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Transverse deuteron spin relaxation in thermotropic liquid crystal polymers: determination of viscoelastic properties
Volume
195. number
4
CHEMICAL
PHYSICS
LETTERS
24 July 1992
Transverse deuteron spin relaxation in thermotropic liquid crystal polymers: determination of viscoelastic properties Nicholas
Heaton,
Dirk Reimer
and Gerd
Kothe
Institut ftir Physikalische Chemie, Universitiit Stuttgart, Pfaaffenwaldring55, W-7000 Stuttgart 80, Germany Received
2 1 April 1992; in final form 19 May 1992
Transverse deuteron spin relaxation times, T 8, from quadrupole echo pulse trains have been measured as a function of pulse spacing, 7, for a thermotropic liquid crystal polymer and dimer in the nematic phase. The relaxation times exhibit a dispersion providing the dominant transverse relaxation process. The observed anlaw, Tgi -T-‘/‘, consistent with director fluctuations isotropy m TFz measured as a function of orientation, S,, for the polymer approximately follows the ( sin2&cos2&)dependence characteristic of this type of low-amplitude motion. Analysis of the experiments is achieved employing a simple hydrodynamic model for director fluctuations, yielding values for the effective viscosity and average elastic constant of the liquid crystal polymer and dimer.
1. Introduction Thermotropic liquid crystal polymers have attracted widespread interest over the past decade. These materi ils combine the anisotropic molecular organisation of the liquid crystalline state together with physical properties typical of standard polymers. This combination gives liquid crystal polymers unique material properties which have been exploited, for example, in the development of highmodulus fibres, information storage technology and non-linear optics [ 1,2]. Although there has been intense activity directed towards the synthetic design and investigation of the macroscopic properties of these materials, relatively little attention has been focused on the dynamic and structural features of the molecular units. Up to now, only limited information has been available about order and dynamics in the different polymer phases [ 3-5 1. Particularly, the occurrence of collective motions is not yet well established in thermotropic liquid crystal polymers. Correspondence to: G. Kothe, Chemie, Stuttgart
448
Universitlt Stuttgart, 80, Germany.
Institut ftir Pfaffenwaldring
Physikalische 55, W-7000
0009-2614/92/$
This dynamic process is of special interest since it directly reflects the viscoelastic nature of the medium. Thus, in principle, characterization of the spectrum of collective motions allows determination of the viscosity parameters and elastic constants. In this Letter, we present the results of orientation and pulse frequency-dependent transverse 2H spin relaxation experiments [ 61 conducted on a thermotropic main chain liquid crystal polymer and a dimeric analogue. It is shown that transverse NMR relaxation measurements performed at standard high Armor frequencies can provide information regarding low frequency collective motions. The experiments are analyzed in terms of a simple hydrodynamic model, which yields values for the effective viscosity and average elastic constant of the liquid crystal polymer and dimeric analogue.
2. Theory Within the Redfield limit, the transverse spin relaxation rate of spin I= 1 nuclei in an ordered medium during a pulse train type experiment [ 7,8] is given by [ 9,101 05.00 0 1992 Elsevier Science Publishers
B.V. All rights reserved.
+$A (WIJ)+Jz(2%)}
(1)
9
where tc is the correlation time for the Markovian motion, o0 is the Larmor frequency, w= l/r is the inverse pulse spacing, and eZqQ is the quadrupole coupling constant. The spectral densities , JM(Mwo), are the single-sided Fourier transforms of the autocorrelation function G,& t) of the fluctuating Hamiltonian. For nematic phases in which deuteron spin relaxation is dominated by director fluctuations, these correlation functions may be written as
XP~d[~(01-<%0))
3
(2)
where D.&$(Q) are second-rank Wigner matrix elements and So, is the order parameter for the particular C-D bond relative to the director. It has been assumed that the quadrupole coupling tensor is symmetric and that its principle axis lies along the C-D bond. The Euler angles, Q(t), then describe the orientation of the instantaneous director in the laboratory frame at time t. If the motion is such that 1/7c CK coo, then the final two terms in eq. ( 1) may be dropped and TFl is determined solely by the secular term proportional to Jo (0). The spectral density functions in eq. ( 1) refer to the laboratory frame where measurements are performed. However, for the purposes of testing particular relaxation models it is convenient to rewrite them in terms of the equivalent functions expressed in a frame determined by the symmetry axis of the motion (i.e. average director orientation). Thus, if this symmetry axis is oriented at an angle t9, to the static magnetic field then the spectral density, Jo(O), is [11,12] Jo(O)= z
24 July 1992
CHEMICAL PHYSICS LETTERS
Volume 195,number 4
[d3(&)12J,iFF~(0) 3
(3)
where dj,$ (19,) and JgF) (0) are the reduced second-rank Wigner matrix elements and the Fourier transforms of the correlation functions, GgF) (t), expressed in the director frame. Thus, in order to calculate T&!‘, we must first evaluate the GPF) (t) for the particular motional model assumed to be responsible for spin relaxation. Collective motions in liquid crystalline phases may
be described by a distribution of modes which characterize fluctuations of the order director relative to its time-average orientation [ 131. Each mode has a mean-square amplitude given by (4)
(@i(q)> =kBTlV&* and a relaxation time, r(q)
=rllKq* ,
(5)
where V is the volume of the uniform sample, K is the average elastic constant, q is the effective viscosity and q is the wavevector of the mode. The overall mean square amplitude of the fluctuations is obtained by summing the individual amplitudes for each mode. This is achieved’ analytically by integrating eq. (4) over q between the limits qg=27c/l, and qcE 2x/A,, where Apand ACare the long and short wavelengths cutoffs of the director modes, respectively. For nematic phases, this yields (6;)=k,T/xM,.
