Domain Structure in an Unoriented Lamellar Lyotropic Liquid Crystal

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Langmuir 2001, 17, 6455-6460

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Domain Structure in an Unoriented Lamellar Lyotropic Liquid Crystal Phase Studied by 2H NMR S. V. Dvinskikh† and I. Furo´* Division of Physical Chemistry, Department of Chemistry, Royal Institute of Technology, SE-10044 Stockholm, Sweden Received May 9, 2001. In Final Form: July 23, 2001 The spatial variation of the phase director in an unoriented lamellar lyotropic liquid crystal is investigated by 2H two-dimensional and one-dimensional exchange and PGSE NMR. Exchange NMR probes the singleparticle orientational correlation function of D2O molecules that diffuse among regions with different director orientations. The obtained correlation time and the water diffusion coefficient, measured by 2H PGSE NMR, provide the persistence length of director orientation that is defined as domain size. The nature of spatial variation is revealed by the decay of the 2H stimulated echo signal recorded with different evolution times. The persistence length of the director is found to be strongly dependent on the rate of cooling the sample from its isotropic phase.

Introduction It is well established that molecules in liquid crystalline phases are preferentially aligned with respect to an axis called the director.1 Although this orientational order can extend up to tens of micrometers, in typical macroscopic samples of ca. millimeter size the director orientation is spatially inhomogeneous within the sample volume unless some external force (e.g., magnetic or electric) is applied. This inhomogeneous structure may take the form of welldefined domains with homogeneous orientation with sharp boundaries between or may instead have a more continuous variation, that is, no sharp domain boundaries. The average domain size, that is just as easy to define for the former case as for any other polycrystalline material, becomes then the persistence length of molecular alignment. The domain size is an important input parameter when interpreting some experimental results concerning the molecular structure and dynamics.2-12 Even more importantly, the feasibility of “nanomaterials”13 produced * Corresponding author: Istva´n Furo´. Tel: +46 8 7908592. Fax: +46 8 7908207. E-mail: [email protected]. † On leave from the Institute of Physics, St. Petersburg State University, 198904 St. Petersburg, Russia. Current address: Division of Physical Chemistry, Arrhenius Laboratory, Stockholm University, SE-10691 Stockholm, Sweden. (1) de Gennes, P. G.; Prost, J. The Physics of Liquid Crystals; Clarendon: Oxford, 1993. (2) Callaghan, P. T.; So¨derman, O. J. Phys. Chem. 1983, 87, 17371744. (3) Blum, F. D.; Padmanabhan, A. S.; Mohebbi, R. Langmuir 1985, 1, 127-131. (4) Blum, F. D.; Franses, E. I.; Rose, K. D.; Bryant, R. G.; Miller, W. G. Langmuir 1987, 3, 448-452. (5) Bloom, M.; Sternin, E. Biochemistry 1987, 26, 2101-2105. (6) Boden, N.; Hedwig, G. R.; Holmes, M. C.; Jolley, K. W.; Parker, D. Liq. Cryst. 1992, 11, 311-324. (7) Ortiz, C.; Wagner, M.; Bhargava, N.; Ober, C. K.; Kramer, E. J. Macromolecules 1998, 31, 8531-8539. (8) Clarke, S. M.; Terentjev, E. M.; Kundler, I.; Finkelmann, H. Macromolecules 1998, 31, 4862-4872. (9) Fridrikh, S. V.; Terentjev, E. M. Phys. Rev. E 1999, 60, 18471857. (10) Erbes, J.; Gabke, A.; Rapp, G.; Winter, R. Phys. Chem. Chem. Phys. 2000, 2, 151-162. (11) Geil, B.; Feiweier, T.; Pospiech, E.-M.; Eisenbla¨tter, J.; Fujara, F.; Winter, R. Chem. Phys. Lipids 2000, 106, 115-126. (12) Feiweier, T.; Geil, B.; Pospiech, E.-M.; Fujara, F.; Winter, R. Phys. Rev. E 2000, 62, 8182-8194.

