摘要:

298 Date 1965 1965 1966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 1971 1972 1972 1973 1973 1974 1974 1975 1975 1976 1977 1977 1977 1978 1978 1979 1979 1980 GENERAL DISCUSSIONS OF THE FARADAY SOCIETY Subject The Kinetics of Proton Transfer Processes Intermolecular Forces The Role of the Adsorbed State in Heterogeneous Catalysis Colloid Stability in Aqueous and Non-Aqueous Media The Structure and Properties of Liquids Molecular Dynamics of the Chemical Reactions of Gases Electrode Reactions of Organic Compounds Homogeneous Catalysis with Special Reference to Hydrogenation and Bonding in Metallo-Organic Compounds Motions in Molecular Crystals Polymer Solutions The Vitreous State Electrical Conduction in Organic Solids Surface Chemistry of Oxides Reactions of Small Molecules in Excited States The Photoelectron Spectroscopy of Molecules Molecular Beam Scattering Intermediates in Electrochemical Reactions Gels and Gelling Processes Photo-effects in Adsorbed Species Physical Adsorption in Condensed Phases Electron Spectroscopy of Solids and Surfaces Precipitation Potential Energy Surfaces Radiation Effects in Liquids and Solids Ion-Ion and Ion-Solvent Interactions Colloid Stability Structure and Motion in Molecular Liquids Kinetics of State Selected Species Organization of Macromolecules in the Condensed Phase Phase Transitions in Molecular Solids Oxidation Volume 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 * 66 67 68 69 * Not available; for current information on prices, etc, of available volumes, please contact the Marketing Oficer, Royal Society of Chemistry, Burlington House, London Wl V OBN stating whether or not you are a member of the Society.298 Date 1965 1965 1966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 1971 1972 1972 1973 1973 1974 1974 1975 1975 1976 1977 1977 1977 1978 1978 1979 1979 1980 GENERAL DISCUSSIONS OF THE FARADAY SOCIETY Subject The Kinetics of Proton Transfer Processes Intermolecular Forces The Role of the Adsorbed State in Heterogeneous Catalysis Colloid Stability in Aqueous and Non-Aqueous Media The Structure and Properties of Liquids Molecular Dynamics of the Chemical Reactions of Gases Electrode Reactions of Organic Compounds Homogeneous Catalysis with Special Reference to Hydrogenation and Bonding in Metallo-Organic Compounds Motions in Molecular Crystals Polymer Solutions The Vitreous State Electrical Conduction in Organic Solids Surface Chemistry of Oxides Reactions of Small Molecules in Excited States The Photoelectron Spectroscopy of Molecules Molecular Beam Scattering Intermediates in Electrochemical Reactions Gels and Gelling Processes Photo-effects in Adsorbed Species Physical Adsorption in Condensed Phases Electron Spectroscopy of Solids and Surfaces Precipitation Potential Energy Surfaces Radiation Effects in Liquids and Solids Ion-Ion and Ion-Solvent Interactions Colloid Stability Structure and Motion in Molecular Liquids Kinetics of State Selected Species Organization of Macromolecules in the Condensed Phase Phase Transitions in Molecular Solids Oxidation Volume 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 * 66 67 68 69 * Not available; for current information on prices, etc, of available volumes, please contact the Marketing Oficer, Royal Society of Chemistry, Burlington House, London Wl V OBN stating whether or not you are a member of the Society.

ISSN:0301-7249    

DOI:10.1039/DC98069FX001

出版商:RSC    

年代:1980

数据来源: RSC

摘要:

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. (b) Soft-mode frequency (non-fitted) as a function of temperature in deuterated biphenyl. these molecules are different from the orientational ordering in an angle dependent periodic potential; more particularly in this last case, the soft-mode behaviour does not exist.The authors are grateful to J. Bouillot, A. Heidemann, F. Mezei and C. Vettier of I.L.L.; also to A. Girard of Groupe de Physique Cristalline for helpful discus- sions and assistance during measurements. 0. Bastiansen, Acta Chem. Scand., 1949, 3, 408. H. Suzuki, Bull. Chem. Soc. Japan, 1959, 32, 1340. G. P. Charbonneau and Y. Delugeard, Acta Cryst. By 1977, 33, 1586. J. L. Baudour, H. Cailleau and W. B. Yelon, Acta Cryst. B, 1977, 33, 1773. Y. Delugeard, J. Desuche and J. L. Baudour, Acta Cryst. B, 1976, 32, 702. H. Cailleau, J. L. Baudour and C. M. E. Zeyen, Acta Cryst. By 1979, 35, 426, ' J. L. Baudour, Y. Delugeard and H. Cailleau, Acta Cryst. B, 1976, 32, 150. * J.L. Baudour, Y. Delugeard and P. Rivet, Acta Cryst. B, 1978, 34, 625. J. C. Messager, M. Sanquer, J. L. Baudour and J. Meinnel, 1st European Crystal Meeting, Bordeaux, 1972. T. Schneider and E. Stoll, Phys. Rev. B, 1976, 13, 1216. G. A. Samara, T. Sakudo and K. Yoshimitsu, Phys. Rev. Letters, 1975, 35, 26. l3 J. Paureau and C. Vettier, Rev. Sci. Instr., 1975, 46, 963. I4 H. Cailleau, J. L. Baudour, A. Girard and W. B. Yelon, Solid State Comm., 1976, 20, 577. I5 J. P. Bachheimer and G. Dolino, Phys. Rev. B, 1975, 11, 3195. l6 H. Cailleau and A. Dworkin, Mol. Cryst. Liq. Cryst., 1979, 50, 217. l o S. Aubry, J. Chem. Phys., 1975, 62, 3217. S . S. Chang, 7th ASME Symp. Thermophys. Prop., 1977. A. Dworkin and H. Cailleau, to be published. l 9 A. Cullick and R. E. Gerkin, Chem. Phys., 1977, 23, 217.18 STRUCTURAL PHASE TRANSITIONS I N POLYPHENYLS ’O H. Cailleau, A. Girard, F. Moussa and C. M. E. Zeyen, Solid State Cornrn., 1979, 29, 259. 21 H. Cailleau, A. Heidemann and C . M. E. Zeyen, J. Phys. C, 1979, 12, L411. 22 A. Girard, H. Cailleau, Y . Marqueton and C. Ecolivet, Chem. Phys. Letters, 1978, 54, 479. 23 B. A. Bolton and P. N. Prasad, Chem. Phys., 1978, 35, 331, 24 A. Bree and M. Edelson, Chem. Phys. Letters, 1977, 46, 500. 25 L. D. Landau and E. M. Lifshitz, Statistical Physics (Pergamon, Oxford, 1968). 26 A. Guyon, Thtse 32rne Cycle (Universite d’Amiens, 1978). 27 V. Dvorak and Y . Ishibashi, J . Phys. SOC. Japan, 1978, 45, 775. 2 9 H. Cailleau, F. Moussa, C. M. E. Zeyen and J. Bouillot, Solid State Cornrn., in press. 30 A. W. Qverhauser, Phys. Rev. B, 1971, 3, 3173. H. Cailleau, F. Moussa and J. Mons, Solid State Cornrn., 1979, 31, 521.

ISSN:0301-7249    

DOI:10.1039/DC9806900007

出版商:RSC    

年代:1980

数据来源: RSC

摘要:

Phase Transitions in Crystals of Chain Molecules Relation between Defect Structures and Molecular Motion in the Four Modifications of n-C33H68 BY BERND EWEN AND GERT R. STROBL Institut fur Physi kalische Chemie, Universitat Mainz, Jakob-Welder-Weg 15, 6500 Mainz, W. Germany AND DIETER RICHTER Institut fur Festkorperforschung, KFA Jiilich, 5170 Julich, W. Germany Received 17th December, 1979 n-Tritriacontane (n-C33H68) exhibits three solid-solid phase transitions before melting. Applying a variety of experimental techniques, including X-ray scattering, Raman spectroscopy, n.m.r., dielec- tric relaxation and quasielastic neutron scattering, it was possible to specify for all four modifications defect structures and the corresponding molecular motions. Each phase transition is accompanied by a step-like decrease in the degree of order resulting from the successive onset of rotational jumps, translational jumps in chain direction and the creation and diffusion of intrachain defects.Molecular crystals may exhibit one or more solid-solid phase transitions before melting1 These transitions in most cases are of the first order and accompanied by the introduction of defects. They can generally be regarded as reactions of the crystal lattice to the onset of specific types of molecular motion. In the special case of oligomeric chain molecules, which crystallize as lamellar systems, defect structures may concern the orientationill order of chains, the lamellar surface or the internal chain conformation. This paper gives an example and presents detailed studies on the solid-phase behaviour of n-tritriacontane (n-C33H68). This crystalline n-alkane passes through four stable modifications between room temperature and the melting point.A variety of experimental techniques was applied in order to characterize in detail the molecular motions and defect structures in the four solid phases with the general aim of gaining a better understanding of phase transitions in chain molecular crystals. SOLID PHASE BEHAVIOUR OF n-C33H68 n-Tritriacontane (n-C33H68), which was synthesized in a manner that excluded the presence of neighbouring homologues, exhibits three solid-solid phase transitions before melting (Tm = 71.8 "C) as can be seen from the d.s.c. diagram (see fig. 1). The four crystalline modifications which appear with increasing temperature are denoted in the following as A, B, C, D.Modification A is the orthorhombic low- temperature form found with all odd-numbered paraffins. Modification D is usually called the " rotator phase ", expressing a view introduced first by Muller3 according20 PHASE TRANSITIONS IN CRYSTALS OF CHAIN MOLECULES , I A --L B - r C - k - D -I- I I I I 54.5 65.5 68.0 71.8 T/"C FIG. 1 .-D.s.c. thermogram of solution crystallized n-CJ3H6*, showing three solid-solid phase transi- tions before melting. A, B, C and D denote the four solid modifications. to which the chains should rotate quasi-freely about their long axes. Hitherto no attention has been directed to the modifications intermediate between A and D. Wide-angle X-ray scattering experiments were used to determine the structures in these four modifications.For the highly disturbed crystal lattice in modification D it was not possible to specify the unit cell parameters. However, essential para- meters like the angle I// between the chain axes and the normal to the lamellar surface, TABLE 1 .-n-CS3Hb8, RESULTS OF WIDE-ANGLE X-RAY SCATTERING EXPERIMENTS q,: electron density in the interior of the lamellae; I,U: angle between chain axis and surface normal. modification subcell crystal structure q, ty remarks A(25 "C) orthorhombic orthorhombic 0.341 0 a = 7.44 A b = 4.96 A c = 87.65 8, B(60 "C) orthorhombic monoclinic 0.334 0 a = 7.57 A b = 4.976 A c = 87.94 A p = 88.8 or 91.2" space group Pcam layer surface parallel to (OOl), plane of the sublattice layer surface parallel to (OOl), plane of the sublattice twin structure C(66 "C) orthorhombic monoclinic 0.332 18.5 layer surface parallel a = 8.02 A to (101), and (lOT), b = 4.985 A planes of the sub- c = 88.0 A lattice = 73.5 or 106.5' twin structure D(69 "C) nearly, but 0.318 19.5 multiple twin struc- definitely not ture completely, hexagonalB .EWEN, G . R . STROBL A N D D . RICHTER 21 the electron density qc in the interior of the lamellae and the cross-section S per chain were obtained with sufficient accuracy. The results of these investigations are sum- marized in table 1 . RESULTS OF SMALL-ANGLE X-RAY SCATTERING A N D RAMAN SPECTROSCOPY For perfect n-alkane lamellae all CH,-end groups of the molecules are slfuated in the planar lamellar surfaces.This situation is expected to change if motions of extended chains with a component in the chain direction become active or if interchain defects, which may diffuse along the chains, lead to a chain shortening. In both cases the originally planar structure of the lamellar surface is perturbed. Perturbations of this kind can be examined by measuring the absolute intensity of the small-angle X- ray scattering (SAXS). Since these investigations play a central role for the following discussion, the principles of evaluation are sketched briefly here. A detailed explana- tion of the procedures is given in ref. (5). As can be seen from fig. 2(a), in a perfect, well ordered n-alkane crystal adjacent FIG. 2.-Examination of the interfacial order in n-alkane crystals with the aid of small angle X-ray scattering experiments.Deviations from the equalities d, = d,,, d,, = D, (a), arising from longi- tudinal shifts (ds = d,,, d,, < D,) (6) and from intrachain defects (d, < d,,, d,, < 0,) (c). lamellae are separated by a plane layer of voids. Its thickness d, is given by the difference of the long period L and the projection of the extended chain on the normal to the lamellar surface d, = L - 1.27 A n cos t , ~ (1) (n is the number of C atoms in the backbone and t , ~ the angle between chain direction and surface normal). This lamellar structure is reflected in a SAXS experiment by a series of (001) reflections, their intensities being solely determined by the electron density profile of the interfacial region. Any disorder in the lamellar interfaces arising from the introduction of defects changes this profile and hence the intensities in a characteristic manner.Two important structure parameters can be directly derived. First, the extrapolated intensity of the zeroth-order reflection I, is proportional to the square of d,,, the average thickness of the voids between adjacent end groups. I, - d;,. (2)22 PHASE TRANSITIONS I N CRYSTALS OF CHAIN MOLECULES Secondly D,, the overall thickness of the void layer, is related to the curvature at the origin %] -0:. 1 = 0 D, is connected with the second moment of the electron density profile, 02, by D, = (120~)"~. (3) (4) Using eqn (2) and (3) one can decide about the occurrence of defects and distinguish between different types.For a well ordered lamellar system all three quantities d,, d,, and D, should be equal [see fig. 2(a)]. If the chains are randomly shifted in chain direction but remain stretched [see fig. 2(b)], D, is increased with respect to d,, the averaged void thickness d,, remaining unchanged. In the event that intrachain defects as kinks6 or Reneker defects7 are introduced [see fig. 2(c)], additional vacancies are created leading to an increase in both d,, and D,, with respect to d,. Thus, by measuring the absolute intensities of SAXS reflections and comparing D, and d,, with ds, it becomes possible to characterize the lamellar surfaces in a quantitative manner. In table 2 the experimental results of the SAXS investigations on n-C&Hs8 are summarized. For all modifications the long period L and the parameters d,, d,, and D, are listed.TABLE 2.-CS3HSS: LONG PERIOD L AND STRUCTURE PARAMETERS d,, d,,, D t , DETERMINED BY SAXS EXPERIMENTS FOR MODIFICATIONS A, B, c AND D A(25 "C) 43.8 + 0.01 1.83 f 0.01 1.82 & 0.03 1.8 5 0.8 B(60 "C) 43.96 jI 0.01 1.96 f 0.01 1.95 5 0.03 2.3 i 0.8 C(66 "C) 42.2 f 0.05 2.37 k 0.05 2.42 5 0.05 7.5 f 0.6 D(69 "C) 42.05 5 0.1 2.46 jI 0.1 3.32 & 0.08 10.1 5 0.6 The data of modification A prove that the crystals, which were obtained from a dilute solution, are perfect at room temperature. The well ordered lamellar surface remains nearly unchanged when passing to modification B. In modification C one still finds coincidence between d,, and d,; however, there is a significant increase in D,. This result clearly signals the occurrence of longitudinal disorder, the chains remaining in the extended all-trans conformation.Finally, in modification D the SAXS data are indicative of additional intrachain defects. Their occurrence could be confirmed by the i.r. spectrum,8 where bands appeared (at 13 10 and 1350 cm-I), which are considered to represent local modes attached to a gauche-trans-gauche sequence in an otherwise stretched paraffin ~ h a i n . ~ r l ~ Additional evidence for the occurrence of intrachain defects follows from the low- frequency Raman spectrum. It can be observed that the accordion mode, which is assigned to the all-trans conformation,ll shows an intensity decrease upon passing to modification D. Fig. 3 shows the accordion modes in the modifications A and D, both being normalized to the integral intensity of the CH,-twisting band (1295 cm-I), which can be used as an internal scattering standard.I2 The result indicates the occur- rence of intrachain defects for ~ 4 0 % of the chains.B.EWEN, G . R . STROBL AND D. RICHTER 23 I I 1 1 I 1 I 90 85 80 75 70 65 60 F/cm-' FIG. 3.-Accordion vibration bands in the low frequency Raman spectrum of n-C33H68, obtained for modifications A and D after normalization with regard to the CH, twisting intensity. RESULTS OF WIDE-LINE N . M . R . In contrast to SAXS, where only small changes of the characteristic parameters occur when passing from A to B (see table 2), major effects can be seen in the wide line n.m.r. spectra.' These are shown in fig. 4, where the line width and the second moment are plotted as a function of temperature between - 150 "C and the melting point.Step-like decreases are observed at the transitions A -+ B and C -+ D, whereas the transition B -+ C is passed continuously. A quite similar decrease was found by Olf and Peterlin', in n.m.r. experiments on the neighbouring homologue n-C32H66 using a uniaxially oriented sample. On the basis of model calculations14 these authors concluded from the orientational de- pendence of the second moment that the motion becoming active at the first step should involve 180" rotational jumps of the extended chains about their long axis. As a mechanism of motion they suggested flip-flop screw jumps, i.e., lS0" rotational jumps coupled with a simultaneous shift in chain direction over one CH2 unit.By such a kind of motion the packing of the methylene groups in the interior of the lamella is unchanged. Similar measurements on n-C33H68 led to the same experimental result.8 At the transition from C + D the angular dependence of the second moment sug- gests a quasi rotation-like motion of the CH, groups about the long axis. The experimentally determined second moment, 4.8 G2,8 is smaller than the value of 5.8 G2 expected for rigid-rod-like rotating chains as was initially assumed by Miiller. DEFECT STRUCTURES I N THE FOUR MODIFICATIONS O F ll-C,,H,8 From the experiments described a molecular picture evolves as sketched in fig. 5. At room temperature the crystals are perfect with the possible exception of a The transition to modification B is few single orientational defects [see fig.5(A)].24 PHASE TRANSITIONS I N CRYSTALS OF CHAIN MOLECULES accompanied by the onset of pure 180"-rotational jumps. Since the lamellar surface remains planar and, furthermore, the long period L is practically unchanged, any simultaneous longitudinal shift can be excluded. In particular, flip-flop screw jumps, as proposed by Olf and Peterlin,l3 appear improbable. Pure 180"-rotational jumps seem possible in principle since the chains indeed reach a relative minimum after a 180" turn.15 However, for energetic reasons it is unreasonable to expect the molecules to perform these jumps individually. Much more likely is a cooperative jump process 2l 20 X I FIG. 4.-Second moment <AH2) and linewidth dH, of the wide-line n.m.r.spectra of an isotropic n-CJ3H6* sample. In modification D a second component with linewidth dH, appears. which preserves an orientational short range order [see fig. 5(B)]. The orientational long range order, present within modification A, is broken down. With the transition to modification C [see fig. 5(C)] the originally planar lamellar surfaces are destroyed. This result indicates that longitudinal disorder is present, resulting from a motion with a component parallel to the chain direction. One possibility would be a change from the pure rotational-jumps of modification B to a flip-flop screw motion. In this case the motional components parallel and perpendi- cular to the chain axes would be strongly correlated. Alternatively one can also imagine a motional behaviour where the 180" rotational jumps, active in modification B, are superposed by independent translational jumps in the chain direction.This question cannot be decided from the experimental data presented so far. The results of quasielastic neutron scattering experiments 16*' described in the next section will permit a decision. The wide-angle X-ray pattern obtained for modification D4 indicates that the crystal lattice is highly distorted, the regular CH, group packing being largely broken down [see fig. 5(D)]. This is supported by the low value of the n.m.r. second moment, which suggests a considerable degree of chain twisting. SAXS intensities, i.r. andB . EWEN, G . R . STROBL AND D . RICHTER 25 Raman spectroscopic observations provide evidence for intrachain defects. Estimates based on the Raman intensity of the accordion mode and, assuming simple kink defects (g+-t-g- sequences), on the SAXS parameter day, give a concentration of defected chains of the order 0.4-0.7.' As reflected by the value measured for D,, there is an additional increase in the motional components parallel to the chain axes.As a consequence the simple rotator model commonly used for describing the high temperature modification of n-alkanes should be modified with two respects : First, one has to include intrachain motions like high amplitude torsional oscillations C D FIG. 5.-Defect structures in the four modifications of n-C33H68, in views perpendicular and parallel to the chain direction. A, single orientational defects; B, coupled 1 80" jumps, orientational short-range order; C, superposition of restricted longitudinal motion; D, hindered rotation, spacial limited diffusion in chain direction, The orientation of the zig-zag planes is indicated by arrows.chain twisting, kink defects. and defect diffusion. Secondly, there is an additional component of motion parallel to the chain axis involving the molecules as a whole. The experiments described so far do not permit a determination of the time scales to be attributed to the different types of motion. For this purpose two other experi- mental methods, namely dielectric relaxation measurements l8 and quasielastic26 PHASE TRANSITIONS I N CRYSTALS OF CHAIN MOLECULES neutron ~ c a t t e r i n g ' ~ ~ ~ ~ were applied.Since the quasielastic neutron scattering is simultaneously sensitive to the motional behaviour in time as well as in space, this method provides additional more detailed information. DIELECTRIC RELAXATION MEASUREMENTS The dielectric relaxation experiments were performed on a sample of n-C3,H6' to which a small amount (5%) of a symmetric ketone (C33H660) with the same chain length was added.'' The results are shown in fig. 6, where the dielectric loss tangent /- / / / / 4 5 6 log,, f 7 FIG. 6.--Solid solution of the symmetric ketone C33H660 (5%) in n-C33H6e. Frequency dependence of the dielectric loss tangent, measured for different temperatures in the modifications A, B and C . (a) 66 "C (C), (6) 64 "C (B), (c) 57.5 "C (B), ( d ) 55 "C (A/B), (e) 50 "C (A), (f) 40 "C, (g) 30 "C, (h) 68, 69 "C (D).is plotted as a function of frequency for some discrete temperatures in the modifica- tions A, B and C. In modification A one observes only a weak absorption signal, which may be caused by rotational jumps of single isolated molecules [see fig. 5(A)]. With the transition to modification B a significant change occurs, leading to a complex lineshape and an increase of the intensity. The observed relaxation strength corre- sponds to 180" rotational jumps between equally occupied orientational positions. This is in agreement with the results of the wide-line n.m.r. measurements.' The jump frequencies are lo5 - lo6 s-l. A further discontinuous change is observed at the transition to modification C . Since the maximum of tan 6 at 66 "C is out of the range of measurement, one can only estimate its position from the shape of the curve on the low frequency side.In this way one is led to a relaxation frequency of ~ ( 1 - 2 ) x lo7 s-l. QUASIELASTIC NEUTRON SCATTERING The scattering of slow neutrons by molecules containing protons is nearly entirely determined by the large incoherent cross section of the protons. The scattering intensity is proportional to the incoherent scattering law Sinc(Q, co), where hQ and hco are the momentum and energy transfer during the scattering process. AccordingB . EWEN, G . R . STROBL AND D. RICHTER 27 to van Hovel9 Sinc(Q, u) can be written as the Fourier transform of the self-correla- tion function Gs(r, t ) of the protons with respect to space and time: Sometimes it is more convenient to use instead of eqn (5a) the intermediate scattering law As can be seen from eqn ( 5 ) it is possible to separate longitudinal and rotational components of motion if uniaxially oriented samples are used with the vector of momentum transfer oriented parallel or perpendicular to the chain axes.Under the assumption of instantaneous uncorrelated jumps with negligible jump times the time evolution of Gs(r, t ) can be calculated from rate equations and in the limiting case of continuous diffusion from the diffusion equation. The resulting scattering law in general is a superposition of Lorentzians with line widths given by the eigenvalues of the rate equation system and weights which are connected with the eigenvectors of the system. Io(Q), the so-called elastic incoherent structure factor, only contributes to eqn (6) if molecular motion is spatially restricted.It represents the Fourier transform of the equilibrium distribu- tion reached for t -+ co. The measurements were performed at the Grenoble high flux reactor using the backscattering spectrometer IN10 and the multichopper time-of-flight machine IN5?717 Spectra were obtained simultaneously at different @values for various orientations. In modification C a distinct broadening was observed for both, Q parallel and Q perpendicular to the chain axis. This result can be taken as confirmation that molecu- lar motion in modification C indeed is composed of both longitudinal and rotational components. In the analysis of spectra two different jump mechanisms were considered which are compatible both with the X-ray and spectroscopic data as described above.The first one is a flip-flop process, where longitudinal and rotational motions are strongly correlated and described by one common jump rate of. The corresponding scattering law has the form of eqn (6); SAXS data suggest one should set N = 5 (see table 2). Secondly, completely independent rotational and longitudinal motions are as- sumed. In this case the intermediate scattering law is given by the product of a longitudinal [Itrans(Q, t ) ] and rotational [Zrot(Q, t ) ] part. Itrans(Q, t ) is equivalent to the intermediate scattering law of the flip-flop model if the jump vectors rm are exchanged in a proper way. N is the number of sites accessible for each proton. The rotational part is very simple2’ and has the form: Irot(Q, t ) = Zo(Q) + Il(Q)e-2L’rott (7) = +(1 + cos Qrrot) + +(1 - cos Qrr,t)e-2L’ott where orot is the rotational jump rate and rrot the corresponding jump vector.The theoretical scattering laws were compared with the experimental results after an averaging process l4 accounting for the uniaxial orientational distribution in the samples.28 PHASE TRANSITIONS I N CRYSTALS OF CHAIN MOLECULES TABLE 3.-JUMP RATES AND DIFFUSION COEFFICIENTS IN THE MODIFICATIONS c AND D OF n-C33H68, AS OBTAINED BY QUASIELASTIC NEUTRON SCATTERING T/OC u ~ , , ~ / s - ~ UtransIS- ' Drotls-' Dtrans/Cm2 s-' 66 ~2 x 1 0 7 (2.1 i- 0.3) x 108 - - 67 (5 2) x lo7 (2.6 f 0.4) x lo8 - - 70 - - (6.3 f 0.1) x 10'' (1.2 f 0.2) x 10 -' From this data analysis it became evident that there was no rigid coupling between the longitudinal and rotational components of motion.Systematically the jump rates uf derived from the spectra with Q perpendicular to the chain axis were smaller than those for a parallel Q-orientation. This observation definitely excludes pure flip-flop jumps. The jump rates obtained assuming independent 180" rotational jumps and longitudinal jumps over distances of 1.27 A are listed in table 3. In fig. 7 some spectra of a series, where Q is mainly parallel to the axis of orienta- tion, are fitted with the corresponding scattering law. The rotational jump rates urot, which lie just on the border of the instrumental resolution, are in good agreement with the results of the dielectric measurements, described above.An explanation for the different jump rates follows from the energy maps calcu- lated by McCullough.zl They show that the hindrance potential for 180" rotational jumps is indeed larger and exhibits a stronger temperature dependence than that of the motion in chain direction. I I 1 - 1 0 1 1 I I I I 1 - 1 0 1 L I I I - 1 0 1 - 1 0 1 PeV FIG. 7.-Quasielastic neutron spectra obtained on uniaxially oriented n-C33H68 at 67 "C. parallel to the axis of orientation. Q = 1.9 A-' y = 0"; (ii) Q = 1.4 A-1, y = 14.4"; (iii) Q = (i) Q = 1.0 1.65 A-', y = 25.6"; (iv) Q = 1.9 A-1, y = 41.8".B . EWEN, G . R . STROBL AND D . RICHTER 29 In modification D the quasi-elastic broadening is considerably larger than in modification C .Here the elastic incoherent structure factor (see fig. 8) could be measured separately using the INlO, while the quasi-elastic spectra were obtained with the IN5. Analysis of the experimental data was based on a dynamic model where the motional process was assumed to be composed of a rotational diffusion of stretched 1 I 0 0.5 1.0 1.5 2:o 0 0.5 1.0 1 5 2.0 QlA-' FIG. 8.-Elastic incoherent structure factor in modification D. (a) Q oriented nearly perpendicular, (6) Q oriented nearly parallel to the axis of orientation. 0, x experimental points before and 0, A after multiple scattering corrections. Solid lines : theoretical curves. Translational amplitude: I, b = 4 A; 11, b = 5 A. chains about the chain axis and of a diffusive motion in chain direction within a limited range 2b, both occurring independent of each other.The rotational part of the scattering law has the form22 03 Irot(Q, t ) = J;(Q, a) + 2 1 J,'(Q, a)e-"'%tt n = l where Ji are the Bessel functions of the ith order, D,,, is the rotational diffusion con- stant and a denotes the radius of the hydrogen rotation.30 PHASE TRANSITIONS I N CRYSTALS OF CHAIN MOLECULES The translational part, calculated for a linear diffusion in a limited range of length 2b may be written as : 23 with D,,,,, is the linear diffusion coefficient. Again the intermediate scattering law be- comes the product of eqn (8) and (9) and has to be averaged with respect to uniaxial orientation of the sample. In fig. 8 the elastic incoherent structure factor is shown for Q parallel and Q perpendicular to the chain axis.The solid lines give the theoretical curves for a rotational radius of a = 1.4 A, which corresponds to a rigid-rod-like rotation, and for a translational diffusion length of 2b = 8 and 10 A. The majority of experimental points lies between these two lines. The spatial extension of the diffusional process in chain direction thus derived is in good agreement with the corresponding SAXS result. The values a = 1.4 A and b = 4.5 8, were then used to evaluate the time-of-flight n c U .I 2 W c 00 -0.75 -0.25 0.25 0.75 -0.50 0 0.50 LA- LA I 1 . . 0.75 -0.25 0.25 0.75 -0.75 -0.25 0.25 0.75 -0.50 0 0.50 -0.50 0 0.50 -0.75 -0.25 0.25 0.75 -0.75 -0.25 0.25 0.75-0.75 -0.25 0.25 0.75 -0.50 0 0.50 -0.50 0 0.50 -0.50 0 0.50 e.nergy ?ransfer, AE I meV FIG.9.-Quasielastic neutron scattering in modification D. Experimental spectra after conversion to the energy scale. (a) Q perpendicular to the chain axes for Q = 1.9 A-', (b) Q parallel to the chain axes for Q = 1.6 A-'. Solid lines: theoretical scattering law for independent rotational and longi- tudinal diffusion (i) y = 28.9", Q = 0.2 A-'; (ii) y = 51.4", Q = 1.0 A-'; (iii) y = 75.9", Q = 1.6 A-'; (iv) y = 47.0", Q = 0.2 A-'; (v) y = 24.5", Q = 1.0 A-'; (vi) y = 0", Q = 1.6 A-'.B . EWEN, G . R . STROBL AND D . RICHTER 31 spectra. In fig. 9 some typical experimental spectra are shown together with corre- sponding theoretical curves. The diffusion coefficients used are given in table 3. In all cases, the theoretical scattering law fits the experimental data very well.This result shows that the molecular dynamics in modification D can be described pre- dominantly as a diffusive process with independent rotational and longitudinal components. The creation of intrachain defects does not lead to a general increase of the radius of rotation of the protons, compared to rotating chains with an extended chain conformation. Otherwise the rotational part of the incoherent structure factor [see fig. 8(a)] would decrease faster with respect to Q. The authors thank their colleagues Dip1.-Phys. K. Malzahn, Dr. W. Piesczek and Dip1.-Phys. T. Trzebiatowski for their important contributions to this work. They also thank Dr. A. Heidemann and Dr. R. Lechner for their advice during the neutron scattering experiments. Thanks are due in particular to Prof.Dr. E. W. Fischer and Prof. Dr. T. Springer for their continuous interest in this work and for many helpful and stimulating discussions. Finally financial support by the Deutsche Forschungsgemeinschaft (Sonder- forschungsbereich 41, Physik und Chemie der Makromolekule, Mainz/Darmstadt) and by the Bundesministerium fur Forschung und Technologie is gratefully acknow- ledged. Physics and Chemistry of the Organic Solid State, ed. D. Fox, M. M. Labes and A. Weissberger (Interscience, New York, London and Sydney, 1963), vol. I. W. Heitz, T. Wirth, R. Peters, G. Strobl and E. W. Fischer, Makromol. Chem., 1972, 162,63. W. Piesczek, G. Strobl and K. Malzahn, Acta Cryst., 1974, B30, 1278. G. Strobl, B. Ewen, E. W. Fischer and W. Piesczek, J. Chem. Phys., 1975, 61, 5257. W. Pechhold, W. Dollhopf and A. Engel, Acustica, 1966, 17, 61. ’ D. H. Reneker, J . Polymer Sci., 1962, 59, 539. B. Ewen, E. W. Fischer, W. Piesczek and G. Strobl, J. Chem. Phys., 1975, 61, 5265. R. G. Snyder, J. Chem. Phys., 1967, 47, 1316. lo G. Zerbi, Pure Appl. Chem., 1971, 26, 499. R. F. Schaufele and T. Shimanouchi, J. Chem. Phys., 1967, 47, 3605. l2 G, Strobl and W. Hagedorn, J. Polymer Sci., Polymer Phys. Ed., 1978, 16, 1181. H. G. Olf and A. Peterlin, J . Polymer Sci. A-2, 1970, 8, 771. l4 H. G. Olf and A. Peterlin, J . Polymer Sci. A-2, 1970, 8, 753. P. E. McMahon, R. L. McCullough and A. A. Schlegel, J. Appl. Phys., 1967, 38, 4123. l6 D. Richter and B. Ewen, Proc. ZAEA Symp. Neutron Inelastic Scattering, Paper M-219135, Vienna 1977. l7 B. Ewen and D. Richter, J . Chem. Phys., 1978, 69, 2954. l* T. Trzebiatowski, Diplomarbeit (Mainz, 1976). l9 L. van Hove, Phys. Rev., 1954, 95, 249. 2o J. D. Barnes, J. Chem. Phys., 1973, 58, 5193. 21 R. L. McCullough, J . Macromol. Sci. Phys., 1974, 9, 97. 22 A. Dianoux, F. Volino and H. Hervet, Mol. Phys., 1975, 30, 1181. 23 P. L. Hall and D. K. Ross, Mol. Phys., 1978, 36, 1549. ’ A. Muller, Proc. Roy. SOC. A , 1932, 138, 514.

