I~VESll( iACIÚ~ REVISTA MEXICANA OE FISICA.U 15l4lí6--H19

A modcl for c1cctrorhcology in polymeric li(lUid crystals

OCTUBR.E IY9X

R.F. Rodrígucz and J.F Call1i.lchoDepartalJlcnto dc Fúinl (jl/lmica, Instituto de Fúiul. lJ"iI'Cf.üdad NacioNa/ ,\utÚf/01/W de México

Apartado postaI20-36.J. 0/000 México. D.F. A1e.ric(1

Rl'dhiJo el 2 de julio dI.' 19()X:aceptado d 24 de agoslo de IIjI)X

\Ve lk\(~I(lr <In anntYlical moJel lo ••.lUdy lhe response 01' a polymerir lIematir I¡quid crystal lo a dc c1ectri(' lIeld. \Ve show that ror thellllal .\talionar\' ...¡nte where the inJlIced reorientation 01' the director has OL'Cllrl'¡J.an aligncd ...Inl(ltm' wlllt a !!reat!y enhanced \'i"'CClSlty(elcl'll"(l[heohl~ic;¡1 cffect) i••.produced. Tite ••.cope and limilJliollS 01'the model and mc[hods employed are abo di ••.cu ••.sed.

A'n'\I'(!1~1.I: ElcL'lrorheological effecl; IIcmalics: Iyotropics

Se propone un modelo analítiro para Jcscrihir la respuesta eleclrnreológicl de una pdícula de crislallíqllido nem,ílico polimérico. Se muestraque Cll el e....lado tinal estacionario alcanzado después de que orurr¡ó 1<1I'curiclllaci6n. aparece una ~...lrllclllra alineada del nell1~ílico para lal'U<l1LJ ,i ...cn....idad oricntacíonal aumellta hasta en tres órdenes de magnilud. dando lugar al dedo c1cctlnrcológico. También se disculen lo....~llcann's y limitaciones del modelo. así (omo las de los mélodos lltili/auos

/)ncriplon'_c [rC(to c1cctroreológico; ncmálico: liotrópico

p.\es: hl.30.-\': hl.30.Gd 62.10.+ ....

1. Introductioll

In Ihe lasl decadc- there has heen a grcal dcal oI' interes! in theuse of rheological materials 10 manufacture a large varic-ty 01'clectrorheological (ER) deviccs. Thesc ll1alerials are essel1-lialy Illlitb which are ill1heJdeJ with particulate solids whichreacl to ;In ('IcClrical field hy prouucing drall1alic and signili-rant changes in t!leir \"iscosily and olher material propcrties.f\.1ore precisely, an ER fluid undc-rgoes a lransilion from a liq-uid :-.I;lIL'i!lID a viscoelastk solid-Iike state upon the applica-lion nI' sll"Ong c-Icctric fields (typically lIleasured in l06Y/Ill),

which is charactcrized hy a large reversihle increase in vis-cosity 111. The usual systellls Ihal exhihit Ihis ER Iransiliollare cOJl1Jlosed 01' individual particles Ihat hecol11c polarizcuhy Ihe applied tield and align Ihel1lselvcs into chains anu tJl-amenls. Tllis lilamentary slruclure O(curs. for example, in(OIKclltrated sllspensions, in soliJ particles in a dielcclric1l1l'diuIll or in sOll1e special polYIllt:ric fluids [2], and is re-spolIsihle 1'01"the gelation-Iike Iransition mentioneJ ahovc.

¡\clually. sorne 01' lhese fealUres are also presenl in ne-11Iatic [iquid crystals. These systelJls arc l1uids thal exhihillllllg-range orientational onler over distances lJlí1ny limeslarger lhan the dimensiollS 01' the lJlolceules of \vhich lheyare C<llllp(}sed [31. The inlrinsic aniS(llropy 01' lheir Illoleculesgive ....rise lo Illacroscopic properties Ihal are also anisotropicIhis is Ihe case ofthe JieleClric conslanl and Ihe magnctic sus-ceplihilil)-'. Bul this intrinsi<.: anisolropy is also responsihle forconsidcring liquid crystals as interesling l1uids for eleclrorhe-ologi<.:al applicalions. In fael. they olfer ohvious advanlagcsover the more cnnventional EH /luids, such as Ihe ahscnce 01"prohlt:lIls associated wilh lhe setlling of Lhe Jisperscd phasein Ihe usual El? systems, sincc ¡iquid erystals are hOlJlogc-fll'OllS phases. The feasihility nI' liquid cryslals to produce a

practical ER ellect has heen shown recenlly by Yang et tll. [,1 J.who have ohservcd <Inorder o magnitude incrcase 01"the vis-cosity 01' a SolUlion 01' a polYlIlcric liquid cryslal when actedupon by an external elcctric lield in a rotational rheollleler.

