Critical behaviour at the nematic–smectic A phase transition in a binary mixture showing induced nematic phase

Abstract

High resolution optical birefringence (Δn) measurements have been performed on thirteen mixtures of two smectogenic compounds, 5-trans-n-pentyl-2-(4-isothiocyanatophenyl)-1,3-dioxane (5DBT) and 4-cyano-4′-n-decyloxy-biphenyl (10OCB), showing an induced nematic phase from the optical transmission method for two different wavelengths. The values of the birefringence have also been compared with those determined from the thin prism method. The critical behaviour of the nematic–smectic A (N–SmA) phase transition has been studied from the birefringence measurements and the nature of the transition in these mixtures has been discussed. The critical exponent values for these mixtures show a definite pattern when plotted both against molar concentration as well as McMillan ratio (T_SN/T_NI). On both sides of the phase diagram, a uniform crossover trend from second order to first order N–SmA phase transition is observed as the tricritical point (TCP) is approached. For both the TCP's there exists a common value of the McMillan ratio = 0.992. In addition, the 3D-XY model is reached almost exactly at x_5DBT = 0.696 for which the McMillan ratio is 0.912.

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1 Introduction

Studies of the mesomorphic properties of liquid crystalline mixtures have received special attention not only due to their technological importance^1–3 but also due to their fundamental interest for understanding the phenomena of phase transitions in soft matter. In order to obtain better materials which can produce desirable values of the different display parameters, often liquid crystalline compounds are mixed together. By mixing several compounds one can control the physical properties of the resulting liquid crystalline mixtures, such as birefringence, dielectric anisotropy, elastic constants and rotational viscosity in accordance with the requirements for display as well as non-display applications.⁴ On the other hand binary mixtures are also used to study the various phase transitions and critical phenomena⁵ in the field of liquid crystal research. The nematic–smectic A (N–SmA) phase transition has been studied extensively in binary mixtures.^6–8

Sometimes mixtures of nematic compounds may induce new higher order liquid crystalline phases which are commonly known as induced phases.^9–12 These types of behaviors are generally observed in binary mixtures of two compounds, one having a strongly polar terminal group and the other being a non-polar one^12–21 or in mixtures of two polar nematic compounds.^22–24 Conversely, smectic phases existing in pure compounds in the binary mixtures decrease their smectic stability and new lower order phases are created. These phases are termed as induced nematic phases.^25–28

Moreover, several types of phase transitions in liquid crystals have been studied for testing the general concepts of phase transitions and critical phenomena. The first-order and second-order character of the phase transitions as well as the universality class of the critical exponents has been investigated widely by a large variety of experimental techniques. The transition between nematic (N) and the smectic A (SmA) phase has been studied extensively but still remains one of the most interesting problems in the area of condensed matter physics. In spite of several efforts, the nature of nematic (N) and the smectic A (SmA) phase transition has been a matter of controversy due to lack of agreement between the different reported results. The one dimensional positional order of the SmA phase can be described in terms of a two component complex order parameter, and thus the nematic–smecticA (N–SmA) phase transition was expected to be in the three dimensional 3D-XY universality class.²⁹ But, different experiments have revealed non universal critical behaviour. Using the mean field theory, Kobayashi,³⁰ McMillan³¹ and de Gennes²⁹ predicted a crossover from second-order transition to first-order one at a tricritical point (TCP). The theoretical limiting value of McMillan ratio (T_SN/T_NI) at the tricritical point (TCP) is expected to be 0.87 where, T_SN and T_NI are the nematic–smectic A and nematic–isotropic phase transition temperatures respectively. So when the McMillan ratio is above the limiting value, the N–SmA transition is first order while below that it is second order transition. Although experimentally the crossover happens at a value higher than that predicted and it strongly depends on the polarity of the systems under study. Alben³² suggested the occurrence of a tricritical point (TCP) in binary liquid crystal mixture for the smectic A to nematic transition. Till now a number of experiments have been performed to find the tricritical point (TCP) in binary mixtures and from those studies it is observed that the tricritical point do not show evidence of a universal value for the McMillan ratio which ranges from 0.942 to 0.994. Moreover, coupling with the nematic order parameter and nematic fluctuations may affect the order of the N–SmA transition. Using the Landau free energy expansion de Gennes²⁹ showed that if the nematic region is small, strong coupling between the nematic and smectic order parameter makes N–SmA transition first order whereas for large nematic range weaker coupling driven away the transition to second order. On the other hand, Halperin, Lubenski and Ma^33,34 predicted that if coupling between the nematic director fluctuation and the smectic A order parameter is taken into account, the N–SmA phase transition should always be a weakly first order in nature, although the latent heat involved might be quite small. So the conventional tricritical behaviour is ruled out as this is a fluctuation driven first order transition. To modify the coupling strength several mechanisms are used and one of the most common methods is to vary the nematic range which can be achieved by preparing the binary mixtures.

