Suppression of the reentrant nematic and stabilization of the smectic phases by carbon nanotubes

Journal of Molecular Liquids 286 (2019) 110858

Contents lists available at ScienceDirect

Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq

Suppression of the reentrant nematic and stabilization of the smectic phases by carbon nanotubes G.V. Varshini a,c, D.S. Shankar Rao a,⁎, P.K. Mukherjee b, S. Krishna Prasad a a b c

Centre for Nano and Soft Matter Sciences, Bengaluru 560013, India Government College of Engineering & Textile Technology, Serampore 712201, India Physics Department, Mangalore University, Mangalagangotri, India

a r t i c l e

i n f o

Article history: Received 19 January 2019 Received in revised form 22 April 2019 Accepted 24 April 2019 Available online 28 April 2019 Keywords: Reentrant nematic Smectic A Carbon nanotube Antiparallel coupling Conductivity switch

a b s t r a c t We have carried out the first optical, dielectric and Xray investigations on composites of a liquid crystal (LC) exhibiting the nematic-smectic-nematic reentrant (N-SmA-RN) sequence, with small concentrations of carbon nanotubes (CNT). Despite the fact that both components are rod-like anisotropic structures, the presence of CNT, even though in the dilute concentration limit, significantly influences the thermal behaviour of the host LC system. The large aspect ratio (~250) CNT favour the smectic phase over the reentrant nematic: for the 0.5% CNT composite the SmA-RN transition temperature gets lowered by 11 K, whereas the high temperature NSmA boundary is hardly affected. The carbon nanotubes encourage a stronger antiparallel coupling between the neighbouring dipoles, a feature that gets reflected in the dielectric data in the mesophase as well as in the isotropic phase, and corroborated by Xray studies. Magnetic field driven Freedericksz transformation measurements bring out the influence of CNT on the splay and bend Frank elastic constants, with the former exhibiting a large enhancement. The magnetic field-changeable electrical conductivity of the medium is proposed as a possible conductivity switch. Two different scenarios are offered to explain the preference for the smectic phase. A Landau-de Gennes model which brings out this prevalence of the layered phase is also presented. © 2018 Elsevier B.V. All rights reserved.

1. Introduction Nematic liquid crystals characterized by long range orientational order without positional order and exhibiting optical and dielectric anisotropies, form the basis of the commercially successful electro-optic display devices [1–4]. At another level, suspensions of anisometric particles in the elastically soft LC environment have created much interest appealing to their mutual influence [5–7]. Examples of nanostructures used for such suspensions are ferromagnetic ferroelectric particles [8–10], as well as carbon nanotubes (CNTs) [11–16]. The highly elongated shape (aspect ratios of 102 to 103) and anomalously high anisotropy of the diamagnetic susceptibility (10−5–10−4) of the CNTs are especially attractive for the purpose. Reports on the LC-CNT composite have shown remarkable features such as, improvement in electrooptic response of LCs, faster response time, lower driving voltage, suppression of parasitic back flow and image sticking, typical for LC devices [5–7,11–16]. While the influence of CNTs on the physical properties of the nematic (N), the layered smectic A (SmA) and the ferroelectric LC

⁎ Corresponding author. E-mail address: [email protected] (D.S. Shankar Rao).

https://doi.org/10.1016/j.molliq.2019.04.135 0167-7322/© 2018 Elsevier B.V. All rights reserved.

phases have been well studied [11–16] surprisingly, the effect on systems exhibiting the re-entrant nematic phase sequence have not been studied. A phase is said to be re-entrant if, upon a monotonic variation of thermodynamic fields such as temperature or pressure, the system returns to the same structure after one or more transitions involving other phases [17–19]. The observation of re-entrant nematic phase in liquid crystals completes the deGennes' analogy [20] between normalsuperconductor and nematic-smectic transitions. The return to the disordered N phase at temperatures below the smectic is argued to be caused by competition between smectic and nematic order parameters, yielding multiple re-entrance as well [21–23]. In these systems, the occurrence of re-entrance has been entirely due to forces responsible for liquid crystallinity. We have recently demonstrated that adding a minute quantity of CNT to a host mixture which exhibits only the nematic phase induces the SmA phase [24]. In the present article, we generalize the influence of CNT and show that if the parent system already has the N-SmA-RN sequence, then the presence of CNT stabilizes the smectic at the expense of the N and RN phase, more drastically for the latter. The nanotubes also have much effect on the Frank elastic constants: not only the magnitudes of the splay and bend constants are increased, their anisotropy as well as the critical thermal variation on

2

G.V. Varshini et al. / Journal of Molecular Liquids 286 (2019) 110858

approaching the smectic from the N phase. The experimental observations of the phase behaviour are further supported by a Landau-de Gennes' model. 2. Experimental The binary phase diagram formed by the parent liquid crystals, nhexyloxy- and n-octyloxy cyanobiphenyl (6OCB and 8OCB), over a partial composition range, is shown in Fig. 1; 6OCB and 8OCB were procured from BDH and Synthon, respectively. A specific mixture of this system, referred to as HLC hereafter, comprising 28% 6OCB in 8OCB, was chosen for the experiments. It exhibits the phase sequence NSmA-RN on cooling from the isotropic phase, with the RN phase stable at ambient temperature. The CNT employed is single walled carbon nanotube (Heji, Hongkong), with a tube diameter of ~2 nm and an aspect ratio of ~250. Thus the LC and CNT exhibit extremes of shape anisotropy, with the LC molecules having a value of ~5 and the CNT is presenting a 50 times larger value. In the Onsager limit, CNT is thus considered a true hard rod. 2.1. Preparation of HLC + CNT composites CNTs being entirely of carbon atoms tend to aggregate very strongly due to the strong van der Waals interactions between adjacent tubes. To realize a homogeneous LC + CNT composite, it is thus imperative to overcome the van der Wall attraction. To achieve this solvent induced mixing method was employed. For this purpose weighed amount CNT was taken in a vial containing HLC and acetone. This solution was stirred continuously for 24 h at the end of which the solvent was evaporated. Subsequently, the material was heated to 100 °C, a temperature well above the isotropic point that ensures removal of any traces of the solvent which could cause undesired solvent effects on the phase transitions. The homogeneity of the composite and the uniform dispersion of CNT were then confirmed by observing a thin film of the sample under a polarizing microscope (POM). Non-observation of sedimentation of CNT in the vial even after extended periods of time indicated a well-prepared mixture, in all the three composites studied. Despite such care, high loading of CNT still results in separation of the two materials, and therefore the highest concentration of CNT studied was limited to 0.5% of CNT by weight.

2.2. Measurements The dielectric measurements were carried out using an impedance analyser (Agilent HP4194A). The samples were sandwiched between two indium tin oxide (ITO) coated glass plates, pre-treated with either a silane (ODSE from Aldrich) solution for homeotropic alignment, or with a polyimide layer (PI2555 from HD Microsystem) rubbed for unidirectional planar alignment, were used. The probing field was kept low (0.5 V) to avoid any influence of electric-field driven effects on the orientation of the LC molecules. For elastic constant measurements the well-known Freedericksz technique was employed. Using an electric field for this purpose, though convenient, was avoided in the present studies to avoid possible dielectric breakdowns [25] at higher voltages needed for the purpose. Instead reorientation caused by an applied magnetic field in samples surface-oriented to be either in a homeotropic or planar configuration, was utilized. A computercontrolled electromagnet (Bruker B-MC1) provided a uniform field (spatial variation b0.01%) over the entire sample cell placed inside a temperature-controlled stage and positioned between the pole pieces; the maximum field achievable was B = 1.46 T. For determining the director orientation as well as to monitor the Freedericksz transformation, the sample capacitance (Cp) was measured. Again a small probing electric field applied normal to the substrate surfaces was used for the purpose. The HLC as well as the composites with CNT exhibit positive dielectric anisotropy, ε|| N ε⊥, || and ⊥ indicating directions parallel and perpendicular to the director. Thus the measured capacitance corresponds to the equilibrium ε||/ε⊥ for silane/polyimide treated cells for fields less than a critical value Bc, and for B N Bc the director deforms and at large fields reaches a limiting capacitance corresponding to ε⊥/ ε||. The Xray diffraction measurements were performed using PANalytical X'Pert PRO MP X-ray diffractometer consisting of a focusing elliptical mirror and a fast high resolution detector (PIXCEL), the wavelength of the radiation is 0.15418 nm [26]. The sample was contained in a glass capillary tube (Capillary Tube Supplies Ltd). The temperature of the sample was varied using a Mettler hot stage/programmer (FP82HT/ FP90), and could be controlled to a precision of 100 mK. The profiles collected using this apparatus were analyzed using Fityk profile fitting software [27].

