DOI 10.1140/epje/i2013-13115-y Regular Article Eur. Phys. J. E (2013) 36: 115 THE EUROPEAN PHYSICAL JOURNAL E Biaxial coherence length in a nematic π-cell R. Hamdi1,a, G. Lombardo2, M.P. de Santo1,2, and R. Barberi1,2 1 Physics Department, University of Calabria, 87036 Rende, Italy 2 CNR-IPCF UOS di Cosenza, c/o University of Calabria, 87036 Rende, Italy Received 8 June 2013 and Received in final form 1 August 2013 Published online: 16 October 2013 – c© EDP Sciences / Societa` Italiana di Fisica / Springer-Verlag 2013 Abstract. In a highly frustrated calamitic nematic phase, the strain can be relaxed by lowering the nematic order: the starting uniaxial symmetry can be broken and it can be replaced locally with transient biaxial domains. Using simple optical retardation measurements, we estimate the length scale over which the biaxial disturbance decays in space within a π-cell submitted to a weak electric field. We also characterise the transition cascade from the uniaxial splay texture to a bend texture through slow defect motion. 1 Introduction The π-cell is the best known fast switching nematic liq- uid crystal (LC) device with a wide viewing angle [1]. The cell admits two topologically distinct equilibrium textures having different optical responses, known as the H (hor- izontal) and V (vertical) states [2]. Originally, this de- vice was investigated in the V -bend state, also known as the Optically Compensated Bend mode [3]. In this con- figuration, the π-cell presents faster switching times than conventional LC displays, capable of a combined on-off re- sponse time of under 5ms for a typical display thickness of about 5μm. With the quite recent advent of liquid crystal flat panel televisions, which require fast switching electro- optical devices to display high quality video information, there has been renewed interest in π-cells [4–8], with par- ticular interest in the possibility of intrinsic electro-optical bistable behaviour [6,7,9]. During the last fifteen years this renewed interest has led scientists to study the fun- damental mechanism of switching between the V and H states, which cannot be described in the framework of the elastic model with constant order parameter as it involves topological variations of the cell textures [7,9–15]. The energy barrier between the two topologically distinct tex- tures, which depends on the tilt angles and on the ratio of the elastic moduli of the nematic material [4], prevents any spontaneous fast relaxation between them, giving rise to an actual topological barrier. Two main techniques have been proposed to overcome this barrier: breaking the sur- face anchoring in the case of weak anchoring energy [16], and reconstruction of the bulk as well as surface order in the case of infinite anchoring energy [9,13]. Anchor- ing breaking, which can be considered as a limited case of surface order reconstruction [13], and bulk order re- a e-mail:
[email protected] construction allow changes in the orientation of the two nematic textures of π/2 without any continuous macro- scopic director rotation. Both phenomena are induced by short, strong electric field pulses in the millisecond range. The nematic strain occurs over a length scale comparable with the nematic biaxial coherence length ξb [11,17,18], and it relaxes by locally lowering the nematic order with the advent of transient biaxial states [7,9–13,19]. In this frame, the starting H nematic texture loses its uniaxial symmetry and is locally and temporarily replaced with biaxial ordered domains, which are able to connect the H configuration with the topologically distinct state V . Recently, much effort has gone into understanding the theoretical foundation of biaxial domains, which can grow within uniaxial LC systems in the bulk as well at the LC interface [20,21]. Recent experiments on LC frustrated systems have shown that local and transient bulk biaxial order can be induced inside a nematic phase [18,22–24], suggesting that biaxial order plays a fundamental role in LC phenomena which take place at the nanometric scale. The typical length over which the biaxial disturbance de- cays in space, the biaxial coherence length [9,18,21,25], is in fact a few tenths of a nanometre. Local biaxial order is, for instance, present close to the core of a disclination line [25]. The nematic is uniaxial everywhere except for a small region around the defect core [18]. A first attempt to measure this critical separation was performed by Atomic Force Microscopy in the force spectroscopy mode by me- chanically squeezing a−1/2 line defect [18], confirming the expected order of magnitude of this phenomenon. Nev- ertheless, the force spectroscopy mode experiments are not easy to perform and require considerable expertise. This paper demonstrates that it is possible to evaluate the electrically induced biaxial π-wall defect scale, and hence the biaxial coherence length, by conventional optical mi- croscopy techniques with polarised light. Page 2 of 6 Eur. Phys. J. E (2013) 36: 115 Fig. 1. π-cell geometry and four distinct possible textures: (a) splay state, H; (b) intermediate state, HW ; (c) vertical state, V and (d) twist state, T . 