Light scattering on liquid crystals

Transcription

1 UNIVERSITY OF LJUBLJANA FACULTY OF MATHEMATICS AND PHYSICS DEPARTMENT OF PHYSICS Jadraska Ljubljaa SEMINAR Ligh scaerig o liuid crsals Maej Pregelj Meors: prof. dr. Mari Čopič dr. Mojca Vilfa Lj Absrac I am goig o describe he behaior of ligh whe passig hrough a liuid crsal (LC. Udersadig his pheomeo is impossible wihou kowig liuid crsal s properies. Therefore I will eplai basic opical properies of liuid crsals such as he ide of refracio ad iroduce he Frak elasic cosas. I will describe he flucuaios of he direcor i he emaic liuid crsal ad relae hem o he scaerig of ligh. A he ed a eample of ligh scaerig eperime will be gie.

2 Coe 1 Iroducio Liuid crsallie phase.1 Molecular properies 4. The refracie ide 4. The elasic cosas 5 Thermal flucuaios i emaic liuid crsal 6 4 Ligh scaerig o he flucuaios i emaic liuid crsal 9 5 Ligh scaerig eperime 11 6 Coclusio 1 7 Bibliograph 1

3 1. Iroducio Liuid crsals are uiue as he ehibi properies of boh solid crsal ad isoropic liuid [1]. A liuid crsal ca flow like a ordiar fluid while oher properies such as birefrigece remid us of a crsallie phase. These mied properies make liuid crsals so ieresig o obsere ad sud. 10µm Figure 1: Liuid crsal as see hrough he polarizig microscope[8] I he liuid crsallie phase he elogaed LC molecules ed o be aliged alog he preferred direcio oherwise he behae jus like a ordiar liuid. Thermal flucuaios ha occur i he order of LC molecules ifluece he dielecric properies of LC ad coseuel srogl scaer icide ligh. Tha makes ligh scaerig oe of he mos used mehods for obserig ad ideifig hese flucuaios which is he ke o comprehedig he properies of LC elasic cosas iscosi flucuaios relaaio imes ad also he surface ieracio of LC ad he aligig subsrae.. Liuid crsallie phase Liuid crsallie phase is a phase bewee he solid crsallie sae ad isoropic liuid sae of maer. Wha eables he formaio of such phase is he shape of he liuid crsal molecules which are elogaed ad herefore srogl aisoropic. I coras o he isoropic liuid ha has o orderig o macroscopic scale liuid crsallie phase ehibis orieaioal ad i some cases parial posiioal order. I his respec wo mai groups of liuid crsallie phases are disiguished. The closes o he isoropic liuid is he emaic phase. I his phase he molecules are radoml disribued ad show o posiioal order while he log rage orieaioal order of log molecular ais is prese. Some pes of LC hae bewee he emaic ad solid sae oe or more differe smecic phases showig liuid srucure wih boh orieaioal ad parial posiioal order. Orieaioal order of molecules i a emaic phase is far from perfec. I a uiaial emaic phase he direcio alog which he elogae molecules ed o be aliged is called direcor wih legh 1. I should be saed ha is o a proper ecor due o he molecular smmer oe ca o disiguish he eds of he molecule. I a acual sample he direcor orieaio ma ar i space ad ime as i is iduced b he boudar codiios ad possibl b he eeral fields.

4 Molecular orieaio deiaio from he local direcor is described wih scalar order parameer S. I measures he aerage deiaio of he molecules log ais awa from he direcor. I is defied as: S 1/ <( cos θ - 1>. (1 The agle θ is he agle bewee he isaaeous direcio of he molecule s log ais ad he direcor. The epressio is aeraged oer all he molecules i he esemble. I a ideal emaic wih all he molecules aliged S 1 ad i isoropic liuid S 0. Bu i he real liuid crsal perfec orieaioal order ca o be achieed herefore alwas holds S < 1. Because he rasiio from isoropic liuid o emaic phase is a firs order rasiio he order parameer abrupl leaps from S 0 o S 04 a rasiio emperaure T N. I he icreases wih decreasig emperaure o S 06 a phase rasiio i o solid ha is agai of firs order where S agai rapidl chages o 1 [1]. I smecic phases more rasiios ca be oed. (a (b (c Figure : The hree phases of a codesed maer (a crsal (b liuid crsal ad (c isoropic liuid.1 Molecular properies Liuid crsals are made of orgaic molecules (Fig.. A pical LC molecule is srogl aisoropic ad highl polarized due o bezee rigs which are i a majori of LC molecules. The form a rigid ceral par o which fleible chais are aached a he eds. Oe ca compare his shape o a m log cigar (some hae a shape of a disk wih a legh o widh raio of approimael fie. Figure : Liuid crsal molecule 5CB. Mos widel used LC wih he liuid crsallie phase a room emperaure. I is used i ma displas ad oher deices.. The refracie ide Opical ais i he uiaial emaic is gie b he orieaio of he direcor herefore i has wo pricipal refracie idices o ad e. The ordiar refracie ide o is obsered whe he polarizaio is perpedicular o he opical ais ad eraordiar ide e holds for a parallel polarizaio. 4

