J. Phys. Chem. 1985,89, 545-550
I
\
m
= 10,
8
CSDSI
= 7 ~ 1 0 -mol ~ dm-3
I
I 0
1
2
4
6
CHITCI I
8
1 10
mol dm-3
Figure 4. Dependence of micelle concentration on dye concentration. [RhdG] is 1/10 of [HITC].
(at 25.0 f 0.1 0C)34and the aggregation number reported to be 62.35 The micelle concentrations obtained from this time-resolved experiment are in good agreement with the calculated value above mol dm-3. [SDS] = 8 X Premicellar Region (below the CMC). The concentration of micelles does not vanish at the cmc, as shown in Figure 3. Moreover, the presence of a small amount of micelles is indicated below the cmc (Le., in the premicellar region). The occurrence of dye-rich induced micells has been reported below the cmc for dye-surfactant systems when the dye and surfactant carry opposite ~ h a r g e ~ ~ as ~ -well ~ ~as *for~ a~neutral * ~ ~ molecule-surfactant - ~ ~ The Occurrence of dye-rich induced micelles is induced by the presence of the dye at surfactant concentrations where the (34) Sato, H.; Kusumoto, Y.; Nakashima, N.; Yoshihara, K. Chem. Phys. L e f f . 1980, 71, 326. (35) Fendler, J . H.; Fendler, E. J. "Catalysts in Micellar and Macromolecular Systems"; Academic Press: New York, 1975. (36) Mukerjee, P.; Mysels, K. J. J . Am. Chem. SOC.1955, 77, 2937. (37) Kusumoto, Y.; Sato, H. Chem. Phys. Left. 1979, 68, 13. (38) Atik, S.; Singer, L. J . A m . Chem. SOC.1979, 101, 6759. (39) Sato, H.; Kawasaki, M.; Kasatani, K. J . Phofochem. 1981, 17, 243. (40) Sato, H.; Kawasaki, M.; Haga, M.; Kasatani, K.; Ban, T.; Suenaga, H.; Kitamura, N. Nippon Kagaku Kaishi 1984, 51. (41) Mandal, K.; Demas, J. N. Chem. Phys. Letf. 1981, 84, 410. (42) Law, K. Y. Phofochem. Photobiol. 1981, 35, 799.
545
surfactant when alone does not micellize or micellizes to a lesser extent. Figure 4 shows the effect of dye concentration on the micelle concentration below the cmc. The donor dye (Rh-6G) concentration was 1/10 of that of the acceptor (HITC). The micelle concentration increases linearly with dye concentration. Baxendale and R ~ d g e r sreported ~ ~ , ~ ~for the R U ( ~ ~ ~ ) , ~ ' - Ssystem D S that the total concentration of dye-rich induced micelles ("clusters" in their notation) was proportional to the dye ( R ~ ( b p y ) ~ ~ ' ) concentration. The dye concentrations they used (2.5 X 10-5-l.0 X mol dm-3) were much higher than those in this paper. Thus, the portion of micelles which increase in proportion to the dye (donor plus acceptor) concentration in Figure 4 must be due to dye-rich induced micelles. The micelle radius seems to be nearly independent of SDS concentration both above and below the cmc (Figure 2a). Atherton et al.45 and Ban et al.46concluded that dye-rich induced micelles are smaller than ordinary micelles. The difference is probably due to the high dye concentrations used in these studies. Incidentally, we stress that the very low concentration of micelles at SDS concentrations below the cmc can be measured quantitatively by the techniques reported here. Acknowledgment. We are grateful to Prof. Keitaro Yoshihara, Institute for Molecular Science, for helpful discussions, to Mr. Masanori Tanaka for his assistance with the experiments and calculations, and to the Instrument Center, Institute for Molecular Science, for the use of the spectrofluorometer. This work was supported partly by the Joint Studies Program (1982-83) of the Institute for Molecular Science, and by a Grant-in-Aid from the ministry of Education, Science, and Culture of Japan. Supplementary Material Available: The proof for p ( R ) = cuR in model I and the details of the Monte Carlo calculation in model I1 (4 pages). Ordering information is given on any current masthead page. (43) Baxendale, J. H.; Rodgers, M. A. J. Chem. Phys. Leu. 1980, 72,424. (44) Baxendale, J. H.; Rodgers, M. A. J. J . Phys. Chem. 1982,86,4906. (45) Atherton, S. J.; Baxendale, J. H.; Hoey, B. M. J . Chem. SOC.,Faraday Trans. 1 1982, 78,2167. (46) Ban, T.; Kasatani, K.;Kawasaki, M.; Sato, H. Phofochem. Phofobiol. 1983, 37, 131.
