Flexoelectrically controlled twist of texture in a nematic liquid crystal

2014 We describe a new flexoelectric volume effect in homogeneous electric field, in the nematic phase of MBBA. An initial splay-bend texture is created by strong antagonistic planar and homeotropic anchoring on two glass plates. This results in a volume flexoelectric polarization. A transverse DC electric field creates a twist of the texture, which can be measured by the resulting wave guided rotation of light polarization. We obtain the volume flexoelectric constant e1 e3 = + 1.0 ± 0.2 10-4 stat C/cm at 20 °C. The continuously controllable light polarization rotation on a total amplitude ~ ± 45° may be useful for applications. J. Physique LETTRES 43 (1982) L-365 L-369 15 mm 1982, Classification Physics Abstracts 61.30 78.20J 78.20H Most nematic liquid crystals are flexoelectrics [1], i.e. they build an electric polarization P when submitted to a curvature strain. As shown by the Bordeaux Group [2], the flexoelectric effect is mostly quadrupolar i.e. P ~ e V. Q, where Q is the electric quadrupole density, proportional to the nematic order parameter. In the « isotropic » case e is a scalar and the coupling of P with an applied uniform electric field E results only in surface effects, which imply the use of weak surface anchoring difficult to control [3]. People have induced flexoelectric texture distortions by use of inhomogeneous electric fields [4, 5]. In the so-called « hydrodynamic » limit, it was argued [2] that there exists no bulk flexoelectric effect In fact this statement was misleading, at least for us. The « hydrodynamic » assumption was really the « homogeneous » assumption [5], i.e., the case of small angular distortions from an (*) On leave from Institute of Solid State Physics, Sofia, Bvd. Lenin 72, Bulgaria. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019820043010036500 L-366 JOURNAL DE PHYSIQUE LETTRES initially homogeneous sample. In non-homogeneous situations, this statement is not valid [6]. The reason is due to the fact that e is a tensor, which rotates in space as the nematic order parameter itself For arbitrary angular distortions, there must be a bulk flexoelectric effect. The volume induced flexoelectric polarization can be coupled to an applied uniform electric field [7, 8]. Such an effect has been indirectly observed from the threshold of a texture instability [9]. In this paper, we describe the first direct observation of a new volume flexoelectrooptic effect, from an external uniform E field, acting on a permanent flexoelectric polarization created by a non uniform bend and splay texture. The result is an E field controlled twist of texture, which may be useful for practical applications. The sample geometry is shown on figure 1. A nematic liquid crystal is placed in between two glass plates a and b, coated to induce respectively a parallel (x) and perpendicular (z) orientation. Fig. 1. Sample geometry. The unperturbed director line is in the xz plane. Under the action of E, this line (dashes) twists continuously along y. The twist ~(z) is visible on the xy projection (dots). An optical beam, linearly polarized along x on plate a, shows at the output of the sample (plate b) a linear polarization along the maximum twist ~(0). A single domain will present a state of permanent splay and bend. The « non isotropic » permanent polarization created by this distortion can be visualized as where n is the director, d the sample thickness and e* = e 1 e3 is the anisotropic flexo volume effect in classical notations [1, 10]. P is parallel to x and localized close to the plate a. Using two parallel electrodes one can apply an electric field E alongy, parallel to the plates, and perpendicular to P. The result is an electric torque proportional to the applied field. At equilibrium, the texture is continuously twisted compared to the rubbing direction on plate a. To calculate this twist we first assume that the volume polarization charges div P are just cancelled by ionic conductivity. We keep only a static problem of torque equilibrium. We call 0 and 4> the usual Euler angles which define the orientation of n compared to the normal to the plates z, and the rubbing direction on plate a (x) (see Fig. 1). The elastic free energy density in absence of field is simply : where K is the assumed isotropic Frank curvature constant. The bend-splay texture is defined by ¢ = 0 and d2B/dzz = 0, or 0 = nz/2 d. The coupling energy between the permanent flexo polaL-367 FLEXOELECTRIC TWIST IN NEMATICS rization and the electric held is P. E = e* E sin2 9 sin de . The e uilibrium e q uations write : fiel E 0 0E equilibrium e uations the azimuthal equation (2) for low E field, will give a twist 0 linear in E. The bend equation (1) describes the change in the initial bend-splay due to the field One sees immediately that this correction is second order in applied field, and can be neglected for low twist We then calculate c/> by linearizing equation (2) in 0 and using for 0 the unperturbed splay-bend 0 = Tcz/2 ~ Equation (2) becomes We are interested by the largest value ~(0) close to the lower plates (z = 0). Equation (3) can be integrated exactly [11] as : ... ~ 11 In practice, to measure ~(0), one can use the wave guide property of the twisted texture. We send for instance a light beam normal to the sample and polarized along x (~ =0) on the upper plate. We observe the polarization of the transmitted beam outside the