Devil’s staircase between antiferroelectric SCA* and ferroelectric SC* phases in liquid crystals observed in free-standing films under temperature gradients Keizo Itoh,a Masaaki Kabe,b Kouichi Miyachi,b Yoichi Takanishi,b Ken Ishikawa,b Hideo Takezoeb and Atsuo Fukuda*b aKashima Oil Co. L td., R & D Department, T owada, Kamisu-machi, Kashima-gun, Ibaraki 314-02, Japan bT okyo Institute of T echnology, Department of Organic and Polymeric Materials, O-okayama,Meguro-ku, T okyo 152, Japan By studying the electro-optical properties and the textures of the subphases successively emerging between antiferroelectric SCA* and ferroelectric SC* (the Devil’s staircase), we have revealed several interface effects in both homogeneous and homeotropic cells; free-standing films are most suitable for making observations almost free from the effects.By applying appropriate temperature gradients to the free-standing films, we can directly see any part of the subphase sequence in the visual field of an optical microscope. The two ferrielectric subphases on the low- and high-temperature sides of ferrielectric SCc* together with another ferrielectric subphase between the antiferroelectric subphase (designated as AF in ref. 9) and SC* were thus confirmed to exist definitely.We have discussed the origin of these successive subphases in terms of the several theoretical models reported so far, concluding that the ANNNI model with the third-nearest-neighbour interaction well describes their Devil’s staircase character. Various tilted, chiral, fluid smectic (SC*-like) phases have been Among these subphases, SCa* is quite different from the other ones between SCA* and SC* in the sense that it is located found in antiferroelectric liquid crystals,1,2 which are shown in Scheme 1 in increasing order of temperature; some of the above SC*.In previous papers,2,22–24 we reported that SCa* forms not only the electric-field-induced staircase, SCa* (qE), phases may not actually occur but, when they do exist, they follow this order in almost all the compounds and mixtures but also the temperature-induced one, SCa* (qT).However, the staircase characters are not so typical to allow us meaningful investigated so far.1–25 The SCA* and SC* phases are the fundamental ones and the others between them, together with comparison between theory and experiment.We have also conjectured that the subphases emerging between SCA* and SCa*, are the subphases. Ferrielectric SCc* and antiferroelectric AF phases seem to be secondarily fundamental.8–11 On the SC* form another temperature-induced staircase describable by the one-dimensional Ising model with long-range repulsive high- and low-temperature sides of SCc*, there may emerge ferrielectric FIH and FIL phases, respectively.8–11 The existence interactions.2,8–11,26–28 Since some other theoretical explanations have also been published,29–53 it is appropriate to of FI, another ferrielectric subphase between AF and SC*, was reported recently by Hatano et al.13 and O’Sullivan et al.21 investigate the subphases experimentally in more detail and to examine the applicability of the proposed theoretical models.Isozaki et al.9 insisted that a few additional subphases seem to emerge in the vicinity of FIH and FIL. Likewise,some subphases We expect that this staircase will appear much more typical so that the investigation and the examination will be performed other than FI are expected in the temperature region between AF and SC*.Consequently, we designated these regions as practically, if thick free-standing films54–56 of suitable materials are prepared carefully. The purposes of this paper are: (1) to spr1, spr2 and spr3, respectively, where spr refers to subphase region. establisha convenient method of studying the staircase between SCA* and SC*; (2) to introduce some suitable materials which There are three factors that may apparently confuse the above sequence.First, the rather stable antiferroelectric AF allow us to characterize unambiguously the subphases in the regions spr3, spr2 and spr1; (3) to discuss the origin of the phase appearing in addition to SCA* may cause inappropriate identification of AF to SCA*. Secondly, ferroelectric tilted staircase in terms of the several theoretical models so far proposed; and (4) to conclude that the ANNNI model hexatic SI* below SCA* may cause inappropriate identification of SI* to SC*.25 Thirdly, the staircase character of SCa* with the third-nearest-neighbour interaction (ANNNI+J3 model)29–34,36 describes well the Devil’s staircase character of described in the following may complicate the situation, particularly when SC* does not emerge.The fourth complexity is the subphases between SCA* and SC*.