J. Phys. Chem. 1996, 100, 19007-19016
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Pattern Forming Instability in Homeotropically Aligned Liquid Crystals Shoichi Kai,* Ken-ichi Hayashi, and Yoshiki Hidaka Department of Applied Physics, Kyushu UniVersity, Fukuoka 812-81, Japan ReceiVed: May 28, 1996; In Final Form: August 15, 1996X
The electrohydrodynamic pattern forming instability (EHC) driven by an ac voltage applied to a homeotropically aligned nematic liquid crystal layer is experimentally studied near onset. By controlling an external magnetic field, many different scenarios become accessible. For finite fields various regular convective structures are observed, which we call, for example, wavy, chain, and bamboo-chevron patterns. The various types are documented with the help of a phase diagram governed by the frequency and the strength of the applied ac voltage. In addition the stability regimes in the voltage-wavenumber plane (the “Busse balloon”) are mapped out. One finds significant differences from the conventional planar case, for which a theoretical analysis is lacking so far. For zero magnetic field, on the other hand, no regular structure is observed, even immediately above the onset of EHC. A new spatiotemporally chaotic pattern called the soft mode turbulence directly appears via a supercritical bifurcation from the nonconvective state. This is due to the presence of the Goldstone mode related to the spontaneously broken rotational symmetry associated with the director component in the plane of the layer.
1. Introduction Electrohydrodynamic convection (EHC) occurs when an electric ac voltage V stronger than a certain threshold Vc is applied to a thin nematic layer with a negative dielectric constant a ) | - ⊥ < 0.1 Here, | and ⊥ are the dielectric constants parallel and perpendicular to the director, respectively, which represents the averaged direction of the molecular orientations of liquid crystals.1 EHC is a typical self-organization phenomenon and one of the most convenient systems to investigate pattern formation in systems far from equilibrium.1-4 In many former investigations on this subject, much attention has been paid to the planar geometry in which the director aligns parallel to glass substrates by proper surface treatments, because the theoretical and as well as the experimental situation is simpler.3-5 EHC in the planar orientation occurs directly as a first instability from a nonstructured state. On the other hand EHC in homeotropic samples, where the director aligns perpendicular to glass substrates, occurs as a secondary instability following the primary one, the so-called Freedericksz transition. The latter leads to a continuously growing planar component of the director near the midplane of the nematic layer. Therefore the onset of convection was considered to be qualitatively not too different from the planar case. Consequently the basic research on EHC in homeotropic nematics was not active and systematic. The situation changed substantially after recent theoretical and experimental studies on the homeotropic EHC. One is actually faced with some important and new findings such as a direct transition to spatiotemporal chaos (STC) at the convection onset and various new patterns like chains and spirals in the weakly nonlinear regime.6-12 In some analogy to the planar case one can have normal or oblique rolls with respect to the planar director orientation determined by the Freedericksz transition and/or the imposed magnetic field H. There exists the so-called Lifshitz point along the threshold curves as a function of H and the frequency f of the applied ac voltage, where the obliqueness of the rolls vanishes. Homeotropic EHC has a great experimental advantage for the study of the critical dynamics near a Lifshitz point X
Abstract published in AdVance ACS Abstracts, November 1, 1996.
S0022-3654(96)01539-0 CCC: $12.00
Figure 1. Schematic drawing of director alignments. (a) Below the threshold VF of the Freedericksz transition, a uniform homeotropic orientation of the director is shown. (b) Above VF a homogeneous Freedericksz distortion of the director is shown. (c) Above the onset of the electrohydrodynamic instability, the orientational distortion modes of the director due to EHC and Freedericksz instability are superimposed (proposed by Cladis and Fradin9).
