**4. Frustration of Ferroelectric and Antiferroelectric Phases; Successive Phase Transition to Several Subphases between Two Phases**

When the antiferroelectric SmCA\* phase was discovered in 4-(1-methylheptyloxycarbonyl) phenyl 4 -octyloxybiphenyl-4-carboxylate (MHPOBC), three phases with a narrow temperature range called "*subphase"* or "SmC\* variants" were already observed between SmA and SmCA\*. Fukui et al. first noticed the existence of subphases by differential scanning calorimetry (DSC) [12]. At almost the same time, Takezoe et al. [13] and Chandani et al. [14] reported three subphases between SmA and SmCA\* in MHPOBC, and these subphases were tentatively designated as SmCα\*, SmCβ\* and SmCγ\* in order of temperature at the beginning, and later found that SmCβ\* was normal ferroelectric SmC\*. The SmCγ\* phase shows four-state switching in the planar cells [15], and, by conoscope observation [16], this phase shows an averagely three-layer periodic structure in which two molecules are tilted to the one side and one molecule is tilted to another side with respect to the layer normal, as shown in Figure 9b. After further study, some other phases are frequently observed between ferroelectric SmC\* and antiferroelectric SmCA\* phases. These phases exist in a very narrow temperature range (several K to 0.2 K) but are clearly distinguished by texture observation [17], conoscope observation [18–20], dielectric [21,22] and thermal measurements [23,24], etc. Later complicated successive phase transition to many subphases is extensively being studied. In particular, subphases are frequently observed in the binary mixture of FLCs and AFLCs [18–20,25].

**Figure 9.** Molecular arrangement of subphase between SmC\* and SmCA\*, and the corresponding qT number. (**a**) antiferroelectric SmCA\*, (**b**) correponds to SmCγ\* phase, (**c**) is the subphase with four-layer periodicity, and (**d**) ferroelectric SmC\*.

First, discovered subphase (SmCγ\*) forms a three-layered periodic structure mentioned above, and four-layered periodic subphases were next discovered in 4-(1-methylheptyloxycarbonyl) phenyl 4 -octylbiphenyl- 4-carboxylate (MHPBC) [26]. Typical molecular arrangements are shown in Figure 9c. In the four-layer periodic phase, two molecules in the nearest layer tilt with the same sense, but the molecules in the next two layers tilt with opposite sense, and such a molecular arrangement is formed repeatedly. Here, molecular ordering in adjacent layers is defined as follows: when the molecules in adjacent layers tilt with opposite sense, the molecular ordering is "A", which means antiferroelectric (or anticlinic) ordering. Additionally, when they tilt with the same sense, the molecular ordering is "F", which means ferroelectric (or synclinic) ordering. Here, any subphase expected to appear is specified by an irreducible rational number, qT [25],

$$\mathbf{q}\mathbf{T} = [\mathbf{F}]/([\mathbf{A}] + [\mathbf{F}])^2$$

where [F] and [A] are the number of synclinic ferroelectric and anticlinic antiferroelectric orderings, respectively. In the three-layered periodic subphase (SmCγ\*), qT = 1/3, while qT = 1/2 in the four-layered periodic subphase. In antiferroelectric SmCA\* and ferroelectric SmC\*, qT = 0 and 1, respectively, so that the qT number of the subphase step by step and gradually increases on heating.

At first, at least five subphases are proposed in the electro-optic and conoscope measurements [25], but because of the experimental difficulty such as the temperature accuracy being less than ±0.1 ◦C due to the narrow temperature range of each subphase and indirect information about the structure obtained by such an experiment, only two subphases got the public's attention. For SmCA\*(qT = 1/3) and SmCA\*(qT = 1/2), the structures are relatively recognized because these subphases are observed more frequently in many liquid crystal compounds and mixtures, and there are many experimental data. Finally, the three-layer or four-layer periodic structures have been decisively recognized world wide by the resonant X-ray scattering technique about 15 years ago [27].

#### **5. Resonant X-ray Scattering and the Determination of Molecular Arrangement of Subphases**

The conventional X-ray scattering comes from the electron density distribution of the materials. Because the spherical symmetry of electron density was supposed in each atom, the X-ray susceptibility of the system is usually regarded as a scholar. In this technique, density periodicity can be detected by the Bragg condition. The smectic phase has a one-dimensional positional order with a two-dimensional liquid order, and the density has a periodicity along the layer normal. Since the X-ray susceptibility is treated as a scholar, the densities of the smectic layer in which tilt of the molecule is left and right are not distinguished. Hence, in synclinic SmC\* and anticlinic SmCA\* (and other suphases), X-ray diffraction profiles reflecting the density distribution along the layer normal are the same.

