High-pressure Phase Transitions in Molecular and Plastic Crystals BY ALBERT WURFLINGER University of Bochum, Institute of Physical Chemistry, 4630 Bochum, West Germany Received 3rd December, 1979 The phase transitions of some molecular and plastic crystals have been studied byp Wand dielectric constant measurements up to 3000 bar. The pVT data show that the density of solid dodecane immediately below the freezing point is comparable with the density of solid undecane immediately below the rotational transition temperature. The phase diagram of acetonitrile exhibits a solid- solid transition (denoted as an a-B solid-solid transition at atmospheric pressure), the pressure de- pendence of which has not yet been reported. Also cyclohexanone has been studied; it exhibits three solid phases (I, I1 and 111), one of which (11) is only observed at elevated pressures.The dielectric results for cyclohexanone are discussed in terms of the Kirkwood-Frohlich-Onsager theory. The Kirkwood correlation g-factor is about unity both for liquid and solid (I) cyclohexanone. How- ever, a small increase in the g-factor is observed at the freezing point and a considerable decrease at the 1-11 solid-solid transition. In recent years the high-pressure phase behaviour of molecular and plastic crystals has been studied by different experimental methods, e.g., differential thermal analysis,l4 dielectric c o n ~ t a n t , ~ * ~ and p VT measurement^.^-^^ Plastic crystals exhibit at least one solid-solid transition, accompanied by a large enthalpy and volume change. This phase transition is associated with the loss of orientational freedom, leading to a distinct break in the permittivity insofar as polar compounds are considered.Some plastic crystals, moreover, have an intermediate phase transition for which the orien- tational order is partially l o ~ t . ~ p ~ ~ * ~ ~ In some cases the intermediate solid-solid transi- tion is only observed at elevated pressure^.^*^ This paper gives new experimental results for the p VT data of undecane and dode- cane, the phase behaviour of which has been studied in earlier investigations.' Also the phase behaviour of acetonitrile is studied with p VT and dielectric measurements. New dielectric results of cyclohexanone are discussed, together with recently de- termined p VT data,l' in terms of the Kirkwood-Frohlich-Onsager theory.This theory has been widely used for the discussion of polar fluids,14*15 but should also be applicable to the plastic phases of polar compounds, where the high value of the dielectric constant is maintained or even increased. Bottcher et aZ.16 have shown that the Kirkwood-Frohlich equation is also valid in the case of molecules with isotropic polarizabilities on a cubic lattice, and Hassel and Sommerfeldt l7 found cyclohexanone to have a face-centred cubic lattice in the high-temperature solid phase (denoted here as solid I). EXPERIMENTAL The pVTmeasurements were carried out with a new high-pressure apparatus developed by Landau Two high-pressure devices have been used for the differential thermalA . WURFLINGER 147 and described in detail el~ewhere.~*’O The substance under investigation is enclosed within a steel capsule that is closed by a special moving piston.The piston consists of two parts, separated by an indium seal. A hollow space between these works like a Bridgman’s un- supported area yielding a very good sealing of the piston, even after solidification of the substance. The displacement of this moving piston is recorded inductively allowing the calculation of volume changes. The high-pressure equipment for the dielectric measurements has been described in ref. (7) where details of the experimental procedure and instruments used are also given. Prelimin- ary results on cyclohexanone were obtained with an open capacitor that consisted of two cylinders.’ A remarkable improvement was achieved in transmitting the pressure with a moving piston similar to that mentioned above for the p VT measurements.The improved capacitor was first applied to the dielectric investigation of acetonitrile and will be described elsewhere. Commercial cyclohexanone (Merck, Darmstadt) was purified as described in ref. (6) and re-examined with the improved capacitor. The frequency used was 1 MHz. No significant frequency dependence or dielectric relaxation was observed. The accuracy in E (dielectric constant) is believed to be between 0.5 and 1%. RESULTS THERMODYNAMIC MEASUREMENTS Fig. 1 shows former results of the phase behaviour of some even (on the left) and odd (on the right) n-alkanes.’” Most of them exhibit a solid-solid transition 403 383 k4 L.363 3 4 3 323 303 I 1 2 233LJ+’*- 213b 1 2 3 plkbar FIG. 1 .