3. Results and discussion 3.1. Phase diagram

Fig. 1. Partial phase diagram of the binary system 6OCB/8OCB with Y6OCB being the weight % of 6OCB in the mixture. The composition (labelled HLC) on which the measurements have been carried out is indicated by the vertical dashed line, and presents the sequence N-SmA-RN on cooling the sample from the isotropic phase.

Fig. 2(a) shows the temperature concentration phase diagram for the HLC-CNT binary system. All the mixtures showed N-SmA-RN phase sequence with characteristic mosaic texture in N and RN phases and fan-shaped focal conic texture in the SmA phase. Microphotographs obtained for the exemplary concentration X = 0.24 (X is the concentration of CNT, by weight%) are shown in Fig. 3. Variation of the CNT content has drastically different influence on the different phase lines even over this small concentration range. While the isotropic-nematic temperature (TIso-N) decreases substantially, the N-SmA temperature (TN-SmA) sees only marginal changes. However, it is the SmA-RN phase boundary that diminishes drastically. Consequently, over the range of concentrations studied, while the N range reduces by ~4 K, the thermal range of the SmA phase gets enhanced by N11 K (Fig. 2(b)). This contradictory behaviour for the N and SmA ranges is intriguing. First, let us look at the reduction of TIso-N. It is well known that increasing the aspect ratio of the LC molecules, longer core or higher members of a homologous series, result in increasing the clearing point (TIso-N). The fact that despite the added CNT having a very large aspect ratio compared to that of the LC molecules TIso-N gets lowered, suggests that a straight forward of the aspect ratio argument cannot be used. From an experimental point of view, the variation of TIso-N can be tagged to changes in the orientational order S. The dielectric anisotropy (Δε = ε|| – ε⊥) of the

G.V. Varshini et al. / Journal of Molecular Liquids 286 (2019) 110858

3

ε Δ

ε/ε

Δ

ε

Fig. 2. (a) Temperature-concentration phase diagram of HLC/CNT with X being the weight % of CNT. The Iso-N and the SmA-RN transition temperatures diminish with increase in X, the latter having a larger change; in contrast, the N-SmA boundary exhibits a marginal change. Consequently, the thermal range of the nematic phase (ΔTN) decreases slightly, that of the smectic phase (ΔTSmA) increases drastically, as depicted in the inset (b), occurring at the expense of the N and RN phases.

system in the mesophases is a measure of S, a fact that is especially true in the case of strongly polar LC molecules such as the ones in the HLC. 3.2. Dielectric behaviour In the background of the association of the dielectric anisotropy with orientational order let us now look at the temperature dependence of the two principal permittivities ε|| and ε⊥, for the host HLC and the three composites. As depicted in Fig. 4(a), wherein the data have been normalized with εIso, all the materials show qualitatively similar behaviour, exhibiting a sharp rise (fall) in ε|| (ε⊥) at TI-N, followed by a gentle variation in the mesophases. While ε⊥ does not vary much with CNT concentration, ε|| diminishes substantially with X. In fact, the normalized ε|| reduces by ~25% for X = 0.24. It may be mentioned that the permittivity data were collected by using cells with the appropriate surface treatment in addition to subjecting the sample to a magnetic field of 1.46 T in the required geometry. As an example of the quality of the

Δε/ε

ε

Fig. 4. (a) Plot of the normalized dielectric constant (ε||, ε⊥) as a function of temperature for pure HLC and the composites X = 0.05 and 0.24; the normalization is done with respect to εIso, the value at the nematic-isotropic transition. While ε⊥ remains approximately the same for all the materials, ε|| diminishes significantly with increase in concentration of CNT. (b) Thermal variation of the normalized dielectric anisotropy Δε/ εIso whose value is lowered with X. (c) Diagram depicting that the average permittivity in the mesophases is lower than εIso for all the three materials, the difference εIso increasing as the CNT content is increased. In each case, the solid line represents the weak thermal behaviour in the isotropic phase extrapolated into the mesophase region.

homeotropic alignment, we show the obtained conoscopic image for the X = 0.24 sample contained in silane treated cells. The clear and sharp uniaxial Maltese cross (inset of Fig. 3, obtained using homeotropically aligned sample), similar to the ones for the other samples, rules out any inadequate alignment to be the cause for the observed decrease in ε|| especially for the X = 0.24 mixture. Hence this behaviour suggests that with increasing X, the orientational order parameter diminishes, causing TIso-N also to decrease.

Fig. 3. Microphotographs showing mosaic and fan shaped texture characteristic of N and RN, and SmA phases, respectively, for the HLC + CNT composite (X = 0.24%). The photographs are taken with unaligned sample in order to unambiguously identify the phase using the textural patterns. Inset: sharp uniaxial maltese cross from a homeotropically aligned sample demonstrating that the presence of CNT does not hamper the quality of alignment in all the three phases.

4

G.V. Varshini et al. / Journal of Molecular Liquids 286 (2019) 110858

The thermal variation of the dielectric anisotropy for HLC and the X = 0.24 CNT composite are shown in Fig. 4(b). The composite showing diminished Δε throughout the mesophase range could be taken to be indicating reduction of the orientational ordering. However, as seen from Fig. 4(a), while ε⊥ varies marginally the change in Δε is dominated by diminution of ε||. This can be interpreted to be due to reduction in the concentration of HLC molecules in monomeric form and a concomitant increase in the population of the dimers formed by these mesogens. The formation of dimers results in a better cancellation of the contribution of the terminal cyano dipoles. Obviously, this would drastically lower the ε|| value, but does not affect the contribution in the lateral direction. A statistical model of antiferroelectric short range order in the nematic [28], to be discussed below, also argues that in polar systems, the anisotropic average permittivity, εavg [=(ε|| + 2ε⊥)/3], is lower than εext, the extrapolated value from the thermal trend in the isotropic phase. Fig. 4 (c) presents the thermal variation of εavg for HLC and two CNT composites: Just as in HLC, εavg is less than εext for the composites as well. Interestingly, the difference δε = εext – εavg increases with increasing concentration of CNT being 0.45, 0.71 and 2.8 at temperatures well below the isotropic point. Since δε is associated with antiparallel association or dimer formation, its increase with X indicates the population of dimers increasing with the concentration of CNT. We find that this conjecture is supported by the behaviour of the permittivity in the isotropic phase, a feature that is described in the following. Consequences of the favour for improved antiparallel ordering in the presence of CNT are also seen in the isotropic phase. Owing to its weak first order character, the I-N transition, exhibits pretransitional effects associated with short-range ordering of the nematic-like regions in the isotropic phase. In polar materials the permittivity in the isotropic liquid phase εIso is expected to be proportional to μ2/kBT, where μ is the dipole moment and kB the Boltzmann constant. Consequently, εIso should monotonically and linearly increase on lowering the temperature. While this is true in several materials including many liquid crystals, strongly terminally polar liquid crystals, such as the ones constituting HLC used in the present studies, present an exception: away from the nematic, εIso increases with diminishing temperature, but in the proximity of TIso-N reverses the trend after passing through εmax, a maximum in the value. First reported by Bradshaw and Raynes [29], this convex shaped anomaly (CSA), can be explained on the basis of antiparallel association of neighbouring permanent dipoles [28] in a wide range of materials [30]. Fig. 5(a) and (b) presents ε(T) vs. temperature for HLC and the X = 0.24 composite; both materials display pronounced CSA feature. It is seen that δε = (εmax-εIN)/εIN and δT = Tmax – TIN, the coordinates of the peak point, are higher for the composite: δε = 9.06 × 10−3 and 12.50 × 10−3, δT = 7.37 and 8.79 for HLC and the composite, respectively. This gives credence to the argument that antiparallel pairing or dimer formation is slightly more favoured in the composite. In fact, Sridevi et al. [30] also found that an increased flexibility of the core region diminishes the magnitude of the CSA. Conversely, it could be argued that a stiffer nematic environment could support a stronger CSA. From this point of view it is possible that the stiffer nematic environment in the composites caused by the presence of CNT is thus directly responsible to the slightly more dominant CSA feature observed. The shape profile of CSA mimics the dielectric behaviour near the critical solution point in fluid binary mixtures has been analyzed by Sengers et al. [31] and described in terms of asymptotic power law behaviour in addition to a background linear temperature dependent factor. This has been deployed by Thoen and Menu [32], and more extensively by Rzoska's group [33,34] and our group [30,35,36] to analyze the CSA feature with the following functional form ϕ