because the system can be easily and precisely brought to the vicinity of such a point by changing H and/or f. This is in contrast to planar EHC, where the existence of a Lifshitz point depends on the appropriate values of material constants such as the dielectric constants and conductivities, which are not easily adjustable. In the case of finite magnetic field, one has stable periodic patterns above threshold, but experimental results on the Busse balloon in homeotropic EHC were missing at all, although its knowledge is decisive for the understanding of pattern selection and further bifurcation scenarios. After consideration of finite H, we address now zero magnetic fields. In a homeotropic sample one has rotational symmetry when the electric field strength V is below the threshold VF of the Freedericksz transition (Figure 1a). For V above VF the equilibrium orientation of the director acquires a nonzero projection onto the plane of the layer, which is characterized by an azimuthal angle φ (see Figure 1b). The angle φ is in © 1996 American Chemical Society
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Figure 2. Frequency f (a) and (b) dependences of wavenumbers of convective modes.
principle arbitrary (0 e φ e 360°). Thus the phase transition to convection is continuously degenerate and there must exist an associated soft mode (“Goldstone mode”). When convection sets in (V g Vc), the continuous translational symmetry is also spontaneously broken, resulting in an second Goldstone mode. Dramatic changes in the pattern formation scenario is then expected due to the coupling between the longwave orientational (φ) modes and short-wave EHC modes in our problem.11,12 The interplay of interacting Goldstone modes has been also discussed for other cases (see ref 13 and further references there). For a specific model (seismic waves),13 it is shown that the slow turbulence directly starts via a supercritical bifurcation from a spatially uniform state and a simple scaling law for the correlation time of the turbulence is described. In homeotropic EHC very recently irregular patterns have been observed close to the convection threshold;6,10 they were interpreted in terms of the additional Goldstone mode. Disordered patterns in experiments near the homeotropic EHC threshold are actually well-known for many years. They were however considered rather as a quite trivial phenomenon due to possibly small nonuniformities in the initial director alignment and not understood in the light of the mechanism described above. In order to be sure that generically a new class of STC is identified, careful and systematic work is essential.6,10 In that respect the previous investigation10 is quite preliminary and insufficient since no details of dynamics and bifurcation types have been reported. In the present paper we will experimentally describe the pattern formation in the weakly nonlinear regime and introduce the pattern morphology of EHC for a homeotropic nematics. We report here for the first time the corresponding stability diagram (the so-called Busse balloon), which will be the basis
Figure 3. Soft mode turbulence in the absence of H (f ) 500 Hz, ) 0.1, Vc ) 8.34 V): (a) real image; (b) power spectrum of a. The outer and inner wavenumber circles in the power spectrum correspond to the inverse of width of one roll and that of a pair of rolls, respectively.
for the discussion of the dynamics of various regular patterns as well as STC. 2. Experimental Section Preparation and Experimental Setup. We use the nematic liquid crystal N-(p-methoxybenzilidene)-p-n-butylaniline (MBBA), which is filled between two parallel glass plates, both surfaces of which are coated with transparent electrodes, indium tin oxide (ITO). The space between the glass plates is maintained with polymer spacers of 50 µm, and the lateral size of the space is of 1×1 cm2. Therefore the aspect ratio of the convective systems in the present study is 200. In order to realize homeotropic alignment, the surface of the glass plates is treated by a surfactant N,N-dimethyl-N-[3-(trimethoxysilyl)propyl]-1octadecanaminium chloride (DMOAP). In the absence of magnetic fields the preferred direction for convective rolls is determined at first by the spontaneously chosen (initially uniform) tilt of the director, which could be defined as the x-axis. When STC sets in, rolls respect only locally the corresponding director orientation, which changes randomly in time. Thus, macroscopically, on the average, there does not exist a preferred direction. For a finite magnetic field applied parallel to the glass plates, the x-axis points along the field direction. As already mentioned, the convection rolls can nucleate in a
Homeotropically Aligned Liquid Crystals
J. Phys. Chem., Vol. 100, No. 49, 1996 19009 magnetic fields: H ) 0 and H ) 1600 G. The threshold for a magnetically induced Freedericksz transition is HF ) 1100 G in our sample. The threshold value for the electrically induced Freedericksz transition, on the other hand, is VF ) 3.92 V. An ac voltage is applied to the sample by use of an electronic synthesizer controlled by a computer. We define the normalized external control parameter as ) (V2 - Vc2)/Vc2. The conductivity is σ| ) 3.30 × 10-7 Ω-1/m and σ⊥ ) 2.34 × 10-7 Ω-1/m which is controlled by using the 0.012 wt % doping of tetra-n-butylammonium bromide (TBAB). The frequency dependence of the wavenumber qx at the onset (actually ) 0.03) and the -dependence of qx for various frequencies of the present sample at H ) 1600 G are shown in Figure 2 as a reference. The critical frequency fc of the applied electric field is about 2600 Hz. The threshold for the onset of the convection is Vc ) 10 V at f ) 500 Hz and H ) 1600 G. The dielectric constant is | ) 4.21 and ⊥ ) 4.70; i.e., a ) -0.49. The temperature is controlled at 30 ( 0.02 °C with a control stage and a doublewall copper cavity. All images are taken by use of a charge coupled device (CCD) camera and stored onto a magnetic tape as well as a magnetic disk for later analysis by a computer. The software used for image analysis is the NIH-image. The procedure to obtain the Busse balloon is the conventional frequency-voltage jump method,14-16 which has already been described elsewhere in detail.17 3. Results and Discussion
Figure 4. Spatiotemporal map for soft mode turbulence: (a) ) 0.104; (b) 0.241; (c) 0.694.
