# Tuning the electro-optic and viscoelastic properties of ferroelectric liquid crystalline materials

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## Abstract

This work deals with determination of the viscoelasticity coefficient in chiral smectic liquid crystals possessing the helical structure and is the continuation of the work, in which the elasticity coefficient was presented [Phase Transitions 89 (2016) 368]. The measurements have been performed using optical detection in the small deformation limit. The viscosity coefficient was measured on two commercially available pure chiral materials, namely, 4-(*n*-hexyloxy phenyl)-1-(2-fuethyl butyl) biphenyl-4-carboxylate and 4-(2-methylbutyl) phenyl-4-*n*-octylbiphenyl-4-carboxylate, and on resulting binary mixture composed from those materials in 50:50 weight ratio. All three liquid crystalline materials exhibit a tilted ferroelectric phase over a reasonably broad temperature range. Each of the two pure liquid crystalline materials have its own disadvantages. However, by design of the binary mixture in definite weight concentration, we tried to improve the behavior and to tune the properties; specifically, the effect of the viscoelastic properties on the mixture composition has been established.

## Keywords

Liquid crystals Ferroelectric liquid crystals Electro-optic behavior Viscoelasticity Viscosity## Introduction

**c**-director. Therefore, the precise determination of this adequate parameter is of great importance for successful design of new optoelectronic applications. Most commonly, the measurements of the viscosity coefficient were done using the switching phenomenon (see, e.g., De Gennes 1974, Takezoe et al. 1984, Lagerwall and Giesselmann 2006, and Marzec et al. 2014). However, this approach might provide incorrect results. The measurement of viscosity is based on the observation of flow caused by an external force. To ensure correctness of the measurements, the flow should be laminar. It is only the case when the flow stimulating factor (in our case the deformation of the helical structure) is weak. For this reason, the switching method can designate only qualitative but not the quantitative results. Up to now, the mechanical properties of the

**c**-director in ferroelectric smectic crystals with non-deformed or weakly deformed helical structure have been unusually investigated (Kuczyński et al. 2009; Dardas et al. 2009). In this work, the electro-optic method has been used for determination of the rotational viscosity coefficient,

*γ*, associated with the smectic

**c-**director in ferroelectric liquid crystalline materials, namely, 4-(

*n*-hexyloxy phenyl)-1-(2-fuethyl butyl) biphenyl-4-carboxylate (denoted here as

*) and 4-(2-methylbutyl) phenyl-4-*

**Ce3***n*-octylbiphenyl-4-carboxylate (denoted here as

*), and also their binary mixture (denoted here as*

**Ce8***); this mixture possesses the helical superstructures originating from two ferroelectric liquid crystalline materials with different character of helicoidal behavior. In principle, to apply mixing of different LC materials in order to calibrate and optimize the response time, the dissipation modes and the appropriate laws of mixtures are to be established (Sengupta 2013). In general, it is demanding and not straightforward procedure which involves the dissipative and non-dissipative constitutive equations for nematic liquid crystals (Parodi 1970; Stewart 2004). In fact, from this procedure, the elastic moduli and/or the viscosity coefficients in various mesophases may be inferred. In the present work, an alternative approach is exploited which is based on the dynamic equation for the azimuthal angle and involves the electro-optical method. It is demonstrated that the viscoelastic properties of the binary LC mixtures may be predicted if the viscoelastic properties of the pure compounds are known in advance. It is demonstrated on the example of pure*

**Ce3/8***C*

*material mixed with pure*

**e3***C*

*material in a 1:1 ratio in this work. The ferroelectric SmC* phase of both materials has a layered structure and a single helicoid in the direction perpendicular to the layer plane. However, probably due to differences resulting mainly from the tilt angle of molecules with respect to the layer normal, one can notice differences in the temperature behavior of the pitch period for both materials. For*

**e8***compound, the tilt angle is 45° and does not depend on the temperature; at the N*-SmC* phase transition, the values of tilt angle jump-up, which clearly indicates the first-order phase transition. On contrary for the*

**Ce3***material, the tilt angle reaches 12° as maximum and decreases with increasing temperature (Dardas 2016). Figure 1 represents the temperature dependence of the helical pitch as p*

**Ce8**^{−2}(T) and p

^{−1}(T) for

*and*

**Ce3***compounds, respectively. This reveals a different behavior and non-linear nature of helicoid in the*

**Ce8***compound compare with that in the*

**Ce8***compound included in the measurement results presented later in this work.*

**Ce3**## Theoretical background

*γ*(Larson 1999; Stewart 2004). The linear combinations of the Leslie coefficients defining three Mięsowicz viscosities, i.e.,

*η*

_{1},

*η*

_{2}, and

*η*

_{3}, and the rotational viscosity

*γ*are schematically presented in Fig. 3.

