Double-well Potentials and Structural Phase Transitions in Polyphenyls BY H E R V ~ CAILLEAU, JEAN-LOUIS BAUDOUR AND JEAN MEINNEL Groupe de Physique Cristalline E.R.A. au C.N.R.S. no. 015, UniversitC de Rennes, Campus de Beaulieu, 35042 Rennes Cedex, France AND ARY DWORKIN Laboratoire de Chimie Physique des Materiaux Amorphes, L.A. au C.N.R.S. no. 75, Universite Paris Sud, Centre Scientifique, 9 1405 Orsay Cedex, France AND FERNANDE MOUSSA Laboratoire Leon Brillouin, Cen Saclay, B.P. 2, 91190 Gif Sur Yvette, France AND CLAUDE M. E. ZEYEN Institut Max von Laue-Paul Langevin, B.P. 156 X, 38042 Grenoble Cedex, France Received 20th December, 1979 In their crystalilne phases, the conformation of non-rigid polyphenyl molecules results from a delicate balance between competing intramolecular and intermolecular forces.At low temperature, the polyphenyl crystals undergo structural phase transitions associated with stabilization of a non- planar conformation, with respect to a torsional angle between the phenyl rings. An interesting feature is the different natures of these phase transitions: displacive in biphenyl and order-disorder in p-terphenyl. In order to illustrate this point we consider the following features: the thermal motion in the high-temperature phases, the influence of hydrostatic pressure on the phase transition in p-terphenyl, the transitional heat capacities, the observation of the critical slowing down of fluctua- tions in p-terphenyl and the existence of incommensurate phases in biphenyl. In a molecular crystal, the intramolecular potentials are generally weakly perturbed by the intermolecular interactions so that the internal vibrations are similar to those of the isolated malecule.In the case of a double-well intramolecular potential, however, very special features may arise if a delicate balance between competing intramolecular and intermolecular forces exists : in particular, the system is able to8 STRUCTURAL PHASE TRANSITIONS I N POLYPHENYLS undergo a structural phase transition. This fact is well illustrated by materials such as non-rigid polyphenyl molecules ;' biphenyl, p-terphenyl, p-quaterphenyl . . . (fig- 1). biphenyl p - terphenyl o-octc> p- quaterphenyl intramolecular potent ial inter molecular potential result in potent i3 FIG. 1 .-Schematic drawing of intramolecular, intermolecular and resulting torsional potential in polyphenyls.NON-RIGID POLYPHENYL MOLECULES The isolated molecule is non-planar, as a torsional angle exists between the planes of the phenyl rings:' the conjugation energy between phenyl rings is not sufficient to overcome the ortho-hydrogen repulsion, and this situation induces the existence of a double-well intramolecular potential. The conformation of these molecules strongly depends on the environment: the torsional angle in biphenyl is ~ 4 0 " in the gas phase and is reduced to 20" in the liquid phase. In the solid state, the different polyphenyls are apparently planar3-' at room temperature and their crystals are isostructural within the usual space group for flat molecules (P2Ja). At low temperature, the polyphenyl crystals undergo antiferrodistorsive structural phase transitions (table 1) associated with a stabilization of the molecules in a non- planar configuration.6-8 This behaviour is an indication of the delicate balance between intramolecular and intermolecular forces.Resulting from crystal forces which tend to place the phenyl rings parallel,' the intermolecular part of the potential TABLE 1 .-TRANSITION TEMPERATURE IN POLYPHENYL CRYSTALLINE PHASES biphenyl p-terphenyl p-quaterphenyl hydrogenated 40 KZ4 191 K16 243 KZ3 deuterated 38 KZ8 179 K16 has a single-well shape and is antagonistic to the intramolecular barrier. A double- well potential, corresponding to the instability of the planar configuration, subsists with a smaller barrier height between two close minima than in the isolated molecule (fig.1): the molecules are no longer rigid with respect to the torsional angle. An interesting feature is that different polyphenyls undergo phase transitions ofCAILLEAU, BAUDOUR, MEINNEL, DWORKIN, MOUSSA, ZEYEN 9 different natures : displacive in biphenyl, it becomes order-disorder in p-terphenyl. The purpose of this paper is to illustrate the difference between the mechanisms of these two types of phase transitions in polyphenyls. DOUBLE-WELL POTENTIAL AND STRUCTURAL PHASE TRANSITION As indicated above in polyphenyl crystalline phases, the double-well potential results essentially from the competition between the ortho-hydrogen repulsion and crystal packing forces. In p-terphenyl the ortho-hydrogen repulsion forces are about twice as large as in biphenyl, hence the distance between the two minima is larger and the barrier height is higher than in biphenyl (fig.2). Consequently, the torsional angle in the low-temperature phase is about 21 O in p-terphenyl ' and only 10" in biphenyl.6 torsional angle/deg FIG. 2.