5 I he case ha he agle bewee he direcor ad he elecric wae ecor of he propagaig ligh is arbirar he refracie ide for eraordiar beam is: 1 ( e cos o si e ( Usig subscrips ad _ _ for he direcios parallel ad perpedicular o he direcor i ca be wrie: o _ _ ad e. The leel of birefrigece is ha gie b: e o _ _. ( I pracice we usuall fid > _ _ ad aries from zero o abou 04 which makes liuid crsals srogl birefrige. Sice is proporioal o he scalar parameer S i depeds o he emperaure. The birefrigece is mos appare whe obserig he colors of liuid crsal uder a polarizig microscope (Fig. 1.. The Frak elasic cosas I a emaic liuid crsal eis o permae forces opposig he displacemes bewee pois like i solids bu here are resorig orueses which direcl oppose orieaioal disorios. If differe orieaios of are iduced a differe par of liuid crsal deformaio i he direcor field will occur. These modificaios of ake place oer he macroscopic disaces (few micros ad are easil obsered opicall. Disorios cos eerg ad his is he basis of he coiuum heor of he emaic liuid crsal which is aalogue o classical elasic heor of a solid. I he coiuum heor deailed srucure o he molecular scale is o impora herefore he defecs i he order parameer S are egleced. The liuid crsal is described wih he spaiall depede orieaio of he direcor (r ad he emperaure depede order parameer S(T. Wih hese assumpios i a uiaial liuid crsal a each poi r he direcio of preferred orieaio is gie b (r whereas sig of has o phsical sigificace. We assume ha (r aries slowl ad is defied a oher pois b coiui. A each poi r we iroduce a local Caresia coordiae ssem wih z ais alog he direcor. Assumig ha free-eerg desi f is a uadraic fucio of he liear compoes of he curaure srais i is relea o cosider also higher order erms. Howeer he ca be omied b akig io accou he smmeries of liuid crsal. As he free eerg is gie b F f dv (4 erms of f which hae he form of u where u(r is a arbirar ecor field ca be rasformed io surface iegrals (Gauss heorem ad herefore omied cosiderig he properies of a bulk emaic liuid crsal. Also he erms which coribue ol o surface eerg ca be omied. I he ed we ge he fudameal euaio of coiuum heor for emaics free-eerg []: F ( [ ] [ ] 1 1( ( ( dr (5 The cosas 1 ad are called he Frak elasic cosas. The correspod o he hree fudameal deformaios of direcor show i Fig. 4. The firs erm describes he spla deformaios where 0. The secod erm is associaed wih he wis disorios ad he hird correspods o he bed deformaios. The eperimeall obsered sabili of a uiform direcor paer demads ha 1 ad are all posiie wha correspods o he 5

