5
I:"JVESTIGACIÓN REVISTA MEXICANA DE FíSICA ~7 (3) 2sl-~85 Asymmetric electrooptic response in a nematic liquid crystal Constanta Dascalu Departmellt of Plzysics, "Polite/mica" Unll'ersity o/ Buclzarest Sp/aill/ /ndependelltei 3/3. R-77206. Bllclzarest, Romania Recibido el 14 de julio de 2000; aceptado el 21 de noviembre de 2000 JUf'IO 2001 An asymmctric elcctrooptic response in ncmatic Iiquid crystal (LC) has becn obtaincd. The Iiquid crystal hybrid ccll was made by using a standard eonfiguration. Onc ofthe ITO (lndillm Tin Oxide) clectrodes was covered with a surf;::.ctant.which induces a homeotropic alignment. The second of the indium tin oxide e¡ectrades was covered by a thin layer of photopolymer. which was previously mixed with an acid. which favours a process of release of protons. Such catioos are responsable of eleclrochemieal process in the Le leading to an asymmetric electrooptic response, which depend on the polarity of the applied e1eclric field. This faet is due to an internal field. which change the effective vohage thresholds for the reorienlation of lhe liquid crystal. During the anodic polarisation. the optical switching is inhibited beeause lhe effective field dccreases bclow the threshold valuc. On contrary for lhe opposite polarisation the elTective field is enough to determine a homeorropic alignment. Keywords: Liquid cryswl; photopolymer; electrooptieal response Se ha obtenido una respuesta electro-óptica asimétrica el1 cristales líquidos nemáticos. La célula híhrida de cristal líquido fue construida utilizando una configuración estándar. Uno de los electrodos [TO fue cubierto con una película delgada de material orgánico para inducir una alineación homeótropa. El otro electrodo no fue cubierto con una película delgada de fotopolímero anteriormente mezclada con un ácido para favorecer la emisión de protones. Estos cationes son responsables del proceso electroquímico en LC. conduciendo a una respuesta electro-óplÍca asimétrica que depende de la polaridad del campo eléctrico aplicado. Este efecto es originado por un campo interno que cambia el umbral efectivo del voltaje para la reorientacion del cristal líquido. Durante la polarización anódica. la conmutación óptica se inhibe debido a que el campo efectivo disminuye abajo del valor del umhral. Por el contrario. para la polarización opuesta el campo efectivo cs suficiente para determinar un alineamiento homeótropo. Descriplores: Crislallíquido; fOlOpolímcro; respuesta electroóptica PACS: 61.30.Gd; 61.30.Cz 1. Inlroduction The liquid crystal state is a state of mattcr in which the elon- gated (typieally eigar-shaped) moleeules have orientational order (like erystal) but laek positional order (Iike liquid). The liquid erystals (LC) already oeeupy a permanent place as electrooptical materials for the display of information. su eh as digital watches and c1ocks. pocket calculators. measure- ment instruments, etc. The liquid erystals are widely reeogni. sed as one of the best elcctrooptical material s for the screens of personal computers in their eompact, "Lap-Top" version. The liquid erystals can be applied in field eontrollable light guides [l,2J. The switching times of liquid crystal wave gui- des. based on nematic Iiquid crystals, are in the range of hu n- dreds of microseconds to milliseconds, since the thickness of the film are ehosen to be less than 10 11m.Furtber progress depends on the development ofnew electrooptical effects and optimalliquid crystal materials. Because of their anisotropic nature. the liquid crystals can be arranged to serve as wave retardcrs, or polarization ro- lators. Liquid erysta!s can be therefore used as eleetrieally controlled optical wave retarders, modulators, and switchers. These devices are particularly useful in display tcchnology. Elcelrooplie effeels in Le are eilher odd (mainly linear) or even (mainly quadratic) with respcct to the external elee- trie field t31. The nematie liquid crystals are centrosymmetric and nematic ceH with symmetric boundary condition has the same syrnrnctry. Sueh cells exhibit only the quadratíc aspect in field eleetrooptie effeets. In the speeial case of a hybrid cel! with asymmetric boundary conditions (the planar orien- tation on the plate and homeotropic on the other limiting pla- te), the flexoclectríc polarization [4] and linear electrooptic effeets [5, G], typieal of polar strueture, may be observed. But the electrooptical response is too weak to be used in practical application. Recently {7] an interesting polarity sensitive electroopti- cal effeet has been observed in nematie liquid erystal-photo- polymer mixture where a gradient of polyrner network den- sity is ereated by strong absorbed ultraviolet light. In the present work, we show another possible way of creating strong electrooptical asymmetry in the liquid crystal eell in a special nematie liquid erystal bybrid eell. A tbin layer of photopolyrner. which was previously rnixed with an acid, eovered one of the ITO eleetrodes. The seeond ITO eleetrode was covered with a surfaetant, which induces a homeotro- pic alignment. The cel! was oriented in such a way that has maximum transmitted light on a microscope stage between cross polarizers. \Vhen a square electric field is applied with E > E tlu (he transition lo homeotropic alignment occurs in the Le eell, but only in tbe case of eathodie polarization of photopolimer layer side of cel!. When an anodic polarization is applied a maximum transmission of Iight occurs. In the