† rather essential and is due to substrate interfaces which sometimes influence the subphase appearances considerably. Experiment Three antiferroelectric liquid crystal compounds were used in this experiment, the structural formulae of which are summar- † The molecular orientational structures are specified by qT in the onedimensional Ising model with long-range repulsive interactions2 and Scheme 1 A possible, most general subphase sequence in antiferro- by q in the ANNNI+J3 model.29–34 Both of the models assign electric liquid crystals essentially the same structures to SCA* (q=1/2, qT=0), SCc* (q=1/3, qT=1/3), AF (q=1/4, qT=1/2) and SC* (q=0, qT=1) but may predict different ones for subphases in spr3, spr2 and spr1. 407 J.Mater. Chem., 1997, 7(3), 407–416408 ized in Scheme 2. Homogeneously aligned samples were pre- using free-standing films under temperature gradients; an eyepiece together with a beam splitter was added, so that by the pared by rubbing polyimide (Toray, SP510) spin-coated on glass substrate plates with indium tin oxide (ITO) electrodes.use of backward illumination, we can pin-point the sample area where the conoscopic observation occurs. Homeotropically aligned samples were also prepared between glass substrate plates coated with silane coupling agents (Toray Dow Corning Silicone, AY 43-021). Polyester (PET) films were Results used as spacers in both homogeneous and homeotropic cells. Free-standing film samples were formed in a 1.5×8 mm2 12BIMF10 homogeneous cells rectangular hole of a glass frame depicted in Fig. 1. The film Plate 1 shows micrographs of a 6 mm thick 12BIMF10 cell thickness was estimated at ca. 100 mm from the upper and aligned homogeneously by polyimide (Toray, SP510) rubbing. lower film surfaces pinpointed with an optical microscope.An When the phase transition from SA to the unidentified SX1* electric field can be applied parallel to the 1.5 mm edges using phase occurs, needle-like defects emerge perpendicular to the two ITO electrodes prepared along the 8 mm edges. The frame smectic layer, but fringe lines parallel to the smectic layer, has another ITO heater electrode on the right side, which can indicating a helicoidal structure, do not appear; the extinction produce a temperature gradient in the free-standing film directions are parallel and perpendicular to the smectic layer. sample.Samples aligned in a homogeneous/homeotropic cell These are the characteristic features of SCa* and hence SX1* or prepared as a free-standing film were mounted in an oven must be SCa*. On cooling to another unidentified phase SX2*, and the temperature was controlled with an accuracy of both focal conics and fringe lines parallel to the smectic layer, ±10 mK.indicating the helicoidal structure, appear and light trans- Texture observation and electro-optical switching investimission occurs slightly even when the crossed polarizers are gation were performed using the same system as described in set at extinction directions parallel and perpendicular to the previous papers.1,2,22–24 The helicoidal pitch multiplied by the smectic layer; this SX2* texture looks like that of SCc*.As the average refractive index was determined by observing the temperature decreases further, SCA* appears. transmittance loss due to selective reflection using a spectro- The switching currents observed in the same cell at various photometer (Hitachi, U-3410).Laser light diffraction patterns temperatures by applying a 0.5 Hz, ±6 V mm-1 triangular were obtained by the same system as used in photon correlation wave are shown in Fig. 3. In the high-temperature region of spectroscopy with a He–Ne laser and a goniometer.57 Fig. 2 SX1*, two current peaks were observed, suggesting the anti- illustrates a system for obtaining conoscopic figures by applyferroelectric character of SX1*; the number of current peaks ing an electric field to unwind the helicoidal structure.Its increases with the decrease of temperature in SX1*. This details have already been reported in ref. 58, apart from one switching behaviour, together with the texture illustrated in improvement which is essential in the present