**3.1 Order parameter**

To quantify amount of the orientational order in the LC phase, the term order parameter has been introduced; it is a second-rank symmetric traceless tensor defined as

**Figure 9.** *(a) Liquid crystal director direction and (b) temperature dependence of order parameter.*

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where θ is the angle between the axis of an individual molecule and the local director *n*^ as shown in **Figure 9(a)**. It is the preferred direction in a volume element of a LC, and the average is taken over the complete ensemble. The bracket denotes both temporal and spatial average. For a completely isotropic sample, S = 0, whereas for a perfectly aligned sample, S = 1. For a typical LC sample, the value of S is 0.3 to 0.9, and for nematic LC, it is 0.5–0.7. **Figure 9(b)** shows the temperature dependence of order parameter (S), which follows an inverse relation [11, 35, 36].

#### **3.2 Anisotropy in liquid crystals**

LCs exhibit uniaxial symmetry around the director, which gives them shape anisotropy. The shape anisotropy of LC and their resulting interactions with the surrounding environment (applied fields) leads to an anisotropy in many other physical properties such as refractive index (RI), dielectric permittivity, magnetic susceptibility, viscosity and conductivity.

#### *3.2.1 Optical anisotropy*

micellar phase (a bulk LC sample with spherical water cavities) [33, 34]. Different

1.Hexagonal phase (hexagonal columnar phase) (middle phase) (**Figure 8(a)**)

By varying concentration, even within the same phases, their self-assembled structures can be tuned. For example, in lamellar phases, distance between the layers increases with the solvent volume. Since lyotropic LCs indirectly depend on a subtle balance of intermolecular interactions, it is difficult to analyse their properties and structures as compared to those of thermotropic LCs. Similar type of phases

To quantify amount of the orientational order in the LC phase, the term order parameter has been introduced; it is a second-rank symmetric traceless tensor

*<sup>θ</sup>* � <sup>1</sup> ; For nematic phase 0*:*5<sup>&</sup>lt; *<sup>S</sup>*<0*:*7 (4)

and properties has been observed in immiscible diblock copolymers.

*(a) Liquid crystal director direction and (b) temperature dependence of order parameter.*

2.Discontinuous cubic phase (micellar cubic phase) (**Figure 8(b)**)

lyotropic phases are listed below:

*Liquid Crystals and Display Technology*

3.Lamellar phase (**Figure 8(c)**)

5.Reverse hexagonal columnar phase

6. Inverse cubic phase (inverse micellar phase)

4.Bicontinuous cubic phase

**3. Properties of liquid crystals**

*<sup>S</sup>* <sup>¼</sup> <sup>1</sup>

<sup>2</sup> 3 cos <sup>2</sup>

**3.1 Order parameter**

defined as

**Figure 9.**

**20**

LCs are optically anisotropic materials and show birefringence. LCs have two direction-dependent refractive indices, ordinary RI ( *no*) and extraordinary RI ( *ne*) with birefringence:

$$
\Delta n = n\_e - n\_o. \tag{5}
$$

Also, the average RI is given by

*nav* ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3 *n*2 *<sup>e</sup>* þ 2*n*<sup>2</sup> *o* � � <sup>r</sup> (6)

**Figure 10.**

*Indicatrix of optically uniaxially material: (a) positive birefringent material and (b) negative birefringent material.*

The value of *Δn* may be positive or negative, which can be represented by indicatrix as shown in **Figure 10**. For uniaxial crystal, it is ellipsoid where the rotational axis is identical to the optical axis [34, 37, 38].

For rod-like molecules (Nematic LC) *ne* >*no*, where Δ*n* is positive and between 0.02 and 0.4. For discotic and chiral nematic molecules *ne* < *no*, and thus negative birefringence is associated with the discotic or columnar phase.

The values of optical anisotropy *Δn* can be increased [39]:


In the LC materials, consisting of non-polar molecules, there is only an induced polarization, which consists of two parts: the electronic polarization (which is also present at optical frequencies) and the ionic polarization. In the LCs with polar molecules, the orientational polarization exists along with the above-mentioned polarization. Considering the uniaxial LC phases in a macroscopic coordinate system, *x*, *y* and *z,* with the *z*-axis parallel to the director *n*^, it is possible to distinguish two principal permittivities, parallel to the director *ε<sup>ǁ</sup>* ¼ *εzz* and perpendicular to the director *<sup>ε</sup>*<sup>⊥</sup> <sup>¼</sup> <sup>1</sup>*=*<sup>2</sup> *<sup>ε</sup>xx* <sup>þ</sup> *<sup>ε</sup>yy* . *<sup>ε</sup><sup>ǁ</sup>* is the characteristic of nematic LCs, as it corresponds to the polarization contribution related to the molecules. Then the dielectric anisotropy *Δε* ¼ *ε<sup>ǁ</sup>* � *ε*<sup>⊥</sup> can take positive or negative values. If the value of *Δε*> 0, then LC molecules align parallel to the field, whereas if the value of *Δε*<0, then the LC molecules tend to align perpendicular to the field (**Figure 12(a)**). The graph of temperature dependence of dielectric permittivity for a typical LC (**Figure 12(b)**) shows that magnitude of *Δε* usually depends on temperature. With the increase in temperature, liquid crystal material behaves as isotropic liquid with dielectric

