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. 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