During the last four decades, many high-resolution heat capacity and X-ray studies have been devoted to the N–SmA phase transition and the centre of attention is the critical exponents for determining the universality class of this transition. But high resolution birefringence data showing critical behaviour in the vicinity of N–SmA transition is scanty. Most of the cases the resolution of the data available in the literature in the vicinity of the nematic–smectic transition from birefringence measurements are insufficient to extract the critical behavior near the phase transition region. In this paper the critical behaviour of the smectic A to nematic transition has been analyzed on the basis of high resolution birefringence measurements. The critical exponent (α) involved with the N–SmA transition provides information about the order character of the transition and therefore enables us to locate the tricritical point (TCP).

In this work a detailed study has been performed for mixtures of two smectogenic compounds 5-trans-n-pentyl-2-(4-isothiocyanatophenyl)-1,3-dioxane (5DBT) and 4-cyano-4′-n-decyloxy-biphenyl (10OCB) showing an induced nematic phase in a certain concentration range of 0.05 < x_5DBT < 0.952 by means of high-resolution optical birefringence measurement. The first component has a smectic A₁ phase and the second component possesses partially bilayer smectic A_d phase. Measurements of optical birefringence have been conducted by two different probing methods viz. thin prism and optical transmission (OT) methods which permit a precise comparison between the two sets of values. The results obtained by these two techniques are in good agreement with a small deviation of about 2–3%.

Temperature dependence of extraordinary and ordinary refractive indices as well as refractive index in the isotropic phase have been measured at a wavelength λ = 632.8 nm by using the thin prism technique.³⁵ As the refractive index measurement was performed on the visual inspection by means of wedge method, it is difficult to take sufficient data points required to extract the critical behaviour near the phase transition. Using the high resolution (in both temperature and birefringence) temperature scanning technique the optical birefringence (Δn = n_e − n_o) have also been determined for all the mixtures at two different wavelengths 532 nm and 632.8 nm and have been compared with the same as obtained from thin prism technique. Phase diagram and dielectric permittivity measurement have been reported by us.³⁶

2 Experimental

2.1 Materials

The compounds 5DBT and 10OCB were purchased from AWAT Co. Ltd., Warsaw, Poland and Merck, U.K. respectively and were used without further purification. The structural formulae and chemical names of the two pure smectogenic liquid crystal compounds are as follows:

Component 1: 5-trans-n-pentyl-2-(4-isothiocyanatophenyl)-1,3-dioxane (5DBT)

Component 2: 4-cyano-4′-n-decyloxy-biphenyl (10OCB)

Thirteen mixtures having molar concentration of 5DBT equal to 0.100, 0.161, 0.203, 0.252, 0.301, 0.447, 0.503, 0.604, 0.696, 0.741, 0.804, 0.887 and 0.952 were prepared. The phase transition temperatures were determined with the help of a polarizing optical microscope (Motic BA 300) equipped with a Mettler FP900 hot stage.