I{(l\\'l'\"er. in spile (JI' the Iarge \'ariety of ER Jcvices Ihatha\'c bcell in\'enled 111. OUT1I1HkrslanJing of lhe hasic I11cl'h.,misms responsiblc for the I:..'R elleel is, in general, rathapOOl".Thc hasic pllrposl' of Ihis \\'ork is lO study analyticallya simple moJel for reorienlalion 01' a Ihin mm ol' a polymericIlemalic sollllion of/JOIY(II-II£'\yl isocywwtej (PHIC) undcr aconstant electric lield, Sincc its molecular wcighl is not loolargc ( 10"), lo describe il.s dynamics we use as a f¡rst ap-proxilllalion, hydrodynalllic cqllalions 01' motion for the di-rt:ctor and lhe velocity lields which \ve have useJ previollslyror thermolropics [:)1. In descrihing Ihe realignment of thedirector, we lakc inlO ;¡ccollnl lhe l'olllhincd action nf hoth,the hacldlows \\'hich are inevilahly induceJ hy reorientation.and lhe hydrmJynamic now prodtl(cd hy an external pressuregradicllt. \Ve sho\\' thal lhe induccd reorientation produces aninnease nf almost Ihrce orders nI" magnitlldc in the apparen!visL'osity 01' PHIC which slJows the existenee orthe EN ellec!.

2. Modc) ami go\'crning C(luatiolls

Consider a quiscenl tlelllatic layer of thickness I conlaincdhel\Veen two rarallel conducting plales. as dcpicted in Fig. l.It is assullled thal the transn'rse dimensions along lhe J' anJ.tI¡¡XCSare largc comparcd [o /. Thus. Ihe cel! has a large aspeclratio hut a finilc VOllllllC \' = I>!.I. Tlle initial orit:ntalion nflhe direclor is planar. so tha[ \vhen lhe cleclric ¡¡eld E is ap-plied along Ihe:: dircction. the director íi \vill rcoricllt insidcIhL' ceH when E is largcr Ihan a nitical lIelJ El"

A tvl(1)EL FOR ELECiRORilEOLOGY IN P{)LY~1EI~IC IJ0lJII) ('I{YSTALS 4ó7

At Ihis poinl jt sh01llJ he slrcsscd once more that. slrictlyspcakill~. Eqs. (S) ami (6) provide ror a hydrodynamic dc-scriplil11l (Ir;¡ lo\\! molecular weighlllematic (thennotropics).sjnce in Ihis case Ihe director is lhe only additional hydmdy-namical \';¡riahlc. apan fmm Ihe usually conservcd variahlesof mas,'; dcnsily. spceitic cntmpy and 1ll0menlUIll densily [7],The ClllTL'sponding descriptinn for a polyrneric nematic (ly-olropie) is lIluch more complicated owing lo the largc nlllll-her of tlcgl"eL's 01 freedom Ihal may conlribute lo the dynam-io in the hydrotlynamic limil. Sincc in generallheir numhcrami nalure are not \••..ell idel1lifled. in Ihe following sections\vc shall apply Ihe rormalislll dcvelopcd in Ihis seclion to Ihepolymcrie nemalic I'HIC. \Ve helive this (o he a reasonahleapproxirnalioll si/lee I'HIC has a sufficienlly low molecularweight (...... l (J'~')and Ihe syslclll is ¡¡lways a\Vay far fmm acritical¡milll. This approximatiol1 has lile greal advanlagc ofh'cping Ihe des(,.'riplion simple cnough so Ihat an analytiraltreatlllL'nl is possihlc.