-Phase diagrams of several even (left-hand) and odd (right-hand) n-alkanes. The hatched areas are confined above by the melting curve and below by the rotational transition line.148 HIGH-PRESSURE PHASE TRANSITIONS (rotational transition) immediately below the melting curve. The high-temperature solid phase is to a certain degree plastic and marked as a hatched area in fig. 1. The rotational transition is observed between nonane and henicosane only for the odd n-alkanes and between docosane and about C40 also for the even n-alkanes. A detailed discussion for the odd n-alkanes has already been given in ref. (1). The even n-alkanes reported by Koppitz et al.' were partially re-examined by Josefiak et aL3 The plot of C36H74 is missing from fig.1 (because it has not yet been redeter- mined), but is most probably similar to the phase behaviour of the adjacent n-alkanes. In all cases the experimentally observed plastic phase of the alkanes is restricted to a limited pressure range. Fig. 2 shows recently determined pVT data of undecane and d ~ d e c a n e . ~ The 1 . 4 1.2 4 I M % . a 1 . 2 1.1 plkbar FIG. 2.-Specific volume as a function of pressure for undecane (left-hand) and dodecane (right-hand) at different constant temperatures (in K). specific volume is plotted against pressure for different constant temperatures. Two distinct steps are observed in the volume against pressure plot of undecane, corre- sponding to the melting (larger step) and the rotational transition (lower step), whereas dodecane shows only one step corresponding to the volume change of melting. The distance between the two steps of undecane diminishes with increasing pressure, indicating the convergence of the transition lines which intersect at the triple point liquid-solid-I-solid-I1 (cf.fig. 1). Above the triple point there is only one step, due to the melting. The volume change of the rotational transition is nearly pressure-independent, but the volume changes of melting decrease considerably with increasing pressure as has also been found by other authors.'* Furthermore the volume change of melting of dodecane (which exhibits no rotational transition) is approximately the sum of the volume changes of melting and the rotational transition of undecane.It seems as if the rotational transition anticipates part of the melting process of the n-alkanes. Fig. 2 shows further that the rotational transition line runs fairly parallelA . WURFLINGER 1 49 to an isochore, a statement that had already been suggested in earlier investiga- tions.' p VT data of acetonitrile (Landa~),~.'~ cyclohexane and cyclohexanone (Wisotzki) have also been determined, both for the liquid and solid states. Fig. 3 shows the 0 1 2 3 plkbar x , pVTmeasurements ; FIG. 3.-Phase diagram of acetonitrile. 0, dielectric constant measurements;6 A, ref. (19); 0, ref. (20). phase diagram of acetonitrile in comparison with literature data. The melting curve has already been reported by other author^,^^,^^ but the pressure dependence of the a-/3 solid-solid transition has not been found in the literature.However, there is an additional paper of interest by Jakobsen and Mikawa21 who believe they have found a third solid phase y, using high-pressure infrared spectroscopy. They report a transition at 30 kbar and room temperature which most probably does not belong to the a-p transition. The volume changes accompanying the phase transitions allow the calculation of the associated enthalpy changes using the Clausius-Clapeyron equation: Melting of dodecane: 35.4 kJ mol-' at I bar rising to 41.4 at 2600 bar. Melting of undecane: 21.5 kJ mol-l, nearly pressure-independent ; rotational transition: 6 kJ mol" at 1 bar rising almost linearly to 12 kJ rno1-I at the triple point (2300 bar).Melting of acetonitrile: 8 kJ mol-' at 1 bar rising to 9.5 kJ mol-' at 3000 bar; or-p solid-solid transition: 0.87 kJ mol-I at 1 bar increasing to z 1 kJ mo1-I at 2700 bar. In the case of the n-alkanes the pressure dependence of the enthalpy changes has been determined for related compounds by Kamphausen et aZ.22 using high-pressure d.s.c. calorimetry. The reported pressure dependence 22 agrees qualitatively with the calculations in this paper.150 HIGH-PRESSURE PHASE TRANSITIONS DIELECTRIC MEASUREMENTS Fig. 4 shows the dielectric constant as a function of temperature for various sub- stances. Liquid acetonitrile has a large value of the permittivity, which decreases by an order of magnitude at the freezing temperature. There is a further small change (A& N" 0.05) at the cc-p solid-solid transition (see arrow) corresponding to the small volume change.On the other hand cyclohexanone shows only a small dielectric step at the freezing point, but a considerable decrease in the dielectric constant at a lower 50 I I I I I I I I I I I I I x-. I - x 20 - 10 - C T K FIG. 4.