2 π-cell The proposed set-up is based on the well-known π-cell ge- ometry, which represents the convenient testing ground for uniaxial escape symmetry [11]. The π-cell has a sandwich geometry, with a thin film of nematic material confined between two flat glass plates. The internal surfaces of the cell are treated to induce uniformly tilted but oppositely directed alignments, i.e. with anti-parallel configuration, to generate an H-splay texture, fig. 1a. This configuration presents the minimum energy state in the absence of an external electric field. If the nematic material presents a positive dielectric anisotropy, the application of a vertical external electric field, stronger than the Fredericks thresh- old, results in a new director distribution, which tends to minimise the electrically induced distortion. The orienta- tion of the director should be considered as varying only along the z-axis (along the thickness of the cell), with- out loss of generality, since the ideal geometry presents a translational symmetry along the other two spatial direc- tions. In the presence of an external electric field in excess of the Fredericks threshold, the nematic director is per- pendicular to the plates everywhere except for three thin regions: the middle of the cell and the two surface layers, where the nematic molecules, on average, remain close to a planar orientation due to the cell symmetry and strong anchoring conditions, respectively. The maximum nematic frustration occurs in the middle of the cell, where a biaxial domain of thickness comparable with ξb starts to connect the homeotropic upper and lower textures with the middle planar orientation, HW (fig. 1b). This intermediate state forms when the external electric field is applied, and is characterised by a thin wall of high distortion in the mid- dle of the layer. Now, if the applied voltage amplitude is above the bulk order reconstruction threshold, then the order reconstruction phenomenon occurs and the H state is uniformly switched to the V state across the middle plane, fig. 1c, in a time scale of some tenths of microsec- onds [7,26]. Furthermore, if the applied voltage is below the order reconstruction threshold, HW can still be locally transformed to a splay state by moving a nematic defect, e.g. a disclination line. The π-wall energy provides the lo- cal barrier preventing spontaneous textural exchange be- tween the HW and V states and, since the energy cost is already paid by the defect domain itself, a very low field is sufficient to favour one or other texture and produce a local switching through the horizontal induced motion of the defect line [27]. The defect motion is usually slow and its textural evolution can be investigated with con- ventional optical polarised microscopy and electro-optical techniques, such as dynamic phase retardation measure- ments. Accordingly, we assembled a π-cell by using two plates already coated with an electrically conductive layer of In- dium Tin Oxide (ITO). The ITO electrodes were pat- terned by a photolithographic treatment that created two ITO strips on each plate. The strips on the two plates were assembled in order to form four regions, with each area about 1mm2. The oblique symmetrical anchoring on the two plates was carried out by spin coating a solution of 20% polyamide acid LQ1800 (from Hitachi) in 1-methyl- 2-pyrrolidione and rubbing the surface in a parallel di- rection. The cell thickness, d, was fixed with calibrated spherical spacers and found to be d = 2.5μm using an interferometry technique. Furthermore, the pre-tilt angle was measured by an optical retardation method in a sym- metric test cell and found to be 8◦ from the surface [28]. The cell was filled with the nematic liquid crystal 5CB (from MERK) in its isotropic phase by capillarity in a vacuum chamber. It was then thermo-stabilised at 25 ◦C using a hot stage and placed between crossed polarisers at a controlled angle with respect to the rubbing direction. We connected the upper and lower electrodes to a voltage generator, and were able to independently adjust the volt- age U and the frequency f , which was manually triggered. To observe textural changes, we placed the sample be- tween crossed polarisers at a controlled angle with respect to the rubbing direction. We also added a λ/4 retardation plate to observe weak birefringence. We recorded our ob- servations by means of a CCD camera connected to the microscope. The transmitted light intensity was measured using a photomultiplier that was also connected to the polarised microscope, and could be captured by a GPIB interface connected to the oscilloscope. 