by mineralization or polymerization of lyotropic liquid crystalline structures depends to a large extent on the size of the produced grains with a homogeneous structure. Hence, the domain size and the nature of domain boundaries, that are known to be influenced by a number of experimental parameters such as temperature, temperature history, mechanical action, and interaction with the container wall and with electric and magnetic fields, should be readily measurable. While optical microscopy14-18 is the method of choice for transparent thin samples with domain sizes of micrometers or larger, turbid samples, thick ones, those with small domains, or those that lack sharp domain boundaries may exhibit a problem. Methods for smaller domain sizes include cryo-electron microscopy19-22 and light23 and X-ray24 scattering. Here, we present a simple NMR procedure to estimate domain sizes and characterize domain boundaries. Twodimensional (2D) 2H and 31P exchange NMR spectroscopy has already been applied to investigate the extension of small domains in gels25 as well as to characterize transport properties and curvature of biomembranes12,25-30 although (13) Antonietti, M.; Go¨ltner, C. Angew. Chem., Int. Ed. Engl. 1997, 36, 910-928. (14) Kle´man, M. Rep. Prog. Phys. 1989, 52, 555-654. (15) Boltenhagen, P.; Lavrentovich, O. D.; Kle´man, M. Phys. Rev. A 1992, 46, R1743-R1746. (16) McGrath, K. M.; Ke´kicheff, P.; Kle´man, M. J. Phys. II France 1993, 3, 903-926. (17) Lyou, D. J.; Kim, S. C. Polym. J. 1997, 29, 279-285. (18) Blanc, C.; Kle´man, M. Eur. Phys. J. B 1999, 10, 53-60. (19) Sammon, M. J.; Zasadzinski, J. A. N.; Kuzma, M. R. Phys. Rev. Lett. 1986, 57, 2834-2837. (20) Lindblom, G.; Rilfors, L. Biochim. Biophys. Acta 1989, 988, 221256. (21) Boltenhagen, P.; Kle´man, M.; Lavrentovich, O. D. J. Phys. II France 1994, 4, 1439-1448. (22) Hamersky, M. W.; Tirrell, M.; Lodge, T. P. Langmuir 1998, 14, 6974-6979. (23) Picken, S. J.; Wijk, R. J. v.; Lichtenbelt, J. W. T.; Westerink, J. B.; Klink, P. J. v. Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 1995, 261, 535-547. (24) Prosa, T. J.; Moulton, J.; Heeger, A. J.; Winokur, M. J. Macromolecules 1999, 32, 4000-4009. (25) Dolainsky, C.; Karakatsanis, P.; Bayerl, T. M. Phys. Rev. E 1997, 55, 4512-4521. (26) Auger, M.; Smith, I. C. P.; Jarrell, H. C. Biophys. J. 1991, 59, 31-38. (27) Fenske, D. B.; Jarrell, H. C. Biophys. J. 1991, 59, 55-69. (28) Macquaire, F.; Bloom, M. Phys. Rev. E 1995, 51, 4735-4742. (29) Dolainsky, C.; Unger, M.; Bloom, M.; Bayerl, T. M. Phys. Rev. E 1995, 51, 4743-4750.

10.1021/la010693c CCC: $20.00 © 2001 American Chemical Society Published on Web 09/20/2001

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Figure 1. 2H NMR spectra of D2O in the lamellar phase of the CsPFO/D2O mixture (42 wt %) at 290 K. Spectrum a is recorded in a homogeneously oriented sample with its director parallel to the magnetic field; spectra b and c are from unoriented samples obtained after slow (b) and fast (c) cooling of the mixture from its isotropic phase outside the magnetic field.

