A deuterium NMR study of orientational order in tertiary-butyl bromide

JOURNAL

OF MAGNETIC

RESONANCE

4,

508-517 (1981)

A Deuterium NMR Study of Orientational Order in Tertiary-Butyl Bromide BRIAN

A. PETTITT,*

JOHN S. LEWIS,~ RODERICK E. WASYLISHEN,* WERNER DANCHURA,* AND E. TOMCHUK?

Departments

*Chemistry and ‘IPhysics, University of Winnipeg, Winnipeg, Manitoba R3B 2E9, Canada

of

AND

E. BOCK Department of Chemistry, University of Manitoba,

Winnipeg, Manitoba R3T 2N2, Canada

Received February 12, 1981 An analysis of NMR lineshapes and spin-lattice relaxation times for the 2H nucleus is reported for the liquid and solid phases I, II, and III of (CD&CBr. The main results are a discontinuous increase of the rotational correlation time on melting (with a concomitant increase of the Arrhenius parameter) and, for phase II, a demonstration of the feasibility of measuring order parameters on the different timescales probed by the absorption profile and the relaxation experiment. INTRODUCTION

As presently understood, the thermodynamic behavior of the condensed phases of t-butyl bromide, (CH,),CBr, can be summarized as solid III + (209 K, AS = 3.3R) solid II + (231.5 K, AS = 0.56R) solid I + (256 K, AS = 0.91R) liquid (I). Solid I is face-centred cubic (a0 = 8.78 ? 0.05 A at 238 + 3 K) (2) and exhibits the type of orientational disorder characteristic of the plastically crystalline state (3,4), whereas the crystal structures of the lower-temperature phases have not, to our knowledge, been reported. The dynamical aspects of the polymorphism of this substance are largely unexplored. The temperature dependence of the proton NMR linewidth (5) implies that the rotation of the molecule about its symmetry axis combined with internal rotation of the CH, groups progressively develops on the approach to the solid III += solid II transition, above which lines characteristic of complete disorder were observed. The observations, however, were not inconsistent with some motion of the molecular symmetry axis. In agreement with this, far-infrared absorption spectra (6) at 90 and 195 K indicate that large-amplitude orientational motion is absent at the lower temperature, where lattice modes are evident, but some slow reorientational process obscures these modes at the higher temperature. The spectra of the other two solid phases, being similar to those for the liquid, were attributed to mixed orientational and translational modes (6). 508 0022-2364/81/090508-10$02.00/O Copyright 0 1981 by Academic Press, Inc. Au tights of reproduction in any form reserved.

ORIENTATIONAL

ORDER

IN

t-BUTYL

BROMIDE

509

In this paper, we present the results of a 2H NMR study of the fully deuterated species of t-butyl bromide, undertaken particularly in the expectation that subtle changes associated with the weak II + I transition would be reflected in the 2H absorption profile, a well-known signature of partially ordered structures (7). EXPERIMENTAL

The sample of (CD,),CBr (purchased from Merck, Sharp & Dohme, Canada) was placed in a 7-mm (outer diameter) tube with a narrow constriction 9 mm from the bottom, degassed by repeated freeze-pump-thaw cycles, and sealed. This tube was placed in a IO-mm tube containing a methanol bath and insulation. The temperature was controlled within 2 1 K by a conventional N2 flow system and measured by inserting a glass-encapsulated thermocouple into the bath. The magnetic field was held constant with a proton lock and spectra were taken at 13.815 M.Hz (with a 7r/2 pulsewidth of 8.1 psec) using a Bruker SXP 4-100 pulsed spectrometer. Spectra for the liquid and solid phases I and II were obtained by the conventional quadrature FFT method, and for solid III by the quadrupolar echo technique (8). The number of scans accumulated varied from 4 in the liquid to 200 in phase III. Spin-lattice relaxation data were obtained by the inversionre.covery method and T,‘s were extracted from these using the three-parameter nonlinear fitting program of a Nicolet 1180 NTCFI package. THEORY

The dominant interaction of the ‘H nuclear spin in a molecule like (CD,),CBr, other than with the magnetic field of a spectrometer, is with the gradient of the local electric field-which, to the level of approximation of this treatment, can be regarded as having axial symmetry with respect to the C-D bond. Thus, an approximate Hamiltonian of the form (9) x = -l$,Z’O’ + f vg