sample. The criterion for the wave guide regime [ 1 ] is An. p > ~, where An is the apparent birefringence of the nematic sample, p is the pitch of the twisted texture and ~, the wave length of the light beam. In our case, p = 2 7c dz and because of the tilt, close to the lower plate, On = Ano 02 with One 0.2. Using d~ ~ ~ o o ° equation (3), one gets the condition : The resulting error for ~(0) is : This error is negligible in the linear regime where ~(0) is smaller than 1, and for thick samples. In practice a sample of thickness d larger than ~, should behave as an optical wave guide for linear polarization, from the upper plate till the lower plate. To observe the effect, we use a nematic sample of MBBA (Methoxy Benzylidene Butyl Aniline) at room temperature (20 °C). The plate a is coated with polyvynil alcohol to induce a planar orientation. The plate b is silane coated, to induce an homeotropic orientation. The electrodes are two Al foils, 2.5 mm apart, oriented along the rubbing direction of plate a. The thickness is d = 40 J1m. The sample is observed on the stage of a polarizing microscope. For convenience, the planar plate a is placed close to the condenser side, so that we have just to rotate the analyser to measure an eventual rotation of polarization. In absence of E field, we observe that the sample behaves optically as a birefringent slab, with the neutral lines along (and perpendicular to) the rubbing direction. Sometimes, domains are observed, delimited by a disclination line. Applying an oblique magnetic field in the xz plane one can make these domains grow independently. One identifies these domains as the expected twins for the splay-bend geometry (see Fig. 2). L-368 JOURNAL DE PHYSIQUE LETTRES Fig. 2. The twin domains of splay-bend. Under the action of the magnetic field H, the ( + ) domain grows. This indicates the polarity n(div n) of the domain (in the opposite direction of n). We now apply a DC electric field E. We obtain at the output of the sample a light linearly polarized along a direction rotated by an angle 4>(0) compared to the rubbing (and polarizer) direction. Rotating the polarizer compared to the rubbing direction results in an elliptical polarization at the output This proves that the director remains aligned along x close to the plate a, Le. that the planar anchoring is strong. We have measured ~(0, E) on one single domain. The results are plotted on figure 3. We do find a linear dependence as expected, with positive and negative values according to the sign of E. We now observe the corresponding rotation on the twin domain. For the same value of the field, the polarization rotation is exactly the opposite, as expected. From the slope of the ~(0, E) plot, we can derive With K ~ 5 x 10 -’ cgs, we find e* ~ ~ 1.2 x 10 4 cgs, which is the correct order of magnitude. Taking into account the anisotropy of the elastic constants [1] (K1 = K3, K3/K2 = 2.5), equation (3) can now be integrated numerically as Fig. 3. ~(0) versus E, for the two twin domains ( + ) and ( ). The linearity and the symmetry of the two plots prove the existence of a volume flexoelectric polarization p = e* n(div n). L-369 FLEXOELECTRIC TWIST IN NEMATICS (K3 is the bend and K2 the twist Frank curvature constant). The flexo constant is now From our magnetic field effect, we can check the direction along which the director lines drop from plate a to plate b, i.e. the direction of n(div n). We have verified that the director lines oriented along n(div n) rotate toward the direction of E, as shown on figure 1, i.e. that e* = ei e3 is positive. An important parameter is the maximum value of 4> that can be achieved. This value is in the range of ± 450. For higher fields, we observe the onset of electrohydrodynamical instabilities, as usual in flexoelectric experiments when effects in E2 dominate the effects linear in E. Within our present accuracy, we have not observed any flow in the linear regime. Other features of this effect are now being studied. We have checked for instance that the maximum twist ~(0) increases with d The response time of the texture twist is nematic-like (slow, of the order of 100 ms in our sample), but can be decreased using stabilizing fields. The effect is also visible in other bend-splay geometries, above a Freedericks transition for instance from homeotropic to planar textures, with a symmetrical distortion resulting in a total twist 2 ~(0). All these points will be described in forthcoming papers. In conclusion, we have observed the volume flexoelectric effect from uniform electric field, using strong plate anchoring, in the nematic liquid crystal MBBA. We have created a permanent splay-bend distortion, which results in a permanent volume density electric polarization. A transverse homogeneous electric field induces a continuous twist of this texture. Using the wave guide property of this twisted texture for linearly polarized light, we have measured the twist. This allows us to give the first direct measurement of the bulk flexoelectric constant This value compares reasonably with the only data available of e* = 1.7 x 10 4 cgs units for BMAOB from the instability experiment of the Russian group [9]. With the accepted result that the quadrupolar volume effect in electric field gradient measures the quantity el + e3, we have now two independent experiments to determine the constants e, and e3, in strong anchoring, i.e. without the problems due to the uncontrolled weak anchoring.