Structure of a hybrid aligned cholesteric liquid crystal cell I. Dozov, I. Penchev
To cite this version: I. Dozov, I. Penchev. Structure of a hybrid aligned cholesteric liquid crystal cell. Journal de Physique, 1986, 47 (3), pp.373-377. �10.1051/jphys:01986004703037300�. �jpa-00210215�
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Physique 47 (1986)
Physics Abstracts 61.30
STRUCTURE OF A HYBRID ALIGNBD CHOLESTERIC LIQUID CRYSTAL CELL
DOZOV and I.
Institute of Solid State Physics, Bvd. Lenin 72, Sofia, Bulgaria
le 9 mai 1985,
forme définitive Ze
Résumé.- La structure d’une cellule hybride d’un crystal liquide cholestérique est étudiée expérimentalement et théoriquement. Pour des petites épaisseurs la cellule présente une structure caractérisée par une
déformation unidimensionelle-structure uniforme (dans le plan xy). Cette structure est décrite théoriquement et des résultats numériques sont présentés et discutés. Pour des grandes épaisseurs de la cellule apparaît une déformation tridimensionnelle caractérisée par une modulation du champ du directeur dans le plan xy et par l’apparition de domaines. Une idée simple concernant cette déformation est présentée et
the recent years there has been
increased interest in the so called hybrid aligned nematic (HAN) structure [1,2]. This structure is characterized by a continuous splay-bend deformation due to the antagonistic orientation imposed on the two glass-liquid crystal interfaces - a homeoan
similar more complex structure - a hybrid aligned cholesteric liquid crystal (HAC). The uniform HAC structure has been already discussed both theoretically  and from the point of view of a possible practical application , but up to now some interesting properties of this complex splay-bendtwist deformation are not discussed in the literature. discussion is Our theoretical based upon the experimental data for a HAC cell containing a long-pitch nematic-cholesteric mixture.
order to observe the structure we use a % (by weight) solution of cholesterin undecylat in methoxybenzylidene butylaniline (MBBA). The natural pitch of the mixture is P = 301im. A 10 mm long wedge-shaped cell with deposited indium-tin oxide electrodes both on glass plates provides a liquid crystal thickness gradient from 0 to 90 am (Fig. 1). The lower plate is treated with
Abstract.of a hybrid aliThe structure gned cholesteric liquid crystal sample is investigated both experimentally and theoretically. At small thickness the cell prea structure sents characterized by unidimensional deformation - uniform (in the xy plane) structure. This structure is described theoretically and numerical results are presented and discussed. At large thickness there more complex three dimenappears a sional deformation characterized by a domain structure due to a modulation of the director field in the xy plane. A simple idea about the type of the deformation involved in this structure is proposed and qualitatively discussed.
polyvynil alcohol and rubbed to give a planar orientation along the x-axis. The upper plate is treated with a silane agent to induce a homeotropic alignment. As
in the HAN case a bend-splay dewould arise in order to satisfy
the boundary conditions imposed on our cell. Due to the intrinsic twist of the cholesteric and to the lack of any azimuthal torque on the homeotropic plate we expect that the equilibrium state would be twisted. The result is the splay-bend-twist
deformation figure 1. The
sample is observed
on the stage To measure the in the wave guide cell we use the property of the twisted structure. We send a light beam normal to the sample and polarized along the x-axis on the lower plate and we observe the polarization of the transmitted beam outside the sample. As we shall see in the next section the criterion for the wave guide regime is well satisfied
of a twist
in our sample except for a small region in the vicinity of the upper plate. Thus the the rotation of polarization through the sample is a good measure of the angle = o(d) near the homeotropic side.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004703037300
Fig. 2.- A set of micrographs of a hybrid aligned cholesteric cell : a) Crossed polarizers ; b) Parallel polarizers ; c) Crossed polarizers and electric field along the The thickness d direction of observation. 90 am from d = 0 increases (left) to d (right). Close to the left edge of the micrograph c) is seen the border between the regions with and without electric field. It is due to the fact that we have removed the conductive layer on one of the plates in order to prevent short-circuit. =
1.- Geometry of the cell and splaybend-twist deformation of the hybrid aligned cholesteric.