(6)
Within the small-angle approximation, it is usual to neglect the correlation functions, GIDF’(t) and GdDF)(t), while the remaining term, CID’)(t) is simply 3 ( 0, (0) 0, (I) > . This function is readily evaluated by integrating over modes assuming that each mode relaxes exponentially with time constant, r(q) [ 141. The result for T$L is then l/T ~~=~(~*~Q/~z)*S~~~~T(A*K)-X sin*& cos2f3N
4e x
I
t(q){1 -or(q)
tanh[or(q)]-‘jdq.
(7)
4n Exact closed form solutions to (7) are not available. However, good approximate solutions are obtained by employing the series expansion, 4c r(q){1 -m(q) s 49
tanh[wr(q)
9c 6, n-* J (
]-‘}dq=&-*
r(q)[l+r2(q)($nxo)2]-‘dq
49
>
. (8)
This series converges rapidly and it is reasonable here 449
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CHEMICAL PHYSICS LETTERS
to retain just the first term. The expression for T$f then has the same general form as that for T,, due to director fluctuations [ 1S- 17 1. Thus, the o dependence of TzZ predicted by eqs. ( 7) and ( 8) may be separated into three distinct regions defined by the low-frequency cutoff, o,=Kq: Jq and highfrequency cutoff, 0, = Kq,Z /q: T$:;‘K2Jt& T:;
o-co,,
x (KZ/rj)“2~“2,
Tcp N ZE -?@~2,
(9) wp
@>a,,
,
(10) (11)
3. Experiments and methods The chemical structures and relevant physical properties of the polymer and dimer compounds are shown in fig. 1. Details of the preparation of the materials are given elsewhere [ 18 1. Transition temperatures were measured by DSC prior to and after the series of experiments. The average molecular weight, H,,, of the polymer was determined by vapour pressure osmometry employing 3$bis(trifluoromethyl)phenol as solvent. The ‘H NMR experiments were performed on a Bruker MSL 300 spectrometer operating at 46.1 MHz using a home-built broad-band probe equipped with a sample spinning mechanism. Using a permanent magnet dc motor and a mechanical drive, spinning
24 July 1992
rates of between 5x 10e3 Hz and 70 Hz could be achieved about an axis perpendicular to the static &, field. Sample spinning rates were measured using photodiode sensors linked to a Hameg frequency counter and were constant to within + loJo.A sample spinning rate of 5.0 Hz was used to obtain the lineshapes presented here. No changes in the lineshape could be detected for spinning rates between 2 and 10 Hz. A further check on possible effects of sample spinning on the ‘H lineshapes was performed by comparing the Tfl values for the 0” director orientation measured by the spinning technique with that determined for a static sample. The two T$z values agreed within experimental error. Transverse ‘H spin relaxation times, T$L, were determined using a quadrupole echo pulse train [ 68,191, (~/2),-r-( (n/2),,-2~),, with an eight step phase cycling scheme. Free induction decays (FIDs) were recorded with quadrature detection and typical dwell times of 1 ps. The x J2 pulse length was set between 2.0 and 2.5 ns. High power (&/2~=40 kI-Iz) and rotating frame [20] proton decoupling was applied in control experiments conducted on static samples using a Bruker double-tuned broadband probe. Use of proton decoupling resulted in measured Tyl values, which were at most 15% greater than those values obtained without proton decoupling, independent of pulse spacing. A low-frequency ( z 500 Hz) modulation of the
POLYMER D
D
s 436 K n 55QK i
c 403 K n 508 K i Fig. 1. Molecular structure and relevant physical properties for the main chain liquid crystal polymer and dimer. i = isotropic, n = nematic, s=smectic, c= crystalline.
450
Volume 195, number 4
24 July 1992
CHEMICAL PHYSICS LETTERS
echo decay curves could be observed due to the effects of 2H-2H dipolar couplings [ 211 which necessarily introduce some additional error in the determination of the echo amplitude decay rates. It was usually possible to detect at least two periods of the dipolar modulation during the echo amplitude decay and for most measurements this additional error is less than 5%. For measurements with longer pulse spacings which provide fewer reliable points in the echo decay curves, the error is estimated to be about 10%.
4. Results and discussion The principal aim of the experiments described here is the characterization of low frequency molecular dynamics in a main chain liquid crystal polymer, using transverse 2H spin relaxation. Initial indications of the occurrence of collective motions in such systems were provided by the very shallow temperature dependence of Tzg in the nematic phase [ 221. These findings are in accordance with the results of recent low frequency proton T,, measurements [23]. The orientation dependence of spin relaxation times can be a highly sensitive probe of molecular motion in anisotropic systems. However, most thermotropic nematics align rapidly in the presence of strong magnetic fields such as those employed in NMR spectroscopy. Consequently, measurements are generally limited to a single director orientation. This limitation may be overcome by spinning samples about an axis oriented at an angle to B,, [24]. Slow spinning of nematic samples about an axis perpendicular to the applied magnetic field induces a distribution of the nematic director centered about some angle, &, relative to the field. The value of 0, depends on the spinning rate and the field strength as well as the diamagnetic anisotropy and viscosity of the sample [ 25 1. If the sample is spun fast enough, a two-dimensional distribution of the director is obtained in a plane perpendicular to the spinning axis [ 26,271. For viscous nematics such as the polymeric liquid crystal examined here, the necessary spinning rates can easily be achieved and it is therefore possible to determine the anisotropy of the transverse 2H spin relaxation using this sample spinning tech-
nique [28]. However, for the dimer at 485 K a sufficiently high spinning rate could not be attained. Fig. 2 shows typical 2H lineshapes recorded using the quadrupole echo pulse train (see section 3) for static and spinning samples of the polymer liquid crystal. The line-shapes refer to T=473 K, a pulse spacing of 7= 150 us and a spinning rate of 5.0 Hz. Note that the outermost edges in the lineshapes with quadrupole splitting, Av,,,=22.5 kHz, correspond to the 0” director orientation, whilst the singularities which appear at fAvmax derive from those domains in which the director is oriented at 90” to B,,. In fact, the situation is complicated slightly by the inequivalence of the aromatic deuterons in the polymer, exhibiting two doublets for each director orientation with maximum splittings of 22.5 and 18.7 kHz, re-
UC
A_
rotating
ti
---P
20
10
Ln=8
0
-10
-20
kHz Fig. 2. Deuterium NMR lineshapes obtained for the liquid crystal polymer at 473 K. The top trace shows the normal static spectrum. The appearance of two distinguishable quadrupole splittings is due to the inequivalence of the aromatic ring deuterons. The peak at v=O derives from a 5% isotropic impurity. The second trace shows the “unrelaxed” quadrupole echo spectrum of the rotating sample (spinning rate 5 Hz) obtained with a 30 ps pulse spacing. The lower three lineshapes were obtained by Fourier transformation of the nth echoes of a quadrupole echo train with r= 150 us.
