Univerza v Ljubljani Fakulteta za matematiko in fiziko Andreja Sarlah Vpliv ograjujoˇce povrˇsine na fluktuacije nematske ureditve v tekoˇcem kristalu disertacija Ljubljana, 2001 University of Ljubljana Faculty of Mathematics and Physics Andreja ˇ arlah Effect of the confining substrates on nematic order-fluctuations in liquid crystals Thesis Ljubljana, 2001 Na raziskovalni poti me je od vseh zaˇcetkov usmerjal in vodil prof. dr. Slobodan Zumer. Za njegove nasvete in ravno pravo mero priganjanja ter usmerjanja, s kom naj se ˇse pogovorim in kateri poster bi si bilo dobro ˇse ogledati, se mu najlepˇse zahvaljujem. Vesela sem, da sem bila od vsega zaˇcetka deleˇzna majhnih raziskovalnih skrivnosti, ki jih je z nami delil Primoˇz. Postale so ˇze kar legendarne in jih z veseljem prenaˇsamo na mlajˇse rodove. Primoˇz, hvala tudi za pripombe, ki si jih imel ob prebiranju ne ˇcisto zadnjih verzij mojih ˇclankov. Za prijetno delovno vzduˇsje ste skrbeli Anamarija, Gregor in Daniel, del ˇcasa pa tudi Jure, Primoˇz, Fahimeh in Matej. Daniel in Grega, hvala za vse naˇse pogovore, zaradi katerih je bil marsikateri ne nujno raziskovalni problem videti precej manjˇsi, ali pa ga sploh ni bilo veˇc. Nenazadnje hvala vam, mami, oˇci in Katarina, ker me vedno podpirate in ste ponosni name. Hvala tudi tebi Grega, ker skrbiˇs za prijetno vzduˇsje, kadar nisem v sluˇzbi, in ˇse posebej zato, ker se te, kot praviˇs, ne da odgnati. Povzetek Delo je posveˇceno ˇstudiju vpliva ograjujoˇce povrˇsine na fluktuacije nematskega ureditvenega parametra - posredno preko spremembe povpreˇcne ravnovesne ureditve in neposredno preko spremembe robnih pogojev za fluktuacije. Posebej se posveˇcam sistemom, v katerih so v bliˇzini faznih in strukturnih prehodov pomembne tudi nedirektorske prostostne stopnje ureditvenega parametra. Za kolektivne fluktuacije ureditvenega parametra v heterofaznem sistemu — nematik v stiku z razurejujoˇco povrˇsino — je znaˇcilen mehek fluktuacijski naˇcin fluktuacij skalarnega ureditvenega parametra, ki predstavlja fluktuacije debeline staljene omoˇcitvene plasti. Nematsko hibridno celico oznaˇcujejo tri razliˇcne urejene strukture. Prehod med njimi je lahko tako nezvezen kot tudi zvezen. Posebej ˇstudiram dvoosno strukturo, v kateri je direktorsko polje nedeformirano na raˇcun staljenega reda in poveˇcane stopnje urejenosti v notranjosti celice. Strukturni prehod iz dvoosne v upognjeno direktorsko strukturo vodi mehek direktoski fluktuacijski naˇcin. Moˇcno upoˇcasnjene so tudi fluktuacije stopnje urejenosti. Spremembe urejenosti, ki so posledica ograditve, povzroˇcajo strukturni in psevdo Casimirjev privlak ali odboj med stenami. Spremenjena simetrija ureditve pa spremeni van der Waalsovo silo, ki jo tekoˇci kristal posreduje med stenama. Kljuˇcne besede: nematski tekoˇci kristali, ograditev, fluktuacije, fazni prehod, strukturni prehod, moˇcenje, van der Waalsova sila, strukturna sila, stabilnost PACS: 61.30.-v, 61.30.Cz, 61.30.Hn, 64.70.Md, 68.15.+e, 68.60.Dv, 78.67.-n Abstract The thesis deals with effects of the confining substrates onto the nematic order parameter fluctuations, both, through changing the equilibrium average order and through changing the boundary conditions. A special attention is paid to systems in which in the vicinity of phase and structural transitions certain degrees of freedom differ significantly from the ones in the bulk. In the analysis of collective excitations in a nematic liquid crystal in contact with disordering substrates, a soft fluctuation mode is discovered. It represents fluctuations of the thickness of the molten wetting layers. In hybrid nematic film, there are three possible ordered structures and the structural transition between them can be either discontinuous or it can become continuous. In biaxial structure, the director field is undistorted at the expense of the decreased order and increased biaxiality of the order in the middle of the film. At the transition to structure with bent director field, the spectrum of collective excitations is characterized by soft director fluctuations and in addition, fluctuations of the degree of order and of the parameter of biaxiality become softer as well. Surface-induced change of the order gives rise to the structural and pseudo-Casimir interaction between the confining walls and the changed symmetry of the order changes the van der Waals interaction. Keywords: nematic liquid crystals, confinement, fluctuations, phase transition, structural transition, wetting, van der Waals force, structural force, stability PACS: 61.30.-v, 61.30.Cz, 61.30.Hn, 64.70.Md, 68.15.+e, 68.60.Dv, 78.67.-n Contents 1 Introduction 13 2 Phenomenological description 19 2.1 Nematic order parameter ......................... 19 2.2 Phenomenological Landau–de Gennes theory .............. 23 2.2.1 Landau theory of phase transitions ............... 23 2.2.2 Phase transition in a nematic liquid crystal ........... 30 2.2.3 Correlation lengths of the nematic order parameter ...... 34 2.3 Dynamics of the ordered fluid ...................... 35 2.3.1 Pretransitional collective dynamics in a nematic liquid crystal 41 2.4 Forces acting on a thin liquid-crystalline film .............. 44 2.4.1 Stability of thin liquid films ................... 47 3 Van der Waals force 53 3.1 Van der Waals interaction ........................ 56 3.2 Electromagnetic field surface modes ................... 58 3.3 The zero-point energy of surface modes ................. 62 3.4 Van der Waals force in a multi-layer system .............. 64 3.5 Uniaxial Hamaker constant ........................ 67 4 Heterophase nematogenic system 73 4.1 Equilibrium profiles ............................ 78 4.2 Pretransitional dynamics ......................... 82 4.3 Structural and pseudo-Casimir forces .................. 93 4.4 Van der Waals force in heterophase liquid-crystalline systems ..... 96 5 Hybrid nematic cell 101 5.1 Equilibrium structures .......................... 106 5.1.1 Monte Carlo simulations of a hybrid cell ............ 116 11 12 CONTENTS 5.2 Pretransitional dynamics ......................... 122 5.3 Structural and pseudo-Casimir forces .................. 128 5.4 Stability of thin hybrid nematic films .................. 136 6 Conclusion 143 Bibliography 147 Razˇsirjeni povzetek Uvod Snovi v naravi obstajajo v treh osnovnih agregatnih stanjih: plinastem, tekoˇcem in trdnem. Pri mnogo snoveh meje med razliˇcnimi agregatnimi stanji niso ostre, ampak obstajajo te tudi v vmesnih stanjih. Tekoˇcekristalna faza oznaˇcuje stanje, v katerem snov teˇce kot tekoˇcina, vendar ima optiˇcne lastnosti podobne kot trdne snovi, saj je delno urejena. Tekoˇcekristalne faze so znaˇcilne za snovi, ki jih tvorijo organske molekule moˇcno anizotropnih oblik: paliˇcaste organske verige z dobro definirano dolgo osjo in diskaste molekule, katerih sestavni del je mreˇza benzenskih obroˇcev. Ureditev molekul v tekoˇcekristalni fazi je lahko odvisna predvsem od temperature — termotropni tekoˇci kristali — ali predvsem od koncentracije tekoˇcekristalne snovi v topilu — liotropni tekoˇci kristali. V doktorskem delu obravnavam prve. Najpreprostejˇsa tekoˇcekristalna faza je nematska. V njej so molekule v povpreˇcju urejene okrog doloˇcene smeri v prostoru, ki ji pravimo direktor, teˇziˇsˇca molekul pa so enakomerno porazdeljena po prostoru. Za nematske tekoˇce kristale je znaˇcilna cilindriˇcna simetrija, tako da sta smeri direktorja n in — n^ enakovredni. Optiˇcna os snovi v enoosni nematski fazi sovpada s smerjo direktorja. Pod vplivom zunanjih dejavnikov ali polj se v nematski ureditvi lahko pojavi nova znaˇcilna smer, ki jo opiˇse sekundarni direktor, simetrija pripadajoˇce ureditve pa je dvoosna. Pri ohlajanju nematskega tekoˇcega kristala lahko ta ali preide v trdno kristalno strukturo ali pa v eno naslednjih, bolj urejenih tekoˇcekristalnih faz [4,5]. Tekoˇce kristale so prviˇc opazili, oziroma o njihovem obstoju poroˇcali, ˇze davnega leta 1888 [2]. Botanik Friedrich Reinitzer je pod mikroskopom opazoval taljenje rastlinskega holesterola. Kristal se je stalil v motno tekoˇcino in Reinitzer je pravilno ugotovil, da njena motnost ni posledica