Nuclear Spin Relaxation of Biphenyl in Nematic and Smectic Liquid Crystalline Media Peter R. Luyten, Free University of Amsterdam, Amsterdam, The Netherlands
Regitze R. Vold, and Robert L. Vold* Department of Chemistry, University of California, San Diego, La Jolla. California 92093 (Received: July 13, 1984; In Final Form: October 2, 1984) Deuterium relaxation data at 9.2 and 38.4 MHz are reported for ortho/meta and para deuterons of biphenyl-dlo in a liquid crystalline mixture consisting of 26 wt % hexyloxycyanobiphenyl and 74 wt %. Spectral densities J l ( w o ) and J2(2w0)were determined at both Larmor frequencies for all deuterons in both the nematic and smectic A mesophases. No reentrant nematic behavior was observed. Biphenyl was treated as a symmetric rotor, and two rotational correlation times 7,, and 7,, were derived from the J2 data. The J1data for para deuterons provide information about director fluctuations in both the nematic and smectic A phases. Parameters for both rotational motion and director fluctuations show no pronounced discontinuity at the nematic smectic A transition.
-
Introduction Molecular motion in fluids with long-rang order is far more complex than in simple isotropic fluids. Time comelation functions for molecular reorientation in ordered fluids decay to nonzero average values which are determined by equilibrium order parameters, and the shapes and decay rates of the correlation 0022-36~4/85/2089-0545$01.50/0
functions depend further upon molecular order as expressed in appropriate conditional probability distribution functions. Simple models of the dynamic effects of ordering include early calculations based on strong c~llisionl-~ and rotational diffusion in a cone.44 (1) Glarum, S. H.; Marshall, J. H. J . Chem. Phys. 1967, 46, 55.
0 1985 American Chemical Society
546 The Journal of Physical Chemistry, Vol. 89, No. 3, 1985
More elaborate calculations involve solving equations for small step rotational diffusion in presence of an anisotropic restoring I pseud~-potential.~-l Modified correlation functions for molecular reorientation are not the only consequence of long-range orientational order; more subtle contributions arise from collective fluctuations involving large numbers of molecules. The most important process of this type involves fluctuations in the instantaneous orientation of the director.12 The major features of director fluctuation theory for nematic m e ~ o p h a s e s ' ~have J ~ been amply confirmed by spin relaxation measurements on protons13-17 and, more recently, on ~-~~ deuterons.18-zoThere is a growing body of e v i d e n ~ e *suggesting that collective modes similar to nematic director fluctuations persist in smectic mesophases, but the precise origin of smectic director Recent theoretical investigation^^^.