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Page 1: Asymmetric electrooptic response in a nematic liquid crystal · 2008-07-05 · Asymmetric electrooptic response in a nematic liquid crystal Constanta Dascalu Departmellt ofPlzysics,

I:"JVESTIGACIÓN REVISTA MEXICANA DE FíSICA ~7 (3) 2sl-~85

Asymmetric electrooptic response in a nematic liquid crystal

Constanta DascaluDepartmellt of Plzysics, "Poli te/mica" Unll'ersity o/ BuclzarestSp/aill/ /ndependelltei 3/3. R-77206. Bllclzarest, Romania

Recibido el 14 de julio de 2000; aceptado el 21 de noviembre de 2000

JUf'IO 2001

An asymmctric elcctrooptic response in ncmatic Iiquid crystal (LC) has becn obtaincd. The Iiquid crystal hybrid ccll was made by using astandard eonfiguration. Onc ofthe ITO (lndillm Tin Oxide) clectrodes was covered with a surf;::.ctant.which induces a homeotropic alignment.The second of the indium tin oxide e¡ectrades was covered by a thin layer of photopolymer. which was previously mixed with an acid.which favours a process of release of protons. Such catioos are responsable of eleclrochemieal process in the Le leading to an asymmetricelectrooptic response, which depend on the polarity of the applied e1eclric field. This faet is due to an internal field. which change the effectivevohage thresholds for the reorienlation of lhe liquid crystal. During the anodic polarisation. the optical switching is inhibited beeause lheeffective field dccreases bclow the threshold valuc. On contrary for lhe opposite polarisation the elTective field is enough to determine ahomeorropic alignment.

Keywords: Liquid cryswl; photopolymer; electrooptieal response

Se ha obtenido una respuesta electro-óptica asimétrica el1 cristales líquidos nemáticos. La célula híhrida de cristal líquido fue construidautilizando una configuración estándar. Uno de los electrodos [TO fue cubierto con una película delgada de material orgánico para induciruna alineación homeótropa. El otro electrodo no fue cubierto con una película delgada de fotopolímero anteriormente mezclada con unácido para favorecer la emisión de protones. Estos cationes son responsables del proceso electroquímico en LC. conduciendo a una respuestaelectro-óplÍca asimétrica que depende de la polaridad del campo eléctrico aplicado. Este efecto es originado por un campo interno que cambiael umbral efectivo del voltaje para la reorientacion del cristal líquido. Durante la polarización anódica. la conmutación óptica se inhibe debidoa que el campo efectivo disminuye abajo del valor del umhral. Por el contrario. para la polarización opuesta el campo efectivo cs suficientepara determinar un alineamiento homeótropo.

Descriplores: Crislallíquido; fOlOpolímcro; respuesta electroóptica

PACS: 61.30.Gd; 61.30.Cz

1. Inlroduction

The liquid crystal state is a state of mattcr in which the elon-gated (typieally eigar-shaped) moleeules have orientationalorder (like erystal) but laek positional order (Iike liquid). Theliquid erystals (LC) already oeeupy a permanent place aselectrooptical materials for the display of information. su ehas digital watches and c1ocks. pocket calculators. measure-ment instruments, etc. The liquid erystals are widely reeogni.sed as one of the best elcctrooptical material s for the screensof personal computers in their eompact, "Lap-Top" version.The liquid erystals can be applied in field eontrollable lightguides [l,2J. The switching times of liquid crystal wave gui-des. based on nematic Iiquid crystals, are in the range of hu n-dreds of microseconds to milliseconds, since the thickness ofthe film are ehosen to be less than 10 11m.Furtber progressdepends on the development ofnew electrooptical effects andoptimalliquid crystal materials.