neˇcistosti vzorca, ampak posebnih fizikalnih lastnosti stanja snovi. Moten videz tekoˇcih kristalov je namreˇc posledica moˇcenga sipanja svetlobe, ki se od sipanja na izotropnih tekoˇcinah razlikuje za kar do 6 velikostnih redov [4,5]. Zanimanje za tekoˇce kristale se je poveˇcalo v drugi polovici prejˇsnega stoletja, ko so spoznali, da so tekoˇci kristali snovi, ki bi se jih dalo s pridom uporabiti v i ii Razˇsirjeni povzetek industriji prikazalnikov. Raziskave so bile najprej vezane na velike vzorce tekoˇcih kristalov, v katerih so povrˇsinski efekti zanemarljivi. V njih je red odvisen le od temperature in zunanjih elektriˇcnih in magnetnih polj. Poleg ravnovesne ureditve snovi v tekoˇcekristalni fazi je vedno veˇc zanimanja deleˇzna tudi dinamika spreminjana ureditve [9–16]. Pri tem se zanimamo za kolektivna gibanja — termiˇcne fluktuacije ureditve —, ki imajo vpliv na povpreˇcen makroskopski red. Kolektivne termiˇcne fluktuacije so predvsem pomembne, ker so takrat najveˇcje, v bliˇzini faznih prehodov. Kasneje so zaradi tehnoloˇskih potreb pa tudi zaradi zanimanja za osnovne fizikalne pojave postali precej bolj zanimivi ograjeni tekoˇci kristali [8]. V njih so zaradi velikega razmerja povrˇsine glede na prostornino sistema vplivi povrˇsine na urejenost tekoˇcega kristala nezanemarljivi. Na efektivno interakcijo tekoˇcekristalne snovi z ograjujoˇco povrˇsino vpliva tako interakcija molekul obeh materialov, ki se razlikuje od interakcije med molekulami tekoˇcega kristala, kot tudi manjkajoˇce ˇstevilo sosedov. V doktorskem delu predstavljam rezultate svojih raziskav vpliva ograjujoˇce povr-ˇsine na fluktuacije nematskega ureditvenega parametra. Povrˇsina vpliva na fluk-tuacije neposredno z robnimi pogoji in posredno preko spremenjene povpreˇcne ravnovesne ureditve, ki predstavlja potencial za fluktuacijske naˇcine. Loˇcim dva primera: (i) sisteme, v katerih je vpliv povrˇsine vezan na njeno neposredno bliˇzino [15,24,23], in (ii) sisteme, v katerih povrˇsina spremeni ureditev v celotnem tekoˇcekristalnem sistemu [21,23]. V obeh primerih me najbolj zanima pojav mehkih fluktuacijskih naˇcinov, ki jih v neograjenih tekoˇcekristalnih sistemih ne zasledimo. Njihov obstoj pomeni, da je fazni oziroma strukturni prehod zvezen, medtem ko je fazni prehod v neograjenih sistemih nezvezen. Nadalje me zanimajo najznaˇcilnejˇse opazljivke v ograjenih sistemih, sile, ki jih spremenjena ureditev povzroˇca med ograjujoˇcimi stenami. Omejim se na strukturne sile, ki so posledica spremenjene povpreˇcne ravnovesne ureditve, psevdo Casimirjeve sile, ki izvirajo v spremenjenem spektru termiˇcnih fluktuacij, in van der Waalsove sile, ki so sicer posledica fluktuacij elektromagnetnega polja, a prav tako zavisijo od nehomogenosti in anizotropije tekoˇcega kristala med stenami [31]. Sile med ograjujoˇcimi stenami lahko merimo z razliˇcnimi spektroskopskimi metodami, naprimer z mikroskopom na atomsko silo [67], pri katerem prevzame vlogo ene od sten kar tipalo mikroskopa, ali preko njihovega vpliva na stabilnost tekoˇcekristalnih nanosov, ki imajo eno prosto povrˇsino [104,158]. V delu obravnavam slednji primer. V doktorskem delu najprej pojasnim nekaj osnovnih pojmov, ki jih potrebujem za opis ureditve v tekoˇcem kristalu in za opis termiˇcnih fluktuacij. Sledi definicija in opis doloˇcitve sil med ograjujoˇcimi povrˇsinami ter komentar vpliva interakcij Razˇsirjeni povzetek iii v tekoˇcekristalnem nanosu na njegovo stabilnost. Nato se podrobneje posvetim van der Waalsovi sili, kjer sledijo novi rezultati za silo med anizotropnimi sredstvi [31]. Konkretni izraˇcuni vpliva ograjujoˇce povrˇsine na fluktuacije nematskega ureditvenega parametra so vezani na heterofazni sistem nematika v stiku s povrˇsino, ki tali red [15,24,23], in na hibridni nematski sistem, v katerem je zaradi nasprotujoˇcih si vplivov ograjujoˇcih povrˇsin nematski red moˇcno deformiran [21,23]. V obeh primerih izraˇcunam strukturno silo [29], medtem ko so rezultati za psevdo Casimirjevo silo le navedeni. V heterofaznem sistemu se zaradi nehomogenosti reda v sistemu pojavi dodatna “stena” — fazna meja med izotropno in nematsko fazo. Povpreˇcno ureditev v heterofaznem sistemu renormaliziram glede na van der Waalsovo silo, ki deluje med fazno mejo in mejo med tekoˇcim kristalom in trdno ograjujoˇco povrˇsino. V hibridnem sistemu doloˇcim sile v plasti tekoˇcega kristala in ˇstudiram njegovo stabilnost napram spinodalnemu razomoˇcenju [30]. Delo zakljuˇcim s pregledom opravljenega in izpostavitvijo odprtih vpraˇsanj. Fenomenoloˇska teorija Fenomenoloˇska teorija opiˇse tekoˇce kristale v bliˇzini faznega prehoda, ko se sistemu nezvezno spremenijo nekateri termodinamski potenciali in zvezno ali nezvezno njegova makroskopska urejenost [38]. Veˇcja urejenost sistema je povezana z zmanjˇsanjem njegove simetrije. Za opis posameznega sistema je pomembna doloˇcitev parametra, ki opiˇse spremembo ureditve — ureditveni parameter. Ta ustreza fizikalni opa-zljivki, ki se ji vrednost ob faznem prehodu spremeni z niˇc na vrednost razliˇcno od niˇc. Izhajajoˇc z mikroskopske porazdelitve molekul predstavlja ureditveni parameter prvi nekonstantni neniˇcelni moment v porazdelitveni funkciji, na makroskopskem nivoju pa se odraˇza z vplivom na fizikalne koliˇcine istega ranga (skalar, vektor, ten-zor,...). Pri enoosnem nematskem tekoˇcem kristalu je f(?) = J2^Lo fnPn(cos ?), kjer je ? kot med dolgo osjo molekule in smerjo direktorja. Zaradi cilindriˇcne simetrije ureditve so vsi lihi momenti porazdelitve enaki niˇc, prvi nekonstantni neniˇcelni moment pa ustreza brezslednemu tenzorju 2. reda 1 Q = —S (3n ® n^ — I) , (1) 2 kjer je S = ((3cos2? — 1)/2) skalarni ureditveni parameter, ki meri stopnjo urejenosti glede na os direktorja [4]. Kadar so vse molekule poravnane vzporedno direktorju, je S = 1, ob popolnem neredu — enakomerni porazdelitvi molekul po kotu ? — pa je S = 0. Najniˇzja vrednost skalarnega ureditvenega parametra S = —1/2 ustreza molekulam enakomerno porazdeljenim v ravnini pravokotni na direktor. V sploˇsnem je nematski red lahko zaradi zunanjih vplivov deformi- iv Razˇsirjeni povzetek Slika 1 Shematska predstavitev odstopanj od enoosnega reda, ki ustrezajo posameznim amplitudam qm. ran. V najsploˇsnejˇsem primeru ga opiˇse 5 neodvisnih parametrov: dva kota, ki doloˇcata smer direktorja, skalarni ureditveni parameter glede na direktor, kot sekun-darnga direktorja in stopnja dvoosnosti P = (3/2) (sin ? cos 2?), kjer je ? kot glede na sekundarni direktor v ravnini pravokotni na direktor, Q = 12S (3n ® n^ - I) + 12P (e^1 ® e^1 - e^2 ® e^2). Vˇcasih je smiselnejˇsa parametrizacija ureditvenega parametra glede na bazne tenzorje brezslednega tenzorja 2. reda Q = J2i=-2qmTm, kjer so [32,33] T0 T1 = T2 = e^1 ® e^1 - e^2 ® e^2 ? 2 e^1 ® n^ + n ® e^1 ? 2 = 3n ® n^ - I ? 6 , T-1 = T -2 e^1 ® e^2 + e^2 ® e^1 v2 n^ +n^ , (2) e^2 ® n^ + n^ ® e^2 = --------------------p---------------. v2 Amplitude v razvoju, qm, predstavljajo skalarni ureditveni parameter (q0), stopnjo dvoosnosti ureditve (q1) in smer sekundarnega direktorja v ravnini, ki jo doloˇcata enotska vektorja e^1 in e^2, (q-1), ter odstopanja direktorja od smeri n v ravnini z vektorjem e^1 (q2) ali z vektorjem e^2 (q-2). Pomen posameznih amplitud je shematsko predstavljen na sliki 1. Urejenost in s tem simetrja nematske tekoˇcekristalne faze se odraˇza na simetriji tenzorskih opazljivk. V primeru enosne ureditve se tenzor magnetne susceptibilnosti zapiˇse kot 2 ? = ~?aQ + ?iI, — 3 (3) kjer je ?a = ?\\ - ?± anizotopija suscepibilnosti in sta ?\\ ter ?± susceptibilnosti popolnoma urejenega sistema v smeri direktorja in pravokotno nanj. ?i = (?