investigation Plate 1, almost unambiguously identifies SX1* as SCa*. After the phase transition to SX2*, five current peaks were observed; the number of current peaks remains five in SX2*.Since three peaks are expected to appear in SCc*, it is not reasonable to simply identify SX2* as SCc*. Fig. 4 summarizes the laser light diffraction patterns obtained at various temperatures covering SA, SX1*, SX2* and SCA* in a 350 mm thick 12BIMF10 cell aligned homogeneously using a 1 T magnetic field.The phase-transition temperatures are different from those in Fig. 3, because they depend on the cell thickness and surface treatment. In both SA and SX1*, no diffraction peaks emerge and the background lines are sufficiently low and almost noiseless; this was particularly true after all our effort to detect the diffraction peaks in SX1* by changing the temperature at 0.1 °C intervals.When the phase Scheme 2 Compounds used and their phase sequences outlined roughly. Note that substrate interfaces sometimes influence not only transition to SX2* occurs at 54.9 °C, the background lines the transition temperatures but also the phase appearances themselves become very high and noisy and two broad diffraction peaks emerge.The dashed line in Fig. 4 shows the zero level line of the diffraction observed at 54.9 °C. The large-angle diffraction peak moves toward the small-angle side with decreasing temperature, while the small-angle diffraction peak scarcely shows any temperature variation.The two diffraction peaks at the highest temperature in SX2* correspond to periodicities 2.1 and 0.8 mm, which are not in the relation of the first- and second-order diffraction peaks. 12BIMF10 homeotropic cells As described above, at least SX2* appears to be affected Fig. 1 Frame for a free-standing film and holder for producing tem- considerably by substrate interfaces in homogeneous cells.perature gradients Hence we tried to observe the Bragg reflection due to theJ . Mate r . Chem., 1997, 7(3), 407–416 409 Fig. 2 Schematic illustration of the optical system for observing conoscopic figures Plate 1 Micrographs of a 6 mm thick, 12BIMF10 cell homogeneously aligned by polyimide (Toray, SP510) rubbing410 Fig. 4 Laser light diffraction patterns obtained at various temperatures in a 350 mm thick, 12BIMF10 cell aligned homogeneously using a magnetic field.The patterns are shown at 0.1°C intervals and their ordinate zeros are shifted upwards constantly by one division Fig. 3 Switching current observed in the same cell as used in Plate 1 at various temperatures by applying a 0.5 Hz,±6 V mm-1 triangular Fig. 5 Temperature variation (#, cooling; $, heating) of Bragg- wave electric field. The peak indicated by : is due to flow of reflected peaks observed in a 100 mm thick, 12BIMF10 cell homeotrop- accidentally contained ions ically aligned by surfactant (Toray Dow Corning Silicone, AY 43-021) the diffraction peak in Fig. 4 which shifts from 50° to 30° with decreasing temperature. helicoidal structure in a 100 mm thick homeotropic cell.The 12BIMF10 free-standing films results exceeded expectation and a beautiful Bragg reflection was observed; Fig. 5 shows the temperature variation of the In this way, homeotropic cells are much more ideal than homogeneous cells from the viewpoint that some subphase reflected peak. The helicoidal pitch in SCA* must be very short so that the corresponding Bragg reflection could not emerge structures are realized easily.Still, the hysteresis and the disappearance of the red reflection in the low-temperature in the transparent region of 12BIMF10. On heating to SX2*, a red colouration was visible and a Bragg refection peaking at region suggest some influence exerted by substrate interfaces. To be as free from this influence as possible, we observed the ca. 600 nm appeared. On further heating, the peak showed a steep increase to ca. 1.5 mm and then decreased slightly; SX2* subphases in a ca. 100 mm thick free-standing film under a temperature gradient and obtained their conoscopic figures by consists of at least two subphases. After the phase transition from SX2* to SX1*, no Bragg reflection was observed.In a applying an electric field. Plate 2 shows a micrograph under crossed polarizers and two conoscopic figures. Between SCA* cooling process, SX2* behaved similarly in the high-temperature region, but hysteresis was observed and the 600 nm Bragg and SCa*, there exist two ferrielectric