6 idea ha he presece of deformaio icreases free eerg. The Frak elasic cosas deped uadraical o he order parameer S [4] ad coseuel icrease wih decreasig emperaure. Figure 4: a spla deformaios b wis deformaios c bed rasformaios Achieig he euilibrium cofiguraio of a amaic ssem is performed as miimizaio of free-eerg desi wih respec o he ariaios i he direcor (r. The Euler-Lagrage euaios ca be wrie as: F F λ ( (6 i j r i h i j ( j whi i beig he i-h compoe of he direcor i beig he i-h Caresia coordiae ad λ he Lagrage muliplier. Here was iroduced he molecular field h which s direcio is i euilibrium parallel o he direcor.. Thermal flucuaios i a emaic liuid crsal So far he molecular damics ad ime depedece of he direcor field has o bee cosidered. Moios of he LC molecules rage oer a wide freuec rage. There is er fas (10-9 o s roaioal ad rasiioal moio of a sigle molecule. There are desi flucuaios of he sample due o he molecular raslaioal mobili which are also prese i isoropic liuid bu he weakl scaer ligh. Specific for he liuid crsal are spaiall correlaed collecie orieaioal flucuaios which appear a lower freuecies are specific for liuid crsallie phase ad are herefore of grea imporace. The iclude he direcor flucuaios describig local flucuaios i he orieaio of he direcor ad he order parameer flucuaios describig he flucuaios i magiude of S. I a emaic liuid crsal he direcor flucuaios are more impora as he are easier o ecie he he flucuaios i order parameer. The laer should be ake i o accou ol close o he emaic o isoropic phase rasiio herefore far from rasiio S ca be assumed a cosa. I he followig par he hermal flucuaios of he direcor i he bulk emaic LC are goig o be furher discussed. 6

7 7 Due o hermal eciaios he acual direcor i he bulk emaic deiaes from he euilibrium orieaio. Tha is wh i ca be wrie as a sum of he euilibrium direcor 0 ad he spaiall ad ime depede erm: (r 0 (r δ(r. (7 The δ(r represes he deiaio of he direcor from euilibrium alue ad is epeced o be small δ << 1. To simplif higs le us assume ha he aerage direcor orieaio i a emaic sample is i z direcio 0 ( The flucuaig par has he i he firs approimaio ol wo o-zero compoes δ ( 0 for he ariaios z parallel o he euilibrium direcor orieaio are of higher order ad ca be egleced. The flucuaioal par of Frak free eerg is [1]: dr z z F 1 1 (8 The mos coeie wa o aalze he flucuaig uaiies (r ad (r is o epad hem i erms of plaar waes wih wae ecor ir e r ( ( (9 ad similarl for (r. The sum is oer all possible wae ecors limied wih he size of he sample o oe side ad wih he reuiremes of he coiuum limi o he oher. Hece oe fids: ( ( z z ( (10 ad likewise for. Figure 5: The ew coordiae ssem: e 1 ad e are iroduced ad he flucuaios are decupled io wo eigemodes 1 ad For each i is suiable o roae he coordiae ssem z aroud he z ais so ha he ew ais coicides wih ui ecor e which is perpedicular o z ais ad o wae ecor. The ais ha coicides wih e 1 which is perpedicular o e ad o z ais. I his ew ssem we hae 0 ad flucuaio eigemodes ( alog e ( 1 (Fig. 5. The flucuaio eigemode 1 ( describes a combiaio of spla (Fig. 6a ad bed (Fig. 6c flucuaios ad he eigemode ( a combiaio of wis (Fig. 6b ad bed (Fig. 6c.The free eerg ca be ha wrie as: [ ] V F ( ( ( ( _ _ (11 where _ _ ad z.

8 Figure 6: Two flucuaio eigemodes are a combiaio of spla ad bed flucuaios ad a combiaio of wis ad bed flucuaios. For simplici he flucuaios are ploed separael: a spla flucuaios where he wae ecor is parallel o he flucuaig par δ ad hus perpedicular o 0 ; b wis flucuaios where he wae ecor he udisored direcor ad he flucuaig par are perpedicular o each oher; c bed flucuaios where he wae ecor is parallel o he udisored direcor 0 I should be oed ha a his poi free eerg is sill ime depede. Oce he hermal aerage of he eigemodes ampliudes ( is assumed his depedece fall off. To calculae he ampliude of he eigemodes oe ca use he euipariio heorem which saes ha for a classical ssem i he hermal euilibrium he aerage eerg per degree of freedom is eual o ½ k B T where k B is he Bolzma cosa. We ca assume ha eer eigemode eciaio o is ow coss ½ k B T. As a resul he hermal aerage of ( is: ( 1 V k B ( _ _ T 1. (1 This is he ceral euaio of flucuaio heor for emaics []. The correspodig relaaio imes τ ( depedig o he wae ecor ad polarizaio mode ca be obaied b balacig he iscous ad he hermodamic orue of he molecular filed [1]: η h λ (1 where η beig he effecie roaioal iscosi ad λ he Lagrage muliplicaor. Takig he io he accou free-eerg (E. 5 ad calculaig he molecular field (E. 6 he euaio of moio for he direcor filed ca be wrie (o simplif: 1 : η ( (14 Iserig E. 7 liearizig for small flucuaios ad solig he damic euaio for icompressible emaics he relaaio rae for gie eigemode ca be obaied []: 1 _ _ 1 (15 τ ( η ( where η are effecie iscosiies. Wih he use of he euipariio heorem a epressio for he ariace of hermal flucuaios of was accomplished. 8

