Huge dielectric constants of the ferroelectric smectic-A phase in bent-shaped dimeric molecules †

We present the dielectric relaxation properties of the ferroelectric smectic-A (SmAP F ) phase formed by a mixture of bent-shaped dimeric molecules a , o -bis(4-alkoxyanilinebenzylidene-4 0 -carbonyloxy)pentanes with diﬀerent alkyl chain lengths. The SmAP F phase shows a Goldstone-like mode at approximately 500 Hz, which can be attributed to the ferroelectricity and huge dielectric strength ( D e ). D e varies dramatically with the cell thickness; in particular, it attains more than 7000 in a 50 m m-thick cell, which is much higher compared to that reported in the bent-shaped molecules.


Introduction
Since the reports on bent-shaped molecules with unique polarity by Watanabe et al., [1][2][3] bent-shaped molecules have become an important field in the investigation of liquid crystals.In particular, the ferroelectric and the antiferroelectric smectic phases that comprise bent-shaped molecules are of great interest because they can be switched between different polar ordered states by electric fields. 2 Various bent-shaped molecules, with ferroelectric and antiferroelectric behaviors, have been synthesized and investigated to clarify the structure-property relationships associated with their profound mesomorphic properties. 4,57][8][9] Two mesogens in a molecule are forced to be relatively tilted to each other by the conformational constraint of the alkyl spacer. 3,10he typical molecules are a,o-bis(4-alkoxyanilinebenzylidene-4 0 -carbonyloxy)pentanes (mOAM5AMOm) [7][8][9] with the formula shown in Fig. 1.These bent-shaped dimeric molecules form three types of smectic liquid crystals (refer to Fig. S1, ESI †), depending on the alkoxy tail group. 7When the tail length was short (m = 4 and 6), the SmCA s phase was formed.Here each mesogen, but not a bentshaped dimeric molecule, participated to form each smectic layer.When the tail length was medium (m = 8-12), the frustrated SmCA f phase was formed.When the tail length was long (m = 14-18), the smectic structure was identified as the antiferroelectric SmC A P A phase, where each bent-shaped dimeric molecule participates in each layer comprising a bilayer of mesogenic groups.Within a layer, the bent-shaped dimeric molecules are packed with significant tilting of the molecular axis and with the same direction of the bent (polar) axis.The polar directions are opposite between the neighboring layers.Furthermore, the mixture of 4OAM5AMO4 forming the SmCA s phase and 16OAM5AMO16 forming the SmC A P A phase with antiferroelectricity formed the ferroelectric smectic-A (SmAP F ) phase in a specific ratio.The ferroelectricity of the SmAP F phase was identified by a single current peak under a triangular wave field (refer to Fig. S2, ESI †) and clear second harmonic generation. 9he SmAP F phase generally shows that the molecules lie perpendicular to the layer, with ferroelectric alignment of the polar directions (refer to Fig. S1(d), ESI †).2][13] In bent-shaped dimeric molecules, mixing of 4OAM5AMO4 and 16OAM5AMO16 with different alkyl chain lengths obviously prevents the tilted association of molecules, although the reason is not clear. 9As one of the properties of the SmAP F phase with a bent-shaped molecule, Guo et al. reported that W586 has high dielectric constants. 14n this study, we examined the dielectric relaxation properties of the SmAP F phase in bent-shaped dimeric molecules and showed extremely large dielectric constants of 2000-7000 attributable to the collective fluctuation of the polar molecules.