εðTÞ ¼ ε þ AðT−T Þ þ BðT−T Þ

ε

ε

Fig. 5. Thermal variation of ε in the isotropic phase for (a) HLC and (b) X = 0.24 composite. Both the materials show convex shaped anomaly with both the (δT, δε) coordinates of the peak position being higher for the composite. The arrows indicate the isotropic-nematic transition point. The solid line represents the fit to Eq. (1) with the exponent ϕ being fixed at 0.5, the fitting parameters given in Table 1.

specific heat critical anomaly. δT is a measure of the discontinuity of the transition or “first orderliness”. It has been shown that the value of ϕ = 0.5 fits the data very well for a large number of materials. This is surprising since α = 0.5 signifies that the system is in the proximity of a tricritical point. Fig. 5(a) and (b) also shows the data fit to Eq. (1) with ϕ fixed at a value of 0.5. The quality of the fitting is excellent in both cases and the best-fit parameters are listed in Table 1. We also performed the fitting by floating all the parameters in Eq. (1), referred to as case 1 below, and by fixing T*, but floating ϕ (case 2). While the quality of the fitting remained equally good – as was judged by the χ2 values – the exponents were essentially the same for HLC in both cases (case 1: 0.50 ± 0.03 and case 2: 0.50 ± 0.01). This is true for the composite as well, but only in case 2 (0.51 ± 0.01) whereas the value is slightly different in case 1 (0.58 ± 0.02). However, a significant difference is seen for the δT parameter: While HLC shows a value of 2.8 K, the composite has a much smaller value of 1.2 K. Since δT is a measure of the first orderliness, it can be considered that the transition is much weaker for the composite.

ð1Þ

where (ε⁎,T⁎) are the coordinates of the hypothetical continuous phase transition, TIso-N = T* + δT. A and B are constants and the parameter ϕ = 1–α, with α being the critical exponent associated with the

3.3. Elastic constants The SmA phase can be considered as a one-dimensional solid whereas the nematic has liquid-like spatial ordering in all three

G.V. Varshini et al. / Journal of Molecular Liquids 286 (2019) 110858

5

Table 1 Parameters obtained by fitting the thermal variation of the dielectric constant (to Eq. (1)) in the isotropic phase of HLC and X = 0.24. X=0 Fixed ϕ T* ε* A (×10−3) B (×10−3) ϕ

75.51 ± 0.04 6.641 ± 0.002 −25.97 ± 0.08 164.88 ± 0.89 0.5

X = 0.24 Floating ϕ — case 1 75.51 ± 0.25 6.64 ± 0.03 −26.01 ± 1.01 164.43 ± 19.07 0.50 ± 0.03

Fixed ϕ

Fixed T* — case 2 75.51 6.643 ± 0.003 −26.08 ± 0.25 163.75± 2.35 0.50 ± 0.01

directions. This difference shows up very well in the elastic behaviour of the two phases. In the nematic phase, the excess energy resulting out of elastic deformation of nematic director can be expressed in terms of three principle Frank elastic constants, namely, splay K11, twist K22 and bend K33. However, owing to the condition that the layer spacing is a conserved quantity, the twist and bend elastic distortions are absent in the SmA phase. Thus, on approaching the smectic from the nematic phase K22 and K33 exhibit a large increase, with a power-law divergence if the transition is second order. The presence of CNT is known to increase the electrical conductivity of the system, even to the point of taking it towards dielectric breakdown [25]. Therefore we employed the magnetic field-driven Freedericksz transformation technique to determine the splay and bend elastic constant on HLC and two CNT composites, viz., X = 0.05 and 0.24. For these studies the samples were contained in a cell made of two indium-tin-oxide (ITO) coated glass plates pre-treated with a silane/polyimide solution to achieve the initial homeotropic/planar director orientation, for the bend/splay measurements. The critical field Bc is related to Kii through B11 ¼


 π μ o K 11 1=2 d χa

ð2Þ

B33 ¼


 π μ o K 33 1=2 d χa

ð3Þ

Here B11 and B33 are the critical fields for the splay and twist geometries, μo is the permeability, d is the sample thickness and χa the diamagnetic susceptibility anisotropy. Thus Bii is a direct measure of Kii. Fig. 6(a)–(f) show the raw profiles of permittivity (ε) vs. B for HLC and the X = 0.05 and X = 0.24 composite at several temperatures in the N and RN phases on approaching the SmA phase in the splay and bend configurations. The threshold field Bii, indicated by the knee point at which ε starts deviating from its equilibrium value, is clearly seen for the CNT composites also. The threshold increases to higher field values as the system moves towards the smectic. The shift with temperature is more drastic in the case of the bend deformation (Fig. 6(b), (d) and (f)), as is to be expected. Fig. 7 shows the thermal variation of B11 for HLC and the X = 0.05 and 0.24 CNT composites. For the first two materials, on approaching the smectic from the high temperature N side there is essentially a linear increase in the value, whereas from the RN side, B11 has a slight non-linear enhancement. This non-linear feature gets better noticed in the case of the CNT composite, with the X = 0.24 composite developing it on the N side as well. In fact, pre-transitional but non-critical increase of K11 is known [37–39]. The more prominent increase seen for the X = 0.24 composite should be indicating that there is much opposition from the system for the splay deformation, especially in the SmA phase. The more important feature, however, is the substantial (~70%) increase in the absolute value for the composite. In fact, this is true for the different composites studied here, and suggests that even deep in the N phase the systems becomes much stiffer for the splay deformation. The situation is

76.32± 0.04 16.52 ± 0.01 −58.32± 0.2 371.03 ± 2.18 0.5

Floating ϕ — case 1 76.82 ± 0.14 16.66 ± 0.03 −65.36 ± 2.44 289.38 ± 17.84 0.58 ± 0.02

Fixed T* — case 2 76.32 16.53 ± 0.01 −59.17 ± 0.65 363.47 ± 5.56 0.51 ± 0.01

somewhat similar to the higher splay value seen for main chain polymer systems [40,41], wherein the larger aspect ratio is argued to be the cause. It is possible that the presence of CNT with its large shape anisotropy, albeit in a small quantity, mimics this behaviour. The enhanced value and the rise on approaching the SmA phase could also be taken to be supportive of the nanophase segregation process, to be discussed later. The Freederiskz experiments carried out on cells with initial homeotropic alignment and field-driven reorientation towards planar alignment yield the bend elastic constant according to the Eq. (3). Fig. 8a depicts the temperature dependence of the bend threshold field, B33, for HLC and the composites with X = 0.05 and 0.24. In all cases, as expected B33 values diverge on approaching the SmA phase from both N and RN phases. The first difference to be noted for the composites is the higher magnitude of B33 than for HLC. Interestingly, the observed increase of ~25% is much smaller than the nearly two-fold enhancement in the splay threshold B11. This feature is in contrast to the behaviour observed for a nematic LC doped with 2% gold nanorods (GNR) leading to doubling of K33 whereas K11 remains unaltered [42], the exact opposite behaviour as in the present studies. This could perhaps be due to the aspect ratio of the nanoparticles employed: in the GNR case the aspect ratio was 3–4, i.e., even lower than that for LC (~5) whereas for the CNTs used here it is large (~250). These observations suggest that a reduction of aspect ratio (but still anisotropic) influences the director deformation in the lateral direction, causing K33 to be influenced more than K11, whereas increasing the aspect ratio of the included nanoparticles has the opposite effect. An indirect evidence of this is in pure LC systems having out-of plane bridging groups in the aromatic part of the molecule, which result in increase of the K33/K11 ratio [43]. The divergence of K33 or equivalently of B33 as the system approaches the SmA phase is captured well by the Landau-de Gennes theory [44]. Explicitly, the dependence of these parameters on the dimensionless reduced temperature t = |T–Tc|/Tc (Tc is the SmA-N/RN transition temperature) is dictated by a power-law whose exponent is the same as that for smectic correlation length. Therefore the data were analyzed using B233 ¼ B1 t −x þ B2

ð4Þ

where B1 and B2 are constants. In all cases, Eq. (3) was found to fit the data quite well, as can be better seen in the double logarithmic representation of Fig. 8b where the slope of the line is the exponent x. Owing to the limitation of the experimental apparatus we could not collect data below ambient temperatures, and thus the reduced temperature range covered for the RN-SmA transition is quite limited and therefore we will discuss the exponent obtained for the NSmA transition: the values are tabulated in Table 2. The exponent, which is close to the value of −0.66 expected for the XY universality class for the pure HLC diminishes to −0.51 and −0.55 for the composites X = 0.05 and 0.23, respectively. In point of fact, the value of −0.5 is expected for a system obeying mean field behaviour with

6

G.V. Varshini et al. / Journal of Molecular Liquids 286 (2019) 110858

Fig. 6. Raw profiles showing magnetic field dependence of the permittivity ε⊥ and ε|| (normalized with respect to the zero field value) in the N and RN phases of (a) HLC, (b) X = 0.05 and (c) X = 0.24 composites; the left and right panels represent, respectively, the Freedericksz splay and bend deformation. The threshold field indicated by the knee point increases on approaching SmA from N and RN phases indicating the enhancement of the corresponding elastic constant.

negligible contributions from fluctuations. The transition has to be approached much closer in order to be certain of this crossover from the XY-class to the mean field behaviour. The elastic constant anisotropy (B233/B211) decreases with increase in X, the concentration of CNT as shown in Fig. 8c.

