•mol) = 188.7 J/(mol•K). The two values differ by S₀ = 0.82±0.05 cal/(K•mol) = 3.4 J/(mol•K). This result was then explained by Linus Pauling, to an excellent approximation, who showed that ice possesses a finite entropy (estimated as 0.81 cal/(K•mol) or 3.4 J/(mol•K)) at zero temperature due to the configurational disorder intrinsic to the protons in ice.

In the hexagonal or cubic ice phase the oxygen ions form a tetrahedral structure with an O-O bond length 2.76 Å (276 pm), while the O-H bond length measures only 0.96 Å (96 pm). Every oxygen (white) ion is surrounded by four hydrogen ions (black) and each hydrogen ion is surrounded by 2 oxygen ions, as shown in Figure 5. Maintaining the internal H₂O molecule structure, the minimum energy position of a proton is not half-way between two adjacent oxygen ions. There are two equivalent positions a hydrogen may occupy on the line of the O-O bond, a far and a near position. Thus a rule leads to the frustration of positions of the proton for a ground state configuration: for each oxygen two of the neighboring protons must reside in the far position and two of them in the near position, so-called ‘Ice rules’. Pauling proposed that the open tetrahedral structure of ice affords many equivalent states satisfying the ice rules.

Pauling went on to compute the configurational entropy in the following way: consider one mole of ice, consisting of N of O2- and 2N of protons. Each O-O bond has two positions for a proton, leading to 22N possible configurations. However, among the 16 possible configurations associated with each oxygen, only 6 are energetically favorable, maintaining the H₂O molecule constraint. Then an upper bound of the numbers that the ground state can take is estimated as Ω<22N(6/16)N. Correspondingly the configurational entropy S₀ = k_Bln(Ω) = Nk_Bln(3/2) = 0.81 cal/(K•mol) = 3.4 J/(mol•K) is in amazing agreement with the missing entropy measured by Giauque and Stout.

Although Pauling’s calculation neglected both the global constraint on the number of protons and the local constraint arising from closed loops on the Wurtzite lattice, the estimate was subsequently shown to be of excellent accuracy.