To clarify both the non-collective and collective motions of the molecules, dielectric dispersion measurements were performed in the SmCA s (m = 4), SmC A P A (m = 16), and SmAP F (mixture of m = 4 and 16) phases.Here, it should be noted that the domain size is small especially in the SmAP F and the SmC A P A phases just as cooled from the isotropic phase.Hence, the domains of SmAP F and SmC A P A were enlarged by applying an AC field of 5 V pp mm À1 and 33 V pp mm À1 , respectively; they were grown by agitation of molecular motion due to ferroelectric switching.In Fig. 2(a) and (b), the real (e 0 ) and imaginary (e 00 ) parts of the dielectric constants are plotted against the frequency from 10 1 to 10 7 Hz, respectively.Two relaxation modes are observed in this frequency range.One is a low frequency mode which is observed at approximately 500 Hz in the SmAP F phase and possesses huge dielectric constants.The low relaxation frequency ( f r ) of the order of 1 kHz or less, [14][15][16] as well as the suppression under DC bias fields (mentioned later), shows that this mode is a Goldstone-like mode, collective fluctuation of polarization that is attributable to the ferroelectric phase.The other is a high frequency mode observed at approximately 100 kHz in the SmCA s and the SmC A P A phases.The isotropic phase also exhibits this mode.It is attributable to non-collective molecular rotation around the short axis of the mesogens as observed in conventional liquid crystal phases. 17,18ig. 2(c) and (d) show the Cole-Cole plots for these two modes.The experimental dielectric spectra were further analyzed by fitting the following Cole-Cole model: 19 eÃ where o is the angular frequency, e s and e N are the ''static frequency'' and ''infinite frequency'' dielectric constants, respectively, De is the dielectric strength, t = 1/(2pf r ) is the relaxation time (where f r is the relaxation frequency), and a is the distribution parameter of the relaxation time.S1.
For the high-frequency relaxation of the SmCA s and the SmC A P A phases, a is around 1.0 Â 10 À3 .For the Goldstone-like mode of the SmAP F phase, on the other hand, a is between 0.04 and 0.14.Most significant is that De is very large (approximately 1800) for the Goldstone-like mode of the SmAP F phase, whereas much smaller but usual values (o10) are observed in the high frequency mode of the SmCA s and the SmC A P A phases.1][22] W586 showed the maximum value of De (B300) as observed in a 3 mm-thick cell. 14Furthermore, a bent-shaped azo molecule consisting of nonsymmetrical molecules with a lateral fluoro substitution on one of the wings achieved the maximum value of De (B400) in a 9 mm-thick cell. 22The present De of 1800 exceeds these values substantially.
Fig. 3 shows the temperature dependence of the characteristic f r of the relaxation processes observed in the SmCA s , SmC A P A , and SmAP F phases in a 3 mm-thick cell.The high frequency mode observed in the SmCA s and SmC A P A phases follows the standard Arrhenius equation 20,22 where f r is the relaxation frequency, A is the pre-exponential factor, E act is the activation energy, k B is the Boltzmann constant and T is the absolute temperature.Eqn (2) is fitted to the experimental data points in order to obtain E act .The estimated E act of the SmCA s and SmC A P A phases is 4 and 40 kJ mol À1 , respectively.In the higher temperature region, f r of the SmAP F phase is somewhat temperature dependent, but seems to be independent in the lower temperature region as expected for the Goldstone mode. 23s another feature of the Goldstone-like mode of the SmAP F phase, f r and De strongly depend on the cell thickness (refer to Fig. 4).As the thickness increases from 3 to 50 mm, f r decreases from 500 to 300 Hz.On the other hand, De almost linearly grows; although De is approximately 1800 in a 3 mm-thick cell, it is 3200, 4500, and 7400 in 10, 25, and 50 mm-thick cells, respectively.
A similar thickness dependence of De for Goldstone-like modes has been observed in the ferroelectric chiral Sm-C* 24 and SmAP F phases. 14Ozaki et al. found that some chiral Sm-C* materials possess dielectric constants larger than 7500 in a 250 mm-thick cell. 24Guo et al. mentioned that by linear extrapolation of De for W586 an even larger value (B20 000) is expected for this cell thickness. 14We estimated De by linear extrapolation similar to Guo et al.The expected De of 30 000 exceeds these values.One may speculate that such a cell thickness dependence is caused by strong anchoring of the molecules to the cell surfaces, which can prevent the collective fluctuations of dipoles near the cell surfaces.However, anchoring can cause significant suppression only in a thin cell, but not lead to a linear increase with the cell's thickness.A theoretical approach to explain this trend has been carried out by Guo et al. 14 They collected De with the cell where the bent direction of the bent-shaped molecules lay perpendicular to the cell's surface by strong surface anchoring.In such a cell, it is speculated that a splay of polarization arises across its thickness that can induce the cell thickness dependence in De and f r in relation with the anchoring strength.Comparing the model and the experimental results, they have estimated the correlation length in the bulk and the surface to be approximately 10 and 1 mm, respectively.