phases which must correspond to SX2*. The red Bragg reflection is clearly seen reflection did not appear in the low-temperature region.The 1.5 mm peak nearly corresponds to the periodicity producing and, within this red region, a conoscopic figure illustrated onJ . Mate r . Chem., 1997, 7(3), 407–416 411 we first confirmed SC* by texture observation and then observed a conoscopic figure, which is clearly different from SC* and SCA*; it looks like ferrielectric at 200 V mm-1 but antiferroelectric at 333 V mm-1 as shown in Plate 3.Consequently, we could not identify unequivocally the subphase in spr1 as ferrielectric. Plate 4 shows micrographs and conoscopic figures of ca. 100 mm thick partially racemized TFMHPBC free-standing films. When the optical purity is ee=(R-S)/(R+S)=92%, both SCc* and another ferrielectric subphase in spr3, FIL, were observed clearly between SCA* and SCa* as seen in Plate 4(a).The dark blue colour on the left side is caused by the Bragg reflection due to the SCA* helicoidal structure; the SCA* texture appears very uniform because the helicoidal pitch is short. The dark area on the right side represents SCa*, the texture of which is always quite uniform in homogeneous cells as well as in free-standing films.The difference between the ferrielectric phases becomes much more clear if we observe conoscopic Fig. 6 Apparent tilt angle vs. temperature determined by measuring figures under an applied electric field of 17 V mm-1 as shown centre-shifts in the conoscopic figures under an applied electric field, in Plate 4(a). When the optical purity was slightly reduced to 267 V mm-1 ee=(R-S)/(R+S)=84%, three ferrielectric subphases, FIL in spr3, SCc* and FIH in spr2, and one antiferroelectric subphase, AF, were observed between SCA* and SCa* as seen in Plate 4(b).the lower left is observed when an electric field high enough As demonstrated in this and the preceding sections, free- to unwind the helicoidal structure is applied. The region above standing films under temperature gradients are very effective (to the right of ) the red one becomes dark because of the for the direct observation of the subphases between SCA* and infrared Bragg reflection; a conoscopic figure illustrated on the SC*.When the helicoidal pitch is long, however, the film right in the lower part is observed when unwinding the appearance may become disturbed and spurious phase bound- helicoidal structure.We saw the boundary between this dark aries may appear as illustrated in Plate 5. Even in such cases, region and SCa*, although it is not clear in the plate. conoscopic observation under an applied electric field can We can determine the apparent tilt angle by measuring discriminate between the real and spurious phase boundaries.centre-shifts in the conoscopic figures as plotted in Fig. 6. The Among the several boundaries in the ferrielectric subphases, tilt angle is 21° in SC* produced from SCA* by applying an in fact, the lowest temperature boundary is the real one, electric field stronger than its threshold,and the two ferrielectric because the conoscopic figures observed on both sides of this phases corresponding to the red and dark regionshave apparent boundary are quite different, as shown in Plate 5.tilt angles of 4.2°#21°/5 and 6.9°#21°/3, respectively. Consequently, it is reasonable to assign the two ferrielectric subphases corresponding to SX2*, which exhibit the red and infrared Bragg reflections, as subphases in spr3 and SCc*, Discussion respectively. Note that the Bragg reflection due to the helicoidal The successive phase transitions observed between ferroelectric structure has not been observed in either of the subphases so far.SC* and antiferroelectric SCA* can be regarded as the formation of large-scale structures in simple physical systems other- Free-standing films of TFMHPBC and MHFPDBC wise dominated by short-range forces.Some type of frustration must be present in those parts of the phase diagram where the To recognize properly the validity and limitation of the method using free-standing films under temperature gradients, we structures are encountered. When the two dominant ordering forces of a system happen to compete with each other, a large introduce two other materials, TFMHPBC and MHFPDBC, listed in Scheme 2, although the results obtained are rather number of alternative structures may have almost the same free energy.This degeneracy can