9 4. Ligh scaerig o he flucuaios i a emaic liuid crsal Le us ow clarif how he flucuaios of ifluece he opical dielecric esor ε ad coseuel scaer icide ligh. The elecric field of he icide ligh beam ca be wrie: ik0r iω Ei ( r i E0e (16a ik ' r ' iω Es ( r ' fa( ( E0 / R e (16b where E 0 is he ampliude ad i a ui ecor perpedicular o he igoig wae ecor k 0 defiig he direcio of polarizaio. The elecric field E i (r iduces i he sample he polarizaio P(r ε 0 [I ε(r]e i (r. Our ieres is o kow he flucuaig par of scaerig field E s (r i some disa poi r r R wih R beig large compared wih he sample size r. The polarizaio of he ougoig wae wih wae ecor k R is gie b ui ecor f ad relea par of he dielecric opical esor is he: ε ( 1 if f ε i ε ε i f (17 where i i ad f f are he compoes of alog he wo polarizaio direcios. The flucuaig par of he opical dielecric esor accordig o he E. 7 euals: δε if ε [ i ( f δ f ( i δ ] (18 where i 0 i ad f 0 f. If δ is spli upo he eigemodes 1 : δ( e 1 1 ( e ( (19 he flucuaig par of he opical dielecric esor ca be wrie as: δε if ( ε ( ( i f i f 1 (0 1 where i i e is he compoe of i alog e ad likewise for he f ad k k 0. We hae show ha he orieaio flucuaios couple ia he opical dielecric esor o he icide elecric field ad ge a flucuaig polarizaio P(r f ε 0 [1 - δε if (r] E i (r. We ca assume ha eer flucuaio represes ad elecric dipole P(r oscillaig a a paricular agular freuec. Summig coribuios oer all sample olume ad resricig ourseles o large R >> r he euaio for a differeial cross-secio per ui solid agle of he ougoig beam ca be wrie usig E.1 ad E.16b as []: dσ dω A( εω V 4πc εω Vk 4πc B 1 T ( i ( 1 ( _ _ ( i f i f f i f. (1 The polarizaio facor i he E. 1 shows ha maimum iesi of scaered ligh is achieed wih crossed polarizaio of icide ad ougoig ligh beams wha is corar o he siuaio i isoropic liuid where scaered ligh has he same polarizaio as he icide. 9

10 Figure 7: Scaerig geomer where opical ais is z ais i ad f are polarizaios of icomig ad ougoig beams e 1 ad e LC eigeecors k 0 is icomig ad k ougoig wae ecor φ scaerig agle ad _ _ k - k 0 Le us ur o eample show i Fig. 7 where he icide ad he scaered beam are boh perpedicular o z he opical ais of he emaic. The icide beam is liearl polarized i he scaerig plae while he scaered beam is polarized alog z ais. Deoig he scaerig agle b ϕ we hae k si( / 0 1 ϕ i cos( / i si( / 1 ϕ ϕ i f f 0 f 1 1 ( Iserig subsiuios io E. 1 gies: 4 dσ ε ω k 4 dω 16π c cos ( ϕ / si ( ϕ / B T co ( / ϕ 1 1 /. This relaio ca also be obaied eperimeall ad used for obaiig 1 / [] (Fig. 7. Similarl we ca obai he raio / b sudig he depedece of cross-secio o he agle bewee ad he opical ais. ( Figure 7: Deermiig he 1 / raio b erapolaig he sraigh lie [] where i i ( 1 ad _ _ is par of he wae ecor perpedicular o z ais e 10