the origin of the method lies with studies of molecular reorientational processes.31-40 The presented procedure is based on a selection of one-dimensional (1D) exchange experiments in the form of stimulated echoes. In combination with experiments that yield the spatial diffusion coefficient of the spin-bearing particle, this approach provides a quicker and more straightforward estimate of the domain structure than that by the original set of 2D exchange experiments.12,25,27-30,36 The method is demonstrated in an unoriented lamellar phase of a lyotropic liquid crystal. Experimental Section Cesium perfluorooctanoate (CsPFO) has been synthesized as described previously.41 The liquid crystal has been produced by mixing CsPFO with D2O at 42 wt % concentration. The isotropicnematic and nematic-lamellar phase transition points were measured via the 2H NMR spectrum to 307 and 301 K, respectively. This is in agreement with the established phase diagram.42 In the nematic phase, the director orients parallel to the applied external magnetic field and keeps this orientation while cooling into the lamellar phase.6 The resulting homogeneous director orientation within the sample is easily validated by the conventional 2H NMR spectrum of D2O that consists of two narrow lines instead of the characteristic powder pattern (Figure 1a). To produce an inhomogeneous director orientation in the lamellar phase, the sample was cooled with the rate of about 0.3 K/min to 290 K from the isotropic phase at 310 K outside the spectrometer magnet. Due to the high viscosity of the lamellar phase, the formed regions of different director orientation do not (30) Picard, F.; Paquet, M.-J.; Dufourc, E. J.; Auger, M. Biophys. J. 1998, 74, 857-868. (31) Spiess, H. W. J. Chem. Phys. 1980, 72, 6755-6762. (32) Lausch, M.; Spiess, H. W. J. Magn. Reson. 1983, 54, 466-479. (33) Fujara, F.; Wefing, S.; Spiess, H. W. J. Chem. Phys. 1986, 84, 4579-4584. (34) Fujara, F.; Petry, W.; Schnauss, W.; Sillescu, H. J. Chem. Phys. 1988, 89, 1801-1806. (35) Dries, T.; Fujara, F.; Kiebel, M.; Ro¨ssler, E.; Sillescu, H. J. Chem. Phys. 1988, 88, 2139-2147. (36) Schmidt-Rohr, K.; Spiess, H. W. Multidimensional Solid-State NMR and Polymers; Academic Press: London, 1994. (37) Ro¨ssler, E.; Eiermann, P. J. Chem. Phys. 1994, 100, 5237-5248. (38) Geil, B.; Fujara, F.; Sillescu, H. J. Magn. Reson. 1998, 130, 1826. (39) Bo¨hmer, R.; Hinze, G. J. Chem. Phys. 1998, 109, 241-248. (40) Tracht, U.; Heuer, A.; Spiess, H. W. J. Chem. Phys. 1999, 111, 3720-3727. (41) Jo´hannesson, H.; Furo´, I.; Halle, B. Phys. Rev. E 1996, 53, 49044917. (42) Boden, N.; Corne, S. A.; Jolley, K. W. J. Phys. Chem. 1987, 91, 4092-4105.

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Figure 2. The rf pulse sequence for the exchange experiments where te and tm are the evolution and mixing times, respectively. In the two-dimensional experiment, the delay t1 between the first two pulses was incremented and the signal was detected during time t2 directly after the third pulse. In the 1D experiments, the amplitude of the echo signal was detected at time te after the third pulse. The pulse angles for the second and third rf pulses were set to φ ) 54.7° in the 2D and to φ ) 45° in the 1D experiments. reorient even when placed in the magnetic field and the 2H spectrum is the expected powder pattern43 as presented in Figure 1b. The spectral width in both cases is determined by the residual quadrupole coupling of 2H in D2O, that arises as a result of fast (picosecond or slower) motional averaging by molecular reorientation.44 Another important consequence of the partial averaging is that the axis of symmetry of the residual coupling tensor coincides with the local phase director.44 One should note that any orientational modulation of the coupling on time scales shorter than the inverse of the duration (∼10 ms) of the timedomain NMR signal of 2H in D2O contributes to this motional averaging and, hence, affects the residual coupling.45 Hence, the methods applied below can only detect orientational modulations on time scales longer than this limit. In contrast to the sample with the spectrum in Figure 1b, a sample was also prepared by cooling more quickly from the isotropic phase (310 K) to the lamellar phase (290 K). With an estimated cooling rate of 10 K/min, the obtained spectrum in Figure 1c differs from that in Figure 1b. As explained below, this difference is connected to small domain sizes resulted from the quick cooling. Note that no change of the spectral shape was observed while the samples were kept in the magnet at the target temperature of 290 K for long (ca. days) periods. The measurements were performed on a Bruker DMX 200 spectrometer, operating at 31 MHz for 2H with a home-built multiple-tuned gradient probe.46,47 The probe was equipped with two interchangeable gradient coils that produced homogeneous magnetic field gradients along either the z (parallel to B0) or the x (perpendicular to B0) axes. The length of the 90° 2H radio frequency (rf) pulse was 5 µs. The gradient coils were calibrated by measuring the diffusion coefficient of pure D2O in the temperature range of 5-45 °C by 2H PGSE NMR and comparing it to literature data.48 The sample temperature, calibrated by Pt100 resistor and regulated by a Bruker BVT-3000 unit, was stable and reproducible within (0.15 K. The sample, contained in a 5 mm o.d. tube, was approximately 20 mm long. The water diffusion in the isotropic phase was measured by conventional spin-echo-type 2H PGSE NMR.49 In the anisotropic phases, the quadrupole-echo-based pulse sequence was used instead.46,50 The exchange among regions with different orientations was measured by the conventional three-pulse 1D stimulated echo sequence51 (Figure 2) with delay t1 fixed and the amplitude of the stimulated echo at t1 ) t2 ≡ te monitored as a (43) Davis, J. H. Biochim. Biophys. Acta 1983, 737, 117-171. (44) Halle, B.; Wennerstro¨m, H. J. Chem. Phys. 1981, 75, 19281943. (45) Halle, B.; Quist, P.-O.; Furo´, I. Phys. Rev. A 1992, 45, 37633777. (46) Furo´, I.; Jo´hannesson, H. J. Magn. Reson., Ser. A 1996, 119, 15-21. (47) Dvinskikh, S. V.; Furo´, I. J. Magn. Reson. 2000, 144, 142-149. (48) Mills, R. J. Phys. Chem. 1973, 77, 685-688. (49) Stejskal, E. O.; Tanner, J. E. J. Chem. Phys. 1965, 42, 288-292. (50) Callaghan, P. T.; Le Gros, M. A.; Pinder, D. N. J. Chem. Phys. 1983, 79, 6372-6381. (51) Schmidt, C.; Blu¨mich, B.; Spiess, H. W. J. Magn. Reson. 1988, 79, 269-290.