\\ + 2?j_)/3 je povpreˇcna magnetna susceptibilnost oziroma njen izotropni del. Razˇsirjeni povzetek v i T > T** ji / T = T */T < T* Slika 2 Prosta energija sistema z nezveznim faznim prehodom kot funkcija ureditvenega parametra za razliˇcne temperature. Krepko je narisana prosta energija pri temperaturi prehoda, ˇcrtkana in pikˇcasta ˇcrta pa ustrezata temperaturam TNI < T < T?? oziroma T? < T < TNI. Do spremembe urejenosti pride pri faznih prehodih. Fenomenoloˇski opis faznih prehodov temelji na definiciji ureditvenega parametra in razvoju proste energije sistema po simetrijsko dovoljenih invariantah ureditvenega parametra [34,38]. Razvoj velja v okolici zveznih faznih prehodov in se ga da uporabiti tudi pri opisu nezveznih prehodov, predvsem ˇsibko nezveznih. Invariante tenzorskega nematskega ureditvenega parametra so potence operatorja sled na tenzorju. V bliˇzini faznega prehoda se gostota proste energije torej zapiˇse kot fhom =fizo + ~ AtrQ 11 - BtrQ3 + C(trQ2)2, 3 4 (4) kjer so A, B in C parametri, ki jih doloˇcimo fenomenoloˇsko. ˇ len v prvi potenci je identiˇcno enak niˇc, saj je nematski ureditveni parameter brezsleden, kar zagotavlja, da je reˇsitev Q = 0 ekstrem proste energije. Clen tretjega reda nakazuje, da bo fazni prehod nezvezen, ta potenca pa je simetrijsko dovoljena, ker reˇsitvi Q in — Q ustrezata razliˇcnim fizikalnim stanjem [enaˇcba (3)]. Kadar ˇclen tretjega reda simetrijsko ni dovoljen, je fazni prehod zvezen. Clen ˇcetrtega reda poskrbi za obstoj globalnega minimuma proste energije, viˇsji ˇcleni v razvoju pa so zanemarljivi, razen kadar je C < 0. Fenomenoloˇski parametri A, B in C so v sploˇsnem temperaturno odvisni. Parameter A mora pri T* menjati predznak, ˇce naj pod to temperaturo izotropna faza ne bo veˇc moˇzna reˇsitev. A je torej funkcija lihih potenc (T — T*), ponavadi pa zadoˇsˇca, da upoˇstevamo le najniˇzji ˇclen, A = A(T — T*). V bliˇzini prehoda lahko privzamemo, da sta parametra B in C konstantna. Za tipiˇcne tekoˇce kristale je A ~ 105 J/m3K in B ~ C ~ 106 J/m3 [4]. Prosta energija sistema z nezveznim faznim prehodom je upodobljena na sliki 2. V obseˇznem tekoˇcem kristalu je povrpreˇcna smer ureditve konstantna in se s vi Razˇsirjeni povzetek temperaturo spreminja le stopnja urejenosti okrog direktorja. Izotropna faza je stabilna nad temperaturo prehoda TNI = T? + B2/27AC in metastabilna v temperaturnem intervalu TNI > T > T?. Ob prehodu sta v ravnovesju izotropna faza in nematska faza s stopnjo urejenosti Sc = 2B/3y6C ~ 0,2 — 0,4. Nematska faza je stabilna pod temperaturo prehoda, ko se stopnja urejenosti spreminja kot S/Sc = 0,75(1 + 1 — 8?/9), kjer je ? = (T — T?)/(Tni — T?), in metastabilna za Tni < T < T?? = T? +B2/24AC (glej sliko 2). Ob prehodu iz izotropne v nematsko fazo se sprosti latentna toplota ql = B4/729C3[TNI/(TNI — T?)] ~ 106 J/m3 [147]; za primerjavo, latentna toplota ob zmrzovanju vode je ql ~ 3 • 108 J/m3. Pod vplivom zunanjih polj, najveˇckrat gre za vpliv ograjujoˇcih sten, je povpreˇcna smer nematske ureditve lahko razliˇcna v razliˇcnih delih sistema. Deformacija ureditvenega parametra zviˇsa prosto energijo. Njen prispevek zapiˇsemo s simetrijsko dovoljenimi gradientnimi ˇcleni. V bliˇzini faznega prehoda je deformacija ureditvenega parametra vezana predvsem na krajevno spreminjanje stopnje urejenosti, zato takrat ne pridejo do izraza vsi vidiki elastiˇcnosti sistema in zadoˇsˇca opis z eno elastiˇcno konstanto, 1 . fel = ~L VQ VQ, (5) 2 kjer je L =~ 10-11 N do 10-10 N [4]. Globoko v nematski fazi, ko je stopnja urejenosti pribliˇzno konstantna po celotnem vzorcu, so deformacije ureditvenega parametra vezane na elastiˇcne deformacije direktorskega polja. Te ponavadi opiˇsemo v okviru Frankove elastiˇcne teorije [42], ki razdeli prispevke k prosti energiji na prispevek pahljaˇcne, zvojne in upogibne deformacije. Spontane deformacije ureditvenega parametra ni, saj ta viˇsa prosto energijo sistema. Seveda pa je ureditveni parameter lahko deformiran zaradi vpliva ograjujoˇcih sten. Na mikroskopskem nivoju so interakcije med nevtralnimi molekulami van der Waalsove interakcije kratkega dosega (1/r6). V fenomenoloˇski teoriji jih zato ponavadi nadomestimo s kontaktnimi interakcijami. V okviru direktorskega opisa nematika opiˇse sklopitev s povrˇsino Rapini-Papoularjev izraz [49], ki sta ga za popolnejˇsi tenzorski opis prilagodila Nobili in Durand [17], 1 2 fS = — G tr (Q — QSi) ? (z — zS), (6) 2 kjer je G moˇc sklopitve s povrˇsino, QS je tenzorski ureditveni parameter, ki ga vsiljuje ograjujoˇca povrˇsina, ta pa se nahaja pri z = zS. Izraz predstavlja prvi ˇclen v razvoju proste energije zaradi interakcije s povrˇsino. Viˇsji ˇcleni ponavadi niso potrebni, razen pri opisu temperaturne odvisnosti ekstrapolacijske dolˇzine. Ta nam pove, ali je energijsko ugodnejˇsa deformacija direktorskega polja, pri ˇcemer je le-to v soglasju z redom, ki ga vsiljuje povrˇsina, ali pa je ugodneje krˇsiti robne pogoje in se pri tem izogniti zviˇsanju proste energije zaradi deformacije, ? ~ L/G. Moˇc vpliva Razˇsirjeni povzetek vii povrˇsine meri razmerje ekstrapolacijske dolˇzine in tipiˇcne dolˇzinske enote v sistemu: sklopitev s povrˇsino je moˇcna, kadar je ?/d —> 0, in je ˇsibka, ˇce je ?/d ^ 1. V nadaljevanju bomo raˇcunali z brezdimenzijskimi koliˇcinami: ureditveni parameter bomo zapisali glede na stopnjo urejenosti ob prehodu v obseˇznem sistemu, Q = Q/Sc, dolˇzine bomo merili v tipiˇcnih enotah sistema d (x^i = xi/d in ^ = dV) ali s korelacijsko dolˇzino ?NI = ?d = J27CL/B2 ~ 10 nm, temperaturo s ? = (T — T*)/(Tnj — T*) in prosto energijo v enotah f = L?N~2(2B2/27C2). Znak •"" bomo v nadaljevanju spustili. Brezdimenzijska gostota proste energije je potem 1 2 r o 1 22 1 -? trQ — V6trQ +(trQ ) + 2 2 2 1 2 r -\ 1 22 1 ^i-ri-r f = fizo+ ? trQ — v 6 trQ +(trQ) + ? VQ .VQ. (7) Povrˇsinski prispevek h gostoti proste energije je 1 2 f s = ~g tr (Q — Qs) ? (z — zs), (8) 2 kjer je g = (?N2I/Ld)G ali g = (3?N2I/2d)?_1, ˇce uporabimo zapis z ekstrapolacijsko dolˇzino. Korelacijske dolˇzine fluktuacij Fenomenoloˇska teorija faznih prehodov se zanima le za povprˇcno vrednost ureditvenega parametra in pri tem zanemarja krajevna in ˇcasovna odstopanja od povpreˇcja. Ta so predvsem pomembna v okolici zveznih faznih prehodov. Tudi v obseˇznem sistemu je ureditveni parameter odvisen od kraja, ?(f) = ?o + ?(r*), a je {?(f)) = ?o in torej (?(r*)) = 0. Tu ? oznaˇcuje vsako od petih prostostnih stopenj nematske ureditve. Vsaj na majhnih podroˇcjih so molekule vedno urejene. Pomembno pa je, kako velika so ta podroˇcja. Povezanost oziroma korelacijo v urejenosti nam opiˇse korelacijska funkcija ?(r) = (?(r)?(0)) ~ i?(r)i?(0)) = (?(r*)?(0)). Razvijemo korelacijsko funkcijo v Fourierovo vrsto po ravnih valovih, ki so v sistemih z zvezno translacijsko simetrijo naravna izbira lastnih funkcij, in dobimo ?(r*) = J2$?(q) e~iq'r, kjer je ?(q) = (|?(q)|2). Amplitude lastnih funkcij ?(q) doloˇcimo iz proste energije, = d r f = J^o + V ??2 + Lq |?(q)| + (9(|?(q)| ), (9) 2 ^ s pomoˇcjo ekviparticijskega teorema, ?(q) = ksT/V(L?~2 + Lq2), kjer je ? = J L/ (?2 f /??2) korelacijska dolˇzina danega fluktuacijskega naˇcina. V direktnem prostoru dobimo namreˇc ?(r) = jBr- e~r'^. Kadar je korelacijska dolˇzina konˇcna, pojema korelacija urejenosti z eksponentom razdalje in fluktuacije niso pomembne. V primeru neskonˇcne korelacijske dolˇzine je korelacija dolgega dosega in pojema obratno sorazmerno z razdaljo. viii Razˇsirjeni povzetek 0 -0.5 >\' fluktuacije dvoosnosti nematska faza \ izotropna faza \_ fluktuacije skalarnega ureditvenega parametra direktorske ^\ fluktuacije "\, vsi tipi fluktuacij _———---- 0.0 0.5 1.0 ? 1.5 2.0 2.5 Slika 3 Temperaturna odvisnost korelacijskih dolˇzin petih prostostnih stopenj nematskega reda v (meta)stabilni izotropni in enoosni nematski fazi. Izpiˇsimo zdaj korelacijske dolˇzine posameznih prostostnih stopenj enoosne ne-matske ureditve, ki jo opiˇse ureditveni parameter Q = a0T0, ?0" /?ni = ? - 6a0 + 6a ?±i /?ni = ? + 6a0 + 2 a = ? + 3a0 + 2a0 sš/? 2 NI (10) V izotropni fazi so ravnovesne vrednosti vseh parametrom qm enake 0, tako da so korelacijske dolˇzine vseh fluktuacijskih naˇcinov enake, ?I7 /?N~f = ?. V nematski fazi se korelacijske dolˇzine posameznih fluktuacij skih naˇcinov s temperaturo spreminjajo kot kaˇze slika 3. Ob prehodu iz izotropne v nematsko fazo ali ob prehodu v nasprotni smeri ostajajo korelacijske dolˇzine vseh fluktuacijskih naˇcinov konˇcne, kar je ˇse ena znaˇcilnost nezveznih faznih prehodov. Zato pa korelacijska dolˇzina ?I divergira na meji stabilnosti izotropne faze (? = 0 oziroma pri T*), korelacijska dolˇzina stopnje urejenosti nematske faze ?n,0 pa divergira pri najviˇsji temperaturi pregrete nematske faze (? = 9/8 oziroma pri T**). V bliˇzini faznega prehoda je ?n,0/? 