After filling the sample we observe delimited by disclination lines. We identify these domains as the expected twins similar to the splay-bend-twist splay-bend twin domains in the HAN cell . Applying an oblique magnetic field and after waiting for an hour at temperature near the cholesteric-isotropic transition we obtain a monodomain sample suitable for observation of the HAC structure. A typical structure is shown in figure 2 : a) Perpendicular polarizers; the black bands correspond to x-polarization of the transmitted b) Parallel n = 0,1,... polarizers ; the black bands correspond to domains
vely simple case when 0 and (e are functions only of the z coordinate we make the following assumptions in order to make the analysis more simple and transparent : We assume that the anchoring is strong, that the i.e. surface anchoring energy is much greater than the volume eleastic enercase the In this boundary conditions gy. for 0 and o are extremely simple :
light -tPd =n’m ;
2 and electric
field applied along the At f = 3000 Hz). 5V ,
(the left part of the
structure with uniform
rotation of the polarization (at fixed x). This region corresponds to a structure characterized by 8 and 9 which are functions only of z (at constant thickness). Hereafter we shall refer to this structure as uniform HAC At (UHAC). large thickness there appears gradually a continuous modulation of the (p angle along a direction approximately perpendicular to the local molecular director at the upper plate (modulated HAC structure - MHAC).
where F is the density of the volume elastic free energy. B.
for the elastic ric :
"two constant" approximation properties of the choleste-
This is the simplest possible approximation conserves the main properties of the which the coupling between UHAC structure, e.g.
8 (z) and o(z) . The modulation wavelength increases slowly with increasing sample thickness starting from about 100 pm at d = 40 pm and reaching about 150 pm at d 90 pjn. The amplitude of the modulation goes from zero at small thickness to about 45 degrees at d = 90 p.m. This modulation seems to be superposed on
the UHAC structure.
2. Discussion. 2.1. UHAC structure.- Even in this relati-
assumptions  :
375 The numerical results are presented in figure 3 for some values of the dimensionless parameter Q and for k=l (the approximate value for our mixture). One sees that the large Q-values favour a planar alignment in the bulk of the sample. A characteristic feature of the UHAC structure is the coupling between 9(z) and 8(z) given by equation (3). As a results the experimentally obtained value for (p d carries some information about 8(z). This coupling explains the displacement of the bands on figure 2c. In fact, as our mixture has negative dielectric anisotropy At = the electric field applied 0, z-axis favours the planar alignment of the molecules. The equation for 0 is
second assumption is the strong Our anchoring. It can easily be satisfied using a sample with large d and small qo. In the of weak case anchoring if d depends also on the surface anchoring energy on both plates and therefore it is possible in this way to these energies. From figure 3 one measure with increasing Q the torque see that can d8/dz decreases on the planar side and slightly increases on the homeotropic one. the independent deThis property enables termination of both surface anchoring energies from the dependence of y d on Q. This treated will be matter rigorously in a forthcoming paper. In conclusion to this
shall demonstrate that Yd is to the polarization rotation. The criterion for wave guide regime being  21t6n > N l-P we have (close to dz the homeotropic plate) :
0.2 is the birefringence of the À and is the wavelength of the light beam. The thickness of the region in which the criterion fails is
0 c-’- 0
(see equation (4)) ::
3.- Numerical results for 8 and 9 functions of z/d for k = 1.
and k’ =
case N -
30x10’4 cm ;
k - 1 and we have 1.5 pm ; rad. This error is reasonable for the present qualitative work, but should be accounted for if the UHAC structure is used to measure the surface energy or elastic anisotropy, especially in the case of weak anchoring on the planar side of the cell.