3.4. Electrical conductivity Fig. 9a presents the temperature dependence of the conductivity measured at 10 kHz along the director (σ||) for HLC and X = 0.24 composite. As expected the composite has a higher conductivity (by an

G.V. Varshini et al. / Journal of Molecular Liquids 286 (2019) 110858

7

Fig. 7. The thermal variation of the square of the threshold field in the splay geometry, which is linearly proportional to the splay elastic constant K11, increases on approaching SmA from both N and RN phases. Substantial increase in the magnitude is seen over the entire temperature range as the CNT component increases. Stronger pre-transition increase is also observed for the X = 0.24 composite. The dotted lines represent the NSmA and SmA-RN transition temperatures.

order of magnitude), but interesting is the fact that the thermal variation is significantly lowered for the composite. Fitting the data to the Arrhenius expression, σ|| ~ exp (-W/kBT) where kB is the Boltzmann constant and W is the activation energy. The data for both materials is well described by this expression, as seen from the line through the data points. The calculated value of W is significantly diminished with increasing concentration of CNT, as shown in the inset (Fig. 9b). These features are in line with earlier observation on LC + CNT composites [45,46]. The anisotropic in conductivity can be conveniently switched between its extreme values (simply by changing the field strength from zero to the maximum (1.46 T)). The response time of this conductivity switch is actually controlled by the time taken for the magnetic coils (of the electromagnet) to react and change the field rather than the intrinsic response time of the material. As seen from Fig. 9c, the switching is highly reproducible between cycles. 3.5. Xray diffraction Both HLC and the composites exhibit significant short-range order in the N and RN phases. Thus Xray diffraction measurements have been carried out over the temperature range covering the N, SmA and the RN phases. As expected, in all cases the wide angle region has a diffuse peak, whose spacing - 0.45 nm – corresponds to the intermolecular distance in the plane perpendicular to the director. The low angle region on the other hand shows a much sharper peak, whose width is quite strongly temperature dependent. Fig. 10a presents the thermal variation of the spacing d associated with this sharp peak it corresponds to the layer thickness in all the three phases, with the difference that in the SmA is long-range ordered, in the two nematics, it would have a local character. Upon lowering the temperature, pure HLC (X = 0) exhibits a concave profile with d decreasing initially and reversing its trend after attaining a minimum. It may be recalled that such a profile has indeed been reported in the few cases wherein detailed thermal variation of the layer thickness have been performed [47] in systems exhibiting the N-SmA-RN sequence. Such a profile is also seen in studies

Fig. 8. The thermal variation of the square of the threshold field in the bend geometry, which is linearly proportional to the bend elastic constant K33, exhibiting diverging behaviour on approaching SmA from both N and RN phases. At any temperature, the inclusion of CNT is seen to result in only a small increase in the bend parameter, unlike in the splay case. (b) The critical behaviour of the bend elastic constant well known in the vicinity of the N-SmA transition seen for the composites as well in the double logarithmic plot of B233 versus reduced temperature. The solid line representing a straight line fit to the data yields the critical exponent x given in Table 2. (c) The elastic constant anisotropy B233/B211 is seen to decrease with increasing X.

wherein the applied pressure is varied and the temperature is kept constant [48]. In fact, the authors of reference 48 argued that the reversal in the thermal expansion slope seen on approaching the RN phase is consistent with the expectations of the Cladis model [49] for re-entrance. Now let us compare the behaviour exhibited by the composite, data for which are also shown in Fig. 10a. While the concave profile is still seen, interestingly the minimum value is stabilized over a range of temperatures in the SmA phase. If the lowering of d in the high temperature region can be associated with the increased interdigitation of the neighbouring molecules owing to improved antiparallel correlation, the “flat bottom” feature for the X = 0.24 composite could be taken to

Table 2 The values of the exponent x obtained by fitting the thermal variation B33 in the N and RN phases to Eq. (4). Transition\CNT concentration

X=0

X = 0.05

X = 0.24

N – Sm A Sm A - RN

−0.60 ± 0.01 −0.85 ± 0.01

−0.51 ± 0.01 −0.49 ± 0.01

−0.545 ± 0.002 0.46 ± 0.01

8

G.V. Varshini et al. / Journal of Molecular Liquids 286 (2019) 110858

Fig. 9. (a) Semi-logarithmic plot of electrical conductivity parallel to the director (σ||) as a function of inverse temperature for HLC and X = 0.24. For the composite, the σ|| value is not only higher but has much weaker temperature dependence. In both the cases the variation is linear indicating Arrhenius behaviour with the slope of the fitted straight line, shown as solid line providing Ea, the activation energy of the associated potential barrier. (b) Ea is seen to be diminishing with X, and gets halved between pure HLC and the X = 0.24 composite. (c) Magnetic field driven temporal switching of conductivity between the low field (0.06 T) σ⊥ value to the high field (1.44 T) σ|| value; the variations are seen to be substantial and very fast and in fact, could be limited by the temporal response of the controller used for the electromagnet.

suggest that the antiparallel correlation are stabilized and held at a certain value. Another important feature to note is that the spacing value is higher for the composite in all the three phases. This corroborates the nanophase segregation model discussed above. Of course, as has been observed in a variety of nanophase segregation situations [50–52], the increase in d is in the range of ~ 0.03 nm. The argument proposed by Glaser et al. [50] that the small change observed is due to the small concentration of the chemical species responsible for the change, is all the more applicable in the present case since the CNT concentration is very low. The thermal variation of the peak width of the low angle peak (Fig. 10b) has profiles similar to that of the layer spacing variation. The “flat bottom” as well as the lower value for the composite suggests better layering in the SmA phase, which even extends to the short range ordering in the RN phase. The mass density profile giving rise to the

Fig. 10. (a) Thermal variation of the layer spacing (d) for HLC and X = 0.24 composite, being the equilibrium value in the SmA and that from the short-range cybotactic regions in the N and RN phases. The higher value for the composite is suggestive of the nanophase segregation mechanism discussed in the text. (b) Thermal variation of full width at half maximum (Δ2θ) for HLC and X = 0.24 composite. The values are smaller with pronounced plateau region for the composite. The vertical lines indicate the N-SmA and SmA-RN transition temperatures.

layering has a purely sinusoidal profile in the alkoxy cyanobiphenyls [53,54]. Owing to this the intensity of the second harmonic peak would be extremely (factor of 104) weak. In view of the sharper profile for the X = 0.24 composite we indeed looked for the possible appearance of the second harmonic, but did not find. It is possible that if higher loading of nanorods/cylinders is possible perhaps this feature can be observed. At this point we would like to recall the recent observation made by Mandle and Goodby [55] that in a similar system the increased pairing of the molecules and the concomitant reduction in the number of free molecules can lead to a more stable smectic A phase. Now we discuss at possible reasons for the suppression of the reentrant nematic phase and the consequent stabilization of the smectic phase. A large number of cases have been reported wherein the induction of the smectic phase is due to hydrogen bonding, charge transfer complex formation, dipole–dipole or dipole-induced dipole interactions [56,57], photo-driven nanosegregation [51]. In the present case, these features either do not exist or can be ignored and the prominent feature is the large shape anisotropy difference between the host LC and the dopant CNT. Thus, we can consider two possible explanations for the stabilization of the SmA of the layered phase. The first of these is referred to as the photo-driven dynamic self-assembly, which presents

G.V. Varshini et al. / Journal of Molecular Liquids 286 (2019) 110858

many parallels between the behaviour observed here and with that seen for the case with azobenzene photoactive dopants [51]. The dopants in their photoisomerized cis state, have an identical influence on the different phase lines as seen here, and thereby enhance the SmA range, while the N range is slightly diminished. Further, the effect on the SmA-RN line is the largest. The role of the cis concentration in this case seems to be played by the composition of CNT in the present investigation. This photo-driven stabilization of the layered phase was explained on the basis of a photosegregation model, wherein the shape altered (from rod-like to bent) azobenzene molecules favour demixing, from the host molecules, although on a local level. This in turn reduces the ease with which the molecules librate along the director direction, thus promoting a molecular-level nanosegregation of the rod-like and bent molecules supporting the formation or stabilization of the layered phase over the fluid mesophase. The situation being very similar in the present case is tempting to argue that a similar nanophase segregation of CNTs and the LC molecules takes place, as schematically shown in Fig. 11a. While the difference between the host and the photoisomerized guest molecules facilitate the segregation, in the azobenzene case, in the present scenario the segregation could be

