J. Phys. Chem. B 1998, 102, 6337-6341
6337
Determination of Splay and Twist Relaxation Modes in Nematic Liquid Crystals from Dynamic Light Scattering Experiments Redouane Borsali,†,‡,§ Do Y. Yoon,† and R. Pecora*,‡ IBM Research DiVision, Almaden Research Center, 650 Harry Road, San Jose, California 95120-6099, and Department of Chemistry, Stanford UniVersity, Stanford, California 94305-5080 ReceiVed: April 24, 1998
The dynamic behavior of a nematic liquid crystal is investigated by photon correlation spectroscopy. Two relaxation modes, the so-called splay and twist, are identified for a specific experimental geometry involving the directions of the wave vector q, the polarizations of the incident and scattered beams, and the vector director of the nematic liquid crystal n. The results obtained in this work are in good agreement with theoretical predictions describing the dynamics in nematic liquid crystals. The relaxation rates are in reasonable agreement with those reported for the splay mode in earlier experimental results using the same optical configuration, as well as with those for the twist mode using a more difficult experimental geometry. We emphasize here that the relaxation times of both modes may be obtained simultaneously using a relatively simple experimental geometry.
Introduction and Theoretical Background In isotropic liquids, scattering of light is caused by fluctuations of the optical dielectric constant, δ(r,t), and are due mainly to density fluctuations, which are in turn caused largely by fluctuations in the temperature. Scattering of light in nematic liquid crystals, in their ordered phases, is roughly 106 times stronger than in ordinary isotropic liquids. The reason, as has been shown by de Gennes and Prost,1 is that an additional and important contribution to δ(r,t) arises from the spontaneous thermal fluctuations of the vector director n of a defect-free nematic domain. Such dynamic distortions of the nematic structure can be characterized in terms of three elastic deformations, namely, twist, splay, and bend. Dynamic light scattering2 is a noninvasive technique that provides a useful tool for determining the relaxation rates of these three viscoelastic deformations from the relaxation frequencies of the thermal fluctuations. Although the molecules in a nematic liquid crystal exhibit a distribution of orientations, they tend to be oriented parallel to a director n. The director, in turn, may differ from its equilibrium orientation n0 in a given region of the liquid crystal by a small fluctuation δn ) n - n0. For the sake of simplicity, we consider a nematic liquid crystal with n0 parallel to the z axis. Then the fluctuations at any point r can be described by nx(r) and ny(r), and the total curvature elastic energy is
Ft )
∫
1 2
{ ( K11
)
(
)
∂nx ∂ny 2 ∂nx ∂ny 2 + K22 + + ∂x ∂y ∂y ∂x ∂nx 2 ∂ny K33 + ∂z ∂z
[( ) ( ) ]} 2
dV (1)
where V is the volume of the sample. In eq 1, the three terms * To whom correspondence should be addressed. † IBM Almaden Research Center. ‡ Stanford University. § On leave from CERMAV-CNRS and Joseph Fourier University, P.O. Box 53, F-38041, Grenoble Cedex 9, France.
represent the splay, twist, and bend free energy densities and K11, K22, and K33 are the “bare” Frank elastic constants.3 It is convenient to write the quantities nx(r) and ny(r) in terms of their spatial Fourier components:
nx(r) )
∑q nx(q) exp(iq‚r)
(2)
Since the Fourier component of wave vector q is
∫
nx(q) ) V-1 nx(r) exp(-iq‚r) dV
(3)
it is evident that
(∂nx/∂x)2 )
∑q |nx(q)|2qx2,
etc.
(4)
The theoretical development of the formalism describing light scattering by liquid crystals1,4-9 can be greatly simplified by properly defining the initial orientation of the director axis n0 with respect to the scattering wave vector q, such that the new x′ axis coincides with the unit vectors e1 (perpendicular to the z axis in the q-z plane) and y′ then coincides with e2. In this new system, qy ) 0 and the components of n(q) along e1,2 are n1,2(q), respectively. Within the framework of such considerations, the total free energy associated with these deformations can be rewritten from relation 1 as
1 Ft ) V 2
∑q |n1(q)|2(K11q⊥2 + K33 q|2) + |n2(q)|2(K22q⊥2 + K33q|2) (5)
where q| and q⊥ are the wave vector components parallel and perpendicular to the director n, respectively. The calculation of the thermal average (〈...〉 ) of |n1(q)|2 is done using the equipartition theorem.10 For a classical system in thermal equilibrium, the average energy, per degree of freedom, is equal to (1/2)kBT, (kBT is the Boltzmann energy). It follows that
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6338 J. Phys. Chem. B, Vol. 102, No. 33, 1998
Borsali et al.