^ fluctuations is less clear.23,25*z6 of director fluctuations have not been concerned with this question, focusing instead on the breakdown of hydrodynamic models applied to collective modes with spatial wavelengths of molecular dimension. Deuterium spin relaxation is well suited for stringent tests of current models of motion in ordered phases. The spectra are simple enough to permit detailed relaxation analysis using suitably m ~ d i f i e dRedfield ~ ~ . ~ ~ theory, and relaxation is dominated by the exclusively intramolecular quadrupolar mechanism. By means of appropriate combinations of pulse techniques, enough relaxation data can be gathered to determine individual spectral density The subscript M refers to a projection parameters index of second-rank spherical harmonics, and the second-rank tensorial nature of the quadrupolar relaxation Hamiltonian implies that J I is monitored at the Larmor frequency wo while J2 is monitored at 2u0. It is important to realize that Ji is not generally equal to J 2 even in the extreme narrowing limit where neither parameter is frequency d e ~ e n d e n t . ~The ~ , ~symmetry ~ properties JM(~).31932
(2) Luckhurst, G. R.; Zannoni, C. J . Magn. Reson. 1976, 23, 275. (3) Jacobsen, J. P.; Bildsoe, H. K.; Schaumburg, K. J . Magn. Reson. 1976, 23, 153. (4) Warchol, M. P.; Vaughan, W. E. Ado. Mol. Relaxation Interact. Processes 1978, 13, 317. ( 5 ) Wang, C. C.; Pecora, R. J . Chem. Phys. 1980, 72, 5333. (6) Lipari, G.; Szabo, A. J . Chem. Phys. 1981, 75, 2971. (7) Nordio, P. L.; Busolin, P. J . Chem. Phys. 1971, 55, 5485. (8) Nordio, P. .; Rigatti, G.; Segre, U. J . Chem. Phys. 1972, 56, 2117. (9) Polnaszek, C. F.; Freed, J. H. J . Phys. Chem. 1975, 79, 2283. (10) Freed, J. H. J . Chem. Phys. 1977,66, 4183. (1 1) Black, E. P.; Bernassau, J. M.; Mayne, C. L.; Grant, D. M. J . Chem. Phys. 1982, 76, 265. (12) Pincus, P. Solid State Commun. 1969, 7, 415. (13) Doane, J. W.; Tarr, C. E.; Nickerson, M. A. Phys. Rev. Lett. 1974, 33, 620. (14) Ukleja, P.; Pirs, J.; Doane, J. W. Phys. Rev. A 1976, 14, 414. (1 5) Wade, C. G. Ann. Rev. Phys. Chem. 1977, 28, 47. (16) Graf, V.; Noack, F.; Stohrer, M. Z . Naturforsch, A 1977, 32A, 61. (17) Visintainer, J. J.; Dong, R. Y.;Bock, E.; Tomchuk, E.; Dewey, D. B.; Kuo, A.-L.; Wade, C. G. J . Chem. Phys. 1977, 66, 3343. (18) Vold, R. R.; Vold, R. L.; Szeverenyi, N. M. J. Phys. Chem. 1981,85, 1934. (19) Dickerson, W. H.; Vold, R. R.; Vold, R. L. J . Phys. Chem. 1983.87, 166. (20). Barbara, T. M.; Vold, R. R.; Vold, R. L.; Neubert, M. E. J . Chem. Phys., In press. (21) Dong, R. Y. J . Chem. Phys. 1981, 75, 2621. (22) Blinc, R.: Vilfan, M.; Luzar, M.; Seliger, J.; Zagar, V. J . Chem. Phys. 1978, 68, 303. (23) Mugele, T.; Graf, V.; Wolfel, W.; Noack, F. Z . Naturforsch., A 1980, 3SA, 924. (24) Dong, R. Y.; Lewis, J. S.; Havelock, M. E.; Tomchuk, E.; Bock, E. J . Magn. Reson. 1981, 45, 223. (25) Brochard, F. J . Phys. (Orsay, Fr.) 1973, 34, 41 1. (26) Blinc, R.; Luzar, M.; Vilfan, M.; Burgar, M. J . Chem. Phys. 1975, 63, 3445. (27) Zientara, G. P.; Freed, J. . J . Chem. Phys. 1983, 79, 3077. (28) Faber, T. E. Proc. R . Soc. London, Ser. A 1977, 353, 277. (29) Werbelow, L. G.; Grant, D. M. Ado. Magn. Reson. 1977, 9, 190. (30) Vold, R. L.; Vold, R. R. Prog. N M R Spectrosc. 1978, Z2, 79. (31) Jacobsen, J. P.; Schaumburg, J . Magn. Reson. 1976, 24, 173. (32) Vold, R. R.; Vold, R. L. J . Chem. Phys. 1977, 66, 4018.