Because of their anisotropic nature. the liquid crystals canbe arranged to serve as wave retardcrs, or polarization ro-lators. Liquid erysta!s can be therefore used as eleetrieallycontrolled optical wave retarders, modulators, and switchers.These devices are particularly useful in display tcchnology.

Elcelrooplie effeels in Le are eilher odd (mainly linear)or even (mainly quadratic) with respcct to the external elee-trie field t31. The nematie liquid crystals are centrosymmetric

and nematic ceH with symmetric boundary condition has thesame syrnrnctry. Sueh cells exhibit only the quadratíc aspectin field eleetrooptie effeets. In the speeial case of a hybridcel! with asymmetric boundary conditions (the planar orien-tation on the plate and homeotropic on the other limiting pla-te), the flexoclectríc polarization [4] and linear electroopticeffeets [5, G], typieal of polar strueture, may be observed. Butthe electrooptical response is too weak to be used in practicalapplication.

Recently {7] an interesting polarity sensitive electroopti-cal effeet has been observed in nematie liquid erystal-photo-polymer mixture where a gradient of polyrner network den-sity is ereated by strong absorbed ultraviolet light.

In the present work, we show another possible way ofcreating strong electrooptical asymmetry in the liquid crystaleell in a special nematie liquid erystal bybrid eell. A tbin layerof photopolyrner. which was previously rnixed with an acid,eovered one of the ITO eleetrodes. The seeond ITO eleetrodewas covered with a surfaetant, which induces a homeotro-pic alignment. The cel! was oriented in such a way that hasmaximum transmitted light on a microscope stage betweencross polarizers. \Vhen a square electric field is applied withE > Etlu (he transition lo homeotropic alignment occurs inthe Le eell, but only in tbe case of eathodie polarization ofphotopolimer layer side of cel!. When an anodic polarizationis applied a maximum transmission of Iight occurs. In the

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282 CONSTANTA DASCALUn

case 01' ¡¡nodie pol¡¡rization a prqccss of interc<!lation of posi-tive hydrogen ion s (protons) wilf inhibit the s\vitching to lhehOlllcotropic statc (from nov..' on we will consider the polari-zation with respcct ro polymcr side of the ceH). The strongdcpcndence of the rcsponse on the sign of E represenls lhemain difference 01' the new NLC cel! as compared with theusual Nl.C cells, where the switching depends on El.

2. Thcorctical considcrations

The liquid crystals are anisotropic fluids. [n the nematic I¡-quid crystals. the molecular positions show only short-rangcarder in aH directions, but the rnolccules posscs a long-rangcorientational order of their long axes, which trend to be para-Hel to a comOlon axis called thc director n.

In slllectic ¡iquid crystals lhe rnolecules arc parallel, bUItheir cenlres are slacked in parallel Iayer ,,-'ilhin they haverandolll position, so that they havc posilion arder in only onedimensiono The cholesleric phase is a distorteJ form of thenernatic phase in which the orientation undergoes helical ro-talion about on axis.

In order to investigate the anisotropy of the properties ofLe anJ the charactcr 01' their c1ectrooptical behavior, it is es-sential to assign a definite oriental ion to thcir molecules atthe boundary walls of the eell.

The orientation of the molecules on the surface is charac-terised by t\VO parameters: the average angle (Jo of the 1110-

lecules to the plane 01' the surfaee and the O1nchoring energyH'.~. Using the angle 80 we can distinguish various orien-tations: homeotropie (00 = O). planar (00 = ,,/2) and tilted(O < 00 < 1[/2).

Mechanical rubbing of the surfaee of polymer thin layer,whieh eovered the eleetrode plate, produces aplanar orienta-tion (8). A good planar orientation of the NLC was obtainedby UV-illumination of photopolymer eoated substrates wit-hout m~ehallieal treatment [9].

The most popular technique for the homeolropie orienta-tion is utilisation of surfactants, all orienting monolayer fromthe solulion, by polymcrisation of the organosilieon films di-rectly onto the surface. The treatmcnt of lhe walls with 01'-

ganoll1~tallic eomplcx also produces a stable homeotropieorientation in nematic Le.

The orientation of the 1l10lecules at a given angle to thesurface is achieved using laycrs of SiO produccd by obliqueevaporation at a very large angle (800 _900) belwccn the nor-mal to the surfaee and the direetion to the so urce [10].