Figure 5. Time correlation of soft mode turbulence. The time correlation functions are averaged over spatially.
perpendicular fashion (normal rolls) or with a definite angle (oblique rolls) with respect to the magnetic field above Vc. In the present study, we have conducted experiments for two
3.1. Spatiotemporal Chaos in the Absence of a Magnetic Field. The experiments in the absence of H have been carried out at f ) 500 Hz, where Vc ) 8.34 V holds. The convective roll orientation could point in any direction in the x-y plane because of the arbitrariness of the initial director tilting at the Freedericksz transition for H ) 0. The observed convective pattern in EHC is irregular and nonperiodic, i.e. STC as shown in Figure 3. However, we must distinguish STC due to the new dynamics in connection with the additional Goldstone mode from that due to initially random deformations triggered, e.g., by thermal fluctuations. This is checked by use of magnetic fields as follows:6 The regular zigzag patterns are observed in the presence of magnetic fields. Then removing the magnetic fields for fixed , the patterns gradually recover to the original chaotic patterns, although the preferred axis is once set up by H. This suggests the spontaneous creation of STC due to an intrinsic instability mechanism in the system. Preliminary results have already been reported by two groups.6 The details will be published elsewhere.18 Such an aspect is obviously observable in an image of the two-dimensional Fourier spectrum in Figure 3, where a circular spectrum is obtained with the wave vector corresponding to the inverse of the width of a pair of rolls.18 The spatiotemporal behavior of the patterns is very complicated; that is, STC is already observed at the onset. In order to obtain quantitative measures, the spatiotemporal map for various is taken as shown in Figure 4. The temporal and spatial scales of the image variation become shorter as is larger; that is, the map becomes more complicated with . However, no sharp and intermittent transitions are observed in Figure 4. Such features of spatiotemporal maps of STC are quite different from the ones usually observed in STC in the planar case; e.g., the defectmediated turbulence where active defect motions are observed and the corresponding map show sharp and intermittent characteristics.19 In order to obtain a better insight into the quantitative features, the time correlation function C(τ) has been calculated in one dimension and further processed by a spatial average. The result is shown in Figure 5. It is checked by
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Figure 6. -dependence of correlation time τ0.