Luyten et al. associated with different values of M imply that JIis much more sensitive than J2 to fluctuations in a restricted angular space, and it follows that the relative magnitudes of J 1 and J z can provide information about restricted motion which transcends any particular model. In this paper we report a deuterium relaxation study of biphenyl-dlo in a liquid crystalline mixture which exhibits both nematic and smectic A mesophases. The order parameter for the long axis of this solute is large, S,, 0.5, and it is of interest to see whether or not the rotation can be described as rotational diffusion in a symmetric restoring potential even though the static -0.1. The relaxation ordering is slightly asymmetric, S, - S, behavior of the ortho and meta biphenyl deuterons is insensitive to collective modes,34 but the relaxation behavior of the para deuterons can be used to investigate how director fluctuations in the smectic phase differ from nematic director fluctuations.
-
-
Theory If the solute motion is viewed as a combination of fast molecular reorientation and slow fluctuations of director orientation, the following expressions pertain for spectral density parameterslo For para deuterons, which lie on the molecular z axis
Here T~ is the correlation time for reorientation of the molecular z axis, and we treat biphenyl as a symmetric top. This is rea-
sonable in view of the molecular shape, which is characterized by a dihedral angle of about 30' between the aromatic planes in liquid crystalline solvent^.^^*^^ For ortho and meta deuterons, which lie at 60' to the z axis, one obtains
where the correlation times
T~
and
72
are given by
an
T ~ is , the correlation time for reorientation about the z axis. The coefficients K~~ in eq 1 and 2 account for effects of static order. They may be obtained by solving the rotational diffusion equation including torque produced by a model restoring potentiaL7-I1 In what follows we assume a Maier-Saupe potential. While scarcely exact (it cannot account for the observed values of S, - S,,,,) this potential is consistent with the approximation of treating biphenyl as a symmetric rotor. K~~ and K~~ can then be obtained directly from convenient tables given by Lin and Freed.37 Our definitions differ slightly from those of Freed and co-workers; we normalize K~~ to unity (rather than 1/5) in the limit of zero order, and we assign the first subscript (M) to projection onto space fixed axes and the second ( K ) to projection onto molecule fixed axes. Determining K factors with K different from zero (needed for off-axis deuterons) is more complicated. Polnaszek and Freed9 have shown that these K factors depend on the anisotropy ratio R = T ~ / T as ~ , well as the restoring potential, and they provide perturbation formulae valid for all values of R but only for low values S, C 0.15. To obtain K factors for the larger values of S,, observed for biphenyl, we have solved the rotational diffusion equation for a symmetric rotor using a method suggested by Moro
(33) Vold, R. R.; Vold, R. L. In "Liquid Crystals and Ordered Fluids"; Griffin, A. C., Johnson, J. F.,Eds.; Plenum: New York, 1984; Vol. 4, 561. (34) Luyten, P. R.; Vold, R. R.; Vold, R. L. J . Magn. Reson., in press. (35) Niederberger, W.; Diehl, P. Mol. Phys. 1973, 26, 571. (36) d Annibale, A.; Lunazzi, L.; Boicelle, A. C.; Macciantelli, D. J . Chem. SOC.,Faraday Trans. 2 1973, 1396. (37) Lin, W.-J.; Freed, J. H. J . Phys. Chem. 1979, 83, 379.
The Journal of Physical Chemistry, Vol. 89, No. 3, 1985 541
N M R of Biphenyl in Liquid Crystalline Media and Nordio3* with a restoring potential of the form
v(p)= CC,(COS" p - 1 )
(4)
n
where the coefficients c, are arbitrary. A one-term potential with n = 2 is equivalent to the symmetric Maier-Saupe form used by Polnaszek and Freed.9 Our computational procedures are described elsewhere in detail39 but we note here, as have that the entire formulation of eq 1 and 2 in terms of K factors which modify the usual W o e s ~ n e r - H u n t r e s sexpressions ~~~~ is valid only in the limit of extreme narrowing. In nematic mesophases, the director fluctuation contribution JIDF(wo) is given bylo I
20
I
1
30
40
50 T
+ tan-'
+
[ ( ~ W , / W ~ ) ~ /I)] ~
(7)
with
F+ = ( w c / w o ) f ( 2 w , / ~ 0 ) + ~ /1~
'C
in nominally identical liquid crystalline mixtures. Solid symbols refer to the sample used for 9.2-MHz relaxation measurements, and open symbols refer to the sample used for 38.4-MHz measurements. minor differences in order parameters, apparent at higher temperatures in the nematic phase, are ignorable for purposes of analyzing the relaxation data.