3 First technique: light intensity Figure 1 shows that there are three possible director con- figurations during the H − V transition: H, HW , and V textures. The V state has the lowest optical retardation, and is therefore the darkest domain under the crossed po- larisers. On the other hand, the H state has the high- est retardation; it is expected to be the brightest domain when the optical axis of the sample forms a π/4 angle with respect to the optical axis of the crossed polarisers. Un- fortunately, we could not visualise the defect propagation due to the brightness of the H state when we applied the Eur. Phys. J. E (2013) 36: 115 Page 3 of 6 Fig. 2. CCD images of the texture cycle under an optical microscope in a π-cell: (a) splay state, H; (b) splay+wall state, HW ; (c,1,2), growth of the V state, darker region, after application of a sinusoidal voltage with amplitude U = 3.5V and frequency f = 1kHz; (d,1,2) twist texture, bright region, and its relaxation after removing the external field. electric field. For that reason, to obtain the pictures pre- sented in fig. 2, we rotated the plate of the microscope in such a way that the optical axis of the sample was nearly parallel to the polariser axis. When no voltage is applied the π-cell is in the H state, the molecules are all aligned along the rubbing direction, and one observes a quite uni- form shade of gray, fig. 2a. When an external voltage is applied, e.g. U = 3.5V with 1 kHz frequency, the optical image shows a light leakage in the edges of the pentagonal pixel, accompanied by a small reduction of light transmis- sion when compared with the optical image captured when no voltage was applied or with the region out of the pixel (fig. 2b). We can easily understand this change in the light intensity, because 3.5 volts is sufficient to vertically align the nematic liquid crystal molecules in the top half and in the bottom half of the cell, creating the HW texture. Due to the symmetric configuration of the π-cell, the two re- orientations have opposing rotational sign, leading to the formation of a π-wall in the middle of the liquid crystal layer where molecules cannot rotate. This is the signature of the HW state. Now, if the voltage is increased further, the V state becomes energetically favourable [29]. Since the V state is topologically distinct from the starting H state, the transition from H to V usually starts from a local inhomogeneity in the cell (fig. 2c), where a defect starts to develop. The V region then grows by moving the defect line that runs across the sample parallel to the containing plates. Figures 2(c1) and 2(c2) were captured at t1 = 2 s and t2 = 10 s, respectively, after the com- mencement of the HW -V transition. The V texture was observed to grow slowly. The defect line separating HW and V expanded smoothly in all directions, eventually fill- ing the whole pixel. This smooth growth is possible only if the defect is moving in the bulk with minimal interaction with boundary surfaces [29]. Figure 2(c1), captured in the xy-plane, shows the closed loop of the moving disclination line, which carries the topological charge since some parts the cross section have winding number +1/2 and other parts have the op- posite winding number. The applied electric field favours the V state, and the HW state tends to disappear. The ve- locity of this disclination line depends on the applied volt- age [5,30]. The V texture transforms into a topologically equivalent π-twisted T state when the external voltage is switched off (fig. 2(d1)) [9], since the T state is ener- getically favoured in 5CB, whose twist elastic constant is lower than the bend one. The T state appears very bright due to the match between the optical anisotropy and the cell thickness. The part of the pixel still in the HW state returns to its ground H state in the absence of an electric field (fig. 2d). The ground H state is obviously more stable than T [9], and hence the T state completely vanishes via propagation of the H state in a few seconds, depending on the cell thickness. This is visible in fig. 2(d2), which was captured 1 s after fig. 2(d1). Page 4 of 6 Eur. Phys. J. E (2013) 36: 115 Fig. 3. Intensity of the light transmitted through a 2.5μm thick π-cell subjected to a sinusoidal electric field with fre- quency f = 1kHz and amplitude U = 3.5V vs. time. To obtain data about the π-wall during the H-V tran- sition, we investigated the residual birefringence of H, HW and V states by measuring the light transmitted through the cell before and during the application of the same ex- ternal electric field used to obtain the textures shown in fig. 2, where U = 3.5V. We rotated the plate of the mi- croscope in such a way that the optical axis of the sample was oriented at π/4 with respect to the optical axis of the crossed polarisers, and then we applied the external electric field. Figure 3 shows that, upon application of the external voltage, the intensity saturates to a first level corresponding to HW state, and then slowly relaxes to a lower value corresponding to the V state, where the wall disappears. Even if the π-wall is destroyed, the transmit- ted intensity does not vanish due to the partial bending of the molecular texture on the surfaces and strong anchoring conditions. The light intensity passing through the cell de- pends on the angle between the polarisation