Deuterium NMR Study of the Glassy Crystal Pentachlorotoluene

J. Phys. Chem. 1996, 100, 15933-15941

15933

Deuterium NMR Study of the Glassy Crystal Pentachlorotoluene. Hadamard Quadrupole-Order Exchange NMR Atsushi Kubo,† Atsuo Yogo, Fumio Imashiro,‡ and Takehiko Terao*,§ Department of Chemistry, Graduate School of Science, Kyoto UniVersity, Kyoto 606-01, Japan ReceiVed: May 8, 1996; In Final Form: July 11, 1996X

Pentachloro[R,R′,R′′-2H3]toluene was studied by deuterium nuclear magnetic resonance (NMR) experiments. Three doublets were observed below 230 K, while a single doublet was observed above 300 K in a single crystal with its long axis perpendicular to the static magnetic field. The three doublets are attributable to the three disordered orientations of the molecule around its pseudo-6-fold axis. Populations in these three orientations were determined from the area intensities of the doublets. Below 150 K, the populations were found to be temperature-independent, indicating that the reorientational motions around the pseudo-6-fold axis are frozen into orientational disorder around this temperature and that the system is brought into a glassy crystalline state. A Hadamard quadrupole-order exchange NMR experiment was developed to determine the rates of the pseudo-6-fold reorientations by one-dimensional NMR. The cross and diagonal peak intensities of the exchange NMR spectra between 170 and 212 K were fitted well by using the three reorientation rates. A distribution of reorientation rates, which is often assumed in glassy materials, was not found. The temperature dependence of the reorientation rates could be described by Arrhenius law.

Introduction Low-frequency molecular motions in glassy materials have been characterized by various physical methods, such as dielectric, calorimetric, and NMR, and quasielastic light scattering measurements.1 The correlation time of the molecular motion often shows a non-Arrhenius temperature dependence and a non-Debye behavior (a broad distribution of the correlation time) near the glass transition temperature.2 Theoretical approaches have been developed to explain these phenomena in nonnetwork forming liquids and polymers using the modecoupling theory.2-5 There is a group of compounds called glassy crystals or orientational glasses;6,7 they form crystals and the molecules are situated at the regular points in a unit cell, while the molecular orientations are frozen in nonequilibrium disorder at low temperatures. In such materials, the motional freedom of a molecule is much less than in glasses formed from supercooled liquids. The behavior of the correlation times of molecular motions in such compounds between crsytals and glassy materials attracts physicochemical interest. Glassy crystals have big advantages for theoretical modeling and computer simulations. Also “single crystal” experiments on glassy crystals enable us to obtain detailed information on molecular motions in these materials. Dielectric and wide line 1H NMR studies have been made on the solid phases of hexasubstituted benzenes (CH3)nCl6-nC6.8-13 These molecules were shown to be dynamically disordered around their pseudo-6-fold axes at room temperature,14-17 which can be explained by the fact that these two substituent groups have nearly the same volume (Cl 20 Å3, CH3 24 Å3). At low temperatures, the compounds with n > 2 show first-order phase transitions, while no discontinuity or anomaly has been found for n ) 1 pentachlorotoluene by dielectric measurements8-12 and differential thermal analysis.9 The heat anomaly due to the glass transition is usually small in glassy †

Email: [email protected]. Email: [email protected]. § Email: [email protected]. Fax: 075-751-2085. X Abstract published in AdVance ACS Abstracts, September 1, 1996.

crystals.6 We expect that pentachlorotoluene forms a glassy crystal at low temperatures. Measurements of the 1H NMR spin-lattice relaxation time T1 have been reported for some glassy crystals.18-21 The reciprocal-temperature dependence of T1 often shows an asymmetric minimum with a gentler slope on the low-temperature side of the minimum. Such asymmetric 1H T1 minima were interpreted by continuous19,20 or discrete distributions21 of the correlation times. Ito et al., studied the motions of the pyridinium ions in (C5NH6)AuCl4 and (C5NH6)AuBr4 by 1H NMR second moment and T1 measurements.21 The pyridnium ions were found to undergo pseudo-6-fold reorientation among the nonequivalent sites. The authors interpreted their T1 results using a three-site jump model between the crystallographically nonequivalent sites and pointed out that some distribution of the reorientation rates may exist. If more than one kind of molecular motion exists, however, it is often difficult to provide separate information by measurements of 1H T1. Deuterium NMR is a powerful method to distinguish various types of motion22-26 or orientational disorder27 in crystals. Especially multidimensional exchange NMR is a powerful tool to obtain clear information on slow motions.28-32 At low temperatures, however, the relaxation time is often very long, so that two- or multi-dimensional exchange NMR experiments are often very time-consuming. In this study, we developed Hadamard quadrupole-order exchange NMR to improve efficiency of measurements by combining deuterium selective-excitation exchange NMR spectroscopy33-37 with the Hadamard method38-40 and studied pentachlorotoluene using this method together with measurements of deuterium spectra and relaxation times. The Hadamard Quadrupole-Order Exchange NMR Experiment If we apply a selective π pulse to the |0〉 T |-1〉 peak of a doublet in a single-crystal deuterium NMR spectrum, quadrupole order can be created as follows:

‡

S0022-3654(96)01312-3 CCC: $12.00

Iz ) 2Iz(1,0) + 2Iz(0,-1) f 2Iz(1,0) - 2Iz(0,-1) ) 2{3Iz2 - I(I + 1)} © 1996 American Chemical Society