*η*

_{1},

*η*

_{2},

*η*

_{3}, or

*γ*), it is necessary to induce a properly controlled flow or deformation using an external force. In ferroelectric liquid crystals, an applied electric field can be used as an external factor. Due to its conjugation with spontaneous polarization, the induction of flow or deformation can be easily realized. When the electric field

**E**is applied parallel to the smectic layers (electric field vector is in the

*x*), then, a moment of force acts on the unit of the C* smectic volume,

*P*

_{0}×

*E*, where

*P*

_{0}is a spontaneous polarization of a single smectic layer and

*φ*is the angle between the electric field vector

**E**and the physical director

**c**(i.e., the unit vector in the direction of the projection of the medium position of the long axis molecules on the plane of the smectic layer). Deformation of the molecular distribution causes a change in the macroscopic properties of the sample. In a non-disturbed chiral smectic C*, the position of

**c**-director in the successive smectic layers is described by \( \varphi =\frac{2\pi \bullet z}{p} \);

*z*is the coordinate in the direction normal to the smectic layers, whereas

*p*is the pitch of the helix. The constitutive equations, used to determine the effect of deformations on the electric field, originate from Orsay LC Group and also were described by Pierański et al. (1977) and by Kuczyński (2003). The moment of force from a small external disturbance is balanced by the elastic and viscous torques that can be expressed with the equation of motion for the azimuthal angle (Panarin et al. 1998),

*K*is the elasticity constant (for cone movement) and

*γ*stands for the above-mentioned rotational viscosity (some more details on the origin of the specified equations are given in Appendix).

**c**-directors in the neighboring smectic layers must be small in comparison to the equilibrium value of this angle (Wojciechowski et al. 2013). Moreover, the systems are assumed to possess helical superstructures with helix axes oriented along the

*z*-axis, perpendicular to smectic planes. The linear electro-optic coefficient

*a*(Dardas et al. 2006) describing the change of optic axis orientation under a weak electric field

**E**which does not destroy the helicoidal structure can be expressed as:

*p*is helix pitch, the tilt angle

*θ*of the molecules in a smectic layer (in the ferroelectric smectic C* phase, chiral molecules are spontaneously tilted at a tilt angle with respect to the layer normal), and

*τ*

_{G}denotes the relaxation time of the Goldstone mode:

*τ*

_{G}<<1, one obtains (see Appendix) the expression for the interlayer elasticity coefficient

*K*(Dardas 2016):

*K*, as soon as parameters

*a*(the linear electro-optic coefficient),

*P*

_{0}(the local spontaneous polarization),

*p*(the helical pitch of structure), and

*θ*(the tilt angle) are known. Additionally, from Eqs. (3) and (4), the rotational viscosity can be obtained if

*τ*

_{G}is known,

## Experimental

Two commercially available materials 4-(*n*-hexyloxy phenyl)-1-(2-fuethyl butyl) biphenyl-4-carboxylate (denoted * Ce3*) and 4-(2-methylbutyl) phenyl-4-

*n*-octylbiphenyl-4-carboxylate (denoted as

*) and resulting binary mixture (denoted here as*

**Ce8***) composed from those materials in 50:50 weight ratio have been used for the investigations. All three FLC materials exhibit ferroelectric properties within a relatively broad temperature range. The mesomorphic properties of pure FLC materials and of the binary mixture were determined by complementary methods: differential scanning calorimetry, electro-optics, and dielectric spectroscopy. The characteristic textures and their changes were observed using polarized optical microscope (POM) equipped by the temperature chamber with temperature controller. The synthetic details and the mesomorphic behavior of those materials were presented earlier (Bone et al. 1984). Ferroelectric liquid crystal, denoted as*

**Ce3/8***, possesses the cholesteric (N*) and the tilted ferroelectric smectic C* (SmC*) phases with phase transition temperatures as follows: Cr 65.0 °C SmC* 77.5 °C N* 162.0 °C Iso. Absence of the orthogonal smectic A* phase for this material caused definite difficulties while obtaining a homogeneous alignment. The second material studied was also ferroelectric liquid crystal, denoted as*

**Ce3***; it possesses a very rich polymorphism between the isotropic and crystal phases: Cr 39.6 °C SmG 56.0 °C SmJ 65.0 °C SmF 67.0 °C SmC* 86.0 °C SmA* 124.0 °C N* 145.5 °C BP 147.0 °C Iso. The resulting binary mixture possesses the following sequence of mesophases: Cr 39.6 °C SmG 56.0 °C SmJ 65.0 °C SmF 67.0 °C SmC* 85.8 °C SmA* 123.2 °C N* 148.7 °C BP 150.0 °C Iso.*

**Ce8**For the measurements, the materials under the study were introduced by means of capillarity action in standard planar cells produced by Linkam Co. (UK) of 5 μm thick with ITO electrodes coated with a planar alignment polymer layers. The cells were placed in a modified Mettler hot stage. Their temperature was stabilized using Digi-Sense TC-9500 temperature controller with the accuracy of about 0.1 K.