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*(a) Alignment of positive and negative dielectric anisotropic LCs in external field, (b) temperature dependence*

The mean dielectric permittivity *ε* is temperature and frequency dependent,

The behaviour of LCs in an external electric field is highly dependent on their viscoelastic properties. While dealing with the elasticity of the nematic LC, we assume that the order parameter S remains invariable throughout the volume of LC at a constant temperature T and only director *n*^ changes with external field. The elastic constants of LCs associated with the restoring torques become apparent when the system is perturbed from its equilibrium configuration. These are of the order of 10�<sup>11</sup> N (especially for nematic and fluid smectic phases), which suggests that a LC can be easily deformed by external forces, such as mechanical, electric or magnetic. The resistance of the LC to the external field gives rise to deformation. Final deformation pattern depends on the contribution of the associated elastic

<sup>3</sup> ð Þ *<sup>ε</sup><sup>ǁ</sup>* � *<sup>ε</sup>*<sup>⊥</sup> (7)

*<sup>ε</sup>* <sup>¼</sup> <sup>1</sup>

permittivity *εiso* [15].

**Figure 12.**

*of dielectric constant.*

**3.3 Elastic properties**

**23**

which can be described as

In general, birefringence *Δn* of LCs decreases as the wavelength of the incident light or the temperature increases. Also, if the temperature of the LC material is raised up to its clearing point/nematic-isotropic temperature (TNI), its internal order gets destroyed, and it behaves like an isotropic liquid with RI *n*iso as shown in **Figure 11**.

**Figure 11.**

*Temperature dependence of refractive index (RI).*

#### *3.2.2 Dielectric anisotropy*

Dielectric properties of LCs are related to the response of LC molecules upon application of an electric field. Permittivity is a property of a material that determines how dielectric medium affects and is affected by an electric field. It is determined by the capability of a material to polarize upon application of an electric field and in turn partially cancels the field induced inside the material [40, 41].

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#### **Figure 12.**

The value of *Δn* may be positive or negative, which can be represented by indicatrix as shown in **Figure 10**. For uniaxial crystal, it is ellipsoid where the

For rod-like molecules (Nematic LC) *ne* >*no*, where Δ*n* is positive and between 0.02 and 0.4. For discotic and chiral nematic molecules *ne* < *no*, and thus negative

2.with the elongation of the conjugation chain parallel to the long molecular axis

4.by shortening the alkyl chain of the end molecular groups in homologous series

In general, birefringence *Δn* of LCs decreases as the wavelength of the incident light or the temperature increases. Also, if the temperature of the LC material is raised up to its clearing point/nematic-isotropic temperature (TNI), its internal order gets destroyed, and it behaves like an isotropic liquid with RI *n*iso as shown in

Dielectric properties of LCs are related to the response of LC molecules upon application of an electric field. Permittivity is a property of a material that determines how dielectric medium affects and is affected by an electric field. It is determined by the capability of a material to polarize upon application of an electric field and in turn partially cancels the field induced inside the material [40, 41].

3.by increasing the values of the order parameter S or decreasing the value of

rotational axis is identical to the optical axis [34, 37, 38].

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in the form of even-odd alternation

temperature

*3.2.2 Dielectric anisotropy*

*Temperature dependence of refractive index (RI).*

**Figure 11.**

**22**

**Figure 11**.

birefringence is associated with the discotic or columnar phase. The values of optical anisotropy *Δn* can be increased [39]:

1.by replacing saturated aromatic rings with the unsaturated ones

*(a) Alignment of positive and negative dielectric anisotropic LCs in external field, (b) temperature dependence of dielectric constant.*

In the LC materials, consisting of non-polar molecules, there is only an induced polarization, which consists of two parts: the electronic polarization (which is also present at optical frequencies) and the ionic polarization. In the LCs with polar molecules, the orientational polarization exists along with the above-mentioned polarization. Considering the uniaxial LC phases in a macroscopic coordinate system, *x*, *y* and *z,* with the *z*-axis parallel to the director *n*^, it is possible to distinguish two principal permittivities, parallel to the director *ε<sup>ǁ</sup>* ¼ *εzz* and perpendicular to the director *<sup>ε</sup>*<sup>⊥</sup> <sup>¼</sup> <sup>1</sup>*=*<sup>2</sup> *<sup>ε</sup>xx* <sup>þ</sup> *<sup>ε</sup>yy* . *<sup>ε</sup><sup>ǁ</sup>* is the characteristic of nematic LCs, as it corresponds to the polarization contribution related to the molecules. Then the dielectric anisotropy *Δε* ¼ *ε<sup>ǁ</sup>* � *ε*<sup>⊥</sup> can take positive or negative values. If the value of *Δε*> 0, then LC molecules align parallel to the field, whereas if the value of *Δε*<0, then the LC molecules tend to align perpendicular to the field (**Figure 12(a)**). The graph of temperature dependence of dielectric permittivity for a typical LC (**Figure 12(b)**) shows that magnitude of *Δε* usually depends on temperature. With the increase in temperature, liquid crystal material behaves as isotropic liquid with dielectric permittivity *εiso* [15].