ISSN:0301-7249    

DOI:10.1039/DC9806900019

出版商:RSC    

年代:1980

数据来源: RSC

摘要:

The Phase Transition in t-Butyl Cyanide [(CH,),CCN] t BY JONATHON C. FROST AND ALAN J. LEADBETTER Chemistry Department, University of Exeter, Exeter EX4 4QD AND ROBERT M. RICHARDSON 8 Institut Laue-Langevin, Grenoble Received 18th January, 1980 Extensive neutron scattering measurements have been carried out on t-butyl cyanide over a wide temperature range. Crystal structures and the potential for methyl-group rotation have been deter- mined and will be reported elsewhere; this paper is concerned with the whole-molecule motions as determined by incoherent quasielastic and inelastic scattering measurements. One transition only has been observed at 233 K and the whole-molecule reorientational motions determined between 190 and 280 K (m.p. 292 K). At all temperatures the molecules undergo rapid reorientational motion about their C-C-N axes in a 3-fold potential which is strongly temperature dependent.The transition is associated with an order of magnitude decrease in the rotational correlation time and the onset of a slower, strongly cooperative motion involving translational as well as rotational displace- ments of the molecules. These probably involve relaxations between different local structures each having essentially the monoclinic structure of the low-temperature phase and related by rotations of 71/2 resulting in the average tetragonal symmetry of the high-temperature phase. The neutron results lead to a self-consistent reinterpretation of previous n.m.r. data. Tertiary butyl cyanide [(CH,),CCN] is an especially interesting member of the (CH3)3CX series which shows a rich solid-state polymorphism.The dielectric studies of Clemett and Davies' indicated that in the highest temperature crystal phase (I) the molecules may rotate about the CCN axis but the dipolar axis itself does not reorientate until the liquid phase is reached. gave the following phase behaviour : Extensive calorimetric measurements (Crystal 111) T = 213 K; A S = 1.1 J K-I mo1-I (Crystal 11) T = 233 K; A S = 7.8 J K-' mol-I (Crystal I) T = 292 K; A S = 31.8 J K-I mol-I (liquid). The transition III/II was not well-defined but the total entropy increment I11 to I is near R In 3 (9.1 J K-' mo1-I) which was taken to imply that I is a uniaxial rotator phase. Whether or not this interpretation of A S is correct the rotation of the molecules about their CCN axes in Crystal I has been confirmed by n.m.r.3 and very recent incoherent neutron quasielastic (INQES) meas~rements.~ t Experiments carried out at I.L.L., Grenoble and A.E.R.E., Harwell.$ Now at Rutherford Laboratory, Chilton, Didcot, Oxon OX1 1 OQX.J . C . FROST, A. J . LEADBETTER AND R . M . RICHARDSON 33 A number of uncertainties remain, however. First, an earlier n.m.r. study' was interpreted as showing only methyl group rotation below the m.p., while in the later work3 the single step decrease in linewidth near 110 K was interpreted (more reasonably) as due to the onset of both methyl and whole-molecule uniaxial rotations. The additional observation of a single TI minimum near 160 K led to the conclusion that the methyl and molecular rotations are coupled and describable by a single correlation time and energy barrier (16.3 kJ mol-I).The motion is established in phase 111 and almost no changes are observed at the transitions (except a slight 41 0.0 0.5 hw/meV FIG. 1 .-Scattering law, S(Q, o) of t-butyl cyanide at 227 K in the Crystal I1 phase measured on IN5 with lo = 10.0 A. The spectra (which are not to the same vertical scale) have elastic Q values (from bottom to top) of 0.17,0.38,0.76 and 1.15 A-'. The solid curve is a fit of the 3-fold rotational jump model [eqn (6), (7) and (ll)] to the experimental data. The dashed line separates the elastic from the quasielastic scattering. hysteresis around III/II as found calorimetrically). This is a puzzling conclusion because the barriers to methyl rotation in t-butyl compounds have been thoroughly studied '--* and are largely intramolecular in origin whereas the whole-molecule re- orientation must be controlled by intermolcular forces and should have a different temperature dependence. Secondly, the correlation time at the II/I transition from the n.m.r.work is z s in I with a jump at the transition to values too great to be measured in I1 ( 2 2 x s). Questions remain, therefore, about the relation between the different experimental results, the nature and time scale of intra- and inter-molecular rotations in the various phases and the nature of the phase transition(s). s whereas the INQES results gave z 5 x34 PHASE TRANSITION I N (CH3)3CCN t-Butyl cyanide was chosen for an extensive set of measurements on a uniaxial rotator phase with the ultimate objective of understanding not only the single molecule motions but also the collective aspects of the reorientational motions and the nature of the transition.It was selected because both hydrogenous and deuterated versions may be obtained and the melting point being near room temperature should facilitate the preparation and handling of single crystals. Furthermore the strong cyano - 1 0 hw/peV 1 FIG. 2.-Scattering law, S(Q, w) of t-butyl cyanide in the Crystal I1 phase measured on the back scattering spectrometer IN1 0. The solid curves are calculated for the 3-fold jump reorientation model [eqn (6), (7) and ( l l ) ] and the dotted curves separate elastic and quasi-elastic components: (a) Q = 0.57 A-', T = 208 K; (b) Q = 1.51 A-', T = 208 K; (c) Q = 1.51 A-l, T = 190 K.dipole and the uniaxial rotations suggested that the dipoles are probably all aligned (along 0 and n) to give essentially a 2-d system of coupled rotors. So far we have completed low-resolution structural determinations of low- and high-temperature phases and INQES plus incoherent neutron inelastic scattering (INIS) on powder samples.* The structure from < 5 to >227 K is monoclinic (P2Jrn) with two mole- cules per cell in a head to tail arrangement, all dipoles colinear and directed along c. No structural or other changes have been found at the reported III/II transition so we hereafter designate the structure existing below 233 K as Crystal 11. The structure ofJ .C . FROST, A . J . LEADBETTER A N D R . M . RICHARDSON 35 0.0 0.5 ho/meV 0.0 0.5 1.0 U L 1 i .I - .._1-- ho/meV 0 0 0.5 10 1.5 2.0 2.5 FIG. 3.-Scattering law, S(Q, o) of t-butyl cyanide in the Crystal I phase. The solid curves are typical fits to the data using eqn (3), ( 5 ) and (1 1) with arbitrary elastic coefficients, the dashed curves separate elastic and quasielastic components and the horizontal straight lines show the flat inelastic back- ground: (a) 270 K, measured using IN5 at 1, = 10 A, Q values (bottom to top) of 0.17, 0.38, 0.76, 0.92 and 1.15 A-'; (6) 240 K measured using 4H5, Q values (bottom to top) of 1.7, 1.1, 0.75 and 0.53 ii-l.36 PHASE TRANSITION I N (CH3)$CN Crystal I is tetragonal (probably P4/n) with 2 molecules per cell, at the origin and body centre, again in a head to tail configuration with dipoles along the 4-fold axis (parallel to c ) and related to I1 by a straightening of the molecules and a greater spatial requirement in the plane perpendicular to the C-C-N axes.* The methyl- group rotations have been fully characterized and are well described in terms of motion in a 3-fold potential well of depth -18 kJ mol-I and with zo - 0.3 x s where the average residence time in the well is z = zo exp(V3 - EJRT.This paper is concerned with the whole-molecule motions as determined by incoherent neutron scattering measurements in both phases. Structural and methyl rotation data will be used as appropriate in analysing the results but will not be discussed further here.8 Since only incoherent scattering measurements have been made (except for the struc- tural work) only a single-particle description of events is obtained directly but this proves to be very informative and indirectly reveals pronounced co-operative phe- nomena involving coupled rotational and translational displacements in Crystal I, the onset of which appears to be the essential feature of the phase transition.EXPERIMENTAL The sample was obtained from B.D.H. and checked for purity by n.m.r. and gas- liquid chromatography: no impurities were detected suggesting a purity of better than 99.8 %. Differential scanning calorimetry using a Perkin-Elmer DSC-2 gave the following results: (Crystal II)T= 232 k 2 K ; AH = 2.0 i 0.3 kJ mol-' (Crystal I) T = 291 f 2 K; AH -= 9.1 $0.5 kJ mol-' (liquid) in satisfactory agreement with the results of Westrum and Ribner.2 The specimens for the neutron experiments were contained in disc-shaped aluminium alloy cans of diameter ~5 cm and with I mm walls.The sample thickness was ~ 0 . 3 mm giving 900/, transmission. Sample temperatures were controlled to f 1 K and gradients across the sample were probably of similar magnitude. Four spectrometers9 were used in this work and measurements made at several temperatures as follows: (i) The chopper time- of-flight spectrometer 4H5 at AERE Harwell: incident wavelength ,lo = 4.8 A; elastic energy resolution AEo % 100 peV (h.w.h.m.); maximum elastic scattering vector Qh' = 1.85 No other transition was detected despite a careful search. A-1 (Qel == % sin 0). Temperatures T/K = 227, 240, 265, 284.(ii) The multichopper A n time-of-flight spectrometer IN5 at I.L.L., Grenoble: (a) A. == 6.45 A; AEo = 24 peV (h.w.h.m.); Qk' = 1.75 A-l; T/K = 227, 240, 250; (b) ;lo = 10.0 A; AEo = 10 peV (h.w.h.m.); Qh' = 1.15 k'; T/K = 201, 218, 227, 239, 270. (iii) The very high resolution backscattering spectrometer IN10 at T.L.L.: A. = 6.28 A; AEo % 0.5 peV (h.w.h.m.); QK' = 1.8 A-'; T/K = 190, 201, 208. (iv) The rotating crystal spectrometer IN4 at I.L.L. This was used exclusively for inelastic measurements with: Eo = 12.5 meV; AEo = 0.3 meV (h.w.h.m.) QK' = 4.8 A-'; T/K = 5 , 150, 201, 218, 227, 239. Any counters which received Bragg scattering were excluded from the quantitative analyses. The experimental resolution function was determined using both a vanadium sample and the t-butyl cyanide sample itself at low enough temperatures for all rotational motion to be frozen.After subtraction of background and empty-can scattering and correction for absorption and self-shielding the results were normalised by comparison with the scattering from a vanadium sample and were converted, using standard procedures, to the symmetrised scattering law, S(Q, co). This involves only experimentally known quantities and S(Q, w ) is a property only of the system, independent of the experiment; it is also the function most * Crystal I1 (227 K)u = 6.45 A, b = 6.95 A, c =I 6.75 A, p = 92.8"; C-C-N axis parallel to c. Crystal I (239 K) u = b = 6.89 A, c = 6.70 A; C-C-N axis parallel to c.J . C. FROST, A. J . LEADBETTER AND R .M . RICHARDSON 37 1.400 1.050 0.700 1.000 0.500 2.500 2.500 0.99 0 0.540 0.090 4 + + + + + + + + t i + + ' I + + t '+ t i t + + + t + t + + + + t t t t I 1 0.0 5.0 10.0 15.0 20.0 tio/meV FIG. 4.--Inelastic S(Q, w) spectra at various temperatures measured using IN5 in energy gain. Similar results were obtained for energy loss using the IN4 spectrometer. simply related to theoretical calculations. Typical quasielastic spectra are shown in fig. 1- 3 and inelastic results in fig. 4. These results show that some bound stochastic motions are occurring in both crystal phases and these turn out to be whole-molecule rotations about the dipole axis. RESULTS AND DISCUSSION (i) QUASIELASTIC DATA ANALYSIS The incoherent scattering law from a proton in molecule undergoing reorienta- tional motions may be written" S,(Q, m) = e - z w [SXQ, m> -1 SXQ, m)I where e-2w is a Debye-Waller factor {2W = <<Q u>~>>,38 PHASE TRANSITION I N (CH3)3CCN Ss(Q, w ) is the inelastic scattering (broadened by the rotational motion) and SF(Q, a) is the rotational scattering which may be written quite generally (assuming exponential time correlation functions) : where 9 ( z i 1 ) = z i ' / ( z i 2 + a*) and the A,(Q) depend upon the type of motion and 2 A l = 1.l = O Ao(Q) = 1 exp iQ.rP(r) dv l2 (4) where P(r) is the (time independent) distribution function for the proton. Ao(Q) is commonly called the elastic incoherent structure factor (EISF) and gives a direct measure of the trajectory of the proton, averaged over " infinite " time, as the mole- cule rotates. For non-equivalent protons the above results must be averaged over the different protons." Infinite " time is determined by the resolution of the experi- ment, so effectively t , z 5h/AE0. For a molecule such as t-butyl cyanide the number of protons is sufficiently large that except at Bragg reflections the observed scattering is generally completely dominated by the incoherent cross section of the protons. The data were first analysed using eqn (1) and (3) with adjustable coefficients Al to determine the EISF and hence the geometry of the reorientational motions. A model was then devised which specifies the Al (see below) and the data fitted to obtain the correlation times T ~ . In all model calculations the known molecular geometry was used.11*12 Provided that the quasielastic scattering is not too broad S:(Q, w ) may be taken as a flat background in the quasielastic region.The EISF was then determined as the ratio of elastic to total (elastic plus quasielastic) scattering after subtraction of a flat inelastic background. The Debye-Waller factor is given by the Q-dependence of this total intensity. A careful analysis of errors in the EISF was made by taking account of the effects of varying (i) the functional form (number of Lorentzians) in eqn (3) for the computer fitting and also separating elastic and quasi- elastic components by eye for the high temperature phase and (ii) the energy range of the fit. The very small effects of the data being at constant scattering angle and not constant Q were also taken into account.A very useful model for analysing the data is that for instantaneous jump motion among m equidistant points on a circle of radius Y for a powder sample,13 for which the scattering law is of the form of eqn (3) with the quasielastic summation from I = 1 to I = rn - 1, and the coefficients for I 3 0 given by: j , ( X ) = sin X / X and z1 = ( 2 / 2 ) sin2 (nZ/rn) where T is the residence time ( 7 - l is the jump frequency). In determining the EISF, m could be varied and the coefficient of the elastic term allowed to be an arbitrary variable: Do(Q). The model scattering law was fitted to the data after convolution with the experimental resolution function. Reorientational motion between three equidistant sites is important both for methyl and whole-molecule rotations and for this eqn (3) becomes, for a powder sample, S,"(Q, a) = Ao(Q)d(a) + 11 - Ao(Q)12(zi1) A,( Q) = [ 1 + 2j0( Qr.t/3)]/3and z1 = 2 ~ / 3 , with z being the residence time.J .C . FROST, A . J . LEADBETTER A N D R . M . RICHARDSON 39 (ii) EFFECT OF METHYL ROTATIONS For compounds like t-butyl cyanide both methyl and whole molecule rotations may occur. Assuming that these are independent then the total scattering law is a convolution of the two separate components which for a powder sample may be the bar denoting the powder average. general form (3) for the whole-molecule motion (B) gives The methyl rotation (A) has been shown to be described by eqn (6) and taking the where the A and B are functions of Q. Explicit expressions for the coefficients have been given by Schlaak14 for the case where the whole-molecule motion is also a 3-fold uniaxial rotation. For the present purpose, however, we note the fortunate circumstance that the correlation times for methyl rotation in t-butyl cyanide are at least one order of magnitude slower than those for the uniaxial molecular rotation for T 2 190 K, which confirms their inde- pendence in this case.This has been established by investigating the apparent elastic peaks such as those of fig. 3 using an order of magnitude higher resolution. The methyl rotation is thus so slow that it is virtually unobservable using IN5 (and 4H5). For a truly unobservable methyl rotation the EISF would be given by EISF = A,B, + (1 - Ao)Bo = Do. In fact, although it made little difference to the value of the EISF, especially in the high T phase, a significantly better fit to S(Q, co) was obtained by taking account of the very small broadening contribution of the methyl rotation (just observable in the wings of the elastic component).This was simply achieved by writing the scattering law as: SdQ, m) = Bo(Q){4Q>a<a> + [1 - a(Q>ln-lz(~A1)) + 1 B ~ ( Q ) ~ - ' ~ ( z E ~ ) (11) I = 1 where the expression in brackets describes the experimental spectrum observed within the limited energy window of the very high resolution spectrometer IN10 so that a(@ and Z, are experimentally determined quantities. This approach may be used with Bo(Q) as a variable to determine the EISF, or to fit a given model [e.g., eqn (6)] to S(Q, co), and means that a self-consistent treatment is used to analyse all data at all resolutions.(iii) MULTIPLE SCATTERING The correction of experimental data for the effects of multiple scattering is very complicated and expensive of computer time. We have adopted the simpler proce- dure of incorporating the effects of multiple scattering into the model calculations. These calculations were based on the Monte Carlo programme of Johnson15 and parameters such as the EISF or Z~ values from the resultant scattering laws can be used to assess the effects of multiple scattering on the experimental results.40 PHASE TRANSITION I N (CH,),CCN CRYSTAL I1 With the 4H5 spectrometer (A& z 100 peV h.w.h.m.) no quasielastic broadening was observed, in agreement with previous low-resolution work on this phase.4 How- ever, with higher resolution a rotational motion is clearly seen and by also using the back-scattering spectrometer this was observed down to -190 K.Analysis of the data as described above yields the EISF values shown in fig. 5. This shows that t 0.0 0.5 1 .o 1.5 2 .o Q1A-l FIG. 5.-EISF results for the Crystal I1 phase. Experimental values from the IN5 data are shown as filled shapes and from IN10 as open shapes, with the following instrumental resolutions and sample temperatures. AE/peV T/K 0 24 227 4 10 227 218 208 rn 10 0 0.5 n 0.5 20 1 c! 0.5 I90 The error bars (some of which have been omitted for clarity) have been estimated by extracting the EISF for the whole body motion with the value of the time constant for methyl reorientations [rA in eqn (1 I)] changed by i 100% of its most probable value.Line A shows the EISF that would be obtained if only methyl group reorientation were being observed. Line B shows the EISF expected from whole body reorientation about the CCN axis between 3 equivalent sites [eqn (6)] and line C represents the same model with the estimated multiple scattering correction. throughout the low temperature phase the average proton trajectory remains to a good approximation the same. Furthermore the EISF is independent of resolution as well as temperature, which suggests immediately that S(Q, a) is of a rather simple form. Model calculations show clearly that the motion observed is a rotation of the molecule about its CCN axis between three equivalent sites.This model fits the EISF dataJ . C . FROST, A . J . LEADBETTER A N D R . M. RICHARDSON 41 very well although the higher resolution and higher Tresults may be slightly low at the higher Q values indicating some additional motion, probably a precursor of that observed in Crystal I to be discussed below. However, any such effect is difficult to quantify uniquely in Crystal 11 and indeed the full S(Q, w) is fitted very well at all Q values by the sipple 3-fold reorientation model (see fig. 1 and 2). If the residence time is allowed to vary with Q in the fitting computations then except for the lowest Q values, where the EISF tends to unity, z is generally constant to better than 5%. For comparison the EISF expected for rotation of the methyl groups alone is shown in fig.5 which clearly demonstrates that this is not the motion being seen. These results unequivocally demonstrate that the molecules of t-butyl cyanide are undergoing rapid reorientation in a 3-fold potential in the ordered low temperature phase. The correlation times z1 (eqn (7)) are plotted in fig. 6 where the correlation times for the methyl group reorientation are also shown for comparison. z1 varies linearly with T-' and the line in the figure gives the relation z,/s = 1 x exp 23.3 kJ mol-'/RT. The pre-exponential factor is totally unphysical since a value of the order of the free rotator correlation time (Z/kT)1'2 z A more reasonable interpretation is possible using the inelastic data shown in fig. 4. By virtue of its high intensity, which also increases with T, the lowest energy s is expected.I6 TABLE PARAMETERS FOR THE MOLECULAR ROTATION IN CRYSTAL 11.COMPARISON OF INELASTIC AND QUASIELASTIC RESULTS Cia BARRIER HEIGHTS AND RESIDENCE TIMES. 5 150 190 201 208 21 8 227 (239 4.2 f 0.2 3.7 & 0.2 (6.5) & 1.0 7.2 36 34 7.5 3.2 & 0.2 5.8 k 1.5 6.5 23 13 7.2 (5.6) & 1.5 5.8 11.4 10.5 5.2 3.0 f 0.3 5.1 f 1.5 5.1 6.9 6.9 4.5 2.8 & 0.3 4.5 3.1 1.5 4.2 3.6 4.5 3.9 2.8 & 0.4 4.5 i 1.5; Crystal I) 10.0 i 1.0 7.7 i 1.0 z(calc)/s = 4.5 x z(exp>/s = 4.5 x and z(exp) is the experimentally determined residence time. exp ( V3 - Eo)/RT exp ( Vge - Eo)/RT = zo(T) exp(V3 - E,)/RT peak at 3-4 meV may be assigned to the libration of the molecule about its CCN axis. The peak is increasingly swamped by multiphonon scattering with increasing T but remains detectable into the high-temperature phase. Assuming a potential energy of the form V(0) = V3(l - cos 30)/2 then in the harmonic approximation the depth of the potential well is given by V3 = 8 n21(E1 - Eo)/9ir2.The reorientation frequency is then given by the relation r1 = TO' exp - (V, -E,)/RT. Using this very simple model the following calculations were made with the results Residence times were calculated using values of (V, - Eo) derived given in table 1.42 PHASE TRANSITION I N (CH,),CCN from the inelastic results and zo = 4.5 x s. [Note that dipole correlation times (7,) not residence times (7) are shown in fig. 6 for ease of comparison with Crystal I results later.] The agreement between zexp and zCalc is as close as could be expected.The difference is shown in two ways by giving (a) r,(T) values and (b) VSe values required to give the solid line shown in the figure. It is obvious that a slight change of V,, well within the uncertainty of that derived from the librational fre- quencies, or a slight temperature dependence of zo gives perfect agreement with experi- ment for z. Hence inelastic and quasielastic results are self-consistent and the ap- parent Arrhenius behaviour for crystal I1 shown in fig. 6 is an artefact caused by a barrier height to reorientation which is a strong function of T. It is in fact approxi- mately linear in T for 190 < T/K< 230 and in precisely this region the unit cell dimen- sions become strongly temperature dependent.-8 t -9 --. 3.0 4 .O 5.0 1 0 3 ~ 1 T FIG. 6.-Correlation times T~ for t-butyl cyanide. Filled circles are for methyl-group reorientations. Open circles are for whole-molecule motions and show all the results obtained with three different spectrometers at 4 different experimental resolutions (see Experimental section). Vertical line shows the transition temperature. The hatched area shows the estimate of the correlation time for the cooperative translation/rotation relaxation discussed in the text. For Crystal I the line through the points is simply the best fit Arrhenius line. For Crystal 11, see text and table 1 . CRYSTAL I The EISF is determined very accurately from the INS data at both resolutions and also with reasonable accuracy from the lower resolution 4H5 data because of the width of the quasielastic components (200-300 peV, h.w.h.m.).The results are shown in fig. 7 and 8 and for the two higher resolution experiments with the IN5 spectro- meter the EISF is independent both of resolution and temperature showing that on a time-scale of wlO-" s the average proton trajectory is constant throughout theJ . C . FROST, A . J . LEADBETTER AND R . M . RICHARDSON 43 temperature range of Crystal I. However, with the lower-resolution instrument 4H5 the EISF is both considerably higher and also temperature dependent, but approaches the IN5 result with increasing temperature. This shows that the molecular motion is more complex than in Crystal 11 and this is confirmed by detailed model calcula- tions. The 4H5 data at the two lower temperatures (fig.7) are in fair agreement with a model of jump reorientation in a 3-fold potential but in much better agreement with 1 I I I 1 I 0.0 0.5 1 .o 1.5 2.0 QlA - FIG. 7.-The EISF for the Crystal I phase of t-butyl cyanide determined from the 4H5 data. 0,284; V, 265 and 0, 240 K. The lines represent the EISF calculated for: A, 3-fold jump reorientation about the CCN axis [eqn (6)]; B, reorientation about the CCN axis between 6 or more equivalent sites [eqn (3) and (5)J; C, uniaxial rotational diffusion in a 3-fold cosine potential” with barrier height, V = 2kT 21 4 kJ mol-’. a model for diffusive motion in such a potential (effectively a strong collision model) with V3 = 4 kJ mo1-I which is in good agreement with the value deduced from the damped librational peak seen at the lowest temperatures in Crystal I, and with the Arrhenius activation energy deduced from the temperature dependence of the correla- tion times (see below).Furthermore, the simple jump model fits S(Q, co) very well, with a residence time z independent of Q to within 10%. Taken all together this evidence confirms that the faster stochastic motions in Crystal I are the molecular rotations in a 3-fold potential. This motion becomes increasingly diffusive in character (as opposed to librational plus jump reorientation) with increasing T be- cause of the small barrier height: V3 = 2RT. At the highest temperature the 4H5 EISF is close to that for diffusion on a circle among rn sites with rn 3 6, but the high resolution results are still lower at all tempera- tures.Thus it seems clear that there must be at least two components of motion present. The additional motion cannot be methyl-group rotation which is much too44 PHASE TRANSITION IN (CH,),CCN slow, and 3-dimensional molecular rotation is excluded by EISF calculations (fig. 8). In searching for the origin of the additional motion two facts must be noted: first the existence of steric hindrance to rotation even in the high-temperature phase and second the fact that the molecular C3 axis lies parallel to the four-fold axis of the tetra- gonal unit cell of Crystal I. The latter can be taken into account by assuming rota- tion about the CCN axis among 12 sites on a circle but this cannot explain the results because motion additional to diffusion on a circle is required.These considerations lead to the suggestion that the local structure in Crystal I is close to that of Crystal I1 with a coherence length (L) of a few unit cells. Four-fold disorder of this structure in steps of n/2 leads to the average tetragonal symmetry of Crystal I. The molecules then rotate very rapidly in a 3-fold potential about their dipole axes as for Crystal I1 but in addition undergo a slower relaxation of four-fold symmetry among the different 0.0 0.5 1.0 1.5 2.0 Q1A-I FIG. 8.-EISF for the Crystal I phase of t-butyl cyanide. Experimental results obtained using the IN5 spectrometer at the following temperatures and resolutions AElpeV T/K 0 10 270 0 10 239 0 24 240 a 24 250 The error bars represent the probable effect of contamination by coherent " elastic " diffuse scattering at Q > 1.3 A-I.A multiple scattering correction has been applied to all the model EISF calcula- tions shown. The lines A, B and C represent the EISFs calculated for the models described in the text: A (dashed) = 3-jump + 4-fold + z = 1.0 A B C = 3-jump + 4-fold -t (it;>+ = 0.30 8, = 3-jump + 4-fold + 1.2 8, additional displacement parallel to a and b of fig. 9. D is the EISF for a model of reorientation about the CCN axis between 6 or more sites [eqn (3) and ( 5 ) ] and E is the EISF for isotropic tumbling of the moleculeJ . C. FROST, A . J . LEADBETTER AND R. M. RICHARDSON 45 local configurations. These may be generated by 7r/2 rotations of the monoclinic Crystal I1 structure about either c* or c: the former results in molecular rotation plus translational (centre of mass) displacement in theab plane and the latter gives additional translational displacement of the molecules along their dipole axes.The actual magnitudes of the average molecular centre of mass displacements will depend on the coherence length of the local structure, and there must also be additional more- or-less random displacements at the boundaries between local structures of different orientation. The dynamic translational/rotational displacements have therefore been modelled as described below using the known crystal and molecular structures to determine a manifold of N sites for each proton, the molecules also being allowed to undergo 3-fold rotation about their CCN axes in all cases. The EISF was then calculated using the formula for a powder sample : where r l j = Iri - rjl, Y being the position of a proton site of occupation probability P, and in general we assumed P = N-I.Ao(Q) was averaged over non-equivalent protons. The starting point was to rotate a unit cell of Crystal I1 about an axis through its centre of mass parallel to c* in steps of 7c/2 and then displace the resultant structure FIG. 9.-Model for translational/rotational disorder in Crystal I. The bold triangles represent the projections of the 3 methyl groups for one of the molecules in the unit cell onto the ab plane in solid I1 (or the ab plane of solid I). The light triangles are the methyl group positions generated by 72/2 rotations about c* and suitable translations to keep the centre of mass of the 2 molecules in the cell constant.This (arbitrary) procedure generates a pseudo-tetragonal local structure and gives both molecules small lateral displacements. The real displacement could be much larger depending on the coherence length of the Crystal I1 local structure in Crystal I (see text). by appropriate combinations of b/2 and/or 4 2 to give the resultant superposition of structures for each of the two molecules in the unit cell shown in fig. 9. This repre- sents the minimum possible set of translational displacements for this model and it does not describe the EISF. A displacement z parallel to the dipole axis (c in Crystal I) which would result from 4 2 rotations about c in Crystal I1 was therefore added as an adjustable parameter to give a total of 8 molecular positions and the EISF for z = 1.0 8, is shown in fig. 8(A).An as alternative, an isotropic Gaussian displacement46 PHASE TRANSITION IN (CH3)3CCN of the molecular centre of mass was added to the minimum set of four positions from the c* rotation (and the %fold molecule rotation) and the EISF for ( ~ f ) l / ~ = 0.3 A is shown in fig. 8(B). Finally, on the grounds that displacements perpendicular to the direction of the opposed dipoles are perhaps the most likely, additional displace- ments x parallel to the unit cell axes a and b were added to the structure shown in fig. 9 and the resultant EISF for x = 1.2 A is shown in fig. 8 (C). The fits in all cases, but especially for C, are reasonable when taking account of uncertainties (included in the error bars) at the highest Q arising from the possible presence of coherence contri- butions due to the strong coherent diffuse scattering in the disordered phase.An elastic coherence contribution of only a few percent would remove the discrepancies with the models at the highest Q. The parameters for models A, B and C imply a coherence length for the local structure in Crystal 1 of not more than w 5 unit cells. The geometrical nature of the molecular motions in the high temperature phase is thus established as rotation in a weak 3-fold well about the molecule C, axis, plus additional rotational/translational motions which are strongly cooperative, generate the correct high temperature symmetry and involve centre of mass displacements of & l A.The dynamics of these two components of motion have also been deter- mined as follows. The fastest and largest amplitude motion is undoubtedly the 3- fold rotation about the dipole axis which dominates the lower resolution data (see above). For the higher resolution data, using a scattering law of the form of eqn (3) and ( 5 ) then at low Q the term in z1 is the totally dominant quasielastic com- ponent. In fact using rn = 10, the values of z1 were only weakly Q-dependent and for Q < 1.5 A-' fell on a straight line enabling an averaged low-Q value to be determined. These values may then be regarded as the dipole correlation time for the motion in the 3-fold potential and the results are given in fig. 6, where it is seen that self-consistent data are obtained from all experiments with experimental resolutions differing by up to an order of magnitude.The temperature dependence is described by z1 = 0.5 x exp 3.4 kJ inol-l/RT. The zo value is totally reasonable and the activation energy implies V, w 3.5 kJ mol-l which is in agreement both with the EISF analysis and the value derived from the librational frequency. The order of magnitude of the correlation time of the slower co-operative relaxa- tion may be estimated from the fact of its observation with the high resolution but not (at least at lower T ) with the lower resolution experiment and this is also shown in fig. 6. COMPARISON WITH OTHER EXPERIMENTS The previous lower resolution INQES experiments4 on Crystal I were interpreted in terms of a simple 3-fold jump reorientation model and gave z values ~40-50% greater than those reported here and an Arrhenius activation energy of ~2 kJ mol-l.In view of the much lower resolution of these experiments the agreement is as good as could be expected. The n.m.r. measurements (at 16 MHz) showed a single step decrease in linewidth between 95 and 120 K and a single TI minimum (measurements made at 42 MHz) at 162 K from which it was concluded that methyl and whole-molecule uniaxial rotations are coupled and describable by a single correlation time z, and a single energy barrier (16.3 4 0.8 kJ mol-'). The INQES results show clearly that this is not correct but also provide an alternative explanation. Extrapolation to lower T of the neutron correlation times for methyl and whole-molecule reorientation (fig.J .C . FROST, A . J . LEADBETTER AND R . M . RICHARDSON 47 6) show that these become equal near 130 K at r1 NN 10-7-10-8 s so that the corre- lation times indeed become equal at temperatures where their value is just that of the n.m.r. frequency; but this is fortuitous and the motions are not strongly coupled. The n.m.r. activation energy is in excellent agreement with the neutron value for methyl reorientation. The n.m.r. correlation time is about a factor of 10 less than the observed r1 values for methyl rotation but a factor of 2-3 is expected since the spin- lattice relaxation experiment gives r2 [P2 (cosO)]. Hence the n.m.r. Tl experiments are certainly dominated by the methyl reorientations. On the other hand an analysis by El Saffar et aL3 of the linewidth transition gave values for a correlation time, for T < 120 K in agreement with the extrapolated INQES values for whole-molecule rotation and a corresponding activation energy (12 & 1 kJ mo1-l) close to the 5 K value from the INIS data suggesting both that this analysis is sensitive to the whole-molecule motion and that the barrier height is not significantly temperature dependent for T .L( 100 K. Hence the onset of the line width transition is determined by the slower (at this T ) molecular rotations but the spin-lattice relaxation is dominated by the methyl-group rotations.THE PHASE TRANSITION The t-butyl cyanide molecules undergo rotation about their 3-fold axes in a 3-fold potential at all temperatures. In the low temperature phase the residence time 2(>2 x s) is more than an order of magnitude longer than the flight time 248 x s) where this is defined by zf = (kT/Z)-1/2 and also about an order of magnitude or more longer than the librational period zL = v ~ l ( z , w 10-l2 s).Hence the model of librational motion interspersed by relatively rare instantaneous jumps between wells is reasonable. The molecules may be envisaged as jumping to a new orientation when allowed by phonon-type thermal fluctuations. In the high-temperature phase z is only w2 or 3 times longer than zf and zL so that the simple reorientational model is no longer adequate, but more importantly neighbouring molecules must frequently be rotating simultaneously. Hence, strongly co-operative displacements of the centres of mass are required to allow this.Con- sideration of the molecule and site symmetry has led to a picture of relaxation among a set of local configurations comprising different 7rn12 orientations of structure similar to that of the low-temperature phase. The correlation time for this motion is slower than that for the 3-fold molecular rotation. The energy barrier for the 3-fold potential in the low temperature phase is strongly temperature dependent but does not collapse at the transition: in fact it changes rather little. Instead the decrease of the barrier, which is linked to a high thermal expansion, increases the rotation frequency until the concomitant growth of the co-operative translational displacements triggers the transition to the disordered phase. Thus it is the presence of the z 1 A co-operative displacements of the molecu- lar centres of mass and much faster molecular rotation about the C-C-N axis which distinguishes the high temperature phase and not the existence of rotation of the molecules about their C-C-N axes. C. Clemett and M. Davies, Trans. Favaday Soc., 1962, 58, 1705. E. F. Westrum and A. Ribner, J. Phys. Chem., 1967, 71, 1216. Z. M. El Saffar, P. Schultz and E. F. Meyer, J. Chem. Phys., 1972, 56, 1477. S. Urban, J. Mayer, I. Natkaniec, J. Sciensinki and W. Nawrocik, Acta Phys. Polon., 1978, A53, 379. S. L. Segel and A. Mansingh, J. Chem. Phys., 1969,51, 4578. J. R. Durig, S. M. Craven and J. Bragin, J. Chem. Phys., 1970, 53, 38.48 PHASE TRANSITION IN (CH3)jCCN ' C. I. Ratcliffe and T. C. Waddington, J.C.S. Furuduy II, 1976, 72, 1821. J. C. Frost, Ph.D. Thesis (University of Exeter, 1979). A. J. Dianoux, R. E. Ghosh, H. Hervet and R. E. Lechner, I.L.L. Internal Report 75 D16T (1975); W. S. Howells, I.L.L. Internal Report 75 H1 30T (1975); H. A. Baston, A.E.R.E. M5270 Harwell (1972); lo A. J. Leadbetter and R. E. Lechner, in The Plastically Crystalline State, ed. J . N. Sherwood (Wiley, Chichester, 1979, chap. 8). l1 R. L. Livingston and C. N. R. Rao, J. Amer. Chem. SOC., 1959, 81, 3584. l2 L. J. Nugent, D. E. Mann and D. R. Lide Jr, J . Chenz. Phys., 1962, 36, 965. l3 J. D. Barnes, J . Chem. Phys., 1973, 58, 5193. l4 M. Schlaak, Mol. Phys., 1977, 33, 125. lS M. W. Johnson, A.E.R.E. Report 7682 (1974). l6 C. Brot, Chem. Phys. Letters, 1969, 3, 319. A. J. Dianoux and F. Volino, Mol. Phys., 1977, 34, 1263.