The liquid crystals used to make electrooptic devices areusually of suffieiently high resislivity that lhey ean be regar-ded as ideal dieleetrie materials. Beeausc of the clongatcdshape of the constituent molecules. and their ordered orien-tation, the Iiquid crystals have anizotropic diclectric proper-ties with uniaxial symmetry. Thc cleetric pCI,nitivity is Eu foreleetric field pointing in the direetion of the moleeules andE.L is the eleetrie pal11itivit)' in the perpendicular directionLe for whieh E

U> E.L (positive uniaxial), are usually selee-

ted for e1ectrooptic applieations.

•E

FI(jUHI: l. The lllolcL'uk of a PO~ilivc uniaxialliquio rryqal rola(C~ano ;~ aligncd with (he c1eclric tic lo.

e

I ~FIGUHE 2. SL'hclllc of the ceH: (3) BK7 glas~: (o) ITO ekclrode~:(e) pholopolymer film; (d) surfaL'lant for horncotropic align: (e) Ji-quid cryslal (SeB) Iayer.

\Vhen a steady (a low frequeney) eleetric field is ap-plied, electric dipoles are induecd and as the result the clec-trie forces exert torque on the Illolecules. The molecules ro-late in a direction sueh that the free electrostatie energy,-!E7D- - - 1 { El + E' E2}.. .. d (h2 - 2" Eu 1 Eij 2 + E.L 3 ,IS IlllnlllllZe, e-

re El, E2, and E';l are eomponents of E in lhe direetion ofthe principal axes). Sinee En > E.l' for a given direetion ofrhe electrie field, the minimum energy is achieved when lhemoleeules are aligned with the field. so that El = £2 = 0,E = (O, O, E), and the energy is lhen ~£IIE'. When the align-ment is complele the molecules axis points in the direction ofthe eleetrie field (Fig. 1). Evidently. a reversal of lhe elee-trie field produces the same molecular rotation. An alternati-ve electric field generated by an ae voltage also has the sameeffeel.

In 1110stpractical applieations and when exarnining Iiquiderystals. sandwieh-type eells are used. A flat eapillary with athiekness of (8-30 lim) is formed from two glass plates witheonductive transparcnt eleetrodes coating (ln203 01' ITO).

3. Expcriment and resull~

As it is shown in Fig. 2, in our investigation special hybridaligned nematie liquid crystal ceUs have been used. In thehybrid eell, the Illoleeules have homeotropic order on oneboundary and planar order on lhe seeond boundary. The ho-meolropie alignment on the one boundary was obtained witha solution (1: J 000) of oetadeeyltrimelhoxysilane.

Thc seeond boundary pI ate was eovered by spin coatingwith a solution eomprised a 90/10 mixture of prepolymeraerilat de butyl and acid itaeomic. The mixture was dopedwith 1% wt. of a photoinitiator. After spin eoating lhe pre-polymer f11m was irradiated for 20 minute by nonfiltred xc-non UV light. The distance between (he lamp and plate was60 em.

Rel'. Met. FiJ . .17 (3) (200 [ ) 2KI-285

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283

CN

ASYM'IETRIC ELI'CTROOPTIC RESPONSE IN A NEMATIC L1QUID CRYSTAL

\Ve made the hybrid eell wilh a lhiekness of 10 11m(usingMayler spaeers), and a slandard sandwich eonfiguration [11].The liquid eryslal sen (see Fig. 3) provided by Merek hasbccn used.

The dieleetrie anisotropy, 6£ = é, = + 9.9 (al roomtemperature, 20°C), being EM and E.L the dielectric conslantsparallel and perpendicular lo the director n, respectively. The

FIGURE J. Liquid Cryst~l 5CB [K 24 N 35.3 11 proviJl:u hy~lcrck. \I,!here K. N. 1 represenl crystallinc. ncmalic and isolropicphasc respectivel}'.

(a) (h)

FIGURE ..L Typical AFf\.1 ¡muge of (he pothopolymcr layer slIrfacc. (a) Top vicw; (h) isomctric vi c\\'.

FIGURE 5. Imagcs ObSCfVCJtu CCD camera: (a) ¡¡nodie polaril;¡.tion of phOlopolyrncr layer; (b) calhooic polaril.<llion of the photo-polymer layer.

cell was filled with 5CB in its isotropic phase. Since no oricll-ting procedure was applied to the plale with lhe polymcr layerand the other plate can induce a homcotropic orientation. bel-ween cross polarizcrs \Vecan sce a degenerale planar oricn-talion.