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CHEMICAL PHYSICS LETTERS
spectively. The central peak in the lineshapes results from a 5% impurity. Inspection of the lineshapes in fig. 2 reveals that the 0 ’ and 90 ’ orientations relax most slowly. A more quantitative analysis of the T$Z anisotropy was performed by fitting series of lineshapes such as those in fig. 2 using the general form,
+co, [M(&)
l’+G2[4:‘(6?v,
12,
(12)
where the C,,,,, are free adjustable parameters. This general form for the anisotropy of T$z is exactly correct within the Redlield limit if non-secular contributions to transverse relaxation are negligible. The orientation dependence of Tff determined in this way for the polymer is shown in fig. 3. Also depicted is a theoretical curve (solid line) showing the (sin ‘6, CO&~,)- ’ dependence expected for collective motions (see eq. (7) ). The curves have been normalized to the respective T$L values at the magic angle (54.7” ) for the purposes of comparison. It is
P
Fig. 3. Experimentally determined orientation of T$z for the liquid crystal polymer (squares). The solid line shows the (sin20NcosZON)-’ dependence predicted by the Redfield relaxation model.
452
24 July 1992
evident that whilst the experimental anisotropies do not display the divergence expected theoretically at the 0” and 90” orientations, the general form of the orientation dependence of T$E is consistent with contributions to transverse relaxation from director fluctuations. Similar T 5: anisotropy curves have also been obtained for different pulse spacings. Signiticantly, this observed form of the orientation dependence cannot be reproduced by standard models for molecular reorientation. The most likely sources of discrepancy between the model and experimental curves are (i) neglect of non-secular terms and contributions from faster molecular motions in the calculation of transverse relaxation rates, (ii) omission of so-called second-order terms, GhDF)( t ) and GiDF) ( t), in the calculation of the director fluctuation correlation function [ 281, and (iii) breakdown of Redlield limit approximations due to slow components of the director motion. Estimates of the first two possible sources of error are readily obtained. First, the contributions to transverse relaxation from J1 (wg) and J2( 20,) may be determined from measurements of spin-lattice relaxation times T,z and T,,. With values of TIZ= 19 ms and T,,= 18 ms at t9,=O” and T=473 K compared with the Tg of 4.3 ms at T= 150 ps the nonsecular contributions to Tfz are certainly not sufficient to explain the observed TyE anisotropy. The second-order terms, GADF)(t) and GiDF) (t), have been evaluated according to the model described by Vold et al. [ 291 assuming reasonable values for the viscoelastic parameters (see table 1). For the polymer, these terms are found to make contributions of the order of just 10% of the measured relaxation rates. Therefore, it is most probable that slow-motional effects are primarily responsible for the discrepancies between theoretical and observed T$z anisotropies. Indeed, model calculations for uncoupled director modes employing the stochastic Liouville approach [ 301 predict TFz orientation dependencies similar to those observed [ 61. Thus, given that slow-motional effects are clearly apparent in the measured Tyz anisotropy, the model outlined in section 2 should then be reqarded as a first-order approximation. A more rigorous analysis is deferred for a later publication. The dependence of T$E on the pulse frequency,
24 July 1992
CHEMICAL PHYSICS LETTERS
Volume 195, number 4
Table 1 Parameters characterizing order director fluctuations in the main chain liquid crystal polymer and dimeric analogue Derived parameters
Determined parameters
polymer (473 K) dimer (485 K)
effective viscosity ‘) rl
average elastic const. K
long wavel. cutoff b, 2*P
short wavel. cutoff C’ AC
mean square amplitude
director order param. SD,
40Pas 0.20 Pa s
6.5X10-“N 3.5x lo-” N
23X IO-‘m 3X 10e6m
1.5x 10W9m 1.5x 10-9m
0.02 rad2 0.04 rad*
0.97 0.94
‘) Values adopted from ref. [ 22 1. b, In the case of the polymer the lower limit of 1, has been estimated from the shortest Tfl value measurable. ‘) Values adopted from ref. [ 231.
i 10*
1 IO4
103
lo5
W(Hz)
Fig, 4. Pulse frequency dependence of Tg for the liquid crystal polymer at 473 K (open squares) and the dimer at 485 K (filled symbols). Absolute Tyf values refer to the 45” director orientation (see text).