(1)

(2)

LE,

dHI- -odI ;=::I::j - .

z' I/ 2 [1 =:;;;;;::=======================::;;;;;:==::J

LP](;llI{¡\ l. Schcmalics of ti planary alingned liquid nyslallilm in[he prt'''cllc(, Oí;l ("(lIl~(antckctric tic Id.

()wing lo Ih~s(' fcalurcs of lhc ccll. it is rcasonablc loa""L1l11e"palial hOlllogcllcily in lhe .,. dircclioll. so tha! '1 =[:-;ill fI (.::. t). o, ('ose (.::. t)]. \Ve aSSlIllle ,hal lhe rcoriclllalion<Ingle H satisfics slrong anchoring conditiolls al lhe pI ates,

11is ;t1"o fe'asihle lo aSSLlIIlCIhal lhe rcsulting hyLlrodynallli-cal 1I0\\'s will (lCcur in Ihe .1'-::: planc. Thcll lhe only relevanlcompollcn! (Ir lhe vclocily I¡cld is I'Jo, which salisfics lIo-sliphOlllllbry conditions al lhe plates.

l' (: = cJ=~) = (J,./ .) (3 )

J. Slationary r~gilll~

The fillal stationary state arter the rcorientation has occurredis deflncd hy selling Ihe len hand sidcs of (5) and (6) cqual to/.ero. l\toreo\'er, ir (he external prcssurc gradient is constanl.dpld.r = 6/)/ L == ¡'OIlSt., from (6) \vc ohlain

Furthcrmorl.', if as usual lhe reoricnlalion is considered lohe ,In isothennal proccss. Ihe associated Helmholtz frec ell-l'rgy luncti(lIlal is ofllle r(mn

IJere 1'.1 is Ihe average vclocily given I1y

(X)

The varalional dcrivalive 6F/68 is given explicitly hy

\vhere (1./2. /':{. with J\ == 11/''1' denole various viscosilYcOl'fiicielHs or Ihe nem<llic. Note that rhe lasl tcrm in (6) isdlle lo an eXlernal pressun.' gradient along lhc ,1' Jircctioll,

Hcre f 11 = f IJ - f 1. stands for lhe diclectrÍC anisolropy ofIhe Ilcmatic and f'(:' f) denotes its mass density. It sl10uld heslressed Ihat in wriling this expression wc l1avc madI..' lhe as-sUl1lplion of ('qual elastil: constants rol' lhe splay, hcnd andt\\'i ..•lcIaSlic dcformalions (1\' == l\".~ :;:: !\"u :;:: /\'T).

FoJlllwing Ihc usual pmccdure to derive nematodynílmiccqualions ¡::{]. rmm (4) \vc arrive at the foltowing set 01' dy-namical equalions [o]

iJH :1 á F DI':,.=----(I-.\)"osO- (5)n: 11 líll ' iJz'

J", V, ¡)1", .\ - 1 D [ 1 liF] 1 DI'¡¡¡ = r; iJz1 + ----:¡¡;- iJz cosli Mi - P ¡J.r' (ó)

(~)

( I 1 )1f j41fJ,

E, = {V ----;;:-

",' = ~ = ¡16.I' [Vl + (.\ - 1)1,"], id /. ~" 4

Here \\'c havc introduced the dirnensionlcss variables ( ~ z/lami t :;::::1:'/ Er .• wh~re

\Vhere q i...•the mass flow rale. By solving (8) ror l!le assulllcdno-slip hOlllldary condiliolls on Vx (3) and upon sllhslilUtionnI' lllis solution inlo (S) with D8/DI :;:: O. \Ve arrive al Ihefollowing nonlinc¡¡r djmensionlcss cqualjon for Ihe final mi-cntarional slale:

fPH .---:c; + '1E' sil< ~H + ti.\' (,-os 11= (J, (1 (J)dV

is lhe critical tield [81 and is such thal reorienlation occurs irE > l. /\" stand s rOl"the claslic constanl of lhe malerial inlhe equal colHanls approximalion.