15934 J. Phys. Chem., Vol. 100, No. 39, 1996

Figure 1. Pulse sequence for the deuterium Hadamard quadrupoleorder exchange NMR experiment for three sites. Three Gaussian selective π pulses are applied at the frequencies (δ1νq[1], δ2νQ[2], δ3νQ[3]) in the first experiment, and (-δ1νQ[1], -δ2νQ[2], -δ3νQ[3]) in the second experiment, where δ ) (1. The second free induction decay is subtracted from the first, so that only the quadrupole order is observed. The sign sequence (δ1, δ2, δ3) is changed in every successive measurement according to the reduced Hadamard matrix. The phase φ is cycled as x, y, -x, -y to eliminate the single and double quantum coherences.

Kubo et al.

Figure 2. 2H NMR spectra of a single crystal of C6Cl5CD3 with its long axis oriented parallel to the goniometer axis and with a goniometer angle φ ) 167°. The spectra were recorded at temperatures 105 K (a) and 322 K (b). The number of acquisitions was 64 (a) and 128 (b).

where Iz(1,0) and Iz(0,-1) are fictitious spin-1/2 operators.34,41,42 If the |1〉 T |0〉 peak is inverted, the sign of the quadrupole order is reversed. By subtracting one spectrum from the other, the Zeeman signal left by incompleteness of the selective pulse and the nonirradiated doublets can be eliminated. During the mixing time, the quadrupole order can transfer from the irradiated doublet to the others by the pseudo-6-fold molecular reorientations. After the mixing time the quadrupole order is converted to a couple of antiphase transverse magnetizations for observation by using in-phase 45° and 90° pulses. Then the transfer of the quadrupole order can be detected by the appearance of antiphase doublets other than the initially irradiated doublet. Repeatedly applying this method to the other doublets, we can obtain all the exchange spectra; this scheme is much less time-consuming than 2D exchange NMR. By incorporating a Hadamard method into the quadrupole-order exchange NMR experiment, however, all the exchange spectra can be obtained simultaneously, further improving the efficiency of the measurements. In Hadamard spectroscopy, the lines of some sites are selectively inverted according to a Hadamard matrix. The individual correlation spectra are encoded to some 1D spectra by the selective inversion. Each correlation spectrum can be decoded by a proper linear combination of the spectra. Hadamard matrices exist for M ) 4n, where n is an integer.43 However, for any number N of sites, where N e M, the same procedure can be applied using any N columns of the Mth-order Hadamard matrix. This method requires only one Nth the time necessary to obtain the N correlation spectra with the same sensitivity one by one using the single selective pulse scheme. A pulse sequence of Hadamard quadrupole exchange NMR for N ) 3 is shown in Figure 1. The Hadamard quadrupole-order exchange NMR experiment has the following advantages: (1) It is less time consuming compared with two-dimensional exchange NMR experiments, being favorable for variable mixing time exchange NMR experiments. It can also be implemented favorably to multidimensional exchange NMR. (2) Since this method enables us to observe only the quadrupole order, we have to consider only the quadrupole-order spin-lattice relaxation time T1Q but not the Zeeman-order spin-lattice relaxation time T1Z in the simulation of experimental results.

toluene were grown by slow evaporation of the benzene solution at about 5 °C. A single crystal with the dimension of 1 × 2.5 × 15 mm and a powder sample packed in a 5 mm NMR tube were used for the NMR measurements. All the NMR experiments were carried out on a CMX-300 NMR spectrometer with a deuterium Larmor frequency of 46.122 MHz. A goniometer was attached to a commercial wideline probe with a 5 mm solenoid coil, which enables us to rotate a single crystal around a goniometer rotation axis perpendicular to the static field. The sample temperature was regulated by temperature-controlled gas flow. The temperature of the gas was monitored by a copper-constantan thermocouple at the gas outlet positioned 1 cm above the coil. The sample temperature was calibrated by measuring the temperature difference between the sample position and the gas outlet under the same gas-flow condition as used in the NMR measurements. The deuterium NMR spectra were recorded using a quadrupole echo sequence with a 90° pulse length of 2.6 µs and a pulse spacing of 20 µs. Typically, 64 signals were accumulated with a repetition time longer than 7T1Z. T1Z and T1Q were measured by using a conventional inversion recovery sequence and an ordinary45 or a broadband46 Jeener-Broekaert excitation sequence, respectively, with a quadrupole echo detection. 2H Hadamard quadrupole-order exchange NMR experiments were performed by applying three Gaussian selective pulses with a length of 400 µs. The radio frequency was switched between the adjacent Gaussian selective pulses within 1 µs using an available function of the CMX spectrometer. The excitation band width was (5 kHz. 60% of the Zeeman order could be converted to the quadrupole order. The signal-to-noise ratio will be improved by x3 compared with a single selective pulse experiment by combining the four spectra recorded with the signs of the quadrupole order, (1, 1, 1), (-1, 1, -1), (1, -1, -1), and (-1, -1, 1) for the three sites, respectively. These four vectors are taken from the right three columns of the fourthorder Hadamard matrix. However, we performed only the former three experiments to reduce the experimental time. By adding the signals of two experiments, we could improve the signal-to-noise ratio by a factor of x2: (1, 1, 1) + (-1, 1, -1) ) (0, 2, 0), etc.