On the other hand, the resonant X-ray scattering (RXS) technique treats the X-ray susceptibility as a tensor but not a scholar at the absorption edge energy of a specific element as surfa, selenium and bromine, etc. Thus, RXS intensity reflects the system symmetry, and, as a result, the prohibited Bragg diffraction in the conventional method can appear as satellite peaks, which reflects the corresponding system symmetry. Hence, the orientational order (local layer structure) of different subphases can be clarified by measuring and analyzing resonant scattering satellite peaks. Using this technique, Mach et al. firstly directly clarified the two-, three-, and four-layer periodic structures in SmCA\*(qT = 0), qT = 1/3 and qT = 1/2 [27]. RXS satellite peaks obtained from the subphase at the resonant condition appear at

$$\mathbf{Q} \mathbf{Q} \mathbf{Q}\_0 = \mathbf{l} \mathbf{ + m} / \mathbf{v} \pm \varepsilon \mathbf{r}$$

where *Q* is the scattering vector and *Q*<sup>0</sup> = 2π/d, d is the smectic layer spacing, *l* = 1, 2, 3 ... is the diffraction order due to the smectic layer spacing, ν is the number of layers in the unit cell of a subphase, *m* = ±1, ±2*,* ... , ±(ν − 1) defines RXS peak positions due to the super-lattice periodicity*.* ε = *d*/*P*, and *P* is the pitch of the macroscopic helix. Later, using this technique, some groups have extensively studied subphases with 3- and 4-layer periodic structures [28–31]. Related to the RXS experiment, Levelut and Pansu calculated the tensorial X-ray structure factor in the smectici phase with the various structure models [32].

The author and his coworkers also performed a new experiment using microbeam resonant X-ray scattering to determine the structure of the novel subpahse except qT = 1/3 and qT = 1/2. Here, we synthesized Br-containing chiral molecule, (*S***,***S*)-*bis*-[4 *-*(1-methylheptyloxycarbonyl)-4-biphenyl] 2-bromo-terephthalate (compound **1**) [33], and measured RXS in the mixture of compound **1** and (*S*,*S*)-α, ω -bis (4-{[4 -(1-methylheptyloxycarbonyl)biphenyl-4-yl] oxycarbonyl}phenoxy)hexane [34]. The results are shown in Figure 10. The most notable result was Figure 10d; 1 ± 0.17 satellite peaks are observed, which clearly indicates a six-layer periodic superstructure just above the SmCA\*(qT = 1/2) phase. Based on the paper written by Osipov and Gorkunov [35], we calculated the satellite peak intensity of six-layered structures. Considering the calculated results and finite dielectric constants caused by a non-zero net spontaneous polarization are observed for the observed six-layer subphase

(qT = 2/3), we conclude that the ferrielectric structure of Figure 10e is considered to be more suitable to explain our experimental results. This structure was theoretically predicted by Emelyanenko and Ishikawa [36]. The smectic phase with six-layer periodicity has been already found between the SmC\* and SmCα\* phases by Wang et al. [37], but this is the first evidence of the subphase except qT = 1/3 and 1/2 between SmC\* and SmCA\*.

**Figure 10.** Two-dimensional microbeam resonant X-ray scattering profiles at various temperatures: (**a**) SmCA\*(qT = 0), (**b**) qT = 1/3, (**c**) qT = 1/2, (**d**) qT = 2/3, and (**e**) SmC\*(qT = 1) [34]. The red arrows indicate the satellite peak positions.

Further experiments were conducted and three new subphases were found by Feng et al. [38]. The results are shown in Figure 11, obtained from the mixture of Se-containing chiral liquid crystalline molecules. Between SmCA\*(qT = 0) and SmCA\*(qT = 1/3), subphases showing 3/8 and 5/8 resonant scattering are observed. Phase transition to this subphase was also observed by in situ texture observation by the polarized microscope equipped in the RXS system. From the calculation of relative intensities of RXS peaks based on Osipov and Gorkunov's paper [35], we could conclude that this subphase is SmCA\*(qT = 1/4) with an eight-layer periodic structure, as shown in Figure 11d. This summary is also consistent with to the simplest Farey sequence number, and the results of electric-field induced optical birefringence measurement. Furthermore, this subphase was universally observed in the different mixture of two Se-containing chiral molecules.