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function of the mixing time tm. To record instead 2D exchange spectra at a fixed mixing time tm,36,51,52 the signal was acquired during time t2 while t1 was incremented by 1 ms in 128 steps, with 8 collected transients at each step. 2D Fourier transformation by the States algorithm53 was applied to obtain pure absorption 2D spectra. Phase cycles,52 that eliminate unwanted signal pathways through single- and double-quantum coherences during tm, were applied in the 1D and 2D experiments.

Results and Discussion 2

1D H NMR Spectra and Diffusion. While both spectra in Figure 1b,c indicate a random director distribution, the obvious difference is the more “smooth” appearance of spectrum c. Since the exchange of water molecules between regions of different director orientation modulates the residual quadrupole coupling (recall the coincidence of its axis of symmetry with the local phase director), a straightforward conclusion is that the time of exchange is in the order of the inverse of the residual quadrupole splitting.4 In other words, we observe the onset of the motional averaging of the residual coupling. This is also supported by the peak splitting in Figure 1c that is somewhat lower than that in Figure 1b. This exchange must be due to water diffusion which is then fast enough to allow water molecules to visit spatial regions with significantly different director orientation during a period that is comparable to the inverse of the time-domain NMR signal (∼10 ms).4 In contrast, there is no such exchange observed in sample b that has, presumably, a longer persistence length of the director. Note that lamellar defects, such as fused layers that have been inferred in other lamellar systems,2 have negligible influence in the present system where the bilayers are pierced by holes.41,42,54-56 In the discussion above, we tacitly disregarded temporal director fluctuations. The slow undulation modes are characterized by an effective diffusion coefficient57 Du ) K11/η, where K11 is the splay modulus and η is the membrane viscosity. Extrapolation from data measured in the nematic phase of the CsPFO-water system58-60 sets Du in the order of 10-12 m2/s. Hence, the diffusing water molecules (D ∼ 10-9 m2/s) see an effectively static landscape of undulated membrane. Note that faster membrane fluctuations on length scales much less than micrometers are included in the experimentally measured water diffusion coefficient.61 Another indication of different persistence lengths of the director comes from the results of the diffusion measurement in the two samples (samples b and c). Water diffusion in oriented (monodomain) lamellar phases of CsPFO/D2O has already been investigated,41,46 and here we repeated the same type of experiments in our particular sample (sample a). The obtained results, presented in Figure 3, are required to analyze the exchange experi(52) Schaefer, D.; Liesen, J.; Spiess, H. W. J. Magn. Reson., Ser. A 1995, 115, 60-79. (53) States, D. J.; Haberkorn, R. A.; Ruben, D. J. J. Magn. Reson. 1982, 48, 286-292. (54) Holmes, M. C.; Smith, A. M.; Leaver, M. S. J. Phys. II (France) 1993, 3, 1357-1370. (55) Holmes, M. C.; Leaver, M. S.; Smith, A. M. Langmuir 1995, 11, 356-365. (56) Leaver, M. S.; Holmes, M. C. J. Phys. II (France) 1993, 3, 105120. (57) Halle, B. Phys. Rev. E 1994, 50, R2415-R2418. (58) Gudilov, S. M.; Sonin, A. S. Sov. Phys. Crystallogr. 1988, 33, 580-581. (59) Gudilov, S. M.; Gagolina, S. Y.; Sonin, A. S. Zh. Fiz. Khim. 1991, 65, 1927-1934. (60) Bajc, J.; Hillig, G.; Saupe, A. J. Chem. Phys. 1997, 106, 73727377. (61) Gustafsson, S.; Halle, B. J. Chem. Phys. 1997, 106, 1880-1887.

