Chemistry Reference
In-Depth Information
If the energy or free energy of the 2
×
2 unit cell was parametrized according to
the value of
I
2
×
2
13
,I
2
×
2
23
,I
2
×
2
12
,I
2
×
2
15
,
and
I
2
×
2
11
, then a first guess for the energy of
configurations of the 4
4 cell would be in terms of the invariants in Eqs. (31)-(36).
At this level of approximation, the parameters needed to describe the many H-bond
isomers of the 4
×
×
4 cell, the
α
's of Eq. (11), would be known from calculations
for the smaller 2
2 cell, and only direct enumeration or Monte Carlo sampling of
topologies required for the 4
×
×
4 cell. Perhaps comparison with more expensive,
×
detailed calculations for the 4
4 cell would indicate reasonable convergence of
the energy. If not, use of invariants involving bond pairs further separated from
each other would be an option to improve the description. This would involve
invariants for the 4
×
4 cell which have no counterpart in the 2
×
2 cell.
III. ANALYSIS OF THE HYDROGEN-BOND
ORDER-DISORDER IN ICE
A.
Ice Ih/XI
As reviewed in Section I, our fundamental notions of H-bond disorder in ice Ih
have been shaped by Linus Pauling's prediction of the residual entropy of ice [2]
and its experimental confirmation by Giauque and Stout [6]. Close to the melting
point of ice, the H-bonds are fully disordered subject to the ice rules. The motion of
protons within the ice lattice occur via Bjerrum (orientational) or ionic (protonic)
defects [98]. As ice is cooled to low temperatures, H-bond rearrangements come
to a halt, and a glassy transition has been observed to occur near 110 K [107]
prohibiting the transition to a proton-ordered phase.
As tabulated in [108], numerous dielectric studies on powder and single-crystal
samples have been performed over the years. Kawada and Niinuma [109] and
Kawada [110] reported results on dielectric studies on single crystals with a Curie-
Weiss temperature of 46 and 55 K for H
2
O and D
2
O [58], respectively, with the
electric field parallel to the
c
-axis. Studies by Johari and Whalley [108] on pow-
dered samples of H
2
O indicate a Curie-Weiss temperature significantly lower, 6.2
K. However, in samples doped with impurities, particularly KOH, a clear calori-
metric signature of a first-order phase transition is observed at 72 K with weak
dependence on the concentration of the KOH impurity [8, 58]. In experiments with
samples of D
2
O, the transition temperature is shifted by 4
◦
and occurs at 76 K.
Antarctic ice samples have been examined with neutron diffraction and Raman
spectroscopy. It is believed that these samples, kept at a constant low tempera-
ture for thousands of years, have equilibrated to a proton-ordered arrangement
[111]. Those studies indicate that a second-order phase transition to an H-bond or-
dered phase of ice occurs at 237 K, which is significantly larger than the observed
transition temperature in KOH doped ice samples. Neutron diffraction spectra
of Greenland ice samples, prepared under similar conditions, showed no distinct