w= 1/r, for the two compounds is depicted in fig. 4. The absolute Tyf values shown refer to the 45” director orientation. In the case of the dimer, it has been assumed that the TFf anisotropy is broadly similar to that of the polymer and TFc values determined for static samples have then been resealed using the anisotropy shown in fig. 3. For both the polymer and the dimer, T zf displays a clear o iI2 dependence over more than one order of magnitude in w. This is precisely the dispersion law expected for collective motions at intermediate frequencies, wp< o c w,, where Tyi is predicted to be proportional to (K3/q)'12x co'/'(see eq. (10)). However, for neither compound was it possible to unambiquously determine both q and K from these results. For this reason we have measured v inde-
pendently using the sample spinning technique [ 22,251 and employed these values to obtain Kfrom the Tz dispersion curves. This procedure yields 1~40 Pa s and K=6.5~10-” N for the polymer whilst for the dimer we obtain ~~0.2 Pa s and K=3.5x lo-” N. For the dimer there appears to be a levelling off of the dispersion at w = 1O3Hz. Assuming that this represents the “true” low frequency cutoff of the director modes, we obtain an estimate of 3 x 1Om6m for il , . Interestingly, this value is of the same order of magnitude as the magnetic coherence length [ 301, suggesting that the long wavelength cutoff in this case reflects magnetic field damping of the director modes
1281.
The experimental error for both K and q is estimated to be about 1O”/b.However, the systematic errors introduced by the approximations concerning our use of the Redtield theory imply that the total error in K is certainly greater than this. However, it is encouraging to note that the evaluated elastic constants are in broad agreement with those reported for similar systems [ 23 1. Using independent estimates for the short wavelength cutoff, II, [ 23 1, together with the K values determined above, the mean-square fluctuation, (0; ) , of the director is calculated to be 0.02 and 0.04 rad2 for the polymer and dimer, respectively, corresponding to director order parameters, Snr, of 0.97 and 0.94. The viscoelastic parameters and derived quantities are summarized in table 1. In conclusion, the anisotropy and pulse frequency dependence of the transverse 2H spin relaxation time from quadrupole echo pulse trains is able to provide reliable infor453
Volume 195, number 4
CHEMICAL PHYSICS LETTERS
mation concerning the hydrodynamic modes in liquid crystalline phases.
References [ 1 ] J.L. White and J.F. Fellers, J. Appl. Polym. Sci. Symp. 33 (1978) 137. [2] M. Eich, J.H. Wendorff, Et. Reck and H. Ringsdorf, Makromol. Chem. Rapid Commun. 8 (1987) 59. [3] K. Mtiller, P. Meier and G. Kothe, Progr. Nucl. Magn. Reson. Spectry. 17 (1985) 211. [4] C. Boeffel and H.W. Spiess, in: Side chain liquid crystal polymers, ed. C.M. McArdle (Blackie, London, 1989) p. 224. [5] K. Mtlller, K.-H. Wassmer and G. Kothe, Advan. Polym. Sci.95(1990) 1. [ 61 J. Stohrer, G. Grijbner, D. Reimer, K. Weisz, C. Mayer and G. Kothe, J. Chem. Phys. 95 (1991) 672. [ 71 H.Y. Carr and E.M. Purcell, Phys. Rev. 94 ( 1976) 630. [8] S.B. Ahmad, K.J. Packer and J.M. Ramsden, Mol. Phys. 33 (1977) 857. [9] J.S. Blichanki, Can. J. Phys. 64 (1986) 733. [IO] A.J. Vega, R. Poupkoand Z. Luz, J. Magn. Reson. 83 (1989) 111. [ 1 I] P. Ukleja, J. Pirs and J.W. Doane, Phys. Rev. A 14 (1976) 414. [ 121 TM. Barbara, R.R. Vold and R.L. Vold, J. Chem. Phys. 79 (1983) 6338.
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[ 13 ] P.G. de Gennes, The physics of liquid crystals (Clarendon Press, Oxford, 1974). [ 141 P. Pincus, Solid State Commun. 7 (1969) 415. [ 151 J.W. Doane, C.E. Tarr and M.A. Nickerson, Phys. Rev. Letters 33 (1974) 620. [ 161 R. Blinc, Nucl. Magn. Reson. Basic Principles Progr. 13 (1976) 97. [ 171 R.R. Vold and R.L. Vold, J. Chem. Phys. 88 (1988) 4655. [ 181 K. Kohlhammer, K. Miiller and G. Kothe, Liquid Cryst. 5 (1989) 1525. [ 191 M. Bloom and E. Stemin, Biochemistry 26 (1987) 2101. 120) E.J. Dufourc, C. Mayer, J. Stohrer and G. Kothe, J. Chim. Phys. 89 (1992) 243. [ 2 1] N. Boden and P.K. Kahol, Mol. Phys. 40 ( 1980) 1117. [ 22) N.J. Heaton, D. Reimer and G. Kothe, unpublished results. [23] U. Zeuner, T. Dippel, F. Noack, K. Mtlller, C. Mayer, N.J. Heaton and G. Kothe, J. Chem. Phys., in press. [24] J. Courtieu, D.W. Alderman, D.M. Grant and J.P. Bayles, J. Chem. Phys. 77 (1982) 723. [ 251 F.M. Leslie, G.R. Luckhurst and H.J. Smith, Chem. Phys. Letters 13 ( 1972) 368. [ 261 S.G. Carr, G.R. Luckhurst, R. Poupko and H.J. Smith, Chem. Phys. 7 (1975) 278. [27]P.J. Colling, D.J. Photinos, P.J. Bos, P. Ukleja and J.W. Doane, Phys. Rev. Letters 42 (1979) 996. [28] N.J. Heaton and G. Kothe, unpublished results. [ 291 R.L. Vold, R.R. Vold and M. Warner, J. Chem. Sot. Faraday Trans. 84 (1988) 997. [30] J.H. Freed, J. Chem. Phys. 66 (1977) 4183.