69d - 0. 4
briefly discuss the case assumptions are not statified. If Ki X R3(as in most liquid crystals) one can obtain numerically o as a function of Q and of the elastic anisotropies k =(Kl-K2)1 We
The is equation (3) unchanged, but o(z) would also depend on E due to the coupling with 8(z). At large E values 0 is different from n/2 only in a small region near the homeotropic side. The rotation of the polarization will be 9 d z Q, which results in displacement of the bands to the left side. This approach enables the direct determination of Q and therefore of the natural cho-
from the correspon-
equilibrium equations. The comparison with the experimental data enables the determination of k and k’ without application of an external field, i.e. in a more direct way than in most known methods for determination of the elastic constants.
MHAC structure. - To obtain the strucof our sample in the MHAC region we need to minimize the elastic energy for the case when 8 and (P are functions of all the coordinates. The equations of state in this case are very complicated and here we shall not try to obtain the general solution (a separate paper on this matter being in progress), but shall give only an idea of the type of deformation involved. Further for
simplicity constant :
376 2 suggests that 0 and y are of the solutions for the UHAC and the small pestructure riodical functions (x,y,z) and p and The wavevectors of 1 p depend only on the thickness, are parallel to one another the xy-plane. The angle a betand lie in ween the wavevectors and the x-axis is also a function only of the thickness. It is to consider a sample with consconvenient tant thickness and to rotate our coordinate system around the z-axis at angle (x so that the wavevectors lie along the new x-axis. In the new coordinate system the polar anand gle is unchanged the azimuth iso + oc. (x,z) p Without great loss of generality we can take and x-dependences of (p1 as pure sine
BO(z), q)o(z) 81 81
6=6 (z) + 81(x,z) (po (z)
elliptical cone whose axis is the -r no corresponding to the UHAC
The semiaxes of the basis of the the amplitudes T(z) and F(z). In is fact this structure only the leading and term in the general x-dependence of with ql = or The terms other q2 = q>,. mq 2 = 2,3,...) can be obtained from no (m,n the terms of order higher than two in the been free energy, which have neglected in structure. cone
equation (7). now Let us briefly consider the z-dependences of the amplitudes T(z) and F(z) in equation (6) for the structure in figure 4. The boundary conditions shown (8) imply that the amplitudes can be expanded in the following Fourier series :
dephasing function of
is in general
81(z) - 82 (z) z
shall see later role in the problem. as we
Under our assumptions free energy density is :
sin 2 is the ) of the UHAC structure and the subs-
F0 = (0’ 0 z - qo
80) K /2
We neglect the terms of order higher than two as and are sup1 posed to be small. The first order terms x
Fig. 4.- Deformation in a thin slab of hybrid aligned cholesteric in the case of modulated HAC structure.
also neglected because in the following averaging on x they vanish. are
Unfortunately in our case we cannot expect that the amplitudes are pure Fourier compothe conditions (9) cannot In fact, nents. is a all z-values as a. be satisfied for is z-dependent. So we have and constant
gain in the twist energy in the slab zi corresponding to cos (P (z1 ) - ix) > at loss some and z2 corresponding to
0. In order to have a (x) than the lower energy MHAC structure with ensures UHAC one, we need a solution which to overcompensate the gain in the loss in the the -region, for example an cos
((p ( z2 ) -
A possible gain in the cholestericlike twist energy of the modulated structure is related to the terms linear in q . After averaging the free energy with respect to x these terms do not vanish only if qi - qz . If we consider a slab of the sample at constant the free z, energy of this structure is minimized at :
resulting deformation MHAC structure
(P (z) - oc) 81, (z) t--- (Po (z) - ac
". 2 We see (z) 2 the minimifrom the above discussion that the free the MHAC zation of energy of demonstration of its and the structure stability is a difficult problem which is beyond the scope of this letter. sing
slab is -
presented in fiure 4. The director
amplitude which is multiplied by cos
n descrithe surfa-
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