Flow properties of a twist-bend nematic liquid crystal

Abstract

We present the first shear alignment studies and rheological measurements in the twist-bend nematic (N_tb) liquid crystal phase of odd numbered flexible dimer molecules. It is found that the N_tb phase is strongly shear-thinning. At shear stresses below 1 Pa the apparent viscosity of N_tb is 1000 times larger than in the nematic phase. At stress above 10 Pa the N_tb viscosity drops by two orders of magnitude and the material exhibits Newtonian fluid behavior. This is consistent with the heliconic axis becoming normal to the shear plane via shear-induced alignment. From measurements of the dynamic modulus we estimate the compression modulus of the pseudo-layers to be B ∼ 2 kPa; this value is discussed within the context of a simple theoretical model based upon a coarse-grained elastic free energy.

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1. Introduction

Conventional low molecular weight nematic liquid crystals used in liquid crystal displays are anisotropic fluids characterized by a Poiseuille flow behavior with low viscosities (η < 1 Pa s) that only slightly depend on the alignment.¹ Chiral nematic (N*) liquid crystals may exhibit a micron-scale helical structure in which the twisting director is perpendicular to the helical axis. Their mechanical behavior is strongly dependent on the orientation of the helical axis with respect to the shear stress: they flow like nematics when the helix axis is normal to the shear stress, but the flow along the helical axis is permeative,² similar to Bingham fluids.³ This flow behavior is very much like that of smectic-A or C liquid crystals, which have a nanoscale density modulation (layering) along one direction but are fluid-like in the two perpendicular directions. Smectics show viscous behavior within the layers, and elastic response normal to the layers.

A novel nematic phase with conical helical structure – termed the twist-bend nematic (N_tb) – was predicted theoretically,^4–6 and observed experimentally only recently both in flexible end-to-end dimers^7–9 and in certain rigid bent-core molecules.¹⁰ Presently they are attracting a significant amount of attention,^11–22 as they form chiral structures even from achiral molecules, such as is similarly observed in tilted polar smectic phases of bent-core molecules.^23,24 Electro-optical studies indicated a very short pitch in the order of 10 nm,²⁵ which was directly confirmed via freeze fracture transmission electron microscopy (FFTEM).^10,17,26,27 This represents a nanoscale pseudo-layered structure,^11,18,28 which also explains the focal-conic and striped structures^{11,12,16,18,28} of the N_tb phase that are normally seen in smectic mesophases.²⁹ However, as revealed by X-ray diffraction,^16,17,30 the N_tb phase lacks the mass density modulation characteristic of a true smectic (hence the designation pseudo-layered).

To the best of our knowledge, no rheological studies have been reported on the flow properties of liquid crystals in the N_tb phase. In this paper, we describe systematic rheological measurements of a six-component mixture KA(0.2) synthesized in Merck Chemicals Ltd. to which 20 mol% of a methylene linked dimer 1′′,9′′-bis(4-cyano-2′-fluorobiphenyl-4′-yl)nonane (CBF9CBF) is added to a base mixture containing five-component odd-membered liquid crystal dimers with ether linkages. The detailed molecular structures are given in ref. 16. The material has monotropic mesophases with the sequence I – 77 °C–N – 37.4 °C–N_tb observed in cooling¹⁶ and a heliconical pitch of p = 10.5 nm at 33 °C in the N_tb phase.²⁷ We studied the shear-induced alignment properties, measured the viscoelastic properties as a function of temperature, shear rate, stress and frequency, and compared the results with the rheological properties of conventional chiral nematic and smectic phases.

2. Results and discussion

2.1. Alignment properties

Polarizing optical microscopy (POM) observations and typical textures of KA(0.2) films under various conditions are shown in Fig. 1.