**Figure 11.** Microbeam resonant X-ray scattering intensity profiles in a binary mixture including a Se-containing chiral molecule for two-layer SmCA\* (**a**), eight-layer qT = 1/4 (**b**), and three-layer qT = 1/3 (**c**) [38]. (**d**) Molecular arrangement of qT = 1/4 with an eight-layer periodic structure obtained from the calculation using the Osipov–Gorkunov formula [35].

Between SmCA\*(qT = 1/3) and SmCA\*(qT = 1/2), other different subphases are confirmed [38]. Figure 12 shows one-dimensional RXS intensity profiles along the layer normal of two Se-containing chiral molecules at various temperatures. Just above SmCA\*(qT = 1/3), RXS satellite peaks were observed at *Q*/*Q*<sup>0</sup> = 0.3 and 0.7, which suggest that the subphase has a ten-layer periodic structure. Calculating RXS relative peak intensities for the respect to ten-layer periodic structures using the Osipov–Gorkunov formula and comparing electric-field induced optical birefringence measurement, we could summarize this subphase is SmCA\*(qT = 2/5) and its molecular arrangement in the ten-layer periodic structure shown in Figure 12b. Furthermore, another RXS satellite peak at about *Q*/*Q*<sup>0</sup> = 0.286 (~2/7) was observed at the temperature between SmCA\*(qT = 2/5) and SmCA\*(qT = 1/2), suggesting the subphase SmCA\*(qT = 3/7) with ferrielectric structure detecting by electric-field induced optical birefringence measurement. However, this satellite peak sometimes overlaps the RXS satellite peak at *Q*/*Q*<sup>0</sup> = 0.3 due to the coexistence of two subphases, and it is difficult to determine the structure decisively. We would like to perform more delicate measurements in the future.

In this way, the existence of suphases, which have not been recognized before, was clarified in addition to the three- and four-layered structures (SmCA\*(qT = 1/3) and SmCA\*(qT = 1/2)). Similar subphases are theoretically predicted by Emelynaneko and Osipov [39]; they considered the frustration between ferroelectric SmC\* and antiferroelectric SmCA\* and resulting degeneracy lifting by long-range interlayer interaction due to the discrete flexoelectric effect, they predicted subphases that could emerge, and, in these subphases, qT = 1/4 is contained between SmCA\*(qT = 0) and SmCA\*(qT = 1/3). They also predicted the emergence of qT = 3/7 (seven-layer periodic structure) but not the emergence of qT = 2/5 (ten-layer periodic structure), because their numerical calculations were limited to nine smectic layers in the unit cell. When the calculations are expanded to ten layers, qT = 2/5 with ten-layer periodicity appears [40].

**Figure 12.** (**a**) Microbeam resonant X-ray scattering intensity profiles along the layer normal as a function of the normalized scattering vector (*Q*/*Q*0, where *Q*0 = 2π/*d* and *d* is a layer spacing) in of the mixture of AS657 (80 wt. %) and AS620 (20 wt. %). Between three-layer (qT = 1/3) and four-layer (qT = 1/2) subphases, ten-layer (qT = 2/5) and seven-layer (qT = 3/7) subphases are observed [38]. (**b**) Molecular arrangement of qT = 2/5 with a ten-layer periodic structure obtained from the calculation using the Osipov–Gorkunov formula [35].

#### **6. Brief Summary: My Future Plan and Expectations for Applications**

Finally, in the future, it is necessary to find additional suphases in different compounds to demonstrate their universality. For this purpose, not only resonant hard X-ray scattering, which requires specific elements, but also resonant soft X-ray scattering (RSoXS) using the absorption K-edge of carbon atom would be a powerful tool [41–43], and we have just started the study using RSoXS. As for applications, large flat panels may be difficult, but applications to small, high-definition display devices [44], optical modulators [45], and microwave and millimeter-wave control devices [46] can still be expected with the advantage of a fast response time (μsec order).

**Funding:** This research received no external funding.

**Acknowledgments:** This work is strongly supported by many collaborators outside, our staff and former and students. I acknowledge all these collaborators. In particular, special thanks are due to A. Fukuda, H. Takezoe, A. Iida, and J. K. Vij who led me to the interesting field and experiments, and made significant discussion. Some of the works was carried out under the approval of the Photon Factory Advisory committee (Proposal No. 2009G586, 2011G581, 2012G105 and No. 2014G154), and the second hutch of SPring-8 BL03XU constructed by the Consortium of Advanced Softmaterial Beamline (FSBL), with Proposal No. 2015A7255.

**Conflicts of Interest:** The author declares no conflict of interest.