be removed either by weak preliminary. The TFMHPBC enantiomer has a simple phase sequence, where only SCa* exists between SCA* and SA, but long-range forces or by thermal effects. The frustration at issue is the one between ferroelectricity and antiferroelectricity, i.e.its racemization complicates the phase sequence.2,10 As far as the authors are aware, MHFPDBC is the only compound in the tilting correlation in adjacent layers, in the SC*-like phase; we would not expect to encounter such frustration, since it which some subphase between AF and SC* has been reported to exist.13 Quite recently, O’Sullivan et al.also reported a seems easy to lift any degeneracy by changing the molecular orientations in some way. In fact, the SC*-like phase has two similar subphase in spr1 in another compound.21 Plate 3 shows a micrograph of a ca. 100 mm thick degrees of freedom, the polar angle, h, and the azimuthal angle, w. Notwithstanding this, several theoretical treatments have MHFPDBC free-standing film under a temperature gradient.We can see clearly the existence of at least one subphase in been developed so far to understand the observed sequence of subphases based on the X–Y model.41–53 spr1. We were unable to observe its conoscope by applying an electric field, because some flow induced by the field occurred Possible antiferroelectric and ferrielectric structures induced by the multilayer tilt ordering from the parent SA have been in SC* on the right side and disturbed the texture considerably.To avoid this flow, we stopped using the temperature gradient constructed systematically on the basis of symmetry analysis. 51,52 To choose realistic structures for the most stable and tried to keep the film temperature uniform. On cooling,412 Plate 2 A micrograph of a ca. 100 mm thick, 12BIMF10 free-standing film under a temperature gradient and two conoscopic figures of a subphase in spr3 (1/2>q>1/3) and SCc* (q=1/3) under an applied electric field, 182 V mm-1, sufficient to unwind the helicoidal structure. The q data presented in this and the following Plates have been determined by comparing the experimental observations with Yamashita’s phase diagram reproduced in Fig. 7. ferrielectric and antiferroelectric subphases, SCc* and AF, we (liquid-crystal-induced circular dichroism) due to the helicoidal structures.60 Consequently, the n-layer (n3) spiral model51,52 naturally have to resort to several experimental facts. The first one is the temperature variation of the smectic layer spacing is not practical for AF, because the apparent n-fold symmetry diminishes biaxiality to such an extent that no optical rotatory observed through the successive phase transitions; the spacing shows only a slight discontinuous change at the transitions, if power could be observed.The bilayer azimuthal mode model for SCc*41,51 is not practical, either, because of the third and any. Moreover, the diffraction peak does not show any change such as splitting.Consequently, the molecular tilt angles are forth experimental facts that the biaxial optical plane orients parallel to the applied field2,61 and that our recent X-ray practically constant, not only in a smectic layer but also from layer to layer; bilayer models with different tilt angles in experiment with synchrotron radiation revealed a Bragg reflection corresponding to three-layer spacing;62 note that the adjacent layers41,44,51 are impractical for SCc*.This fact is in accord with our intuition that smectics are one-dimensional bilayer azimuthal mode model needs to presuppose an azimuthal angle difference of ca.±80° in adjacent layers and is crystal and the layer spacing change accompanies a large energy increase and hence seldom occurs.Empirically, once a unrealistic. Although the low-frequency dielectric properties have been reported to be well understood by the bilayer tilt angle as large as 10° or more has been established, the electroclinic effect59 is hardly observed. azimuthal mode model,47 these characteristic properties can also be explained by the three-layer Ising model.63 The second experimental fact is that AF shows an LCICDJ .Mate r . Chem., 1997, 7(3), 407–416 413 Plate 3 A micrograph of a ca. 100 mm thick, MHFPDBC free-standing film under a temperature gradient, and two conoscopic figures of a ferrielectric subphase in spr1 (1/4>q>0) under 200 and 333 V mm-1; ferrielectric behaviour is shown at 200 V mm-1 and antiferroelectric