2.2 Optical birefringence measurements

2.2.1 Thin prism method. The refractive indices (n_o, n_e) for a wavelength of λ = 632.8 nm were measured within ±0.0006 by thin prism technique.³⁵ A hollow glass prism (refracting angle <2°) was constructed by placing the rubbed surfaces inside, with the rubbing direction parallel to the refracting edge of the prism. The temperature of the sample filled prism was controlled by a temperature controller (Eurotherm 2404) with an accuracy of ±0.1 K. A magnetic field of about 0.8 Tesla was also applied to align the liquid crystalline sample. When a He–Ne laser beam passes through the aligned sample it splits into two rays i.e. ordinary and extraordinary rays. From the high resolution photograph of the two images the ordinary (n_o) and extra ordinary refractive indices (n_e) have been calculated using the prism angle.

2.2.2 Optical transmission (OT) method. The high resolution optical birefringence (Δn) measurements (accuracy ±10⁻⁵) were also performed by measuring the intensity of a laser beam transmitted through a homogeneously aligned LC cell of thickness 5 μm. To probe its phase retardation a solid state green laser (λ = 532 nm) and a He–Ne laser (λ = 632.8 nm) beam were used. The temperature of the cell was regulated and measured with a temperature controller (Eurotherm PID 2404) with an accuracy of ±0.1 K by placing the cell in a custom built heater made of brass. The transmitted light intensity was measured by a photo diode at an interval of 3 seconds. When the heater temperature is varied at a rate of 0.5 K min⁻¹, this translates into a temperature difference of 0.025 K between two readings. The experimental details for the birefringence measurement from optical transmission method in the nematic and SmA phases have been described in our earlier publications.^37,38

3 Results and discussions

3.1 Phase diagram

The phase diagram of the binary system 5DBT + 10OCB (Fig. 1) has been discussed in detail in our earlier publication.³⁶ The nematic–isotropic (T_NI) and nematic–smectic A (T_SN) phase transition temperatures were recorded by us from polarizing optical microscopy during cooling. It is observed that in mixtures of two polar smectogenic compounds, where one of them possesses the monolayer smectic A (SmA₁) and the other partially bilayer smectic A (SmA_d) phase, a depression of smectic phase stability occurs with the appearance of a nematic gap within the concentration range 0.05 < x_5DBT < 0.952. The smectic phase stability is higher in the region where the concentration of any one of the pure components is high. In the concentration range 0.252 < x_5DBT < 0.696 only nematic phase is present. Typical marbled type textures were observed in the induced nematic phases whereas in the smectic phases texture patterns were either fan shaped³⁹ or focal conic.

Fig. 1 Phase diagram for the binary system of 5DBT + 10OCB. x_5DBT is the mole fraction of 5DBT. I – isotropic phase, N – nematic phase and SmA – smectic A phase. ○ – nematic to isotropic (or smectic to isotropic) phase transition temperature; ■ – nematic to smectic A phase transition temperature and □ – melting temperature.

3.2 Refractive index measurements

Temperature dependence of the principal refractive indices n_o and n_e in the N and SmA phases and the refractive index in the isotropic phase (n_iso) at a wavelength of λ = 632.8 nm of all the thirteen mixtures obtained from the thin prism method are shown in Fig. 2. For all the mixtures, the values of the extraordinary refractive index decreases with increase in temperature, while the values of ordinary refractive index remains almost constant in lower temperature region but increases with increase in temperature near N–I transition. On cooling from isotropic to nematic phase a pronounced change in the refractive index is observed whereas near the nematic to smectic A phase transition the change is not so prominent except for the two mixtures x_5DBT = 0.252 and 0.741. For these two concentrations, the extraordinary and ordinary components of refractive indices show a noticeable change at the nematic–smectic A phase transition.