195. number
4
CHEMICAL
PHYSICS
LETTERS
24 July 1992
Transverse deuteron spin relaxation in thermotropic liquid crystal polymers: determination of viscoelastic properties Nicholas
Heaton,
Dirk Reimer
and Gerd
Kothe
Institut ftir Physikalische Chemie, Universitiit Stuttgart, Pfaaffenwaldring55, W-7000 Stuttgart 80, Germany Received
2 1 April 1992; in final form 19 May 1992
Transverse deuteron spin relaxation times, T 8, from quadrupole echo pulse trains have been measured as a function of pulse spacing, 7, for a thermotropic liquid crystal polymer and dimer in the nematic phase. The relaxation times exhibit a dispersion providing the dominant transverse relaxation process. The observed anlaw, Tgi -T-‘/‘, consistent with director fluctuations isotropy m TFz measured as a function of orientation, S,, for the polymer approximately follows the ( sin2&cos2&)dependence characteristic of this type of low-amplitude motion. Analysis of the experiments is achieved employing a simple hydrodynamic model for director fluctuations, yielding values for the effective viscosity and average elastic constant of the liquid crystal polymer and dimer.
1. Introduction Thermotropic liquid crystal polymers have attracted widespread interest over the past decade. These materi ils combine the anisotropic molecular organisation of the liquid crystalline state together with physical properties typical of standard polymers. This combination gives liquid crystal polymers unique material properties which have been exploited, for example, in the development of highmodulus fibres, information storage technology and non-linear optics [ 1,2]. Although there has been intense activity directed towards the synthetic design and investigation of the macroscopic properties of these materials, relatively little attention has been focused on the dynamic and structural features of the molecular units. Up to now, only limited information has been available about order and dynamics in the different polymer phases [ 3-5 1. Particularly, the occurrence of collective motions is not yet well established in thermotropic liquid crystal polymers. Correspondence to: G. Kothe, Chemie, Stuttgart
448
Universitlt Stuttgart, 80, Germany.
Institut ftir Pfaffenwaldring
Physikalische 55, W-7000
0009-2614/92/$
This dynamic process is of special interest since it directly reflects the viscoelastic nature of the medium. Thus, in principle, characterization of the spectrum of collective motions allows determination of the viscosity parameters and elastic constants. In this Letter, we present the results of orientation and pulse frequency-dependent transverse 2H spin relaxation experiments [ 61 conducted on a thermotropic main chain liquid crystal polymer and a dimeric analogue. It is shown that transverse NMR relaxation measurements performed at standard high Armor frequencies can provide information regarding low frequency collective motions. The experiments are analyzed in terms of a simple hydrodynamic model, which yields values for the effective viscosity and average elastic constant of the liquid crystal polymer and dimeric analogue.
2. Theory Within the Redfield limit, the transverse spin relaxation rate of spin I= 1 nuclei in an ordered medium during a pulse train type experiment [ 7,8] is given by [ 9,101 05.00 0 1992 Elsevier Science Publishers
B.V. All rights reserved.
+$A (WIJ)+Jz(2%)}
(1)
9
where tc is the correlation time for the Markovian motion, o0 is the Larmor frequency, w= l/r is the inverse pulse spacing, and eZqQ is the quadrupole coupling constant. The spectral densities , JM(Mwo), are the single-sided Fourier transforms of the autocorrelation function G,& t) of the fluctuating Hamiltonian. For nematic phases in which deuteron spin relaxation is dominated by director fluctuations, these correlation functions may be written as
XP~d[~(01-<%0))
3
(2)
where D.&$(Q) are second-rank Wigner matrix elements and So, is the order parameter for the particular C-D bond relative to the director. It has been assumed that the quadrupole coupling tensor is symmetric and that its principle axis lies along the C-D bond. The Euler angles, Q(t), then describe the orientation of the instantaneous director in the laboratory frame at time t. If the motion is such that 1/7c CK coo, then the final two terms in eq. ( 1) may be dropped and TFl is determined solely by the secular term proportional to Jo (0). The spectral density functions in eq. ( 1) refer to the laboratory frame where measurements are performed. However, for the purposes of testing particular relaxation models it is convenient to rewrite them in terms of the equivalent functions expressed in a frame determined by the symmetry axis of the motion (i.e. average director orientation). Thus, if this symmetry axis is oriented at an angle t9, to the static magnetic field then the spectral density, Jo(O), is [11,12] Jo(O)= z
24 July 1992
CHEMICAL PHYSICS LETTERS
Volume 195,number 4
[d3(&)12J,iFF~(0) 3
(3)
where dj,$ (19,) and JgF) (0) are the reduced second-rank Wigner matrix elements and the Fourier transforms of the correlation functions, GgF) (t), expressed in the director frame. Thus, in order to calculate T&!‘, we must first evaluate the GPF) (t) for the particular motional model assumed to be responsible for spin relaxation. Collective motions in liquid crystalline phases may
be described by a distribution of modes which characterize fluctuations of the order director relative to its time-average orientation [ 131. Each mode has a mean-square amplitude given by (4)
(@i(q)> =kBTlV&* and a relaxation time, r(q)
=rllKq* ,
(5)
where V is the volume of the uniform sample, K is the average elastic constant, q is the effective viscosity and q is the wavevector of the mode. The overall mean square amplitude of the fluctuations is obtained by summing the individual amplitudes for each mode. This is achieved’ analytically by integrating eq. (4) over q between the limits qg=27c/l, and qcE 2x/A,, where Apand ACare the long and short wavelengths cutoffs of the director modes, respectively. For nematic phases, this yields (6;)=k,T/xM,.