Experiments

Results and Discussion

2,3,4,5,6-Pentachloro[R,R′,R′′-2H3]toluene

was obtained by chlorination of commercially available [99.6% 2H3]toluene according to the procedure described by Harvey et al.44 Repeated recrystallization from benzene yielded pentachlorotoluene as colorless needles. Single crystals of pentachloro-

Spectral Features. Figure 2 shows the 2H NMR spectra observed at 105 and 322 K for a single crystal. All the singlecrystal NMR experiments were performed with its long axis oriented parallel to the goniometer axis. Three doublets were observed in the spectra recorded below 230 K, while a single

Glassy Crystal Pentachlorotoluene

Figure 3. Rotation patterns of the half 2H quadrupole splitting of a single-crystal C6Cl5CD3 recorded at 189 K (a) and 322 (b). 4, O, and ] denote sites 1, 2, and 3, respectively. The site numbers correspond to the numbers in Figure 2a. The lines are the least-squares fitting curves with the six-site disorder model described in the text.

Figure 4. Single-crystal 2H NMR spectra of C6Cl5CD3 recorded for the orientation φ ) 167° at 235 K (a), 245 K (b), 263 K (c), and 282 K (d). The dotted lines are the best fit curves based on the three-site jump model.

doublet was observed above 300 K. The spectrum recorded at 105 K is much broader than that recorded at 300 K. When the crystal long axis was oriented perpendicular to the goniometer axis, the number of lines was doubled. The half quadrupole splittings at 189 and 322 K in the parallel orientation of a single crystal are plotted in Figure 3 as a function of a rotation angle φ measured from a certain axis perpendicular to the crystal long axis. At each of the four temperatures 235, 245, 263, and 282 K, spectra were recorded for the three different orientations φ ) 167°, 108°, and 13°; the spectra for φ ) 167° are shown in Figure 4. Figure 5 shows the powder spectra observed at 157 and 363 K. Variable-temperature measurements of powder spectra indicated that the quadrupole coupling tensor is almost axially symmetric below 230 K and nonaxially symmetric above 300 K. The principal values were determined from the powder spectra: (νQ1 , νQ2 , νQ3 )/kHz ) (20.4, 18.2, -38.6), (-11.4, -5.7, 17.1), and (-10.8, -6.1, 16.9) at 157, 317, and 363 K, respectively, where |νQ2 | e |νQ1 | e |νQ3 |. Pentachlorotoluene crystallizes in the monoclinic space group P21/c with Z ) 216 and is isostructural with such halogenomethylbenzenes17 as 1,2,4-trichloro-3,5,6-trimethylbenzene,14 1,2,3,5-tetrachloro-4,6-dimethylbenzene,16 1,2,4,5-tetrachloro3,6-dimethylbenzene,16 and pentabromotoluene.15 The crystal

J. Phys. Chem., Vol. 100, No. 39, 1996 15935

Figure 5. 2H NMR spectra for a powder sample of C6Cl5CD3, recorded at 157 K (a) and 363 K (b). The number of acquisition was 64 (a) and 512 (b).

of 1,2,4-trichloro-3,5,6-trimethylbenzene was reported to be a needle elongated along the crystalline b axis.14 The number of quadrupole doublets observed in the 2H NMR spectra of pentachlorotoluene can be interpreted if the crystal long axis of our sample is aligned parallel to the b axis: When the b axis is aligned parallel to the goniometer axis, the molecules related by the twofold symmetry cannot be distinguished from each other by 2H NMR. Also, a pair of molecules related by a 180° rotation around the pseudo-6-fold axis cannot be distinguished. Therefore, three doublets are expected to be observed when the molecular reorientation around the pseudo-6-fold axis is frozen into orientational disorder, while a single doublet should be observed when the molecule reorients rapidly. Figures 2-5 show that the reorientation rate is much lower below ca. 230 K and much higher above ca. 280 K than 104 s-1. Detailed analyses of these spectra will be given below. We analyze the rotation pattern obtained at 189 K in Figure 3a by this pseudo-6-fold disorder model, assuming that the quadrupole coupling tensor is axially symmetric around the CD3-C direction. The half-quadrupolesplitting for the k-th molecular orientation, νQ[k], is given by the following equation

νQ[k] ) νQ3 D2* 00(ΩLfP[k])

(1)

where νQ3 is the major principal value of the quadrupole coupling tensor. ΩLfP represents the Euler angles for the transformation from the laboratory frame (L) to the principal 2 (Ω) is the axis system (P) of the quadrupole tensor. Dnm second-order Wigner matrix according to the definition by Rose.24,45 The Euler angle transformation ΩLfP is expressed by the three successive transformations ΩLfG, ΩGfM, and ΩMfP. G denotes the coordinate system fixed to the goniometer whose z-axis is parallel to the rotation axis. M denotes the coordinate system fixed to the pentachlorotoluene molecule, whose z axis and x axis are chosen to be parallel to the pseudo6-fold axis and one of the CD3-C directions, respectively. The Wigner matrices are expressed as 2

D2* n0(ΩLfP[k]) )

2* 2* 2* Dnm (ΩLfG)Dml (ΩGfM)Dl0 (ΩMfP[k]) ∑ m,l)-2

2

)

2* 2* 2* Dnm (π/2,π/2,φ)Dml (R,β,γ)Dl0 (kπ/3,π/2,0) ∑ m,l)-2

(2)