Fig. 2 Variation of extraordinary and ordinary refractive indices with temperature for (a) ○ – x_5DBT = 0.100; ● – x_5DBT = 0.161; Δ – x_5DBT = 0.203; □ – x_5DBT = 0.252, (b) ○ – x_5DBT = 0.301; ● – x_5DBT = 0.447; Δ – x_5DBT = 0.503; □ – x_5DBT = 0.604, (c) ○ – x_5DBT = 0.696; ● – x_5DBT = 0.741; Δ – x_5DBT = 0.804; □ – x_5DBT = 0.887; ▲ – x_5DBT = 0.952. Solid arrow denotes nematic–isotropic transition (T_NI) or smectic A–isotropic (T_SI) transition (only for 0.952) and dashed arrow represents nematic–smectic A transition (T_SN) respectively. Molar concentrations of different mixtures are indicated in the figure.

3.3 Optical birefringence measurements

The optical birefringence (Δn) of the mixtures have been measured for two different wavelengths (532 nm and 632.8 nm) using high resolution temperature scanning technique. These two sets of values have been compared with those obtained from thin prism technique and is shown in Fig. 3. It is seen that the Δn values obtained from thin prism technique are slightly lower than the values obtained from optical transmission (OT) method. The possible reason for this discrepancy in the two sets of measurements is due to the fact that in case of thin prism the sample thickness is very much higher (40–80 times) than 5.0 μm cell which is used in the transmission method. Therefore, the surface anchoring is much better for the thin cells in comparison to the bulk samples in thin prism,³⁷ which causes a little high birefringence in the transmission method. The small differences in the Δn value obtained from transmission method have been reported by others.⁴⁰ The birefringence data covers the nematic as well as smectic A phase of the mixtures. The Δn values drop rapidly near the nematic–isotropic transition. From Fig. 3 it is seen that most of the mixtures exhibit continuous changes in the birefringence at the N–SmA transition except x_5DBT = 0.100. Within the temperature range of about 1–3 K above and below the smectic–nematic transition the pretransitional effect for N–SmA coupling has clearly been observed. Optical birefringence determined for the wavelength 532 nm is found to be slightly higher than that for 632.8 nm. This type of variations of optical anisotropy with wavelength has also been reported by Blinov et al.⁴¹ for well known liquid crystals PAA and MBBA. If we plot the birefringence of the mixtures against the molar concentration of 5DBT as shown in Fig. 4, then Δn values obtained from two different wavelengths show the similar variations. In the lower concentration region of 5DBT Δn remains almost constant and then decreases with increase in concentration, showing a local minimum near 0.604.

Fig. 3 Birefringence (Δn = n_e − n_o) as a function of temperature for different mixtures. (a) x_5DBT = 0.100, (b) x_5DBT = 0.252, (c) x_5DBT = 0.447 and (d) x_5DBT = 0.887. Δ and □ represents birefringence from optical transmission (OT) method with λ = 632.8 nm and λ = 532.0 nm respectively whereas the solid circles (•) represents the same obtained from thin prism technique for λ = 632.8 nm. Solid arrow denotes nematic–isotropic transition (T_NI) and dashed arrow represents nematic–smectic A transition (T_SN) temperatures respectively. In the inset of (a), b and d) the variation of birefringence (Δn) in the vicinity of N–SmA phase transition are shown.

Fig. 4 Concentration dependence of birefringence at T = 330 K. Δ – birefringence from optical transmission method with λ = 632.8 nm, □ – birefringence from optical transmission method with λ = 532.0 nm and • – represents the same obtained from thin prism technique with λ = 632.8 nm.