2 NI 6 - 5? in ?n ±i/?NvI ? 18 - 9?, korelacijska dolˇzina direktorskih fluktuacij pa je neskonˇcna na celotnem obmoˇcju nematske faze. Dinamika ureditvenega parametra Termodinamsko ravnovesje sistema je doloˇceno z minimumom njegove proste energije, F = JdVf. V okviru fenomenoloˇske teorije ˇstejemo v prosto energijo le prispevke povpreˇcnega makroskopskega reda, ki se krajevno spreminja kveˇcejmu na razdaljah veˇcjih od nekaj tipiˇcnih dolˇzin molekul. Z minimizacijo dobimo Euler-Lagrangeve enaˇcbe ?f ?Q = 0, (11) Razˇsirjeni povzetek ix kjer je funkcionalni odvod ?/?Q = ?/?Q — V • (?/?VQ) in k f prispevajo volumski ˇcleni v prosti energiji. Povrˇsinski ˇcleni doloˇcajo robne pogoje. Zdaj pa si predstavljajmo sistem, ki ga malo izmaknemo iz ravnovesja. Takrat je ?f/?Q razliˇcen od 0 in predstavlja generalizirano silo, ki vleˇce sistem nazaj v ravnovesje. Pribliˇzevanje ravnovesju je proces, pri katerem se zgublja energija. Opiˇsemo ga z viskozno silo, ki je po analogiji z viskozno silo v mehaniki sorazmerna s hitrostjo, tokrat s hitrostjo spreminjanja ureditvenega parametra [7,54,52], ?Q ?t = —? , (12) kjer je ?-1 posploˇseni viskozni koeficient. V primeru nematskega tekoˇcega kristala, ko je ureditveni parameter tenzor drugega reda, je posploˇseni viskozni koeficient ten-zor 3. reda. V obravnavi ga nadomestim z izotropnim tenzorjem in s tem s skalarnim koeficientom. V brezdimenzijski obliki doloˇca generalizirana viskoznost tipiˇcni ˇcas za reorientacije direktorja, ?a = 27C?-1/B2 ~ 10-8 s [32,4]. Enaˇcba (12) je znana kot Landau-Halatnikova enaˇcba ali Ginzburg-Landauov ˇcasovno odvisni model, ki sta ga leta 1954 prva predlagala Landau in Halatnikov [7]. Do istega rezultata pridemo s striktno obravnavo disipativnega anizotropnega sistema z zanemaritvijo makroskopskih masnih tokov [52,53]. Casovno spreminjanje fluktuacijskih naˇcinov opiˇse v tem najpreprostejˇsem relak-sacijskem opisu eksponentno pojemanje, bi oc e-µt, kjer je Q(F, t) = A(r) + B(r*, t), ||B|| 0 (v konˇcnem sistemu si mislimo limito zvezne funkcije, ki bi v ustreznih diskretnih vrednostih ustrezala naˇsi funkciji). Relaksacijska hitrost mehkega fluktuacijskega naˇcina pade od prehodu v limiti q —> 0 na 0, medtem ko je stran od prehoda razliˇcna od niˇc. Za Goldstoneov fluktuacijski naˇcin pa je znaˇcilno, daje njegova relaksacijska hitrost enaka 0 na celotnem obmoˇcju urejene faze. Omenimo ˇse, da je Goldstoneov fluktuacijski naˇcin posledica zlomljene rotacijske simetrije izotropne faze [36,37]. Goldstoneove fluktuacije v nematskem tekoˇcem kristalu so direktorske fluktuacije. Te so odgovorne za moˇcno sipanje svetlobe na nematikih, o ˇcemer smo govorili v uvodu. Mehkih fluktuacijskih naˇcinov v neograjenih nematikih ni, saj spremljajo zvezne fazne prehode, lahko pa se pojavi zaradi vpliva povrˇsin. x Razˇsirjeni povzetek T 1 Slika 4 Shema ograjenega tekoˇcekristalnega sistema za ˇstudij sil: pri spremembi razmika med ograjujoˇcima stenama se prostornina in povrˇsina tekoˇcega kristala ne spremenita. Strukturna in psevdo Casimirjeva sila Zakljuˇcimo predstavitev fenomenoloˇske teorije opisa faznega prehoda pri tekoˇcih kristalih s predstavitvijo sil, ki delujejo na tanko plast nematika, kadar je ta v stiku s stenami. Zaradi prisotnosti sten se spremeni tako povpreˇcna ravnovesna ureditev v tekoˇcem kristalu, kot tudi spekter fluktuacij. Spremenjeni ureditvi ustreza drugaˇcna prosta energija, kot enaki prostornini neograjenega tekoˇcega kristala (glej sliko 4). V sploˇsnem je prirastek proste energije odvisen od razdalje med stenama, zaradi ˇcesar se steni ali privlaˇcita ali odbijata. Termodinamska definicija sile je t^ . (13) ?r ,, , Strukturna sila imenujemo del sile, ki izhaja iz spremenjenega povpreˇcnega ravnovesnega reda, F = -??Fmf/?d = -?FMF/?d + fneoA, kjer je Fmf prosta energija sistema v okviru fenomenoloˇske teorije povpreˇcnega polja, enaki prostornini neograjenega tekoˇcega kristala pa znotraj iste teorije ustreza prosta energija fneoA. Psevdo Casimirjeva sila izhaja iz spremenjene proste energije fluktuacij. To doloˇca fazni integral po vseh moˇznih konfiguracijah fluktuirajoˇcih polj [54,38], 0 Db e L J/ e i (14) kjer je kB Boltzmannova konstanta in T temperatura. H[b] = L/2{f[?~2b2 + (?b)2]dV + J2i ?i~ / b2dA} je Hamiltonian za fluktuacije v harmonskem pribliˇzku. Izraˇcun proste energije fluktuacij je ponavadi precej zapleten, predvsem pa moti njena divergenca, ki za odpravo katere je razvitih veˇc metod [155]. Za obˇcutek povejmo nekaj o sploˇsnih lastnosti psevdo Casimirjeve sile, ki temeljijo na robnih pogojih za fluktuacije. V primeru moˇcne sklopitve s stenami, ?i,2 <^i d, ali ˇsibke sklo-pitve, ?i,2 3> d, je psevdo Casimirjeva interakcija med stenami privlaˇcna. Nasprotno Fcas = -kBT Razˇsirjeni povzetek xi (a) d (b) J Slika 5 Shematski prikaz (a) oslabitve kapilarnih valov v primeru odbojne sile na plast tekoˇcega kristala in (b) ojaˇcitve kapilarnih valov. je interakcija v primeru meˇsanih robnih pogojev, torej ˇsibke sklopitve z eno steno in moˇcne z drugo, odbojna [27,63,64]. ˇ tudiju strukturnih in psevdo Casimirjevih sil se v delu posveˇcam zaradi ˇstudija stabilnosti tankih tekoˇcekristalnih plasti. Plast tekoˇcega kristala je v stiku s trdno steno, eno povrˇsino pa ima prosto, v stiku z zrakom. Zaradi termiˇcnih fluktuacij prosta povrˇsina ni ravna, ampak so na njej vzbujeni kapilarni valovi, ki povzroˇcijo, da se debelina filma s krajem spreminja. Na plast zaradi njene strukture, termiˇcnih fluktuacij ureditvenih in elektromagnetnih polj, deluje skupna sila, odvisna od debeline filma, ?(d). (Tudi, kadar bomo govorili o sili, bomo v resnici mislili silo na enoto povrˇsine, torej na dodatni tlak v plasti.) Ko se debelina filma veˇca, se mora velikost sil, ki delujejo nanj, zmanjˇsevati. Slednje ne velja le pri debelinah, ko se spreminja znaˇcaj sile iz odbojne v privlaˇcno ali obratno. Za zdaj imejmo v mislih le monotono padajoˇce odbojne sile in monotono rastoˇce privlaˇcne sile, sploˇsni izrazi pa bodo veljali tudi za prevojna obmoˇcja. Ce torej deluje na plast odbojna sila, bo odboj v stanjˇsanih delih veˇcji kot odboj v odebeljenih delih in bo sila povzroˇcila zmanjˇsanje razlik v debelini. Nasprotno bo veˇcji privlak v tanjˇsih delih in manjˇsi privlak v debelejˇsih povzroˇcil ˇse dodatno poveˇcanje razlik v debelini, dokler se ne bo na stanjˇsanem delu prosta povrˇsina dotaknila trdne podlage in se bo film razgradil v kapljice tekoˇcega kristala in suha podroˇcja. Opisanemu mehanizmu razomoˇcenja trdne podlage preko ojaˇcitve kapilarnih valov pravimo spinodalno razomoˇcenje; she-matsko je predstavljen na sliki 5. Razmiˇsljanje lahko posploimo v pravilo: film je stabilen napram spinodalnemu razomoˇcenju, ˇce je v okolici povpreˇcne debeline filma sila padajoˇca funkcija debeline (?'(d) < 0), medtem ko je v nasprotnem primeru (?'(d) > 0) film nestabilen. Povedano zapiˇsemo formalno z relaksacijskim ˇcasom za xii Razˇsirjeni povzetek kapilarne valove [73,74], 1 Lr 4 2/ -i — = — ?q — q ?