(6)
Within the small-angle approximation, it is usual to neglect the correlation functions, GIDF’(t) and GdDF)(t), while the remaining term, CID’)(t) is simply 3 ( 0, (0) 0, (I) > . This function is readily evaluated by integrating over modes assuming that each mode relaxes exponentially with time constant, r(q) [ 141. The result for T$L is then l/T ~~=~(~*~Q/~z)*S~~~~T(A*K)-X sin*& cos2f3N
4e x
I
t(q){1 -or(q)
tanh[or(q)]-‘jdq.
(7)
4n Exact closed form solutions to (7) are not available. However, good approximate solutions are obtained by employing the series expansion, 4c r(q){1 -m(q) s 49
tanh[wr(q)
9c 6, n-* J (
]-‘}dq=&-*
r(q)[l+r2(q)($nxo)2]-‘dq
49
>
. (8)
This series converges rapidly and it is reasonable here 449
Volume 195, number 4
CHEMICAL PHYSICS LETTERS
to retain just the first term. The expression for T$f then has the same general form as that for T,, due to director fluctuations [ 1S- 17 1. Thus, the o dependence of TzZ predicted by eqs. ( 7) and ( 8) may be separated into three distinct regions defined by the low-frequency cutoff, o,=Kq: Jq and highfrequency cutoff, 0, = Kq,Z /q: T$:;‘K2Jt& T:;
o-co,,
x (KZ/rj)“2~“2,
Tcp N ZE -?@~2,
(9) wp
@>a,,
,
(10) (11)
3. Experiments and methods The chemical structures and relevant physical properties of the polymer and dimer compounds are shown in fig. 1. Details of the preparation of the materials are given elsewhere [ 18 1. Transition temperatures were measured by DSC prior to and after the series of experiments. The average molecular weight, H,,, of the polymer was determined by vapour pressure osmometry employing 3$bis(trifluoromethyl)phenol as solvent. The ‘H NMR experiments were performed on a Bruker MSL 300 spectrometer operating at 46.1 MHz using a home-built broad-band probe equipped with a sample spinning mechanism. Using a permanent magnet dc motor and a mechanical drive, spinning
24 July 1992
rates of between 5x 10e3 Hz and 70 Hz could be achieved about an axis perpendicular to the static &, field. Sample spinning rates were measured using photodiode sensors linked to a Hameg frequency counter and were constant to within + loJo.A sample spinning rate of 5.0 Hz was used to obtain the lineshapes presented here. No changes in the lineshape could be detected for spinning rates between 2 and 10 Hz. A further check on possible effects of sample spinning on the ‘H lineshapes was performed by comparing the Tfl values for the 0” director orientation measured by the spinning technique with that determined for a static sample. The two T$z values agreed within experimental error. Transverse ‘H spin relaxation times, T$L, were determined using a quadrupole echo pulse train [ 68,191, (~/2),-r-( (n/2),,-2~),, with an eight step phase cycling scheme. Free induction decays (FIDs) were recorded with quadrature detection and typical dwell times of 1 ps. The x J2 pulse length was set between 2.0 and 2.5 ns. High power (&/2~=40 kI-Iz) and rotating frame [20] proton decoupling was applied in control experiments conducted on static samples using a Bruker double-tuned broadband probe. Use of proton decoupling resulted in measured Tyl values, which were at most 15% greater than those values obtained without proton decoupling, independent of pulse spacing. A low-frequency ( z 500 Hz) modulation of the
POLYMER D
D
s 436 K n 55QK i
c 403 K n 508 K i Fig. 1. Molecular structure and relevant physical properties for the main chain liquid crystal polymer and dimer. i = isotropic, n = nematic, s=smectic, c= crystalline.
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echo decay curves could be observed due to the effects of 2H-2H dipolar couplings [ 211 which necessarily introduce some additional error in the determination of the echo amplitude decay rates. It was usually possible to detect at least two periods of the dipolar modulation during the echo amplitude decay and for most measurements this additional error is less than 5%. For measurements with longer pulse spacings which provide fewer reliable points in the echo decay curves, the error is estimated to be about 10%.
4. Results and discussion The principal aim of the experiments described here is the characterization of low frequency molecular dynamics in a main chain liquid crystal polymer, using transverse 2H spin relaxation. Initial indications of the occurrence of collective motions in such systems were provided by the very shallow temperature dependence of Tzg in the nematic phase [ 221. These findings are in accordance with the results of recent low frequency proton T,, measurements [23]. The orientation dependence of spin relaxation times can be a highly sensitive probe of molecular motion in anisotropic systems. However, most thermotropic nematics align rapidly in the presence of strong magnetic fields such as those employed in NMR spectroscopy. Consequently, measurements are generally limited to a single director orientation. This limitation may be overcome by spinning samples about an axis oriented at an angle to B,, [24]. Slow spinning of nematic samples about an axis perpendicular to the applied magnetic field induces a distribution of the nematic director centered about some angle, &, relative to the field. The value of 0, depends on the spinning rate and the field strength as well as the diamagnetic anisotropy and viscosity of the sample [ 25 1. If the sample is spun fast enough, a two-dimensional distribution of the director is obtained in a plane perpendicular to the spinning axis [ 26,271. For viscous nematics such as the polymeric liquid crystal examined here, the necessary spinning rates can easily be achieved and it is therefore possible to determine the anisotropy of the transverse 2H spin relaxation using this sample spinning tech-
nique [28]. However, for the dimer at 485 K a sufficiently high spinning rate could not be attained. Fig. 2 shows typical 2H lineshapes recorded using the quadrupole echo pulse train (see section 3) for static and spinning samples of the polymer liquid crystal. The line-shapes refer to T=473 K, a pulse spacing of 7= 150 us and a spinning rate of 5.0 Hz. Note that the outermost edges in the lineshapes with quadrupole splitting, Av,,,=22.5 kHz, correspond to the 0” director orientation, whilst the singularities which appear at fAvmax derive from those domains in which the director is oriented at 90” to B,,. In fact, the situation is complicated slightly by the inequivalence of the aromatic deuterons in the polymer, exhibiting two doublets for each director orientation with maximum splittings of 22.5 and 18.7 kHz, re-
UC
A_
rotating
ti
---P
20
10
Ln=8
0
-10
-20
kHz Fig. 2. Deuterium NMR lineshapes obtained for the liquid crystal polymer at 473 K. The top trace shows the normal static spectrum. The appearance of two distinguishable quadrupole splittings is due to the inequivalence of the aromatic ring deuterons. The peak at v=O derives from a 5% isotropic impurity. The second trace shows the “unrelaxed” quadrupole echo spectrum of the rotating sample (spinning rate 5 Hz) obtained with a 30 ps pulse spacing. The lower three lineshapes were obtained by Fourier transformation of the nth echoes of a quadrupole echo train with r= 150 us.