and the cutoff function U(w,/wo) is 1 U(w,/wo) = - In IF-/F+I + 2n { ( ~ W , / W ~ )' / ~ 1)
70
Figure 1. Temperature dependence of order parameters for biphenyl-dlo
where the amplitude factor A may be written
1 -[tan-' n
I
60
(8) (9)
Here X is a spatial wavelength, thought to be on the order of a molecular length,42 below which the medium cannot support collective fluctuations. When the Larmor frequency wo is much larger than the cutoff frequency o,,U tends to zero and director fluctuations no longer produce spin relaxation. The negative cross term between director fluctuations and molecular tumbling was first derived by Freed,Io who noted that there could be additional cross terms of similar magnitude. 7,K,and D in eq 5-7 stand for operational averages of anisotropic viscosities, elastic constants, and translational diffusion coefficients, respectively. Equations 4-7 are not supposed to apply to smectic phases. Because of smectic layer formation, the elastic constant K3 (corresponding to deformations which bend the layers) di~erges.4~~" When the "one-constant" approximations for K and 9 are replaced by more elaborate calculations,26 it is found that divergence of the bend constant K3 makes A in eq 6 go to zero. However, the ~ existence of smectic order parameters42has been s h 0 w n ~ ~to9lead, in certain limiting situations, to formulae with identical frequency dependence as eq 5-9 but a reduced value of A corresponding to "renormalized" elastic constants. Blinc and co-workers22p26have identified possible additional collective fluctuation modes in smectic phases. Equations 4-7 may fail near the N-SA phase transition where some elastic constants and viscosity coefficients di~erge.4~9" Other types of collective modes have been considered for smectic phase^,^^^^^ leading in certain limiting cases to spectral densities with a frequency dependence identical with eq 5-9.
Experimental Procedures A mixture consisting of 26 wt % hexyloxycyanobiphenyl and 74 wt % octyloxycyanobiphenyl (BDH Chemicals, Ltd, Poole, (38) Moro, G.; Nordio, P. L. Chem. Phys. Lett. 1983, 96, 192. (39) Vold, R. L.; Vold, R. R., to be published. (40) Woessner, D. E. J . Chem. Phys. 1962, 37, 647. (41) Huntress, Jr., W. T. Adu. Mag.Reson. 1970, 4, 1 . (42) DeGennes, P. "The Physics of Liquid Crystals"; Oxford University Press: London, 1974. (43) De Gennes, P. Solid State Commun. 1972, IO, 753. (44) Cladis, P. E. Phys. Rev. Lett. 1975, 35, 48.