(d0) , (15) ?g 3? ki je pozitiven — in s tem zagotavlja eksponentno pojemanje vzbujenih stanj — za poljuben val, ˇce je ?' < 0, oziroma postane negativen za doloˇcene vrednosti valovnega vektorja q za ?' > 0; takrat se dani val ojaˇci. Van der Waalsova sila Van der Waalsova sila med dvema nevtralnima molekulama zdruˇzuje disperzijsko interakcijo med fluktuirajoˇcimi dipoli, ki nastanejo zaradi trenutne prerazporeditve elektronov v molekulah, in orientacijsko interakcijo med fluktuirajoˇcimi stalnimi dipoli. Pripadajoˇca energija pada z razdaljo kot 1/r6, pri velikih oddaljenostih med molekulama pa se disperzijski prispevek zmanjˇsa (za velike razdalje se pribliˇzuje odvisnosti 1/r7 [91,68]). Zmanjˇsanje disperzijske interakcijske energije je posledica konˇcne hitrosti svetlobe — ko postane ˇcas, ki ga elektromagnetno polje vzbujenega dipola potrebuje, da doseˇze drugo molekulo primerljiv tipiˇcnemu ˇzivljenskemu ˇcasu danega vzbujenega stanja, se zgublja fazna povezava med interagirajoˇcima molekulama. Pojav imenujemo retardacija. Kadar interagirajo med seboj molekule, ki tvorijo makroskopska telesa, radi prevedemo mikroskopske interakcije na makroskopske. Najenostavnejˇsi naˇcin je, da kar seˇstejemo dvodelˇcne interakcije med vsemi pari molekul, kot da bi bili izolirani. Za ravninsko geometrijo pada energija pripadajoˇce interakcije v pribliˇzku brez upoˇstevanja retardacije kot 1/d2 (v limiti moˇcne retardacije 1/d3). V trdnih snoveh je gostota molekul precej veˇcja, da bi upraviˇcila pribliˇzek “idealnega plina”. Molekule vplivajo druga na drugo, kar bi morali upoˇstevati z veˇcdelˇcnimi interakcijami. Na mezoskopskih in makroskopskih razdaljah lahko pozabimo na “zrnatost” snovi in jo obravnavamo kot kontinuum. V tem pribliˇzku lahko o van der Waalsovi interakciji med makroskopskimi telesi govorimo v smislu spremembe proste energije sistema, ker smo telesi iz neskonˇcne oddaljenosti pribliˇzali na konˇcno razdaljo. S tem smo spremenili spekter in pripadajoˇco prosto energijo elektromagnetnega valovanja v dielektriˇcni snovi. Spremenjeni fluktuacijski naˇcini v sploˇsnem zavisijo od razdalje med telesoma, kar vodi do privlaka ali odboja med njima. Z opisom van der Waalsove sile med makroskopskimi telesi se je na nivoju opisa idealnega plina prvi ukvarjal Hamaker [92]. Kasneje je Lifshitz opis za trdna telesa popravil s kontinuumskim opisom [93], na nivoju kvantne teorije polja pa so se z njim ukvarjali Dzyaloshinskii in sodelavci [94,95]. Vsi opisi so vezani na izotropna makroskoska telesa. Kihara in Honda sta v 60-ih letih predstavila van der Waalsovo silo med optiˇcno enoosnimi telesi, vendar rezultata nista komentirala. Tako se ˇse Razˇsirjeni povzetek xiii (a) (b) Slika 6 Shema sistema, v katerem ˇstudiram van der Waalsovo silo: (a) dve polneskonˇcni makroskopski telesi, ki ju loˇcuje plast tretje snovi z debelino d; (b) pogosto je na trdni podlagi ˇse dodatna plast oksida, vode,... z debelino t. Puˇsˇcice oznaˇcujejo smer optiˇcnih osi. vedno tudi za opis van der Waalsove sile med moˇcno anizotropnimi sredstvi uporabljajo izrazi za izotropna sredstva. Eden od razlogov za to, je poleg neznanosti izraza za enoosna sredstva tudi to, da za silo med izotropnimi sredstvi obstajajo poenostavljeni izrazi za izraˇcun sile [89] za anizotropna sredstva pa ne. Ker je v tekoˇcih kristalih dielektriˇcna in optiˇcna anizotropija precej velika, sem se lotila ˇstudija van der Waalsove sile med optiˇcno anizotropnimi sredstvi. Omejila sem se na edino geometrijo in simetrijo sistema, ki nas privede do konˇcnega analitiˇcnega rezultata za van der Waalsovo silo: dve polneskonˇcni makroskopski telesi, med katerima je ravninska plast tretje snovi. Telesa so v sploˇsnem optiˇcno enoosna, optiˇcne osi pa so v vseh snoveh pravokotne na mejne povrˇsine, tako da je tudi simetrija celotnega sistema enoosna. Shema ˇstudiranega sistema je predstavljena na sliki 6. ˇ eprav se morda zdi, da smo za ˇstudiran sistem postavili preveˇc omejitev, da bi lahko govorili o kakˇsnih sploˇsnejˇsih vplivih dielektriˇcne anizotropije na van der Waalsovo silo, pa ne smemo pozabiti, da je to edini sistem, glede katerega lahko kaj povemo, poleg tega pa opiˇse tudi precej tekoˇcekristalnih sistemov, ki jih opisujem v tem delu, pa tudi sistemov, ki so sicer predmet raziskav [77-85]. Van der Waalsova interakcija je posledica spremenjenih elektromagnetnih fluk-tuacijskih naˇcinov. Te doloˇcajo Maxwellove enaˇcbe. V enoosnem sistemu, ki ga ˇstudiram, lahko elektromagnetne fluktuacijske naˇcine razdelimo na transverzalne magnetne (TM) in transverzalne elektriˇcne (TE). Naˇcini v polneskonˇcnih sredstvih eksponentno pojemajo z vdorno globino Pt M (u) = e_i_ / 2 ui ---- K------ell L|| c2 / 2 /2 W pTE(oj) = k------e± i 1/2 (16) , xiv Razˇsirjeni povzetek kjer je ? valovni vektor v ravnini meje med sredstvoma. Z upoˇstevanjem robnih pogojev na meji sredstev, dobimo sekularno enaˇcbo, katere reˇsitve so frekvence elektromagnetnih fluktuacijskih naˇcinov. Te doloˇcajo prosto energijo sistema, i T^ ln (2 sinh —ht% ) - T^ ln ( 2 sinh t% ) , (17) L__• 2kg T L__• 2kgT i i kjer teˇce vsota i po frekvencah v sistemu s konˇcno razdaljo med polneskonˇcnima makroskopskima telesoma d, vsota i' pa po frekvencah sistema d —> oo, ko je celoten prostor napolnjen samo z vmesno snovjo. Sekularne enaˇcbe nam ni treba eksplicitno reˇsiti, ker lahko vsote v enaˇcbi (17) prevedemo z integracijo po kompleksni ravnini [102,100] na ?T 2? f 0 kBTA f°° ^ D(i?n) =-------- d?? 2_^ ln , (18) D0(i?n) kjer je A povrˇsina mejnih ploskev, D(?) = 0 sekularna enaˇcba za sestavljen sistem in D0(?) = 0 sekularna enaˇcba za homogen sistem vmesne snovi; ?n = 2?kBTn/h. Silo, ki izhaja iz poveˇcanja proste energije zaradi prisotnosti sten, zraˇcunamo po enaˇcbi (13). V triplastnem sistemu [glej sliko 6 (a)] je DTM(?) = 1 + ?12(?)?23(?)e 2?—2(?) = 0, (19) kjer je in kjer je R ei-i-(?)?—j(?) - ej±(?)?—i(?) —j-(?) = —-----------, (20) ei±(?)?—j(?) + ej±(?)?—i(?) D R TE (?) = 1 + ?R12(?)?23(?)e 2?2(?)d = 0, (21) R ?i(?) - ?j(?) ?i j(?) =---------. (22) ?i(?) + ?j(?) Za izraˇcun van der Waalsove sile moramo poznati ˇse funkcijsko odvisnost dielektriˇcne konstante od frekvence. Dielektriˇcna konstanta je povezana s polarizabilnostjo molekul ? kot e = 1 + n?/t0, kjer je n gostota molekul. K polarizabilnosti molekule prispeva veˇc mehanizmov: reorientacija permanentnih dipolov, reorientacija ionov in deformacija elektronskega oblaka. Permenentni dipoli sledijo zunanjemu polju le pri zelo nizkih frekvencah, meja za reorientacijo ionov je v infrardeˇcem obmoˇcju, pri frekvencah v obmoˇcju vidne svetlobe pa zunanjemu polju sledi le ˇse elektronski oblak, ki se deformira zaradi skokov elektronov v vzbujena stanja. Pri izraˇcunu van der Waalsove sile je pomembna vrednost statiˇcne dielektriˇcne konstante, e(0), in pa Razˇsirjeni povzetek xv konstante pri ? = in?1. Ker je ?1 = 2,5 • 1014 s 1 v obmoˇcju, kjer je polarizabilnost le ˇse posledica elektronskih prehodov, vzamemo za e(i?n) = 1 + 1+n?t/1?^. Privzeli smo, da je v snovi en tipiˇcen elektronski prehod, ki je v vseh snoveh (pribliˇzno) enak; ?e ~ 2? • 3 • 1015s_1 [89]. Zdaj lahko izraˇcunamo van der Waalsovo silo kBT L2|i (0) r°° 2 ?—12?—23e_x 16?d3 L2± (0) 0 1 + ?12 ?23 e~ kBT ^ 3 °° / ? R 2? R 3 e~2pd ^2« ?(d,T) = + ?d3 1 J2d dp p2 (23) 1 + ? ? ? 12 23 ~ -2pd ~ + 2± ? 12 ? 23 e R —R 12 ? 23 ~ 2pd 1 + ? ~ 2pd , kjer smo upoˇstevali, da je e soda funkcija argumenta. Prvi ˇclen v izrazu predstavlja statiˇcen odziv sistema, oziroma prispevek stalnih dipolov, medtem ko je drugi ˇclen posledica dinamiˇcnega odziva. Funkcije ?ij = ?ij(0), ?Ri j = ?Ri j(i?n) in ?Ri j = ? (i?n) so doloˇcene v enaˇcbah (20) in (22); d = d ^?n/c V enaˇcbi (23) smo integracijo po valovnem vektorju ? nadomestili z integracijo po brezdimenzijskem parametru p = ?2(?)c/(?n^/t2±), tako da so ? ij = ?i j = si — s j si + s j ?ij = —is—j , , = Qi t^, si = p2 — 1+ ^i±_ /^2±_, (24) = p2 — 1+ t-iii / 2|| , in ti = ti(i?n), ˇce ni navedeno drugaˇce. Enoosnost zunanjih polneskonˇcnih makro- li ,____ skopskih teles renormazira dielektriˇcno konstanto, q —> -/Li,, q±. Po drugi strani enoosnost snovi, ki je vrinjena med makroskopski telesi ne renormalizira le dielektriˇcne konstante, ampak tudi samo moˇc interakcije. Izraˇcun van der Waalsove sile iz izraza v enaˇcbi (24) je precej zamuden, odvisnost sile od parametrov snovi pa precej nejasna. Izraz se poenostavi v neretardiranem pribliˇzku, c —> oo. Pribliˇzek velja dobro za majhen razmik med makroskopskima telesoma (d <• ?e = 2?c/?e ~ 100 nm). Pri veˇcjih razdaljah ne dobimo priˇcakovanega zmanjˇsanja dosega sile. Velikokrat nas natanˇcna odvisnost od razmika med telesoma niti ne zanima, saj je v sistemu ena od sil dominantna, druge pa predstavljajo le popravek. Dominantna je lahko van der Waalsova sila ali pa morebitne elektrostatiˇcne sile ali druge. V tem primeru je pomemben predvsem znaˇcaj sile (odbojna ali privlaˇcna) in njena velikost. Vse te podatke nam v zadostni natanˇcnosti nudi tudi pribliˇzek brez retardacije. V pribliˇzku brez retardacije je ? = —A/6?d3, kjer je 3kBT ~ e2|| (i?n) f°°, 2 ?—12(i?n)?