-Dielectric constant as a function of temperature at different pressures (in bar): cyclo- hexanone (-), acetonitrile (- - - -), and cyclohexane (- - - - - - -). AB = liquid-solid I, CD, 1-11 solid-solid and EF, 11-111 solid-solid transition of cyclohexanone. 0, Corfield and Davies." solid-solid transition. The completely different dielectric behaviour of these two substances indicates clearly that cyclohexanone must possess orientational freedom within the high temperature solid phase, whereas acetonitrile obviously does not belong to the class of " reorientational solids ".Cyclohexanone is a typical represen- tative of the " plastic crystals " which usually have low enthalpy- and volume-changes of melting, as has been confirmed in high pressure calorimetric6 and volumetric" measurements. Fig. 4 also shows atmospheric pressure data of Corfield and Davies" for comparison. The agreement is good at room temperature and a t w -50 "C, but some deviations occur near the melting temperature. Even low concentrations of water impurities cause a sluggish melting transition and values of E that are too low in solid phase I, which might perhaps be the reason for the deviations.A . WURFLINGER 151 Furthermore fig. 4 shows that cyclohexanone has an additional dielectric step at 2000 bar, due to a second solid-solid transition recently found by differential thermal analysk6 Preliminary dielectric measurements of cyclohexanone7 have now been extended to 3000 bar and are summarized in table 1.The dielectric results of aceto- nitrile will be discussed in detail elsewhere.8 TABLE STATIC DIELECTRIC CONSTANT, E, OF CYCLOHEXANONE AS A FUNCTION OF PRESSURE AND TEMPERATURE. THE COLUMNS ARE SUBDIVIDED, REFERRING TO THE LIQUID, SOLID I AND SOLID 11, RESPECTIVELY. THE DATA WERE SMOOTHED USING A POLYNOMIAL E = a + bp 3. cpz AT CONSTANT TEMPERATURE FOR EACH PHASE SEPARATELY. T/K plbar 303 293 283 273 263 253 243 233 223 1 250 500 750 1000 1250 1500 1750 2000 2000 2250 15.54 15.77 16.00 16.20 16.40 16.59 16.76 16.92 17.07 17.21 16.17 16.85 17.57 18.35 19.16 20.00 22.10 23.1 16.35 17.05 17.79 18.52 19.32 21.10 22.20 16.53 17.26 18.00 18.70 20.40 21.30 20.86 16.72 17.46 18.19 19.78 20.60 21.50 16.91 17.65 18.37 19.96 20.80 20.38 17.10 17.83 19.30 20.15 19.75 20.5 17.29 18.68 19.50 20.34 19.87 17.48 18.87 19.68 19.33 20.0 18.30 19.04 19.87" 18.84" 19.44 18.47 19.22 18.94 19.54 2500 17.94 18.65 18.40 19.05 19.65 2750 18.15 18.80 18.50 19.15 3000 18.22 18.03 18.60 19.25 * Both values refer to the 1-11 solid-solid transition which takes place at 2000 bar.Cyclohexane is a further representative of the " plastic crystals " and has also been studied by calorimetri~~*~~ and pYT measurements.11 However, this compound does not possess a permanent dielectric dipole moment ; therefore, its dielectric constant is small and does not change much (A& x 0.05) at the phase transitions (see arrows).The temperature dependence of E agrees on the whole with the literature data of Chew and Chan.24 Fig. 5 shows another representation of the dielectric results of cyclohexanone. The dielectric constant is plotted as a function of pressure along the phase transition lines of cyclohexanone. The curves A, B, C . . . correspond to the edges A, B, C . . . of fig. 4. The distance AB shows the increase in E at the freezing temperature and the distance CD refers to the decrease in E at the solid-I-solid-I1 transition, respectively. The low value of E in the solid phase I11 (edge F) is not shown in fig. 5, because it was152 HIGH-PRESSURE PHASE TRANSITIONS rather difficult to perform dielectric measurements in that brittle and hard solid phase.The value at atmospheric pressure is ~ 2 . 8 , in accordance with data of Corfield and Davies.12 At higher pressures the accuracy of E in the solid phase I11 is less, but E does seem to be below 3.0. Fig. 5 further shows that the transition lines E and D diverge with increasing pres- sure according to the increasing solid phase I1 region. They intersect at atmospheric w I I I I I I 0 . 