long-time measurements that all the correlation functions decay virtually to zero at sufficiently large τ. The correlation time τ0 (in seconds) is obtained for various from these correlation functions and fits well to the following analytical expression:
τ0-1() ) (0.61 ( 0.01)
(1)
as is shown in Figure 6 by the solid line. It is evident that STC starts continuously at the first bifurcation point (the convective onset ) 0). This tendency is certainly different from STC in the planar case, where the onset of STC does not coincide with the convective onset but appears clearly when the convection is already sufficiently developed, for example at ) 0.45 in Figure 3 in ref 19. Relation 1 obviously indicates the softening of irregular modes (macroscopic fluctuation) toward the bifurcation point. We call this therefore the softmode turbulence (SMT),18,20 which is due to the particular dynamics in connection with the additional Goldstone mode. Some aspects of this scenario were recently predicted theoretically for EHC7,11,12 and discussed in a more general framework in ref 13. In refs 7, 11, and 12 it was shown in detail (for the material parameters of the nematic MBBA) that an unbalanced viscous torque on the director is the driving force behind the destabilization of the periodic patterns which leads eventually to STC. In ref 13, the scaling relation (1) was explicitly predicted for the first time.21 A route to turbulence and chaos via a single supercritical bifurcation has been observed so far in very few experiments as far as we know,22 whereas direct transition to STC via subcritical bifurcations from a nonconvective (or nonstructural) state have been observed more often. With respect to STC in homeotropic EHC the authors of refs 6 and 10 pretended to have found sufficient experimental evidence for the conjecture in ref 11, but a convincing quantitative analysis was missing. The present experimental results, in particular the behavior of the correlation time τ0 ∝ -1 indicating supercritical bifurcation and softening, must be considered as the first unambiguous proof for the peculiar dynamics associated with a soft mode. 3.2. Patterns in the Presence of Magnetic Field. As is already mentioned in the previous section, the application of magnetic fields leads to the preferred axis for the convective roll and regular EHC patterns by suppressing the additional Goldstone mode. In this section a variety of patterns and their dynamics are described, for a particular magnetic field of H ) 1600 G. Figure 7 shows the various patterns after the occurrence of EHC in the frequency-voltage plane ranging from f ) 100 Hz to 1900 Hz.23 In the low-frequency regime (f ) 100-600 Hz), at the first bifurcation point the zigzag pattern is observed which
Figure 7. Phase diagram in the presence of magnetic fields (H ) 1600 G).
is shown in Figure 8a, as well as the frequency dependence of its angle with the magnetic field (obliqueness). This dependence does not agree well with theoretical predictions,11 and its tendency is rather similar to the previous experimental result by Richter.6,7 Probably this may be due to the experimental conditions; e.g., the measurement is not performed exactly at onset ( ) 0, linear regime) but for ) 0.1 (weakly nonlinear regime) in order to facilitate optical detection. In addition the material parameters of the actual experiment had to be used. Even if the theory may well describe the obliqueness in the planar EHC,24 its reliability for homeotropic EHC remains to be shown. After continuously increasing the driving voltage V, at first the wavy patterns form above a second threshold (Figure 8b) until one finds finally turbulence. On the other hand, in the high-frequency regime (f ) 14001900 HZ), normal rolls are at first observed at the convective threshold, as shown in Figure 8c, and then successively the bamboo-chevron (BC, Figure 8d) and the chain patterns (Figure 8e)9 show up with an increase of V through transitions with the well-defined thresholds before reaching fully developed turbulence. In the regime between them (f ) 700-1300 Hz), transition from the normal rolls to the zigzag patterns takes place as V is raised from the convective onset. Then the chain pattern forms. Turbulence starts beyond the chain pattern regime. These various patterns form at well-defined thresholds and are summarized in Figure 7. In Figure 9 a typical pattern transition with change of V (i.e., ) is shown at f ) 1700 Hz and H ) 1600 G. In this regime the usual normal rolls first appear at the onset of convection (Figure 9a). Then the zigzaglike patterns are found but not of the conventional type with increase of V (Figure 9b,c). The roll deformation in Figure 9b seems to occur in the x-direction normal to the roll axis, while in the common zigzag instability it occurs in the y-direction. We call it a bamboo-chevron (BC) pattern. Then by increasing V, the distance between the periodic deformation of the BC pattern becomes shorter, as shown in Figure 9c,d. After further increasing the voltage, the normal rolls gradually appear again (see Figure 9e,f); that is, thereentrant normal (R-normal) roll phenomenon occurs. The chain pattern forms from the R-normal rolls by increase of V. The threshold of the transition between the R-normal rolls and the BC pattern
Figure 8. Typical patterns in the presence of H ) 1600 G. The observation area is 400 × 400 µm2: (a) zigzag pattern (The upper figure shows the frequency dependence of the oblique angle of the zigzag; (b) wavy pattern; (c) normal roll; (d) bamboo-chevron (BC) pattern; (e) chain pattern.