Fig. 1 Polarizing optical microscope textures of KA(0.2). Top row: 5 μm-thick homeotropic cell. (a) N at 55 °C, (b) N_TB at 33 °C. Second row: 5 μm-thick planar cell. (c) N at 55 °C; (d) N_TB at 33 °C. Rubbing is along the stripes in (d). Third row: 30 μm shear cell at 33 °C. (e) Before shearing; (f) after shearing with [small gamma, Greek, dot above] = 10 s⁻¹ for 10 s; arrow shows the shearing direction. Bottom row: 100 μm film (g) rubbed polyimide surfaces (rubbing direction is indicated by dotted arrow); (h) no alignment layer. Scale bars indicate 100 μm length. Double arrows show the directions of the crossed polarizers.

The top row shows a 5 μm film between glass substrates treated with homeotropic alignment layers in the nematic phase at 55 °C (a), and in the twist-bend nematic phase at 33 °C (b). In the N phase a strongly fluctuating schlieren texture appears with two-brush defects indicating either a tilted director or a biaxial structure. In the N_tb phase the fluctuations become frozen and a randomly oriented pattern of stripes appears.

The second row of Fig. 1 shows a 5 μm film with planar alignment layers in the nematic phase at 55 °C (c), and in the twist-bend nematic phase at 33 °C (d). The planar alignment in the N phase is uniform, whereas in the N_tb phase regular stripes appear along the rubbing direction with a periodicity comparable to the film thickness, as described in previous studies.^16,20

The third row of Fig. 1 shows the textures in the twist-bend nematic phase at 33 °C of a sample contained in a 30 μm-thick shearing cell before shear (e) and after displacing the top plate at [small gamma, Greek, dot above] = 10 s⁻¹ shear-rate for 10 s (f). As expected, the texture is inhomogeneous, containing small domains full of defects and stripes. At lower shear rates (but with [small gamma, Greek, dot above] > 1 s⁻¹), partial alignment develops, and the texture becomes dark between crossed polarizers along the shear direction. This texture brightens as the sample is rotated between crossed polarizers indicating planar alignment with optic axis parallel to either the polarizer or analyzer.

Finally, the bottom row of Fig. 1 shows the textures in the N_tb phase at 33 °C in 100 μm films, which have the same thickness as the samples used in the rheological measurements discussed below. Fig. 1(g) and (h) show the textures in cells with uniformly rubbed polyimide alignment layers and without any surface treatment, respectively. Both textures reveal elongated focal conic defects of width comparable to the film thickness. In the cell with rubbed polymer substrates, additional stripes form perpendicular to the rubbing direction. The period of these stripes is not regular; its average value is about an order of magnitude smaller than the film thickness. These stripes do not appear in the film with untreated glass substrates, which indicates that they are related either to the strength of the surface anchoring or to some other aspect of the rubbing process.

2.2. Rheological measurements

We performed rheological studies using a commercial cone-and-plate rheometer HAAKE MARS II. Results for the temperature dependence of the viscosity, measured at a constant shear stress 0.75 Pa (which is lower than that needed to align the N_tb phase), and at constant shear rate 100 s⁻¹ (which provides high enough shear stress for alignment in the entire N_tb phase) are shown in Fig. 2. In the isotropic and nematic phase both measurements give the same results, as the alignment is missing in the isotropic phase, and because even 0.75 Pa stress could align the director in the nematic phase as well as the 100 s⁻¹ shear rate. In the nematic phase the measured viscosity values can be fitted by a straight line, which indicates an Arrhenius behavior with an activation energy of E_a ∼ 60 kJ mol⁻¹. At the transition to the N_tb at 0.75 Pa stress that leads to phase a focal conic texture shown in Fig. 1(h), the measured apparent viscosity increases rapidly (by 2 orders of magnitude over 1 °C). On the other hand, the shear alignment obtained by 100 s⁻¹shear rate leads to a two order of magnitude decrease of the viscosity, due to the onset of director alignment above 1 Pa stress. The activation energy in this aligned N_tb phase is found to be E_a = 110 kJ mol⁻¹, almost twice that in the N phase. Comparison of the two viscosity measurements, at constant stress and at constant shear rate, dramatically demonstrates the non-Newtonian behavior at the onset of the N_tb phase.

Fig. 2 Inverse absolute temperature dependence of the apparent viscosity measured with the cone and plate rheometer at constant τ = 0.75 Pa stress (pink triangle) and at constant [small gamma, Greek, dot above] = 100 s⁻¹ shear rate (black open squares).