9

caused by the difference in the aspect ratio of the HLC molecules and CNT. The second possibility that could be responsible for the stabilization of the smectic phase, is the frustration caused in the packing of the LC molecules in the presence of the carbon nanotubes. Berker and coworkers [58] have successfully used the “Frustrated Spin-gas” model to explain the appearance of reentrant nematic and smectic phases in molecules with a strongly polar terminal group (as is true for the host LC molecules in the present studies). The theory considers intermolecular steric hindrance and van der Waals attraction but emphasizes the importance of dipole-dipole forces. The model (schematically shown in Fig. 11b) takes into account a two-body dipolar potential with ferroelectric and antiferroelectric interactions, and then considers the influence of a third dipole, restricting the interactions to the director direction (also the dipole direction). For packing reasons, the system is approximated to a triangular structure which creates frustrated situations among the polar molecules. When dipolar forces between two neighbours cancel, a third dipole is free to diffuse from layer to layer “frustrating” smectic order, favouring the reentrance of the nematic phase. If the cancellation is not complete, short-range dipolar interactions dominate and the triplets stabilize layering. With the concept of ‘permeation of minima’, or “notches” stability of a particular triplet configuration and thus of nematic or smectic phase is accounted for. Applying this model to the present case, it may be argued that frustration is reduced in the presence of CNT favouring short-range interactions and thus promoting the layered phase over the re-entrant nematic. However, in the case of the high temperature N-SmA transition, this favour has to compete with thermal fluctuations and thus drastically reducing its influence. Consequently the N-SmA transition temperature doesnot decrease as is true for I-N transition. It may be noted that the situation is neither biased to enhance the range of the smectic (as is true for the SmA-RN boundary). The Raman spectroscopy evidence that we have obtained shows that in the region of the SmA phase, the intensity of the C\\C band (1590 cm−1) is indeed higher normal to the director direction than along it (see Fig. 12). This suggests that the nanophase segregation model may be more applicable. It would be interesting to check whether the influence of CNT can be tuned by varying the aspect ratio of the employed CNTs. 4. Theory We consider the mixture of liquid crystal and carbon nanotubes. Pure form of liquid crystal exhibits the reentrance sequence of NSmA-RN. When mixing with carbon nanotube, the experiments show

Fig. 11. (a) Schematic representation of the possible CNT disposition in the N/RN and SmA phases. While in the nematic the CNTs take a direction essentially parallel to the LC director, in the SmA phase they are favoured to be in the orthogonal direction causing a nanophase segregation of the CNT and LC regions. The individual LC molecules are shown as arrows with the arrowhead indicating the terminal CN dipole. As expected in such systems there is a preference for the formation of dimers indicated here by the green entities. (b) Depiction of the frustration caused by dipoles in a triangular formulation. While a pair formed by oppositely (X,●) pointed dipoles minimize the free energy, the presence of a third one (shown with a question mark) frustrates the system, which could be relieved by translations along the director direction as shown on the right.

Fig. 12. Raman spectra for X = 0.24 composite taken at representative temperature in the SmA phase. Left and right panels depict, respectively, situations with the director n parallel (||) and perpendicular (⊥) to the polarization of the incoming laser beam. The downward arrow corresponds to the signal from the biphenyl moiety of the LC, whereas the upward arrow indicates the G band signal from CNT, occurring at ~1590 cm−1. This peak is prominent for the n ⊥ laser polarization configuration.

10

G.V. Varshini et al. / Journal of Molecular Liquids 286 (2019) 110858

the disfavour caused to the reentrant nematic phase. To describe the experimental results, we use the combination of Flory-Huggins theory and Landau-de Gennes theory. The total free energy per unit volume of the liquid crystal‑carbon nanotubes mixture can be written as F ¼ F mix þ F CNT þ F LC þ F int

ð5Þ

where Fmix represents the free energy density of isotropic mixing, FCNT is the contribution of the CNT dispersed in liquid crystal, FLC corresponds to the free energy density of liquid crystal and Fint is the free energy density associated with the interaction between liquid crystal and CNT. The free energy density Fmix can be approximated in terms of the Flory-Huggins theory [59] as F mix −1 −1 ¼ v−1 CNT φ lnφ þ vLC ð1−φÞ ln ð1−φÞ þ v0 χφð1−φÞ kB T

ð6Þ

where kB is the Boltzman constant and T is absolute temperature. φ and (1-φ) describe the volume fractions of CNT and liquid crystal. χ is the Flory-Huggins interaction parameter. Here vCNT ≈ π4 LD2 2 occupied by CNT of length L and diameter D. vLC ≈ π4 ld

is the volume

is the volume occupied LC of length l and diameter d. v0 is the volume of cell of the Flory lattice. The contribution of the CNT free energy density can be expressed [60].   F CNT φ 1 u u u 1− S2CNT − S3CNT þ S4CNT ¼ vCNT 2 3 9 6 kB T

ð7Þ

where u ¼ φL D is related to the volume fraction of CNT. SCNT is the degree of orientational ordering of CNT [60]. Expanding FLC in powers of nematic and SmA order parameters S and ψ0 has the form [61].   1 1 1 1 1 1 F LC ¼ ð1−φÞ aS2 − bS3 þ cS4 þ αψ20 þ βψ40 þ δψ20 S2 2 3 4 2 4 2 T1∗)

ð8Þ

and α = α0(T − with a0 N 0, α0 N 0. We assume, a = a0(T − T1∗ and T2∗ are the virtual transition temperatures. We choose b N 0, c N 0 and β N 0 for the stability of the free energy (6). The coupling constant δ is chosen negative to favour the SmA phase over the nematic phase. δ is responsible for the appearance of the re-entrant nematic phase [62–64]. Finally, the contribution of free energy density due to the interaction between LC and CNT can be written as ð9Þ

The coupling constants γ and η are chosen positive. Substitution of Eqs. (6)–(9) into Eq. (5), the total free energy per unit volume of the LC + CNT mixture can be written as  −1 Þ ln ð1−φÞ þ v−1 F ¼ kB T v−1 CNT φ lnφ 0 χφ  þvLC ð1−φ  ð1−φÞ  φ 1 u u u 1− S2CNT − S3CNT þ S4CNT þkB T vCNT 2 3 9 6 1 1 3 1 1 1 þð1−φÞ aS2 − bS þ cS4 þ αψ20 þ βψ40 2  3 4
2  4  1 1 1 1 þ δψ20 S2 þ φð1−φÞ − γS2 SCNT 1− SCNT þ ηψ20 S2CNT 2 2 2 2

 −1 F ¼ kB T v−1 Þ ln ð1−φÞ þ v−1 ð1−φÞ CNT φ ln φ 0 χφ  þvLC ð1−φ  φ 1 u u u 1− S2 − S3 þ S4 þkB T vCNT 2  3 CNT 9 CNT 6 CNT 1 1 3 1 þð1−φÞ aS2 − bS þ cS4 2 3
4  1 2 1 −φð1−φÞ− γS SCNT 1− SCNT 2 2

0 S CNT

1

γ ð1−φÞ γ ð1−φÞ B 2 C    S4 A ¼ u @S − −1 1− u 2kB Tφv−1 2k 1− Tv B CNT CNT 3 3

ð12Þ

Substitution of Eq. (12) into the free energy density (11) reads as  −1 −1 F N ¼ kB T v−1 CNT φ lnφ þvLC ð1−φÞ ln ð1−φÞ þv0 χφð1−φÞ 1 1 3 1 þð1−φÞ aS2 − bS þ c1 S4 2 3 4 2

v2CNT 2

γ vCNT ð where c1 ¼ c þ φð1−φÞ kB Tð1−uÞ 2

3

ð13Þ

−1Þ.

The first order I-N phase transition temperature can be calculated using the conditions F N ðS; φÞ ¼ F iso ðφÞ; F 0N ðS; φÞ ¼ 0; F 00N ðS; φÞ ≥0

ð14Þ

For the phase equilibrium, the chemical potentials in the isotropic and nematic phases are equivalent i.e. μiso = μN. After simplification of Eq. (14), the I-N transition temperature can be expressed as ð15Þ

where A1 ¼ T 1 þ 2b 9c , 2

2

v2CNT 2

−1Þ,

v2CNT

−1Þ.