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spectively. The central peak in the lineshapes results from a 5% impurity. Inspection of the lineshapes in fig. 2 reveals that the 0 ’ and 90 ’ orientations relax most slowly. A more quantitative analysis of the T$Z anisotropy was performed by fitting series of lineshapes such as those in fig. 2 using the general form,
+co, [M(&)
l’+G2[4:‘(6?v,
12,
(12)
where the C,,,,, are free adjustable parameters. This general form for the anisotropy of T$z is exactly correct within the Redlield limit if non-secular contributions to transverse relaxation are negligible. The orientation dependence of Tff determined in this way for the polymer is shown in fig. 3. Also depicted is a theoretical curve (solid line) showing the (sin ‘6, CO&~,)- ’ dependence expected for collective motions (see eq. (7) ). The curves have been normalized to the respective T$L values at the magic angle (54.7” ) for the purposes of comparison. It is
P
Fig. 3. Experimentally determined orientation of T$z for the liquid crystal polymer (squares). The solid line shows the (sin20NcosZON)-’ dependence predicted by the Redfield relaxation model.
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evident that whilst the experimental anisotropies do not display the divergence expected theoretically at the 0” and 90” orientations, the general form of the orientation dependence of T$E is consistent with contributions to transverse relaxation from director fluctuations. Similar T 5: anisotropy curves have also been obtained for different pulse spacings. Signiticantly, this observed form of the orientation dependence cannot be reproduced by standard models for molecular reorientation. The most likely sources of discrepancy between the model and experimental curves are (i) neglect of non-secular terms and contributions from faster molecular motions in the calculation of transverse relaxation rates, (ii) omission of so-called second-order terms, GhDF)( t ) and GiDF) ( t), in the calculation of the director fluctuation correlation function [ 281, and (iii) breakdown of Redlield limit approximations due to slow components of the director motion. Estimates of the first two possible sources of error are readily obtained. First, the contributions to transverse relaxation from J1 (wg) and J2( 20,) may be determined from measurements of spin-lattice relaxation times T,z and T,,. With values of TIZ= 19 ms and T,,= 18 ms at t9,=O” and T=473 K compared with the Tg of 4.3 ms at T= 150 ps the nonsecular contributions to Tfz are certainly not sufficient to explain the observed TyE anisotropy. The second-order terms, GADF)(t) and GiDF) (t), have been evaluated according to the model described by Vold et al. [ 291 assuming reasonable values for the viscoelastic parameters (see table 1). For the polymer, these terms are found to make contributions of the order of just 10% of the measured relaxation rates. Therefore, it is most probable that slow-motional effects are primarily responsible for the discrepancies between theoretical and observed T$z anisotropies. Indeed, model calculations for uncoupled director modes employing the stochastic Liouville approach [ 301 predict TFz orientation dependencies similar to those observed [ 61. Thus, given that slow-motional effects are clearly apparent in the measured Tyz anisotropy, the model outlined in section 2 should then be reqarded as a first-order approximation. A more rigorous analysis is deferred for a later publication. The dependence of T$E on the pulse frequency,
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Table 1 Parameters characterizing order director fluctuations in the main chain liquid crystal polymer and dimeric analogue Derived parameters
Determined parameters
polymer (473 K) dimer (485 K)
effective viscosity ‘) rl
average elastic const. K
long wavel. cutoff b, 2*P
short wavel. cutoff C’ AC
mean square amplitude
director order param. SD,
40Pas 0.20 Pa s
6.5X10-“N 3.5x lo-” N
23X IO-‘m 3X 10e6m
1.5x 10W9m 1.5x 10-9m
0.02 rad2 0.04 rad*
0.97 0.94
‘) Values adopted from ref. [ 22 1. b, In the case of the polymer the lower limit of 1, has been estimated from the shortest Tfl value measurable. ‘) Values adopted from ref. [ 231.
i 10*
1 IO4
103
lo5
W(Hz)
Fig, 4. Pulse frequency dependence of Tg for the liquid crystal polymer at 473 K (open squares) and the dimer at 485 K (filled symbols). Absolute Tyf values refer to the 45” director orientation (see text).