5 1 1.5 2 2 . 5 3 I plkbar FIG. 5.-Dielectric constant of cyclohexanone as a function of pressure along the phase transition lines. The letters A, B, C . . .refer to fig. 4. 0, smoothed values according to a polynomial E = a + bp + cp’; A and +, experimental points except curve E which was also smoothed. pressure at the triple point solid-I-solid-11-solid-I11 ; for details of the phase diagram see ref.(6), where the phase-transition temperatures as a function of pressure are also given. DISCUSSION n-A L KANES One of the striking features of the n-alkanes of higher chain length is the occurrence of the “rotational transition”. The nature of this phase transition has been in- vestigated i n t e n s i ~ e l y , ~ ~ - ~ ~ and will certainly be discussed by Ewen et aZ.29 Apart from the difficulties of explaining the mechanism of the rotational transition the question arises: why does this phase transition occur at all? Bople and Karasz3’ have suggested a model that allows one to a certain extent to predict the existence of plastic phase transitions.A good review of this model is given by Findenegg31 The essen- tial parameters of the model are two energy barriers w and w’, concerning the repulsion of a positionally disordered molecule in the lattice and the energy of an orientationally disordered molecule, respectively. Then the ratio w’/w is a measure of the anisotropy of the non-spherical molecule. Assuming a certain volume de- pendence of w and w’ (but such that the ratio w ’ / ~ is independent of temperature and pressure) it is possible to predict the pressure dependence of the phase transitions. The main result is that the high-temperature solid phase region should be increased with increasing pressure. This is indeed true for many plastic ~ r y s t a l s , ~ , ~ but is not observed in the case of the n-alkanes (see fig.l), as has already been pointed out by KO hler .32 This different high-pressure phase behaviour of the n-alkanes may perhaps be qualitatively understood if a pressure dependence of w’/w is assumed. According to early X-ray results of Muller 33 the compressibility of tricosane is anisotropic. There-A . WURFLINGER 153 fore the ratio w’/w is most probably pressure-dependent and should increase with increasing pressure. However, an increasing ratio of w’/w is unfavourable for the existence of a plastic phase. According to Karasz and Pople3’ a plastic phase does not occur for w’/w > 0.66, occurs only at elevated pressures for 0.325 < w’/w < 0.66, and at atmospheric pressure for W’/W < 0.325. It is possible that, in the case of the n-alkanes, the ratio w’/w is considerably enlarged with increasing pressure and leads to a disappearance of the plastic phase region (hatched areas in fig.1) for a sufficiently high pressure (triple-point pressure). However, the theory of Pople and Karasz should not be taken too seriously in the case of the n-alkanes, because certainly more than two orientations, as well as more than two energy barriers, are involved with these comparatively long chain molecules, not to mention that more than one solid-solid transition is observed in some case^.^^^^ CYCLOHEXAN ONE Fig. 4 shows that the dielectric constant of cyclohexanone increases with de- creasing temperature. There is a further increase at the freezing point, probably caused by the increase in the density.It is the main purpose of this section to discuss whether the changes in E with temperature and pressure correspond exactly to changes in the density. An eventual deviation may be caused by the interaction of the individual molecular dipoles which is often described in terms of the Kirkwood- Frohlich-Onsager theory using the well-known Kirkwood g-factor : 34 ( E - EW)(~E + ~m)9kTM = E ( E ~ + 2)24~N0pp2 ’ E is the measured static dielectric constant and ,ndex (n, = 1.451 at 20 “C and 1 atm)35 according to the relation36 is calculated from the refractive E , = 1.05 n2. The change in with temperature and pressure was calculated with the help of the Lorenz-Lorentz equation. The necessary density data, p, have been taken from Wisotzki.’l p is the permanent electric dipole moment of an individual molecule that is usually obtained from dielectric measurements of dilute gases.However, most of the reported literature data refer to dilute solutions of cyclohexanone, p being 2.8,37 2.9,38*39 2.9540 and 3.01 D.41*42 Another value of 2.87 D has been obtained from microwave spectros~opy.