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Figure 9. Pattern transitions with an increase in applied voltages at f ) 1700 Hz with H ) 1600 G. The magnetic field was applied to the x-direction.
is not too well-defined at this moment. The measurements are still in progress. Finally we would like to mention that the turbulenceturbulence (T1-T2) transition is observed in all regimes when extremely high V is applied beyond the first turbulent state.3,25 However the T1 state in the homeotropic EHC is slightly different from that in the planar EHC. The background large scale flow in T1 in the homeotropic EHC propagates in the
direction of the original roll orientation, while no such a flow is observed in the planar EHC. We call it the traveling DSM (the dynamic scattering mode). Here DSM is a traditional name for fully developed turbulence in EHC in liquid crystals.1 The detail will be reported elsewhere.26 3.3. Busse Balloon. In EHC for planar alignments, there exist several publications related to the Busse balloon.15-17,27 However, there has been no report for EHC in homeotropic
Homeotropically Aligned Liquid Crystals
Figure 10. Busse balloon in homeotropic EHC with H ) 1600 G: NS, neutral stability line; E, Eckhaus boundary; EZ, Eckhaus-zigzag line; ZZ, zigzag line (for detail see text).
alignments because a regular pattern is not observable at the onset in the absence of magnetic fields. It seems to be unclear at the moment how to identify secondary instabilities in the case of the STC scenario. Therefore the study here is conducted when a magnetic field H ) 1600 G is applied at the standard frequency f0 ) 800 Hz with the help of the frequency-voltage jump method,14-17 where regular normal rolls are obtained at the onset. In a two dimensional planar system, one observes basically three different selection dynamics for EHC pattern formation such as (1) simple decay of rolls, (2) elastic relaxation of rolls (in some cases by defects), and (3) relaxation of rolls via an additional instability. The corresponding transition lines define the Busse balloon.17 In a two dimensional homeotropic system, however, the situation is slightly different. In addition to the three mechanisms explained above there exists a relaxation process of rolls via passing transiently two successive instabilities. Moreover an unusual (backward bending) Eckhaus boundary is obtained. Therefore the Busse balloon becomes more complicated and looks quite different in comparison to the conventional type. In Figure 10 such a Busse balloon of the new type is shown. The finally stable pattern are normal rolls near the onset at the band center Q ) 0 under the given conditions. Here NS, E, ZZ, and EZ indicate respectively the neutral stability, the Eckhaus, the zigzag, and the Eckhaus-zigzag boundaries. Below NS, the rest state is stable. In region I, the final pattern is selected through the Eckhaus instability when the frequencyvoltage jump is performed. In region II, the elastic relaxation of rolls and/or defect motions takes place during selection of a final pattern. In region III, the zigzag pattern appears via the zigzag instability. In region IV, the Eckhaus instability occurs at first, then the zigzag instability starts after a certain time delay after the Eckhaus process, and finally the steady zigzag patterns form. These transient processes are shown in Figure 11. In Figure 11a the pattern selection process in the region below the NS line is shown. Figure 11b shows the Eckhaus instability (region I) where the periodic defect creation/annihilation is observed (Figure 11b(2)). In region II, no remarkable change of normal rolls is observed (see Figure 11c) and small wavenumber adjustments take place through elastic modulations. In region III, no remarkable change of the zigzag pattern is observed after the frequency-voltage jump.