kBT (K q 2 + K33q|2)-1 V ββ ⊥
〈|nβ(q)|2〉 )
(6)
This is the central equation of nematic fluctuation theory.1 The expression for the scattering cross-section (scattered intensity) of nematic liquid crystals can be derived by relating the fluctuations of n to the fluctuations of the optical dielectric tensor . If we have an incident plane electromagnetic wave with polarization i and wave vector qi, and we observe the f component of the scattered wave, we find that the contribution to the fluctuation in the dielectric constant due to the fluctuations of the director is
δif ) ∆[i| (f‚δn) + f|(i‚δn)]
(7)
where i| ) n0‚i, and f| ) n0‚f are the projections of, respectively, the incident light wave polarization direction and scattered wave polarization directions on the equilibrium director. ∆ ) | ⊥ is the anisotropy of the dielectric constant. The Fourier components of the director fluctuations may be written in terms of the coordinate system defined previously as δn(q) ) e1n1(q) + e2n2(q). Equation 7 may be further rewritten as
∑
δif ) ∆
nβ(q)(iβf| + i|fβ)
β)1,2
dΩ
) 〈|A| 〉 ) 2
( ) ∆ω2V
2
4πc2
∑ 〈|nβ(q)|2〉 (iβf| + i|fβ)2 β)1,2
(9)
where 〈...〉 denotes the thermal average given by eq 6 and A is the scattering amplitude. The intensity of the scattering by a nematic liquid crystal per unit volume V is then given by
dσ
)
dΩ
( ) π∆2 λ02
∑
kBT
β)1,2
(iβf| + i|fβ)2 Kββq⊥2 + K33q|2
(10)
The autocorrelation function measured in dynamic light scattering from a nematic liquid crystal has been shown,1,4-9,11-17 to exhibit two relaxation modes: -Γ1(q)t
ST(q,t) ) A1(q) e
-Γ2(q)t
+ A2(q)e
(11)
where A1(q) and A2(q) are the amplitudes of the normal modes having frequencies Γ1(q) and Γ2(q). These are the so-called splay-bend and twist-bend modes.10-16 The general expressions for these two frequencies or normal modes are given in terms of the elastic constants given in eq 5 and the viscosities
Γβ(q)(β)1,2) )
K33q| + Kββq⊥2 ηβ(q)
(12)
The viscosities ηβ(q)(β)1,2) are defined as
η1(q) ) γ1 -
(q⊥2R3 - q|2R2)2 q⊥4ηb
+
q⊥2q|4(R1
η2(q) ) γ1 -
+ R3 + R4 + R5) + q| ηc (13)
q⊥2ηa+ q|4ηc
A2 )
( ) ( )
(n0 - ne cos θ)2 dσ ∆ 2 ) kBT dΩ2 2λ0 K22[n02 sin2 θ + (ne - n0 cos θ)2]2 (16)
where ∆ ) | - ⊥ and q⊥ ) (2π/λ0)[n02 sin2 θ + (ne - n0 cos θ)2]1/2 with n0 and ne being the temperature-dependent ordinary and extraordinary refractive indices of the nematic liquid crystal. λ0 is the wavelength of the incident beam in a vacuum (in this case λ0 ) 4880 Å) and θ the scattering angle inside the nematic sample related to the scattering angle (in the laboratory frame) θlab by Snell’s law21 (i.e., n0 sin θ ) nm sin θlab), where nm is the refractive index of the matching liquid. When q| ) 0, the frequencies may be found from eqs 1214:
Γ1(q) ) Γsplay(q) )
K11q⊥2 γ1 - (R32/ηb)
)
K11q⊥2 ηsplay
K22q⊥2 K22q⊥2 Γ2(q) ) Γtwist(q) ) ) γ1 ηtwist
(17)
(18)
We show below that the relaxation times in eqs 17 and 18 for the nematic liquid crystal pentylcyanobiphenyl (5CB) may be obtained by dynamic light scattering in the simple geometry described. These relaxation times (or viscoelastic parameters) will provide a basis of comparison for further studies of the cholesteric liquid crystals formed when 5CB is mixed with a chiral agent and also when a polymer is added to the system to stabilize the cholesteric phase. These systems, in addition to their fundamental importance, are of interest as components of display devices.22 Experimental Section
4
q|2R2
( ) ( )
ne2 sin2 θ dσ ∆ 2 A1 ) ) k T dΩ1 2λ0 B K [n 2 sin2 θ + (n - n cos θ)2]2 11 0 e 0 (15)
(8)
where iβ ) i‚eβ is the component of i along eβ. The differential scattering cross-section may then be shown to be
dσ