;

c(m, m’)o~~,,,o(Sl)z’~‘z’m”

[II

m.m’=-1

can be used to describe the equivalent deuterons. Here vL is the Larmor frequency, vq = (3/4) (quadrupolar coupling constant = e’&/Zz), and I(“’ (m = - 1, 0, + 1) denotes a spherical component (in a space-fixed frame with the magnetic field along the z axis) of the deuteron (I = 1) dimensionless spin operator (10). The quantity D&(fi) denotes an element of the Wigner rotation matrix in the Euler angles R = ((II, /3, $ of a transformation (10) from the space-fixed frame to a principal-axis system of the electric field gradient (the z axis of which is along the C-D bond), and (9) c(m, m’) =

[2 + (m + m’)]![2 - (m + m’)]! l@ (1 + m)!(l - m)!(l + m’)!(l - m’)! I .

1

PI

To first order in perturbation theory, the energy levels of this Hamiltonian have associated NMR absorption frequencies at v+

=

VL

-c

%(4lTb(P)>,

[31

PEl-l-ITT

510

ET AL.

where (&t&(P)) = ((l/2)(3 co,+ /3 - 1)) is an average over all fluctuations of the angle p between the magnetic field and the C-D bond. If the processes underlying these fluctuations impart cylindrical symmetry to the order, then (4%(P)> = s’4%(Poh

[41

where p0 is the preferred orientation (with respect to the field) of the motional symmetry axis, and S’ is an effective order parameter (7). It appears from the proton NMR linewidths (5) and far-infrared studies (6) that, except in the lowtemperature region of phase III, internal methyl rotation and reorientation of the molecule about its symmetry axis (the C-Br bond) occurs. If it is assumed that these two processes are statistically independent and of at least threefold symmetry about their respective axes, then S’ = (l/9) < @,(8@) > =(S/9), where the order parameter, S, results from an average over fluctuations of the C-Br bond though angle Sp from a preferred orientation at angle PO, and it has been assumed that the C-C-Brand C-C-D angles are tetrahedral. These assumptions lead to the idealized absorption profile (7) f(v) = N 1

r dpo sin p0 +O”dv’p(/l,)6{v’ I --m

-

c+,-jI 0

VL

*

S’%$4w%J)g(~

-

v’),

[a

where p(po) is a distribution function characterizing the preferred orientations in the sample, g(v - v’) is a phenomenological broadening function, and N is an arbitrary scale factor. If it is further assumed that the energy shifts attributable to the residual quadrupolar interaction can be ignored in comparison with vL, then application of the standard semiclassical relaxation theory (I I ) based on Eq. [l] results in the following expressions for the relaxation times (T2 is included for completeness): 1

FM

- = 2 ~~(e~qQ/h)~~.J~~‘(wL) + 4J$*)(20L)} T, 2

and -

1

T2

= 2 n-*(e*qQlh)*{3J~*‘(O) 4

+ 5J~2’(~L) + 2J$*)(20L)}

where wL = 27~~. The spectral densities are defined by

Jgy(mdL) =I Ocdt cos (mwLt)(

d $$;;(~o)b~!o(f&))

[71

0

with where Sz, and fi, are the orientation fl at some arbitrary time and at t later. The correlation functions in Eq. [7] can be written in the form (12) ml;(~o)~~~o(Q))

= it lR2o(WI”)

- 1(Gilo(W)

121G)(0~

[91

where Cg’(O) = 1 and C~)(M) = 0, but the behavior at intermediate times depends on the detailed dynamics of the reorientational processes, about which very little

ORIENTATIONAL

ORDER

IN

t-BUTYL

BROMIDE

511

can be stated with confidence at present. A simplification, attractive for its tractability, results from assuming that C$‘(t) = C(‘)(t) for m = 0, 1, 2 and that the dominant reorientational processes are faster than tic1 (extreme narrowing). In such a case, it is a relatively straightforward matter to show that 1 -=-