vector of the incident beam and the initial orientation of the director of the nematic liquid crystal [31]. Since the sample forms an angle π/4 with respect to the optical axis of the crossed polarisers, the transmitted intensity I can be written as I = I0 sin2 ( Δϕ 2 ) , (1) where Δϕ is the phase difference Δϕ = 2πdΔnλ , and d, Δn, λ and I0 are the cell thickness, the average nematic bire- fringence, which depends on the texture, the light wave- length and the intensity of the polarised light incident on the cell, respectively. By using eq. (1), one can obtain a rough determination of the thickness of the π-wall. In fact, to simplify the picture, we can assume that in the H state all molecules are oriented almost horizontally, and hence Δn = n‖−n⊥ across the thickness of the cell. Moreover if we assume that, in the HW state, molecules are oriented mainly horizontal inside the wall of thickness l = 2ξb [29], where ξb is the biaxial coherence length, and close to the boundary surfaces on a length ξb, due to the strong an- choring conditions, then one has Δn = n‖ − n⊥ within the π-wall and also close to the containing plates. Out- side these regions, the molecular orientation is considered vertical and hence Δn = 0. Consequently, the phase differ- ences of the H and HW states become 2πdΔnλ and 8πξbΔn λ , respectively. The above assumptions are based on the tex- ture dynamics shown in fig. 1 and on the experimental observations of the liquid crystal director field evolution in the π-cell under an electric field described in ref. [29]. If we equate I0 in the two configurations H and HW by eq. (1), we obtain sin2 ( 4πξbΔn λ ) = IHW IH sin2 ( πdΔn λ ) , where IH and IHW correspond to the transmitted intensity in the H and HW states, respectively. In our sample the cell thickness d was chosen in such way that sin2(πdΔnλ ) ≈ 1. Furthermore, according to refs. [11,17,18], the biaxial coherence length ξb is of the order of tens of nanometers, and hence for the small angle approximation, sin2( 4πξbΔnλ ) ≈ (4πξbΔnλ )2. Therefore, IHW = IH ( 4πξbΔn λ )2 ⇒ ξb = λ16πΔn √ IHW IH . (2) If we substitute into eq. (2) d = 2.5μm, λ = (0.6 ± 0.02)μm, Δn = 0.17, IH = 2.90± 0.08 and IHW = 0.65± 0.05 as shown in fig. 3, we can estimate ξb = 33± 9 nm. 4 Second technique: optical retardation Another evaluation of the π-wall thickness l, and hence of ξb, can be obtained by means of the crystal rotation method [32]. This allows local observation of the optical properties on an area given by the size of a laser spot (He- Ne laser, λ = 0.633μm) of the order of 1mm2. Sprang [28] described the experimental set-up of this technique, which was used to measure the tilt angle of the sample by con- sidering the optical phase difference, δ, as a function of the rotation angle. Here we used the same procedure as Sprang, but we measured the optical phase shift as a func- tion of the applied voltage. We fixed the cell in such a way that the incident linear polarised light was parallel to the director. The transmitted incident light is, in general, el- liptically polarised. A quarter plate, placed after the cell, transforms it into linear polarised light, whose rotation of the plane of polarisation gives δ. This value is then mea- sured using a rotating analyser as described in ref. [33]. Figure 4 shows that, for an applied voltage weaker than the Fredericks threshold, δ does not exhibit any change, whereas for a voltage between 1.2 and 2.7V, as the di- rector located in the upper and lower regions of the cell (fig. 1b) tends to continuously rotate to align with the direction of the vertical field, δ reduces continuously. If Eur. Phys. J. E (2013) 36: 115 Page 5 of 6 Fig. 4. Optical phase difference δ as a function of the external voltage. the voltage is increased a little then the π-wall can break, starting for instance from a surface inhomogeneity, and the texture V starts to develop (fig. 2c), moving a line de- fect. After several seconds the π-wall has completely van- ished in the area illuminated by the incident laser beam. The discontinuity of the molecular reorientation due to the moving defect leads to a jump in δ from 19 ± 1◦ to 14±1◦, as shown in fig. 4 (δ = 5◦±1◦). One can estimate the thickness of the biaxial wall from the phase difference δ = 2πlΔnλ , obtaining l = λδ 2ππΔ = (55 ± 11) nm. Con- sequently, the biaxial coherence length can be estimated to be ξb = (28 ± 6) nm. If the intensity of the external electric field is increased further, more and more nematic molecules are aligned along the vertical field and δ con- tinues to reduce. 5 Conclusion Up to now, the indirect measurement of the biaxial coher- ence length ξb using atomic force spectroscopy [18] and its theoretical estimation indicated that ξb is of the order of 10 nm. Our direct optical observations using conventional optical set-ups and based on standard observations using polarised light, together with the simplified optical mod- els adopted in this study, probably tend to slightly over- estimate ξb. 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