The measurement of optic axis deviation *a* in a weak electric field is essential for determination of the elastic constant *K* when the electro-optic method is used. This quantity was measured by detection of electro-optical response with a photodiode connected to a lock-in amplifier SR850 (Stanford Research) followed by the calibration procedure as described by Dardas et al. 2011. This calibration procedure allows expressing the experimental results as angular quantities independent on experimental conditions. The remaining quantities can be found using further methods and techniques presented in more details by Diamant et al. (1957), Dąbrowski et al. (1992), Kuczyński et al. (2002), Dardas and Kuczyński (2004), Jeżewski et al. (2008) Kuczyński (2010), Kuczyński et al. (2010) and Kuczyński et al. (2012). The relaxation time was determined from the measurement of electro-optic response versus frequency in order to determine the rotational viscosity of the system.

## Results and discussion

*K*using Eq. (4) in the weak external field limit, from the static properties of the light modulation depth. After that, in order to determine the rotational viscosity coefficient

*γ*using Eq. (5), the frequency of Goldstone mode relaxation by the electro-optic method was measured. The frequency dependence of the real (corresponds to a dielectric dispersion) and imaginary part of the optical response (corresponds to a dielectric absorption curve) for

*and*

**Ce3***is presented in Fig. 4.*

**Ce8***compound, the relaxation time decreases with temperature, whereas the relaxation time is not sensitive to temperature change in case of*

**Ce3***compound. On the basis of the obtained results, the coefficients of rotational viscosity of the tested materials were derived. The temperature dependence of the rotational viscosity,*

**Ce8***γ*, for

*and*

**Ce3***pure compounds and for the resulting*

**Ce8***binary mixture are presented in Fig. 5. The rotational viscosity coefficient*

**Ce3/8***γ*vanishes at the phase transition from the SmC* phase to the paraelectric phase without helically structure (SmA*) or, in our case, with a different kind of helix (N*). This can be expected because the rotational viscosity

*γ*parameter describes the properties of the

**c**-director, which vanish at this transition. It is important to mention that from the application point of view, as predicted, the coefficient of rotational viscosity of the binary mixture runs practically between the curves obtained for pure materials. The value of the viscosity coefficient in the ferroelectric SmC* phase of the tested materials at 72 °C was 22 mPa s for the

*material, 38 mPa s for*

**Ce3***, and 29 mPa s for*

**Ce8***binary mixture, respectively. The viscosity coefficient is the smallest for material lacking the smectic A* phase. In the binary mixture, the smectic A* phase was induced and shows the viscous consequence. The highest viscosity has been detected for*

**Ce3/8***compound. For all three studied materials, the viscosity coefficients considerably decreasing with temperature increase.*

**Ce8***lnγ*versus inverse temperature is presented in Fig. 6 and shows a linear behavior in a wide temperature range far from the transition to the paraelectric phase. The obtained activation energy for the material

*is equal to 0.67 eV within the ferroelectric SmC* phase and is comparable with the results obtained for other similar materials (see for example refs. Wojciechowski et al. 2013). The activation energy values obtained for*

**Ce8***binary mixture increase to about 1.6 eV. In case of*

**Ce3/8***compound with uncommon, direct transition from the twisted ferroelectric phase to the cholesteric phase, the activation energy is significantly increased. In this case, the increase in activation energy is probably caused by the slowing down of the reaction time and frustration of the helix.*

**Ce3**## Conclusion

The main objective of this work was to check the viscoelastic behavior of a binary ferroelectric mixture and to compare the values with those obtained on the pure FLC materials—the original components of the mixture. In the specific system studied here, the experimental results reveal that if we know the viscoelastic properties, specifically the viscosity, of the pure components, it is possible to predict it for the resulting binary mixture, e.g., the data presented in this work clearly show that the viscoelastic properties of new binary * Ce3/8* mixture can be pre-determined by well-known viscoelastic properties of

*and*

**Ce3***materials. Very promising, from the potentials application point of view in information visualization devices is the fact, that knowing the coefficients of pure liquid crystal material with the ferroelectric properties, we could to anticipate the viscoelastic properties in newly designed liquid crystalline mixtures. In case of ferroelectric liquid crystalline*

**Ce8***binary mixture, the estimated coefficients of elasticity (Dardas 2016) and, in this work, the viscosity are practically between the elasticity and viscosity coefficients of the original*

**Ce3/8***and*

**Ce3***pure materials. Having available a database of viscoelastic properties of FLC materials, it will be quite possible to produce a binary or a multicomponent mixture with the desired viscoelastic behavior. In the future, it would be worthwhile to verify this hypothesis on different subsequent FLC materials and to expend the area of ferroelectric materials/mixtures to other polar mesophases, like antiferroelectric and ferrielectric smectic mesophases. An essential difference in the evolution of the rotational viscosity seems to have now its grounds on helicoidal features and the nature of the neighboring liquid crystalline phases.*

**Ce8**## Notes

### Funding information

This work was supported by the National Science Centre (NCN) Poland under Grant 2017/25/B/ST3/00564.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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