Ih and ice Ic

Pauling has shown as early as 1935, the value of this residual entropy can be easily evaluated (13). For each 0-atom, four energy boxes exist, which correspond to two covalent bonds and two H-bonds, and there are C4 = (4 /2 2 ) = 6 possibilities to distribute these two covalent and H-bonds among these four energy boxes (C4 is the number of possible combinations of four objects taken two by two). With no ice-rule, which means with a complete random distribution of the H-atoms between the two positions they can take 

Using the Boltzmann s definition of entropy, we find the residual entropy S, corresponding to this number of degenerate states for one mole of H2O molecules to be, with R, the gas constant equal to 8.31 J deg  

This is equal to the measured experimental value. Let us note that the Pauling s evaluation is a first approximation, as it treats all 0-atoms as being independent and neglects correlations of configuration between neighbour 0-atoms. This correlation enters for a small part only in the value of the residual entropy, but it has up to now not been calculated. The existence of such a residual entropy has no important consequences, except that if it is not properly taken into account, thermodynamical values may differ when measured from different experiments, thus lacking consistency. 

These librations are not free rotations, as in water vapour, but hindered rotations that are governed by force constants due to H-bonds between H2O molecules. The fact that Ph o appears at such high a wavenumber is due to the particularly small moment of inertia of the H2O molecule. In ice these librations have a limited amplitude, and may consequently be considered as vibrations. This will not be the case in liquid water, where librations will be seen to display great amplitudes that are at the origin of the dramatic differences between the properties of ice and liquid water. 

Infrared spectroscopy on thin samples of ice about 10-20 nm thick have also shown that the H-bond network on the surface of ice in contact with water vapour is not exactly the same as in the bulk, but more hquid-hke, indistinguishable from that of liquid water (21). This transition layer extends over more than 10 HjO layers at 0 °C but rapidly shrinks (22), becoming hardly a single layer at 10 °C. The structure of this layer on the surface of ice is not precisely known. Thus X-ray diffraction indicates that its density is more in the vicinity of 1.2gcm than around the expected value 1 (23). It at least shows that the arrangements of H2O molecules on the ice surface are definitely different from those in the bulk. 

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Fig. 3. The upper two diagrams are the calculated dispersion curves for ice Ih and ice Ic based on a simple LD model containing O atoms only. An 0-0-0 bending and an 0-0 stretching force constants, G = 0.33 eV/Rad and K = 1.1 eV/A were used. The diagram second below shows three curves of g(co) for the three particular reciprocal directions in ice Ic. The lowest diagram is the completely BZ integrated g(o)) for both ice Ic and Ih. It differ considerably from the curves above, indicating the incomplete BZ integration can be misleading.

Figure 15. (a) Temperature dependence of the Gibbs free energy for ice Ih (solid line) and ice Ic (dotted line) at atmospheric pressure, where contributions from the configurational entropy and anharmonic vibrations are omitted, (b) Temperature dependence of the energy which is defined as the sum of the interaction energy at its minimum structure and the vibrational energy for ice Ih (solid line) and ice Ic (dotted line). 

Figure 84 Tetrahedral orderings in ice Ih (left diagrams) and ice Ic (right diagrams). In both bottom drawings the axis defined by O-atoms 1 and 5 is the vertical axis of the paper sheet. The two npper diagrams are obtained from the lower ones by a 90° rotation around the horizontal axis of the paper sheet. The tetrahedra defined by O-atoms 1, 2, 3, 4 and 5 are the same in all drawings.

Figure 4. Voltage (microvolts) generated by the differential tho-mopile upon heating 2.7g HDA sample at 1 bar. From left to right, exothermic peaks correspond to the HDA-to-LDA ( 117K), LDA-to-ice L ( 152K), and ice Ic-to-ice Ih ( 225K arrow) transition, respectively. The broad endothermic peak at T< 117K, indicates continuous structural relaxation of HDA prior to the HDA-to-LDA transition. Adapted from Ref. [48].