With this choice of the Euler angles ΩLfG, the xG axis is parallel

15936 J. Phys. Chem., Vol. 100, No. 39, 1996

Kubo et al.

to the magnetic field when φ ) 0. The orientations of the pseudo-6-fold axis and one of the CD3-C vectors are given by (sin β cos R, sin β sin R, cos β) and (cos γ cos β cos R - sin γ sin R, cos γ cos β sin R + sin γ cos R, - cos γ sin β) in the goniometer-fixed coordinate system (xG,yG,zG). By a least-squares fitting, we determined from the rotation pattern the Euler angles ΩGfM to be R ) 65 ( 7°, β ) -158 ( 2°, and γ ) 37 ( 7°. We used the value of 38.3 kHz as νQ3 , which was obtained from the powder pattern observed at 192 K. The calculated curves are shown in Figure 3a. The major principal axes of the deuterium quadrupole tensors for sites 1, 2, and 3 (the site numbers correspond to the numbers of the doublets in Figure 2a) were found to make angles of 87°, 70°, and 73° with the crystallographic b axis, respectively. The atomic coordinates are given for the isomorphous compounds of pentabromotoluene and 1,2,4,5-tetrachloro-3,6-dimethylbenzene. The angles between the three C-X vectors (X ) Br, Cl, or CH3) and the b axis are 69°, 86°, and 72° in pentabromotoluene and 72°, 88°, and 71° in 1,2,4,5-tetrachloro3,6-dimethylbenzene, respectively. Although the atomic coordinates of pentachlorotoluene have not been determined, the resemblance of the major principal axis directions to the C-X directions suggests that the pentachlorotoluene molecules occupy positions similar to those in pentabromotoluene and 1,2,4,5tetrachloro-3,6-dimethylbenzene, and site 1 is assigned to the second C-X direction. The above Euler angles R, β, and γ will be used for the analyses of the spin-lattice relaxation data. Populations The fractional populations pj of the sites j (j ) 1, 2, 3) were directly determined from the fractional area intensities of the individual doublets in the single-crystal spectra observed between 105 and 208 K. For the accurate determination of the fractional populations from the quadrupole echo spectra, one has to estimate the echo decay due to homonuclear and heteronuclear dipolar interactions during the refocusing period. However, although the 2H echo amplitude of a CD2 group has been shown to be modulated as a function of the pulse interval by the homonuclear 2H-2H dipolar interactions,48 our calculation for the rotating CD3 group showed that this effect is negligible under our experimental conditions, i.e., the pulse interval of 20 µs. We can also neglect 2H-35Cl (or 37Cl) dipolar couplings, which are only about 60 Hz. We also determined the populations of the three sites at 322 K by fitting the angular dependence of the quadrupole splittings shown in Figure 3b using the following equation: 322 K

νQ

(φ) ) r{νQ[1]

189 K

(φ)p1

322 K

+ νQ[2]

189 K

(φ)p2

189 K

νQ[3]

322 K

+

} (3)

Figure 7. Temperature dependence of the asymmetry parameter ηQ ) (νQ2 - νQ1 )/νQ3 obtained from the powder spectra. The solid and the broken lines are the theoretical curves calculated for νQzLT ) 18.2 and 20.4 kHz, respectively.

sites 2 and 1 can be estimated to be 0.82 ( 0.02 kJ mol-1 from the solid line in Figure 6. Below 150 K, the plots of ln(p1/p2) deviate from the solid line and become temperature independent. This indicates that the molecular reorientations freeze and the system is in the glassy crystalline state below 150 K. Although the plots of ln(p1/p3) do not show a linear dependence below 200 K, the energy difference of the potential minima between sites 3 and 1 can roughly be estimated from the dotted line: E3-E1 ) 0.7 ( 0.1 kJ mol-1. The fractional populations at 105 K are p1 ) 47%, p2 ) 23%, and p3 ) 30%. The quadrupole-coupling principal values obtained from the powder spectra between 309 and 363 K are slightly temperature dependent. Figure 7 shows the temperature dependence of the asymmetry parameters ηQ ) (νQ2 - νQ1 )/νQ3 obtained from the powder spectra. If the pseudo-6-fold reorientations are fast enough, ηQ is given using the principal values νQxLT, νQyLT, and νQzLT at the low-temperature limit:

ηQ ) {(νQyLT - νQxLT)/2νQzLT} × x(3p1 - 1)2 + 3(p2 - p3)2 (5)

(φ)p3

322 K

where νQ322 K(φ) and νQ[j]189 K(φ) are the half splitting at 322 K and that at 189 K for the j-th site, respectively. pj322 K is the fractional equilibrium population of the j-th site at 322 K, and r is the reduction factor of the quadrupole coupling constant due to the difference of vibrational motions between 189 and 322 K.49 We obtained p1322 K ) 39 ( 1%, p2322 K ) 30 ( 1%, p3322 K ) 31 ( 1%, and r ) 0.98 ( 0.05, from the least-squares fitting. The populations obtained are summarized in Figure 6 as plots of ln(p1/p2) and ln(p1/p3) versus the reciprocal temperature. Above 150 K ln(p1/p2) obeys Boltzmann’s law:

ln(p1/p2) ) (E2 - E1)/RT

Figure 6. Plots of ln(p1/p2) (O) and ln(p1/p3) (2) versus the reciprocal of temperature, where pj is the fractional population of the j-th site. The solid and the dotted lines are the theoretical lines for E2 - E1 ) 0.82 kJ mol-1, and E3 - E1 ) 0.7 kJ mol-1, respectively.

