2.4 Conclusions and outlook
6.3.2 Crystalline phases in the water–tert-butylamine
There are eleven known crystalline phases in the water–tBA system, including two anhydrous forms and the three structures reported in this work (Table 6.5). Six of these structures were reported by Dobrzycki et al., along with a relationship for the variation of density with the hydration number of each phase.5 Such relationship takes the form of an inverted Morse potential:
ρ(N) =s−de(1−e−a(N−re))2 (6.3) Were ρis the density of each crystalline phaseN (denoted by its hydra-tion number), normalised with respect to ice Ih, sis 1.076 g/cm3,de is 0.163 g/cm3,ais 0.179, andre is 4.32. Values for the four parameterss, de,a, and re were obtained by fitting the equation to the experimental densities.5 The validity of the proposed relationship was confirmed by its ability to predict the densities of the tert-butanol water clathrates.5 An important observation by Dobrzycki et al. is the existence of a region of hydration numbers between 1 and 7¼ for which no crystal structures could be obtained. The maximum density, predicted for a po-tential 4.32-hydrate, falls in this region, hence structures with hydration numbers between 4 and 5 are expected to show the most efficient pack-ings. It was stated that “this gap should almost be expected because it represents a fuzzy guest/host transition area without nucleation/crystal growth”,5 based on the possible interactions between water molecules in the structure. For the lowest hydration numbers, water molecules cannot interact between them, as they are located in the intermolecular spaces in the packing of tBA molecules. In theory, as the hydration number
in-Table 6.5: Crystalline phases of the water–tBA system.
N Structure type T orPb ρ Norm.ρ
Authors (g/cm3) (g/cm3)
0 Anhydrate 133 K 0.860 0.857 Dobrzycki et al.5
0 Anhydrate 0.78 GPa 0.925 − This work
0.25 Simple hydrate 273 K 0.884 0.884 Dobrzycki et al.5 1 Simple hydrate 173 K 0.975 0.975 Dobrzycki et al.5 5.65 Semi-clathrate 0.76 GPa 1.117 1.059 This work 5.8 Semi-clathrate 0.61 GPa 1.133 1.086 This work
7.25 Semi-clathrate 248 K 1.038c 1.041 St¨aben and Mootz4 7.75 Semi-clathrate 203 K 1.035 1.041 Dobrzycki et al.5 9.75 Semi-clathratea 271 K 1.005c 1.010 McMullan et al.2 11 Semi-clathrate 250 K 1.004 1.014 Dobrzycki et al.5 17 Semi-clathrate 243 K 0.932 0.940 Dobrzycki et al.5
a Although formally a semi-clathrate, this structure does not exhibit permanent defects in the water network, hence it resembles a true clathrate.
b Structures for which a temperature is given were obtained at ambient pressure.
Structure for which a pressure is given were obtained at ambient temperature.
c From the redeterminations by Dobrzycki et al. rather than the original author.
creases, the water network changes from this 0D state (0.25-hydrate) to chains (1D, mono-hydrate), layers (2D), and finally cages (3D, 7.25- and higher hydrates). In practice, considering the whole continuum, a lay-ered water network has only been observed fortert-butanol hydrates.30 FortBA, a 2D arrangement has not been observed; the gap in hydration numbers probably exists because the jump from a 2D to a 3D arrange-ment requires a considerable increase in water molecules.
The densities of the two high-pressure semi-clathrates reported in this work have been normalised (§6.3.2.1) and the relationship between the density and the hydration number has been recalculated using Fi-tyk31(Fig. 6.9). The new parameters for Eq. 6.3 are in Table 6.6, along
Table 6.6: Parameters of Eq. 6.3, as determined originally by Dobrzycki et al. (top) and redetermined in this work (bottom). s anddein g/cm3,aandre
with the original parameters. Overall, only a small change is observed, meaning that the two new structures behave as expected.
The use of high pressure shows that it is possible to obtain crystal structures which are closer to the maximum possible density. Neverthe-less, the transition between simple hydrates and clathrate-like structures is still unclear. Without neglecting the idea of the stability of the struc-ture associated with the water network proposed by Dobrzycki et al., the 5.65- and 5.8-hydrates point towards a more complex scenario, in which a 3D water network still exists at low hydration numbers, but cannot be interpreted without considering the intercalation of amine groups.