J. Phys. Chem., Vol. 100, No. 49, 1996 19013 The most interesting fact is observed in region IV as shown in Figure 11e. Here Figure 11e(1) shows the image immediate after the frequency-voltage jump (tjp ) 0). After 18 s (tjp ) 18 s) in Figure 11e(2), the Eckhaus instability can be observed. Then it is over for tjp ) 28 s in Figure 11e(3) and simultaneously the zigzag instability starts. Finally the stable zigzag pattern appears. Figure 11e(4) indictes the steady state of the zigzag pattern (tjp ) 170 s). Therefore in region IV transiently two instabilities, the Eckhaus and the zigzag ones, occur until the finally stable pattern forms. Region IV is surrounded by the E, EZ, and NS lines. One may notice however very strange aspects in these lines. It is worth mentioning that, for example, the Eckhaus (E) line bends toward Q ) 0 for large and the EZ line is parallel to the Q axis. The former aspect might be related to the presence of the R-normal rolls. No related theoretical results are available so far. One might say at least for the two successive instabilities that the growth rate of the Eckhaus mode is larger than that of the zigzag one at the beginning, but the zigzag mode can grow as the wavenumber reaches Q ) 0 because the band center is zigzag unstable.28 The EZ line is not the extrapolation of the ZZ line and independent of that. This can also be explained by the scenario in ref 28. It is therefore interesting to study dependences of the Busse balloon on f0 and H related to changes of these instability lines. 4. Summary Finally we summarize the results obtained in the present study as follows. For zero magnetic field, the soft mode turbulence (new STC) directly appears via a supercritical bifurcation from the nonconvective state. This suggests that the Goldstone mode related to the additional rotational symmetry plays an important role for the formation mechanism of STC. On the other hand, for finite magnetic fields, the various regular patterns are observed as well as their stability regimes. Some new pattern types (wavy, reentrant normal rolls, BC patterns, propagating DSM) are found in this work. The phase diagram determined for the homeotropic EHC is quite different from that for planar EHC. In Figure 12 we schematically draw the phase diagram obtained from the present study in the frequency-voltage plane. The lines indicate the thresholds for the onset of the corresponding patterns. The formation mechanism of the reentrant normal roll is interesting. This might be related to the backward bending of the Eckhaus boundary. The detailed study will be necessary. The Busse balloon obtained in the presence of H is not a conventional balloon. Especially the Eckhaus boundary has not a simple profile and bends toward the center Q ) 0 for large . No theory predicts such an aspect. In some regions in the balloon two successive instabilities (the Eckhaus and the zigzag instabilities) occur for the pattern selection dynamics from initial to final patterns after the frequency-voltage jump is made. The zigzag instability occurs when the wavenumber approaches Q ) 0 after the Eckhaus instability because the band center of the Busse balloon in homeotropic EHC is zigzag unstable.28 It is therefore very interesting to investigate the profile of the Busse balloon as the frequency f0 is changed. Each boundary must change because the observed patterns change with f, as shown in Figure 12. This is now in progress and will be reported elsewhere in the near future. Acknowledgment. It is a great pleasure for the authors to dedicate the present article to Professor John Ross on the occasion of his 70 years old anniversary. S.K. was working
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Figure 11. Typical selection processes after frequency-voltage jump.
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J. Phys. Chem., Vol. 100, No. 49, 1996 19015
Figure 11. (continued)
of the manuscript. This work is partly supported by the Grantin-Aid for Scientific Research from the Ministry of Education, Sport, Science and Culture in Japan. References and Notes
Figure 12. Schematic drawing of the phase diagram as shown in Figure 7. The R-normal indicates the reentrant normal rolls. The dotted lines are not definite thresholds yet and still continued to be measured.