where Ri refer to the five Leslie viscosity coefficients,18,19 γ1 is the twist viscosity, and ηa, ηb, and ηc are Miesowicz viscosities.20 It has been shown that, using different optical geometries, one can have access to the three pure deformations of nematic liquid crystals. There are different configurations which can be utilized in order to extract separately the three deformations (splay, twist, and bend). In this paper, however, we shall focus on one configuration called hereafter the “splay-twist geometry” and show that its dynamic behavior is described by two relaxation modes, splay and twist, which can be observed simultaneously. In this configuration the incident beam is vertically polarized and the scattered light horizontally polarized with the vector director n, perpendicular to the plane of the incident and scattered directions. The direction of a planar monodomain is oriented parallel to the incident polarization. For such an experimental geometry, the above relations are greatly simplified (q| ) 0), and the amplitudes of the two light scattering modes in eq 11 may be written as
(14)
Sample Preparation and Equipment. Pentylcyanobiphenyl (5CB) was purchased from Aldrich Chemical and used as received. This liquid crystal is nematic at room temperature, and the nematic-isotropic transition temperature is about T ) 35 °C. The sample preparation was made with great care since light scattering is extremely sensitive to defects such as dust
Relaxation Modes in Nematic Liquid Crystals
J. Phys. Chem. B, Vol. 102, No. 33, 1998 6339
Figure 1. Variation of the splay and twist mode amplitudes normalized to (∆/2λ0))2kBTK11 as a function of the scattering angle at the temperature T ) 32 °C, calculated using eqs 15 and 16, where K11/K22 ) 1.5.13
and disclination lines in liquid crystals. The glass substrates which consist of two parallel planes were sonicated in refined deionized water. We prepared a homogeneously aligned nematic cell with buffed polyimide alignment layers on the inner surfaces of the glass plate substrates. The rubbing directions of the top and bottom plates were parallel, and the cell gap was controlled by a 25 µm film (Ablefilm Corp.). The sample (5CB) was then filled under vacuum, and the cell was sealed with epoxy resin. The dynamic light scattering apparatus used in these experiments utilized a Spectra Physics Model 165 argon ion laser operating at 4880 Å. The laser power was used at 200 mW (to ensure stability) and with filters to reduce the incident beam. The autocorrelation function of the scattered intensity was obtained by using a Brookhaven BI9000 autocorrelator. The scattered light of a vertically polarized laser is measured at several angles in the range (between 15 and 65°). The total intermediate scattering function ST(q,t) is related to the measured homodyne intensity autocorrelation function G(2)(q,t) by the Siegert relation:23
G(2)(q,t) ) B[1 + R|ST(q,t)|2]
(19)
where B is the baseline and R is the spatial coherence factor that depends on the detection geometry. Cumulant24 and single and double exponential fits were used to fit the data and to resolve the relaxation modes. To avoid operating in the heterodyne case, the cell was always positioned in the index matching bath such that R reached a maximum value generally ranging between 0.7 and 0.8, an acceptable value for homodyne detection. The sample was temperature controlled at (1 °C. Results and Discussion The relations that are given in the Introduction may be used to determine the optimal experimental conditions necessary to simultaneously observe the splay and twist modes in nematic liquid crystals. The variation of the amplitudes of the splay and the twist modes normalized to (∆/2λ0)2kBTK11 as a function of the scattering angle at T ) 32 °C is plotted in Figure 1 (see eqs 15 and 16), where K11 ) 1.5K22 according to experimental results obtained by Gu et al.13 The amplitude of the twist mode vanishes at the “magic angle”, θlab, about 23° at the temperature T ) 32 °C, and, therefore, the dynamics is described by the splay deformation only. At other angles, however, the amplitudes of these two modes come close to each other at the
Figure 2. (a) Typical dynamic light scattering autocorrelation function of the nematic liquid crystal 5CB at T ) 32 °C and θ ) 55°. The dots represent the experimental data and the solid line a single exponential fit. (b) Same system as in a with a double exponential fit.