In our case, however, the molecular orientation reported by Guo et al. 14 is unlikely.Fig. 5(a, c, and e) show the optical microscopic textures of the SmAP F phase in 3, 10, and 25 mmthick cells, respectively.In all these cells, well-developed fanshaped textures are observed, indicating the homogeneous alignment of the molecules, in other words, the alignment of the smectic layers perpendicular to the cell's surface.Fig. 5(b, d, and f) indicate the textures under DC bias fields.The textures do not change at all, but the birefringence colors change.This indicates that the bent (polar) direction of the molecules that are initially parallel to the cell's surface become perpendicular to it because of the ferroelectric response.On the basis of the retardation obtained from the general polarization color chart, the birefringence (Dn) is estimated to be 0.07 at 0 V mm À1 and 0.18 at 1 V mm À1 , which are well expected from  In such a molecular alignment, the splay of polarization does not arise across the cell's thickness as reported by Guo et al. 14 Fig. 5 shows the clear trend of the fan-shaped domain size increasing with the cell's thickness.Further, its average size roughly corresponds to the cell's thickness, meaning that the domain is uniform across the cell thickness.Thus, the coherence length of the molecular dipoles that cooperatively orient in the AC applied field increases with the increase of the cell's thickness.This increment, that is the increase of the number of molecules participating in the collective motion, may be responsible for the increase of De and the decrease of f r .
The close relationship between the domain size and De can be seen even in the same cell.The domain size of the SmAP F phase as-cooled from the isotropic melt is fairly small and, as mentioned above, grows under AC applied fields.The estimated De in the former cell is very small.Examples, as observed in a 25 mm-thick cell, are shown in Fig. 6.De is 1900 while it becomes 4500 after the AC field treatment.
The relationship between the cooperative motion of dipoles and De has been treated theoretically 25 and experimentally. 26,27irkwood 25 has extended Onsager theory by treating interacting dipoles, and has shown that De is proportional to where N is the number of dipoles per unit volume, m is the dipole moment of the molecule, z is the coordination number (the number of interacting dipoles), and y is the angle between the interacting dipoles.This was successfully applied for liquid water; its large dielectric constant of 80 is reproducible by z = 4 and hcos yi = 0.5.In a ferroelectric copolymer based on vinylidene cyanide, z = 30 and hcos yi = 0.62 were elucidated to explain the high dielectric constant of 130. 25 In analogy, we suppose that the huge De of the present molecular system may be attributable to the cooperative fluctuation of the large number of molecules z = B103 which increases with the increase of the domain size.The effect of DC bias fields on the Goldstone-like mode of the SmAP F phase was also studied.As shown in Fig. 7, by applying DC bias fields, the Goldstone-like mode is quickly quenched.As a result, the high frequency mode can be clearly detected at 100 kHz.The disappearance of the Goldstone-like mode is explained by the suppression of the dipoles' fluctuation due to the DC bias fields.Here, a cell thickness dependence is also observed in the critical voltage that suppresses the Goldstone-like mode.The critical voltage is 0.40 V mm À1 for a 3 mm-thick cell, 0.14 V mm À1 for a 10 mm-thick cell, and 0.06 V mm À1 for a 25 mm-thick cell (refer to Fig. S4, ESI †).This trend is the same as that reported by Guo et al. 14 but opposite to that reported by Shimbo et al. 29