By measuring the steady-state shear rate [small gamma, Greek, dot above] as the function of shear stress τ, we obtained the shear stress dependent apparent viscosity η from the relation . These results are plotted in Fig. 3 for different temperatures between 29 °C and 80 °C. Both in the isotropic and nematic phases the viscosity is independent of the shear stress, indicating Newtonian flow behavior. In the N_tb phase and well below a critical shear stress τ_c, the apparent viscosity is almost 2 orders of magnitude larger than at τ > τ_c, which is consistent with director realignment and agrees with the results in Fig. 2.

Fig. 3 Steady-state apparent viscosity η as a function of the applied shear stress τ at various fixed temperatures. Solid lines indicate the best fit using the equation where η₀(η_∞) is the apparent viscosity at zero (infinite) stress, and τ₀ is the stress where the viscosity starts decreasing. The temperature dependence of τ₀ is indicated by the dotted line. The temperature dependence of the critical shear stress τ_c, which shows the critical stress for alignment, is shown by the dashed line.

The dotted line in Fig. 3 shows the onset stress τ₀, at which the viscosity starts decreasing, whereas the dashed line shows the critical stress τ_c at which the material is effectively shear aligned. Both τ₀ and τ_c increase in cooling from the N–N_tb transition. For example, at 33 °C, the alignment is completed at τ_c > 10 Pa, where the apparent viscosity is η ∼7 Pa s. This corresponds to [small gamma, Greek, dot above] > 1 s⁻¹, which is in agreement with Fig. 1(f) that shows complete shear alignment at [small gamma, Greek, dot above] = 10 s⁻¹.

In Fig. 4(a) and (b) we show the dynamic moduli G′ and G′′, measured at constant strain of γ = 0.1, as a function of angular frequency ω; these were measured after applying pre-shear stresses above τ_c for 5 minutes. The data in Fig 4(a) were obtained in the N phase at 55 °C; Fig. 4(b) corresponds to the N_tb at 33 °C. In the nematic phase the moduli are independent of the pre-shear stress, and both G′ and G′′ are proportional to the angular velocity. In the N_tb phase G′′ > G′ across the whole frequency range measured, which indicates fluid-type behavior. Additionally we find that the scaling G′(ω) ∝ ω^0.77, G′′(ω) ∝ ω^0.70, which is very different from the G′(ω) ∝ ω², G′′(ω) ∝ ω scaling found in nematic and in uniform smectic LC phases with the shear plane along the layers.³¹ The behavior is closer to that found in smectics with focal conic defects (G′(ω) ∝ ω^1/2, G′′(ω) ∝ ω^1/2),^32,33 which may indicate the presence of residual defects in the shear aligned samples even though the POM textures appear quite uniform. This also explains why G′ and G′′ increase for increasing shear stresses in the N_tb phase, again similar to smectics with focal-conic defects.³²

Fig. 4 Dynamic storage modulus G′ (solid symbols) and loss modulus G′′ (open symbols) as the function of angular frequency ω at 55 °C (a) and 33 °C (b) at pre-shear stresses 10, 40, 60, 80 Pa applied for 5 minutes prior to measurements.

Finally, we measured G′ and G′′ for increasing strain at ω = 0.06 s⁻¹. An example of the results after the sample is cooled to 33 °C in the N_tb phase is shown in Fig. 5. Both G′ and G′′ are constant (∼2000 Pa) below γ₀ = 0.01, then decrease with increasing strains due to shear alignment. This response is consistent with a model of inhomogeneous sample orientation at low stresses, where, over most of the sample, the shear is not parallel to the pseudo-layers formed by the nanoscale pitch. This leads to relative changes in the spacing of the pseudo-layers on the order of γ, resulting in a stress of the order of B·γ and a modulus of the order of B. From Fig. 5 we estimate the pseudo-layer compression modulus as , and the critical stress, where the alignment occurs, is τ_c = B·γ₀ ∼ 2 × 10³ × 10⁻² = 20 Pa. The latter agrees with the critical stress deduced from the stress dependence of the apparent viscosity (see Fig. 3). For cholesteric liquid crystals the pseudo-layer compression modulus B is related to the twist elastic constant K₂ and the pitch p of the helix as .³⁴

Fig. 5 Strain amplitude dependence of the storage G′ (solid circles) and loss G′′ (open circles) moduli measured after the sample was cooled to 33 °C at the N_tb phase.