3.4 Determination of critical exponent at the N–SmA phase transition

The birefringence data have been used to explore the critical behaviour of the N–SmA phase transition. At the nematic–smA phase transition Δn do not show any sharp discontinuity, therefore to locate the exact transition temperature the quantity d(Δn)/dT has been used, which is related to specific heat anomaly.⁴² However, the first order temperature derivative of Δn obtained numerically is too scattered due to small temperature intervals.⁴³ Therefore, Erkan et al.⁴³ have defined a new parameter which has the following form:where Δn(T_SN) is the birefringence value at the transition temperature T_SN as obtained by differentiating Δn. If −d(Δn)/dT follows a power law nature with a critical exponent α then the parameter Q(T) also follows the power law behaviour with the same critical exponent but with a different critical amplitude.⁴³ So Q(T) also shows a critical nature at the N–SmA phase transition. To study the critical behaviour of the parameter Q(T) at T_SN, we have fitted the data of Q(T) as obtained from eqn (1) with the following equation⁴³

Q(T) = A_±|t|^−α + B_± (2)

where A₊ and A₋ are the critical amplitudes and B₊ and B₋ are the background terms above and below the nematic–smectic A phase transition temperature (T_SN), α is the critical exponent similar to specific heat critical exponent and t = |(T − T_SN)/(T_SN)| is the reduced temperature. Fig. 5(a) and (b) depict the temperature variation of Q(T) showing the critical behaviour in the vicinity of N–SmA phase transition for the eight mixtures.

Fig. 5 (a) and (b) Temperature variation of the parameter Q(T) in the vicinity of nematic–smectic A (N–SmA) phase transition. Open circles (○) represent the calculated Q(T) values (for λ = 532 nm) and the solid lines are fit to eqn (2). Different concentrations are indicated in the figures.

Analyzing the optical birefringence data on both sides of the N–SmA transition, the values of the critical exponent associated with the parameter Q(T) has been determined. This method has been found to be very accurate and offers an easy way to determine the order of the N–SmA transition as compared to calorimetric methods.⁴³ The value of this critical exponent (α), which is similar to the specific heat critical exponent, reflects the nature of N–SmA transition. Table 1 lists the values of the critical exponent (α), the critical amplitude and the background term along with the errors associated with the parameters obtained from eqn (2). During fitting some data points near the N–SmA phase transition are excluded to minimize the error and to get a better fit to the data. The fitting quality has been tested by determining the reduced χ² value on both sides of the transition temperature. For an ideal fit χ² value is unity but generally the values in between 1 to 1.5 yield a good fit. The reduced χ² is defined by the ratio of variance of the fit (s²) and the variance of the experimental data (σ²), and can be written as follows:⁴⁴

where N is the total number of data points, p is the number of adjustable parameter and f_i is the i^th fit value corresponding to the measurement y_i. All the parameter sets represent well enough the Q(T) data, as indicated by χ² value. We obtained the values of χ² in between 1 to 1.3.

Table 1 The best fitted parameter values for Q(T) near N–SmA transition obtained from eqn (2) and the corresponding χ² associated with the fit

x5DBT		A− or A+	α	B− or B+	χ^2	No. of points
0.100	T < TSN	0.00065 ± 0.00004	0.511 ± 0.016	−0.00205 ± 0.00008	1.212	110
	T > TSN	0.00043 ± 0.00014	0.509 ± 0.036	0.01627 ± 0.00099	1.269	20
0.161	T < TSN	0.00053 ± 0.00014	0.454 ± 0.020	−0.00074 ± 0.00021	1.080	87
	T > TSN	0.00031 ± 0.00006	0.446 ± 0.057	0.00435 ± 0.00026	1.195	24
0.203	T < TSN	0.00027 ± 0.00007	0.365 ± 0.031	0.00228 ± 0.00018	1.055	44
	T > TSN	0.00035 ± 0.00017	0.358 ± 0.057	0.00186 ± 0.00043	1.025	31
0.252	T < TSN	0.0005 ± 0.00007	0.221 ± 0.014	0.00047 ± 0.00011	1.117	44
	T > TSN	0.00044 ± 0.00012	0.212 ± 0.048	0.0007 ± 0.00002	1.002	63
0.696	T < TSN	0.0822 ± 0.0062	−0.0068 ± 0.04	−0.0823 ± 0.005	1.107	66
	T > TSN	0.0777 ± 0.0097	−0.0067 ± 0.03	−0.0778 ± 0.002	1.036	33
0.741	T < TSN	0.017 ± 0.0014	0.102 ± 0.027	−0.026 ± 0.0015	1.228	55
	T > TSN	0.013 ± 0.004	0.103 ± 0.014	−0.019 ± 0.005	1.075	45
0.804	T < TSN	0.00096 ± 0.00002	0.284 ± 0.047	0.00043 ± 0.0001	1.009	70
	T > TSN	0.0010 ± 0.00012	0.289 ± 0.010	−0.00025 ± 0.0001	1.136	65
0.887	T < TSN	0.00113 ± 0.00047	0.484 ± 0.035	−0.00123 ± 0.00078	1.152	78
	T > TSN	0.00140 ± 0.00015	0.478 ± 0.015	0.00267 ± 0.00054	1.042	70