ISSN:0301-7249    

DOI:10.1039/DC9806900032

出版商:RSC    

年代:1980

数据来源: RSC

摘要:

A Crysiallographic Study of the Second-order Phase Transition in Bis(p-toluene sulphonate) Diacetylene Polymer Crystals B Y RICHARD L. WILLIAMS, DAVID BLOOR AND DAVID N. BATCHELDER Department of Physics, Queen Mary College, Mile End Road, London E l 4NS AND MICHAEL B. HURSTHOUSE Department of Chemistry, Queen Mary College, Mile End Road, London El 4NS AND WILLIAM B. DANIELS Department of Physics, University of Delaware, Newark, Delaware 1971 1, U.S.A. Received 10th January, 1980 The second-order phase transition which occurs in bis(p-toluene sulphonate) diacetylene polymer crystals has been studied by X-ray diffraction. Measurements of the temperature dependence of the integrated intensity of Bragg reflections from several crystals provide evidence that the phase transition is very sensitive to the presence of defects.For one crystal several Friedel pair reflections have integrated intensities which are very asymmetric in the vicinity of the transition temperature. This suggests that the crystal was not centrosymmetric in this region and provides a possible explanation for the previous observation of pyroelectricity in pTS crystals. The second-order phase transition which occurs in crystals of the polymer of the bis (p-toluene sulphonate) of 2,4-hexadiyne-l,6 diol (pTS) has been studied by a variety of experimental techniques. The crystal structures of both the high- ' and the low- t e r n p e r a t ~ r e ~ ? ~ phases have been determined and the gradual onset of order in the low-temperature phase has been observed in X-ray s t u d i e ~ .~ , ~ Order parameter analysis has been applied to the splittings of the optical absorption and Raman lines observed in the low-temperature phase and to the intensities of " hard " vibrational modes observed by far-infrared spectrocopy.6 The order parameter fitting suggested that at low temperatures the transition had a two-dimensional character while closer to the transition temperature it appeared to be three-dimensional. There were indica- tions of a broad transition region which might be expected for a predominantly two- dimensional phase transition in which fluctuations dominate the behaviour of the system. Fig. 1 shows a projection of the P2Jc unit cell of pTS onto the ac plane at 300 K in the high temperature phase.6 The polymer chains have the chemical structure with R = CH20S02C6H4CH3 and are oriented parallel [.>C-C=C-C CR],,50 DIACETYLENE POLYMER CRYSTALS FIG.1.-Projection of the P2Jc unit cell of pTS crystals at 300 K onto the uc crystallographic plane. The polymer backbone is parallel to the b axis which is perpendicular to the uc plane. The atoms, which are identified in fig. 2, are represented by thermal ellipsoids. The solid lines outline the unit cell in the high-temperature phase. In the low-temperature phase with P21/n structure, the unit cell is doubled in the Q direction (dotted lines) and there are two inequivalent polymer chains, A and B, in the unit cell. to the b axis. The doubling of the unit cell in the a direction, which occurs in the low- temperature P2Jn structure, and the two types of polymer chain (those with type A and those with type B side-groups), are also indicated. Fig.2 shows the anisotropic thermal motion of the non-hydrogen atoms in PTS.~ The central figure is a plot of a backbone segment and one sidegroup at 300 K, and the plots labelled A and B are similar parts of the two distinct types of polymer chain at 120 K. Despite the fact FIG. 2.-Comparison of a pTS side-group at 300 K (centre diagram) with that of the two structures (top and bottom diagrams labelled A and B) in which the side-groups are found in the low-tempera- ture phase at 120 K. The anisotropic thermal motion is depicted by 50% probability ellipsoids for the non-hydrogen atoms. The atomic labels are: C, carbon; 0, oxygen; S, sulphur. Atoms C(l)’, C(1), C(2), and C(2)’ are on the backbone with the vector b defining the polymer chain direction.WILLIAMS, BLOOR, BATCHELDER, HURSTHOUSE AND DANIELS 51 that the atoms in the side-groups labelled A and B occupy positions which are slightly displaced from the positions they would have occupied at room temperature, the ther- mal ellipsoids of side-groups A and B could be completely enveloped within the thermal ellipsoids of the room-temperature molecule.A model for the phase transition has been suggested in which ordering in two dimensions predominates.6 The side-groups along a single side of a polymer chain are considered to form an ordered stack which has either the A or B structure. In the low temperature phase these stacks are ordered in two dimensions (ac plane) while the high temperature phase would consist of a random array of A and B stacks.The third dimension would only be important for a stack in the process of changing from A to B or vice versa. This model would appear to satisfy in a qualitative way the requirements of both the order parameter analysis and the X-ray structural data. The present X-ray study was undertaken in order to provide further evidence against which this model could be tested. In addition it was hoped to find a possible explanation for the observation of the pyroelectric effect in pTS crystal^.^ The struc- tures of both the low- and high-temperature phases determined by X-ray analysis are centrosymmetric, while the pyroelectric effect can only be observed in crystals which lack a centre of symmetry.EXPERIMENTAL High-purity TS monomer crystals were grown by evaporation from acetone solution.' Small as-grown crystals of typical dimensions 0.2 x 0.8 x 0.2 mm were selected for study on a four-circle X-ray diffractometer. Larger crystals typically 2 x 6 x 4 mm in size were cleaved parallel to the (100) face to provide a clean surface for study by a rotating back- reflection X-ray camera. Both types of crystal were then thermally polymerized to 100% conversion at 333 K.' INTENSITY MEASUREMENTS O N THE BACK-REFLECTION CAMERA The large crystals were glued along one edge with low-temperature varnish to the alu- minium cold finger of a continuous flow cryostat. The temperature was controlled to f0.02 K and measured by an NBS calibrated platinum resistance thermometer located only 1 mm from the crystal. With the crystal fixed in the cryostat a rotating back-reflection cameralo*" was used to scan the (D,O, 3) Bragg reflection of the low-temperature phase. This difference reflection does not appear in the high-temperature phase. The normal to the (33,0, 3) planes lies at about 12" to the (100) face and the Bragg angle was approximately 68.6" with Cu Koc radiation.The pinhole X-ray collimator had an angular divergence of &20' and was 19 cm from the crystal. A scintillation counter with a 1.5 cm square NaI crystal was placed 20 cm from the crystal to measure the intensity of diffracted X-rays. Rotating the camera over a 2" range produced a profile of the diffracted Kal and Kaz peaks on a chart recorder." The integrated intensities at each temperature were then determined graphically.INTENSITY MEASUREMENTS O N THE FOUR-CIRCLE DIFFRACTOMETER The smaller crystals were studied on a Nonius CAD-4 X-ray diffractometer with com- puter control. Intensities were collected with Ni-filtered Cu Koc radiation and an 0 / 2 8 scan mode. In each 96-step scan the outer 16 steps on each side constituted left (B,) and right (B,) backgrounds, and the central 64 steps the peak count (C). The integrated in- tensity (lo) of a reflection was calculated from the equation lo = IC - 2(B1 + &)]. The temperature of the crystals was held constant during the measurements to f0.5 K in a stream of cold nitrogen gas. The accuracy of temperature measurement by a thermocouple in the52 DIACETYLENE POLYMER CRYSTALS 1.0 0.8 gas flow was estimated to be 5 2 K.At each new temperature the lattice parameters were calculated by least-squares analysis of setting angles for 25 reflections which were auto- matically centred. I l l I I l l I I I 1 I I I I -OP - 0 - 0 - a 0 - a 0 - RESULTS INTENSITY ME AS U RE ME N TS ON THE B A C K-RE F LE CTI 0 N CAMERA Fig. 3 shows the integrated intensity of the (=,O, 3) Bragg reflection of the low- temperature phase, relative to the value at 80 K, as a function of temperature for two different pTS crystals. The intensities have been ratioed against those of the (12,0,6) 0.0 a 0 . I I I I I I l l I I l l I l l 1 80 100 120 110 160 180 200 220 tempera t ure/K FIG. 3.-Relative integrated intensity of the ( n , O , 3) Bragg reflection of the low-temperature phase as a function of temperature for two different crystals (open and closed circles).The data have been scaled to remove intensity changes associated with the Debye-Waller factor. reflection, which appears in both the high-temperature and low-temperature phases, in order to remove the intensity changes associated with the Debye-Waller factor. The measurements were made with both increasing and decreasing temperature with no evidence of significant hysteresis. The estimated uncertainty in the intensity measurements increased continuously from 3% at 80 K to 20% at the highest tem- perature s. The intensity data in fig. 3 suggest that the two pTS crystals have transition tem- peratures which differ by nearly 10 K. For both crystals the transition temperature is not well defined as the intensity has no sharp cut-off.There is even significant intensity in the (%,0,3) reflection for one crystal as high as 210 K, 15 K above the transition temperature previously estimated from a photographic X-ray inve~tigation.~ For both crystals the Ka, and Ka2 Bragg peaks were clearly resolved with no significant differences in peak width at 80 K between the two crystals. Thus it is not possible to say from these measurements alone which was the " better " crystal.WILLIAMS, BLOOR, BATCHELDER, HURSTHOUSE AND DANIELS 53 INTENSITY MEASUREMENTS ON THE FOUR-CIRCLE DIFFRACTOMETER Fig. 4 illustrates the temperature dependences of the profiles of the (5,0,1) Bragg reflection of the low temperature phase which were recorded on the four-circle diffractometer for three different crystals.Photographic studies of the three crystals T K 250 230 210 2 00 190 170 150 Crystal 1 5 L 5 L Crystal 2 5 4- 5 j - - - - - - - - 54- _ - - - - - - - - __ 5 i -- I Crystal 3 51- - I LO]- -/ FIG. 4.-X-ray diffraction profiles of the (5,0,1) Bragg reflection for three different pTS crystals at several temperatures. Solid (dotted) lines represent the smoothed data taken with decreasing (increasing) temperature. The setting angles for crystals 1 and 3 were not centred between tempera- tures. Shifts in the diffraction maxima along the Bragg angle axis probably represent small rotations of the crystal in its mount. at 300 K had not shown any significant difference in their quality. Crystal 1 would appear to have been the best of the three as the onset of the (5,0,1) difference reflec- tion with decreasing temperature was sharp and the profile symmetric.The onset for crystal 2 is less sharp and the profiles showed some asymmetry. Crystal 3 showed a weak maximum in the (5,0,1) reflection at 235 K and the profiles are very asym- metric. There was some indication of hysteresis of the profiles for crystals 2 and 3 during temperature cycling while those of crystal 1 were reproducible. Similar results54 DIACETYLENE POLYMER CRYSTALS to fig. 4 were observed for another difference reflection, the (7,2,1), for the three crystals. With crystal 2 on the diffractometer a study of the intensities of Friedel pairs was made. Fig. 5 is a comparison of typical profiles for the (5,0,1) and (5,O,T) at 185 K.It can be seen that the latter has much greater intensity with a narrower line-width. In fig. 6 data are plotted for the Bijvoet ratio A which is defined by the relation l3 A = 2(1+ - I - ) / ( I + + l - ) where I + and I - are the integrated intensities of the greater and less intense Friedel related reflections, respectively. The open (closed) circles in the upper part of the 9.5 10.0 10.5 11.0 11.5 12.0 Bragg angle/deg FIG. 5.-X-ray diffraction profiles for the (5,0,1) (lower points) and S,O,T) (upper points) Friedel pair related Bragg reflections at 185 K. The diffracted intensity has been plotted as a function of Bragg angle. Note that the scale for the (5,0,1) points has been expanded by a factor of four relative to those of the (S,O,T) points.figure refer to the (5,0,1)-(5,0,T) pair with decreasing - - - (increasing) temperature- Similar data have been obtained for the (7,2,1)-(7,2,1) pair. In the lower part of fig. 6, data are plotted for the (20,2,4)-(20,2,4) pair of Bragg reflections with de- creasing temperature which appear in both the high- and low-temperature phases. DISCUSSION Considerable variation in the temperature dependence of the profiles and inte- grated intensities of reflections at both high and low Bragg angles has been observed for different pTS crystals. Both large crystals and small crystals exhibit inconsistencies in their X-ray diffraction properties which suggest that the second-order phase transi- tion in pTS crystals is strongly affected by the presence of crystalline defects. Further- more, these defects need only be present in relatively low concentrations as their effect on the X-ray diffraction properties of the crystals at 300 K, far from the transi- tion temperature, was not readily apparent.WILLIAMS, BLOOR, BATCHELDER, HURSTHOUSE AND DANIELS 55 The sensitivity of the phase transition in pTS crystals to the presence of defects which is observed by X-ray diffraction should also be apparent in other properties.Thus far the various spectroscopic techniques which have been applied to the problem have not been sufficiently sensitive to show up any consistent differences among the variety of crystals studied. It is clear, however, that great caution must be used when applying order parameter analysis to properties of crystals which have not been characterized by X-ray diffraction.The extraordinarily large asymmetry between the diffracted intensities of various Friedel pairs as illustrated in fig. 5 could be due to one of two reasons. The Ren- ninger effect,14 whereby the diffracted beam from one set of planes is the incident beam 0 0 0 0 a "1 A A A . A -201 ' I I I I I I I I ' I I I I I I 120 110 160 180 200 220 210 260 280 temperature/K FIG. 6.-Temperature dependence of the Bijvoet ratio, A = 2(1+ - l - ) / ( I + + I-), where I + and I - are the integrated intensities of the greater and less intense Friedel related reflections, respectively. The open (closed) circles in the upper part of the figure refer to the (3,0,1)-(5,0,1) pair with de- creasing (increasing) temperature.Above 220 K the reflections were so weak that the data points do not have too much significance. In the lower part of the figure the triangles refer to the (20,2,4) (%,2,4) pair of Bragg reflections which appear in both the high- and low-temperature phases. for another, can be ruled out as a possible explanation since the scattering geometry was not appropriate. Thus the intensity asymmetry must have been caused by anomalous scattering which occurs in non-centrosymmetric crystals when absorp- tion by one or more atoms in the unit cell causes a phase shift in the X-rays scattered by those atoms. For most Bragg reflections inpTS crystals this effect should be very small as the 1.54 A Cu Koc radiation is very far from the 5.02 K absorption edge of56 DIACETYLENE POLYMER CRYSTALS sulphur, the heaviest atom in the unit cell.For difference reflections like the (5,0,1), however, the scattering amplitude becomes nearly zero in the vicinity of the phase transition and the effects of anomalous dispersion are greatly magnified. From currently available data it is impossible to tell whether this lack of a centre of sym- metry is an intrinsic property of pTS crystals or associated with the presence of de- fects in crystal 2. The pyroelectric effect observed inpTS crystals must be closely associated with the loss of the centre of symmetry. A possible origin for the required temperature de- pendent dipole moment can be found in the A-B stack model of the phase transition described above. If a single polymer chain has the stack of side-groups on one side in the A configuration and the stack on the other side in B then this A-B chain: (a) no longer has a centre of symmetry and (b) has a net dipole moment.The latter occurs since the dipole moments of the strongly polar sulphonyl groups in the A and B stacks are no longer anti-parallel. Even with A-B polymer chains the crystal would still be centrosymmetric if there were a corresponding number of B-A chains with dipole moments in the opposite direction. The model suggests that defects must be present which cause an imbalance between the number of A-B and B-A chains. Such a defect could conceivably be a screw dislocation since it has directional sense. Monomer crystals studied by etching reveal emergent screw dislocations parallel to the polymer axis.Often these are sufficiently numerous to render the end-facets non-planar due to the high density of growth spirals. A few crystals, however, show specular end facets indicating a much lower dislocation density.16 The peak in the Bijvoet ratio A in fig. 6 occurs over roughly the same temperature range in which the pyroelectric constant changes sign.7 At lower temperatures the mean value of A for the (5,0,1) reflection is about 0.13 suggesting that the crystal was non-centrosymmetric even away from the transition region. This would be in agreement with the pyroelectric data but the X-ray measurements need to be repeated on crystals where absorption corrections will be easier to make in order to confirm this result. The same conclusion holds for the (20,2,4) reflection which has a mean value for A of 0.13 between 120 and 300 K.At very low temperatures the X-ray data show that pTS crystals consist of a regular array of A-A and B-B polymer chains, that is the chains have a centre of symmetry with stacks of either type A or type B side-groups on both sides of the chain. The loss of order which occurs with increasing temperature could be due to the random formation of A-B and B-A chains. If defects with a directional sense were present then the numbers of A-B and B-A chains might be unequal and the crystal would no longer appear centrosymmetric. Well above the transition the X-ray structural data could be interpreted as arising from a random array of A-A, A-B, B-A, and B-B chains. This qualitative model is of assistance in the understanding of many of the effects which have been observed in pyroelectric, spectroscopic and X-ray diffraction investigations of the second-order phase transition in pTS crystals.Considerable further X-ray diffraction studies will be required to determine the role of defects in the phase transition and the origin of the unusually large asymmetry in the intensities of Bragg reflections from Friedel pairs. This research was supported by grants from the S.R.C., the National Science Foundation and the University of Delaware. The authors are grateful to Mr. D. J. Ando for the preparation of the polymer specimens and to Mr. S. Mehta for assistance with data processing.WILLIAMS, BLOOR, BATCHELDER, HURSTHOUSE AND DANIELS 57 D. Kobelt and E. F. Paulus, Acta Cryst. B, 1974, 30, 232. V. Enklemann and G. Wegner, Makromol. Chem., 1977,178, 635. V. Enklemann, Acta Cryst. B, 1977, 33, 2842. B. Reimer, H. Bassler, and T. Debaerdemaker, Chem. Phys. Letters, 1976, 43, 85. R. Clarke, personal communication. D. Bloor, D. A. Fisher, D. N. Batchelder, R. Kennedy, A. C. Cottle and W. F. Lewis, Mol. Cryst. Liquid Cryst., 1979, 52, 83. H. Kiess and R. Clarke, Phys. Stat. Solidi a, 1978, 49, 133. G. C. Stevens, D. J. Ando, D. Bloor, and J. S. Ghotra, Polymer, 1976, 17, 623. G. C. Stevens and D. Bloor, J. Polymer Sci., Polymer Phys. Ed., 1975, 13, 241 1. lo D. N. Batchelder, J . Polymer Sci., Polymer. Phys. Ed., 1976, 14, 1235. D. N. Batchelder and R. 0. Simmons, J. Appl. Phys., 1965, 36, 2864. l2 0. G. Peterson, D. N. Batchelder and R. 0. Simmons, Phil. Mag., 1965, 12, 1193. l3 S. Parthasarathy, Acta Cryst., 1967, 22, 98. l4 M. Renninger, Z. Phys., 1937, 106, 141. j5 A. F. Peerdeman, Anomalous Scattering, ed. S . Ramaseshan and S. C. Abrahams (International Union of Crystallography, Munksgaard International Publishers Ltd, Copenhagen, 1973, p. 3. l6 D. Bloor, J. Mat. Sci., 1979, 14, 248.

ISSN:0301-7249    

DOI:10.1039/DC9806900049

出版商:RSC    

年代:1980

数据来源: RSC

摘要:

13C Nuclear Magnetic Resonance Study of Phase Transitions in a Lipid Bilayer Embedded in a Crystalline Matrix: (C,,H,,NH,),CdCl, and (C,,H,,NH,)CuCl, B Y ROBERT BLINC, MATJA~ KO~ELJ, VENCESLAV RUTAR, IVAN ZUPANICIC AND BO~TJAN ZECg J. Stefan Institute, E. Kardelj University of Ljubljana, Ljubljana, Yugoslavia AND HONZA AREND AND RAYMOND KIND Laboratory of Solid State Physics, Swiss Federal Institute of Technology, Honggerberg, CH-8093 Zurich, Switzerland AND GERHARD CHAPUIS Institute of Crystallography, BSP, University of Lausanne, 101 5 Lausanne, Switzerland Received 10th December, 1979 The hydrocarbon parts of (C10HZ1NH3)LCdC14 and (C10H21NH3)2C~C14 represent smectic lipid bi- layers exhibiting two structural phase transitions analogous to those in biomembranes. A 13C nuclear magnetic resonance study showed that in the Cd compound the low-temperature transition is connected with a disordering of the polar heads whereas the high-temperature transition corresponds to a partial melting of the alkyl chains.In the Cu compound the sequence of the two successive transi- tions is reversed. A Landau theory describing the two transitions in terms of order parameters used in the theory of smectic liquid crystals is presented. The occurrence of phase transitions in cell membranes,'V2 which are one of the principal organizational structures of living matter, has recently attracted a great deal of attention. It has been convincingly demonstrated3p4 that the transitions in living cell membranes are due exclusively to the lipid component of the bilayer mem- brane.Most phase transition studies have therefore been performed on " model lipid membranes " which are similar in their structural properties to biomembranes, but can be much better characterized physically and exhibit sharper phase transitions. Lipid bilayers of dipalmitoyl phosphatidyl choline (DPPC) in particular clearly exhibit two first order-phase transitions,' the so-called main transition at Tc2 = 42 "C con- nected with a melting of the chains and the pretransition at T,, = 33 "C. Below T,, the lipid chains are well ordered and tilted with respect to the bilayer normal. X-ray scattering,6 cal~rimetric,~ n.m.r.'-'O and e.p.r." studies have provided much information on the microscopic nature of these transitions, but several important questions are still unanswered.One of them is the nature of the pretransition at Tcl. Another is the detailed state of motion of the hydrocarbon chains in the high (T > Tc2) and the intermediate temperature phase (Tcl < T < Tc2) and the origin of the differ- ence12 in the values of the nematic order parameter s,, = 3(3 cos2 O,, - 1) where 8, is the angular deviation of the nth C--C bond from its orientation in a hydrocarbonR. BLINC et al. 59 chain oriented parallel to a preferred direction, as measured by n.m.r.9J0 and e.p.r." Here we report on a I3C n.m.r. study of the phase transitions in the pseudo two- dimensional layer structure perovskite l3 (C10H21NH3)2CdC14 (henceforth designated C 10Cd) and (C,oH21NH3)2CuC14 (designated ClOCu) which we believe represent the first known examples of a smectic lipid bilayer embedded in a crystalline matrix. The two first-order phase transitions in ClOCd at T,, = 35 "C and Tc2 = 39 "C are closely analogous to the ones in DPPC whereas the nature of the two successive transitions in ClOCu seems to be reversed (Tcl = 36 "C, T,, = 42 "C). The results obtained yield a detailed microscopic picture of the two phase transitions in bilayer membranes and allow for the construction of a Landau theory of the " melting " of membranes in terms of order parameters used in the theory of liquid crystals.Perovskite-type layer structures of the general formula (C,H2, + 1NH3)2MC14 (short notation CnM) show a large variety of structural and magnetic phase transi- t i o n ~ . ' ~ The inorganic layers contain corner-sharing C16 octahedra with divalent metal ions in their centres (M = Cu2+, Mn2+, Cd2+, .. .) thus forming a rigid two- dimensional crystalline matrix. The NH3 groups of the alkyl ammonium chains are attached to the layers by weak N-H - C1 hydrogen bonds and, depending on the length of the chain, different packings are rea1i~ed.I~ For each packing scheme there are several different equivalent orientations of the keys (i.e. NH, groups) in the octa- hedral cavities. The structural phase transitions observed in these systems can be divided into two classes : (a) order-disorder transitions of the rigid alkyl ammonium chains, (b) con- formational transitions of the chains. For short-chain systems (n \< 3) the hindering potential is mainly determined by the key (rpolar head): key hole interaction, where- as for long chains (n > 3) the interaction among neighbouring chains becomes more important.The inorganic layers behave like a stable but elastic matrix and affect the phase transitions only indirectly through the coupling between the MCI, octahedra and the NH3 " heads " of the chains. The projection of the structure of ClOCd on the bc plane at room temperature (T< Tcl) is shown in fig. l(a). The structure consists of CdC14 layers sandwiched between well ordered alkylammonium chains which are tilted by 40" with respect to the normal to the layer. The ammonium end of each chain is linked to the layer by N-H*** C1 hydrogen bonds and each chain is coordinated to six others. In ClOCd the chains in subsequent bilayers form a zig-zag arrangement parallel to the crystal c-axis [fig.l(a)], whereas all the chains are parallel to a single direction in ClOCu and ClOMn [fig. l(b)]. In ClOCd the layers consist of nearly ideal, corner- sharing CdCl octahedra, whereas they consist of bipyramids in ClOCu. The entropy change at T,, [AS = (0.9 0.3)R per mole chains] can be explained by an order- disorder transition of rigid chains between two equivalent sites. The entropy change at Tc2, on the other hand, corresponds to A S = 0.8 R per R-C-C-R bond and can be only explained by a " melting " of the chains, or equivalently, by rapid chain isomerization via kink diffusion. In the intermediate phase of ClOCd the polar NH3 groups are flipping between two possible orientation^,'^ p = 5 1.The alkyl chains also flip by 90" around their chain axes so that neighbouring chains move in opposite directions like a 2-dimensional array of connected gears. The onset of the disordering of the polar heads in the low-temperature phase of ClOCd can be quantitatively followed l4 by measuring the temperature dependence of the I4N quadrupole coupling constant e2gQ/h and asymmetry parameter q. Since the motion between the two sites with e.f.g. tensors T(l) and T(2) is fast on the n.q.r.60 PHASE TRANSITIONS IN A L I P I D BILAYER a c (2) c (3) r B --l C FIG. 1 .-Projection of the low-temperature structure of (CloHz7NH3)zCdCl, (a) and (C10H21NH3)2- CuCl, (b) on the bc plane. The projection of the electron density map of ClOCd on the ac plane in the high-temperature phase is shown in fig.l(c). Due to 90" flips there is a dynamical dkorder between the two displayed positions of the chain which are occupied with a 50% probability.R. BLINC et al. 61 40 time-scale, the 14N n.q.r. spectrum is determined by the time averaged value of the e.f.g. tensor: (W)) = 3(1 + P)T(l) + 30 - p)T(2) - (1) - Knowing the e.f.g. tensors T(1) and T(2) from the completely ordered state, the temperature dependence of p = ( p ( t ) ) can be determined from (T(t)). The Fourier map in the high-temperature phase of ClOCd [fig. l(c)] shows in addition to the CdC1, octahedra well defined N and C( 1) positions whereas the remain- ing part of the chain is parallel to the c-axis and splits symmetrically on both sides of the mirror plane with a statistical weight 0.5 for each chain due to the 90" flipping.The average positions of C(2) and C(3) are weakly defined, whereas the terminal part of the chain [C(4) to C(lO)] forms a continuous distribution parallel to the c-axis. There is no evidence for a cone-like motion around the layer normal. The I3C spectra of the low-temperature and the intermediate-temperature phases of ClOCd are nearly identical but there is a sharp change in the 13C spectrum on going into the high-temperature phase. The angular dependence of the -CH, group spectra in the low-temperature phase is characteristic of two sets of rigid -(CH2)n- chains which are tilted by &40" with respect to the layer normal. The chain methylene groups display the full chemical shift anisotropy similar to that observed in n-eicosan15 or polyethylene.lb In the intermediate-temperature phase the chains are still tilted by &40" and almost rigid (fig.2). The bulk of the data can be described by an axially symmetric 13C chemical shift tensor: a1 = +(Sll + S,,) = 43 3 p.p.m., 811 = S,, = 15 + 5 p.p.m. with respect to TMS. is parallel to the chain direction. The axial symmetry of the l3 C tensor is due to the 90" flips around the chain axis which result in a rapid exchange of Sll and a,, but do not The symmetry axis 611 = 0 90 180 0 45 90 13 5 180 Ull" FIG. 2.-Angular dependence of the 13C spectra in the high temperature (42 "C) and intermediate temperature (36 "C, insert) phases of ClOCd. v, stands for the angle between the c axis and the direction of the external magnetic field.vP3C) = 67.9 MHz, a I H.62 PHASE TRANSITIONS I N A LIPID BILAYER 1.0 0.8 s 0.6 0.4 0.2 affect ~ 3 ~ ~ . The nematic order parameter S with respect to t,he preferred chain direction, 0 = &40°, is here about one, S x 0.95-0.99. There is another set of I3C lines belong- ing to the less rigid terminal methylene groups which is also axially symmetric and can be described by S x 0.75. The angular dependence of the 13C spectra in the high-temperature phase is com- pletely different (fig. 2) from that in the low- and intermediate-temperature phases. It shows that all chains are equivalent and normal to the layers. Three different sets of 13C lines can be resolved. The most intense line belongs to the bulk of the -CH2 groups.It can be described by 611 = 20 2 p.p.m. and d1 = &S,, + dz2) = 34 p.p.m. The anisotropy of the chemical shift tensor is only partially removed by the molecular motion (chain isomerization via fast kink diffusion) and the nematic order is still significant : I ---o--- I I I la) n - - I I I - I< I - I I (bl I , 1 1 - 1 I - - lcl I ----- Another weak line belongs to the terminal methylene group, -CH2-CH3, and can be described by 611, = 20 p.p.m., 61, = 25 & 1 p.p.m. The third line is due to the methyl group. Here ( T I I ~ ~ ~ = 10 & 1 p.p.m., BICHs = 14 It 1 p.p.m. With the help of the rigid lattice 13C shift tensors of n-eicosane l5 (n-C20H42) we find that S(cc-CHJ x 0.25 & 0.05 and S(-CH3) - 0.20 x 0.05. The above results and the absence of a significant flexibility gradient along the main part of the chain is at vari- ance with the more " liquid like " picture of the bilayer interior suggested by e.p.r.spin label studies,11 but agrees with the deuteron resonance data.9 The interpretation of the high-resolution 13C spectra of ClOCu is complicated by the fact that the paramagnetic contribution of the Cu2+ ions is superimposed on the 13C chemical-shift anisotropy. The contribution of the Cu2+ ions to the magnetic field at a given 13C position has been evaluated and subtracted from the experimental data to yield the I3C chemical-shift tensors. In the high-temperature phase the results agree with those of ClOCd within the limits of experimental error. This demonstrates that, as far as the chain dynamics is concerned, the high-temperature phase is the same in both ClOCd and ClOCu.On cooling down through T,, = 42 "C into the intermediate temperature phase r c 2 I IR. BLINC etal. 63 the spectra change only very little. The angular dependence is the same as in the high temperature phase at Tc2 > 46 "C > Tcl. This result is significantly different from the case of ClOCd where the chains are well ordered and tilted in the intermediate temperature phase. At Tc2 = 36 "C the I3C spectra of ClOCu abruptly change. The lines broaden and the angular dependence of the 13C chemical shifts qualitatively agrees with the data obtained in the low temperature phase of ClOCd if we suppose that all chains are parallel to a single direction and tilted by 40" with respect to the c-axis.The temperature dependence of the nematic order parameter S of ClOCu is plotted as a function of temperature in fig. 3 for the bulk -CH2 groups, the terminal methylene group and the methyl group (fig. 3). The hydrocarbon parts of ClOCd and ClOCu thus represents a smectic liquid crystal with structure similar to the interior of the bilayer lipid membrane^.'^-'^ Let us therefore try to describe the two phase transitions in ClOCd and ClOCu by a Landau type expansion of the non-equilibrium free energy P in terms of order parameters 8, S - Sc andp, used in the theory of smectic liquid crystals: l7 I; = +AB2 + $B04 + u(T - Tc2)(S - Sc) - $c(S - SC)' + $d(S - Sc)4 + + +ap2 + $bp4 - 5;A'Q2(S - Sc) - +ap2(S - Sc). (3) S S ClOCd \! FIG. 4.-Schematic temperature dependences of the order parameters S, p and 8 for ClOCd and ClOCu.The d.t.a. curves for these two systems are shown for comparison.64 PHASE TRANSITIONS I N A LIMPID BILAYER Here S stands for the average nematic order parameter and A , B, a, c, d, b, A' and a' are all positive constants whereas a = p(T-T,). The smectic-C order parameter 8 gives the average tilt of the molecules with respect to the layer normal whereas p is the orientational order parameter for the 90" flipping of the chains and the terminal NH3 groups between the two equilibrium orientations corresponding to p = rt 1. S, is the critical value of the average nematic order parameter S, i.e., the arithmetic average of S in the melted and rigid phase coexisting at the first-order transition tem- perature Tc2 (fig.4). In deriving expression (3) we assumed that there are two import- ant driving mechanisms which induce the transitions: the melting of the chains and the disordering of the keys. If T,, & T,., as seems to be the case in ClOCd, a z const. > 0 and the melting of the chains is the important driving mechanism which induces the transitions in S as well as in 8 and p. In contrast to the isotropic-nematic transi- tion, here S is not a symmetry-breaking order parameter and is not zero, both above and below Tc2. Because of that the linear term in the expansion of F in powers of (S - S,) is not identically equal to zero as in the case of nematic liquid crystals l7 but has the same structure as the free energy at liquid-gas phase transitions.The first coupling term -&A'02(S - Sc) accounts for the fact that rigid chains show a stronger tendency for tilting because of the attractive van der Waals interaction between the chains. The second coupling term -3a'p2(S - S,) describes the fact that " melted " molecules are axially symmetric and the potential hindering the rotation around the long axis is smaller above T,, than in the completely rigid state. Minimizing the total free energy with respect to p and 8 we obtain the following stable solutions for the symmetry-breaking order parameters 8 andp as functions of S : (i) 9 = 0, S < S, = A/A' + Sc (4) and (iii) p = 0, S < S2 = a/a' + S, (6) The temperature dependence of S is obtained by inserting expressions (4)-(7) into eqn (3) and minimizing with respect to S.The observed sequence of phases in ClOCd is obtained by choosing S, > S1, i.e., a/a' > 0 so that - Inthiscasewehavefor: T > T C 2 : S < S , < S c , 8 = O , p = 0 , (94 T,, < T < Tc2:S > s,, 0 # 0 , p = 0 (9b) T < T,,:S> s,, B # 0 , p # 0. (94 and for The high-temperature transition at Tc2 corresponds to a partial melting of the chains which simultaneously destroys the tilting of the molecules, whereas the low-tempera- ture transition at T,, corresponds to an orientational transition and a disordering of the polar " heads " in analogy to DPPC and biomembranes. If, however, a/a' < 0, so that S2 < S, andR. BLINC et al. 65 we get the reversed order of the two phase changes. With increasing temperature we get first the chain melting, and at still higher temperatures the disordering of the keys as observed in ClOCu (fig.4). Here we have: T < T c , : S < S 2 < S c , 8 = 0 , p = 0 (1 la> Tc, < T < Tc, : S < Sc, 8 = 0, p # 0 (1 1 4 T < Tc2: S > Sc, 0 # 0 , p + 0 . (1 1 4 The reversed order of successive phase transitions in ClOCd and ClOCu is thus the result of a difference in the key-key-hole interaction. Until now we have analysed only one bilayer. To get the observed ordering of the whole crystal one has to introduce an interaction between the two neighbouring hydrocarbon bilayers on the opposite sides of the same MC14 layer. The additional term in the free energy can be written as F' = (12) where 0, and 82 are the smectic-C order parameters on the opposite sides of the MC14 layer.For K > 0 one obtains 8, = 8, as observed in ClOCu and for K < 0, 8, = -02 as observed in ClOCd (fig. I). D. Chapman, Quart. Rev. Biophys., 1975, 8, 185. J. F. Nagle, J . Membrane Biol., 1976, 27, 233; J . Chem. Phys., 1973, 58, 252; Proc. Roy. SOC. A, 1974,337, 569. D. L. Melchior and J. M. Stein, Biochim. Biophys. Acta, 1977, 466, 148. S. MarEelja, Biochim. Biophys. Acta, 1974, 367, 165. A. Tardieu, V. Luzzati and F. C. Reman, J . Mol. Biol., 1973, 75, 71 1 . M. C. Philips, R. M. Williams and D. Chapman, Chetn. ghys. Lipids, 1969, 3, 234. J. Charvolin, P. Manneville and B. Deloche, Chem. Phys. Letters, 1973, 23, 345. A. Seelig and J. Seelig, Biochemistry, 1974, 13, 4839. B. J. Gaffney and H. M. McConnell, J. Magnetic Resonance, 1974, 16, 1 . N. 0. Petersen and S. I. Chan, Biochemistry, 1977, 16, 2657. ' E. Sackmann, Ber. Bunsenges. phys. Chem., 1978, 82, 891 and references therein. lo J. Urbina and J. S . Waugh, Proc. Nut. Acad. Sci., 1974, 71, 5062. l3 R. Blinc, B. ZekS and R. Kind, Phys. Rev. B, 1978, 17, 3409 and references therein. l4 J. Seliger, R. Blinc, H. Arend and R. Kind, 2. Phys., 1976, B25, 189. l6 S. J. Opella and J. S. Waugh, J . Chem. Phys., 1977, 66,4919. l7 P. G. de Gennes, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1974). D. L. Van der Hart, J. Chem. Phys., 1976, 64, 830. R. Kind, S. PleSko, H. Arend, R. Blinc, B. ZekS, J. Seliger, B. Loiar, J. Slak, A. Levstik, C. FilipiE, V. Zagar, G. Lahajnar, F. Milia and G. Chapuis, J. Chem. Phys., 1979, 71, 2118. l9 R. Blinc, M. I. Burger, V. Rutar, B. &kS, R. Kind, H. Arend and G . Chapius, Phys. Rev. Letters, 1979, 43, 1679.

ISSN:0301-7249    

DOI:10.1039/DC9806900058

出版商:RSC    

年代:1980

数据来源: RSC

摘要:

Intermolecular Forces and Orientational Phase Transitions in Molecular Crystals BY ALFRED HULLER Institut fur Festkorperforschung, Julich, West Germany Received 1 1 th December, 1979 The conditions under which one expects orientational phase transitions are discussed in the introduction. Then we consider the interactions between molecules and construct a Hamiltonian which is compared with the spin Hamiltonians that are in common use in the field of critical phe- nomena. A definition of the order parameter is given with special attention to the ordering of higher multipoles and with a discussion of its experimental determination. Finally the oppor- tunities of obtaining information about critical exponents are considered, but the prospect is rather unpromising. 1. INTRODUCTION In this paper we consider crystals which consist of molecules, polyatomic ions or side groups.These units will be denoted by the common name " molecules " and the crystals they form by the name " molecular crystals ". Methane (CH,) is an example of a proper molecular crystal, whereas in the ammonium halides (e.g., NHZCl-) the " molecule " is an ammonium ion, and in dimethylacetylene the " molecule " is a methyl group. If the molecules are tightly bound units, one may distinguish between the high-frequency internal modes and the low-frequency external m0des.l For our purposes the internal modes may be neglected, whereby the number of degrees of freedom per molecule is reduced to three translational and one, two or three rotational degrees of freedom: one rotation angle for a side group like CH,, two for a linear molecule like N2 and three for a three-dimensional molecule like CH,.Depending on the question as to whether the translational and rotational degrees of freedom are ordered or disordered, we distinguish four main classes of condensed molecular phases : (1) isotropic liquids with translational and rotational disorder, (2) liquid crystals which are rotationally ordered but translationally disordered, (3) orientationally disordered crystals (formerly also called plastic crystals) with trans- lational order and (4) proper crystals (brittle crystals) where both translational and rotational degrees of freedom are ordered. We are concerned with the orientationally disordered phases (3) and transitions towards complete order. Order may be achieved in several stages.Then we observe several orientational phase transitions with a stepwise reduction of the orientational disorder. Orientationally disordered phases are characteristic of crystals consisting of globu- lar molecules.2 The globular shape is evident for cage-like molecules such as ada- mantane or the carbonanes. Small and highly symmetric molecules such as methane may also be considered to be spherical. Electron density contours in the outer parts of the electron shell (where the overlap between neighbours occurs) are veryA . HULLER 67 closely spherical. The weak non-isotropic interactions mainly originate in the relatively weak multipole-multipole interactions. The same argument holds true for hydrogen, but in addition to that the quantum-mechanical uncertainty principle prevents a sharp localization of the molecular axis, thereby amplifying the globular appearance of H,.For globular molecules the angular average of the intermolecular forces is much stronger than the variation of these forces with respect to rotations of the molecules at a fixed distance. The radical part of the potential is responsible for the translational order, i.e., for the formation of a regular lattice of the molecular centres of mass. The angle-dependent part of the potential is responsible for the orientational ordering of the molecules. Consequently, for globular molecules orientational order sets in at lower temperatures than translational order. One observes the phenomenon of orientational disorder in crystals (ODIC). Unfortunately, in most of the phase transitions between crystal phases of different orientational order, the centre of mass structure also changes.In hydrogen, for example, the orientationally disordered phase is h.c.p. whereas the ordered phase is f.c.c. The simultaneous transformation of the centre-of-mass structure distorts the picture of a rotational phase transition. Therefore, the few examples where the symmetry of the centre-of-mass structure is not changed at the phase transition are of crucial importance. Our discussion will be based on the 1/11 phase transition in solid methane and on the II/IV transition in the ammonium halides (see fig. 1). In both cases the centre of mass structure is cubic above and below the phase transition.I 2 1 -- 2 O - 1 160 200 240 T / K FIG. 1 .