(4)

The energy difference E2-E1 of the potential minima between

where we assumed that the principal x axis of the quadrupole tensor lies along the C-CD3 vector of site 1, while the z axis is parallel to the pseudo-6-fold axis of the molecule. The derivation of eq 5 is shown in the Appendix. If we assume that (νQyLT - νQxLT)/2νQzLT is constant between 309 and 363 K, the temperature dependence of ηQ is attributable to the temperature variation of the fractional populations. To confirm the validity of the energy differences obtained from the single crystal experiments, we calculate the temperature dependence of ηQ by using eqs 4 and 5. We used the principal values (20.4, 18.2, -38.6 kHz) obtained from the powder spectrum at 157 K for νQx,y,zLT. While the value of -38.6 kHz is assigned to νQxLT, there are two choices for νQzLT. The solid and the dashed lines of Figure 7 were calculated for νQzLT ) 18.2 and 20.4 kHz, respectively. The solid line is close to the experimental values, and thus we conclude that the energy

Glassy Crystal Pentachlorotoluene

J. Phys. Chem., Vol. 100, No. 39, 1996 15937

differences obtained before are reasonable and that the z component of the quadrupole coupling tensor at 157 K is given by νQzLT ) 18.2 kHz. Although νQzLT is unchanged by the pseudo-6-fold rotation, the major principal values obtained from the powder spectra above 300 K were slightly smaller than νQzLT ) 18.2 kHz. We attribute this decrease of the quadrupole-coupling constant with increasing temperature to vibrational motions and obtain the reduction factor of the quadrupole-coupling constant for a temperature change from Ta to Tb: -4

r ) {νQ(Tb)/νQ(Ta)} ) {1 - 3.2 × 10

spin-lattice relaxation times. All of the results will be summarized as plots of the rates versus reciprocal temperatures. The sites related by the center of symmetry can be distinguished from each other by neither spectral measurements nor exchange NMR experiments, so that the three-site jump model and the six-site jump model give exactly the same results. Therefore, in the analyses of the NMR spectra or the exchange NMR experiments, we use the master equation of the threesite jump model:

[][

p1 -W21-W31 d p 2 ) W21 dt p W31 3

× (Tb - Ta)} (6)

rνQzLT gives quadrupole-coupling constants only a little smaller than those determined from the single-crystal rotation patterns. We use eq 6 for the further analysis.

W12 -W12-W32 W32

][ ]

W13 p1 W23 p2 -W13-W23 p3 (7)

where Wjk is the transition rate from the k-th site to the j-th site. In equilibrium, the following relation is satisfied:

Molecular Reorientations

Wjkpk ) Wkjpj

In this section, the rate of the pseudo-6-fold reorientations will be determined from the Hadamard-exchange NMR results, the spectral simulations, and the angular dependence of the

(8)

There are only three independent rate parameters W21, W31, and W32, if the pj are given.

[]

In the analysis of the spin-lattice relaxation times, however, we should use the master equation of the pseudo-6-fold reorientation, since the 3-fold rotation and the 6-fold rotation can be distinguished from the spin-lattice relaxation times.50

[

-W21-W31 W21 0 ) 0 0 W31

p1 p2 dp b d p3 )W ˆ ‚p b ≡ dt dt p4 p5 p6 W12 -W12-W32 W32 0 0 0

0 W23 -W23-W13 W13 0 0

0 0 W31 -W21-W31 W21 0

0 0 0 W12 -W12-W32 W32

W13 0 0 0 W23 -W23-W13

][ ] p1 p2 p3 p4 p5 p6

(9)

We assumed here that molecular reorientational jumps take place only between adjacent sites and that the sites related by the inversion center are equivalent: the transition rates and the equilibrium probabilities satisfy Wij ) Wij+3 ) Wi+3j ) Wi+3j+3 and 〈pj(t)〉 ) 〈pj+3(t)〉, respectively. In this pseudo-6-fold reorientation model, however, two correlation functions 〈pj(0)pk(t)〉 and 〈pj(0)pk+3(t)〉, which are expressed in terms of the three rate parameters W21, W31, and W32, are different from each other. The sum of the probabilities for the two sites related by the inversion center, p′j ) pj + pj+3 still satisfies eq 7, so that the definition of the transition rate Wjk is consistent between eqs 7 and 9. The Hadamard quadrupole-order exchange NMR experiments can be analyzed by using the following equation:

ˆ - Rˆ 1Q)tm]‚aˆ (0) aˆ (tm) ) exp[(W

(10)

where aˆ (tm) is a 3 × 3 matrix as a function of the mixing time tm, whose element ajk(tm) corresponds to the cross- or diagonalpeak intensity at (ν1, ν2) ) (νQ[k], νQ[j]) in a 2D exchange NMR spectrum and, in a 1D exchange NMR experiment, to the j-th peak intensity of the spectrum in which initially the k-th peak is selectively excited. In ideal selective-excitation experiments, the initial matrix aˆ (0) is given by ajk(0) ) δjk. Rˆ 1Q is the quadrupolar-relaxation rate matrix, of which the jk component (j) (j) is given by (Rˆ 1Q)jk ) δjk/T1Q is the quadrupole, where T1Q order relaxation time of the j-th site. (j) For simplicity, we assume that T1Q is the same for all the (j) three sites. T1Q may not influence the exchange curves except