The mean dielectric permittivity *ε* is temperature and frequency dependent, which can be described as

$$\overline{\varepsilon} = \frac{1}{3} \left( \varepsilon\_{l} - \varepsilon\_{\perp} \right) \tag{7}$$

#### **3.3 Elastic properties**

The behaviour of LCs in an external electric field is highly dependent on their viscoelastic properties. While dealing with the elasticity of the nematic LC, we assume that the order parameter S remains invariable throughout the volume of LC at a constant temperature T and only director *n*^ changes with external field. The elastic constants of LCs associated with the restoring torques become apparent when the system is perturbed from its equilibrium configuration. These are of the order of 10�<sup>11</sup> N (especially for nematic and fluid smectic phases), which suggests that a LC can be easily deformed by external forces, such as mechanical, electric or magnetic. The resistance of the LC to the external field gives rise to deformation. Final deformation pattern depends on the contribution of the associated elastic

5.Replacement of phenyl ring by a trans-cyclohexane ring results in reduced

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The most useful compounds for reducing viscosity in LC materials are cyclohexane derivatives due to their high clearing temperature, good solubility and low

The dependence of the free energy "*F*" of nematic LC on gradients of the director field is a unique property of LC. Therefore, the measurement of elastic constants in LC is a very crucial part in LC studies. The idea behind *Kii* measurement is related to the registration of spatial distortions in structure induced by different factors such as electric field, magnetic field and thermal and surface

Out of which the optical method based on Freedericksz transition is the simplest

In the absence of any surface alignment or external field, LC directors of nematic

Consider two, coated and rubbed (along *X* direction), conducting glass plates,

separated by a distance "*d*". Due to this, LC director tends to align along the direction parallel to the flat surface (*X* direction). Now we consider the following

cases which give rise to splay, bend and twist geometries [20].

molecules are free to point in any direction. However, it is possible to force the director to point in a specific direction by introducing an outside agent to the system. For example, when a thin layer of polymer (usually a polyimide (PI)) is coated on a glass substrate and rubbed in a single direction with a velvet cloth, it is observed that LC molecules in contact with that surface get aligned along the rubbing direction and achieve uniform director configuration. Upon application of magnetic or electric field for any distortion to occur (to overcome the elastic and viscoelastic forces of LC), the strength of the applied field has to be larger than certain threshold value [21]. Initially, when electric field is low, no change in alignment occurs. However, as we increase electric field above threshold, the LC director changes its orientation from one molecule to the next, and deformation occurs. This threshold is called the Freedericksz threshold, and the transition from a uniform director configuration to deformed director configuration is named as Freedericksz transition. To find out various elastic constants, we need to understand geometry of confined LC molecules and applied external field. The external field may be either electric or magnetic; it is more convenient and accurate to record electric field because measurement of magnetic field near to the sample is a tricky procedure due to the field inhomogeneity, and temperature dependence of

fluctuations for which the following methods can be employed:

and most significant from the application point of view [40].

1.Optical method (Freedericksz transition)

viscosity values.

2.Light scattering

*3.5.1 Freedericksz transition*

Hall probe, etc.

**25**

3.Alignment inversion walls

4.Cholesteric-nematic transition

**3.5 LC in electric and magnetic field**

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viscosity.

**Figure 13.** *(a) Splay, (b) twist and (c) bend deformation.*

constants in elastic energy. For nematic LCs, it is assumed that change in elastic energy is only due to splay, twist and bend type deformation (**Figure 13**). The increase of free energy F due to these deformations is described by the continuum theory. This theory was first developed by Oseen and Zocher and later reformulated by Frank. It was based on the balance laws for linear and angular momentum [4, 42]. The contribution of each deformation to the overall energy F is given by

$$F = \frac{1}{2} \left[ K\_{11} (\nabla \cdot \boldsymbol{n})^2 + K\_{22} (\boldsymbol{n} \cdot \nabla \times \boldsymbol{n})^2 + K\_{33} (\boldsymbol{n} \times \nabla \times \boldsymbol{n})^2 \right] \tag{8}$$

where *K*11, *K*<sup>22</sup> and *K*<sup>33</sup> are proportionality constants of splay, twist and bend deformations, respectively, often known as Frank elastic constants [19]. They were forced to splay, twist and bend until equilibrium. When the system is in equilibrium, it is in minimum energy state [15]. Other types of deformation are forbidden due to the symmetry and absent polarity.