3

3 +?(ezqQlh)2(1 - S”) 1 + 2 S” sin* PO T* l 1 2

T,

[W

and .-1 = 3 n*(e*qQ/h)*(l

- S”) 1 + t S”(coG PO + 1) 72r Wbl 1 I where r2 = J; C’*‘(t)& is the correlation time, and S” is defined as in Eq. [4] but distinguished from S’ in that the statistical average is strictly over those fluctuations governing the decay of C(*)(f). These equations imply that deuterons in molecules with their symmetry axes at different angles from the magnetic field relax at different rates. Under conditions of their applicability, qualitative inspection of Eq. [lOa] in conjunction with Eqs. [3] to [5], ignoring the effects of broadening, indicates that only the center of the spectrum (magic angles d$,(&,) = 0) and the wings ((l/2) 5 c&$,&J s 1; i.e., the shoulders of Fig. 2) can be characterized by individual TX’s, whereas other orientations contribute two distinct T,‘s to each frequency within the bandshape between the center and the w:ings (i.e., the peaks of Fig. 2). However, this theory, which is similar to that us,ed in some liquid crystal and lipid bilayer studies (22,13), is reasonable only for weakly ordered states because of the assumption of simple and rapid decay of the correlation functions. We should thus expect weak order to have negligible effect on the relaxation times and that the spectrum would retain its shape during recovery from an inverting pulse. For states of high order and slow reorientation, the analytical simplification of the correlation functions by the methods of classical probability theory is difficult (and of questionable validity), but the nature of relaxation in ordered phases is poorly enough understood at present to warrant investigations based on the simplest assumptions consistent with experimental observations. T2

2

RESULTS

AND

DISCUSSION

Liquid and Solid 1 T, values throughout phase I and at temperatures up to 60 K above the melting point, for which sharp lines characteristic of complete orientational disorder were observed, are listed in Table 1. Arrhenius behavior of the corresponding correlation times calculated via Eq. [lOa] using S” = 0 and (e*qQ/h) = 172 kHz [an assumed value, based on data for (CD&CC1 (14)] is illustrated in Fig. 1. These data exhibit the unusual feature that the reorientation of the C-D bond appears to be retarded by the melting process, because the Arrhenius parameter (E,) increases from 1.7 to 1.9 kcaYmole and the correlation times in solid I on the approach to melting are shorter than those in the liquid just above the melting region. The same effect was observed in PD, by Boden and Folland (15), and we are inclined to agree with them that an alternative explanation in terms of an

1.366 1.260 1.158 1.038 0.929 0.889 0.884

291 282 275 268 260 256 255

f 5%

T, (set)

1.728 1.560

Liquid

313 r 1 302

T W

1

248 245 242 238 235 233 232

252 f 251

T W) 1 0.8% 0.879 0.829 0.781 0.750 0.720 0.713

k 5%

Tl (s-3

I

0.924 0.944

Solid

225 224 223 221 216 214 212 210

231 + 1 228

T W

0.225 0.203 0.191 0.181 0.147 0.137 0.124 0.113

0.254 0.236

2 5%

T, (set)

Solid

II

0.0033 0.0035 0.0036 0.0037 0.0039 0.0040 0.0042 0.0044

0.0031 0.0032

k 0.0001

S’

198

208 k 1 203

T W

2.23

1.82 1.92

+ 5%

T, (msec)

Solid

III

*H SPIN-LATTICE RELAXATION TIMES, AND EFFECTIVE ORDERPARAMETERS FROM SPECTRAL SIMULATION,FOR(CD&CBI

TABLE

0.090

0.090 0.091

k 0.001

S’

z 9 .r

d cl

z

ORIENTATIONAL 30

ORDER IN t-BUTYL

513

BROMIDE

r

10 9 a 7i

3.2

3.6

4.4

4.0 10e3

4.8

TCKI-l

FIG. 1. Temperature dependence of ~~(1 - S’*) for the C-D bonds in (CD,),CBr. liquid (L) and phase I (S,), and is defined by Eq. [lOa] for phase II (S,,).