-Schematic drawing of the crystalline torsional potential in (a) p-terphenyl (order-disorder transition) and (b) biphenyl (displacive transition). In the high-temperature phase of p-terphenyl each molecule is well localized in one of the two bottoms of its double-well (fig. 2), and the disorder consists of an equal orientational distribution over the two wells in the crystal. The jump from one well to the other one is a single-molecule relaxation process governed by an activation energy. Between two jumps, vibrations also exist within the well and they are not strongly temperature dependent.On approaching the phase transition from above, pretransitional effects appear with the growth of short-range clusters. The mechanism of the displacive phase transition in biphenyl is different. At room temperature the barrier energy is negligible compared with the thermal energy (fig. 2) and large-amplitude torsional vibration occurs. On lowering the temperature, the influence of the barrier height becomes more and more important and the fre- quency of the torsional mode is lowered. This soft mode condenses in a superlattice reflection at the transition. The mechanisms of structural phase transitions presented here are oversimplified and more elaborated models may be needed."." THERMAL MOTION I N THE HIGH-TEMPERATURE PHASE Evidence for differences between the behaviour of biphenyl and p-terphenyl can be obtained via the determination of libration tensor from diffraction data.Table 2 shows the mean-square librational amplitudes of phenyl rings around the long molecular axis obtained with an harmonic single-well model at different temperatures. In biphenyl this amplitude decreases with temperature and is smaller than in p - terphenyl for which this amplitude is almost temperature independent. Inp-terphenyl the picture of a double-well potential is rather suitable. Such a description is simply obtained by halving the atomic occupations outside the molecular axis on either side10 STRUCTURAL PHASE TRANSITIONS I N POLYPHENYLS TABLE 2.-MEAN-SQUARE LIBRATIONAL AMPLITUDES OF THE PHENYL RING AROUND THE LONG MOLECULAR AXIS WITH AN HARMONIC MODEL temp./K biphenyl p-terphenyl p-quaterphenyl (central rings) (central rings) 300 105.9(deg2) 260.3(degz) 1 78.3(deg2) 200 248 .O 110 45.7 19.3 of the mean plane.The reliability factor values, compared with those for single-well models, are thus significantly i m p r ~ v e d . ~ The resulting double-peaked structure in the probability density function is characteristic of disorder (fig. 3). 0 L angleideg FIG. 3.-Probability density function for the p-terphenyl central ring librations around the long molecular axis from neutron data at 200 K (broken line) and X-ray data at 300 K (full line).INFLUENCE OF HYDROSTATIC PRESSURE ON THE PHASE TRANSITION IN p-TERPHENYL The compressibility of aromatic molecular crystals is very high compared with that of inorganic materials, and the investigations of phase transitions in polyphenyls under high hydrostatic pressure may be very fruitful. When pressure is increased, the polyphenyl intermolecular potential becomes steeper, whereas the intramolecular one remains unchanged : thence in the resulting double-well potential, the barrier height becomes smaller between two closer minima. Therefore the transition temperature is expected to decrease with increasing pressure. This behaviour is different from that observed for most inorganic materials in which high pressure affects the balance between competing short-range and long-range forces.12 An experiment has been performed on a neutron diffractometer at the Institut Laue -Langevin.A deuterated p-terphenyl crystal was mounted within an helium high-pressure cell13 and a standard cryostat. At high pressure, the intensity of a superlattice reflection presents a discontinuity step at the transition, so the determina-CAILLEAU, BAUDOUR, MEINNEL, DWORKIN, MOUSSA, ZEYEN 1 1 tion of the transition temperature is easy. As expected, the transition is shifted to lower temperature when pressure is increased (fig. 4). This shift is very large and is not linear with pressure. In addition, at low temperature a new phase appears and a triple point exists at ~ 7 0 K and 3.6 kbar. This new phase is associated with an im- portant change in the unit cell, but its structure has not yet been determined.The magnitude in the discontinuity step observed at high pressure increases quickly as the pressure increases. At atmospheric pressure the intensity variation with temperature of a superlattice reflection was found to be continuous or very close to continu~us.