(j) in the low-temperature region, where 1/T1Q becomes comparable to Wjk. We treated the 15 parameters ajk(0), W21, W31, W32, p3/p1, p3/p2, and T1Q as unknowns in the least-squares fitting. We measured the mixing-time dependence of the quadrupoleorder exchange NMR spectra with the single-crystal orientation φ ) 167° at 193 K. While the selectively excited quadrupoleorder 1D exchange spectra were simultaneously obtained for all sites by the Hadamard method, only those for site 3 are shown for some mixing times in Figure 8, and the area intensities of all ajk peaks are plotted as a function of the mixing time in Figure 9. In the spectrum for the shortest mixing time in Figure 8a, the antiphase doublet with the largest splitting (site 3) is selectively excited. As the mixing time tm is increased, the intensities of the other doublets become stronger. The doublet with the middle splitting has the opposite sign compared

15938 J. Phys. Chem., Vol. 100, No. 39, 1996

Kubo et al.

Figure 8. 2H Hadamard quadrupole-order exchange NMR spectra for a single-crystal of C6Cl5CD3 at 193 K with the orientation φ ) 167° and with the mixing time tm ) (a) 2 ms, (b) 30 ms, (c) 80 ms, and (d) 300 ms. Only selective-excitation spectra of site 3 are shown here. The length of a Gaussian selective π-pulse was 400 µs. Eight free induction decays were accumulated for each measurement.

matrices are diagonal and their jj components are given by Ω ˆ jj (j) ) νQ[j] and (Rˆ 2)jj ) 1/T2 . B 1 is a vector given by B 1 ) [1 1 1] and Eˆ is a unit matrix. We used the equilibrium population given by eq 4 with the energy differences E2 - E1 ) 0.82 kJ mol-1 and E3 - E1 ) 0.7 kJ mol-1 for simulating the spectra in Figure 4. The quadrupole frequencies νQ[j] at each temperature were calculated by using the values at 208 K and the reduction factor of eq 6. The three parameters W21, W31, and W32 were determined to obtain the best agreement of the simulated and the experimental spectra for the three orientations of φ ) 13°, 108°, and 167°. The simulated spectra for φ ) 167° are shown in Figure 4 by dotted lines, well reproducing the experimental spectra. Thus, the rates Wjk were determined in the temperature range from 235 to 282 K. Figure 10 shows the angular dependence of T1Z and T1Q at four temperatures between 305 and 361 K. Each relaxation curve could be fitted by a single exponential decay with a leastsquares error of 2% for the relaxation time. The relaxation times can be calculated, based on the pseudo-6-site reorientation model expressed by eq 9 as follows:24-26,50,51

1/T1Z ) (πνQ3 )2/6{J1(ω0) + 4J2(2ω0)}

(13)

1/T1Q ) (πνQ3 )2/2J1(ω0)

(14)

with

Jq(qω0) ) (-1)qRe{∫0 [D2* q0(ΩLfP(τ)) ∞

2* 2* -iqω0τ 〈D2* dτ} q0〉][D-q0(ΩLfP(0)) - 〈D-q0〉]e 6

)-

2* 2 pj{D2* ∑ q0(ΩLfP[k]) - 〈Dq0〉}{Dq0(ΩLfP[j]) j,k)1

ˆ (q2ω02Eˆ + W ˆ 2)-1]kj (15) 〈D2q0〉}[W

Figure 9. Mixing time tm dependence of the area intensities ajk of the three doublets in the 2H Hadamard quadrupole-order exchange NMR spectra observed at 193 K with the single-crystal orientation φ ) 167°. The normalized peak intensities a1k, a2k, and a3k for the three sites are shown by ], 2, and O, respectively. The solid lines are the leastsquares fitting curves.

with the other doublets, showing that the quadrupole coupling of the middle doublet is opposite in sign to the others. The lines in Figure 9 were calculated from eq 10 using the best fit parameters. The theoretical curves agree very well with the experimental results. It is not necessary to introduce a distribution of W ˆ for reproducing the experimental data as is often assumed in motional models of glassy materials. The single-crystal quadrupole echo spectra in Figure 4, which reflect the pseudo-6-fold molecular reorientation, can be simulated by the three-site jump model using the following equation:22,23

S(ω) ) Re{1 B‚(Aˆ - iωEˆ )-1 exp(Aˆ τ) exp(Aˆ *τ)‚p b} + Re{1 B‚(Aˆ * - iωEˆ )-1 exp(Aˆ *τ) exp(Aˆ τ)‚p b} (11) with a pulse spacing τ and