scattering angles (θ < 15° and θ > 40°). In this case, one may expect that the measured autocorrelation function would be described by two relaxation modes. Such dynamics can be resolved if the ratio of the frequencies is large enough. Similar curves were obtained at different temperatures (T ) 22, 25, 30, and 32 °C) with, however, a slight variation of the magic angle (about 1-2°). On the basis of the above theoretical considerations, we have investigated the dynamics of 5CB using depolarized dynamic light scattering at different temperatures (T ) 22, 25, 30, and 32 °C) and angles from 15 to 65° in steps of 5°. Our results show that the dynamics is described by a single relaxation mode for 15° < θ < 40° (cumulant and single exponential fits gave approximately the same results except for θ ) 40°, where deviations were observed) and that two relaxation processes are identified at higher scattering angles. A typical dynamic light scattering intensity autocorrelation function from 5CB at scattering angle θ ) 55° and T ) 32 °C is displayed in Figure 2a. The dots represent the experimental data, and the solid line gives a single exponential fit. The single exponential fit is clearly not satisfactory. Consequently, the data were analyzed as the sum of two exponential functions, in accordance with eq 11. As shown in Figure 2b, a satisfactory description is achieved. The same data treatment was done for the experiments performed at the other angles and temperatures. For angles θ > 65°, the dynamics of the splay mode becomes too fast to be measured using photon correlation techniques. Figure 3 shows the variations of the amplitudes of the two relaxation processes as a function of the scattering angle at the temperature T ) 32 °C. It is shown that for 15° < θ < 40°,
6340 J. Phys. Chem. B, Vol. 102, No. 33, 1998
Borsali et al.
Figure 3. Variation of the amplitudes of the splay and twist modes as a function of the scattering angle at the temperature T ) 32 °C.
the scattering plane (so-called configuration B in ref 13). For the splay mode, however, our results are slightly different from those reported by Gu et al.13 but still in reasonable agreement. The difference is likely due to the fact that, in the cumulant analysis used by Gu et al.,13 an average relaxation time is determined; i.e., 〈Γ〉 ) (asplayΓsplay + atwistΓtwist)/(asplay + atwist), and this difference increases with the scattering angle. This was observed in their results where it was stated that systematic deviations were reported for scattering angles θ > 35° and are due to an increasing contribution of the twist mode under configuration A (splay-twist), the same geometry used in this work. It is this contribution which has been measured and is at the origin of the twist mode measured simultaneously with the splay deformation. Other possibilities for the discrepancy arise from different sample preparation methodsssample thickness, defects, etc. It is also possible that since the two modes observed in the dynamic light scattering are so close together in relaxation time that the separation is giving large errors in the fits. Usually, however, it is the weaker mode (in this case the twist mode) that would have the large error, while in our case, it is the stronger mode relaxation time that is different from that of Gu et al.13 Conclusion
Figure 4. Variation of the frequencies of the splay and twist modes as a function of q⊥2 for temperatures T ) 25 °C and T ) 32 °C.
TABLE 1: Temperature Variation of the Twist Mode Γtwist/q⊥2 ) K22/ηtwist) Parameter (K22/ηtwist), 10-7cm2/s Sefton et al.11 Gu et al.13 this work
t ) 22 °C
t ) 25 °C
t ) 30 °C
t ) 32 °C
4.4
5.0 5.18 4.75
5.9 6.10 6.15
6.5 6.83 6.75
4.3
TABLE 2: Temperature Variation of the Splay Mode Γsplay/q⊥2 ) K11/ηsplay Parameter K11/ηsplay, 10-6cm2/s t ) 22 °C
t ) 25 °C
t ) 30 °C
t ) 32 °C
0.91
0.75 1.11
0.93 1.40
1.03 1.50
al.13
Gu et this work