Figure 3. Water diffusion in the isotropic and anisotropic phases of the CsPFO/D2O mixture (42 wt %) measured with gradient orientation parallel (squares) or perpendicular (circles) to the magnetic field. In the anisotropic phases, the sample is homogeneously oriented with its director (the surface normal of the lamellar layers) parallel to the magnetic field.

ments in terms of domain size. The main feature is a pronounced diffusion anisotropy D⊥ > D| in the nematic and lamellar phases. In particular, D| ) (0.71 ( 0.02) × 10-9 m2/s and D⊥ ) (1.26 ( 0.04) × 10-9 m2/s were obtained at 290 K (note that D⊥ measures the diffusion along the lamellar layers, the surface normal of which is the phase director). With these data, the diffusion anisotropy is σ ) D⊥/D| ) 1.80 ( 0.07. Provided that the magnetic field gradient is set parallel to B0 and diffusion path length is small compared to the average domain size, D⊥ and D| can also be measured by observing the decay of intensity in different parts in the powder spectrum with increasing gradient strength.50,62 Particularly, the decays at the spectral edges and peaks provide the diffusion coefficients D| and D⊥, respectively. At 290 K, we obtain D| ) (0.72 ( 0.05) × 10-9 and D⊥ ) (1.1 ( 0.1) × 10-9 m2/s in the slowly cooled sample (sample b) at the diffusion time of 15 ms. These values as well as the diffusion anisotropy σ ) 1.53 ( 0.15 are comparable to those measured in the oriented sample. In contrast, the same experiment yields σ ) 1.18 ( 0.18 [from D| ) (0.80 ( 0.1) × 10-9 and D⊥ ) (0.95 ( 0.10) × 10-9 m2/s] in the quickly cooled sample, indicating the convergence of two diffusion coefficients to the mean value 〈D〉 ) (D| + 2D⊥)/3 ) 1.05 × 10-9 m2/s and σ to 1, reached in the limit of fast water exchange among regions. 2D 2H Exchange NMR Spectra. 2D exchange experiments can provide detailed information on molecular reorientations that are slow on the time scale of the NMR free induction decay signal.36 For 2H, the molecular orientation can be probed via the angular dependence of the quadrupole frequency,

ω(θ) ) (δ(3 cos2 θ - 1)/2 ≡ (δP2(cos(θ))

(1)

where δ is the magnitude of (residual) quadrupole coupling. In a 2D exchange spectrum S(ω1,ω2;tm), the orientation-dependent frequencies ω(θ(t1)) and ω(θ(t2)) before and after the mixing time tm are correlated so that S(ω1,ω2;tm) describes the joint probability density of finding a molecule at frequency ω1 during the evolution period t1 and at frequency ω2 during the detection period t2 (for simplicity, one assumes that tm . t1, t2 and therefore there is no significant exchange during the evolution and (62) Dvinskikh, S. V.; Furo´, I. J. Magn. Reson. 2001, 148, 73-77.