Therc are two important physical and independent paralll-l:ters in ( 1()). On lhe (lne hand. q :;::f'l £11/8rr /\'. is propor-liollal to the ratio hel\l,.'Celllhe energy of lhe incident t1eld amiIhe nematic's elastic cnergy. Thcrefore. it is a measure 01"theslrength 01"the l'oupling hetwecll Ihe exlernal lielt.l and Iheinduced oricntalional conligllralion.On lhe olher hand. Ihe

(4)

(7)

H('I'. lHex. Fi.\'. 44 (5) ( 19lJH) -lh6--4()l)

R.E RODRIGUEZ ANO J.F CAt\IACHO

e 1.5 Lr-;----=~-----

0.5

-0.04

'- ,,,I

FI(JURA 2. Oricntational configuration (J V.f, (of PHIC for dif.k1L'1l1vaJuc\ (Jf lhe nlllpling constan! (l. (-) 1/ = )(1-1: (--)(/ = 10-'2: (. _.) (/ = lO-:l; (- - _.) 1/= J()-1: (_ ... ) 11 = I()-~).

oO 20 40 50 80 100

E

p;¡ralllL'ICr .Y = liU"J-'1 (,\ - l)j 1\'1' COlllains Ihe L'lfectsdul' lo lhe hydrody'namie tlo\Vs through Ihe RcynolJs IlUIll-hl'r N = //11'.1'//':1.

Among aH lhe possihlc noncquilibrilllll statcs lhe syslcJIIlIlighl llave. \\'C only consider Ihe linal slationary stalc l.:har-al'lcril.cd hy lhe f:1L'1thal Ihe induccd llows llave praclically\'anished. Thi:-. slale is dclincd by Ihe condition .Y :::::::(l. InIhi ... l'ase (10) has till' (01'111nI' lhe cqualioll nf l11otioll for asimple pcndululIl which Illay he sol ved analyticnlly in lcrmsti!" l'Iliplie in(cgrals [!J, 101. AClually. hy dcfining H=- dO/d(;tnl!..\ =- r¡EY. (10) may be rcwrittcn as

FIGURA J. OriL'lllalion dependl'1I1 \-i~(o•.•i1y '1 :::= 1 '1 (E)11(E 1) 11',1, t!"or I'I/Ir alld for lll~ salll~ V¡dUl•..•01' fJ asin Fig. "l

4. Oricntalion \'iscosity

The \'i"icllsity 1'ulll'lion or apparent viseosily relales lileshearing forcc Ik'r unil arca to the lIIa~nillldc 01 lhe toealshcar 11:n It depends UPOll lhe orientatioll 01' the directorlhroll~h 11:\1

I/("E) = 2 "ITI;1I1 [exp(JEiE) e] -Í. (1))

1:(11"Ihe b(llll1dary cllndiliollS (1) and (2). fmm (12) il hll-lo\\"s Iha( Ho = ,,/'2 and J,' = l. Upon inlcgrating (12) \Ve Illl'l1¡¡rri\'l' al

The final slaliol1ary oricnlalional l:onfiguration H ((; J:.,')as a fUl1ction 01" lile applied field t. is ohlaineu by inven.ing (1.+). Thi~ is easily accomplished by using Ihe hyperhuticillllpliludc fUJ1ctioll or (;ut/erfll(Jllflian 1111. wilh Ihe result

I/(f)) =:-IlI(.()~:2f-} ..;ill:!f-}+IJ, + (fl:!+o:¡)sill"2H. (I())

Hcre (11, (1:!.II:~arc Lcslie \.'lIL'flicients anJ 'le is lhe Irans-verse t\-1icsowio. \'iseosity [1,11, The orienlatioll angle f) isgiven hy (15). Note that Ihis drpendellce 01' 'Ion H indicalesthatlhe SYSI!.:llIis non-Newtonian in its behavior. in lile sensclhal '1 is slrongly dependen( 0[\ Ihe driving force.

In (ln!L'r lo exllihil the electrorheological ellcCI we tirslsubstitute (15) illlO (16) and average Ihe result over Ih!.: cel!.¡.c., I"mm ( =:- -I/~ lo (, = 1/'2.This yields Ihe avcraged

apparent viscosity ,¡(E), Sel'olHlly. \ve substillllC lhe ma-

lerial paralllelers of lhe polymcric ncmalic ¡'file. Ilamely.111 = -II()() Paso 11:2 =:- -:~i()(}ras, r1;{ = :t~()Paso

'le = ~J()() I'a .. and f(J = (j.I~) X 1O-9."/V1 141 inlO the

exrressioll lor ,¡(E). IIIVe rloll ,¡(£)- ry(E = 1) I 1m lhesamc valtles uf (1 ror which H (e) \Vas calculalcd in Fig. 2.\Ve arrive at lile curves shown in Fig . .3. \Vc llave taken[\' .......,IO-11S rtll'lhc elastic l"Illlsl;lnl amll = "1X I()-l; 1Il rOl'¡he scparation dislílnce belv/l'l'n lhc plalcs: ror Ihese valuesEq.(ll))icld"F,::: IlI ..JV/lIl.