S” = 0 for the

increase in the electric field gradient on melting is less likely. However, the large, polarizable electron density on the bromine makes it difficult to rule out an intermolecular effect on this parameter. This behavior is not observed in (CD&Cl (14), for which the relaxation times appear to be continuous across the melting transition with E, = 1.5 kcal/mole. It is worth noting that molecular dynamics calculations on the rough sphere model at its melting density predicted greater rotational freedom in the solid than in the liquid (16). This model allows, in a crude way, for exchange of linear and angular momentum by instantaneous, binary collisions, but it would be foolhardy to press it into quantitative service to account for the effect observed here. A simple comparison of the distance between molecular centers [6.2 A at 238 K (2)] and any reasonable hard sphere diameter (>7 A) indicates that steric effects are important (which is true of plastic crystals and liquids in general) and that motion involving translational-rotational coupling is probably significant. Little is known about such processes, but they may be operative in phase I of (CH,),CCN (27) and phase II of (CH,),CCl (18), in both of which the molecular symmetry axes are highly ordered. Any realistic model for the rotational dynamics of this molecule requires at 1e:ast three characteristic times on which 72 depends-for reorientation of and about the symmetry axis and for reorientation of the methyl groups. A correlation function, Cg’(t), for the C-Br bond reorientation was constructed from light-

514

PETTITT ET AL.

scattering experiments on the liquid at 298 K by Constant and co-workers (29). It is exponential for t z 0.5 psec with distinct inertial effects at shorter times, and an approximate integration of Cf’(t) gives 72 = 1.3 psec, which is close to the value (1.5 psec) for the C-D bond (fig. 1). Solid II

The spectra of phase II exhibit the Pake doublet structure of a polycrystalline distribution with temperature-dependent splittings around 400 Hz, indicating that the C-D bonds are weakly ordered in a space-fixed coordinate system. Satisfactory fits to these spectra were obtained by use of Eq. [5] with a uniform distribution (p = a constant), as can be judged from the example illustrated in Fig. 2. This spectrum was obtained from 100 scans of the NMR signal after the sample had been rapidly cooled from the liquid into phase III and heated to 224 K. It was found that slow cooling from the liquid and phase I resulted in irreproducible features, amounting to a few percent of the intensity, particularly in the central part of the spectrum. This effect is well known and can be used to study the nonuniform distribution of domains in a crystal formed by freezing a liquid in an NMR tube (20). The influence of the sample’s thermal history was checked carefully during two cooling-and-heating ‘runs and found not to be responsible for the important spectral parameters. The theoretical spectrum in Fig. 2 corresponds to S’ = 0.0035 and Gaussian broadening with rms width equal to 22 Hz. Experience with the simulation procedure at various temperatures suggested that a combination of Lorentzian and Gaussian broadening would give a better fit, but S’ is insensitive to the details of the broadening mechanism. Effective order parameters (S’) determined by this procedure are listed in Table 1, from which it can be observed that the solid I + solid II transition results in the onset of weakly preferred orientations of some molecular axis (almost certainly the axis passing through the C-Br bond), a preference that slightly strengthens before the solid II + solid III transition intervenes. According to the simple theory summarized by Eqs. [ lOa] and [lob], order this weak should not have a significant effect on the lineshape during recovery from an inverting pulse; but it was observed that the shoulders were slower to recover than the center, and slower still than the peaks. This observation is illustrated in Fig. 2, where the relaxation rate (l/T,) is shown for bands 20 Hz wide as a function of the frequency at the band center. This rate pertains to the recovery of components of the longitudinal magnetization arising from molecules with their preferred orientations within approximately 10” of a given angle from the magnetic field. Also shown is a plot of Eq. [lOa] for the unbroadened analog of the observed spectrum, with 72 = 13 psec and S” = 0.37 chosen to give a good fit on the shoulders and at the center. In the region of the spectrum characterized by two relaxation rates, the recovery was observed to be adequately described by a single exponential, and from the figure it can be seen that the larger rate dominates (because of the large density of orientational states around &, = r/2). Thus, the interesting implication emerges that the degree of order on the timescale probed by the relaxation experiment is much greater than that on the timescale of complete secularization of the 2H energy levels. It is reasonable to assume

ORIENTATIONAL

I I I

ORDER IN t-BUTYL

515

BROMIDE

.