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A(tm, tef0) ∝ (teδ)2〈P2[cos(θ(0))] P2[cos(θ(tm))]〉 (2)

Figure 4. 2D exchange NMR spectra of water in the lamellar phase of CsPFO/D2O at 290 K with mixing time tm ) 100 ms: (a) slowly cooled sample and (b) quickly cooled sample. f1 and f2 denote frequencies present during the t1 and t2 intervals (see Figure 2), respectively.

detection times). The joint probabilities and the corresponding 2D exchange spectra have been calculated for various molecular motions.36 Hence, detailed information on the time scale and geometry of slow molecular reorientations can be readily obtained by simulating, within a certain model of motions, a series of 2D spectra measured at different mixing times. Alternative simplified analyses of 2D spectra have been also reported.25,27 Here, however, we limit ourselves to a qualitative analysis of 2D spectra with quantitative information obtained from 1D time-domain exchange experiments below. In the slowly cooled sample, the off-diagonal intensity in the 2D spectra remains low even at large mixing times. As an example, the spectrum recorded at tm ) 100 ms is shown in Figure 4a. In contrast, the 2D spectrum in Figure 4b, recorded in the quickly cooled sample, corresponds to nearly complete exchange with off-diagonal intensities comparable to diagonal ones. 2D spectra recorded at different tm values (not shown) indicate a gradual spreading of the intensity into the off-diagonal area, that is characteristic for the isotropic rotational diffusion model.36 This suggests, as also confirmed by the 1D experiments below, a gradual change of the director reorientation, that is, the lack of domains with sharp boundaries. 1D Exchange Experiments by 2H Stimulated Echo. In the 1D time-domain exchange experiment, the singleparticle reorientational correlation function is probed by observing the intensity of the stimulated echo signal. We have chosen to observe the so-called sine-modulated echo that is produced when the phases of the first and second pulses (Figure 2) differ by 90°. For spin I ) 1 nuclei, the maximum signal is obtained with pulse flip angles36 φ ) 45°. In the limit of small evolution delay te , 1/δ, the echo intensity is proportional to the correlation function of the second Legendre polynomial:33-40

For long tm, the signal is reduced by longitudinal spin relaxation that sets the upper limit of the accessible time window. Since the echo amplitude is proportional to te2, this delay cannot be set arbitrarily short. For both samples, no significant change of the decay curves was observed for evolution times te smaller than 1 ms. For longer te values, the decay becomes faster (see below) and the signal decays to a plateau level f∞ at tm f ∞ that is small but significantly differs from zero. The results for te ) 1 ms are demonstrated in Figure 5 for the two samples obtained by slow and fast cooling. The decay of the signal with increasing tm is slow for the slowly cooled sample and is, as shown in Figure 5, characterized by a decay time that is comparable to relaxation time of the quadrupolar order63 T1Q ) (610 ( 20) ms estimated with the same pulse sequence but in the oriented sample. Since in the oriented sample water diffusion does not modulate the residual quadrupole coupling, the decay of the signal is solely due to spin relaxation. Note that in the following analysis we neglect the possible anisotropy64 of T1Q. For the quickly cooled sample, the decay is considerably faster as shown in Figure 5. The geometrical information about the molecular motions leading to the loss of orientational correlation that is stored in the observed decays can be best extracted by a numerical simulation33-40 of the stimulated echo decay S(tm) measured at different evolution times te. As concerning a model-independent evaluation, the most common approach is instead the parametrization of the experimental decays by a stretched exponential Kohlrausch-Williams-Watts (KWW) function.65 Explicitly, the function

A(tm, te) ∝ [(1 - f∞(te)) exp(-(tm/τkww)β) + f∞(te)] exp(-tm/T1Q) (3) fitted well to the experimental data as demonstrated in Figure 5 where longitudinal relaxation time was set to T1Q ) 610 ms. The characteristic or mean decay time can then be obtained as66

τ ) τkww Γ(1/β)/β

(4)

where Γ is the gamma function. The experimental decays measured at various evolution times te were all fitted by eq 3, and the extracted τ and f∞ data are summarized in Figure 6. The obtained value β ) 0.51 ( 0.05 was found to be independent of te within the experimental error. Note that this approach to quantitative analysis is only valid in the slow motion limit, that is, in the absence of exchange during the evolution and detection periods. Hence, data from experiments with te comparable to or longer than the estimated decay time τ were not evaluated. Another assumption behind this evaluation procedure is random orientational distribution of the director.38,39 First, this can be justified by the inspection of the spectra in Figure 1 that display regular powder patterns. Second, we combined the stimulated-echo experiment with slice (63) Vold, R. R. Nuclear spin relaxation; Emsley, J. W., Ed.; Reidel: Dordrecht, 1985; pp 253-288. (64) Hoatson, G. L.; Vold, R. L. NMR: Basic Princ. Prog. 1994, 32, 1-67. (65) Williams, G.; Watts, D. C. Trans. Faraday Soc. 1970, 66, 80-85. (66) Lindsey, C. P.; Patterson, G. D. J. Chem. Phys. 1980, 73, 33483357.