The optical obscrvation \Vas made by lIsing a polarisingmicroscope (Axioskop, Zciss) to observe Ihe alignrnenl oftheLC between cross polarizers. The polarising microscopc wasequipped with an image acquisition systcm consisting of aCCD Colour Camera connected to a pe and a videorecorder.A function generator provided square wave voltage pulses ofamplitude belween 0-10 V and various frecuencies. The mea-surements of lhe discharge clectric currcnt in the cell havebeen made.

(a) Inl

By using AFt\1 (Atomic Force Microscopy) techniquc thestructure of photopolymer surfacc ¡ayer have becn analysed.As can be sccn in Fig. 4 the surface of photopolimcr la-yer shows a cuasiplanar characler on a smal! scale explorcd(lIP lo S /"n). For lhat, the liquid erystal moleeules will behaving a degencrate planar alignmcnt at the interfacc LC-pholopolyrner layer.

The application of a low frequcncy clcctric field (squarewave shape) perpendicular to the elcctrode surfaces inducesan asymmetric optical switching as it can be seen dircctly inlhe video-microscope images shown in Fig. 5. Inilially. thecel! was oriented in such a way to have maximum transmil-tcd light on a microscope stage bctween cross polarizers. InFig. 5a. when Ihe calhodic polarization of the squarc elec.lrie field is applied of the photopolimer layer side of eel!,the black image shows a homeotropic alignment of Ihe mo-lecules inside of lhe cel!. The imagc is black. bccause theoptical observation is made between Ihe cross polarizers. andthe transmission lhrough a homeotropic alignment cell gocspraclically lo zero.

In roigo5b. when the anodic polarisation of the sqll<lreeleetrie field is applied of lhe photopolymer layer side of ee-11,for a very short time one can sec a strong diffusion of lheoplieal field and a deerease of the eleelrie field in the bulkbellow lhe threshold force the illlage lo become bright. Thedegenerated planar lcxture in lhe video image is due to an in-ternal field, which change the effective voltage threshold furthe reorienlation of the Le. This inlerna! field appears becall-

Nn'. Me\'. Frs. ~7 (3) (200 1) 281-285

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284 CONSTANTA DASCALU

., ,..,_.~--,-.,...,-.,.-~..,..,..•..t., .'...1..' ..•.• -."""1"" .., ..,. ""'--'--'.--'-"""'.-""."1'" "'-'--'--"--"-'--'1--'--<'

cum:ntvoltage

14.000 tirtp.t1 tilo"

"

-6.000 ti 4.000 s.Ayg _ 2.00 "13\10'

1 2.00 VI 2 200 mVI-18.6000 mv 0.00000 v

FIGURE 6. Voltagc form and currenl oscillograms laken at [he vol.'"ge amplitud 2 V (IDO mllz) for an 10 ¡,m eell (SeS).

se the external posilive field will push out the protons whichwere released by pholopolymer layer inlo the LC. This beha-vior was obtain for frequencies in the range 50 mHz-l kHz,and for square electric voltage in the range, 3-9 V.

From lhe electric measurernents we can scc that lhe asym-metric electrooptic behavior of our ceU is connected withasyrnmetry of discharge electric current. Ir lhe amplitude ofthe square eleetrie voitage is low, for example 2 V (f'ig. 6),the cel! exhibits a homeotrotic texture for both. anodie andcathodic polarization. The discharge current have lhe sameamplitude which means thal lhe electric field licd is lo low loextraet a significant number of the protoos fmm photopoly-mer layer.

\Vilh the inerease of the eleetrie voitage lo 3 V overo(Fig. 7b), the diseharge eurrent following lhe anodie poia-rization of plate with photopolymer. is larger (in absolute va-lues) than the diseharge eurrent whieh follows the eathodiepolarization (Fig. 7a).

L1'-

. ",2

1 5,00 VI 2 soo mvl016,0000 rnV 0.00000 V

1 5.00 VI 2 500 mVI46.0000 mV o .00000 V

-1 0001 s -80.00 U$_Ayg~ 200 ms/OlYVp-p ( 1) 7.81250 V

-1 0001 5 -eo.oo US_':;vg ~ 200 ms/OlvVp-p ( 1 ) 7el250 v

999.9:- msr¡>pe \ 1 \ l"''?