11 5. Ligh scaerig eperime Figure 8: Eperimeal seup: icomig laser is polarized P focused wih les L1 ad direced oo sample S. The scaered ligh passes aalzer A ad is direced oo deecor D hrough a focusig les L ad he opical fiber. The correlaor (ALV-5000 calculaes auocorrelaio fucio g ( [5] The ampliude of he scaered ligh is ime depeda due o he flucuaios i he opical dielecric esor (E. 1 which are relaed o he orieaioal flucuaios of he direcor. The mos reliable wa o sud he damics i he liuid crsal ad o obai iformaio o he flucuaio specrum of he sample is b measurig he auocorrelaio fucio of he scaered ligh. The ormalized iesi auocorrelaio fucio g ( ( is defied as [6] r r ( r I( ' I( ' g ( r (4 I( ' where I( is he iesi of he scaered ligh proporioal o E s ( E sa ( where E sa deoes he elecric field of he saicall scaered ligh. I geeral wo differe regimes of measureme ca be cosidered: homode ad heerode. I he former case he iesi of scaered ligh b flucuaios is much larger ha he scaered ligh which is idepede of ime. I he laer case he iesi coribuio of damicall scaered ligh is small E s ( << E sa ( 1 ad as he coribuios are saisicall idepede he auocorrelaio fucio is * r r Es Es ( r ( ' ( ' iω g ( 1 R e r (5 Esa ( ' wherer deoes he real par. The secod erm i E. 5 is proporioal o he damicall scaered ligh ad i is i his case small compared o 1. Eperimeall his happes whe he ampliudes of flucuaios are small or if he amou of scaerig media is small (e.g. hi cells. This regime ca also be achieed b spliig he ligh beam io wo direcie oe beam o he sample ad ha correlae i wih he direc oe. Usig he E. 5 ad E. 1 he auocorrelaio fucio of he ligh scaered o flucuaios i a emaic liuid crsal ca be calculaed: g ( r εω E0 ( i f i f ( 1 Vk r BT Esa ( 4πc R 1 _ _ / τ ( e. (6 ( From E. 6 ca be see ha i properl chose scaerig geomer ol oe flucuaio mode is obsered: 11

12 g ( r ( E0 dσ / ( 1 e r τ (7 Esa ( R dω where depeds o chose eigemode. This meas ha b measurig he auocorrelaio fucio he relaaio imes of flucuaio eigemodes ad he differeial cross-secio per ui solid agle of he ougoig beam ca be obaied. As a resul we ca obai elasic cosa raios elasic cosa o iscosi raio /η ad heir hermal depedece. I has bee proe ha he mehod is paricularl suiable for hi liuid crsal samples up o a few micros. I such samples his mehod is for eample used o measure he scaed ligh specrum he sregh of achorig eerg which deermies how srog he liuid crsal molecules are achored i he direcio imposed b he surface. 6. Coclusio I his semiar I hae passed o a brief heoreical eplaaio how does he icide ligh scaer o liuid crsals. I order o achiee ha I had o gie eplaaio of he basic liuid crsal properies. I hae also shed ligh o he ligh scaerig eperimeal seup ad describe some applicaios of his eperime. Ligh scaerig o liuid crsals is he basic pheomeo which helps us o disiguish he liuid crsals from he isoropic liuid. The ligh scaerig eperime is mos commol used o deermie he relaaio imes of differe flucuaio modes of LC. Usig hese daa elasic cosa raios elasic cosa o iscosi raio /η ad heir hermal depedece ca be deermied. 1

13 7. Bibliograph [1] P.G. de Gees The Phsics of Liuid Crsals Claredo Oford 1974 [] W. H. de Jeu Phsical Properies of Liuid Crsallie Maerials" Gordo ad Breach New York 1980 [] I. Haller ad J. D. Liser Temperaure depedece of ormal modes i a emaic liuid crsal Phs. Re. Le. ol. 5 pp [4] I. Haller ad J. D. Liser Ligh scaerig of a emaic liuid Mol. Li. Crs. ol. 1 pp [5] M. Vilfa Ifluece of Surface Properies o he Flucuaio Specrum i a Liuid Crsal Disseraio Ljubljaa 001 [6] B. J. Bere ad R. Pecora»Damic ligh scaerig Doer 000 [7] hp:// [8] hp://bl.colorado.edu/lc/flc.hml [9] hp:// [10] hp://www-sci.sci.ku.l/rim/lcl/research/chiral/chiral.hm 1