4.4 Entropy Rule for Room Temperature Structures
4.5.2 Ice rules type model for superprotonic transitions
In the early stages of formulating this work’s model, discussions with other researchers led to the discovery that the evolving rules governing these calculations were very similar to those used in the evaluation of the residual entropy of ice by Linus
Pauling135. Based on the observations of Bernal and Fowler, hexagonal ice (ice Ih) is composed of oxygen ions and protons, with each oxygen atom coordinated by the four closest oxygens residing on the corners of a regular tetrahedron. Hydrogen bonds connect the oxygen atoms with O−O and O−H distances of 2.76 and 1.0 Å, respectively. Each
oxygen atom is surrounded by four potential proton sites, the distance between proton sites on the same hydrogen bond being 0.76 Å, Figure 4.1648.
Figure 4.16 Hexagonal ice: each oxygen in tetrahedrally surrounded by four oxygens and four potential proton sites48.
Bernal and Fowler also concluded that the structures of individual water
molecules in ice were not that different from the those in steam and therefore must satisfy two rules48:
i) two and only two protons are bonded to each oxygen ii) one and only one proton is allowed per hydrogen bond.
To these so called ice rules, Pauling added that135
iii) the hydrogen bonds must be directed approximately towards two of the four neighboring oxygen atoms
iv) the interaction of non-neighboring water molecules does not energetically favor one possible arrangement of protons with respect to other possible configurations so long as they all satisfy i)-iii).
Using these four rules Pauling estimated the number of configurations for a molecule to be
( )
2 24 6*( )
143 2
2 * 4
# *
#
=
=
=
=
Ω
open is site
proton a
y probabilit ions
configurat proton
of protons
of
(4-4)
giving ice a residual molar entropy of R*ln(3/2) = 3.37 J/(mol*K), in extremely good agreement with the experimental data149. Now these rules were first applied to the
relatively static structure of ice, but others found that it equally well explains the increase in entropy of order-disorder transitions in ice polymorphs, clathrate hydrates, and many other water containing compounds such as SnCl2•2H2O, Cu(HCO2)2•4H2O, and
[H31O14][CdCu2(CN)7]150-153. This is in spite of the fact that the compounds vary greatly in both the extent and dimensionality of their hydrogen bonded networks. The application of Pauling’s rules to systems with reorientational disorder is then well documented.
The only remaining logical leap is to apply these ice rules to the tetrahedra found in the compounds in question. A literature search showed that this exact step was
performed by Slater to describe the ferroelectric transition of KH2PO4154. This further application of the ice rules would seem trivial, since each oxygen atom in ice is tetrahedrally coordinated by four oxygen atoms and each phosphate in KH2PO4 is similarly surrounded by four other phosphate groups. However, in ice the six allowed configurations of the protons are crystallographically identical, whereas in KH2PO4 two of the arrangements are different from the other four. The two special configurations
result in a dipole pointing either in the positive or negative c-axis (the preferred axis of the crystal since KH2PO4 is tetragonal) while the other four give polarizations in the plane perpendicular to the c-axis.
The two configurations aligned with crcan therefore have different energies from those perpendicular to cr. This is actually the cause of the spontaneous polarization of the ferroelectric phase, dielectric measurements having shown the c-axis aligned
configurations to have a lower energy than the other four. Thus, a crystal should be completely polarized (ordered) at zero temperature and completely random at high temperatures with the configurational entropy difference between the two states equal to R*ln(3/2) as T ⇒ ∞154. It turns out that the measured entropy change for the ferroelectric transition of KH2PO4 has an excess entropy when compared to R*ln(3/2). This was recently explained by way of local excitation of phosphate defects (HPO4-2 and H3PO4), the formulation of which was given by Takagi in 1948155. These defect pairs add
significantly to the entropy of both the ferroelectric and paraelectric phases and lead to the 12 % increase in the measured transition entropy compared the ice rules value156.
The ice rules have then successfully described the entropy changes of both
disordered ice-like systems and compounds containing tetrahedral groups. This makes the step of applying them to the disordered tetrahedra of the high temperature phases more like a hop. However, this is certainly the first time they have been applied to the superprotonic phases of solid acids, resulting in a very compelling description of the entropic driving force for these transitions, found lacking in the current literature. The ice rules applied to the compounds under consideration are very similar to those given by Slater for KH2PO4, but with the additional complexity that there are now both sulfate and
phosphates in the structure. Besides adding the obvious entropy associated with mixing, calculated with Eq. (4-1), this will also cause the average number of hydrogen atoms per tetrahedron to change from compound to compound. This changes the first ice rule to:
i) only one or two protons will be associated with a tetrahedron (4-5a) There will therefore be two types of tetrahedra in these disordered phases, differentiated not by their central cation, but by the number of protons bonded to their oxygen atoms.
This will add to the entropy of these phases as there will be different possible
configurations associated with the ordering of the one and two proton laden tetrahedra.
The other rules will remain relatively unchanged:
ii) only one proton per hydrogen bond (4-5b)
iii) hydrogen bonds are directed towards oxygen atoms of (4-5c) neighboring tetrahedra
iv) interactions of non-adjacent tetrahedra do not effect the (4-5d) possible configurations of a tetrahedron and its protons
The reference to the configurations of a tetrahedron in Eq (4-5d) is necessary to include the entropic contribution from the crystallographically identical orientations of the tetrahedra.
With this formulation we can adjust Eq. (4-4) to calculate the entropy of these high temperature phases:
=
Ω
positions oxygen
of ts
arrangemen l tetrahdera
of open
is site
proton a
y probabilit ions
configurat proton
of protons
of
#
*
#
*
*
#
#
(4-6)
where the first two terms are evaluated exactly as with ice, only here the coordination need not be tetrahedral. The third term arises from the distinguishable arrangements of one and two proton laden tetrahedra. For example, there are three distinguishable arrangements of one HXO4 and two H2XO4 groups, ten distinguishable arrangements of two HXO4 and three H2XO4 groups, and so on. The final term is caused by the librations of the tetrahedra between their possible orientations that result in multiple oxygen positions for the same hydrogen bond direction, hence increasing the number of
configurations. With Eq. (4-6) we are now ready to calculate the configurational entropy of the high temperature phases!