-Generalized Stevenson plot for the p-T phase diagram in the ammonium halides which has been introduced by Press et al.14 The roman numbers denote the three phases which differ in the orientational order of the ammonium ions: I1 is disordered, IV is ordered in a parallel way (ferro), and in I11 there is antiparallel order (anti-ferro). TCP denotes the tricritical point. For pressures higher than the tricritical pressure the II/IV phase transition is continuous. All other lines denote first-order transitions. 2. INTERMOLECULAR FORCES The angle-dependent intermolecular forces are of the same nature as the central forces between atoms. They arise from the interactions between the permanent electrostatic monopole and multipole moments (electrostatic forces), the interactions between mutually induced multipoles (dispersion forces) and the overlap between68 INTERMOLECULAR FORCES A N D ORIENTATIONAL PHASE TRANSITIONS the electron shells (valence forces).With the exception of the interaction between two hydrogen molecule~,~ the intermolecular forces cannot be calculated from first principles. Therefore, with the exception mentioned, all the potential functions found in the literature, as e.g., atom-atom potentials4 and the Kihara core m0de1,~ are phenomenological expressions containing a number of adjustable parameters. Here we only discuss the expansion of the intermolecular potential V(Rij, mi, mj) between two molecules i and j in a twofold multipole series : 6 , V(Rjj, mi, CO~) zzz 2 A ~ ~ ; m , n , ( R i j ) D ~ A ( m i ) D ~ ; ~ , ( ~ j ) ; (2.1) 1 , m , n .l',m',n' Rij denotes the vector distance between the centres of mass of the two molecules, mi collectively denotes the set of Euler angles (ti, T i , q i ) which define the orientation of molecule i in space. The functions DgA(m) are a complete set of orthonormal func- tions in the Euler angle space, e.g., the Wigner functions. Then 0 I I < co and -1 rn, n I 1 where I stands for the total angular momentum, rn and n are projec- tions of 1. The coefficients Aii',m,,,(Ri j ) are phenomenological functions of Ri which comprise the effect of all three types of interactions. For purely electrostatic forces the distance dependence is [Rij1-'-''-'. In many cases it is advantageous to replace DgA(co) by a complete and orthonormal set of symmetry-adapted functions, e.g., the cubic rotator functions UF;(co) where I is still an angular momentum, but p and v do not have a physical interpretation. High molecular symmetry and a corresponding choice of the proper basis set Uh$(m) greatly reduces the number of non-vanishing coefficients in expansion (2.1).Let us consider expansion (2.1) for two neutral methane molecules with electrostatic interactions only. Modifi- cations due to dispersion and valence forces will be discussed afterwards. The lowest order non-vanishing terms are of order (I, 1') = (3, 3). The octopole moment ( I =I= 3) is the lowest-order multipole moment of a neutral tetrahedral molecule. Therefore there are no terms of order 1 = 0, 1 and 2.Terms of order 1 = 0 are not excluded by symmetry, but they are absent for neutral molecules. The functions A:?,;),, (R) have been calculated by James and Keenan : (2.3) I = A:?,;?, (R) = 2 ~,,P*(R/R) where Z3 is the octopole moment of CH4. The dimensionless matrix Dfi,&, only de- pends on the direction of R. Eqn (2.2) with terms of order (I, Z') = (3,4), (4,4) etc., neglected constitutes the James and Keenan' model for methane. If valence and dispersion forces are important the l/R7 dependence of the octopole term is changed. The new R dependence of the potential cannot be calculated but it can be determined from tunnelling experiments under pressure. This has recently been demonstrated for (NH4+),SnCl; -.* An experiment on CH4 is under way.' More important than the different R-dependence of the coefficients Ai;,';l,vp is the fact that now all symmetry- allowed coefficients are non-vanishing, as e.g., the octopole-monopole interaction As U ~ ~ ) ( c o j ) = 1 is a constant V(390) does not depend on the orientation of molecule j .Summing eqn (2.4) over all neighbours j of molecule i one obtains a contribution to the crystal field experienced by molecule i:A . HULLER 69 where the angular brackets denote an average over the intermolecular distances. The crystal field persists into the disordered phases. It is distinguished from the molecular field V,(co,,) which is obtained from a summation of the interactions which depend on the orientation of both partners. The octopole-octopole term then reads : where the angular brackets denote an average over molecular distances and over w j the orientations of the neighbours of molecule i.In disordered phases the angular average vanishes; therefore the molecular field is zero. Our discussion of V, and V, has been based on the molecular-field approximation which neglects correlations of the intermolecular distances and of the orientations of the molecule j with the orientation of molecule i. The crystal field in methane seems to be rather weak.’O This can be understood in terms of the nearly spherical shape of the outer parts of the electron shell and the charge neutrality of the molecules. The eight nearest- neighbour Cl- ions of a NHZ tetrahedron create a strong crystal field with pronounced minima. The strong crystal field leaves the molecule a choice between two different orientations (more precisely between two groups of 12 equivalent orientations each).Around the two equilibrium orientations, given by Euler angles CU, = (O,O,O) and m2 = (O,n/2,0) the molecule performs small librations with 8” amplitude.” The ammonium-ammonium interaction also contains an (I, Z‘) = (3,O)-term which contributes to the crystal field. The (3, 3)-term (2.2) is the same as in CH4. The electrostatic approximation should be excellent as the NHZ ions do not overlap. For the two equilibrium orientations allowed by the crystal field six of the seven functions U$)(w) are zero. For the libra- tional ground state we obtain <Ui”,(w)) = sc with 5 = 0.893 for NHZ and < = 0.908 for N D p s = & l for wavefunctions centred around col and w2.The octopole- octopole interaction [eqn (2.2)] in the librational ground state thus reduces to : This is different in NHZCl-. Ui:)(co) is equal to 1 for co, and equal to - 1 for co2. For the nearest-neighbour distance R i j = a(1, 0,O) one finds’ Ill,, = -6 and which shows that the orientational ordering in NHZC1- can be described in terms of a pseudo-spin Ising model with a nearest-neighbour ferromagnetic exchange constant J = 6c21$/R7. This is, however, not the whole story: apart from direct interactions to more distant neighbours there are indirect interactions mediated by the halide ions. There are two contributions to the effect, namely the spin-phonon coupling12 and the coupling via the electronic polarizabilities l3 of the halide ions.The indirect interactions favour the antiferromagnetic ordering which is found in NHZ Br- and NH,+I- for zero and moderate pressures. All these interactions contain a factor c2. Therefore phase transitions in the deuterated ammonium halides should generally occur at temperatures which are higher by a factor (0.908/0.893)2 = 1.034 than in protonated samples. This is indeed true for disordered to ferro-ordered transition with transition temperatures of 242 and 249 K for NH4Cl and ND4Cl, respectively. At the disordered to anti-ferro-ordered transition the isotope effect is reversed. In deuterated samples the transition is 30 K lower (!) than in protonated s a m p l e ~ . ~ ~ * l ~ It is the opinion of the author that this reversed isotope effect is not understood on a70 INTERMOLECULAR FORCES AND ORIENTATIONAL PHASE TRANSITIONS microscopic basis, but that there are two approaches towards an understanding. The first one starts from the generalized Stevenson plot', for the ammonium halides (see fig.1). One finds that the replacement of Br- in NHZBr- by Cl - corresponds to the application of an external pressure of 1.8 kbar. Deuteration corresponds to a pressure of 1.3 kbar both in NHZBr- and in NHZCl-. A shift of the zero-pressure line on deuteration not only explains the different sign of the isotope effect for the transition into the ferro-ordered and antiferro-ordered phases, but also its size which depends on the slope of the transition line in thep-Tdiagram. The problem with this explanation is that we do not understand on a microscopic basis why deuteration should correspond to the application of a pressure of 1.3 kbar.The lattice constant change on deuteration is Aa/a z 0.0005, an effect which can be achieved by an externally applied pressure of 0.25 kbar. The second explanation of the isotope effect relies on the difference in the hydrogen bond with protons and deuterons.16 The reasons for a reduction of the hydrogen bonding on deuteration are dynamic in nature. The theory of hydrogen bonding is, however, not developed to the point where a quantitative comparison is possible. In molecular crystals the interactions typically fall off like l/Rn with n = 5, 7 etc. This is in sharp contrast to magnetic systems where the exchange interaction falls off much more rapidly with distance.In many orientational phase transitions, interactions with distant neighbours are important, and if these interactions are com- peting as in the ammonium halides they lead to a complicated phase diagram and thus may influence the critical behaviour. We have given a detailed discussion of the octo- pole-octopole interaction which is responsible for the ordering in CH, and NH: C1-, two important examples for rotational phase transitions. In the case of dumb- bell molecules as, e.g., H2 or N2 the interaction is a little less complicated than eqn (2.2). It retains, however, its basic quality-the tensorial character. The orienta- tion of a dumb-bell is defined by two angles, the polar coordinates 8 and q. Therefore, the interaction is expanded into a double series of functions (the spherical harmonics) defined in that space.The quadrupole moment has only five components, so the 7 x 7 matrix D,,.(R/R) is replaced by a 5 x 5 matrix. Apart from these fine details the tensorial character of the interaction implies that D,,. depends on the direction of the intermolecular distance. In this respect the interaction is radically different from the isotropic exchange interaction in a Heisenberg ferromagnet. The inter- action not only depends on the relative orientation of the two molecules, but on the orientation of the molecules with respect to R i j . Therefore the phase transition in molecular crystals cannot be mapped on a Heisenberg model. Contrary to the Heisen- berg case the transverse excitations (librations) in the ordered phase always have a gap.This is true even for the case of vanishing crystal field which is almost ideally realized in solid hydrogen. On the other hand, for strong crystal fields (NHZCl-, adamantane, K+CN-, etc.) an Ising model represents the phase-transition properties almost ideally. 3. THE ORDER PARAMETER ITS DEFINITION In the Ising limit (strong crystal field) the definition of the order parameter q is obvious. For a ferromagnetically ordered phase we define where <si) is the expectation value of the ith pseudo-spin. In cases with weak crystalA . HULLER 71 fields where we cannot construct a pseudo-spin Ising model such a definition is im- possible. We first claim that a purely rotational phase transition (i.e., without rearrangement of the centre-of-mass structure) can only take place when the molecular symmetry is lower than the symmetry of the centre-of-mass structure.This statement will be discussed afterwards. Methane molecules with tetrahedral symmetry on a cubic (f.c.c.) lattice are an example. The hamiltonian [kinetic energy plus the interaction given in eqn (2.2)] then has the full symmetry of the lattice; in our example this symmetry is cubic. In the high-temperature phase the symmetry of the hamiltonian is not broken by the orientational order of the molecules. If we expandf(co) the orientational distribution function (also called a probability distribution function p.d.f.) for a molecule into a series of symmetry-adapted functions (adapted to the symmetry of the lattice) only those functions are allowed which are simultaneously invariant under the symmetry operations of the molecule and of the lattice.For the CH4 example such an expansion reads: 7 ~ 1 7 co 21 4- 1 the cubic rotator functions U$&(co) have been introduced in Section 2. There are (21 + 1) functions of order 1 but only very few are allowed functions. In our example the non-vanishing coefficients are: AiO,), Ai:), A::), etc. For vanishing crystal field f(o) is a constant, then A::', A::), . . . are all zero. The functions Uii)(co), Uit)(co), . . . modulate the orientational probability density f (co), their coefficients Ai;), Ai:), . . . depend on the strength of the crystal field and for finite crystal fields they are non-zero at all temperatures. When the cubic symmetry of the Hamiltonian is broken, the symmetry at a methane site is spontaneously reduced from cubic to tetrahedral.Then the functions Uii)(co), Ui:)(co), . . . become symmetry-allowed. Their coefficients A!;), Ai;), . . . are zero above the phase-transition temperature and non-zero below. Normally the lowest order coefficient A$;) is identified with the order parameter q. The new definition of q is valid both for weak and strong crystal fields. For strong crystal fields it is identical with the former definition by eqn (3.1). When the molecular symmetry and the site symmetry are identical, spontaneous symmetry breaking is not possible. EXPERIMENTAL DETERMINATION OF THE ORDER PARAMETER In a hypothetical example of polar molecules that undergo a phase transition into a phase with parallel orientation of the dipoles, one would observe a macroscopic electric moment which is proportional to the order parameter.If higher-order multipoles such as quadrupoles, octopoles, . . . order, no macroscopic moment is observed. The same is true for antiparallel dipole order. Therefore, apart from the hypothetical case of parallel dipole orientation, there is no means of observing the order parameter macroscopically. Neutron and X-ray diffraction is the only possi- bility for its determination. A diffraction experiment determines S(Q) = J dcoS(Q, co) which is the Fourier transform of the equal-time density-density correlation function (p(rl, t)p(rz, t ) ) . More precisely stated, p(rl, t ) is the density of the scattering power, i.e., the scattering length density if one considers the scattering of neutrons by nuclei, the electronic spin density in the case of magnetic neutron scattering and the electron density in the case of X-ray scattering.72 INTERMOLECULAR FORCES AND ORIENTATIONAL PHASE TRANSITIONS We divide the density into its contributions from the N molecules in the crystal: Then the correlation function contains terms of the type (pi(rl, t)pj(r2, t ) ) which we separate (pl(r1, t ) ~ j ( r 2 , t>> = <pi(rd><pj(rJ> + <bi(ri, t ) - pi(r)I[~j(~z, t ) - ~j(r)l>* (3.4) The first term comes from the time-independent single-molecule distribution (it leads to Bragg diffraction, the second term contains the correlations); it is responsible for the diffuse intensity between the Bragg points.The Bragg intensity is connected with the probability distribution of scattering centres in a coordinate system fixed in the crystal. To relate (p,(r)) with the expansion (3.2) which defines the order parameter, we introduce a second (primed) coordinate system fixed in the molecule which rotates with it. If the molecular structure is known, b(r‘), the scattering length density in the primed coordinate system is also known, and can be expanded in terms of sym- metry-adapted surface harmonics : b(r’) = C brrm~(r‘)K1,,*(8‘, 9’) (3.5) I’m’ with known coefficients b,.,,(r’). In our examples the proper set of functions are the cubic harmonics. K31- x’y’z’ and K41 - xt4 + f 4 + zr4 - 3/5 are the lowest-order functions with tetrahedral and cubic symmetry, respectively.Using the definition of the cubic rotator functions the scattering-length density in the crystal frame for a given orientation co of the mole- cule. b(r) is averaged with the orientational p.d.f. from eqn (3.2) to yield: (p(r)> = I d(?f(co,)b(r) = C alm(r)K1m(6 9) (3.8) on follows from the orthogonality relations of the cubic rotator functions. The coeffi- cients alm(r) can be measured in a diffraction experiment; they determine the order parameter. If AC) is the order parameter then ~~31(r) is zero above T, and finite below. Its temperature dependence, i.e., the critical exponent p, can be determined from the intensity of the Bragg points (with h, k, I all odd) to which it contributes. The diffuse scattering caused by the second term in eqn (3.4) can be used to obtain information about correlations, in particular about critical correlations in the vicinity of a phase transition.The intensity of the critical scattering is related to the magni- tude of the fluctuations, i.e., to the critical exponent y. The width of the scattering in Q-space is inversely proportional to the correlation length <, i.e., to the critical ex- ponent v. Information about the critical slowing-down is obtained from the energy depend- ence of the critical scattering, that is to say from a determination of S(Q, co).I8A . HULLER 73 4. CRITICAL PHENOMENA Towards the end of the preceding Section, methods to determine critical exponents for orientational phase transitions in molecular crystals have been discussed.The presentation has perhaps been misleading. It may have left the impression that there is a wealth of examples of continuous phase transitions accompanied by critical phenomena. The phase transitions in molecular crystals are almost exclusively discontinuous phase transitions. The use of critical laws like AC) w [(T, - T)/T,]/p is mi~leading,l~-~~ since one obtains spurious values of the critical exponents (e.g., p = 1/6) which have nothing to do with critical fluctuations or low dimensionality. The arguments why a critical exponent p should not be defined at a discontinuous phase transition have been given in ref. (22) andwill not be repeated here. So far the only example where a continuous orientational phase transition has definitely been found is the II/III phase transition of NH,+Cl- and ND,+Cl- under pressure.Here measurements of critical exponents are very difficult. This is seen from the two determinations 23 of /3 (for p = const and for T = const) which differ by 30%. The phase transitions in molecular crystals are not interesting from the point of view of critical phenomena. These systems do not belong to the domain where scaling laws or universality can be tested. The study of phase transitions in molecular crystals is important for a better understanding of intermolecular forces. Very little is known about these forces, which are the basis for all properties of condensed molecular phases. Phase dia- grams are a very sensitive test for the angle dependence of these forces. For the description of the phase transitions themselves a molecular-field theory is mostly sufficient.A very elegant and extremely successful example of such a calcula- tion is the James and Keenan’ theory of solid CH,. The reason for the applicability of molecular are-field theory to slightly discontinuous phase transitions is the fact that the incipient phase transitions are interrupted by the discontinuity before the fluctuations become divergent. The non-divergent fluctuations that occur may be treated within the Ornstein-Zernicke approximation. This is not the case. G. Venkataraman and V. C . Sahni, Rev. Mod. Phys., 1970, 42, 409. J. Timmermans, J. Phys. Chem. Sol., 1961, 18, 1 . I. Silvera, Rev. Mod. Phys., to be published. A. I. Kitaigorodosky, Molecular Crystals and Molecules (Academic Press, New York, 1973). T. B. MacRury and W. A. Steele, J. Chern. Phys., 1976,64, 1288 and references therein. J. 0. Hirschfelder, C. F. Curtis, and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1954). H. M. James and T. A. Keenan, J. Chern. Phys., 1959,31, 12. W. Press and M. Prager, to be published. W. Press, personal communication. l o W. Press, J . Chem. Phys., 1972, 56, 2597. l1 A. Huller and J. W. Kane, J . Chern. Phys., 1974, 61, 3599. l2 Y . Yamada, M. Mori and Y. Noda, J. Phys. SOC. Japan, 1972, 32, 1565. l3 A. Huller, 2. Phys., 1972, 254, 456. l4 W. Press, J. Eckert, D. E. Cox, C. Rotter and W. Kamitakahara, Phys. Rev., 1976, B14,1983. l5 P. Brauer and I. R. Jahn, Z . Naturforsch., 1978, 33a, 1093. l6 C. W. Garland, K. J. Lushington and R. C. Leung, J. Chern. Phys., 1979, 71, 3165. l8 W. Press, A. Huller, H. Stiller, W. Stirling and R. Currat, Phys. Rev. Letters, 1974, 32, 1354. l9 B. Dorner, J. D. Axe, and G. Shirane, Phys. Rev., 1972, B6, 1950. 2o E. Banda, R. A. Craven, R. D. Parks, P. M. Horn and M. Blume, Solid State Cornrn., 1974, W. Press and A. Huller, Acta Cryst., 1973, A29, 252. 1 7 , l l .74 INTERMOLECULAR FORCES AND ORIENTATIONAL PHASE TRANSITIONS 21 J. P. Bachheimer and G. Dolino, Phys. Rev. B, 1975, 11, 3195. 22 A. Huller and W. Press, Theoretical Aspects of Solid Rotator phases, in The Plastically Crystal- 23 W. B. Yelon, D. E. Cox, P. J. Kortman and W. B. Daniels, Phys, Rev. B, 1974,9,4843. 24 A. Huller and W. Press, Phys. Rev. Letters, 1972, 29, 266. line State (Wiley, London, 1978).