w= 1/r, for the two compounds is depicted in fig. 4. The absolute Tyf values shown refer to the 45” director orientation. In the case of the dimer, it has been assumed that the TFf anisotropy is broadly similar to that of the polymer and TFc values determined for static samples have then been resealed using the anisotropy shown in fig. 3. For both the polymer and the dimer, T zf displays a clear o iI2 dependence over more than one order of magnitude in w. This is precisely the dispersion law expected for collective motions at intermediate frequencies, wp< o c w,, where Tyi is predicted to be proportional to (K3/q)'12x co'/'(see eq. (10)). However, for neither compound was it possible to unambiquously determine both q and K from these results. For this reason we have measured v inde-
pendently using the sample spinning technique [ 22,251 and employed these values to obtain Kfrom the Tz dispersion curves. This procedure yields 1~40 Pa s and K=6.5~10-” N for the polymer whilst for the dimer we obtain ~~0.2 Pa s and K=3.5x lo-” N. For the dimer there appears to be a levelling off of the dispersion at w = 1O3Hz. Assuming that this represents the “true” low frequency cutoff of the director modes, we obtain an estimate of 3 x 1Om6m for il , . Interestingly, this value is of the same order of magnitude as the magnetic coherence length [ 301, suggesting that the long wavelength cutoff in this case reflects magnetic field damping of the director modes
1281.
The experimental error for both K and q is estimated to be about 1O”/b.However, the systematic errors introduced by the approximations concerning our use of the Redtield theory imply that the total error in K is certainly greater than this. However, it is encouraging to note that the evaluated elastic constants are in broad agreement with those reported for similar systems [ 23 1. Using independent estimates for the short wavelength cutoff, II, [ 23 1, together with the K values determined above, the mean-square fluctuation, (0; ) , of the director is calculated to be 0.02 and 0.04 rad2 for the polymer and dimer, respectively, corresponding to director order parameters, Snr, of 0.97 and 0.94. The viscoelastic parameters and derived quantities are summarized in table 1. In conclusion, the anisotropy and pulse frequency dependence of the transverse 2H spin relaxation time from quadrupole echo pulse trains is able to provide reliable infor453
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mation concerning the hydrodynamic modes in liquid crystalline phases.
References [ 1 ] J.L. White and J.F. Fellers, J. Appl. Polym. Sci. Symp. 33 (1978) 137. [2] M. Eich, J.H. Wendorff, Et. Reck and H. Ringsdorf, Makromol. Chem. Rapid Commun. 8 (1987) 59. [3] K. Mtiller, P. Meier and G. Kothe, Progr. Nucl. Magn. Reson. Spectry. 17 (1985) 211. [4] C. Boeffel and H.W. Spiess, in: Side chain liquid crystal polymers, ed. C.M. McArdle (Blackie, London, 1989) p. 224. [5] K. Mtlller, K.-H. Wassmer and G. Kothe, Advan. Polym. Sci.95(1990) 1. [ 61 J. Stohrer, G. Grijbner, D. Reimer, K. Weisz, C. Mayer and G. Kothe, J. Chem. Phys. 95 (1991) 672. [ 71 H.Y. Carr and E.M. Purcell, Phys. Rev. 94 ( 1976) 630. [8] S.B. Ahmad, K.J. Packer and J.M. Ramsden, Mol. Phys. 33 (1977) 857. [9] J.S. Blichanki, Can. J. Phys. 64 (1986) 733. [IO] A.J. Vega, R. Poupkoand Z. Luz, J. Magn. Reson. 83 (1989) 111. [ 1 I] P. Ukleja, J. Pirs and J.W. Doane, Phys. Rev. A 14 (1976) 414. [ 121 TM. Barbara, R.R. Vold and R.L. Vold, J. Chem. Phys. 79 (1983) 6338.
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[ 13 ] P.G. de Gennes, The physics of liquid crystals (Clarendon Press, Oxford, 1974). [ 141 P. Pincus, Solid State Commun. 7 (1969) 415. [ 151 J.W. Doane, C.E. Tarr and M.A. Nickerson, Phys. Rev. Letters 33 (1974) 620. [ 161 R. Blinc, Nucl. Magn. Reson. Basic Principles Progr. 13 (1976) 97. [ 171 R.R. Vold and R.L. Vold, J. Chem. Phys. 88 (1988) 4655. [ 181 K. Kohlhammer, K. Miiller and G. Kothe, Liquid Cryst. 5 (1989) 1525. [ 191 M. Bloom and E. Stemin, Biochemistry 26 (1987) 2101. 120) E.J. Dufourc, C. Mayer, J. Stohrer and G. Kothe, J. Chim. Phys. 89 (1992) 243. [ 2 1] N. Boden and P.K. Kahol, Mol. Phys. 40 ( 1980) 1117. [ 22) N.J. Heaton, D. Reimer and G. Kothe, unpublished results. [23] U. Zeuner, T. Dippel, F. Noack, K. Mtlller, C. Mayer, N.J. Heaton and G. Kothe, J. Chem. Phys., in press. [24] J. Courtieu, D.W. Alderman, D.M. Grant and J.P. Bayles, J. Chem. Phys. 77 (1982) 723. [ 251 F.M. Leslie, G.R. Luckhurst and H.J. Smith, Chem. Phys. Letters 13 ( 1972) 368. [ 261 S.G. Carr, G.R. Luckhurst, R. Poupko and H.J. Smith, Chem. Phys. 7 (1975) 278. [27]P.J. Colling, D.J. Photinos, P.J. Bos, P. Ukleja and J.W. Doane, Phys. Rev. Letters 42 (1979) 996. [28] N.J. Heaton and G. Kothe, unpublished results. [ 291 R.L. Vold, R.R. Vold and M. Warner, J. Chem. Sot. Faraday Trans. 84 (1988) 997. [30] J.H. Freed, J. Chem. Phys. 66 (1977) 4183.