Taking K₂ ∼ 1 pN and p = 15 μm (as in case of Ramos et al.³¹), we get B ∼ 0.2 Pa. If we scale this down to the p = 10.5 nm pitch²⁷ of our N_tb material, we find B ∼ 1.8 × 10⁶ Pa, three orders of magnitude larger than the value B ∼ 2 × 10³ Pa we deduced from our experimental results. As we will now show, this large difference can be attributed to the much smaller θ = 16° cone angle of the N_tb phase in KA(0.2)²⁰ compared to θ = 90° in the N* phase.

To find how the compression modulus of the pseudo-layers is related to θ, one needs to generalize the coarse-grained theory³⁴ developed for cholesterics to the case of twist-bend nematics. As it was pointed out by Challa et al.,²⁰ this extension should also include a complex interplay of domains of opposite chirality and domain walls separating them, so in general it is a complicated task. However, previous FFTEM studies on the same material showed that the width of the domain walls is typically less than 10 nm while the smallest distance between them is over 50 nm (see Fig. 1 of ref. 27); therefore, the domains with uniform heliconical twist director structure are much larger than the defect areas. The free energy density functional F that has a minimum (F = 0) for a nematic with a uniform heliconical director structure, n⃑(z) = (sin θ₀ cos(q·z), sin θ₀ sin(q·z), cos θ₀) can be written as

The coarse graining procedure re-expresses this energy density in a form appropriate to a one-dimensional stack of pseudo-layers:

Here u(x) is the displacement of the pseudo-layers, which we consider without loss of generality to vary along the x direction in the x–y layer plane. The displacement u can be related to the director n⃑ as ∂u/∂x = −n_x. In eqn (2), B is the pseudo-layer compression modulus and K is the effective splay constant for the heliconical axis. Then, following the procedure described for a cholesteric in ref. 1, we find that B = K₂q² sin⁴ θ. This reproduces the result B = K₂q² for cholesterics with θ = 90°, and with θ = 16° and q = 2π/10.5 nm⁻¹, gives the B ∼ 2000 Pa for K₂ ∼ 1 pN, which is in excellent agreement with the value of B deduced from our rheological measurements. This supports our simple coarse-grained model for a twist-bend nematic liquid crystal, and suggests that the effect of domains with different helical senses is negligible at least for a determination of B.

3. Conclusions

We presented the first detailed studies of the flow properties of the twist-bend nematic phase in a mixture of odd numbered flexible dimer molecules. It is found that the N_tb phase has three orders of magnitude larger flow viscosity at low shear stresses than in the nematic phase. At large (>10 Pa) shear stresses this viscosity decreases by two orders of magnitude due to shear-induced alignment of the heliconical axis perpendicular to the shear plane. These observations are consistent with the behavior of a system with a pseudo-layer structure with layer spacing determined by the heliconical pitch. The measured storage modulus values at low strains provide an estimate of the pseudo-layer compression modulus B as B ∼ 2 kPa, which we could relate to the director's twist elastic constant K₂, pitch p and conical angle θ by a simple coarse-graining of the elastic free energy of a heliconical nematic phase.

4. Methods

Textural observations were made with a Polarizing Optical Microscope (BX60 from Olympus) on 5 μm, 30 μm and 100 μm thick films. The inner surfaces of the transparent indium tin oxide (ITO) coated glass substrates of 5 μm films were treated with either a unidirectional rubbed polyimide PI2555 (HD Micro Systems) that promotes molecular alignment parallel to the substrates (planar alignment) and along the rubbing direction, or by a polymer SE-1211 (Nissan Chemical Industries, Ltd) that produces vertical (homeotropic) alignment. To test the effect of shear on alignment, a d = 30 μm thick shearing cell was used where the top plate could be shifted parallel to the fixed bottom plate by a micro positioner. Finally, to test the alignment in the conditions used in the rheological measurements, we also made 100 μm-thick cells with and without planar alignment layers. The liquid crystal cells were placed in a computer controlled precision Hot Stage (STC200F from INSTEC) that regulated the temperature with a resolution better than 0.1 °C. The microphotographs of the textures were captured using a Sony CCD camera.