behaviour is shown at 333 V mm-1 The final experimental fact is well known but is very same as the Ising model which we have already proposed;2,9,65 note that the tilt (polar angle) is practically the same in the important.The helical pitch of antiferroelectric liquid crystals is fairly short compared with conventional ferroelectric liquid three layers (Phase III in Fig. 4) and Phase II in Fig. 3 exists practically even in the chiral case, because the helical structure crystals. However, the chiral interaction is still so weak that the helicoidal pitch is very long as compared to the smectic is only a small perturbation caused by a weak interlayer chiral interaction. layer spacing, i.e. the molecular length.64 Consequently, the subphase sequence analysis based on the Landau-type phenom- What we would like to emphasize is the mechanism by which the tilting direction is restricted parallel to a plane both enological models should be performed carefully by taking account of the short-range interactions to produce large azi- in ferroelectric SC* (w=0) and antiferroelectric SCA* (w=0 or p).As mentioned above, we neglect the slight precession of at muthal angle changes between adjacent layers;44,49,50 the existence of the short-range interaction has not been supported by most a few degrees per layer caused by chirality.The excluded volume effect (the packing entropy effect) in a smectic layer any experimental evidence up until now. In this way, the Ising model appears to be most realistic. It should be noted that structure preserving the density wave character must be the main factor that causes the molecules to tilt in the same Phase II in Fig. 3 of the four-layer model and Phase III in Fig. 4 of the three-layer model in ref. 52 are effectively the direction and sense in SC*. Either of the two models based on414 Plate 4 Micrographs of ca. 100 mm thick, TFMHPBC free-standing films under temperature gradients, and conoscopic figures of a subphase in spr3 (1/2>q>1/3) and SCc* (q=1/3) under applied electric fields, 17 V mm-1; (a) for R5S=9654 and (b) for R5S=9258.A subphase in spr2 (1/3>q>1/4) is also observed on the right side of (b), although no detailed study was performed. the electric interaction between permanent dipole moments peting nearest and next-nearest neighbour coupling proposed by Bak and von Boem.35 proposed so far for the stabilization of SCA*, the pairing model by Takanishi et al.22 and the Px model by Miyachi et al.,66 Trying to simply interpret the observed sequence of the subphases in terms of the Bak–Bruinsma Ising model with the assure that the molecular tilting occurs in the same direction but in the opposite senses in adjacent layers, although the long-range repulsive interactions, we assigned Ising spins to the orderings, ferroelectric (F) and antiferroelectric (A), but third model based on the steric interaction in adjacent layers67 may not be able to do so.In this way, it seems to be not to the tilting senses, right (R) and left (L);2,9,22 we also considered that, following Bruinsma and Prost,28 fluctuations well founded to treat the observed sequence of subphases in terms of the frustration between ferroelectricity and antiferro- of C-directors and hence of spontaneous polarizations cause the long-range repulsive interactions.However, the repulsive electricity based on the Ising model. Statistical mechanics models illustrating two different ways of lifting the degeneracy interactions between separate F orderings seem to be rather artificial and several difficulties have been noted so far.2 In have been developed: the one by weak long-range forces is the one-dimensional Ising model proposed by Bak and fact, Bruinsma and Prost,28 based on the fluctuation forces, actually showed the emergence of the electric-field-induced Bruinsma26,27 and the other by thermal effects is the so-called ANNNI (axial next-nearest neighbour Ising) model with com- Devil’s staircase which can be described by the tilting senses,J .Mate r . Chem., 1997, 7(3), 407–416 415 Plate 4 (continued) R and L , but not that of the temperature-induced one. for the first, second and third neighbouring pairs in the axial direction parallel to the layer normal; the second-nearest Moreover, the stability of subphases changes critically from material to material, although the Bak–Bruinsma Ising model neighbour interaction J2 should be negative to ensure competition, and the