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. 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