NOle lha!. indeed. Ihe curvcs in Fig. J sho\\' Iha( lhe I'HICnClIlalie solUlioll L'xhihilS a signilieallt clcctrorheologieal ef-ree!. This is cvidL'need hy lhl' sharp inL"rease nI' Lwo ordcrs 01"ll1a~nillldl' in 11ll' viscosiLy when al! elcclric tield 01'.......,20 E,.

is applicd pcrpl'ndicularly lo Ihe plalcs of (he cel!.

( 14)

( 1.1)

( 121

sin H 1sill (.~= -.-- =- - Sill f).

Slll eo J"

1 ¡." d"" 1 (11 ,,)-- --.-=--IIl/t¡ -+- ,\/'D. (1 {"osO vID . :2 -l

, (1/) =

wllL'rlo'\\'lo'havlo' inlrOllueel! the variable </> by

A plol of H I'S. (is sho\l,:n in Fig. 2 rol' dilferent valucs 01Ihe cOllpling paramelrr r¡. This shows Ihal H ((: E) is <IncvcnlUIlClioll around the middle of the samplc (( = 1/2).;¡ fae(Ihal was 10 he cxpected lo be true whcll lhe possibility 01"lorllling disclinations in Ihe nemalk has bren disrcgarded.

A ~10DEL FOR ELECTRORIIEoI.O(j"" IN POLY\lERIC L1QUIl> CRYSTALS

5. Discússiun

\Ve han: SIHl\\:n (hal <ln ckclrorhcological L'lTcct may he pro-dllccd in a small ceH nI"a Ihickncss 01' a fcw microns l'ontaill-¡ug a !'flle Ilcmalic. To <-:larify and clahoralc on Ihis rcsult\Ve c10sc t!ll.' papel' \vitll lhe following cOllllllcnls.

Firsl. Ihe model shows lhal lhe Illcchanislll Ihal produn:sIhe incrcasl' in lhe arp~IrCnl \"¡scosíly is lhe rcoriclllalioll 01'lhe Jin:clOr intluccJ hy lhe applicd c1cclric f¡cld. This (!L'.

cms hccausc the pCfmtlncnl and induccd dipolc IllOlllcnts of¡'IIIC aligll thclllsclvcs with lhe licld giving r¡se lo <In Cll-

hanccd ViSL"Osity. As a maller 01' rae!. Ihe induccd orienta-I i¡1I1aldisltlrlioll also maní fesls itselr as IHHllincar optical pnl-l'L,S. Thc cx[crnailicld givcs risc to a rcfraclion index gradi-l'1l1which produccs self-focollssing and \\'aveguiding ellcctsin lile ceH I.~)I.i\1thollgh lhe exislence 01';111 elcclrorhl'ologi-<:al L'lfcCI for I'HIC \\'as already sho\\'1l to exisl in a rolalionalrhl'olllell'r hy )(11/.1: mld Sltil1c [.1). here we havc showll lh;¡(lhe sallll' L'lll'cl can he induced in a small planar el'll \\'ithunlya kw lIlicrOIlS scraralion hClwcen ils plales.

Sccondly. il shollld he l'lIlphasized thal our analysis \Vasn:slrictcd lo lhe linal stationary statc of lhe reoricnlalion.\\'hcre thc induccd flows havc praclical1y \'anished. Ho\\'c\'cr.il is possihk to considcr 11<)\\: crfeets in tltis slationar) slale

l. R 'I;IO. (l'dilor). Ul'dmrlU'ologiml FI/lidl. (\Vmld Sl'il'lICCPuhJishl'l's. Ncw York. 1()I)2)

.) 11.Block ;md J.P. KcJly. J 1'11\'.1'.D: AI)I'/. I'hl's. 21 (19XX) Ihhl.