—23(i?n)e" 4 ^ 0 L2 (i?n) 0 e2± (i?n) dx x 1 + ?12(i?n)?23(i?n)e , (25) , xvi Razˇsirjeni povzetek Hamakerjeva konstanta. ˇ rtica nad vsoto oznaˇcuje, da moramo ˇclen z n = 0 mnoˇziti z 1/2; funkcijo ?ij = —y(i?n) smo definirali v enaˇcbi (24). Integral v enaˇcbi (25) je analitiˇcno izraˇcunljiv, vsoto po n pa izvedemo z nekaj nadaljnimi pribliˇzki: upoˇstvamo, da je |?ij| = —kBT—--------------- (26) 4 L2? —1 + —2 —3 + —2 + -----/=(n—21 - n—22 )(n—23 - n—2 2) 8\/2 v2(n22-n22) n2? (2n22 - n—21 - n—22)(2n22? - n—23 - n—22) - 2n22 - n—21 - n— 2 2n2 - n—3 2 - n—2 2 ------------------------------------------------,-------------------------------------------- 2 + n—22(2n22 - n—21 - n—22)(n—21 - n—23 ) Vn—3 + n—2(2n22 - n—23 - n—22)(n—2 1 - n—23 ) —------------------------------------------------------- \ /n—21 + n—22(2n22? - n—21 - n—22)(n21 - n—23) \r Y y - n—2 - n—22)(n21 - n—32) n—23 + n—22(2riŽ - n—23 - n—22)(n—21 - 1 / ? — ,_______ ˇ kjer je ai = -/a^ai? in je a ali statiˇcna dielektricna konstanta e ali lomni koliˇcnik v podroˇcju vidne svetlobe n. Prvi ˇclen v enaˇcbi (26), A?=0, ponovno predstavlja statiˇcni odziv sistema, drugi ˇclen, A?>0, pa ustreza dinamiˇcnemu odzivu. Kot ˇze pri sploˇsnem izrazu, je tudi zdaj jasno razvidno, da anizotropija okolnih sredstev le renormalizira dielektriˇcno konstanto in lomni koliˇcnik, medtem ko je vloga vmesnega sredstva veˇcja. V primeru, da bi bila vsa tri sredstva izotropna, se izraz v enaˇcbi (26) poenostavi v znan izraz [89] 3 2ll (—1 — 2) 3h—?e 2 22 — kBT — — ------ —2 +--------~ž=(n—1 - 4 L2? (ei + L2) 8y2 211 ( 1 2) 3h—?e 2 22 7) A = —kBT —------ 2 +-------~/=(n—1 - n—2) (2 X v2(n22 - n22 ) (n2 + n22)2 + 4n—2 - 2(n2 + n—22)(3n22 - n22 ) + n2?(2n22? - n—21 - n—22)2 2(n—21 + n22)3/2(2n22? - n—21 - n—22)2 . Znaˇcaj van der Waalsove interakcije je odvisen od medsebojne relacije med statiˇcnimi dielektriˇcnimi konstantami in lomnimi koliˇcniki interagirajoˇcih snovi. Znak statiˇcne Hamakerjeve konstante je doloˇcen z relacijami med statiˇcnimi dielektriˇcnimi konstantami: za —2 < —1,—3 ali —2 > —1,—3 je konstanta pozitivna zato je statiˇcni del van der Waalsove interakcije privlaˇcen, za —1 < —2 < —3 ali —1 > —2 > —3 je konstanta negativna in ustrezni del van der Waalsove interakcije odbojen. Podobni pogoji veljajo za dinamiˇcen del Hamakerjeve konstante. Izkaˇze se, daje del izraza (26) v oglatih oklepajih pozitivno definiten in je torej znak konstante doloˇcen z znakom produkta (n—21 - n—22)(n23 - n—22). Za n2 < n1, n3 ali n2 > n1, n3 je dinamiˇcni , Razˇsirjeni povzetek xvii 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 0.0 0.5 1.0 ß 1.5 2.0 Slika 7 Dinamiˇcni del Hamakerjeve konstante za enoosna sredstva kot funkcija parametra ß = n1/n2±. Polna ˇcrta oznaˇcuje konstanto izraˇcunano po enaˇcbi (26) za enoosna sredstva, ˇcrtkana ˇcrta pa oznaˇcuje konstanto v pribliˇzku izotropnih sredstev, kjer je ( n iso 2 = (n2i + 2n2i )/3. Hamakerjeva konstanta je merjena v enotah A0 = 3—h?en2±/8\^2, n21 /n2j_ = 1,2 in n3/n2j_ = 0,67. del Hamakerjeve konstante pozitiven in interakcija privlaˇcna, za n—1 < n—2 < n3 ali n—1 > n—2 > n3 pa je konstanta negativna in interakcija odbojna. Enaki pogoji veljajo za znaˇcaj izotopne Hamakerjeve konstante [enaˇcba (27)], ˇce renormalizirane parametre nadomestimo z izotropnimi. Kot je bilo omenjeno ˇze na zaˇcetku, do sedaj ni bil znan enostaven izraz za van der Waalsovo silo med enoosnimi sredstvi. Zato je bila ta sila doloˇcena iz izrazov za izotropna sredstva, kjer so bili vstopni parametri za Hamakerjevo konstanto izotropni deli ustreznih tenzorjev, čso = (ey + 2ej_)/3 oziroma (niso)2 = (n2, + 2n2L)/3. Kot je razvidno iz pogojev, ki smo jih ravnokar zapisali in kot se vidi na sliki 7, se lahko tako doloˇcena van der Waalsova sila od prave precej razlikuje po velikosti, v ozkem intervalu anizotorije pa celo po znaˇcaju interakcije. V nadaljevanju se bomo posvetili neposrednim vplivom ograjujoˇcih sten na ne-matski red in fluktuacije nematskega ureditvenega parametra. Najprej se bomo posvetili lokalnim vplivom povrˇsin na nematski tekoˇcekristalni red. Lokalizirane variacije reda so znaˇcilne za spreminjanje stopnje nematskega reda, medtem ko se deformacije, povezane s frustracijami direktorskega polja, ponavadi raztezajo po celotnem tekoˇcekristalnem vzorcu. Heterofazni nematski sistem Kadar je nematski tekoˇci kristal v stiku s povrˇsino, ki vsiljuje moˇcan nematski red, se tik ob povrˇsini tudi pri temperaturah nad prehodom v urejeno nematsko fazo xviii Razˇsirjeni povzetek z=0 Slika 8 Shema ureditve molekul v nematskem tekoˇcem kristalu v stiku z razurejujoˇco steno. pojavi mezoskopska urejena plast. Pojav imenujemo (orientacijsko) moˇcenje, opisani sistem pa paranematski sistem. Podoben pojav opazimo pod temperaturo prehoda, ko se ob povrˇsini, ki vsiljuje moˇcan nered, med steno in nematsko fazo pojavi dobro definirana plast izotropne faze (glej sliko 8). Tu opisujem predvsem slednji primer, ki ga imenujem povrˇsinsko staljeni nematski sistem. Orientacijsko moˇcenje je lahko delno ali popolno. Pri delnem moˇcenju je povrˇsinska plast le delno (raz)urejena, njena debelina pa je konˇcna tudi tik ob prehodu v fazo, ki jo vsiljuje tudi povrˇsina. Fazni prehod je nezvezen. Pri popolnem moˇcenju opazimo pred faznim prehodom povrˇsinski prehod, kjer se delno (raz)urejena povrˇsinska plast skokovito (raz)uredi, nato pa njena debelina ob pribliˇzevanju prehodu moˇcno naraˇsˇca in v polneskonˇcnem sistemu ob faznem prehodu divergira. Fazni prehod je v primeru popolnega moˇcenja zvezen. Moˇcenje v polneskonˇcnem sistemu, predvsem paranematskem, so od prvega zapisa leta 1976 ˇstudirali v mnogih skupinah [18,50,108–111,59,60,14]. Tu predstavljam predvsem ˇstudij dinamike v nematski plasti s staljenim redom ob povrˇsinah. Obravnava je vezana na Landau-de Gennesovo teorijo in opis tekoˇcega kristala s ten-zorskim nematskim ureditvenim parametrom. Smektiˇcnega urejanja tik ob povrˇsini ne upoˇstevam, saj je to, posebej v izotropni fazi, zelo majhno. Homeotropno smer nematskega direktorja doloˇca ˇsibko zunanje magnetno polje, ki pa je dovolj moˇcno, da premaga morebitno drugaˇcno preferenˇcno smer direktorja na meji obeh faz [120-125]. Njegovega vpliva na stopnjo urejenosti v obravnavi ni potrebno upoˇstevati. Prav tako ne upoˇstevamo ˇsibke sklopitve med variacijo stopnje urejenosti in deformacijami direktorskega polja [119]. Ravnovesni povpreˇcni ureditveni parameter torej zapiˇsemo kot Q = a0T0, kjer je n = ez in je os z v smeri normale na plast, ureditveni parameter, ki ga vsiljuje povrˇsina pa je QS = aST0, kjer je aS < 1. Na sliki 9 so predstavljeni profili stopnje nematske urejenosti pri razliˇcnih temperaturah in moˇceh vpliva povrˇsine. V primeru nematika v stiku z razurejujoˇcimi stenami pride do popolnega omoˇcenja stene z izotropno fazo le, ˇce stena vsiljuje popoln nered (a S = 0) in je sklopitev dovolj moˇcna (G > 0,0023 J/m2). V obratnem paranematskem sistemu lahko dobimo popolno omoˇcenje za vse vrednosti vsiljevanega ne- Razˇsirjeni povzetek xix 1.2 1.2 1.0 0.8 1.0 0.8 .0.9/ /""/'" r 0.6 /0.999/ 0.6 ~ / / 0.4 /0.99999 0.4 / 0.2 0.2 -/' / 0.0 0. 