ISSN:0301-7249    

DOI:10.1039/DC9806900066

出版商:RSC    

年代:1980

数据来源: RSC

摘要:

Static and Dynamic Aspects of Order and Disorder in CBr, BY MARCEL MORE AND RENE FOUKET Laboratoire de Dynamique des Cristaux Molkculaires (E.R.A. 465), U.E.R. de Physique Fondamentale, Universitk de Lille I, 59655 Villeneuve D'Ascq Cedex, France Received 3rd December, 1979 In the first part of this paper is presented a review of experimental work carried out on CBr4: X-ray and neutron diffraction, neutron diffuse scattering and neutron coherent inelastic scattering. Comparison is made with CD4, and it is deduced that translational and rotational motions are coupled in CBr4. Then in the second part, some consideration is given to the development of a model of translation-rotation coupling using the theoretical work of Yasuda on CD4. 1. INTRODUCTION Below its melting point (365 K), carbon tetrabromide (CRr,) has an orientationally disordered cubic phase (phase I).A first-order transition occurs near 320 K and leads to an ordered monoclinic phase (phase 11). Owing to the similarity of their molecular symmetry (Td), it is interesting to compare CBr, with methane, which has been extensively studied : structure~,~-~ critical ~ c a t t e r i n g , ~ ~ ~ orientational order ~ a r a m e t e r , ~ phonon and librational theoretically etc. The fact that the transition does not appear at the same temperature for CBr, and CD4 is significant. The phenomena are not identical in the two cases mainly because of differences in molecular interaction. The solid-solid transition which occurs at 320 K is of first order and although phase I1 keeps some " memory " of phase I, the orientational order remains unexplained.Pretransitional effects have been seen in methane: critical slowing down for example; they have not been clearly recognized in CBr,. The potential seems to contain components which vary in a different manner with temperature, leading to a few com- peting order parameters. It thus appears interesting to study in more detail the high-temperature phase itself before trying to understand the transition mechanism. Comparison with methane will be made as often as possible. Phases I1 and I, ob- tained by X-ray and neutron diffraction respectively, are briefly schematized in Section 2(a) while in 2(b), the short-range order is analysed in time and space using neutron scattering measurements. The pattern obtained by a neutron diffuse scat- tering experiment is compared with that calculated from a hard-core potential model.Time analysis indicates that correlations have a complicated dynamical origin. Section 2(c) is devoted to neutron coherent inelastic scattering. Examples are given of acoustical phonons measured in the [I 111 direction. They show that acoustic velocities disperse with frequency. The scattering cross-section has a " three-peak- like " soft-mode structure. This probably results from a translation-rotation coup- ling similar to the spin-lattice coupling of the compressible Ising m ~ d e l . ~ ~ . ' ~ In The main experimental results are summarised in Section 2. Section 3 provides a basis for a possible explanation of experimental results.76 ORDER AND DISORDER I N CBr, particular a potential model based on a calculation by Yasudal' on CD, is outlined.This model is adapted to take account of translation-rotation coupling. Although calculations are long and sometimes tedious such a model can account for the com- plexity of the observed phenomena. 2. EXPERIMENTAL 2(a). s T R u c T u R E s The-structures of phases I1 and I have already been described.'6-18 We give only an Low-temperature phase I1 is monoclinic C2,= (a = 21.43, b = 12.12, c = 21.02 A, #? = 110.88", Z = 32). The structure was determined by X-ray diffraction. The 4 molecules in the assym- metric unit are in general positions. Vibrational motions have been analysed in terms of rigid-body tensors T, L and S . Tensor S , which takes account of correlations between translational and rotational motions has been shown to have no significant value. The final R factor has been found equal to R = R, = 0.07.Molecular centres of mass lay approximately on a face-centred cubic lattice, the molecules being tilted a little from the 42m symmetry. The structure of phase I was obtained by neutron diffraction. This phase, extending from 320 to 365 K, is face-centred cubic (a = 8.82 A, Z = 4). The refinement was made with a 6-orientation Frenkel model corresponding to a 42m local symmetry. It gives R = 0.16 with (u$> = 0.2 A2 (isotropic) and an isotropic rotational Debye-Waller factor (ol) = 0.05 Tad2. Introduction of anisotropic coefficients does not improve the fit. A new refinement has been made where the bromine coherent-scattering-length density2 a(r) is expanded as a set of symmetry-adapted functions.This gives a weighted reliability factor R, = 0.07. The probability of orientation of the C-Br bond is a maximum in the [110] direction. Dur- ing reorientation motions, molecules then have a maximum probability of being in a local orientation close to 42m. Nevertheless, Huller and Press l9 have shown that the orientational distribution function f(o) calculated from structure parameters is not always positive. Constraints should be imposed upon the expansion coefficients. The authors prove that f(o) is automatically positive when coefficients are deduced from an appropriate physical potential. To a first approximation it would be an angular potential but, owing to the large value of the trans- lational Debye-Waller factor (a:> = 0.2 A2, translation-rotation coupling must be con- sidered.Such a coupling has been introduced phenomenologically20 and has given new coefficients for the expansion. Nevertheless, we do not know if f(o) is definitely positive, and related coefficients are not connected to a potential. Apart from the fact that molecular centres of mass remain approximately on a f.c.c. lattice, the correspondence between the two phases is not evident. Particularly, orientational order in phase I1 remains unexplained. It could result from lattice distortion leading to a vanishing translation-rotation coupling, as suggested by the zero value of tensor S . outline . The R factor is then reduced to R = 0.037. 2(b).SHORT-RANGE ORDER Measurements made on powder samples by Dolling et aZ." in the cubic phase have re- vealed in the spectrum a diffuse peak for lQl N 2.1 A-'. This peak is not explained by the structure and the authors assign it to correlations. A neutron diffuse-scattering experiment conducted by More et aZ.2' shows that the scattering pattern has cigar-shaped contours near the (220) Bragg peak. The long axes of the cigars point in the [11 13 directions (fig. 1). In this paper,21 we show that scattering cannot be explained by structure-factor fluctuations (AF)2 = <F2> - <F>2. These fluctuations contribute to the scattering intensity in the same region of reciprocal space but the corresponding intensity would be lower and broader.M. MORE AND R .FOURET 77 Coulon and Descamps22 have given an explanation for the existence of this peak. They show that neighbouring molecules are correlated via a potential corresponding to steric hindrance between bromine atoms. This potential has a short-range action and correlates only nearest-neighbour molecules. The authors choose it to be in the form of a static hard- core potential. They use a 6-orientation Frenkel model for the structure. The use of a hard-core potential is justified: among the 6 x 6 possible molecular configurations only 32 give an averaged attractive energy of 2.4 kcal mo1-'. The last 4 give a repulsive energy of 10 kcal mo1-l. The results of their calculation are in good agreement with experiment. Calculated intensity contours are practically identical to those shown in fig.1. 2.E 2 2 .c 1 .E 0 0.5 51 1 FIG. 1 .-Diffuse scattering in the (1, f , 2) plane. The scattering vector is Q = cl(l, 1, 1) + c2( 1 ,I ,O). (Sample 4 cm3, background 400 counts, i.r. = instrumental resolution). However, this model can describe nothing but static disorder with molecules in selected orientations taking account of their surroundings. On the other hand one can imagine that a reorientating molecule translates and rotates in order to find a more stable equilibrium. Then correlations are dynamical. Because of dissipative processes, orientational fluctuations decrease in time as e - r r and give rise to quasi- elastic scattering. Fig. 2 and 3 show energy scans for diffuse scattering at points Q = (2.15, 2.15, 0.) and Q = (2.5, 2.5, 0).We estimate the instrumental resolution width to be 0.054 THz at Ai = 4.05 A. Intensity comprises a purely elastic component arising from the quartz container of the crystal and one (or several) quasielastic component(s). This result is fitted with a Gaussian function for instrumental resolution and a Lorentzian centred at zero energy for the quasi- elastic peak. We then obtain: The corresponding experiments have been made on spectrometer H1 at Saclay. rl = 0.03 0.005 T H ~ (fig. 2) rz = 0.21 i 0.03 THZ (fig. 3). rl is the inverse of the relaxation time of correlations responsible for intensity shown in fig. 1. In making this experiment (fig. 1) our instrumental resolution width was much larger than rl and the result was an integration over energy of correlation functions, that is to say78 ORDER AND DISORDER IN CBr, the static susceptibility.The origin of the (I?,) Lorentzian is not understood at the present. In the case of methane (CD4, phase I), quasielastic diffuse scattering5 additionally reveals anisotropic correlations. As with CBr4, correlations are stronger in [l 111 planes than out of these planes. When the temperature is decreased toward the transition temperature T,, orientational fluctuations slow down,6 the intensity increases, becomes critical and finally we obtain a Bragg peak corresponding to a new order in phase 11. Such critical scattering has not been observed in CBr4. We have only noticed a significant hysteresis of 5 K when 8CO d *z i3 s 600 %+ v) * 1 400 200 A . - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - b -0.1 -0.05 0 0.05 0.1 energy transfer/THz FIG.2.-Constant Q scan (at T = 325 K) at the point Q == (2.1 5,2.15,0). The elastic peak is mainly due to the quartz container. The Lorentzian has half width at half maximum = 0.03 THz. Ai = 5 A. the temperature is lowered toward the transition. At this temperature the crystal breaks. However, when temperature is increased, rl increases and the quasi-elastic intensity decreases (for example: rl = 0.05 THz and intensity is divided by two at T = 358 K). While the transition is first-order, a second-order transition could take place but at To < T , . 2(C). PHONONS Neutron coherent inelastic scattering experiments have been made at Saclay by M. More and B. Hennion. Acoustic phonons were measured in high-symmetry directions and with different polarizations.A few neutron groups are shown in fig. 4 and 5. They corre- spond to the longitudinal branch [ l l l ] . Fig. 5 is a constant Q-scan made at point Q = (1.085, 1.085 and 1.085) with an incoming wave- length Ai = 5 A. It shows a three-peak structure with maxima at fm and 0. At zero energy, an elastic peak due to the quartz container is superimposed. The remaining in- Intensities decrease rapidly and intrinsic widths increase when q increases (fig. 4).M . MORE AND R. FOURET 79 tensity is clearly broader than the instrumental width (0.025 THz in this experiment). In a soft-mode study23 as well as in a theoretical paper worked out for KCN by Michel and N a ~ d t s , ~ ~ we find an expression for the scattering law: where q represents phonon-phonon damping, p= - Cl’, i2 is the phonon frequency with interactions, oo is the phonon frequency without interactions, A-’ is the relaxation time of interactions andf(o) = l/(oz + A2).Using this empirical expression, we have been able to take into account the observed results and then obtain for CBr4 q = 0, 1 21 0.04 THz, n/o0 = 0.6, oo = 0.24 THz. A parametrization of this function for constant-energy scans has allowed us to fit the neutron groups of fig. 4. The full lines in fig. 4 and 5 represent this. A is of the same order as rl previously determined to take account of dynamical correlations. The scattering law [eqn (2. l)] describes soft-mode behaviour. In CBr4, longitudinal acoustic modes behave like this.In fact acoustic velocities are softening with frequency as presented in fig. 6 . A -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 energy transfer/THz FIG. 3,-Constant Q scan (at T = 325 K) at the point Q = (2.5,2.5,0). The Lorentzian has h.w.h.m. = 0.21 THz. 11 = 4.05 A. The anisotropy which has already been seen in diffuse scattering is now characterized by the fact that velocities of longitudinal modes disperse more than transverse ones. The softening of velocities often appears in plastic crystals, for instance in KCN,25 succinonitrileZ6 or ada- man t ane.2 Spectra obtained at several points in reciprocal space (such as zone centres or zone boundaries) are all of the same kind. They show a more or less intense “ background ” centred at o = 0. A de- convolution with the experimental resolution function plus a Lorentzian give an intrinsic half Attempts to measure librational phonons have always failed in CBr4.80 ORDER AND DISORDER I N CBr, width at half maximum of M 1 THz.There seem to exist in CBr4 nothing but diffusive modes characteristic of fluctuations exponentially decreasing with time. In methane,8 acoustic modes broaden out when the frequency increases, but the velocities do not seem to disperse. In the two cases, Librations have not been measured either. 250 200 h Y .-. 1 g - 150 x +J M .4 0) +J .- 100 50 energy transfer/THr f I 1 / I I b 0.1 0.2 0.3 momentum transfer (reduced units) FIG. 4.-Longitudinal acoustic neutron groups in the direction [11 11. molecules are reorientating with too high a frequency and neutrons cannot " see " their oscillations around temporary equilibrium positions.Libration frequencies would be of the order of 1 THz while correlation relaxation frequencies are close to 0.03 THz in CBr, and 0.09 THz in CD4.6 3. THE MODEL In order to explain these experimental features we develop in this Section a Potential energy can be calculated from atom-atom potentials between un- Using a 6-12 Lennard-Jones potential, Yasuda l1 has potential which takes account of translation-rotation coupling. bonded atoms or molecules.M. MORE AND R . FOURET 81 shown that the interaction potential between two tetrahedral molecules (I and 2) such as CBr, or CD, is given by: Vl, = U(R) + V R , 8, q ; w ) + V(R, n - 0, q + ; 7 1 4 + W R , 079; a 1 , m2) (3.0 where (R, 0, q) are the centre-of-mass polar coordinates of molecule 2 with respect to 100 1 O J I I I I I I b -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 energy transfer/THz width: 0.025 THz.FIG. 5.-Constant Q scan (at T = 325 K) at the point Q = (1.085, 1.085, 1.085). Instrumental the centre of mass of molecule 1, w1(co2) are three Euler angles setting the orientation of molecule l(2) in a standard frame. We define our standard orientation for the molecule like Yasuda.ll - U(R) is the isotropic part of the intermolecular potential. To obtain this term, we need an ex- pansion in series of powers of r/R [CA(r/R)p+4] where r = I Y ~ - ~ ~ ~ or In CD,, we have r/R = 0.26 and convergence for the series is correct whenp + q = 24. In CBr,, we have r/R = 0.31 and series has to be taken up to p + q = 36.These expansions have been calculated numerically on a computer. P A v = (R, 0901; m) = 2 u(R, I ) 2 K r n ( 0 , ~ ) Tlrn(w) (3 * 2) 1 2 3 m where the coefficients u(R, 1) given in ref. (1 1) are numerically calculated. are the spherical harmonics, and Tlrn(w) are defined by: YJO, q) 4 (B,, yk) are polar angles for a position of a molecule at a given moment with respect Tetrahedral symmetry of the molecule implies that Al,rn are different from zero to the original position (standard frame). only for I = 0, 3,4, 6, 7, 8, . . . 9(2,,, (co) are the usual Wigner rotation matrices. q, A l , , , are given in ref. (11). W(R, 8, q ; tol, 0.1,) = 2 w(R; 11'1'') 2 C(1I'I";mm') Y&,,,,$ (0, q) (3 -4) 1,1'>3 m,m' l"20 x Tl rn(m 1) Tl rn ( ~ 2 )82 ORDER AND DISORDER IN CBr, 3.0 2 .o 1 .o n 0 .cd - x 3.0 0 .I W w I rn E + 2.0 x 5 4 1.0 s c( w 0 W E c VT A A A A ' A A 500 3.0 7- 0 VT - 500 2 . 0 1 1.04 r1101 1 I I 0 0.1 0.2 0.3 0 . 4 0.5 AEITHz FIG. 6.-Dispersion of the wave propagation velocities.M. MORE AND R . FOURET 83 w(R; ZZ’Z‘’) are numerical coefficients.ll C(ZZ’Z’’; mm’) are the Clebsch-Gordan co- efficients in Rose’s notation.28 This term will not be used here but we can make some remarks about it: (i) The number of relevant coefficients is much greater in CBr4 than in CD, where only a few are preponderant: in particular we notice that James and Keenan” have only used the (3, 3, 6) term leading to the octopole-octopole interaction. (ii) Many of these coefficients have opposite signs and can then cancel each other.(iii) We assume that, in spite of some differences, this term will give results not very different from those obtained in methane. Numerical values (table 1) have been obtained using only interactions between TABLE NUMERICAL VALUES OF u(R, I ) IN EQN (3.2) AND w(R, ll’i”) IN EQN (3.4) IN kcal mol-I v(R, 4) = 1.22 v(R, 6) = 0.688 u(R, 8) = 0.08 I I’ 1” w(R,II‘Z“) 3 3 6 3 3 4 3 3 0 3 4 7 3 4 5 3 4 3 3 4 1 3 6 9 3 6 7 3 6 5 3 6 3 4 4 8 4 4 6 4 4 4 4 4 2 4 4 0 4 6 1 0 4 6 8 4 6 6 4 6 4 4 6 2 - 1.96 - 1.23 1.275 1.13 0.547 -0.649 -0.720 0.762 0.254 -0.362 -0.268 0.665 0.254 0.448 -0.342 -0.253 0.476 0.128 0.124 -0.206 -0.104 bromine atoms (Br-Br). screened by the size of Br atoms. for a Br-Br Buckingham potential Indeed we can assume that C-Br and C-C interactions are We use the coefficients given by Scott and Sheraga2’ VBr-Br = A exp (- Cr) - B/r6 with C = 2.78 A, A = 34 600 and B = 5180 if Yis expressed in kcal mol-’.Yasuda’s calculation needs this potential to be changed into a 6-12 Lennard-Jones potential V = a/r12 - b/r6 where a = 4.374 x lo6 and b = 3924 with V in kcal mol-I.84 ORDER AND DISORDER I N CBr, The crystal potential will now be expanded in series of displacements ui of the centre of mass of the molecule with respect to the averaged equilibrium positions. At zero order, with respect to the displacements, the potential can be written: R i j is the “ equilibrium ” distance between molecules i and j and the second term corresponds to the crystalline field acting on molecule i. Due to disorder, the averaged equilibrium positions are not the true equilibrium positions of the molecules and there are terms of greater order in the expansion. However, owing to cubic symmetry the linear term cancels for the isotropic potential. The second and third terms of eqn (3.1) give: dV = - C ( F [ + Fi) ~ i .(3.7) i If the summation is limited to nearest-neighbour molecules ( r ) at a distance R from molecule i, we obtain : F; = 2 grad V(R, o r , p r ; mi) Fj = 2 grad V(R, n - Or, n + q,; co,). (3.8) (3.9) r r We neglect the linear term arising from the 4th term of eqn (3.1). gradient formula28 to calculate eqn (3.8) and (3.9). Let us use Rose’s (3.10) grad ( u ( R I ) Yirn(or, ~ r ) ) = 2 Elol v(Rt l)TilJrn(or, 9,). I’ Here we have: --I for I’ = I + 1 I + 1 for I‘ = I - 1 0 for I’ # I & 1 and (3.1 1) (3.12) (3.13) where tYl,,rn+p are constants tabulated in ref.(1 1) and differ from zero only if I’ = 4, 6 , 8 . . . and Then we obtain: (3.14) (3.15)M. MORE AND R . FOURET 85 Molecular symmetry gives us I = 0, 3, 4, 6, 7, 8, . . . : crystalline symmetry gives I’ = 4, 6, 8, . . . . Taking eqn (3.10) into account we have I = I’ 1. Then non-zero terms will correspond to : l = 3 , 2 ’ = 4 I = 7, I’ = 6 1 = 7 , 1 ’ = 8 . . . . where the %:’(co,) are the (0,) symmetry-adapted functions belonging to the irre- ducible representation TI,, for I = 3. They have been tabulated by Altmann and Cracknell : 30 This result can be extended to any value of I. bouring molecules. form : The second term Fi implies a summation over all the orientations (co,) of neigh- We can drop out this summation by taking the Fourier trans- (3.18) 12 6 Vlnrn*(q) = 2 eiqR Ylern@r, qr) = 2 [eiqR +.(- 1)”e-’qR] Yl#rnl(Or, qr). (3.19) r = l r = 1 Then : (3.20) where ‘%‘ll.rn(q) is obtained by a generalization of eqn (3.14)-(3.18). For q = 0, we obtain Fi(R, 0) = Fi(R, w). For q # 0, expression (3.19) cannot be reduced like eqn (3.13) and terms of the expansion will be: ((I = 3, I’ = 2 and I’ = 4), ( I = 4, I’ = 3 and I’ = 5), . . .}. Fi(R, q) can be expanded on sets of symmetry-adapted functiQns in the same manner as Fi(R, w). A few of them have large values and for further calculations we will only take these main terms into account. Let us call UB(q) a vector defined with Fourier components %(T)(q) of symmetry adapted angular functions.I varies over all irreducible representations of the rotation group. z numbers all the functions which are classified in irreducible representations of the Oh group. Geometrical factors and coupling constants can be Some coefficients iir.rv(R, 1) have been calculated (table 2).86 ORDER AND DISORDER IN CBr, put into a matrix vorD(q). ments, We denote by ua(q) the Fourier components of displace- Then, the translation-rotation potential can be written in a bilinear form: (3.21) The hamiltonian can be written as where K is the kinetic energy of translation, VT the translational potential energy, Vc the crystalline field and VTR the translation-rotation potential. We have neglected the rotational kinetic energy, assuming that reorientations are slow enough. We have also neglected rotation-rotation coupling W(wl, w,).One can see that V, can easily be included in eqn (3.22). H = C {+P.'(qIPu(q) + +Mufi(q)u,+ (q)%3(q) + NP(!I)US(4)). (3.24) Pu(q) are the conjugate momenta of uu(q) and Mub(q) is the usual dynamical matrix for translational motions. Then we obtain 4 TABLE 2.-NUMERICAL VALUES OF &lie7 U(R, r) IN EQN (3.15) IN kcal m0l-l A-' ~ Z ~ U ( R , 3) = -7.34 &43~(R, 3) = 10.32 634~(R, 4) = -3.64 &54~(R, 4) = 5.36 &56U(R, 6) = 1.84 B76U(R, 6) = 3.04 B ~ ~ v ( R , 7) = 1.83 &67V(R, 7 ) = - 1.06 Let us denote: ua(4) = u;'(q) + U5'"d (3.25) where ui'(q) is an elastic deformation supposed to vary slowly with time and uiib(q) the usual vibration coordinates. From eqn (3.24) we derive the condition for instantaneous equilibrium which is: &'(q) = --(M-')UDNi(q).(3.26) Using (3.27) The first two terms contain translational kinetic and potential energies, the third one has only terms of rotational origin. This result and the fundamental transformation denoted by eqn (3.25) are a generalization of the compressible king model already presented in ref. (14) and (15). This relation connects elastic displacements with angular vector NB(q). (3.24)-(3.26) we get: H = 2 {+PZ ( d P a ( d + 3MuD(q)U2b + ( q ) ~ " % Z ) + 3Nu(q)u,'(d). 4 In this form, the hamiltonian is a sum of independent components.M . MORE AND R. FOURET 87 Although some numerical results have already been obtained, calculations have not yet been completed. However, we think that the theory presented here can explain some experimental features in CBr,.As the translation-rotation potential is bilinear in the rotation and translation coordinates, the dynamical theory written for KCN24 can be applied. In particular, rapid variations with q of matrix ua&) could explain the softening of wave-propagation velocities when q --+ 0. At q = 0, the potential is made of terms of order I = 3 and 7. This could account for additional terms found by Press2* for the structure. The scattering law can be expanded using the hamiltonian derived from the generalized compressible Ising model. So we obtain the translation-translation, translation-rotation and rotation-rotation parts of scattered intensity. Special account of rotation-rotation terms W(R, 8, p; mi, w2) should certainly be taken if we wanted to explain changes occurring at transition point in CBr, as it has been done in CD4.We thank J. Lefebvre (ILL., Grenoble), B. Hennion (L.L.B., Saclay), B. M. Powell (Chalk Rivet, Canada) and M. Bee, J. L. Sauvajol and J. C . Damien (Lille) for their help in experiments and discussions. W. Press, J. Chem. Phys., 1972, 56, 2597. W. Press and A. Huller, Acta Cryst. A , 1973, 29, 252. W. Press, Acta Cryst. A , 1973, 29, 257. W. Press and A. Huller, in Anharmonic Lattices, Structural Transitions and Melting, Nato Advanced Study Institute Series (Noordhoft-Leiden, 1974), p. 185. A. Huller and W. Press, Phys. Rev. Letters, 1972, 29, 266. W. Press, A. Huller, H. Stiller, W. Stirling and R. Currat, Phys. Rev. Letters, 1974, 32, 1354. W. Press and A. Huller, Phys. Rev. Letters, 1973, 30, 1207. W. G. Stirling, W. Press and H. Stiller, J . Phys. C, 1977, 10, 3959. A. Huller, Phys. Rev. B, 1974, 10, 4403. lo H. M. James and T. A. Keenan, J . Chem. Phys., 1959, 31, 12. l1 H. Yasuda, Prugr. Theur. Phys., 1 O 7 1 , 45, 1361. l2 T. Yamamoto, Y. Kataoka and K. Okada, J . Chem. Phys., 1977, 66,2701. l3 K. Maki, Y. Kataoka and T. Yamamoto, J . Chem. Phys., 1979, 70, 655. l4 H. Wagner and J. Swift, 2. Phys.. 1970,239, 182. l5 K. H. Michel and J. Naudts, J . Chem. Phys., 1977, 67, 547. l6 M. More, F. Baert and J. Lefebvre, Acta Cryst. A , 1977, 33, 3681. l7 M. More, J. Lefebvre and R. Fouret, Acta Cryst. A , 1977, 33, 3862. l9 A. Huller and W. Press, Acta Cryst., A 1979, 35, 876. 2o W. Press and A. Huller, Acta Cryst., A 1979, 35, 881. 21 M. More, J. Lefebvre, B. Hennion, B. M. Powell and C. M. E. Zeyen, J. Phys. C, 1980, in press. 22 G . Coulon and M. Descamps, J . Phys. C, to be published. 23 J. D. Axe, S. M. Shapiro, G . Shirane and T. Riste, in Anharmonic Lattices, Structural Transi- tions and Melting. Nato Advanced Study Institutes Series. (Noordhoft-Leiden, 1974), p. 23. G . Dolling, B. M. Powell and V. F. Sears, Mol. Phys., 1979, 37, 1859. 24 K. H. Michel and J. Naudts, J . Chem. Phys., 1978,68, 216. 25 J. M. Rowe, J. J. Rush, N. J. Chesser, K. H. Michel and J. Naudts, Phys. Rev. Letters, 1978, 26 H. Fontaine, R. Fouret, L. Boyer and R. Vacher, J . Physique, 1972,33, 1 1 15. 27 J. C. Damien and G. Deprez, Solid State Cumm., 1976, 20, 161. 28 M. E. Rose, Elementary Theory of Angular Momentum (John Wiley, New York, 1957). 29 R. A. Scott and H. A. Sheraga, J . Chem. Phys., 1965,42,2209. 30 S . L. Altmann and A. P. Cracknell, Rev. Mud. Phys., 1965, 37, 19. 40, 455.

