Freedericksz transition threshold in nematic liquid crystals filled with ferroelectric nanoparticles

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1 Feedeicksz tnsition theshold in nemtic liquid cystls filled with feoelectic nnopticles.yu. Reshetnyk, S.M. Shelestiuk nd T.J. Sluckin b Physics Fculty, Kyiv Ntionl Ts Shevchenko Univesity Pospekt Glushkov, 368 Kyiv, Ukine phone : , fx , e-mils: eshet@iop.kiev.u, segi@u.fm b School of Mthemtics, Univesity of Southmpton, Southmpton SO7 BJ, UK phone: , fx: , e-mil: T.J.Sluckin@soton.c.uk Abstct A key liquid cystlline popety fo electo-optic pplictions is the Fedeiks theshold electic field. Thee hs been ecent expeimentl inteest in liquid cystl-bsed colloidl suspensions in which the colloidl nnopticles both possess pemnent electic poliztion nd povide stong diecto nchoing on the pticle sufce. Such suspensions e sometimes known s Filled iquid Cystls. Ou clcultions suggest, in qulittive geement with expeiment, tht filling the nemtic liquid cystl with feoelectic nnopticles cn significntly decese the electic Fedeiks tnsition theshold field. Keywods: liquid cystl suspensions, Fedeiks tnsition, feoelectic pticles

2 . INTRODUCTION In ecent yes thee hs been much inteest in the physics nd possible technologicl pplictions of colloidl suspensions in liquid cystl host []. As ely s 97, Bochd nd de Gennes [] pointed out tht if the colloidl pticles possess pemnent mgnetic moment, then the oienttion of the locl moments nd the nemtic diecto would be coupled, nd this would give ise to some new nd inteesting physicl effects. These systems e known s feonemtics, nd futhe wok, both expeimentl nd theoeticl, seems to confim the pictue pedicted by Bochd nd de Gennes [3-5]. A system nlogous to this, but in which potentilly much moe dmtic effects might be expected, involves feoelectic the thn feomgnetic colloidl pticles. Such systems hve been fbicted by Reznikov et l [6], who hve shown tht t low concenttions, t lest in some cses, these submicon colloids ppe simil to pue liquid cystl. In pticul, the colloidl pticles do not sctte light in the mnne tht we might expect, nd e theefoe invisible. The pticles e engineeed so s to ncho the liquid cystl diecto stongly t the sufce. It is this mechnism which seems to pemit the intinsic popeties of the colloidl pticles to influence the liquid cystl mtix. A futhe unexpected obsevtion is tht in these suspensions the Fedeiks tnsition theshold voltge deceses. If this phenomenon could be eplicted, thee would be significnt implictions fo the mnufctue of vey low powe liquid cystl displys. In this bief epot we pesent vey simple theoeticl model fo electic Fedeiks tnsition theshold in the feoelectic liquid cystl suspension. The theoy uses the following miniml postultes: () the colloidl pticles do not ffect the elstic nd dielectic popeties of liquid cystlline host; (b) ech pticle possesses pemnent poliztion which cn only be pllel o ntipllel to the locl C diecto; (c) thee is no diect inte-pticle intection; (d) the field-induced poliztion of the colloidl pticles cn be neglected. Condition (b) is sometimes known s stong diecto nchoing in elted contexts. Hypotheses (-d) cn be elxed in pinciple, nd should be

3 elxed in complete theoy. Nevetheless they contin sufficient physics tht they my be egded s sensible stting point. Ou simple theoy pedicts tht filling the nemtic liquid cystl with feoelectic nnopticles my significntly decese the electic Fedeicks tnsition theshold voltge.. BASIC EQUATIONS We conside unifom colloidl suspension of feoelectic submicon pticles embedded in liquid cystlline host. The liquid cystl is plced in cell of thickness with identicl homogeneous (i.e. pln nd unifom) boundy conditions t ech wll. A voltge is pplied coss the cell. We ssume the coupling between the pemnent poliztion of the colloidl pticle nd the liquid cystlline diecto to be sufficiently stong tht the pticle oienttion is esticted to diections pllel o ntipllel to the locl diecto. The theoy is n extension of the bsic ides used by de Gennes nd cowokes [], nd is elted to n nlogous theoy fo feonemtics [5], with n dditionl compliction of locl field effects. A fee enegy functionl fo the suspension is minimized with espect to elevnt vibles. The functionl includes elstic, electic nd entopy tems nd tkes the fom: whee F = Fel FE Fent, () F el ( = K( div n) K ( n culn) K 3[ n culn ] ) dz, FE = D Edz P E dz, F ent kt = f v ( ρ ln ρ ρ ln ρ ) dz. The mening of the quntities in eq.() is s follows

4 Definition Desciption D = ˆ E Electic displcement due to liquid cystlline effects lone. D = D P E λ = λe Totl electic displcement, including both colloidl nd liquid cystlline effects ocl electic field felt by colloidl pticle, nd which entes pticle pemnent poliztion expession ocl field coection fcto P = fdn ( ρ ρ ) Pemnent poliztion pe unit volume in the colloid d f ρ, ρ v K i Pemnent poliztion of pticle Pticle volume fction in the suspension Fctions of feoelectic pticles oiented pllel nd ntipllel to the locl C diecto espectively Pticle volume Fnk-Oseen elstic constnts of liquid cystl Tble : Menings of quntities used in this ppe. The poblem hee is to minimize the functionl () subject to the constint of given voltge pplied coss the cells: = Supposing the nemtic to be n idel dielectic, we wite div D D = o z = z E z ( z) dz () (thee is no dependence on x nd y coodintes.) Solving this, we obtin D = D = const. ikewise, combining z the conditions x y cul E = the boundy conditions E = E = t the cell wlls, yields the intui- tively obvious E =,, E z. ( ( )) Minimizing functionl () with espect to ρ ± subject to the constint ρ ρ = yields:

5 ρ± = exp ( ± λdvβ( E n) ) ( ) exp exp, (3) ( λdvβ E n ) ( λdvβ( E n) ) hee β =. kt Now we conside the sply Fedeiks tnsition in the electic field. In this cse the diecto field is given by n = (cos θ,,sinθ ). Fom Tble, the dielectic displcement cn be witten s: ( ) z( ) ( ) D= sin θ E θ fd ρ ρ sinθ (4) Combining expessions (3) nd the fee enegy functionl () gives the following explicit expession fo the fee enegy functionl (), s functionl of the locl diecto θ (z) nd the locl () electic field z : E z ( cos 3sin ) ( sin ) z ( ) K θ K θ θ θ E θ F = d f z, (5) ln exp( λdvβez( θ) sinθ) exp( λdvβez( θ) sinθ) βv whee eq.() connects the field E z ( z) to the voltge constint. In wht follows we ssume tht feoelectic pticles e smll enough, so tht λ dvβe <<. Then the totl fee enegy functionl (5) to be minimized becomes F = ( Kcos θ K3sin θ) θ ( sin ) ( ) θ Ez θ dz const %, (6) ( ) λ λ fd vβ whee % = is the effective dielectic function nisotopy. Using eq. (4) we ewite the constint () in the fom D = ( sin θ ) λfd vβ sin dz. (7) θ Fo smll ngles θ we hve:

6 E z ( θ ) The fee enegy functionl simplifies to: K D D λfd vβ ( ) λ λ θ. (8) fd vβ θ, (9) F = θ dz const subject to the constint: = D λfd vβ θ dz. () Tivilly, eq. () lso gives n expession fo the displcement field s function of nd the diecto pofile: λfd vβ D = θ dz () We now substitute D into the fee enegy functionl (9) nd minimize the totl fee enegy functionl with espect to the ngle θ. The coesponding fist integl of Eule-gnge eqution hs the fom: K θ ~ θ = const ; () the dditionl conditions on the mid-plne of the cell, θ = θ mx nd θ =, e due to the cell eflection symmety of the cell with espect to z = /. The stndd method of solving such poblems is to chnge the integtion ove the vible z in () to integtion ove θ using eq. (). The esult is the following fomul fo the pplied voltge: = K ~ D θ mx λfd vβ θ θ dθ mx θ

7 whee D is given by () with the sme chnge in the integtion vible. The Fedeiks tnsition theshold voltge is obtined by tking the limit θ mx. It is esy to evlute the integls, find the limit nd obtin: K K th = π = π (3) βλ ~ fd v ( λ ) We note tht, s we expect, in the bsence of feoelectic pticles, nd we then ecove the clssicl fomul fo Fedeiks tnsition theshold voltge [7]: f = clssicl th K = π (4) 3. ESTIMATES We use expeimentl dt fo feoelectic pticles of Sn P [6] diluted in the C mixtue S6 ZI48 (Meck) to mke some estimtes. Reznikov et l [6] epot tht thei Sn P colloidl S6 pticles possess spontneous poliztion of 4 µc/cm t oom tempetue. Fo the liquid cystl mixtue ZI48, the dielectic function nisotopy C = 5.. Substituting these pmetes clssicl into eqs. (3) nd (4) yields /.7 fo feoelectic liquid cystl suspension with th th 6 3 pticle volume fction f =.3 %, pticle volume v = m, nd supposing locl field coection fcto λ =. CONCUSIONS In this ppe we hve pesented simple theoy of the Fedeiks tnsition in nemtic liquid cystl filled with feoelectic nnopticles. In ou simple model, in the feoelectic liquid cystl suspension, the dependence of the ngul distotion on the electic field is simil to tht in the pue system with effective dielectic function nisotopy given by ( ) λ λ fd vβ % =. The theoy is consistent with the expeimentlly obseved decese of electic Fedeiks tnsition theshold

8 voltge [6]. We shll pesent elsewhee moe complete genel effective medium theoy fo feoelectic liquid cystl suspension. This theoy tkes ccount of the shpe, polizbility nd locl field nisotopy ssocited with feoelectic pticles. ACKNOWEDGMENT This wok hs been ptilly suppoted by the Royl Society of ondon though joint Anglo- Ukinin coopetion gnt. REFERENCES [] Stk, H., Phys. Rep. 35, (). [] Bochd F. nd de Gennes P.-G., J. Physique (Fnce), 3, 69 (97) [3] ing, B.J., Chen, S.-H., Phys. Rev. A, 39, 44 (989). [4] Buylov, S.., Zdoozhnii,.I., Pinkevich, I.P., Reshetnyk,.Yu. nd Sluckin, T.J. JMMM, 5, 53 (). [5] Zdoozhnii,.I., sil ev, A.N, Reshetnyk,.Yu., Thoms K.S. nd Sluckin, T.J. Euophysics ett. (to be published 6) [6] Reznikov, Yu., Buchnev, O., Teeshchenko, O., Reshetnyk,., Glushchenko, A. nd West, J., Applied Physics ettes, 8, 97-99, (3) [7] de Gennes, P.-G., The Physics of iquid Cystls, Clendon Pess: Oxfod, (974)