Rheological measurements were carried out in a cone and plate HAAKE MARS II advanced rheometer. It has a diameter of 2R = 20 mm and a cone angle of α = 0.0349 rad (2). The minimum gap between the plate and cone is d = 100 μm. The plate and cone are not treated for any alignment. Measurements were performed using three different modes:

In the controlled-stress (CS) mode the rotor is driven by the motor connected to the cone. The electrically-controlled torque T on the motor shaft results in a shear stress .³⁵ The shear stress results in a rotor angular velocity Ω and shear rate , which is inversely proportional to the viscosity of the sample as . Ω, and the torsion angle φ are measured by means of an optical sensor with 3.6 × 10⁻⁴ degree resolution. The shear strain is constant in the whole volume.

In the controlled (shear)-rate (CR) mode Ω is controlled and the torque induced by the resistance of the sheared material between the stationary bottom plate and the rotating cone³⁵ is measured.

In the dynamic controlled shear rate (DCR) mode an oscillating strain is applied, so that the time dependence of the angle φ of a point on the rotor is φ(t) = φ₀ sin(ωt). The resulting strain is and the shear stress is τ = τ₀ sin(ωt + δ), where the phase shift δ < 90° for a viscoelastic material. The complex modulus is G* = G′ + iG′′, where the storage modulus is , and the loss modulus is . In DCR mode, G′ and G′′ can be measured at the function of angular frequency, ω. To control the alignment of the liquid crystal, prior to the DCR measurements the samples were usually pre-sheared. Each pre-shear stress was applied for 300 seconds, which is long enough to reach the steady state. All measurements have been performed such that the temperature was set to 80 °C (above isotropic) for 300 seconds and then it was ramped down at 1 °C min⁻¹ to the desired test temperature.

Acknowledgements

This work was supported by the NSF under grant DMR 1307674.