-600

+ (u-u,)

600

Hz

FIG. 2. Experimental NMR absorption and spin-lattice relaxation rate profiles for the 2H nuclei of (CD&CBr at 224 K (Solid II) compared with theoretical profiles. Dotted line in upper trace is for Eq. [5] with S’ = 0.0035 (Gaussian rms broadening parameter = 22 Hz). Dots in lower trace are l/T, for 20-Hz bands in the experimental spectrum and the broken line is Eq. [lOa] for the unbroadened analog of the experimental spectrum with Q = 13 psec and S” = 0.37.

that (CD,),CBr resembles (CH&XI (18) and (CH&CCN (17) to the extent that internal methyl rotations are much slower than reorientation of the molecule about its symmetry axis. It is plausible to posit this latter motion as being responsible for the 13-psec correlation time and to regard S” = 0.37 as reflecting a strong hindrance to motion of the symmetry axis itself. Slower methyl jumps combined with the axial rotation would lead to S’ = l/9 (as indicated under Theory), and to e:xplain the apparent order observed in the splitting pattern of the lineshape (S’ = 0.0035) it is necessary to assume the presence of an additional slow process leading to large-amplitude, weakly anisotropic fluctuations of the symmetry axis (S = 9s’ = 0.032, or (cos2 Sp) 1’2 = 0.6). The timescale of this motion must be in the range between that of the fast motion observed in the relaxation and the

516

PETTITT

ET AL.

reciprocal of the rigid lattice linewidth (lo-l1 to 10F6 set) in order to cause the line narrowing. It is most likely associated with jumps of the C-Br axis between preferred orientations determined by the molecular and crystallographic symmetries, in which case 5’ is an index on the extent to which the distribution of preferred orientations is weighted by intermolecular forces along a particular crystallographic axis. Because this procedure for determining both r2 and S” from the relaxation data is time-consuming (and requires a very good spectrum), we have not applied it to the full temperature range of phase II. Instead, we note that T, for the integral over the absorption profile is equal to T, at the center and report the former in Table 1. Equality of these two parameters is consistent with Eq. [ lOa]. The quantity calculated from this T, by the procedure used for the liquid and phase I (where integral and peak height T,‘s were within 1% of each other) is ~~(1 - P). The temperature dependence of this parameter, which can be characterized by E, = 4.0 kcaYmole, is shown in Fig. 1. Solid ZZZ Spectra recorded at temperatures down to 10 K below the II + III transition were split by 10.8 kHz, and the shoulders were less pronounced than those on the narrow spectra of phase II, because of distortions associated with the quadrupolar echo technique (21). Values of S’ determined by spectral simulation and integral T,‘s at three temperatures are listed in Table 1. The order parameter, S = 95’ = 0.81 can be interpreted as reflecting the high ordering of the C-Br bond in this state of the material. The increase of T1 with decreasing temperature implies that the effective reorientational processes are slower than w:l. CONCLUSIONS

The quantitative procedures used in this work combine rotational correlation times and order parameters as means of obtaining information about the effective ordering potential in condensed (CD,),CBr; and it is hoped, insofar as 2H NMR is applicable, that other molecular crystals with weakly ordered states can be studied in this manner. This general technique is familiar from studies of the various liquid crystalline mesophases (12, 13), but the variation of T, within a powder pattern (Fig. 2) has not been a feature of such investigations. It should be interesting to find out if this behavior is exhibited by a variety of partially ordered structures. Other NMR experiments may be of value in sorting out the rotational dynamics of this substance, particularly use of single crystals or variously proportioned ‘H and 2H combinations, but the liquid-like timescale of the axial rotation is likely to preclude a frequency dependence or a distinction between T, and T,,. Longitudinal relaxation of the central 13C was briefly examined in this work but not reported because slow, nonexponential recovery of broad lines led to severe problems of interpretation. Other techniques, such as those of ir, Raman, and Rayleigh-Brillouin spectroscopy, are likely to be more helpful in this matter; and when it is recalled that the crystal structures of phases II and III are unknown,

ORIENTATIONAL

ORDER

IN

t-BUTYL

a combined neutron diffraction and quasi-elastic particularly worthy of consideration.

517

BROMIDE

scattering

experiment

seems

ACKNOWLEDGMENT

This project was supported Re,search Council of Canada.

financially

by grants from the National

Science and Engineering

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18. J. C. FROST, A. J. LEADBETTER, AND R. M. RICHARDSON, Phil. Trans. R. Sot. London B 290, 567 (1980). 19. M. CONSTANT, R. FAUQUEMBERQUE, AND P. DESCHEERDER, J. Chem. Phys. 64,667 (1976). 26'. R. HENTSCHEL, J. SCHLITTER, H. SILLESCU, AND H. W. SPEISS, J. Chem. Phys. 68, 56 (1978). 21. M. BLOOM, J. H. DAVIS, AND M. 1. VALIC, Can. J. Phys. 58, 1510 (1980).