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The concentration dependence of the critical exponent (α) is shown in the Fig. 6 taking the averages over the two values obtained above and below the N–SmA phase transition. The α values provide information about the order character of the transition and therefore enable us to locate the tricritical composition (TCP). Within the concentration range 0.1 < x_5DBT < 0.952, it is observed that the critical exponent (α) values is less than the tricritical value of 0.5, clearly reflecting a second order phase transition for these mixtures at the N–SmA phase boundary. However, for x_5DBT = 0.100, the critical exponent value approaches the tricritical value implying a first order nature of the N–SmA phase transition. Thus, the possible crossover between the second order to first order nature of N–SmA transition in this induced nematic system occurs somewhere in between x_5DBT = 0.100 to 0.165. On the other hand, for x_5DBT = 0.887 the exponent α has the value 0.481 and since there is no nematic phase present at x_5DBT = 0.952, therefore another tricritical point must be present within 0.887 < x_5DBT < 0.952 for this binary system. As indicated in Fig. 6, the extrapolated values of the concentration at which a changeover in the N–SmA phase transition from second order to first order i.e. the tricritical concentrations are found to be x_5DBT = 0.123 and 0.894, on the two sides of the phase diagram.

Fig. 6 Concentration dependence of the critical exponent (α) obtained by fitting Q(T) to eqn (2). The plotted data are taken by averaging the two α values obtained by fitting for T < T_SN and T > T_SN. The vertical dashed lines are corresponds to the tricritical points (TCP). The solid lines are 2nd order polynomial fit to the data.

The critical exponent values for these mixtures show a definite pattern when plotted against the McMillan ratio as shown in Fig. 7. A second order polynomial fit to the data points yields a tricritical value equal to 0.992 of the McMillan ratio for which α = 0.5. For McMillan ratio greater than 0.992, the coupling between the nematic and smectic order parameters is strong enough to obtain a weak first-order N to SmA transition. In addition, with a critical exponent value of α = −0.0068 ± 0.04, the 3D-XY model is reached (the theoretical 3D-XY value of α = −0.007) almost exactly at x_5DBT = 0.696 for which the McMillan ratio is 0.912. An inspection of Fig. 6 and 7 reveals three facts: (i) there is a correlation between the critical exponent α and the McMillan ratio, (ii) at a similar value of T_SN/T_NI = 0.992, the tricritical point (TCP) occurs for this binary system on both sides of the phase diagram. Above this limiting value T_SN/T_NI = 0.992, the N–SmA phase transition is first order and below this it turns out to be second order (iii) both the tricritical and 3D-XY concentrations are present in this binary system.

Fig. 7 Variation of critical exponent (α) with McMillan ratio (T_SN/T_NI). The solid line is polynomial fit to the data.