:1 p.ei. lk (il'llIlCS. TI//' PhYJic.\ o/ Liquid Cry.\'faIJ. (Clarl'ndoll.

Oxford. 196-l).

l. 1.1\, Y;mgaml A.D. Shillc.J RI1('OI. ](.(Il)(J2) 1079

.•. R.F. Rodrigue/. amI J.A. Reyes. 1 ¡Vol/lill. Opl. 1'11.\'.1'.¡\J1l1. •.•

119l)5) 9-l3.

(i R.F. Rodrlguez. P. Ortega. anJ R. Día/.-Urihc. f'1Iys/¡'(/ A 2.\U(1()9()) IIX.

l. R.F. RodrIgue/.. ,1Ild J.A. Reyes. Mol. ("n'.I'I. Liq. ('1'\'.1'1, 2X2(1 ()9h) 2X7.

by solving ( I ()) analytiL:ally in an approximalc fmm hy us-ing houndary layer IIlCll1ods. an c.\tcnsion which is prescnllylindel' way. FuI' (!lis CISC lhe calculation of r!lcoJogical prop-ertil's such as lhc Visl'ollH.'lric funclions can ht: carried out cx-plicilly. ¡\ 1l1orCgl'lll'ral allll physically richcr silualion wouldbe 10 l'ollsider 11J1' full El[. (10) \\'ilh Ni- O. In Ihis case. (heequ:llioll can nol be slll\'cd analylically and olle has lo resorlto the use (Jf nlllllerical lIlcthods. lJowe\'cr. in tltis case thesilllplil'Íly alllllhe analYlicallreallllenl of lhc Illodel prescnlCdhere is losl. It should be strcssed lhat in spite of its simplicilY.Ihis llludcl predicts ,\Il already cxpcrilllcntally verilied ellectl"tlra ditTcrel11 bUI ne\\' pllysicaJ siluatioll.

Finally. il should bl.' pointeJ OUIoncc lllorC that ollr theo-retic:d dt:SlTiption is appmximatl'. AII tilt: crrecls ;t'••sllL'iatedwitll IHlssiblc addilillllal hydrodYIl:llllic variables Ileedcd todcsnibe properly Ihc hydmdynal1lics of I'HIC havc hccll en-tirely Ilcglectcd, II{JwcveL (IUr dt:sl,:ription is simple. analYlicand p!':.'dicls Ihe L'Xistcllce uf;m inleresting: physical t.'Ifccl.

Al"kl)()wledgments

Fin:lllL'ial SUpporl fmm Granl ()(Jt\PA-UNAf\l INI05797.f\1c.\ico. i.s ~raterully ackno\\'ledgcd.

S. le'. 1\11\10 in I',-og,-{'\\ ill O{'lics. c:dílCd hy E. \\'011". (Norlh 1101-Iand. Alllslenblll. 1()XX) Vol. 2fl

!I. L.\1. .\lilnl'-Tl1otllson . .I(/('o/li(/1/ IJIi{'lic FllI/clÚJII.\' Tabfes.¡Don'r. :'\ew Yml-. 1t)50l.

lO. (iR. Fowks. t\/w/rticlI{ .\1{'c1l1l11in. (11011. Rincl1arl and Win-slOIl Nc\\' Yml- 1970¡.

11. I.S. (iraushteYIl :lnd 1.:-'1 Ryzhik. Jilh!es (~nllll'gral.\'. Saie.\' l/l/d{'Hldlld.\'. (Ac;lt!L'lllic Prcss. Nl'\V york. 1(65). Sec I.-l().

11. F.:-'1 Lt'slíl'. in ,\dl'llllcn ill LÚ/I/id Crysllll.\. cJl\ed hy G.IIBIOWll. (i\cadclIlic I'rl'ss. ;-\cw YOlk. 1(79) Vol.-l,

lJ. T. ClIlssOll. ¡\Jo!.Crn'l.l,Íl{. ('ry,\I. H14 (19X4) .107.

1.1 .\1 .\tíl'snwil'l .. Nallm' 17 (llJJ5) :U1I.

Un. ,\ln. rú, -1-1(5) (I()I)X) -l6h---I6t)