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f 0. The second one, the angular jump correlation time τc0 is provided by the time to which the τ variation levels off at large te. This correlation time describes the average time between subsequent significant changes in the direction of the residual quadrupole coupling. Water, the spin-bearing particle of this study, diffuses over length scales of ∼10 µm during the 10 ms limit of resolution of the experiment (see above). Hence, in the present system the angular jump correlation time measures the average difference R between the average orientation of adjacent regions of ∼10 µm size. For the simplest reorientational model of isotropic jumps by a fixed angle R, these two correlation times are connected,68 Figure 5. The intensity of the stimulated echo signal at different mixing times, as obtained via the pulse sequence in Figure 2 with the evolution time te set to 1 ms. Solid and open symbols are results from the quickly and slowly cooled samples, respectively. The solid lines are fits of eq 3 to the experimental points with parameters τkww ) 36 ms, βkww ) 0.53, and f∞ ) 0.077 (quickly cooled sample) and τkww ) 540 ms, βkww ) 0.50, and f∞ ) 0.02 (slowly cooled sample). The relaxation time T1Q is set to 610 ms in both fits. The effect of relaxation is illustrated by the dashed line that displays the decay of echo intensity in the absence of exchange.

τc0 ) τc[(3/2) sin2 R]

Clearly, the rotational correlation times (about 2 s in the slowly cooled and about 65 ms in the quickly cooled samples) differ for the two samples by more than 1 order of magnitude (Figure 6a). Hence, the director reorientation is much slower in the slowly cooled sample which indicates a longer persistence length of the director. Looking at the long-time behavior of τ, we can only conclude that the asymptotic value of τ must be smaller than 60 ms for the slowly cooled sample. Hence, from eq 5 one obtains the R < 8° upper limit for the variation of the jump angle. In other words, the director orientation changes not suddenly but instead gradually across the sample. The data from the quickly cooled sample are not sufficient to get such an estimate. Additional information is obtained from the variation of the f∞ values with increasing te. Our experimental findings can be sufficiently well described by assuming isotropic reorientation as shown by Figure 6b where the result of eqs 4-6 of the isotropic reorientational model of ref 39 is plotted together with the experimental data. Finally, β values in the vicinity of 0.5 indicate a broad distribution (of about 1 order of magnitude) of the correlation times τc.38,69 The average domain size r, that is, the persistence length of director orientation, can be readily estimated from the diffusion coefficient (measured by PGSE NMR) and from the average reorientation correlation time (measured by the stimulated echo experiment) as

〈r2〉1/2 ) (6Dτc)1/2

Figure 6. The mean decay time τ (a) and the plateau value of the decay f∞ (b) as a function of the evolution time te. Solid and open symbols are results from the quickly and slowly cooled samples, respectively. The solid line in (b) is the calculated variation of f∞(te) for isotropic rotational diffusion (ref 39) with δ ) 270 Hz residual quadrupole coupling.

selection to show that the wall effect60,67 on the director distribution is negligible. The results yielded by a 1 mm thick central slice of the approximately 20 mm long cylindrical sample in a 4.4 mm i.d. tube were identical to those obtained from the full sample volume. Spatial Variation of the Director. Information about the geometry and time scale of the molecular motions is stored in the dependence of the fitted parameters τ, f∞, and β on the evolution time te. In particular, from the variation of τ, presented in Figure 6a, one can obtain two correlation times.33-40 The first one, the rotational correlation time τc, is yielded by the limiting value of τ at te (67) Kaeder, U.; Hiltrop, K. Prog. Colloid Polym. Sci. 1991, 84, 250252.