4 Summary and conclusions

High resolution optical birefringence (Δn) measurements have been undertaken on a binary system of smectogenic compounds exhibiting an induced nematic phase. This system appears to be an excellent system to study the phase transition and critical phenomena in the field of soft condensed matter. As we go towards the higher or lower concentration region of any of the two pure compounds the width of the nematic region decreases and smectic phase stability increases. Near the centre of the phase diagram the smectic phase completely vanishes and only the nematic phase appears. Measurements of optical birefringence have been conducted by two different probing methods viz. thin prism and optical transmission (OT) methods and the two sets of values are in good agreement with a small deviation of about 2–3%. For all the mixtures the ordinary (n_o) and extraordinary (n_e) refractive indices show normal temperature variation. Near the nematic–isotropic (N–I) transition both the components of refractive indices show a rapid change with respect to small variation of the temperature.

A simple and precise technique for the determination of the optical birefringence as a function of temperature for two different wavelengths was used in this work. It was possible to measure the birefringence Δn with reasonably good accuracy (better than ±10⁻⁵) in all the liquid crystalline phases under study. Particular emphasis was given to the birefringence value along the N–SmA phase transition line to access the order of the phase transition in the binary system under study. The N–SmA phase transition of most of the mixtures are found to be second order in nature except for x_5DBT = 0.100 for which the N–SmA phase transition is of first order. The parameter Q(T) which was calculated from the high resolution birefringence data shows a power law divergence with a critical exponent α at the N–SmA transition. It is found that as the nematic region of the mixtures decreases the critical exponent (α) increases. It is also observed that the critical exponent (α) values lie between the 3D-XY universality class and tricritical point. One of the noticeable aspect of this binary system is the existence of two tricritical points (TCP) one on either side of the phase diagram for x_5DBT = 0.123 and 0.894 for the N–SmA transition, at which a crossover from second order to first order transition takes place. Interestingly, the McMillan ratios of both the TCP compositions occur at T_SN/T_NI = 0.992. In addition, with a critical exponent value of α = −0.0068 ± 0.04, the 3D-XY model is reached almost exactly at x_5DBT = 0.696 for which the McMillan ratio is 0.912.

Acknowledgements

The authors gratefully acknowledge the financial support provided by the Department of Science and Technology, New Delhi (project no. SB-EMEQ-290-2013).