~~ A mean value of 2.9 D is used for the calculation of the Kirkwood g-factor. Also Corfield and Davies” report 2.9 D calculated from Onsager’s equation; but of course this cannot be taken into account, because the application of Onsager’s equation would mean g = 1. On the other hand the results of Corfield and Davies show that the g-factor of cyclohexanone must be about unity, provided that the mean value of p = 2.9 D37-43 truly refers to an individual molecular dipole.The latter statement is supported by the close value of 2.87 D43 obtained without using any of the assumptions of the Kirkwood-Frohlish-Onsager theory. The other constants of the Kirkwood-Frohlich equation are explained in ref. (34); very similar calculations have also been done by F r a n ~ k . ’ ~ . ~ ~ Fig. 6 shows the Kirkwood g-factor as a function of pressure at three different temperatures in comparison with three isotherms of acetonitrile.* Whereas the g- factor of acetonitrile is distinctly less than 1, indicating preferred antiparallel correla- tion, the g-factor of cyclohexanone is very close to unity, in accordance with the above remarks concerning the results o f Corfield and Davies.12 The pressure and temperature dependences of g are similar for liquid cyclohexanone and liquid aceto-1 54 HIGH-PRESSURE PHASE TRANSITIONS nitrile.At the freezing point of cyclohexanone (2360 bar at 30 "C, 1080 bar at 0 "C, 360 bar at -20 "C), however, a small increase in the g-factor is observed which means that the increase in E at the freezing temperature does not exactly correspond to the increase in the density on freezing. Although this small change in g ( z 1.5%) during the melting transition is comparable with the sum of the relative uncertainties of the measured quantities involved, the increase in g was observed in all cases, also at atmospheric pressure: g = 0.99 in solid cyclohexanone at -35 "C, compared with g = 0.97 in liquid cyclohexanone at -20 "C. The higher value of the g-factor in the solid phase I is perhaps caused by the greater ease of molecular rotation of the cyclo- hexanone molecules compared with the liquid state.This assumption is supported plkbar FIG. 6.-Kirkwood g-factor of cyclohexanone (liquid, solid I, and solid 11) as a function of pressure for 3 different temperatures (in K) in comparison with 3 isotherms of liquid acetonitrile. by Corfield and Davies12 who have shown that the solid rotator phase has an even lower activation energy for dipole relaxation than the liquid. Furthermore Bottcher et aZ.44 report g = 0.66 for solid cyclohexanone at -40 "C, in considerable disagreement with the present paper. The discrepancy arises from the different quantities used, for example p = 1.12 instead of 1.03 g cm-3 [ref.(45)], p = 3.08 instead of 2.9 D, etc. As far as the solid phase I1 which exists only at elevated pressures is concerned, no structural information is available. If solid I1 is also characterized by a cubic lattice the Kirkwood-Frohlich equation may be applied to this phase as well. Reserv- ing this limitation, the calculations have been extended to the solid phase 11. Fig. 6 shows that the g-factor decreases considerably at the 1-11 solid-solid transition (2000 bar at 0 "C, 1240 bar at -20 "C), demonstrating that some part of the reorientational freedom must be frozen in. It does not seem reasonable to describe the lower g-value of the solid phase I1 in terms of antiparallel oriented dipoles or dimers, as has been done with certain success in the case of liquid acetonitrile and related com- poUnd~.8,15,46.47A .WURFLINGER 155 Fig. 6 further shows that the steps of the Kirkwood g-factor are nearly independent of temperature and pressure. Obviously the change in g along a phase transition line is much less (almost negligible) than along an isotherm: g being about 0.95 (A), 0.965 (B), 0.955 (C), 0.89 (D), and 0.88 (E), respectively, the letters A-D referring to fig. 4 and 5. Similar behaviour was found with liquid acetonitrile, which has an approximately constant value of g z 0.7 at the melting curve.' The author thanks the Deutsche Forschungsgemeinschaft for financial support. A. Wiirflinger and G. M. Schneider, Ber. Bunsenges. phys. Chem., 1973,77, 121. B. Koppitz and A. Wiirflinger, Colloid Polymer Sci., 1974, 252, 999.C. Josefiak, A. Wiirflinger and G. 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