(5)

(6)

Equation 6 is not generally valid for anisotropic diffusion, but for small anisotropy, which is the case in the present system, one can approximate D by the isotropic average 〈D〉 ) (2D⊥ + D|)/3. Hence, at 290 K we obtain, from 〈D〉 ) 1.05 × 10-9 m2/s (Figure 3) and τc ) 2 s (Figure 6b), 〈r2〉1/2 ) 110 µm for the slowly cooled sample. In analogous way, 〈r2〉1/2 ) 20 µm can be estimated for the quickly cooled sample. Conclusion We present a simple approach to estimate the persistence length of director orientation in an unoriented lamellar phase. The method, based on monitoring the space-time correlation function by means of 2H exchange NMR, is sensitive and quick. The range of distances which can be probed is determined by the size of the residual quadrupole coupling, the spin relaxation rates, and the diffusion coefficient of the spin-bearing particle. In our present experiments, the choice of rapidly diffusing water (68) Anderson, J. E. Faraday Symp. Chem. Soc. 1972, 3, 82-88. (69) Wachner, A. M.; Jeffrey, K. R. J. Chem. Phys. 1999, 111, 1061110616.

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as the orientational probe sets the probed domain sizes in the range of tens of microns. Although this range can be readily accessed by optical methods, one should recall that (i) we performed the experiment in bulk and thereby avoided effects of surface orientation67,70 and (ii) we detected a gradual variation of the director orientation that might be more difficult to identify from images than an arrangement of domains with sharp boundaries. Smaller sizes in the micron and submicron range, which are not accessible by optical techniques although suitable for electron microscopy toward the lower part of that range, can be probed by choosing the spin-bearing particle that diffuses slowly, for example, deuterated surfactant molecules.25,26,29 In the present sample, no deuteration is feasible but neither is it required since the exchange experiment can also be performed by exploiting the large chemical shift anisotropy36 of the 19F and 13C nuclei71 in the fluorosurfactant molecule. In the method, chemical shift anisotropy substitutes quadrupole coupling while the rest of the procedure can progress analogously.27,37,40,72 Note that in contrast to quadrupole coupling the spectral broadening by chemical shift depends on the magnetic field and thereby the accessible length scales can also be influenced by the choice of the magnetic field. Since the diffusion of CsPFO molecules is approximately 2 orders of magnitude slower than that of water,47,73 correspondingly smaller domain sizes27 can be probed (cf. eq 6). Although some experimental complication may arise from

applying dipolar decoupling during the evolution and detection periods62 (required to avoid the 19F-19F or 13C19F dipole couplings obscuring the anisotropic chemical shift interactions), preliminary 19F 2D exchange spectra (to be communicated elsewhere) have indeed clearly shown the exchange due to surfactant diffusion. The obtained spatial variation of the director orientation is gradual (and could therefore also be described as curvature) instead of sudden. This result is not unexpected. Usual polycrystalline materials, such as metals, are created by crystallization from their liquid phase. The grains that grow from different points have different crystal orientations, and hence sharp grain boundaries are formed when contact is established among them. In contrast, the lamellar liquid crystal of this study is cooled from its isotropic phase through the intermittent nematic region. Orientational correlation is already established in the nematic phase where the spatial variation of the director is determined by the elastic properties of the phase.59,60 Passing from the nematic phase to the lamellar one, the elastic constants increase which imposes a slower spatial variation of the director. When cooling the sample slowly through its nematic phase with a low rotational viscosity,59,60 this slow spatial variation can indeed be built up. On the other hand, in the quickly cooled sample the faster spatial variation of the director, characteristic of the nematic phase with lower elastic constants, is preserved.

(70) Swislow, G.; Schwartz, D.; Ocko, B. M.; Pershan, P. S.; Litster, J. D. Phys. Rev. A 1991, 43, 6815-6825. (71) Dvinskikh, S. V.; Furo´, I. Langmuir 2000, 16, 2962-2967. (72) Dvinskikh, S.; Benini, G.; Senker, J.; Vogel, M.; Wiedersich, J.; Kudlik, A.; Ro¨ssler, E. J. Phys. Chem. B 1999, 103, 1727-1737. (73) Dvinskikh, S. V.; Sitnikov, R.; Furo´, I. J. Magn. Reson. 2000, 142, 102-110.

Acknowledgment. This work has been supported by the Swedish Natural Science (NFR) and Engineering Science (TFR) Research Councils. S.V.D. thanks the Wenner-Gren Foundations for a scholarship. LA010693C