Conclusions
The dielectric relaxation properties were studied for the SmAP F phase in a bent-shaped dimeric molecule mixture of 4OAM5AMO4 (forming the SmCA s phase) and 16OAM5AMO16 (forming the SmC A P A phase).Within a cell of the SmAP F phase, the bent-shaped dimeric molecules lie with their bent plane parallel to the surface.The SmAP F phase shows a Goldstone-like mode at approximately 500 Hz characteristic of the ferroelectric phase and a huge De of over 7000 in a 50 mm-thick cell, which is much higher compared to almost all the reported values in bent-shaped molecules.The highly  cooperative orientation of the bent-shaped dimeric molecule is considered to bring about the giant value of De.It is also characteristic of the Goldstone-like mode that the cooperative fluctuation of dipoles is easily quenched under DC bias fields.The extremely large value of the dielectric constants of the ferroelectric bent-shaped dimeric molecules suggests high potential for application in highly processible elements in electromechanical energy conversion devices.

Experimental
The mOAM5AMOm dimers with m values of 4 and 16 were synthesized following the methods reported in a previous report. 7he bent-shaped dimeric molecule mixture of 4OAM5AMO4 (65 mol%) + 16OAM5AMO16 (35 mol%) was prepared by evaporating a chloroform solution of the two molecules at a predetermined molar ratio. 9The phase transition temperatures were determined by DSC (SHIMADZU DSC-60 Plus) at a rate of 5 1C min À1 under cooling and heating runs.Texture observation and the identification of the mesophases were carried out using a polarizing optical microscope (OLYMPUS BX53) equipped with a hot stage and a temperature controller (Mettler Toledo FP 82HT).Electrooptical and dielectric investigations were performed using glass cells, having indium tin oxide (ITO) electrodes commercially available from EHC Co., Ltd.The area of the ITO electrodes was 100 mm 2 , and the cell thicknesses of the ITO electrodes were 3, 10, 25, and 50 mm.In these cells, the bent-shaped dimeric molecules was homogeneously aligned with the bent plane parallel to the cell plane.Before the measurements, AC applied fields of 33 and 5 V pp mm À1 were applied to 16OAM5AMO16 and 4OAM5AMO4 (65 mol%) + 16OAM5AMO16 (35 mol%) for the growth of liquid crystals, respectively.The complex dielectric constants were measured in the frequency range between 10 1 and 10 7 Hz with an impedance analyzer (FRA51615).The electric field was 1 V pp .

Fig. 1
Fig. 1 Molecular structures and phases sequence for mOAM5AMOm (m = 4, 16, and 4 + 16).The transition temperatures are taken from the DSC cooling run and the switching current curve.9

9
Fig. 1 Molecular structures and phases sequence for mOAM5AMOm (m = 4, 16, and 4 + 16).The transition temperatures are taken from the DSC cooling run and the switching current curve.9

Fig. 2
Fig. 2 (a) and (b) Frequency dependence of the real (e 0 ) and imaginary (e 00 ) parts of the complex dielectric constants of the SmC A P A (at 114 1C), SmCA s (at 90 1C) and SmAP F (at 98 1C) phases, measured in a 3 mm-thick cell.In the insets, the vertical axis is expanded to clarify the high frequency mode.The dashed curves are obtained by fitting eqn (1).(c) Cole-Cole plot for the high frequency mode of the SmC A P A and SmCA s phases, and (d) Cole-Cole plot for the Goldstone-like mode of the SmAP F phase, based on the data of Fig. 2(a and b).The dashed curves are obtained by fitting eqn (1).
For a = 0 the Cole-Cole model reduces to the Debye model.The fitting according to the Cole-Cole equation is shown in Fig. 2(c and d) and Fig. 4(c), and the determined parameters are listed in Table

Fig. 4
Fig. 4 (a) and (b) Frequency dependence of the real (e 0 ) and imaginary (e 00 ) parts of the complex dielectric constants of the SmAP F phase in 3, 10, 25, and 50 mm-thick cells.The dashed curves are obtained by fitting eqn (1).(c) Cole-Cole plot of the SmAP F phase in 3, 10, 25, and 50 mm-thick cells.The dashed curves are obtained by fitting eqn (1).(d) Cell thickness dependence of De and the f r in the SmAP F phase.The solid and dashed curves are guides for the eye.

Fig. 5
Fig. 5 Optical textures in the SmAP F phase (a) in a 3 mm-thick cell at 0 V mm À1 , (b) in a 3 mm-thick cell at 1 V mm À1 , (c) in a 10 mm-thick cell at 0 V mm À1 , (d) in a 10 mm-thick cell at 1 V mm À1 , (e) in a 25 mm-thick cell at 0 V mm À1 and (f) in a 25 mm-thick cell at 1 V mm À1 .

Fig. 6
Fig. 6 Optical textures in the SmAP F phase in a 25 mm-thick cell (a) before application of an AC field and (b) after application of an AC field of 5 V pp mm À1 .(c) Cole-Cole plot for the dielectric data of their SmAP F phases; De is 1900 and 4500 before and after AC application, respectively.

Fig. 7
Fig.7The imaginary (e 00 ) part of the dielectric constants in the SmAP F phase measured under DC bias fields in a 3 mm-thick cell.Their values are presented on a log scale.The critical voltage is 1.2 V (0.40 V mm À1 ) in a 3 mm-thick cell.The small depression observed at 6.0 Â 10 6 Hz is the resonance effect due to the DC bias fields.28