the splay mode dominates the dynamics, and for θ > 40°, a second mode emerges (twist mode). Its amplitude is very small compared to that of the splay relaxation, but it is measurable and becomes more important as the angle is raised. This behavior resembles very much that described in Figure 1 and is in good agreement, at least qualitatively, with the theoretical prediction. Figure 4 gives the variations of the two frequencies as a function of q⊥2 (see eqs 17 and 18) for two temperatures, T ) 25 and 32 °C. From the slopes of plots similar to Figure 4, at different temperatures, we obtain the viscoelastic parameters K11/ηsplay and K22/ηtwist, and the results are listed in Tables 1 and 2. For the twist relaxation mode, K22/ηtwist, excellent agreement is found with the results reported by Sefton et al.11 and those reported by Gu et al.13 measured using another VH optical geometry, namely, that with the nematic director n in
We have investigated the dynamics of a nematic liquid crystal using dynamic light scattering in a VH geometry in which the vector director n of the nematic liquid crystal is perpendicular to the scattering plane. For this optical configuration, theoretical considerations predict that two relaxation modes, corresponding to the splay and the twist modes, may appear in the measured time autocorrelation functions. We have shown that for 5CB one relaxation mode (splay) characterizes the autocorrelation function for angles θ < 40° and that two relaxation modes (splay and twist) are observed at scattering angles 40° < θ < 65°. Thus, only one relatively easy to attain scattering geometry is needed to obtain both relaxation frequencies. Acknowledgment. The authors are grateful to Prof. A. Jamieson for many helpful discussions, to Edward Hanson for his assistance with the light scattering experiments, and to Dr. U. P. Schroeder for his help with the cell preparation during the initial stages of this work. This work was supported by National Science Foundation Grant CHE-9520845 to R.P., by the National Science Foundation through the Stanford Materials Research Science and Engineering Center, and by grants to R.B. by NATO and the IBM Corp. References and Notes (1) De Gennes, P.-G.; Prost, J. The Physics of Liquid Crystals, 2nd ed.; Oxford University Press: Oxford, U.K., 1993. (2) Berne, B. J.; Pecora, R. Dynamic Light Scattering with Applications to Chemistry, Biology and Physics; Wiley: New York, 1976. (3) Frank, F. C. Discuss. Faraday Soc. 1958, 25, 19. (4) De Jen, W. H. Physical Properties of Liquid Crystalline Materials; Gordon & Breach: New York, 1980; Chapters 6 and 7. (5) De Gennes, P.-G. Mol. Cryst. Liq. Cryst. Lett. 1977, 34, 177. (6) Khoo I.-C. Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena; Wiley: 1995; Chapters 3 and 5. (7) Orsay Liquid Crystal Group. J. Chem. Phys. 1969, 51, 816. (8) Orsay Liquid Crystal Group. Phys. ReV. Lett. 1969, 22, 1361. (9) Belyakov, V. A.; Kats, E. I. In Light scattering near Phase Transitions; Cummins, H. Z., Levanyuk, A. P., Eds.; North-Holland: Amsterdam, 1983; Chapter 4. (10) Landau, L. D.; Lifshitz, E. M. Statistical Physics, Part 1; Nauka: Moscow, 1976. (11) Sefton, M. S.; Bowdler, A. R.; Coles, H. Mol. Cryst. Liq. Cryst. 1985, 129, 1. (12) Da, X.; Paul, G. L. Mol. Cryst. Liq. Cryst. 1985, 124, 287.
Relaxation Modes in Nematic Liquid Crystals (13) Gu, D.; Jamieson, A. M.; Rosenblatt, C.; Tomazos, D.; Lee, M.; Percec, V. Macromolecules 1991, 24, 2385. (14) Gu, D.; Jamieson, A. M.; Kawasumi, M.; Lee, M.; Percec, V. Macromolecules 1992, 25, 2155. (15) Chen, F.-L.; Jamieson, A. M. Macromolecules 1994, 27, 4691. (16) Zink, H.; De Jeux, W. H. Mol. Cryst. Liq. Cryst. 1987, 177, 1506. (17) Coles, H. J.; Sefton, M. S. Mol. Cryst. Liq. Cryst. Lett. 1985, 1, 151. (18) Leslie, F. M. Q. J. Mech. Appl. Math. 1966, 19, 357.
J. Phys. Chem. B, Vol. 102, No. 33, 1998 6341 (19) Leslie, F. M.; Waters, C. M. Mol. Cryst. Liq. Cryst. 1985, 123, 101. (20) Miesowicz, M. Bull. Int. Acad. Pol. Sci. Lett. Ser. A 1936, 28, 228. (21) Born, M.; Wolf, E. Principles of Optics; Oxford University Press: Oxford, U.K., 1980; Chapter 14. (22) Borsali, R.; Schroeder, U. P.; Yoon, D. Y.; Pecora, R. Phys. ReV. E, in press. (23) Siegert A. MIT Rad. Lab. Rep. 1943, 465. (24) Koppel, D. E. J. Chem. Phys. 1972, 57, 4814.