ISSN:0301-7249    

DOI:10.1039/DC9806900075

出版商:RSC    

年代:1980

数据来源: RSC

摘要:

Orientational Relaxation in Molecular Solids with Translation Rotation Coupling BY BART DE RAEDT* AND KARL H. MICHEL? Institut fur Theoretische Physik, Universitat des Saarlandes, D66 Saarbrucken, Germany Received 3rd December, 1979 Starting from a Hamiltonian with bilinear coupling of translations and rotations and with single- particle orientational potential, we derive a closed transport equation for the dynamic displacement- displacement correlation function. The coupling to the rotation leads to a softening of the phonons and to an instability in the lattice at the orientational phase transition. The memory kernel which is due to the orientational velocity correlations is calculated in a systematic and consistent way. This leads to additional resonances in the displacement-displacement correlation function.1. INTRODUCTION In molecular liquids, the coupling of translational and rotational degrees of free- dom is a dynamic effect' which manifests itself in the occurrence of mixed transport coefficients. On the other hand, in molecular solids there occurs a static coupling between translations and rotations. This coupling is of particular interest near a structural phase transition where an ordering in molecular orientations and a change in lattice structure occurs simultaneously. Dynamic equations for coupled pseudo- spin-phonon models have been proposed for the description of the dynamics of the ammonium halides.2 A set of coupled equations for the lattice displacements and the anisotropic molecular polarizability has been put forward by court en^,^ following the lines of irreversible hydrodynamics.Recently a coupled system of dynamic equations for translations and rotations has been derived from a microscopic Hamiltonian.' In this Hamiltonian, the rota- tional degrees of freedom are described in terms of angular momenta and symmetry- adapted functions. In principle, all static and dynamic correlations can be calculated in terms of the microscopic model. In practice, only the calculation of static correla- tions is relativeIy easy. Previously one was not able to calculate the frequency de- pendent orientation-orientation transport coefficients. In fact these transport coefficients were replaced by frequency independent constants, which had to be determined from experiment.6 The resulting dynamic equations then have the same structure (frequency dependence) as those of ref.(2) and (3). The main advantage of the microscopic approach4 consists of the fact that it allows one to go beyond the assumption of a constant transport coefficient. Since we know the commutation rules of the angular momenta and the symmetry-adapted functions, we should be able to perform a systematic derivation of kinetic equations and to * Present address : Institut fur Festkorperforschung, KFA Jiilich, D5170 Jiilich, Germany. 7 Present address : Department Natuurkunde, University of Antwerp, B2610 Wilrijk, Belgium.B . D E RAEDT A N D K . H . MICHEL 89 calculate explicitly the non-secular currents which enter the orientational dissipation function. We have done a similar calculation recently in order to describe the orientational dynamics of molecular impurities in crystals.' If the concentration of impurities is sufficiently small (< 1 %), it should be sufficient to consider a single molecule and to neglect the indirect interaction between the impurities which is mediated by the lattice. Therefore, we have considered in ref.(7) the motion of a single impurity in a Devonshire-type potential. This model corresponds to CN' impurities in alkali halides. In the present calculation we consider a model which corresponds to KCN and therefore we take into account a bilinear coupling of the rotations to the translations in addition to the single-particle Devonshire potential. The underlying model is specified in Section 2 where we give also results for the static correlations.In Section 3 we use the method of ref. (9) to calculate the transport coefficients and we apply it to the present model. A discussion of the resonances of the dynamic dis- placement-displacement correlation function is given in Section 4. 2.THE MODEL We start from the Hamiltonian' H = HT + H R + HTR. Here HT describes the pure translational part where s,(k) is the Fourier-transformed centre-of-mass displacement of mass m (we will take rn = 1) per unit cell, and where p , ( k ) is the conjugate momentum with The matrix M denotes the harmonic coupling coefficients. Hamiltonian reads [sf(k), pj(q)] = idi, j d k q . (2.3) The rotational part of the Here Li(k) is the angular momentum of the (linear) molecule with moment of inertia I.The single particle potential VR is of octahedral symmetry;' it admits an expression in terms of cubic harmonics V"(l2) = a, + a4K4(Sz) + . . . (2.5) Here n(n) = (@), ~ ( n ) ) stands for the angular coordinates of the molecule at lattice site n. The coupling between translations and rotations is of the form HTR = i 2 via(k)YJ(k)si(k). (2.6) k Here the rotational coordinates Y,, a = 1-5 are represented by the Fourier-trans- formed linear combinations of spherical harmonics Y,(Q) = Y,O(R) ; Yz(st) = Yi*c(R) ; Y,(n) = Y:*s(Q); Y,(SZ) = Yi9s(12); Y5(Q) = Yi9c(12), where we have used the definitions lo (2.7a) (2.7b) y m . C = 1 (Y?+ YT")/dZ- YF*' = -i( Y y - Y i m ) / d 2 .90 ORIENTATIONAL RELAXATION IN MOLECULAR SOLIDS The functions Y, are of Es symmetry for O! = 1 , 2 and of T2, symmetry for O! = 3, 4, 5.The coupling matrix v reads (2.8) 0 0 k,B k,B k,A -k,A kyB kyA k,A k,B k,B 0 , -2k,A where the elements A and B are given in terms of the microscopic potential5 and where 2a is the lattice constant. Previously, this model was used for the calculation of static and dynamic correla- tion functions. We define a scalar product by the static susceptibility xpv(z = 0): (4, A") = X",(O>, (2.9a) where l1 (2.9b) Then the inverse of the static displacement-displacement susceptibility is given by (2.10) For many experimental purposes, it is convenient and suficient to consider particular symmetry directions in k space, say k = (O,O, k). Then in a cubic crystal (NaC1- type structure), the matrices M and D are diagonal with elements (2.1 la) (2.1 1 b) We have Ml1 = V , ~ ' C : ~ = C O ~ ; M33 = Vzk2c:l = C O ~ , (2.12) where co are the bare elastic constants and where V, = 2a3 is the volume per unit cell. In ref.(5) it was shown that Dll = M,,[1 - &?3] = Q;, (2.13n) and 0 3 3 == kf33[1 - YXFi] Q2. (2.13b) Here we have defined 6 = 8a2B2/Vz~:4, y = 16a2A2/V,ct4. x!& and xFl stand for the single-particle susceptibilities of T2, and E, symmetry respectively, defined with the single-particle potential VR : xto: = {Tr[exp (- VR/r> Yoc(Q) Ya(Q)]/Tr exp (- VR/T)}/T. (2.14) In eqn (2.14), Tr stands for trace and denotes the integration over the angular co- ordinates while T denotes the temperature (in energy units). We recall that x:, has E, symmetry and x:3 has T2, symmetry.Numerical evaluations of these quantities are found in ref. (7). 3 . DYNAMIC RESPONSE THEORY It is convenient to introduce Kubo's relaxation function a&) 55 (A,, (z - q-%) = [x&) - XC("(O)l/Z.B . D E RAEDT AND K . H . MICHEL 91 Here we used the definition of the scalar product ( 2 . 9 ~ ~ ) . The Liouville operator 2 is defined by A ( t ) = exp ( i Y t ) A , where 9 A = [ H , A ] . [lz - s2 + C(z)]@(z) = X(0). (3.2) (3 * 3) Given a set of secular variables {A“}, MoriI2 has derived the matrix equation Here LR is the matrix of restoring forces and C(z) stands for the frequency-dependent dissipation matrix (transport-coefficient matrix) : (3.5) C d z ) = - ( Q Z A p , ( Z - QZQ)-’QPAp)xP;’ (0). Here Q is the projector onto the space of non-secular variables, Q = 1 - P, PA = 2 AVx&l(0)(A,, A). “i, Since C, eqn ( 3 .9 , is again a relaxation function, it satisfies in turn an equation of type (3.3). Continuing this process one obtains a continued-fraction representation l3 for @(z). In order to get closed expressions, one has to terminate this series by some approximation. In ref. (4) we chose as set of dynamic variables {AY) ={si(k) pi(k), ya(k)} with = Y&) - Dij(k)(sj(k), Y,(k))si(k). The dissipation matrix 2:; was approximated by a frequency-independent constant Aag. Since here we want to go beyond this simple approximation, we have to use a systematic method for calculating frequency-dependent transport coefficients.’ We consider a set of orthogonalized dynamic variables {AY} = {Ao, Al, .. . A i , . . . A,,} where A,, is related to A. = A6 by A1 = P A o , A2 = Y A 1 - (A1, A,)(AO, Ao)-’ Ao, . . . A,, = 9 A , , - 1 - (A,,-1, An-1)(An-2, An-d-’ An-2. (3.7) Then only the transport coefficient C,,,,(z) is different from zero because Q 9 A i = 0 for i rf n. The solution of eqn (3.3) then yields the continued fraction representation A; 2-- A; z - - z - . . -A: + where A; = (A”, A ) ( A ” - l , A”-l)-l. (3.9) We use the method of ref. (9) to determine Cnn by means of sum rules. reads The result (3.10)92 ORIENTATIONAL RELAXATION I N MOLECULAR SOLIDS We apply this scheme to the model which has been specified in Section 2. use of the identity We start with Ah = si(k) and determine A:, A;, A$ according to eqn (3.7). ( 2 4 B) = ([At, m, Making (3.1 1) (3.12a) (3.12b) (3.12~) (3.1 3a) (3.13b) (3.134 (3.14) which is independent of the lattice site n.cells and Y&) stands for Ya[Q(n)]. within the single-particle approximation with potential VR : Here N denotes the total number of unit The quantity (YYoc, 2Ya) is now calculated T(LFYa,9Ya)R = Tr[exp (- VR/T)(9Ya)’ 9Ya]/Tr(exp - VR/T). (3.15) Taking again k = (0, 0, k), M and D become diagonal [see eqn (2.1 la)-(2.13b)] and eqn (3.13~) reduces to (A;, A:) = (A;, A;) = 8k2a2B2(9 Y3, 2’ Y3)R (3.16) (A:, A:) = 16k2a2A2(9 Yl, 9 YJR. (3.17) Writing A:,i for A; defined with A; according to eqn (3.9), we obtain for i = 1, 2: = D,, Q;2g = M - D 11 = P I 2 - 2-29 11 (3.18~) (3.18b) and A!,3 = 0 3 3 Qig (3.19a) Ag,3 = M33 - D 33 = p l 2 - Eg (3.19b) (3.19~) The quantities pi2, Y = Eg or T2g are completely determined by means of eqn (2.12), (2.13~) and (2.13b).One finds, respectively, pg = 16k2a2A2( Y,, Yl)R and pFzg = 8 k2a2 B3( Y3 , T3)R. Eqn (2.13~) and (3.18a)-(3.18~) show that displacements in x and y directions (9 YIP 9 YhR = ___ (co2),,. (Yl, YJR- A3,3 =B . DE RAEDT AND K . H . MICHEL 93 (i = 1,2) couple to T2g orientations (a = 3) while eqn (2.13b) and (3.19a)-(3.19~) show that displacements in the z-direction (i = 3) couple to Eg orientations (a = 1). Having determined all relevant coefficients, we obtain from eqn (3.8) where Here we understand Y = T2g for i = 1, 2 and Y = Eg for i = 3. 4. DISCUSSION Eqn (3.20) looks more familiar in the form (3.20) (3.2 1) Eqn (3.20) with the frequency dependent dissipation C(z), eqn (3.21), is the central We first make a comparison with the corresponding result of result of our work.ref. (4). Eqn (6.7) of ref. (4) is directly rewritten as: where 1, was the frequency independent orientational relaxation coefficient : illaa = -(Q9Fa, (iE - Q.YQ)-’QA?Fa)/(FaYa), (4.3) with E -+ 0. or i = 3. (4.1) and (3.22) Here a = 3 corresponds to Y = T2, or i = 1,2 in @if and a = 1 to Eg Putting z = iE -+ 0 in eqn (3.21), we then obtain by identification of eqn )baa = < a 2 > r / ( ( a 2 > r + p:2)’. (4.4) The second moments (co2), entering this equation have been calculated in ref. (7). We rewrite eqn (A.24) and (A.27) of ref. (7) 12T 1 - 3(x4) (W2)Eg = -- - I 0 9(x4> - 1 ’ (4.5~) (4.5b) Here we only took into account the cubic harmonic K4 in the expansion (2.5).Note that in Cartesian coordinates, K4(x, y , z) K (x4 + y4 + 2‘). The thermal averages ( x 4 ) are then evaluated numerically as a function of T/a4. This situation describes relaxational behaviour of orientational correlations. Depending on Eqn (4.1), with A independent of frequency, has been discussed in ref. (4).94 ORIENTATIONAL RELAXATION I N MOLECULAR SOLIDS the magnitude of ;1 in comparison with 4 M EE mi, one distinguishes the cases (a) of fast relaxation, il > mi and (b) of slow relaxation, il < mi. In case (a), the in- elastic spectrum exhibits a soft-mode Brillouin doublet of frequencies &ln, on top of a broad central peak which disappears in the experimental background. This is the situation found experimentally6 in KCN at temperatures above 168 K (orientationally disordered cubic phase).On the other hand, in the slow-relaxation case (b), one has a well defined three-peak structure, where in addition to a narrow central peak, a non- soft phonon doublet is located at z = &ai. This situation is experimentally realized in NH4Br.2*14 We now turn to the more general situation and take into account the frequency dependence of Cr(z). We are interested in the correlation function &(m) = - lim - [@Dff(co + iE) - @Dff(m - i ~ ) ] (4.6) 1 &+O 27c1 which enters directly into the neutron or Brillouin scattering law. We shall discuss eqn (4.1) as a function of the coupling /?L2 and of the moment (a2),. 1. /3L2 -+ 0, vanishing coupling between translations and rotations.Then we see from eqn (3.18b) and (3.19b) that ln; reduces to cof, and eqn (4.1) becomes It is easy to consider several limiting cases : (4.7a) and Consequently we have a Brillouin doublet with bare phonons at co = *tor. y or 6, depending on whether Y = Eg or T2g. transition, the coupling to the rotation leads to an instability in the l a t t i ~ e . ~ ~ * ~ 2. an;?, +- 0, or equivalently [see eqn (2.13a and b)]xF = c-‘, where < stands for This situation corresponds to the phase 1 1 @,f:(z) = - - Yff.(co) = - 6(m) z ln;’ n; 1 All scattering intensity is concentrated in an infinitely sharp central peak. 3. (a2), + 0, the second-order orientational moment vanishes. Then 1 zz - p i 2 1 @ff(z) = - - - nz, 0 z 2 - cof 2’ and a partial fraction decomposition yields : The correlation function reads Then (4.8a) (4.8b) (4.9a) (4.9b) (4.94B .DE RAEDT AND K . H . MICHEL 95 In addition to a Brillouin doublet at co = hco,., there is a sharp central resonance due to orientational relaxation. 4. (02), co:. Here we find 1 - 1 L Off(z) = - n; (z - a):' (4.10a) The rotational frequencies are too fast, the dynamics of translations and rotations are decoupled but there is a shift of the phonon frequencies from w, to a,. We find Let us now discuss some intermediate cases, where we have to evaluate Yff (co) numeri- cally. Using (3.20) and (3.21) we can write n,-2Co,- cpff(z) = 1 - B, Br U - U - Cr U - + i(Br + cr)' (4.1 1) with u = Z/W, (4.12a) Br 2 dxk for r = T2g (4.12b) Br = Y X ? ~ for r = Eg (4.12~) c, = (co2),/co~. (4.12d) In eqn (4.11) the frequency is scaled by the bare phonon frequency co,.We see that the shape of iDfi(z) only depends on the two parameters, B, and C,. If we specify the parameters of the microscopic Hamiltonian [eqn (2.1)-(2.4)] and if we take an explicit form for the one particle potential VR [eqn (2.5)], we can determine B, and Cr.5*7 Instead of doing this, we merely discuss the shape of the correlation function for some realistic values of B, and C,. The physical meaning of the parameters is clear: B, close to zero means that we are far away from the phase transition; Br close to one means that one is in the neighbourhood of the phase transition. C, is simply a measure of the rotational frequency in terms of the phonon frequency. From figs. 1-3 we draw the following main conclusions : The weight of the central resonance, which has its origin in the translation-rotation coupling, becomes more important as we approach the phase transition.A two-peak structure is observed for some values of the parameters: in particular when the phonon frequency is comparable with the rotational frequency this phenomenon is likely to occur. In these cases the two resonances are always broad. This, however, might be an artefact of our approxima- tion. It is easy to verify that a four-pole approximation such as eqn (4.1 1) can never give two sharp resonances at non-zero frequencies. As far as we know, two sharp resonances have not been observed experimentally until now. Theoretically one could answer the question, whether such two sharp peak structures can occur, if one could go one step further in the continued fraction.This, however, is not possible at the moment.96 ORIENTATIONAL RELAXATION I N MOLECULAR SOLIDS 0.1 0.5 1.0 1.5 0 FIG. 1 .-Normalized dynamic phonon correlation function. The frequency is measured in units of the bare phonon frequency. (1) B, = 0.3, C, = 0.3; (2) B, = 0.3, C, = 0.9. 0 0.5 1 .O 1.5 w FIG. 2.-See fig. 1. (1) B, = 0.5, C, = 0.3; (2) B, = 0,5, C, = 0.7; (3) B, = 0.5, C = 0.9.B . D E RAEDT A N D K . H . MICHEL 97 0 0 . 5 1.0 1.5 w __t FIG. 3.-see fig. 1. ( 1 ) B, = 0.7, C, = 0.3; (2) B, = 0.7, C, = 0.9. For a review, see, e.g., B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley-Interscience, New York, 1976). Y . Yamada, H. Takatera and D. Humber, J. Phys. SOC. Japan, 1974,36,641. E. Courtens, J. Physique Lettres, 1976, 37, L-21. K. H. Michel and J. Naudts, J . Chem. Phys., 1978, 68, 216. K. H. Michel and J. Naudts, Phys. Rev. Letters, 1977, 39, 212; J . Chem. Phys., 1977, 67, 547. ti J. M. Rowe, J. J. Rush, N. J. Chesser, K. H. Michel and J. Naudts, Phys. Rev. Letters, 1978, 40,455. B. De Raedt and K. H. Michel, Phys. Rev., 1979, B19,767. A. F. Devonshire, Proc. Roy. SOC. A , 1936,53 601 ; P. Sauer, Z . Phys., 1966,194,360. H. De Raedt and B. De Raedt, Phys. Rev., 1977, B15, 5379. lo C. J. Bradley and A. P. Cracknell, The Mathematical Theory ofSyrnrnetry in Solids (Clarendon, Oxford, 1972). R. Kubo, J . Phys. SOC. Japan, 1957, 12, 570; L. P. Kadanoff and P. C. Martin, Ann. Phys. (N. Y.), 1963, 24, 419. l2 H. Mori, Progr. Theor. Phys., 1965, 33, 423; Here we follow the formulation by W. Gotze and K. H. Michel, in Dynamical Properties of Solids, ed. G. K. Horton and A. A. Maradudin (North-Holland, Amsterdam, 1974). l3 H. Mori, Progr. Theor. Phys., 1965, 34, 399. l4 Y . Yamada, Y . Noda, J. D. Axe and G. Shirane, Phys. Rev., 1974, B9, 4429. l5 S. Haussuhl, Solid State Comm., 1973, 13, 147.