2b γ vCNT A2 ¼ 9ck u ð B Tð1− Þ 2

3

A3 ¼

2

4b γ2 vCNT 9ckB Tð1−u3Þ

ð

2

4.2. Nematic – Smectic A phase transition The value of the smectic ordering and SCNT in the SmA phase near the nematic-smectic-A (N-SmA) phase transition can be expressed as (neglecting the higher order terms like S3CNT and S4CNT) ψ20 ¼ −

SCNT ¼ ð10Þ

ð11Þ

The value of SCNT in the nematic phase near the I-N transition can be expressed as (neglecting the higher order terms like S3CNT and S4CNT)

T I−N ¼ A1 −A2 φ þ A3 φ2

T2∗)


  1 1 1 F int ¼ φð1−φÞ − γS2 SCNT 1− SCNT þ ηS2CNT ψ20 2 2 2

from the free energy density as

 1 α þ δS2 þ ηφS2CNT β

γð1−φÞvCNT u1 kB T


S2 −

U ð1−φÞvCNT 4 S u1 kB T

ð16Þ  ð17Þ

where U ¼ ðγ2 − δη β Þ, . u1 ¼ ð1− u3Þ− ηαð1−φÞ β

4.1. Nematic – isotropic phase transition First we discuss the nematic – isotropic (N-I) phase transition. The dispersion of CNTs in the nematic phase of liquid crystals can be studied

It is clear from Eq. (16) that a nonzero real value of ψ0 exists only when (α + δS2 + ηφS2CNT) b 0. Since η is positive, δ should be positive to favour the SmA phase.

G.V. Varshini et al. / Journal of Molecular Liquids 286 (2019) 110858

By the substitution of ψ0 and SCNT from Eqs. (16) and (17) into Eq. (10), we obtain  −1 −1 F SmA ¼ kB T v−1 CNT φ ln φ þ  vLC ð1−φÞ ln ð1−φÞ þ  v0 χφð1−φÞ 1 1 1 þð1−φÞ a S2 − bS3 þ c S4 2 3 4

ð18Þ

where a ¼ a− δα β ð1−φÞ, 2

c ¼ c− δβ ð1−φÞ. Free energy density (18) shows that the cubic coefficient b does not renormalize with the dispersion of the CNT in liquid crystals. This means that the N-SmA phase transition is still a first order transition even in the presence of CNT. The first order N-SmA phase transition temperature can be calculated from the conditions F SmA ðS; φÞ ¼ F N ðS; φÞ; F 0SmA ðS; φÞ ¼ 0; F 00SmA ðS; φÞ≥0

ð19Þ

ð20Þ

where 2α 0 c A4 ¼ a1 ða0 T 1 þ 2b 3c Þð1− δ ðg−1ÞÞ þ 0

4α 0 cg 2α 0 δ A5 ¼ a1 ða0 T 1 þ 2b 3c Þð δ þ β ðg þ 1ÞÞ− 2

0

2

2δ b 0δ þ 2α ðT 2 −gÞ þ 3βa  c2 − β2 a 2 2 0

0

A6 ¼

1 a0

ða0 T 1

þ

2 4α 0 δg 2α 0 cg 2b 3c Þð β − δ Þ

g¼

þ

2α 0 cgT 2 βa0

ðg−1Þ,

2α 0 cT 2 βa0 δα 0 T 2 βa0

ðα 0 þ

T SmA−RN ¼ E1 þ E2 φ 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H1 þ H 2 φ þ H 3 φ2

B3 E1 ¼ 2D , 1 −D2 B3 E2 ¼ B4 D12D , 2 1

H1 ¼

D1 E21 −G1 , D1

H2 ¼

2E1 E2 D21 −G2 D1 þG1 D2 , D21

H3 ¼

2E1 E3 D21 þE22 D21 −G3 D1 þG1 D3 þG2 D2 , D21 2

δα 0 δ a0 B1 ¼ ð4δ βc −3 þ β þ β ÞM 2 , 3

:

,

2

δα 0 δ a0 4δ B2 ¼ δβα2 0 þ 3− 12δ βc þ ð β þ β Þð βc −3Þ, 2

2

2

2 1Þ− 4αβ20agδ 0

2

0 δb − 8α 3βca0

;

M2 ¼ 14Mc21 b , M1 ¼

ηγ2 δ , ð1−u3Þ2 4c2 ðkB TÞ2 v−2 CNT

2  7b B3 ¼ a0 ð1− 2δ c Þ−2M 1 ða0 T 1 þ c Þ, 2

2  B4 ¼ B1 − 2δ βc ðδa0 −α 0 Þ þ 2M 1 ð6a0 T 1 − 2

4.3. Reentrant nematic phase and SmA-RN phase transition

4

To get the nematic phase stable, we must have δ N 0 and TN−SmA ≤ T2∗. For ψ0 = 0, the renormalized N-SmA transition temperature TN−SmA can be calculated from the equation: αþ

δS2SmA

þ

ηφS2CNT

¼0

ð21Þ

where SSmA

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4a c 1 þ 1− 2 b

ð22Þ

ð23Þ

where 2

2

M ¼ ð4c2 Þ Rða20 þ ðδαβ 0 Þ ð1−φÞ2 −ð2δαβ0 a0 Þð1−φÞÞ, b

2

2

V ¼ α 0 þ R½ð4c2 Þ ð−2a20 T 1 −2ðδαβ 0 Þ T 2 ð1−φÞ2 þ ð2δαβ0 a0 ÞðT 1 þ T 2 Þð1 

b



01 −φÞÞ−ð56c 2a01 Þ−ð2δa c Þ,

b

2

2

W ¼ −α 0 T 2 þ R½16 þ ð4c2 Þ ða20 T 1 þ ðδαβ 0 Þ T 2 ð1−φÞ2 þ ð2δαβ0 a0 ÞT 1 T 2 ð 

b

2

Þ½4−ð4c2 Þ ð−2a0 T 1 þ 2 1−φÞÞ−ð4c2 Þð−14a0 T 1 þ 14ðδαβ 0 T 2 φÞÞ þ ðδb c2 2



b

ηγ2 δb4 2 16c4 ðkB TÞ v−2 u2 CNT 1



b

ðδαβ 0 T 2 φÞÞ, R¼

þ

ð1−φÞ3 ,

þ

2δα 0 a0 ðT 1 þT 2 Þ Þ, β

2

2

4

4δ3 α 0 a0 ðT 1 þT 2 Þ Þ, β2 c

0 δ G2 ¼ M1 ð2δa β ðc −α 0 ÞÞ,

2

4

2

3

G2 ¼ M1 ð3a0 þ ðδαβ 0 Þ þ βa02δc2 − 6δβca0 þ 6δαβ0 a0 − 4δβα2 c0 a0 Þ, 2

4

2

14a0 T 1 b2 Þ, c

D2 ¼ bβcδ2 þ 2δβcα 0 ðδT 1 −T 2 Þ þ M4c1 b2 ð4δβ −3Þ− b 2 3

þ VT SmA−RN þ W ¼ 0



β2

 2  b 0 −α 0 T 2 þ 2δa D1 ¼ δb c T 1 þ M 1 ða0 T 1 þ 4c2 − c2

Substitution the values of SSmA and SCNT from Eqs. (22) and (17) into Eq. (21), Eq. (21) can be rewritten as MT 2SmA−RN

4δ2 a20 T 1

 δα 0 δ B5 ¼ −B2 þ 2δβ2α0 þ 2M1 ð2a20 T 1 ð6δ βc þ β 2 c2 −3Þ−2ð β Þ T 2

2

b ¼  2c

ð24Þ

where

2

ð1 þ gð2 þ α 0 ÞÞ

v2 βγ 2 vCNT ð CNT −1Þ; kB Tδ2 ð1−u3Þ 2 a0 ¼ a0 − δαβ 0 :

Now for δ ≠ 0, η ≠ 0, Eq. (23) is a quadratic equation of TSmA−RN. Then Eq. (23) describes the re-entrant phase boundary within the reasonable limit of the material parameters. Naturally the coupling constant δ is responsible for the existence of the re-entrant nematic phase. In principal δ should be the function of the concentration of CNTs. The suitable choice of the functional form of δ will provide better ideas for the description of re-entrant nematic phase. The solution of Eq. (23) to the lowest order of φ can be written in a simplified form

1

T N−SmA ¼ A4 þ A5 φ þ A6 φ2

2α 0 cT 2 βa0

a01 ¼ a0 −ðδαβ 0 Þð1−φÞ.