England) with additional 5 mol % biphenyl-dlowas found by visual inspection with a polarizing microscope to exhibit the following S A (45.0 "C); S A K phase changes: I N (76.0 "C); N (10 "C).The reported transitions refer to the onset of the lowtemperature phase upon slow cooling; biphasic regions ca. 1 "C wide were observed at each transition. Neither visual inspection nor N M R spectra obtained after sample rotation in a 1.4-T electromagnet revealed any evidence for reentrant nematic behavior, even though the solvent mixture without biphenyl became reentrant nematic at ca. 36 "C. Two different samples were prepared for relaxation studies, both in 10-mm-0.d. N M R tubes. After repeated freeze-pump-thaw cycles, the tubes were sealed close to the liquid surface, leaving just enough room for thermal expansion of the solution. One sample was used for relaxation measurements at 38.4 MHz and the other for measurements at 9.2 MHz with apparatus described e l ~ e w h e r e . ~Figure ~ . ~ ~ 1 shows order parameters for each sample as a function of temperature, and the close correspondence in ordering of the two samples indicates that their respective compositions are sufficiently similar to permit a combined analysis of their relaxation behavior. The data in Figure 1 were obtained by analyzing N M R spectra as a function of temperature, calibrating before and after measurement with a built-in thermocouple near the sample coil. Thermal gradients in our probes are small, well under 0.01 0C:7 and the absolute temperature measurements are accurate to ca. 0.2 "C. Curves similar to Figure 1 were used in subsequent relaxation measurements to determine the sample temperature from the observed splitting. Procedures for measuring relaxation of ortho and meta deuterons, based on Jeener-Broekaert pulse sequences, are described e1~ewhere.j~ The technique yields simultaneous values for the &man relaxation rate RIZand the decay rate R,, of quadrupolar order, from which the two spectral density parameters Jl(wo)and J 2 ( b O )can be derived by means of the relations46
-
-+
J I ~ o=) R ~ Q / ~ C
-+
(1Oa)
where C = ( 3 ~ ~ / 2 ) ( e ~ q Q and / hthe ) ~quadrupole coupling constant ezqQ/h is 183 kHz for biphenyl deuterons.48 Rather different methods were used for the para deuterons, whose large quadrupolar splitting (ca. 100 kHz) precluded application of nonselective pulses in a Jeener-Broekaert sequence. (45) Vold, R. R.; Vold, R. . J . Magn. Reson. 1975, 19, 365. (46) Vold, R. L.; Dickerson, W. H.; Vold, R. R. J . Mugn. Reson. 1981, 43, 213. (47) Vold, R. L.; Vold, R. R. J . Magn. Reson. 1983, 55, 78. (48) Millett, F. S.; Dailey, B. P. J. Chem. Phys. 1972, 56, 3249.
548
The Journal of Physical Chemistry, Vol. 89, No. 3, 1985
Luyten et al. TABLE II: Relaxation h t n for Biphenyl Ortho/Meta Deuterons
TABLE I: Relaxation Data for Biphenyl Para Deuterons
2.922 2.941 2.970 3.006 3.016 3.062 3.110 3.133 3.161 3.214 3.235 3.255 3.266 3.281 3.321 3.376
45.6 f 1.5 47.0 f 0.5 46.8 f 1.2 49.3 f 0.8 47.5 f 1.5 53.2 f 2.3 58.4 f 0.6 61.3 f 0.4 64.0 f 0.6 72.1 f 0.5 75.5 f 0.7 78.9 f 0.9 81.1 f 1.0 83.9 f 1.0 91.4 f 1.1 103 f 1.0
56.7 f 2.2 56.9 f 2.1 60.7 f 2.0 66.6 f 1.8 73.4 f 3.1 78.3 f 2.1 84.7 f 2.1 88.6 f 4.2 89.5 f 2.5 101 f 3.0 103 f 4.0 111 f 5.1 115 f 5.1 114 f 8.5 133 f 4.6 148 f 3.4
40.6 f 2.0 40.5 f 1.9 43.9 f 1.8 48.6 f 1.6 55.5 f 2.8 58.2 f 1.9 62.8 f 1.9 65.7 f 3.8 65.9 f 2.1 74.6 f 2.6 75.3 f 3.8 82.2 f 4.6 85.1 f 4.6 83.2 f 7.6 98.7 f 4.1 109 f 3.1
