Chemistry Reference
In-Depth Information
mechanics suffice? Is the statistical mechanics controlled by relative stabilities
E
i
of the different H-bond isomers, or by the vibrational free energies
A
vib
.i
? The an-
swers to these questions will determine the best practical methods for calculating
or simulating order-disorder transitions in ice. Presently, these questions cannot
be answered definitively for all ice phases, although cases studied to date indicate
that classical statistical mechanics is adequate as long as the region of H-bond
symmetrization at very high pressure is excluded, and that despite the small dif-
ferences in
E
i
between H-bond configurations, the differences in
A
vib
,i
are even
smaller. We briefly review the justification for these statements.
To date, there is no evidence that nuclear quantum effects affect order-disorder
transitions in ice. According to classical statistical mechanics, nuclear mass has no
effect on the location of phase boundaries, aside from the very small mass effects
on the Born-Oppenheimer potential energy surface. The very small difference in
freezing temperature of H
2
O and D
2
O at ambient pressures, (0
.
0vs3
.
8
◦
C) is evi-
dence that quantum effects, although present, are still minor. Similarly, the ice Ih/XI
order-disorder transition has been measured [8, 9, 58-60] to take place at 72 K
in H
2
Oand76KinD
2
O. This is one of the ice order-disorder transitions that
occurs at the lowest temperature, yet isotope effects are minor. With the exception
of tunneling that occurs as the H-bonds tend toward symmetric in ice X, to date
it appears that classical statistical mechanics can safely be used to study order-
disorder transitions in ice. As more detailed calculations and experiments emerge,
we may learn of cases where quantum effects must be treated.
An immediate practical question arises as to the relative importance of the
E
i
or
A
vib
.i
in the evaluation of the parti
ti
on function. Differences in
E
i
are known to
be
s
mall. The transition enthalpy
TS
can be estimated using the Pauling entropy
S
3
2
, which sho
uld
be a
n
upper bound to the transition entropy (see dis-
cussion above), giving
H
≈
R
ln
E
in the range of several hundred joules per mole
J mol
−
1
for the known order-disorder transitions in ice. This number is far smaller
than, say, the zero-point energy for a water stretching vibration. Therefore, it may
seem like vibrational effects might be dominant. However, the statistical mechan-
ics of H-bond arrangements is controlled by
differences
in the vibrational free
energy for the H-bond isomers. Here, the evidence indicates that the vibrational
free energy of the various H-bond isomers is so similar, that the effect of vibrations
is secondary. First, the absence of large isotope effects suggests that large zero-
point energies are nearly canceling. Second, the H-bond arrangement calculated
to have the lowest
E
i
has turned out to be the observed low-temperature phase for
ice XI, VIII, IX, XIII, and XIV. The lone exception is ice XV, the low-temperature
form of ice VI, where calculations disagree with experiment (discussed below in
Section III.2.E).
The third piece of evidence that tells us that differences in vibrational free
energy
A
vib
.i
of various H-bond isomers are smaller than the differences of their
equilibrium energy
E
i
is recent calculations performed by Beck and Singer for
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