, .

r

",1

999_9: msrepetlll ~

W (WFIGURE 7. Voltage form and currenl oscillograms t<lkenal the voltagc amplitud 8 V (lOO rnHz) for an 10 11m ceH (5eR): (a) polarity (-) 3lpholOpolymerlayer; (b) polarily (+) at pholopolymer layer.

In the case of anodie polarization the eleetrie field insideof eell is positive, oriented from pholopolymer layer towardthe liquid crystal interface, whereas for the cathodic polariza-tion it is negative. The positive field will tend to aUract elec-trons from the ITO-liquid crystal interface and also to pushprotons from the pholopolymer layer into the Iiquid eryslal.80th currents having the same direction give birth to a ratherstrong current, which inhibits the homeotropic alignment.

If we nO\\-'have the cathodic polarization. the electric fieldwill tend lo altrael eleetrons from pholopolymer layer but lhe-se cannot be found in it. also ITa cannot injeet positivc char-ges into the liquid crystal because ITa is n-type semiconduc-tor. The electric field will induce a homeotropic alignment in-lO the cell. Such a homeotropic texture is black bctwccn theerossed polarizers.

Similar measuremcnts made with a hybrid eell filIed wilhSeB but where the photopolymer was not mixed with an acid

Ibefore the UV exposurc. shown. of course. a complete sym-metrical optical and elcctrical response with rcspcct to thepolarity of c1ectric field.

4. Conclusions

\Ve have cxamined the asymmetric electrooptic response ofa LC layer in a hybrid eell. One of lhe eleetrodes was eove-red with a thin film of photopolymer mixed with an aeid, amaterial tilat relcases protons. Such eations are responsibleof clcetrochemical process in the Le leading to an asymme-trie elcctrooptic response. whieh depend on the polarity ofthe applied eleelrie field. This faet is due to an internal field,which change the effective voltage thresholds for the reorien-tation of the liquid crystal. During the anodie polarisation.lhe oplieal switehing is inhibiled beeause lhe effective field

Re•..Me.<. Fís. 47 (3) (2001) 2HI-2HS

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ASYMMETR¡C ELECTROOPTIC RESPONSE IN A NEMATIC L1QUID CRYSTAL 285

decreases below ¡he threshold value. On conlrary for the op-posite polarisation the effective field is enough to determinea homeotropic alignment.

In ¡he similar liquid crystal cell, but where the photopoly-mee is nol mixed with the acid, the aplical response is qua-

1. 1'.G. GialJorenzi, l.A. Weiss. and J.P. Shcridan, 1. Appl. Phys.47 (1976) 1820.

2. C. Veilleux. J. Lapierre. and J. Bures. Opl. 1£11. 11 (1986) 733.3. L.M. Blinov and v.G. Chigrinov. Eleclrooplic l:.1fects in Liquid

C')'stals Materia/s, (Springcr. Ncw York, 1993).4. L.A. Bcrcsnev el al .. Pis 'ma Zh. Eksp. Tear. Fiz. '¡S (1987) 592.5. 1. Dosov, Ph. Martinot-Lagarde, and G. Durand, 1. Phys. (Frw¡.ce) 1£11.44 (1983) 817.

6. R. Barberi and G. Durand, Handbook. o/ Liquid Cr)'stal Resear-cit. editcd by P. Colling and J. Patel. (Oxford Universily Press.Oxrord. 1997).

dratic l't'rSUS the exciting field and no asyrnmetric behaviorwas observed.

Further studies are planned. involving other mixtures ofphotoplymer and acid to optimise lhe in¡eraction with ¡he Ji-quid crystal films.

7. R. Bartolino el al.. J. Appl. ['[¡ys. 85 (1999) 2870.

8. S.A. Pikin, Slructural Tralls/onnalion in Liquid Cr)'srals. (Gor-don and Brcaeh. New York. t 991).

9. 1\1. Sehadt. K. Sehmitt, V. Kozinkov, and V. Chigrinov, J,m. 1.Appl. ['[¡ys. 31 (1992) 2155.

10. W. Urbach, M. Molx. and E. Guyon, Appl. [,hys. Lell. 25 (1974)479.

11. L.M. Blinov, EleClrooptical ami Magneto Optical Properties o/l.iquid Crysrals. (\Viley. Ncw York, 1993).

R", Mex. Fis. 47 (3) (2001) 2RI-285