Aˆ ) iΩ ˆ +W ˆ - Rˆ 2

(12)

where Ω ˆ and Rˆ 2 are the quadrupole frequency matrix and the spin-spin relaxation matrix, respectively. Both of these

where P(τ) denotes the principal axis system at time τ. ω0 is the Larmor frequency. 〈 〉 means the average over all the sites. Here the Wigner matrices D2q0(ΩLfP[j]) are given by eq 2. We used the Euler angles (R,β,γ) obtained from the rotation pattern, the quadrupole coupling constant at 157 K of 38.6 kHz, and the reduction factor of eq 6. The equilibrium populations were assumed to obey eq 4 with the energy differences E2 - E1 ) 0.82 kJ mol-1 and E3 - E1 ) 0.7 kJ mol-1. W21, W31, and W32 were determined by the least-squares fitting of eqs 13 and 14 to the experimental T1Z and T1Q, respectively. The program Powell52 was used for the minimization. Figure 11 shows the temperature dependence of T1Z and T1Q measured between 118 and 154 K, where the rates of the pseudo6-fold molecular reorientation are much slower than the spinlattice relaxation rates. The rotation of the CD3 group around its C3 axis may be the dominant relaxation mechanism in this low-temperature region. The activation energies of the CD3 rotation were determined for the three sites: 4.8 ( 0.5, 3.3 ( 0.6, and 3.5 ( 0.5 kJ mol-1 from T1Z, 5.3 ( 0.5, 2.5 ( 0.6, and 2.5 ( 0.6 kJ mol-1 from T1Q. Site 1, the most populated site, has an activation energy about twice as high as those in the other sites. All the values W21, W31, and W32 determined by Hadamard exchange NMR, spectral simulations, and the angular dependence of the spin-lattice relaxation times are plotted as a function of the reciprocal temperature in Figure 12. The exchange rate W obeys the Arrhenius law in the temperature ranges between 170 and 212 K, and between 235 and 361 K; the slope is slightly gentler in the former temperature region.

Glassy Crystal Pentachlorotoluene

J. Phys. Chem., Vol. 100, No. 39, 1996 15939

Figure 10. Angular dependence of T1Z (open) and T1Q (closed) of 2H in a single-crystal C6Cl5CD3 at 305 K(O, b), 322 K(4, 2), 338 K(], [), and 361 K(3, 1). The solid and the broken lines are the leastsquares fitting curves of T1Z and T1Q, respectively. Figure 12. Temperature dependence of the pseudo-6-fold reorientations rates, W2 (O), W31 (2), and W32 (]). The solid, the single-dot-dashed, and the broken lines are obtained by the least-squares fitting for W21, W31, and W32, respectively, which were separately fitted in the two temperature regions of 170-212 K and 235-361 K.

Figure 11. Temperature dependence of T1Z (open) and T1Q (closed) of 2H in a single-crystal C6Cl5CD3 with the orientation φ ) 167°. T1Z and T1Q of sites 1, 2, and 3 are shown by (O, b), (4, 2), and (], [), respectively. The solid and the broken lines are obtained from the least-squares fitting for T1Z and T1Q, respectively.

TABLE 1: Activation Energies (Ea) and Reorientation Rates at Infinite Temperature (W0)a reorientationb

(1 f 2)

(1 f 3)

(2 f 3)

EaLT/kJ mol-1 -1 log(WLT 0 /s ) EaHT/kJ mol-1 -1 log(WHT 0 /s )

49.9 ( 1.0 14.4 ( 0.3 59.7 ( 1.0 16.6 ( 0.2

45.6 ( 1.4 13.2 ( 0.4 57.8 ( 2.0 16.4 ( 0.4

48.5 ( 1.0 14.0 ( 1.0 59.3 ( 1.7 16.7 ( 0.3

a Reorientation rate W is given by W ) W0 exp{-Ea/RT}. W0 is the reorientation rate at the infinite temperature. b Reorientation from the j-th site to the k-th site is denoted by (j f k).

We determined the activation energies and the transition rates at the infinite temperature for these two temperature regions using the least-squares fitting method. The results are shown in Table 1. The activation energies are larger than the values in the other hexasubstituted-benzene (CH3)nCl6-nC6 with fewer chlorines which were obtained by early dielectric measurements.8 The reorientation rates at the infinite temperature WHT in 0 Table 1 are anomalously large. W0 has a tendency to be large when the activation energy Ea is also large.53 Some fluorides with reorientational motions show very large values of W0 and Ea, which may be interpreted by the large moments of inertia of the molecules or the large interatomic interactions between heavy atoms.53-57 Therefore, polychlorinated compounds such as pentachlorotoluene may also have large values of W0 and

Ea. Such an empirical relationship between W0 and Ea has not been theoretically proved; W0 is desired to be calculated by a time-dependent perturbation theory58 or a molecular dynamics method.59 If a phase transition exists, the temperature dependence of W can deviate from the Arrhenius law, influencing the determination of Ea from the slope. The slope of the log W vs 1/T curve is often more gentle in the higher temperature side of the transition temperature. At a first-order phase transition temperature, the log W vs 1/T curve often shows a discrete jump, while a second-order transition usually has a critical region in both the higher and the lower sides of the transition temperature, where the slope of the log W vs 1/T curve becomes steeper as the temperature approaches the transition temperature.20 Apparent W0 and Ea are larger near second-order phase transition temperatures than those in the regions far from the transition temperature. Such a temperature dependence of W cannot be seen in Figure 12, indicating that no phase transition exists between 170 and 361 K. We examined if there are any phase transitions above 350 K by high temperature DSC (differential scanning calorimetry), where no NMR measurements were made because of the fast sublimation of the sample. Below the melting point at 495 K, a small peak was detected at 489 K with a longer tail in the low-temperature side and a small transition enthalpy of 450 ( 100 J mol-1. The shape of the peak was almost independent of the sweep rate of temperature, so that this transition may be a second-order phase transition. A similar heat anomaly was found at 438 K by our measurements on an isostructural compound of 1,2,3,5-tetrachloro-4,6dimethylbenzene recrystallized from petroleum ether with a transition enthalpy of 900 ( 100 J mol-1. Since the transition temperature of 489 K is far from the temperature region where the NMR experiments were made, the influence of this phase transition on the log W vs 1/T curve shown in Figure 12 can be ignored. Since the relative orientations of neighboring molecules are temperature dependent, the intermolecular electric dipole-dipole interactions can be changed, depending on temperature. Pentachlorotoluene having the large electric dipole moment of 1.55 D,9 the intermolecular dipole-dipole interactions may cause a variation of the pseudo-6-fold rotation barrier of the molecule. This may be the reason of the observed change of the pseudo-