The effects of pressure on the phase transition of liquid water to ice (and within the ice phase itself) are complicated by the formation of several pressure-dependent ice polymorphs (Chaplin, 2004 Franks, 1984, 2000 Kalichevsky et al., 1995 Ludwig, 2001). Thirteen crystalline forms of ice have been reported to date Ih (hexagonal or normal or regular ice), Ic (cubic... 

We are, therefore, led to -propose that low temperatures Hrandom network structure of mixed lattice parentage, i. e. that the structure is characterized by locally random 000 angle deviations from ice Ih or Ic), ice II and ice III type lattices. 

At high pressures a number of polymorphic forms of ice exist. They are coded as ice(II). .. ice(IX), ordinary ice being (Ih). The letter h is added to specify that this form belong to the hexagonal system, and to distinguish it from ice (Ic), which belongs to the... 

Ordinary hexagonal ice Ih and cubic ice Ic are formed at normal pressures, the latter being stable below — 120°C. Their hydrogen bonding patterns are... 

In all instances where this disorder was actually observed by X-ray or neutron diffraction methods, X and A are hydroxyl oxygen atoms, as in the ices Ih and Ic, in certain of the high pressure ices, and in the cyclodextrin hydrates, see Parts III and IV. In ice Ih the disorder gives rise to the well-known residual entropy of 0.82 0.05 cal deg-1 [114 to 117]. 

Here we should note that the dispersion curve calculation has provided all the information required to obtain the response from a single crystal sample aligned along a specific direction in Q. Indeed, if such an experiment were realistically feasible it would be the preferred technique. This is because the dispersion curves would be measured directly and the detailed information about the force field could be extracted. However, this is often not practical, at least for the exotic phases of ice and powdered samples were used. For ice Ih, single crystals are widely available (but a large crystal of ice Ic has not been obtained), after many attempts [49,55], reliable dispersion curves have yet to be obtained. This is due to the proton disordering in the structure of ice Ih and hence all the optic modes are localised. 

As an example of the information lost by exploring the DOS of powdered samples we compare the calculated dispersion curves of ice Ih and Ic. They have quite different dispersion curves in the translational region due to their different symmetries, but these are inaccessible to neutron dispersion measurements due to the proton disorder in the structures. Only the DOS can be measured. As a result, the detail of the information in the dispersion curves is lost, or at least degraded, by comparing only the DOS. From both experiments [22,55] and calculations, we have found that the two ices share an identical spectrum as shown in Fig. 3. This is because they share the same local structure in their lattices (the tetrahedral symmetry) and the same local force field. If the integration over the first BZ is incomplete (i.e. if too few q-points were used), there would be a considerable difference between calculation and observation spectra. 

At high pressure, there are many kinds of ice polymorphs and the phase diagram of water is complicated. In ice VlII and XI, protons are ordered while most of ice phases have proton disordered forms. In ambient condition, satisfied is the ice rule water exists as an H2O molecule and a proton sits between two adjacent oxygens. In ice Ih, the number of configurations arising from the proton-disordering is approximately (3/2) " for Nw molecule system[16. This is also true for ice Ic and some other ices except for proton-ordered forms. 

The densities of state of only low frequency modes for ice Ih and Ic are shown in Figure 14 since no difference is seen in high frequency region... 

The molar volumes of the two ice forms are plotted in Figure 16. The volumes for ice Ih and Ic increases in the temperature range above 60 K as temperature is raised. However, both have negative slopes below that temperature. The thermal expansivity of ice Ih is plotted against temperature in Figure 17. The calculated thermal expansivity for ice Ih,... 

The identification of the isomers is unequivocal only in the tetrahedral structures (Ih, Ic, VII, VIII, X). Nevertheless, in some phases we can replace the unequivocal formalism of the tetrahedral structures with a looser definition of dimer orientations. In Figure 3 we show the distribution of angles for ice III and ice IX at 5 K. These dimer angles (see Figure 3) represent the input orientations created from the measurement data. At high temperatures the peaks of the ice III distribution smooth out and only four peaks remain. 

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