12.8 f 1.0 13.6 f 0.5 12.6 f 0.8 12.7 f 0.6 10.2 f 1.1 12.3 f 1.4 13.7 f 0.6 14.5 f 1.0 15.8 f 0.6 17.7 f 0.7 19.2 f 1.0 19.2 f 1.2 19.6 f 1.3 21.5 f 2.0 21.4 f 1.2 24.7 f 0.9
2.951 2.998 3.025 3.046 3.095 3.120 3.139 3.147 3.158 3.178 3.193 3.219 3.244 3.273 3.291 3.329 3.355 3.368
67.0 f 6.3 74.2 f 4.1 79.6 f 4.1 82.7 f 3.5 89.4 f 7.9 94.3 f 3.3 95.4 f 2.2 99.9 f 5.2 99.9 f 3.3 99.1 f 5.1 113 f 5 118 f 3 125 f 4 132 f 5 140 f 4 158 f 5 166 f 6 185 f 7
9.2 MHz 104 f 8 125 f 10 136 f 8 139 f 8 157 f 10 152 f 7 157 f 6 155 f 12 171 f 5 171 f 6 173 f 8 186 f 9 197 f 11 221 f 9 232 f 10 266 f 14 291 f 11 307 f 13
78.0 f 7.3 95.8 f 8.6 104 f 8 106 f 7 121 f 9 115 f 7 120 f 6 117 f 11 131 f 4 131 f 5 130 f 7 140 f 8 148 f 10 168 f 8 177 f 9 203 f 13 233 f 10 234 f 11
14.3 f 3.9 13.5 f 3.1 14.0 f 2.9 15.3 f 2.7 14.8 f 5.0 18.7 f 2.5 18.2 f 1.9 21.2 f 4.0 17.6 f 2.1 17.2 f 3.1 24.5 f 3.4 24.5 f 2.7 25.8 f 3.2 24.3 f 3.6 26.1 f 3.2 28.8 f 4.3 27.6 f 4.4 35.1 f 5.0
Spectral densities calculated from nonselective relaxation rates R,(ns) and selective relaxation rates RIo(sel) according to eq l l a and l l b , with (e29Q/h)= 183 kHz.
Instead, selective inversion-recovery experiments were carried out by reducing the pulse power so that the 90’ pulse width (for one line of the quadrupolar doublet, 2 1 / 2 ~ l t=, 7r/2) was -200 ws. To avoid zerefrequency artifacts in the Fourier transform spectra, the carrier frequency was displaced several kHz from the peak, and audiofrequency modulation was applied during the pulse. No evidence of nonexponential recovery was observed, but the selective relaxation rates are nevertheless to be regarded as initial rates,’* given by
Nonselective inversion recovery experiments were performed with ca. 360-ws 180’ inversion pulses, audiomodulated at half the quadrupolar splitting frequency with a Wavetek Model 184 gated oscillator driving a double balanced mixer. The nonselective relaxation rate is given by the familiar expression 2 -2
Rl(ns) = T ( e 2 q Q / h ) 2 [ J 1 ( w 0+) 4J2(2wO)] ( l l b )
Results and Discussion Relaxation data are presented in Tables I and I1 for para deuterons and ortho/meta deuterons, respectively. The error limits for measured rates represent one standard deviation, and these uncertainties were propagated through eq 10 or 11 to yield the indicated error limits for the spectral densities. Below ca.40 O C strong dipolar coupling precludes determining reliable spectral densities for ortho/meta deuterons,34 but for any temperature between 40 and 70 O C interpolation in Tables I and I1 yields a set of eight spectral densities. Disregarding for the moment the small frequency dependence of J2(2w0)(which is barely outside
2.923 2.949 2.970 3.008 3.013 3.063 3.1 10 3.133 3.161 3.211
21.1 f 0.2 24.2 f 0.3 26.8 f 0.2 30.9 f 0.6 32.1 f 0.7 39.9 f 0.5 49.0 f 0.7 54.7 f 0.7 68.2 f 1.2 75.9 f 1.6
38.4 MHz 14.2 f 0.2 16.0 f 0.2 17.4 f 0.2 20.2 f 0.4 22.2 f 0.3 28.6 f 0.2 35.6 f 0.4 40.0 f 0.4 38.9 f 0.8 59.8 f 1.0
9.6 f 0.1 10.7 f 0.1 11.7 f 0.1 13.5 f 0.3 14.9 f 0.2 19.3 f 0.1 23.9 f 0.3 26.8 f 0.3 26.0 f 0.5 40.2 f 0.7
8.8 f 0.1 10.2 f 0.2 11.3 f 0.1 13.0 f 0.2 13.4 f 0.4 16.5 f 0.3 20.2 f 0.4 22.5 f 0.4 29.5 f 0.6 30.7 f 0.8
2.913 2.982 3.001 3.070 3.119 3.142 3.167 3.192
20.3 f 0.9 29.7 f 2.1 28.6 f 2.5 41.8 f 2.5 54.7 f 2.4 52.9f2.2 63.3 f 1.6 67.8 f 3.1