15940 J. Phys. Chem., Vol. 100, No. 39, 1996 6-fold rotation barrier. We will not proceed here with any quantitative estimate of the temperature dependence of W or Ea, since the atom-atom potential and the crystal structure must be known to calculate them. Conclusion The dynamics of molecular reorientations in crystals of pentachlorotoluene was studied by deuterium NMR experiments. The populations in the three molecular sites around its pseudo6-fold axis were determined as a function of temperature from single-crystal spectra, giving two distinct energy differences among the three potential minima. Below 150 K, the populations were found to be temperature-independent, indicating that the reorientational motions are frozen in nonequilibrium disorder with the populations of 47, 23, and 30% for the three sites. This fact gives evidence that the material is a glassy crystal. The pseudo-6-fold reorientation was investigated by measurements of Hadamard exchange NMR, spectra, T1Z, and T1Q. All the results could be interpreted by introducing three different reorientation rates between the nonequivalent sites. Each reorientation rate was found to obey Arrhenius law with slightly different activation energies between the higher temperature region (235-362 K) and the lower temperature region (170212 K). There was no need to introduce a distribution of reorientation rates. An Arrhenius behavior near the glass transition temperature was found in TlNO2 studied by Moriya et al. using calorimetric and dielectric measurements.6,60 Typical non-Arrhenius behavior was reported for cyanoadamantane by Amoureux et al. using dielectric measurements.61 However, these dielectric studies can not distinguish the different modes of reorientational motions such as 90° and 180° jumps. We believe that the deuterium exchange NMR method may provide more detailed insight for the origin of the distribution of reorientation rates in glassy crystals, distinguishing different molecular reorientations. The Hadamard exchange NMR method developed in this study is especially useful to efficiently collect many data on the mixing time and temperature dependence. In this study, we determined the energies of the potential minima at site 2 and site 3 to be 0.82 and 0.7 kJ mol-1, respectively, higher than at site 1. These energy differences may arise from the intermolecular Cl‚‚‚Cl, Cl‚‚‚CD3, and CD3‚‚‚CD3 van der Waals interactions and the electrostatic interactions.7 The activation energy of the CD3 rotation was found to be the largest at the most populated site 1, which suggests that this site is most tightly packed by the other molecules. The Cl‚‚‚Cl interactions have been found to be attractive and anisotropic;62,63 the present data can be useful to understand such weak intermolecular interactions which play an important role for forming the structures of molecular crystals.64 Acknowledgment. This work was supported by a Grantin-Aid for Science Research. The authors acknowledge Professor C. A. McDowell (the University of British Columbia) for informing them of ref 55, Professor N. Nakamura, Professor T. Eguchi (Osaka University) for their helpful discussions about W0 and informing them of refs 16 and 50, Professor T. Kawamoto and Mr. H. Nakao (Osaka University) for the hightemperature DSC measurements, and Dr. M. Kato (Kyoto University) for the high-temperature TG-DTA measurements. Appendix: Asymmetry Parameter of the 2H Quadrupole Coupling Tensor in the 3-Site Jump Model Here we describe the asymmetry parameter of the 2H quadrupole coupling tensor of perdeuterated pentachlorotoluene

Kubo et al. in fast reorientation around the pseudo-6-fold axis using the principal values at the low-temperature limit and the fractional populations of the three disordered sites. The quadrupole coupling tensor of site 1 at the lowtemperature (LT) limit is diagonal when it is expressed by the (x,y,z) coordinates fixed to the pentachlorotoluene molecule:

[

νQxLT Q ˆ1 ) 0 0

0 νQyLT 0

0 0 νQzLT

]

(A1)

The quadrupole coupling tensor of the other two sites are given by (60° rotations of Q ˆ 1 around the z axis:

[

Q ˆ 2,3 ) νQxLT/4 + 3νQyLT/4

(x3/4(νQyLT - νQxLT)

(x3/4(νQyLT

3νQxLT/4

-

νQxLT)

0

+

νQyLT/4

0

0 0 νQzLT

]

(A2)

The quadrupole tensor of the fast motion limit Q ˆ is given by the average of these three tensors:

ˆ 1 + p2Q ˆ 2 + p3Q ˆ3 Q ˆ ) p 1Q

(A3)

The eigenvalues of Q ˆ are given by

νQ3 ) νQzLT

(A4)

νQ1,2 ) 1/2(νQxLT + νQyLT) ( 1/2(νQyLT - νQxLT)x(p1 - p2/2 - p3/2)2 + 3/4(p2 - p3)2 (A5) The asymmetry parameter of the quadrupole coupling tensor is given by

ηQ ) (νQ2 - νQ1 )/νQ3 ) {(νQxLT - νQyLT)/2νQzLT}x(3p1 - 1)2 + 3(p2 - p3)2

(A6)

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