0.0 20 __^y 0 0.1 0. .0 0.1 0. z/d z/d (a) (b) Slika 9 Stopnja nematske urejenosti v nematiku s staljenima povrˇsinskima plastema v bliˇzini prehoda v izotropno fazo. (a) Temperaturna odvisnost: aS = 0, G —> 00, profili pa so oznaˇceni z vrednostjo ustrezne brezdimenzijske temperature. (b) Odvisnost od moˇci sklopitve s povrˇsino: ? = 1 - 10-5, aS = 0 ter G —> 00 (polna ˇcrta), 0,001 J/m2 (ˇcrtkana ˇcrta) in 0,0003 J/m2 (pikˇcasta ˇcrta). matskega reda, ki je veˇcji od reda ob prehodu v neograjenem sistemu (aS > 1), mejna vrednost sklopitve G, ki dovoljuje popolno omoˇcenje pa je odvisna od aS; na primer G (aS = 1,1) = 0,0006 J/m2. Eden od znakov, da je omoˇcenje popolno, je rast omoˇcitvene plasti, ko se pribliˇzujemo prehodu v izotropno fazo. Na sliki 10 (a) lahko jasno razloˇcimo med divergentnim naraˇsˇcanjem debeline omoˇcitvene plasti v primeru popolnega omoˇcenja in obnaˇsanjem, znaˇcilnim za delno omoˇcenje, ko doseˇze povrˇsinska plast konˇcno obliko ˇze pred prehodom. V konˇcnem vzorcu nematskega tekoˇcega kristala debelina omoˇcitvene plasti seveda ne more divergirati, ker se obe omoˇcitveni plasti prej zdruˇzita, vendar kaˇze temperaturna odvisnost dW(T - TNI) tipiˇcno logaritemsko odvisnost. Zaradi konˇcne preˇcne dimenzije je fazni prehod tudi v primeru popolnega omoˇcenja stene z izotropno fazo rahlo nezvezen. Poleg tega nastopi prehod pri niˇzji temperaturi (viˇsji v primeru paranematskega sistema), vendar je razlika komaj zaznavna — v celici debeline 792 nm je ?ni = 0,99274, v debelejˇsih celicah pa ˇse manj. Predvsem v neposredni bliˇzini faznega prehoda sestavljata heterofazni sistem povrˇsinska plast z redom, kot ga vsiljuje povrˇsina, in notranji del sistema, v katerem se vpliv povrˇsine ne pozna in je tekoˇci kristal urejen tako, kot bi bil pri dani temperaturi v neograjenem sistemu. Obe podroˇcji loˇci dobro definirana fazna meja, tem bolj, ˇcim bliˇzje smo faznemu prehodu. Obe fazi imata sicer enake izotropne dielektriˇcne lastnosti, vendar pa se razlikujeta v simetriji. Pri obravnavi van der Waalsove sile v anizotropnih sredstvih smo ugotovili, da sta taki snovi z vidika van der Waalsove interakcije razliˇcni. Zato fazna meja med njima predstavlja novo steno v sistemu, v omoˇcitveni plasti pa k tlaku prispeva tudi van der Waalsov tlak. Ker je omoˇcitvena xx Razˇsirjeni povzetek 60 50 40 30 20 10 30 25 20 15 10 5 1E-5 1E-4 1E-3 0.01 |? - ?NI| (a) 0.1 1E-5 1E-4 1E-3 |e-ej (b) 0.01 0.1 Slika 10 (a) Temperaturna odvisnost debeline staljene povrˇsinske plasti v primeru popolnega in delnega moˇcenja. Parametri raˇcuna: aS = 0, d = 792 nm ter G —> oo v primeru popolnega moˇcenja in G = 0,001 J/m2 v primeru delnega moˇcenja. (b) Renormalizacija debeline omoˇcitvene plasti zaradi van der Waalsove sile na fazno mejo. Logaritemsko divergenco s kritiˇcnim eksponentom 0 zamenja potenˇcna divergenca ?dW oc (T — Tni)~? s kritiˇcnim eksponentom ? = 0,5. plast precej tanjˇsa od celotne debeline celice, sem obravnavala tridelni sistem trdne stene, omoˇcitvene plasti in nezmotenega tekoˇcega kristala. Blizu prehoda, kjer je fazna meja dobro definirana, nadomestimo profil skalarnega ureditvenega parametra s stopniˇcasto funkcijo in izraˇcunamo prispevek k prosti energiji zaradi van der Waalsove interakcije. Ta je v primeru nematika s staljenimi povrˇsinami privlaˇcna in prispeva k stanjˇsanju omoˇcitvene plasti, v primeru paranematske celice pa odbojna in poveˇca ravnovesno debelino plasti. Temperaturna odvisnost van der Waalsovega popravka k debelini omoˇcitvene plasti je narisana na sliki 10 (b). V podroˇcju dobro definirane fazne meje ima popravek potenˇcno odvisnost od razlike temperatur T — TNI, kritiˇcni eksponent doloˇcen iz prilagoditve pa da vrednost blizu 0,5. Fluktuacije ureditvenega parametra Obravnavamo sistem z enakimi robnimi pogoji na obeh ograjujoˇcih stenah, zato je ravnovesni povpreˇcni profil simetriˇcen glede na sredino celice in ima enako simetrijo tudi potencial za fluktuacije. Kot je znano, so lastne funkcije sodega operatorja bodisi sode ali lihe glede na dano simetrijsko ravnino [130], kar velja tudi za har-monski operator za fluktuacije ureditvenega parametra. Zato reˇsujem enˇcbe le na eni polovici celice, pri ˇcemer za fluktuacijske naˇcine velja, da ima na sredini niˇclo bodisi fluktuacijski profil (lihi naˇcini) ali njegov odvod (sodi naˇcini). Vpliv povrˇsin lahko ˇstudiramo ob primerjavi dimenzijsko in geometrijsko enakega sistema, v katerem pa ograjujoˇce stene vsiljujejo prav tak red, kot bi ga imel neogra-jeni tekoˇci kristal. V takem sistemu je stopnja urejenosti v celici konstantna. Fluk- 1 0 1 Razˇsirjeni povzetek xxi 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.5 0.0 -0.5 0.90 0.92 0.94 0.96 0.98 1.00 ? (a) -1.0 0.0 0.1 0.2 0.3 z/d (b) 0.4 0.5 Slika 11 (a) Spekter fluktuacij stopnje urejenosti, za katerega je znaˇcilen mehki osnovni fluktuacijski naˇcin, in (b) upodobitev profila dveh fluktuacijskih naˇcinov — ˇstevilka, ki oznaˇcuje posamezni naˇcin, pove ˇstevilo vozlov med obema stenama — skupaj z ravnovesnim povpreˇcnim profilom za ? = 1 -10-5. (G › ? in aS = 0) tuacijski naˇcini v konstantnem (ˇskatlastem) potencialu so sinusi, sinqzz, kjer je qz = ?(n + 1)?, pripadajoˇce relaksacijske hitrosti pa so ? ?-2 /?-2 ?2 [( + 1 ) ]2 (28) kjer so ?N,i korelacijske dolˇzine definirane v enaˇcbi (10) z a0 = 0,75(1 + 1 - 8?/9). V bliˇzini faznega prehoda je ?N-,0/?-i ? 6 - 5? in ?N-,±1/?-i ? 18 - 9?, medtem ko je ?N-,±2/?-i = 0 na celotnem temperaturnem intervalu nematske ureditve. Spekter fluktuacij v nematskem sistemu s razurejujoˇcimi povrˇsinami se od homogenega sistema najbolj razlikuje po obstoju mehkega fluktuacijskega naˇcina, s tipiˇcno temperaturno odvisnostjo ?0,0 = ±C|? - ?NI| (Cnem = 3,0 in Cpara = 5,6). Ta ustreza fluktuacijam debeline omoˇcitvene plasti. Pri osnovnem naˇcinu fluktuirata debelini obeh plasti z nasprotno fazo in povrzroˇcata krˇcenje oziroma ˇsirjenje osrednjega nematskega dela. Prvi lihi naˇcin je po energiji enak osnovnemu (prvemu sodemu) kadar je debelina celice dovolj velika, da omoˇcitveni plasti ne “ˇcutita” povezave. Naˇcin predstavlja fluktuacije poloˇzaja osrednjega dela. Od homogenega spektra se loˇcita ˇse prva naslednja lihi in sodi naˇcin, ki spreminjata obliko fazne meje. Viˇsji naˇcini skorajda ne ˇcutijo razurejujoˇcega vpliva sten. Spekter fluktuacij stopnje urejenosti in profili nekaj fluktuacijskih naˇcinov stopnje urejenosti so predstavljeni na sliki 11. Mehki fluktuacijski naˇcin je ˇse en znak za zveznost faznega prehoda, ki spremlja sisteme s popolnim omoˇcenjem. Kadar je moˇcenje le delno, je ?0,0 > 0 in naraˇsˇca z oddaljenostjo od reˇzima popolnega omoˇcenja. Na sliki 12 je predstavljena odvisnost relaksacijske hitrosti mehkega naˇcina od moˇci povrˇsinske sklopitve za aS = 0 v primeru nematika v stiku z razurejujoˇcima stenama in aS = 1,1 v primeru parane- xxii Razˇsirjeni povzetek 1.0 0.8 0.6 0.4 0.2 0.0 10-5 10-4 10-3 G [J/m2] 10-2 Slika 12 Relaksacijska hitrost osnovnega naˇcina fluktuacij stopnje urejenosti v sistemu nematika v stiku z razurejujoˇcima stenama (polna ˇcrta) in v parane-matskem sistemu (ˇcrtkana ˇcrta). V obmoˇcju delnega moˇcenja, G < Gc, je ?0,0 > 0. Kritiˇcna vrednost G je za nematik s staljenimi povrˇsinami 0,0023 J/m2 (aS = 0) in v paranematskem sistemu Gc = 0,0006 J/m2 (aS = 1,1). V paranematskem sistemu sta oba reˇzima loˇcena z omoˇcitvenim prehodom, medtem ko je v primeru nematika v stiku z razurejujoˇcimi povrˇsinami prehod nekoliko zabrisan. matika. Reˇzima popolnega in delnega omoˇcenja loˇci kritiˇcna vrednost sklopitve, sistem pa iz enega reˇzima preide v drugega z omoˇcitvenim prehodom. Ta je v primeru nematika v stiku z razurejujoˇcimi stenami nekoliko zabrisan, ker je popolno omoˇcenje vezano na sam rob faznega diagrama (G, aS), na aS = 0. V nematiku v stiku z razurejujoˇcimi stenami ni drugih fluktuacijskih naˇcinov, katerih relaksacijska hitrost bi ob prehodu padla na 0, medtem ko je v paranematiku tak ˇse najniˇzji direktorski naˇcin, ki je lokaliziran znotraj nematske omoˇcitvene plasti. Zmanjˇsanje njegove relaksacijske hitrosti na 0 je posledica dejstva, da so direktorske fluktuacije v nematski fazi, ki postane ob prehodu stabilna, Goldstoneove. V hetero-faznem nematiku se zaradi poveˇcanja debeline omoˇcitvene plasti ob prehodu moˇcno poveˇca relaksacijska hitrost Goldstoneovih direktorskih fluktuacij. Te so strogo vezane na osrednji nematski del, kjer je njihova korelacijska dolˇzina neskonˇcna, tako da je poveˇcanje relaksacijske hitrosti posledica efektivnega zmanjˇsanja debeline “sistema”. V nasprotju z direktorskimi fluktuacijami so fluktuacije dvoosnosti energijsko zelo neugodne v nematski fazi. Tako je za nekaj najniˇzjih naˇcinov ugodneje, da so vezani na tanko izotropno omoˇcitveno plast in se ˇsele viˇsji naˇcini raztezajo tudi po osrednjem delu. ˇ tevilo fluktuacijskih naˇcinov, katerih relaksacijska hitrost se pribliˇzuje niˇzji vrednosti v izotropni fazi, je odvisna od debeline omoˇcitvene plasti in s tem od temperature. Spektra fluktuacij dvoosnosti in direktorskih fluktuacij sta narisana na sliki 13. Razˇsirjeni povzetek xxiii 10 8 0.10 0.08 0.06 0.04 0.02 :------------------------:-----------^ 6 ___,, 4 ___, _ ----------------------:-----------------------:-----------------------:-----------------------i----------------------- 0.90 0.92 0.94 0.96 ? 