third-nearest neighbour interaction J3 (>0 or predicts rather universal stability.26,27 Another issue raised is that no finite temperature effect is taken into account and <0) is included for the possible wide stability of SCc*. Although they did not show any realistic physical grounds for hence the model can describe only the ground states.50 The ANNNI+J3 model36 was applied to this problem by these rather long-range interactions initially, Yamashita32–34 quite recently claimed an important role played by the sense Yamashita and Miyazima29 and by Yamashita.30,31 The Hamiltonian they assumed is of the molecular long axis, decimated in the partition function the pseudo-spins describing the senses of molecular long axes, and eventually obtained the effective long-range interactions, H=-J .(i,j) sisj-J1 .i A sisi+1-J2 .i A sisi+2-J3 .i A sisi+3 J2, J3 , etc. Such a freedom was already introduced by Koda and where the Ising spin si takes a value of ±1 corresponding to Kimura37,38 to induce negative J2, who also quite recently the molecular tilting senses of the ith smectic layer, the first extended their theoretical treatment and tried to interpret the summation is taken all over nearest-neighbouring pairs (i, j) in the same smectic layer, and other summations SA are only observedsequence of subphasesand the stabilityranges.39 Their416 Plate 5 A micrograph of a TFMHPBC (R5S=88512) free-standing film under a temperature gradient, and two conoscopic figures of a subphase in spr3 (1/2>q>1/3) and SCc* (q=1/3), respectively. Some spurious phase boundaries appear; the boundary characterizing a phase in spr2 (1/3>q>1/4) on the right side seems to be real, although a conoscopic study was not performed.method is essentially equivalent to the ANNNI+J3 model. to be very small for q=2/9 and 1/5, and this smallness may explain the characteristic field dependence of the conoscopic Yamashita34 showed that four ground states are SCA* (q=1/2), SCc* (q=1/3), AF (q=1/4) and SC* (q=0) as illustrated in figure for the subphase in spr1 observed in Plate 3.It exhibits ferrielectric-like behaviour at low fields, but a secondary inter- Fig. 7. He predicted rather stable ferrielectric phases q=2/5 and 4/11 in spr3, q=4/13 and 2/7 in spr2, and q=2/9 and 1/5 in action through dielectric anisotropy prevails at high fields, resulting in the antiferroelectric conoscopic figure. Yamashita spr1, estimating the average of the saturated ordering, also predicted antiferroelectric phases, q=3/8, q=3/10 and q= s=.p i=1 si p 3/14, in spr3, spr2 and spr1, respectively.which is considered to be proportional to the apparent tilt Conclusions angle, i.e. the spontaneous polarization. The estimated ratio of this value to that in SCc* (q=1/3) is ca. 0.6 for q=2/5 and ca. In this way, the ANNNI+J3 model,29–34,36 is flexible enough to explain a variety of observed phase sequences between SCA* 0.27 for q=4/11.The subphase in spr3 observed in Plate 2 and Fig. 6 is therefore identified as q=2/5. The ratio is suggested and SC*. For detailed comparison of theory with experiment,J . Mate r . Chem., 1997, 7(3), 407–416 417 13 H. Hatano, Y. Hanakai, H. Furue, H. Uehara, S.Saito and K. Muraoka, Jpn. J. Appl. Phys., 1994, 33, 5498. 14 H. Moritake, N. Shigeno, M. Ozaki and K. Yoshino, L iq. Cryst., 1993, 14, 1283. 15 H. Moritake, M. Ozaki, H. Taniguchi, K. Satoh and K. Yoshino, Jpn. J. Appl. Phys., 1994, 33, 5503. 16 P. Gisse, J. Pavel, H. T. Nguyen and V. L. Lorman, Ferroelectrics, 1993, 147, 27. 17 P. Cluzeau, H. T. Nguyen, Ch. Destrade, N.Isaert, P. Barois and A. Babeau, Mol. Cryst. L iq. Cryst., 1995, 260, 69. 18 M. Glogarova, H. Svorenyak, H. T. Nguen and Ch. Destrade, Ferroelectrics, 1993, 147, 37. 19 T. Sako, Y. Kimura, R. Hayakawa, N. Okabe and Y. Suzuki, Jpn. J. Appl. Phys., 1996, 35, L114. 20 Yu. P. Panarin, H. Xu, S. T. MacLughadha, J. K. Vij, A. J. Seed, M. Hird and J. W. Goodby, J. Phys.: Condens. Matter, 1995, 7, L351. 21 J.W. O’Sullivan, Yu. P. Panarin and J. K. Vij, Poster Presentation at 16th Int. L iq. Cryst. Conf. (Kent, 1996), C1P.15 (P-125). 22 Y. Takanishi, K. Hiraoka, V. K. Agrawal, H. Takezoe, A. Fukuda and M. Matsushita, Jpn. J. Appl. Phys., 1991, 30, 2023. Fig. 7 A phase diagram obtained by the ANNNI+J3 model for 23 K. Hiraoka, Y. Takanishi, K. Skarp, H. Takezoe and A.Fukuda, J1/|J2|=1 and J3/|J2|=0.3. By courtesy of M. Yamashita.29–34 Jpn. 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