References

1. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, Clarendon Press, Oxford, 2nd edn, 1993.
2. W. Helfrich, Phys. Rev. Lett., 1969, 23, 372–374.
3. R. S. Porter, E. M. Barrall and J. F. Johnson, J. Chem. Phys., 1966, 45, 1452.
4. R. B. Meyer, Les Houches Lect., 1973, 271–343.
5. I. Dozov, Europhys. Lett., 2001, 56, 247.
6. R. Memmer, Liq. Cryst., 2002, 29, 483–496.
7. P. J. Barnes, A. G. Douglass, S. K. Heeks and G. R. Luckhurst, Liq. Cryst., 1993, 13, 603–613.
8. G. R. Luckhurst, Macromol. Symp., 1995, 96, 1–26.
9. A. P. J. Emerson and G. R. Luckhurst, Liq. Cryst., 1991, 10, 861–868.
10. D. Chen, M. Nakata, R. Shao, M. R. Tuchband, M. Shuai, U. Baumeister, W. Weissflog, D. M. Walba, M. A. Glaser, J. E. Maclennan and N. A. Clark, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2014, 89, 022506.
11. V. P. Panov, R. Balachandran, M. Nagaraj, J. K. Vij, M. G. Tamba, A. Kohlmeier and G. H. Mehl, Appl. Phys. Lett., 2011, 99, 261903.
12. M. Cestari, S. Diez-Berart, D. A. Dunmur, A. Ferrarini, M. R. de la Fuente, D. J. B. Jackson, D. O. Lopez, G. R. Luckhurst, M. A. Perez-Jubindo, R. M. Richardson, J. Salud, B. A. Timimi and H. Zimmermann, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2011, 84, 031704.
13. J. W. Emsley, P. Lesot, G. R. Luckhurst, A. Meddour and D. Merlet, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2013, 87, 040501.
14. J. W. Emsley, M. Lelli, A. Lesage and G. R. Luckhurst, J. Phys. Chem. B, 2013, 117, 6547–6557.
15. L. Beguin, J. W. Emsley, M. Lelli, A. Lesage, G. R. Luckhurst, B. A. Timimi and H. Zimmermann, J. Phys. Chem. B, 2012, 116, 7940–7951.
16. K. Adlem, M. Čopič, G. R. Luckhurst, A. Mertelj, O. Parri, R. M. Richardson, B. D. Snow, B. a. Timimi, R. P. Tuffin and D. Wilkes, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2013, 88, 022503.
17. V. Borshch, Y.-K. Kim, J. Xiang, M. Gao, A. Jákli, V. P. Panov, J. K. Vij, C. T. Imrie, M. G. Tamba, G. H. Mehl and O. D. Lavrentovich, Nat. Commun., 2013, 4, 2635.
18. V. P. Panov, R. Balachandran, J. K. Vij, M. G. Tamba, A. Kohlmeier and G. H. Mehl, Appl. Phys. Lett., 2012, 101, 234106.
19. E. G. Virga, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2014, 89, 052502.
20. P. K. Challa, V. Borshch, O. Parri, C. T. Imrie, S. N. Sprunt, J. T. Gleeson, O. D. Lavrentovich and A. Jákli, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2014, 89, 060501.
21. N. Vaupotič, M. Čepič, M. A. Osipov and E. Gorecka, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2014, 89, 030501.
22. J. Xiang, S. V. Shiyanovskii, C. Imrie and O. D. Lavrentovich, Phys. Rev. Lett., 2014, 112, 217801.
23. D. R. Link, G. Natale, R. Shao, J. E. Maclennan, N. A. Clark, E. Körblova and D. M. Walba, Science, 1997, 278, 1924–1927.
24. A. Eremin and A. Jákli, Soft Matter, 2013, 9, 615.
25. C. Meyer, G. R. Luckhurst and I. Dozov, Phys. Rev. Lett., 2013, 111, 067801.
26. D. Chen, J. H. Porada, J. B. Hooper, A. Klittnick, Y. Shen, M. R. Tuchband, E. Korblova, D. Bedrov, D. M. Walba, M. A. Glaser, J. E. Maclennan and N. A. Clark, Proc. Natl. Acad. Sci. U. S. A., 2013, 110, 15931–15936.
27. R. R. Ribeiro de Almeida, C. Zhang, O. Parri, S. N. Sprunt and A. Jákli, Liq. Cryst., 2014 DOI:10.1080/02678292.2014.947346 , in print.
28. V. Panov, M. Nagaraj, J. Vij, Y. Panarin, A. Kohlmeier, M. Tamba, R. Lewis and G. Mehl, Phys. Rev. Lett., 2010, 105, 167801.
29. R. J. Mandle, E. J. Davis, C. T. Archbold, S. J. Cowling and J. W. Goodby, J. Mater. Chem. C, 2014, 2, 556.
30. D. Chen, J. H. Porada, J. B. Hooper, A. Klittnick, Y. Shen, E. Korblova, D. Bedrov, D. M. Walba, M. A. Glaser, J. E. Maclennan, and N. A. Clark, http://arXiv.org/abs/1306.5504, 2013.
31. L. Ramos, M. Zapotocky, T. Lubensky and D. Weitz, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2002, 66, 031711.
32. S. Fujii, S. Komura, Y. Ishii and C.-Y. D. Lu, J. Phys.: Condens. Matter, 2011, 23, 235105.
33. R. Bandyopadhyay, D. Liang, R. Colby, J. Harden and R. Leheny, Phys. Rev. Lett., 2005, 94, 107801.
34. T. C. Lubensky, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 1972, 6, 452–470.
35. G. Schramm, A Practical Approach to Rheology and Rheometry, Gebrueder HAAKE GmbH, Karlsruhe, 2nd edn, 2000.

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