9.2 MHz 11.8 f 0.3 16.5 f 1.0 14.5 f 0.7 24.1 & 0.8 30.6 f 2.0 30.1f1.8 43.1 f 1.1 57.4 f 1.9
7.9 f 0.2 11.1 f 0.7 12.5 f 0.5 16.2 f 0.5 20.6 f 1.3 20.2f1.2 29.0 f 0.7 38.6 f 1.3
8.7 f 0.5 12.9 f 1.1 9.8 f 1.3 18.0 f 1.3 23.7 f 1.2 22.8f1.1 26.5 f 0.8 26.9 f 1.6
“Spectral densities calculated from Zeeman relaxation rates R I Zand relaxation rates R I Qof quadrupolar order according to eq 10a and lob.
experimental error), there are then six independent (smoothed) experimental measurements at each temperature. According to eq 1, 2, and 5 the spectral densities depend on four independent parameters in addition to the K ~ T ,~, , 71, : A, and wc. In principle, different K factors should be used to treat data obtained from the low-field and high-field samples because the static order parameters are not quite the same. However, simultaneous analysis of the data obtained at both fields is facilitated by ignoring this refinement and averaging the order parameters for both samples at each temperature. Rotational correlation times were determined at eight temperatures between 34 and 67 O C from the smoothed data by the following sequence of steps: ( a ) determine values of K~~ (and K ~ which are consistent with (interpolated) values of Szz;(b) use K~~ and J2(2w0)for para deuterons to derive 71 from eq 1; (c) guess a value of the anisotropy ratio R, calculate K21 and K22, use these values with T~ and the experimental values of J2(2w0) for ortho/meta deuterons to estimate a new value of R, and (d) iterate step (c) until both R and the K factors no longer change. No Jl(wo) data are used in this procedure, and the values of this spectral density for ortho/meta deuterons may therefore be used to check the numerous assumptions involved in the fitting procedure. J,(w0) for para deuterons includes contributions from director fluctuations, which may be analyzed after subtracting the (calculated) rotational contribution. Best-fit calculated spectral densities are compared with (unsmoothed) experimental values in Figures 2 and 3. Since the motional parameters were adjusted to fit J1 and J2 for para deuterons as well as J2 for ortho/meta deuterons, it is no surprise that the calculated values of these spectral densities agree well with experimental data. However, calculated values of J,(wo) for ortho/meta deuterons are up to -60% too large at the lowest temperatures. Possible sources of this discrepancy include (a) ignoring the nonzero values of S, - S,, when estimating K factors, (b) requiring that the K factors be consistent with static (equilibrium) values of S,, rather than values, (c) using a Maier-Saupe restoring potential (other forms yield different K values), and (d) treating biphenyl as a symmetric rotor when there are in fact three different prinicipal rotational correlation times. We have made limited efforts to check assumption (a) by repeating the data fitting using different sets of K factors. With a two-term (asymmetric) potential capable of reproducing all the static order parameters,% the final calculated values of Jl(wo) for
.
(49) Bemassau, J. .; Black, E. P.; Grant, D. J . Chem. Phys. 1982,76,253.
~ )
The Journal of Physical Chemistry, Vol. 89, No. 3, 1985 549
N M R of Biphenyl in Liquid Crystalline Media
I
.
I
I
I
295
30
305
4001
1
400
I
I
I
31
315
32
300
I
9.2 MHz
a
100 90 80 ps 7 0 60
-
40 -
50
28
29
30
31
32
IO3/ T
33