Notes and references

1. M. Roushdy, Liq. Cryst., 2004, 31, 371.
2. B. Bahadur, Mol. Cryst. Liq. Cryst., 1984, 109, 1.
3. J. Kirton and E. P. Raynes, Endeavour, 1973, 32, 71.
4. F. V. Allan, Inf. Disp., 1983, 10, 14.
5. M. A. Anisimov, in Critical Phenomena in Liquids and Liquid Crystals, Gordon and Breach Science Publishers, UK, 1991, ch. 10, pp. 305–408.
6. M. B. Sied, J. Salud, D. O. Lopez, M. Barrio and J. L. Tamarit, Phys. Chem. Chem. Phys., 2002, 4, 2587.
7. M. B. Sied, S. Diez, J. Salud, D. O. Lopez, P. Cusmin, J. L. Tamarit and M. Barrio, J. Phys. Chem. B, 2005, 109, 16284.
8. M. G. Lafouresse, M. B. Sied, H. Allouchi, D. O. Lopez, J. Salud and J. L. Tamarit, Chem. Phys. Lett., 2003, 376, 188.
9. H. Sackmann and D. Demus, Z. Phys. Chem., 1965, 230, 285.
10. J. S. Dave, P. R. Patel and K. L. Vasanth, Indian J. Chem., 1966, 4, 505.
11. W. H. de Jeu, L. Longa and D. Demus, J. Chem. Phys., 1986, 84, 6410.
12. C. S. Oh, Mol. Cryst. Liq. Cryst., 1977, 42, 1.
13. M. K. Das, R. Paul and D. A. Dunmur, Mol. Cryst. Liq. Cryst., 1995, 258, 239.
14. M. Brodzik and R. Dabrowski, Mol. Cryst. Liq. Cryst., 1995, 260, 361.
15. P. E. Cladis, Mol. Cryst. Liq. Cryst., 1981, 67, 177.
16. P. D. Roy, S. Paul and M. K. Das, Phase Transitions, 2006, 79, 323.
17. M. K. Das and R. Paul, Phase Transitions, 1994, 46, 185.
18. S. Garg and T. Spears, Mol. Cryst. Liq. Cryst., 2004, 409, 335.
19. P. D. Roy, A. Prasad and M. K. Das, J. Phys.: Condens. Matter, 2009, 21, 075106.
20. R. Dabrowski and K. Czuprynski, in Modern Topics in Liquid Crystal, from Neutron Scattering to Ferroelectricity, ed. A. Buka, World Scientific, London, 1993, pp. 125–160.
21. D. A. Dunmur, R. G. Walker and P. Palffy-Muhoray, Mol. Cryst. Liq. Cryst., 1985, 122, 321.
22. M. Brodzik and R. Dabrowski, Liq. Cryst., 1995, 18, 61.
23. S. Wrobel, M. Brodzik, R. Dabrowski, B. Gestblom, H. Hasse and S. Hiller, Mol. Cryst. Liq. Cryst., 1997, 302, 223.
24. M. Tykarska, B. Wazynska and I. Ulbin, Proceedings of the SPIE, Poland, 2000, vol. 4147, p. 55.
25. R. Dabrowski and J. Szulc, J. Phys., 1984, 45, 1213.
26. R. Dabrowski and K. Czuprynski, Mol. Cryst. Liq. Cryst., 1987, 146, 341.
27. K. Czuprynski, R. Dabrowski and J. Przedmojski, Mol. Cryst. Liq. Cryst., Lett. Sect., 1989, 4, 429.
28. K. Czuprynski, R. Dabrowski, J. Baran, A. Zywocinski and J. Przedmojski, J. Phys., 1986, 47, 1577.
29. P. G. de Gennes and J. Prost, in The Physics of Liquid Crystals, Clarendon, Oxford, 2nd edn, 1993.
30. K. K. Kobayashi, Phys. Lett. A, 1970, 31, 125.
31. W. L. McMillan, Phys. Rev. A: At., Mol., Opt. Phys., 1971, 4, 1238.
32. R. Alben, Solid State Commun., 1973, 13, 1783.
33. B. I. Halperin, T. C. Lubensky and S. K. Ma, Phys. Rev. Lett., 1974, 32, 292.
34. B. I. Halperin and T. C. Lubensky, Solid State Commun., 1974, 14, 997.
35. A. Prasad and M. K. Das, Phase Transitions, 2010, 83, 1072.
36. S. K. Sarkar and M. K. Das, Fluid Phase Equilib., 2014, 365, 41.
37. A. Prasad and M. K. Das, J. Phys.: Condens. Matter, 2010, 22, 195106.
38. G. Sarkar, B. Das, M. K. Das, U. Baumeister and W. Weissflog, Mol. Cryst. Liq. Cryst., 2011, 540, 188.
39. I. Dierking, in Textures of Liquid Crystals, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2003, ch. 7, pp. 91–93.
40. S. Dhara and N. V. Madhusudana, Phase Transitions, 2008, 81, 561.
41. L. M. Blinov, V. A. Kizel, V. G. Rumyantsev and V. V. Titov, J. Phys., 1975, 36, C1–C69.
42. A. V. Kityk and P. Huber, Appl. Phys. Lett., 2010, 97, 153124.
43. S. Erkan, M. Cetinkaya, S. Yildiz and H. Ozbek, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2012, 86, 041705.
44. P. R. Bevington, in Data reduction and error analysis for the physical sciences, McGraw-Hill, New York, 1st edn, 1969.

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