3 B1 −D2 B4 , E3 ¼ B5 D1 −D2D 2

From which get the N-SmA transition temperature

2

11

3δ2 β

2δα 0 a0 T 1 T 2

−3ÞÞ þ M 1 ð

D3 ¼ −

2δ4 α 0 T 2 β2 c

β

4

 M1 14δα 0 T 2 c ð β

−14a0 T 1 ð

2

2

−

2

þ a20 T 1 ð2δ βc þ 1ÞÞ, 2

b þ M4c1 b2 ð3− 12δ β Þþ

4δ3 α 0 a0 T 1 T 2 β

−3ÞÞ þ M 1 ð

2

6δα 0 a0 T 1 T 2 β

M1 c

ð14a0 T 1 ð3− 9δβ Þ− 2

2

þ ðδαβ 0 Þ þ

δ4 a0 T 1 1 βc ðβc

14δα 0 T 2 3δ2 ðβ β

þ 6ÞÞ.

Solving Eqs. (15), (20) and (24) simultaneously one can observe the phase diagram showing I-N, N-SmA and SmA-RN phase transitions respectively. According to Eqs. (15), (20) and (24), the I-N, N-SmA and SmA-RN transition temperatures decrease with increasing concentration of the CNTs as observed in experiment. In order to check Eqs. (15), (20) and (24), the measured T vs φ for the I-N, N-SmA and SmA-RN transitions of the HLC-CNT mixture is plotted in Fig. 13. The fit yields A1=78.32 °C, A2=4.13 °C wt%−1, A3= −21.95 °C wt%−2, A4=43.54 °C, A5=6.75 °C wt%−1, A6=−11.92 °C wt %−2, E2=−957.20 °C wt%−1, H2=2.63×103 °C wt%−1 and H3= 8.83×105 °C wt%−2. For the best fit, we take E1 = H1=29.89 °C. The agreement of the theory with experiment is very good considering the scattered experimental data. Thus by taking into account of the

12

G.V. Varshini et al. / Journal of Molecular Liquids 286 (2019) 110858

References

Fig. 13. The temperature concentration phase diagram showing the best fit of the experimental data (open symbols) to the theoretical expressions (solid line), Eqs. (11), (16) and (20), for the three phase lines.

interactions between the nematic and SmA order parameters and the concentration of CNTs, we present a unified discussion of the SmA-RN phase transition as well as I-N and N-SmA phase transitions. 5. Conclusion The influence of a stiff large aspect ratio component on the phase stability of the different mesophases including the reentrant nematic has been investigated. Very low concentrations (0.5% maximum) carbon nanotubes, having a length to diameter ratio of ~250, are employed for the purpose. The presence of the CNTs enhances the thermal range of the layered smectic phase at the cost of the reentrant nematic phase. Dielectric measurements indicate that the inherent favour towards antiparallel correlation between neighbouring liquid crystal molecules gets further increased when the CNTs are present. The inferred reduction in the nematic order parameter, but preference for layering could be an attractive path to realize “de Vries smectic” structures. In the light of the fact that these observations are in line with our recent observation that introduction of CNT causes induction of the smectic phase in an otherwise nematic-only material, it could perhaps be generalized that the high aspect of the nanotubes ratio has a preference for layered phases. Xray diffraction data corroborate these features. The Frank elastic constant measurements complement these observations with the splay, but not the bend, elastic constant increasing substantially on adding CNT to the host liquid crystal system. This is to be expected since CNTs could be stiffening the system along the director direction. A Landau-de Gennes formalism is also presented that explains some of these features. The studies thus open up a new dimension wherein nanostructures can be used to enrich phase diagrams of liquid crystalline materials, and thus have potential to be employed to investigate multiple re-entrant sequences and frustrated systems. Acknowledgment Funding support from the Thematic project (SR/NM/TP-2 5/2016), Nano Mission, DST, New Delhi, India, is gratefully acknowledged.

[1] S. Chandrasekhar, Liquid Crystals, Cambridge University Press, Cambridge, England, 1992. [2] D.K. Yang, S.T. Wu, Fundamentals of Liquid Crystal Devices, Second edition John Wiley & Sons Ltd, West Sussex, England, 2015. [3] B. Bahadur, Liquid Crystals Applications and Uses, World Scientific, 1992. [4] H.W. Chen, J.H. Lee, B.Y. Lin, S. Chen, S.T. Wu, Liquid crystal display and organic lightemitting diode display: present status and future perspectives, Light. Sci. Appl. 7 (2018), 17168. [5] T. Hegmann, H. Qi, V.M. Marx, Nanoparticles in liquid crystals: synthesis, selfassembly, defect formation and potential applications, J. Inorg. Organomet. Polym. Mater. 17 (2007) 483–508. [6] Q. Li (Ed.), Nanoscience With Liquid Crystals From Self-organized Nanostructures to Applications, Springer, 2013. [7] J.P.F. Lagerwall, G. Scalia (Eds.), Liquid Crystals With Nano and Microparticles, World Scientific, 2016. [8] I. Dierking, M. Heberle, M.A. Osipov, F. Giesselmann, Ordering of ferromagnetic nanoparticles in nematic liquid crystals, Soft Matter 13 (2017) 4636–4643. [9] M.F. Prodanov, E.V. Popova, S.A. Gamzaeva, A.P. Fedoryako, V.V. Vashchenko, Ferromagnetic nanoparticles in a ferroelectric liquid crystal: properties of stable colloids in homogeneous cells, J. Mol. Liq. 267 (2018) 353–362. [10] Y. Reznikov, A. Glushchenko and Y. Garbovskiy, Ferromagnetic and ferroelectric nanoparticles in liquid crystals, chapter 19, in book Liquid Crystals With Nano and Microparticles, J. P. F. Lagerwall and G. Scalia, World Scientific 2016. [11] C. Zakri, Carbon nanotubes and liquid crystalline phases, Liq. Cryst. Today 16 (2007) 1–11. [12] N.I. Lebovka, L.N. Lisetski, A.I. Goncharuk, S.S. Minenko, V.V. Ponevchinsky, M.S. Soskin, Phase transitions in smectogenic liquid crystal 4-butoxybenzylidene-4′butylaniline (BBBA) doped by multiwalled carbon nanotubes, Phase Transit. 86 (2013) 463–476. [13] I. Dierking, in: V.A. Basiuk, E. Basiuk (Eds.), Chemistry of Carbon Nanotubes, vol. 2, American Scientific Publishers, Los Angeles, USA 2008, p. 247. [14] J.P.F. Lagerwall, G. Scalia, Carbon nanotubes in liquid crystals, J. Mater. Chem. 18 (2008) 2890–2898. [15] S.P. Yadav, S. Singh, Carbon nanotube dispersion in nematic liquid crystals: an overview, Prog. Mater. Sci. 80 (2016) 38–76. [16] I. Dierking, G. Scalia, P. Morales, Liquid crystal–carbon nanotube dispersions, J. Appl. Phys. 97 (2005) (044309-1-5). [17] L. Chen, A.A. Bokov, W. Zhu, H. Wu, J. Zhuang, N. Zhang, H.N. Tailor, W. Ren, Z.G. Ye, Magnetoelectric relaxor and reentrant behaviours in multiferroic Pb(Fe2/3W1/3)O3 crystal, Sci. Rep. 6 (2016) 22327. [18] Y.B. Simons, O.E. Wohlman, Y. Oreg, Y. Imry, Phase diagram of reentrant and magnetic-field-induced superconducting states with Kondo impurities in bulk and proximity-coupled compounds, Phys. Rev. B 86 (2012) (064509-1-5). [19] T. Narayanan, A. Kumar, Reentrant phase transitions in multicomponent liquid mixtures, Phys. Rep. 249 (1994) 135–218. [20] P.G. de Gennes, An analogy between superconductors and smectics A, Solid State Commun. 10 (1972) 753–756. [21] , For a review, see e.g.. P.E. Cladis, in: D. Demus, J. Goodby, G.W. Gray, H.-W. Spiess, V. Vill (Eds.), Physical Properties of Liquid Crystals, Wiley VCH, Weinheim, 1999. [22] A. Prasad, M.K. Das, Static dielectric properties of two nematogenic compounds and their binary mixtures showing induced smectic Ad and re-entrant nematic phases, Phys. Scr. 84 (2011) (015603-1-9). [23] V.N. Raja, B.R. Ratna, R. Shashidhar, G. Heppke, Ch. Bahr, J.F. Marko, J.O. Indekeu, A.N. Berker, Pressure studies on phase transitions in 4-alkoxyphenyl-4′nitrobenzoyloxybenzoates, Phys. Rev. A 39 (1989) 4341–4344. [24] G.V. Varshini, D.S. Shankar Rao, P.K. Mukherjee, S. Krishna Prasad, Nanophase segregation of nanostructures: induction of smectic A and re-entrance in a carbon nanotube/nematic liquid crystal composite, J. Phys. Chem. B 122 (2018) 10774–10781. [25] V. Jayalakshmi, S. Krishna Prasad, Understanding the observation of large electrical conductivity in liquid crystal-carbon nanotube composites, Appl. Phys. Lett. 94 (2009) (202106-1-3). [26] D.S. Shankar Rao, M. Vijay Kumar, S. Krishna Prasad, U.S. Hiremath, M. Sarvamangala, S. Basavaraja, Novel columnar–calamitic phase sequences in a binary system of bent-core and rod-like mesogens, J. Mater. Chem. C 1 (2013) 7488–7497. [27] M. Wojdyr, Fityk, Fityk: a general-purpose peak fitting program, J. Appl. Crystallogr. 43 (2010) 1126–1128. [28] N.V. Madhusudana, S. Chandrasekhar, The role of permanent dipoles in nematic order, Pramana Suppl. 1 (1975) 57–68. [29] M.J. Bradshaw, E.P. Raynes, Pre-transitional effects in the electric permittivity of cyano nematics, Mol. Cryst. Liq. Cryst. 72 (1981) 73–78. [30] S. Sridevi, S. Krishna Prasad, D.S. Shankar Rao, C.V. Yelamaggad, Pretransitional behaviour in the vicinity of the isotropic–nematic transition of strongly polar compounds, J. Phys. Condens. Matter 20 (2008) (465106-1-7). [31] J.V. Sengers, D. Bedeaux, P. Mazur, S.C. Greer, Behavior of the dielectric constant of fluids near a critical point, Phys. A 104 (1980) 573–594. [32] J. Thoen, G. Menu, Temperature dependence of the static relative permittivity of octylcyanobiphenyl (8CB), Mol. Cryst. Liq. Cryst. 97 (1983) 163–176. [33] A.D. Rzoska, S.J. Rzoska, K. Czupryñski, Phase transitions from the isotropic liquid to liquid crystalline mesophases studied by linear and nonlinear static dielectric permittivity, Phys. Rev. E 61 (2000) 5355–5360. [34] S.J. Rzoska, S. Starzonek, A.D. Rzoska, K. Czupryński, K. Chmiel, G. Gaura, A. Michulec, B. Szczypek, W. Walas, Impact of BaTiO3 nanoparticles on pretransitional effects in liquid crystalline dodecylcyanobiphenyl, Phys. Rev. E 93 (2016) (020701-1-6).