~ Moreover, the pseudo critical exponent ,8 obtained by a plot of intensity data according to I cc (T, - T)’P is small: This small value can be associated with a classical Landau expansion of the free energy for a first-order transition, the square of the order parameter having a similar temperature dependence near the transition.” 2: 0.15. r 1 1 0 1 2 3 L 5 pressure/ kbar FIG. 4.-Phase diagram of deuterated p-terphenyl. TRANSITIONAL HEAT CAPACITY The heat capacities of hydrogenated biphenyl and p-terphenyl were measured in the vicinity of the transition temperature (fig.5). In p-terphenyl a thermal anomaly extending over 20 K was found16 with a maximum of Cp located at 193.33 K. The overall transition enthalpy was found as AH, = 86.0 J mo1-I and the transition entropy (obtained by graphical integration of a CJT against T curve) was AS, = 0.49 J K-’ mol-’. This transition seemed to be continuous at the temperature resolution used ( ~ 0 . 8 K) and independent work performed by Chang” with a better resolution (0.01 K) confirmed this point. No thermal anomaly was found in bi- phenyl” either near 40 K or near 15 K where Cullick and Gerkin discovered an- other phase transition by e.p.r. meas~rements.’~ An upper limit to the transition en- thalpy may be given as 10 J mol-’.The fact that AH, is hardly noticeable in biphenyl whereas it is easily measured in p-terphenyl appears as a confirmation of the different nature of the phase transitions in these two compounds. In the displacive case the phase transition takes place between two relatively ordered structures and one may expect a smaller change of entropy than in order-disorder case. In this instance however, it has to be remarked12 40 30 4 I - 8 * I b4 c, 20- --- L7 STRUCTURAL PHASE TRANSITIONS I N POLYPHENYLS -- - . . . . . 0 0 I I I 20 30 40 TlK I 1 I 1 i '7 ( B) 0 0 0. . o 0 ' . .a 01 1 1 1 1 I I 1 3 200 TlK FIG. 5.-Heat capacity of hydrogenated biphenyl (A) and p-terphenyl (B).CAILLEAU, BAUDOUR, MEINNEL, DWORKIN, MOUSSA, ZEYEN 13 that ASt of p-terphenyl is very small compared with R In 2 = 5.76 J K-l mo1-I which would happen in the case of a pure order-disorder transition. This behaviour can be understood by assuming the existence of strong short-range correlations between disordered molecules.CRITICAL SLOWING DOWN I N p-TERPHENYL In p-terphenyl, as in biphenyl, the pretransitional short-range correlation effects are located close to a Brillouin zone boundary point, and neutron scattering investiga- tions are particularly suitable. A triple-axis experiment on deuterated p-terphenyl has shown the presence of critical quasielastic scattering close to the C(+,*,O) point.'' This quasi-elastic component is due to the formation of clusters of the ordered structure within the disordered phase.The energy width is proportional to the inverse of the cluster relaxation time 7,: this lifetime is about 2 x lo-" s at room temperature. The energy width could not be followed down to the phase transition because of the ).A + .- , ' resolution energy transferlp eV energy transfer/ peV FIG. 6.-Energy spectra of neutrons scattered in deuterated p-terphenyl at the superlattice point (3, 3, 0). The full curves represent a fit of a convolution of a single Lorentzian scattering, law with the resolution function of the spectrometer. The fitted values of the h.w.h.m. obtained are 0.36, 0.73 and 1.67 peV at, respectively, T, + 0.3, T, + 0.8 and T, + 1.6 K, which yield the cluster lifetimes given in the text. limited energy resolution. Also two very high resolution experiments were performed at the Institut Laue-Langevin, the first one on the backscattering spectrometer21 IN 10 and a second one, very recently, on the spin-echo machine IN 1 1.Fig. 6 shows the intensity of scattered neutrons, in the first experiment, as a function of energy and temperature. As for magnetic phase transitions, a critical slowing down of fluctua-14 STRUCTURAL PHASE TRANSITIONS I N POLYPHENYLS tions is clearly observed: the intensity diverges and the energy width decreases on approaching the phase transition. The incoherent scattering, which at room tempera- ture is of the same order of magnitude as the coherent scattering, is negligible close to the transition temperature T,. Resolution corrections have been performed using a single Lorentzian scattering law.Values of the lifetime z, of 2.5 x and 11.5 x s were obtained at, respectively, T, + 1.6, T, + 0.8 and T, + 0.3 K. The neutron spin-echo data are not yet completely analysed, but they confirm the preceding results. Furthermore, an indication of the existence of a central elastic peak was found close to the transition temperature. On the other hand, it is interesting to notice that, as expected in an order-disorder phase transition, the