ISSN:0301-7249    

DOI:10.1039/DC9806900088

出版商:RSC    

年代:1980

数据来源: RSC

摘要:

Structural Phase Transitions in Malononitrile BY MARTIN T. DOVE AND ALASTAIR I. M. RAE Department of Physics, University of Birmingham, P.O. Box 363, Birmingham B15 2TT Received 3rd December, 1979 Previous studies using nuclear quadrupole resonance and calorimetric methods have shown that there are two second-order phase transitions in malononitrile at 141 and 294.7 K and that the low- and high-temperature phases have the same symmetry. In this paper the crystal structure of the inter- mediate phase is reported and compared with a previous determination of the high-temperature phase. The unit cell dimensions of the two structures are almost identical and the space groups are Pi and P21/n, respectively. The structures are related by relative translations and rotations of the molecules and this motion is identified with the eigenvector of the soft mode associated with the transition.The basis of a theory for such a system is developed using the quasi-harmonic approxi- mation, Malononitrile, CH,(CN),, exists in at least four phases in the solid state. Previous investigations using calorimetric and nuclear quadrupole resonance (n.q.r.) tech- niques have demonstrated the existence of continuous (second-order) transitions at 141 and 294.7 K and a very slow, probably first-order, transition at 260 K. The n.q.r. data are exemplified in fig. 1 from which it is seen that in the low-temperature 2 80 5 s 2 2 70 100 200 300 TIK FIG. 1.-Frequency of one of tne 14N nuclear quadrupole lines (v,) as a function of temperature ( T ) in malononitrile as measured in ref.(2). The splittings in the different phases (a, /I, y, 6) relate to the number of distinguishable nitrogen sites in the crystal. (a) phase and the high-temperature (7) phase there are two distinct nitrogen sites in the crystal while in the intermediate (p) phase there are four. The similarity of the n.q.r. spectra in the a and y phases and the fact that the frequencies change con-M. T. DOVE AND A . I . M . RAE 99 TABLE CELL DIMENSIONS OF THE /? AND y PHASES OF MALONONITRILE /? phase (present work) alnm 0.782( 1) blnm 0.763( 1) clnm 0.615(1) a/" 89.68(5) PI" 96.8(1) a/" 90.23(5) y phase [from ref. (4)] 0.784 0.763 0.61 8 90 96.2 90 Estimated errors of measurement are given in brackets. No corrections have been made for systematic error.tinuously with temperature between these two phases strongly imply that the a and y phases are crystallographically identical. Heat capacity measurements on malononitrile show a very small anomaly at the ct-p transition and no detectable anomaly at the /?-y transition although the accuracy of the latter measurement is adversely affected by the proximity of the melting point (304.9 K). These measurements also demonstrated the existence of a fourth (6) phase: when a sample of malononitrile which had been held at a temperature around 250 K for several days was heated, a large specific heat anomaly was observed at 260 K indicating a first-order transition at this point. N.q.r. measurements confirmed the existence of the 6 phase and it was found that the frequencies from all the nitrogen atoms were identical, indicating that the symmetry of the 6 phase is higher than that of any of the other three.The 6 phase is clearly the equilibrium phase below 260 K, the cc and /3 phases being metastable in this temperature region; however, the time constant for a transition into the 6 phase is very long (although the reverse 6 to p change is fast) so that the a and /? phases are both accessible for experimental investiga- tion. Previous investigations of the crystal structure of malononitrile have resulted in a FIG. 2.-Room temperature ( y phase) structure of malononitrile as reported in ref. (4) shown in projec- tion down the c axis of the unit cell. Molecules I and I1 lie near the plane z = 3c/4 and molecules I11 and IV lie near the plane z = c/4.The structure of the j9 phase appears nearly identical to that of the y phase in this projection. Hydrogen atoms are not shown.100 MALONONITRILE : PHASE TRANSITIONS determination of the complete structure of the y phase at room temperature3v4 and a report of the unit-cell dimensions and space group of the p phase.3 The y phase has been shown to be monoclinic with space group P2,/n and cell dimensions as listed in table 1 . The crystal structure is illustrated in fig. 2 and the atomic coordinates are listed in table 2. The space group of the p phase was reported to be P21,3 but this does not agree with the results reported below. The present investigation has the aim of determining the crystal structures of all TABLE 2.-cOORDINATES OF THE CARBON AND NITROGEN ATOMS IN THE o! AND y PHASES OF MALONONITRILE p phase y phase p phase y phase c3 0.595(4) 0.120(4) 0.7 3 7( 4) 0.464(4) 0.274(4) 0.700( 5) 0.283( 3) 0.212(3) 0.706(4) 0.68 l(4) 0.009(5) 0.766(5) 0.139(3) 0.178(4) 0.706(4) 0.590 0.124 0.726 0.468 0.269 0.691 0.292 0.210 0.720 0.687 0.012 0.746 0.157 0.170 0.733 xla -0.099(4) c4 y/b 0.630(4) z/c 0.793(4) x/a 0.029(4) c5 y / b 0.775(4) z/c 0.816(4) xla 0.195(5) ylb 0.719(4) z/c 0.773(5) x / a - 0.1 96(4) N3 y/b 0.516(4) z/c 0.773(4) x/a 0.327(4) N4 y/b 0.674(4) z/c 0.751(4) - 0.090 0.624 0.774 0.032 0.769 0.809 0.208 0.710 0.780 -0.187 0.512 0.754 0.343 0.670 0.767 Notes: The B phase parameters were obtained in the present work; the y phase parameters are from ref.(4). The numbers in brackets in the phase represent standard deviations; corresponding quantities in the y phase were not published.In the y phase atoms C4 to N4 are related to atoms C1 to N2 by a two-fold screw axis. the other three phases and obtaining as much information as possible concerning the nature of the transitions. This paper reports the crystal structure of the j? phase and a preliminary theoretical study of the phase transitions in this unusual system. EXPERIMENTAL Crystals of malononitrile were grown by sublimation onto a cold (273 K) surface and suitable crystals of approximate dimensions 1.0 x 0.3 x 0.3 mm3 were mounted on glass fibres and prevented from evaporating by thin walled Lindemann glass capillary tubes. Zero to sixth layer data were collected from a crystal rotating about the u axis using a Weissen- berg goniometer and copper Ku X-radiation.The crystal temperature was maintained at 273 As the preparation and mounting of the crystals were performed in a cold room, the data could in principle have been collected without the crystal ever being warmed out of the p phase, but due to a failure of the cool- ing apparatus, the crystal was inadvertently heated to room temperature for a short time be- tween the recording of the third and fourth layers. The intensities of 468 reflections were measured from the films by the S. R. C. microdensitometer service. 1 K throughout by a stream of cold dry n i t r ~ g e n . ~M. T . DOVE AND A . I . M . RAE 101 RESULTS Inspection of the films showed that the cell dimensions of the p phase are very nearly equal to those of the o! phase, but that the condition h + 1 = 2n in the (hOl) zone no longer holds; on the other hand reflections with k odd along the (OkO) reciprocal axis are absent, apparently implying that the space group is P21 as previously r e p ~ r t e d .~ Careful examination of the intensity data, however, showed that the intensities of the hkl and hgl reflections were not in general equal. It follows that the symmetry of the crystal must be triclinic rather than monoclinic and the absence of the OkO reflections with k odd must therefore be accidental rather than systematic. This possibility had also been noted by earlier workers.6 The space group was taken to be Pi as this is the only triclinic space group which can be generated from P24n by a second-order change obeying the Landau rules' and as the n.q.r.measurements clearly indicate that the number of symmetry elements in the /? phase should be half that in the y phase. The unit-cell dimensions measured from the Weissenberg films are shown in table 1 where it is seen that they are indeed very near to the room temperature values: in particular, the CI and y angles are extremely close to the monoclinic values of 90". This close relationship between the two unit cells raises the possibility that the tri- clinic crystals will be twinned, as crystallites with their b axes in opposite directions generate the same monoclinic cell when heated above the transition. The X-ray reflection from the hkl planes of one twin will be at almost exactly the same place as that from the hfil planes of the other, so the observed intensities will in general be a linear combination of the true intensities, corresponding to these two points in reci- procal space.The presence of twinning was confirmed by a careful inspection of the films : a few spots, mostly at very high Bragg angles, were resolved into two components and the values of the c( and y cell angles given in table 1 were in fact obtained from measurements of these splittings. Methods for analysing the structures of such pseudo-merohedrally twinned crystals have been discussed by Britton * and Murray- Rust.' Using the statistical methods described in these papers the twinning fraction was estimated to be 0.18 ; no significant difference was detected between the value of this parameter appropriate to the layers h = 0 to 3 and that for the layers h = 4 to 6 despite the fact that the crystal had inadvertently been heated into the y phase between the recording of these two data subsets.The true intensities were then re- covered from the measured intensities of pairs corresponding to the hkl and h&l planes. Unfortunately, this was not possible for all the reflections as the importance of these pairs had not been realised when the data were collected, and as a result only 298 independent intensities were included in the final data set. Structure factors derived from these intensities were compared with those calculated on the basis of the atomic coordinates of the high temperature phase and an overall temperature factor exponent of 0.02 nm2.The R-factor (defined as CIIF,I-IF,II /CIFo)l was 357<. Ten cycles of least squares refinement were performed in which the coordinates and individual temperature factors of the carbon and nitrogen atoms were refined along with a set of layer scaIes. These were followed by three cycles of refinement of the anisotropic temperature factors of the non-hydrogen atoms ; the hydrogen atoms were included at positions derived from the molecular stereochemistry with isotropic temperature factors the same as those of the central carbon atoms, but these parameters were not varied in the refinement. The R-factor was now 18%. The final atomic coordinates are listed in table 2 and the intramolecular bond lengths and angles are given in table 3.102 MALONONITRILE : PHASE TRANSITIONS TABLE 3 .-INTRAMOLECULAR BOND LENGTHS AND ANGLES IN MALONONITRILE AS MEASURED IN THE p AND 7 PHASES atoms D phase y phase 0.156(4) nm 0.149(4) O.lOS(5) 0.1 16(4) 0.149(4) 0.142( 5) 0.1 15(4) 0.1 1 1 (5) 11 l(2)" 1 12(2) 177(3) 175(3) 179(3) 176(3) 0.148 nm 0.146 0.115 0.1 10 0.148 0.146 0.115 0.1 10 111" 177 180 111 177 180 The jl phase quantities were obtained in the present work; the y phase quantities are from ref, (4).Standard deviations (where known) are in brackets. DISCUSSION The final R factor of 18% is relatively high and this is attributed to inaccuracies in the intensity data, particularly those associated with the crystal twinning; improve- ments could probably have been made if the twinning fraction had been refined as a parameter, but routines to do this were not readily available.Nevertheless, the final atomic positions are believed to be reliable and free from systematic error. This is confirmed by the good agreement between the measured values of the intramolecular bond lengths and angles and those determined in the y phase (cf. table 3) which in turn agree well with standard values. The final values of the temperature factors, on the other hand, have such large standard deviations that they are of only qualitative significance and are not reported in this paper. The relationship between the p and y phases is illustrated in fig. 3 where two mole- cules, which are related by a screw axis in the y phase, are shown in projection down the b axis. The transition to the p phase is accomplished by translating the centres of I I 1 fl- I1 I FIG.3.-Structure of the /I (open circles) and y (filled circles) phases of malononitrile shown in pro- jection down the b axis. Hydrogen atoms are not shown.M . T. DOVE AND A . I . M . RAE 103 mass of each molecule 0.007 nm in the x direction and rotating the molecules in opposite senses by 3.5" about axes parallel to the b axes which pass through these centres. The other two molecules in the unit cell undergo similar changes so that the centre of symmetry is retained. Such displacements leave the y coordinates of all the atoms unchanged which explains why the (OkO) reflections with k odd are apparently systematically absent and why the a and y unit cell angles remain so close to 90" in the p phase.All the significant differences between the structures at room temperature and 273 K are contained in the above symmetry-breaking motions; if the atomic coordinates of one molecule in the p phase are transformed by the two-fold screw axis of P2Jn and then averaged with the corresponding coordinates of the other, the result is identical to the high-temperature structure within experimental error. The P-y transition in malononitrile is therefore a structural transition driven by a soft mode whose eigenvector can be characterised by the molecular motions described in the previous paragraph. This mode is clearly a zone-centre optic mode and its symmetry is that of the Bs irreducible representation of the point group C2h which is the point group associated with the monoclinic space group P2,/n.Although the malononitrile molecule is polar, the space groups of both phases are centrosymmetric, implying that neither is ferroelectric. Although the crystal structure of the cc phase has yet to be determined, it is almost certainly the same as that of the y phase. This statement is based on the fact that the n.q.r. measurements show the nitrogen atom environments in the cc and y phase to be very similar and the cc-p and p-y transitions to be both continuous, which proves that the a and y phases have the same symmetry. If the cc-phase structure were to be sig- nificantly different from that of the y phase, while maintaining the symmetry, then differences between the y-phase structure and the average structure of the /3 phase would be expected, but these were not observed (see above).Examples of substances which change from a high-symmetry, low-temperature phase to an intermediate low- symmetry phase and back again at high temperature are rare. Rochelle salt10-12 is probably the best known example of such a system, but in this case the transitions have been shown to be of the order-disorder type. Malononitrile appears to be the only case where such a sequence of structuraz phase transitions has been observed. The basis of a theory to describe such a system has been developed and will now be described. The form of the temperature dependence of the n.q.r. frequencies close to the /?-y transition indicates that fluctuation effects are negligible and that the system should therefore be capable of being described by mean-field theory.Landau ' has described such a theory which provides a phenomenological explanation of the properties of many phase transitions : the free energy ( F ) is expressed as a power series in the order parameter ( P ) , i.e. F = AP2 + CP4. If A is positive, there is only one minimum in F and this is at the point P = 0, while if A is negative, the equivalent configuration corresponds to P = *(-A/2C)*. A second-order phase change is therefore produced if A is temperature dependent and changes sign at the transition temperature (Tc). That is A = A'(T - Tc) where A' is a constant or slowly varying function of temperature. theory to the present case we replace eqn (2) by the expression To extend Landau1 04 MALONONITRILE : PHASE TRANSITIONS where T2 > Tl.A is now positive if T < Tl or >T2 and negative for T between TI and T2. We thus have P = 0, T < Tl or T > T2 i- (4) P = [--A "(T - T,)(T - T,)/(2C]+, T1 < T < T2 In the case of malononitrile, the order parameter, P, is the amplitude of the " frozen in " part of the soft mode in the p phase, and this has been measured directly at only one temperature. However, the n.q.r. frequency splittings are expected to be pro- portional to P, at least to first order, and these have been measured over a wide range of temperature., Fig. 4 shows the average splitting as a function of temperature T/K FIG. 4.-Average of the n.q.r. frequency splittings (Au) in malononitrile as a function of temperature (T) (continuous line) along with a quadratic curve predicted by the version of Landau theory de- scribed in the text (broken line).along with a curve of the form of eqn (3). The agreement is reasonably good, the dis- crepancies being about what would be expected from the neglect of higher-order effects such as the weak temperature dependence of A" and C. Landau theory is, however, a phenomenological theory and does not provide a microscopic explanation for a phase transition. In the case of a structural phase change from a low-temperature, low-symmetry structure to a high-temperature, high-symmetry phase, a more fundamental explanation can be given in terms of anharmonic interactions between ph0n0ns.l~ The temperature-independent term in eqn (2) corresponds to the harmonic contribution to the energy of the soft mode giving AT^ = -+m;(q, a) (5) where mo(q, A) is the harmonic frequency of the soft mode (whose wavevector is q and polarisation A) in the high-symmetry phase and is imaginary because this mode is unstable at low temperatures.The term proportional to Tin eqn (2) arises from the anharmonicity and if the quasi-harmonic approximation is applied,13 it gives A' = kB z: s'nL(q, k)/&P, k ) k, P * In this paper, Boltzmann's Constant is printed kg, to avoid confusion with wavevector k.M. T. DOVE AND A . I . M. RAE 105 where gi>(q, k) is the fourth-order anharmonic coupling constant between the soft mode and another phonon of wavevector k and polarisation p. To extend this theory to malononitrile, comparison of eqn (2) and (3) shows that three changes from the above are required : the temperature independent term must be positive, the term proportional to Tmust be negative, and there must be an addi- tional positive term proportional to T2.To fulfil the first condition, the only require- ment is that the soft mode be stable at low temperatures and its harmonic energy consequently positive. For the term proportional to T to be negative, the right-hand side of eqn (6) must be negative which can be achieved provided some or all of the fourth-order anharmonic coupling constants are negative. There would seem to be no physical reason why this should not be the case, and it seems surprising that struc- tural phase transitions where the low-temperature phase has a higher symmetry than the high-temperature phase are not more common.Finally the required term pro- portional to T2 will result from a consideration of higher-order anharmonic inter- actions. If we apply the quasi-harmonic approximation in the usual way,I3 but extended to include sixth order as well as fourth order terms, a contribution to A which is proportional to T2 is obtained which corresponds to an expression for A" in eqn (3) as where gl$T(q, k , k') is the sixth-order anharmonic coupling constant connecting the soft mode with the two phonons whose wavevectors and polarisations are (k, p) and The basis of a microscopic theory for a series of phase transitions such as those observed in malononitrile has therefore been obtained and it can be concluded that the occurrence of such a pattern of phase transitions is evidence for the importance of sixth-order three-phonon interactions in this case.(k', pr>. CONCLUSION The crystal structure of the p phase of malononitrile has been successfully obtained from X-ray diffraction measurements. The unit cell is almost identical to that of the y phase, but the symmetry has been lowered from monoclinic P2,/n to triclinic Pi. The eigenvector of the soft mode associated with the transitions consists mainly of opposite rotations of the two molecules, which are related by the screw axis in the high-symmetry structure, about axes parallel to the b axis of the unit cell. On the assumption that the a phase is identical to the y phase, this system of phase transitions can be understood on the basis of the quasi-harmonic approximation, provided the contribution from sixth-order anharmonic three-phonon interactions is included.Further work planned on malononitrile includes an extension of the measure- ments on the p phase to improve the accuracy of the analysis and to study the tem- perature dependence of the order parameter, and the determination of the crystal structures of the CI and 6 phases. It is hoped to measure the changes in other physical quantities associated with the transition and to develop the theoretical analysis further by the calculation of some relevant quantities on the basis of an intermolecular force model. Technical assistance by Mr. R. Pflaumer is gratefully acknowledged as is help from the staffs of the S.R.C. microdensitometer service and the computer centre of the106 MALONONITRILE : PHASE TRANSITIONS University of Birmingham. M.T.D. Financial support was received from the S.R.C. for H. L. Girdhar, E. F. Westrum Jr and C. A. Wulff, J . Chem. Eng. Data, 1968,13,239. A. Zussman and S. Alexander, J. Chem. Phys., 1968. 49. 3792. N. Nakamura, S. Tanisaki and K. Obatake, Phys. Letters, 1971, 34A, 372. K. Obatake and S. Tanisaki, Phys. Letters, 1973, 44A, 341. R. A. Young, J . Sci. Instr., 1966, 43,449. S. Tanisaki, personal communication, 1978. D. Britton, Acta Cryst, 1972, A28, 296. P. Murray-Rust, Acta Cryst, 1973, B29, 2559. B. C. Frazer, M. McKeown and R. Pepinsky, Phys. Rev., 1954,94, 1435. T. Mitsui, Phys. Rev., 1958, 111, 1259. B. %kS, G. C. Shukla and R. Blinc, Phys. Rev, 1971, B3,2306. Amsterdam, 1974). ’ L. D. Landau and E. M. Lifschitz, StatisticaZ Physics (Pergamon, Oxford, 1969). l3 R. Blinc and B. ZekS, Soft Modes in Ferroelectrics and Antiferroelectrics (North Holland,

ISSN:0301-7249    

DOI:10.1039/DC9806900098

出版商:RSC    

年代:1980

数据来源: RSC