G.V. Varshini et al. / Journal of Molecular Liquids 286 (2019) 110858 [35] M. Sarvamangala, M. Vijay Kumar, S.M. Khened, S. Basavaraja, D.S. Shankar Rao, S. Krishna Prasad, Anomalous dielectric behavior in the nematic and isotropic phases of a strongly polar–weakly polar binary system, Phase Transit. 86 (2013) 454–462. [36] M. Sarvamangala, M. Vijay Kumar, S.M. Khened, D.S. Shankar Rao, S. Krishna Prasad, Dielectric behavior in the nematic and isotropic phases of a strongly polar–weakly polar binary system, AIP Conf. Proc. 1447 (2012) 77–78. [37] D. Dhumières, L. Léger, Critical behaviour above a nematic to smectic a phase transition, J. Phys. Colloq. 36 (1975) (C1-113-116). [38] G. Czechowski, J. Jadzyn, Singular elastic properties of mixtures with the re-entrant nematic phase, Acta Phys. Pol. A 106 (2004) 475–485. [39] A. Prasad, M.K. Das, Determination of elastic constants of a binary system (7CPB+9. CN) showing nematic, induced smectic Ad and re-entrant nematic phases, Liq. Cryst. 41 (2014) 1261–1268. [40] A.F. Martins, P. Esnault, F. Volino, Measurement of the viscoelastic coefficients of main-chain nematic polymers by an NMR technique, Phys. Rev. Lett. 57 (1986) 1745–1748. [41] S. Zheng-Min, M. Kleman, Measurement of the three elastic constants and the shear viscosity γ1 in a main-chain nematic polymer, Mol. Cryst. Liq. Cryst. 11 (1984) 321–328. [42] S. Sridevi, S. Krishna Prasad, G.G. Nair, V. D'Britto, B.L.V. Prasad, Enhancement of anisotropic conductivity, elastic, and dielectric constants in a liquid crystal-gold nanorod system, Appl. Phys. Lett. 97 (2010) (151913-1-3). [43] G. Vertogen, W.H. de Jeu, Thermotropic Liquid Crystals, Springer, Berlin, 1988. [44] e.g. See, F. Jähnig, F. Brochard, Critical elastic constants and viscosities above a nematic-smectic a transition of second order, J. Phys. France 35 (1974) 301–313. [45] S. Krishna Prasad, M.V. Kumar, C.V. Yelamaggad, Dual frequency conductivity switching in a carbon nanotube/liquid crystal composite, Carbon 59 (2013) 512. [46] N. Lebovka, T. Dadakova, L. Lysetskiy, O. Melezhyk, G. Puchkovska, T. Gavrilko, J. Baran, M. Drozd, Phase transitions, intermolecular interactions and electrical conductivity behavior in carbon multiwalled nanotubes/nematic liquid crystal composites, J. Mol. Struct. 887 (2008) 135–143. [47] S. Chandrasekhar, K.A. Suresh, K.V. Rao, Liquid crystals, in: S. Chandrasekyhar (Ed.), Proceedings of an International Conference Held at the Raman Research Institute, Bangalore, Dec. 3-8-1979, Heyden and Sons Ltd. 1980, p. 131. [48] S. Chandrasekhar, R. Shashidhar, K.V. Rao, in: L. Bata (Ed.), Advances in Liquid Crystal Research and Applications, vol. 123, Pergamon Press, Oxford 1980, p. 1980.

13

[49] P.E. Cladis, R.K. Bogardus, W.B. Daniels, G.N. Taylor, High-pressure investigation of the reentrant nematic—bilayer-smectic-a transition, Phys. Rev. Lett. 39 (1977) 720–723. [50] Y. Lansac, M.A. Glaser, N.A. Clark, O.D. Lavrentovich, Photocontrolled nanophase segregation in a liquid-crystal solvent, Nature 398 (1999) 54–57. [51] S. Krishna Prasad, G.G. Nair, G. Hegde, Dynamic self-assembly of the liquidcrystalline smectic A phase, Adv. Mater. 17 (2005) 2086–2091. [52] C.A. Guymon, E.N. Hoggan, N.A. Clark, T.P. Rieker, D.M. Walba, C.N. Bowman, Effects of monomer structure on their organization and polymerization in a smectic liquid crystal, Science 275 (1997) 57–59. [53] R. Schatzing, J.D. Litster, in: G.H. Brown (Ed.), Advances in Liquid Crystals, vol. 4, Academic Press 1979, pp. 147–206. [54] J.D. Litster, R.J. Birgeneau, Phases and phase transitions, Phys. Today 35 (1982) 26–33. [55] R.J. Mandle, J.W. Goodby, An interplay between molecular pairing, smectic layer spacing, dielectric anisotropy and re-entrant phenomena in ω-alkenyloxy cyanobiphenyls, Liq. Cryst. 44 (2017) 656–665. [56] S. Sugisawa, Y. Tabe, Induced smectic phases of stoichiometric liquid crystal mixtures, Soft Matter 12 (2016) 3103–3109. [57] J. Szabon, S. Diele, Studies on induced smectic phases (II) transition enthalpies, Xray patterns and layer spacings of orthogonal phases of binary mixtures of nematogens with contrasting dielectric anisotropies, Cryst. Res. Technol. 17 (1982) 1315–1320. [58] J.O. Indekeu, A.N. Berker, Molecular structure and reentrant phases in polar liquid crystals, J. Phys. France 49 (1988) 353–362. [59] P.J. Flory, Principles of Polymer Chemistry, Cornell University, Ithaca, 1953. [60] V. Popa-Nita, S. Kralj, Liquid crystal-carbon nanotubes mixtures, J. Chem. Phys. 132 (2010) (024902-1-8). [61] P.G. de Gennes, J. Prost, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1993. [62] D. Gullion, P.E. Cladis, J. Stamatoff, X-ray study and microscopic study of the reentrant nematic phase, Phys. Rev. Lett. 41 (1978) 1598–1601. [63] P.E. Cladis, New liquid-crystal phase diagram, Phys. Rev. Lett. 35 (1975) 48–51. [64] P.S. Pershan, J. Prost, Landau theory of the reentrant nematic-smectic A phase transition, J. Phys. Lett. 40 (1979) (Paris). (L-27-30).