6.3. Results and discussion data point is labelled with the corresponding hydra-tion number; the two HP semi-clathrates are high-lighted in red. The orange curve corresponds to the recalculated relationship;
the dashed blue curve to the original calculation.5
Form II of pure tBA has not been included in this calculation, be-cause there is not an appropriate reference against which to normalise its density at ambient pressure.
6.3.2.1 Density normalisation
The densities of the structures determined by Dobrzycki et al.5 (Ta-ble 6.5) were normalised to 173 K taking non-deuterated ice Ih as a reference. Unit cell volumes at each temperature of interest were calcu-lated using the V-T equation of state determined by R¨ottger et al.,32 normalisation factorsfT were calculated according to Eq. 6.4, and then the densities were normalised according to Eq. 6.5.
fT= V(T)
V(173) (6.4) ρ(173) =fTρ(T) (6.5)
fP =V(P)
V(0) (6.6)
A similar procedure was applied to normalise the densities of the high-pressure 5.65- and 5.8-hydrates to ambient pressure. A quadratic expression for the pressure dependence of the density of non-deuterated ice Ih was proposed by Gagnon et al. based on Brillouin spectroscopy data collected at 237.65 K up to 0.28 GPa.33 Densities were calculated using this relationship, they were converted into volumes afterwards, and all volumes were finally normalised to 173 K:
ρ(173) =fTρ(T) ⇒ m
Normalisation factors calculated according to Eq. 6.6 have been com-puted using these data (Table 6.7).
Table 6.7: Normalisation factors calculated from data corresponding to non-deuterated ice Ih at 237.65 K.
P (GPa) ρ(g/cm3) V (˚A3) fT T-norm.V (˚A3) fP
0.00 0.9228 129.67 1.0076 128.69
0.61 0.9783 122.32 1.0076 121.40 0.943
0.76 0.9897 120.91 1.0076 119.99 0.932
The normalisation factors in Table 6.7 have been used for all purposes in this study. Nevertheless, acknowledging the inherent risk of extrapo-lating the polynomial curve from 0.28 to 0.61 and 0.76 GPa, two tests have been performed to ensure that the previous procedure performs appropriately.
A P-V equation of state was proposed by Str¨assle et al. for deuter-ated ice Ih at 145 K.34 These data cannot be used to compute the nor-malisation factors in this study, because the compressibilities of deuter-ated and non-deuterdeuter-ated ice Ih are slightly different and the isotope effect has not been evaluated at the pressures used in this study.35 Notwithstanding, the isotope effect is expected to be small and the P -V relationship for the deuterated compound takes the form of a Mur-naghan equation of state, which can be extrapolated more reliably than a quadratic equation. Normalisation factors were calculated using these data, to check that they do not differ considerably from those calcu-lated previously. This indicates that the extrapolation on the quadratic equation does not introduce large errors. The same procedure as before was used (Eqs. 6.7 and 6.6), although the step involving the densities was not required. The normalisation factors obtained were indeed very close to the ones obtained from non-deuterated ice Ih (Table 6.8). See Fig. 6.10 for a comparison of both equations of state.
As second test, a quadratic equation of the same form as proposed by Gagnon et al. was fitted to 11 data points generated from the Mur-naghanP-V equation of state. Both the quadratic and Murnaghan equa-tions overlap below 0.5 GPa, but diverge at higher pressures, albeit just slightly (Fig. 6.10). Translating this observation into the data for deuter-ated ice Ih, it is safe to assume that, in the absence of a more robust equation, the error in the normalisation factors due to the extrapolation of the quadratic equation should be small.
6.3. Results and discussion
P (GPa) V (˚A3) fT T-norm.V (˚A3) fP
0.00 128.18 1.0035 127.73
0.61 121.69 1.0035 121.27 0.949
0.76 120.43 1.0035 120.01 0.940
Table 6.8: Normalisation ice Ih. Data points are represented in the inter-val for which data is avail-able (up to 0.28 GPa for non-deuterated ice and up to 0.5 GPa for deuterated ice). The curves are drawn to 1.0 GPa as solid lines to show the crossing point at 0.75 GPa. The quadratic fit of the data for deuter-ated ice is shown as a dashed line.
6.3.3 Computational interpretation of pure