together with John as a postdoctoral fellow from 1979 to 1981 at Stanford University. He studied about chemical instabilities related to the Liesegang ring formation.29 John was a great teacher for a young scientist who was just starting his life as a scientist. S.K. would like to express his sincere thanks to John for continuous encouragement since then. The authors would like to thank M. I. Tribelsky (University of Tokyo) and W. Pesch and A. Rossberg (University of Bayreuth) for valuable comments and stimulating discussions. We would like to express especially our hearty thanks to W. Pesch for his critical reading
(1) de Gennes, P. G. The Physics of Liquid Crystals; Clerendon: Oxford, U.K., 1974. (2) Kai, S. Pattern Formation in Complex DissipatiVe Systems; World Science: Singapore, 1991. (3) Kai, S.; Hirakawa, K. Prog. Theor. Phys. 1978, Suppl. 68, 212. Kai, S.; Zimmermann, W. Prog. Theor. Phys. supp. 99, 458 1989, Suppl. 99, 458. (4) Kramer, L.; Bodenshatz, E.; Pesch, W.; Thom, W.; Zimmermann, W. Liq. Cryst. 1989, 5, 699. (5) Kramer, L.; Pesch, W. Annu. ReV. Fluid Mech. 1995, 27, 515. (6) Richter, H.; Kloepper, N.; Hertrich, A.; Buka, A. Europhys. Lett. 1995, 30, 37. Tribelsky, M. I.; Hayashi, K.; Hidaka, Y.; Kai, S. Proceedings of the First Tohwa UniVersity Statistical Physics Meeting; Tohwa University: Fukuoka, Japan, 1995. (7) Hertrich, A. Thesis University of Bayreuth, 1995. (8) Kai, S.; Adachi, Y.; Nasuno, S. Spatio-Temporal Patterns; Cladis, P. E., Palffy-Muhoray, P., Eds.; SFI Studies in the Sciences of Complexity; Addison-Wesley: 1995. (9) Cladis, P.; Fradin, C. Private communication, 1995. Cladis and Fradin first observed and named it as a chain pattern. (10) Richter, H.; Buka, A.; Rehberg, I. Phys. ReV. 1995, E51, 5886. (11) Hertrich, A.; Decker, W.; Pesch, W.; Kramer, L. J. Phys. (Paris) II-2 1992, 1915. (12) Rossberg, A.; Hertrich, A.; Kramer, L.; Pesch, W. Phys. ReV. Lett. 1996, 76, 4729. (13) Tribelsky, M. I. Int. J. Bif. Chaos, in press. Tribelsky, M. I.; Tsuboi, K. Phys. ReV. Lett. 1996, 76, 1631. (14) Goren, G.; Procaccia, I.; Rasenat, S.; Steinberg, V. Phys. ReV. Lett. 1989, 63, 1237. (15) Braun, E.; Rasenat, S.; Steinberg, V. Europhys. Lett. 1991, 15, 597. (16) Nasuno, S.; Kai, S. Europhys. Lett. 1991, 14, 779. (17) Nasuno, S.; Sasaki, O.; Kai, S.; Zimmermann, W. Phys. ReV. A 1992, 46, 4954. (18) Hidaka, Y.; Hayashi, K.; Tribelsky, M. I.; Kai, S. Manuscript in preparation.
19016 J. Phys. Chem., Vol. 100, No. 49, 1996 (19) Kai, S.; Kohno, M.; Andoh, M.; Imasaki, M.; Zimmermann, W. Mol. Cryst. Liq. Cryst. 1991, 198, 247. (20) Hidaka, Y.; Hayashi, K.; Kai, S. J. Jpn. Soc. Fluid Dyn. (in Japanese), 1996, 15 (No. 3), 163. (21) For the particular case (EHC) the similar relation has been also obtained theoretically.12 (22) E.g.: Hu, Y.; Ecke, B.; Ahlers, G. Phys. ReV. Lett. 1995, 74, 5040. (23) Hayashi, K.; Hidaka, Y.; Kai, S. Proceedings of the First Tohwa UniVersity Statistical Physics Meeting; Tohwa University: Fukuoka, Japan, 1995. (24) Dennin, M.; Cannell, D. S.; Ahlers, G. Mol. Cryst. Liq. Cryst. 1995, 261, 337. (25) Kai, S.; Zimmermann, W.; Andoh, M.; Chizumi, N. Phys. ReV. Lett. 1990, 64, 1111.
Kai et al. (26) Hidaka, Y.; Hayashi, K.; Kai, S. Manuscript in preparation. (27) Lowe, M.; Gollub, J. P. Phys. ReV. Lett. 1985, 55, 2575; Phys. ReV. A 1985, 31, 3893. (28) A. Rossberg proposed a possible interpretation of this behavior (private communication). In region IV, rolls are actually only Eckhaus unstable and near the band center Q ) 0 zigzag unstable. Therefore the zigzag instability will set in only after the wavenumber reduction via the Eckhaus is achieved. This leads to the successive two instabilities. (29) Kai, S.; Mueller, S. C.; Ross, J. J. Chem. Phys. 1982, 76, 1392. Mueller, S. C.; Kai, S.; Ross, J. J. Phys. Chem. 1982, 86, 4078. Kai, S.; Mueller, S. C.; Ross, J. J. Phys. Chem. 1983, 87, 806.
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