frequencies of torsional internal modes do not depend very much on temperature, as shown by Raman scattering s t ~ d i e s . ~ ’ * ~ ~ 5.7 x INCOMMENSURATE PHASES I N BIPHENYL The dynamical properties associated with the displacive transition of biphenyl are completely different. Raman spectroscopy experiments allowed Bree and EdelsonZ4 to conclude torsional soft modes to be present in the low-temperature phases.In the high-temperature phase the soft mode is located around the B(O,*, 0) Brillouin zone boundary point. The space group P2,/a is non-symmorphic and at the B point the Lifshitz condition is not satisfied:25*26 two modes which are degenerate at the zone boundary come in with opposite but finite slope.27 Also, on the lower phonon branch the minimum is away from the B point [fig. 7(a)]. So, the two low- temperature phases are incommensurate as we have recently shown in deuterated biphenyl.28 In phase 11, which exists between TI, = 21 K and TI = 38 K, the wave vectors characterizing the incommensurate modulation are qs = &da a* $( 1 - 6,) b* [fig.7(b)]; no higher-order satellites could be observed in this phase. At T&, a partial lock-in phase transition takes place and below T,, the satellite positions be- come qs = &$(1 - &) b* [fig. 7(b)]; in this phase 111 we have been able to measure higher-order satellites up to third-order. The variations of 6, and 6, with temperature are The soft-phonon dispersion surface has been observed in the high-temperature phase with triple-axis spectrometers working respectively with thermal and cold neutrons. The shape of this dispersion surface is very anisotropic: the minimum is well pronounced in the b* direction, and it has not been observed in the a* direc- tion (fig. 8). These features are in agreement with the maintenance of incommensura- bility in the direction b and the weak temperature dependence of B,,, the incommensur- ate wave vector remaining close to that corresponding to the deep minimum of the soft phonon branch in this direction.In the high-temperature phase, the mode which corresponds to the minimum of the appropriate dispersion surface is easily resolved in the temperature range 50-200 K and shows a pronounced softening [fig. 9(b)]. Above 200 K the observation of this mode is difficult, the damping being important as expected in aromatic molecular crystals; close to the transition temperature, the mode becomes overdamped [fig. 9(a)]. On the other hand, we have very recently observed a new excitation superimposed on the overdamped mode in the incommensurate phases. The experimental results are discussed and reported elsewhere.29 The dispersion of this excitation is found to follow a linear law originating at the satellite reflection with a slope very much lower than that of the lowest acoustic mode. This new excitation branch 6, lies between 0.04 and 0.05, while 6, falls in the range 0.07-0.085.CAILLEAU, BAUDOUR, MEINNEL, DWORKIN, MOUSSA, ZEYEN 15 I FIG.7. (a) Schematic drawing of the torsional mode dispersion curves close to the B (0, 3, 0) point in biphenyl. (b) Locations of satellite reflections in the (h k 0) scattering plane for phase I1 and phase 111. Dotted lines correspond to the limits of the first Brillouin zone. FIG. 8.-Soft mode dispersion surface of deuterated biphenyl at 49 K.16 STRUCTURAL PHASE TRANSITIONS I N POLYPHENYLS could be the phase-mode one.A phase mode, called a phason by Overhau~er,~~ is a Goldstone mode corresponding to the continuous broken phase symmetry: a uniform shift in phase of the low temperature modulation requiring no energy, it should give rise to a new " acoustic-like " branch. More extensive studies on bi- phenyl and other materials will probably be necessary before a final conclusion can be reached. CONCLUSION It appears that polyphenyl crystals are attractive candidates to illustrate the differ- ence between the mechanisms of displacive and order-disorder transitions : the struc- tural changes in biphenyl and p-terphenyl, two isostructural compounds from the same chemical family, are caused by the same forces but, owing to a different balance between competing internal and external forces, the nature of their phase transitions is not the same.It may be interesting to remark that the transition mechanisms of I I I - 0.3 0.0 0.3 -0.1 0.0 energy transfer/THz (a) 0.1CAILLEAU, BAUDOUR, MEINNEL, DWORKIN, MOUSSA, ZEYEN 17 0 0 50 100 150 200 temperature/K (b) FIG. 9. (a) Constant-Q scans in deuterated biphenyl at two different temperatures (55 K above and 38 K below) at a point where a satellite reflection appears below 38 K. The broken line represents roughly the overdamped mode subtracted from the elastic part. The full lines are guides to the eye. 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