0.98 1.00 0.90 0.92 0.98 1.00 (a) 0.94 0.96 9 (b) Slika 13 Spektra fluktuacij (a) dvoosnosti in (b) direktorja v sistemu nematika v stiku z razurejujoˇcima stenama. (? = 1 — 10-5, aS = 0 in G —> oo) Ograjujoˇce stene, ki jih izotropna oziroma nematska faza v tekoˇcekristalnem sistemu popolnoma omoˇci, moˇcno spremenijo tako povrpeˇcni red kot tudi fluktuacije v bliˇzini prehoda. Zaradi nenasprotujoˇcih si robnih pogojev se steni privlaˇcita. Ravnovesna ureditev je zaznamovana z lokalizirano spremembo ureditvenega parametra, ki je vezana na fazno mejo. Strukturni privlak je zato kratkega dosega. K fluk-tuacijski psevdo Casimirjevi sili prispeva fluktuacijska interakcija med steno in fazno mejo ter interakcija med obema faznima mejama. Najpomembnejˇsi je vpliv Gold-stoneovih direktorskih fluktuacij v nematskem delu sistema, ki vodi do sile dolgega dosega. V paranematskem sistemu je dolgi red interakcije prikrit z eksponentnim padanjem debeline nematskega dela tekoˇcega kristala, tako da je psevdo Casimir-jeva sila enakega dosega kot strukturna sila. Nasprotno je psevdo Casmirjeva sila v nematskem sistemu s staljenimi povrˇsinami dolgega dosega. Hibridna nematska celica V hibridni nematski celici je nematski tekoˇci kristal ograjen s stenama, ki vsiljujeta enoosni nematski red v razliˇcnih smereh. Ponavadi sta vsiljevani smeri pravokotni druga na drugo, tako da je na eni povrˇsini vsiljevani red v smeri pravokotno na steno (na primer v smeri osi z) na drugi pa v doloˇceni smeri v ravnini stene (recimo, v smeri osi x). Ker steni vsiljujeta nematski red v razliˇcnih smereh, je ureditev med njima vedno deformirana v primerjavi s spontano nematsko ureditvijo v neograjenem sistemu. Podobni nasprotujoˇci si pogoji pa niso nujno le posledica vsiljevanega reda na ograjujoˇcih povrˇsinah, ampak pride do njih tudi v bliˇzini defektov in zaradi geometije sistema. Tako v cilindriˇcni geometriji, kjer ograjujoˇce stene cilindra vsiljujejo red v radialni smeri, takemu redu nasprotuje simetrija cilindra, ki daje prednost ureditvi vzdolˇz dolge osi cilindra. xxiv Razˇsirjeni povzetek z=0 (a) z=d z=0 z=d (b) Si A Kj— ex G2 dc elastiˇcna deformacija direktorskega polja ugodnejˇsa od moˇcnega krˇsenja robnih pogojev na eni od sten — upognjena struktura. Direktorsko polje v celici je upognjeno, smer direktorja na obeh stenah pa se v sploˇsnem razlikuje od vsilje-vane smeri. Shema ureditve molekul v obeh opisanih strukturah je predstavljena na sliki 14. Blizu prehoda v izotropno fazo in, kadar je sidranje na povrˇsinah zelo moˇcno ekstrapolacijski dolˇzini pa pribliˇzno enaki, so poleg direktorja pomembne tudi ostale prostostne stopnje ureditvenega parametra. Nekaj raziskav ravnovesnih struktur v hibridni celici, kadar so pomembne tudi nedirektorske prostostne stopnje je ˇze bilo narejenih [20,139], z variacijo razliˇcnih parametrov, ki vplivajo na stabilnost razliˇcnih struktur in fazne prehode med njimi, pa smo vedenje o hibridni celici poglobili s ˇstudijo tudi mi [21]. Ker je za vse opisane strukture znaˇcilno, da leˇzi direktorsko polje v ravnini vsiljevanih smeri x in z, opiˇsemo ureditveni parameter s konstantno trojico enotskih vektorjev n = ey, e^1 = ez in e^2 = ex, na katerih raz-pnemo bazne tenzorje Ti [enaˇcba (2)]. Tako z upoˇstevanjem vseh prostostnih stopenj nematskega reda pri viˇsjih temperaturah popravimo predstavo o ˇze znanih struktu- Razˇsirjeni povzetek xxv 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 a 0 x\ \S P a 0 z A-—-. ^. _^^ ^-^ 0.45 0.50 zid 0.55 0.0 0.2 0.8 1.0 0.4 0.6 z/d (a) (b) Slika 15 (a) Ravnovesni profil neniˇcelnih prostostnih stopenj nematskega ureditvenega parametra: stopnja urejenosti glede na osi x [ax0 = — (a0 + V/3a1)/2] ali z [az0 = (—a0 + V/3a1)/2] — krepki polni ˇcrti — ali glede na os y (a0) — tanka polna ˇcrta. ˇ rtkani ˇcrti oznaˇcujeta absolutno vrednost stopnje urejenosti [S = (v6/2)|Qii|, kjer je predznak Qii nasproten predznaku drugih dveh lastnih vrednosti Q] in dvoosnosti (P = |Qjj — Qkk|/2, kjer j, k = i). (b) Poveˇcan del na obmoˇcju izmenjave direktorja. aS = 1,1 in g1 = g2 -*¦ oo) (? = 0,9, ?2 = 0,01258, rah, najdemo pa ˇse dodatno strukturo, ki jo imenujemo dvoosna struktura. Shemo ureditve molekul v dvoosni strukturi kaˇze slika 14. V dvoosni strukturi se direktor ob stenah ujema z vsiljevano smerjo. Red ob stenah je enoosen, z oddaljevanjem od sten se zmanjˇsuje, poveˇcuje pa se dvoosnost. Do preklopa direktorja iz ene smeri v drugo pride na sredini celice, kjer je red moˇcno staljen. V osrednjem, slabih 10 nm debelem, delu hibridne celice so molekule urejene vedno bolj enakomerno v ravnini obeh direktorjev, torej kot bi bil direktor pravokoten na znaˇcilno ravnino, stopnja urejenosti pa negativna. Odvisnost ureditvenih parametrov od oddaljenosti od ene od sten kaˇze slika 15. Z zmanjˇsevanjem moˇci sidranja na eni od sten preide dvoosna struktura v homogeno strukturo s homogenim direktorskim poljem, medtem ko stopnja urejenosti pada v smeri proti steni, kjer se direktor in vsiljevana smer direktorja ne ujemata. V isti smeri naraˇsˇca dvoosnost ureditve. Preobrazba ene strukture v drugo ni povezana s strukturnim ali faznim prehodom, ampak je stvar dogovora o poimenovanju faz. Pojav dvoosne sturkture je vezan na dovolj veliko frustracijo, kvantitativno jo ocenimo z moˇcjo sidranja G > 10-4 J/m2. Po drugi strani pa ob nespremenjenih enakih (velikih) moˇceh sidranja na obeh ograjujoˇcih stenah z niˇzanjem temperature ali z veˇcanjem debeline hibridne celice pridemo do strukturnega prehoda iz dvoosne strukture v upognjeno strukturo. Prehod je v sploˇsnem ˇsibko nezvezen, kar je razvidno iz slike 16 (a). Ob prehodu, kjer se vrednosti prostih energij ujemata, xxvi Razˇsirjeni povzetek 0.10 0.08 0.06 0.04 0.02 0.00 -0.02 -0.04 e t . s dvoosna 0.4 0.3 ^^ ,^-/ 0.2 0.1 0.0 - ^\; -/upognj ena 0.8 0.9 1.0 ? (a) 1.1 0.0 0.2 0.4 0.6 ? (b) 0.8 Slika 16 (a) Temperaturna odvisnost proste energije dvoosne in upognjene strukture. Do strukturnega prehoda pride pri ?t = 0,951, dvoosna struktura pa je metastabilna v temperaturnem intervalu ?t > ? > ?s = 0,869. (?2 = 0,02, aS = 1,1 in g1 = g2 —*¦ oo) (b) Temperaturna odvisnost proste energije dvoosne in upognjene strukture pri debelini, ki ustreza meji stabilnosti dvoosne strukture ?s = ?s(?). Pod trikritiˇcno toˇcko ?tp = 0,746 in ?T2 P = 0,054 je strukturni prehod zvezen, nad njo pa postane postopoma vedno bolj nezvezen. sta naklona obeh funkcij razliˇcna, ˇcrtkano nadaljevanje funkcij pa ustreza prostim energijam metastabilnega stanja dvoosne oziroma upognjene strukture. Pri debelini 56 nm je dvoosna struktura metastabilna v temperatunem intervalu 0,09 K, do strukturnega prehoda pa pride 0,04 K pod temperaturo faznega prehoda v neogra-jenem sistemu. Latentna toplota povezana s strukturnim prehodom je velikostnega reda 104J/m3, kar je za dva velikostna reda manj kot pri faznem prehodu v neogra-jenem sistemu. Nezvezni strukturni prehod preide pod trikritiˇcno toˇcko v zvezni prehod [glej sliko 16 (b)]. Zgornja meja za trikritiˇcno toˇcko je v limiti neskonˇcno moˇcnega sidranja dTP = 34 nm in TNI - TTP = 0,28 K. Obstoj dvoosne strukture smo preverili s primerjavo rezultatov, ki jih da raˇcunal-niˇska simulacija z metodo Monte Carlo [148,149]. Mikroskopske interakcije med molekulami “tekoˇcega kristala” smo modelirali s preizkuˇsenim Lebwohl-Lasherjevim potencialom [150], ki dobro opiˇse nematski sistem. Samo simulacijo na kubiˇcni mreˇzi so izvedli v skupini dr. Pasinija in prof. Zannonija v Bologni, obdelavo podatkov in primerjavo z rezultati fenomenoloˇskega opisa pa smo izvedli v naˇsi skupini. Na rezultatih simulacije so bila izraˇcunana razliˇcna makroskopska povpreˇcja, ki ustrezajo ureditvenim parametrom, ki jih doloˇcamo s fenomenoloˇsko teorijo. Poseben poudarek smo posvetili doloˇcitvi parametra a-1, ki meri upogib direktorskega polja v celici. Z njim smo razloˇcili med dvoosno in upognjeno strukturo, ki imata v bliˇzini strukturnega prehoda kvalitativno podobno obnaˇsanje ostalih ureditvenih parametrov. Razˇsirjeni povzetek xxvii -2.0 -2.2 -2.4 -2.6 -2.8 -3.0 -3.2 -3.4 P* : °V ••• p ; \ • P\ ^: y^ MC MC \H 1 1.30 1.25 1.20 1.15 1.10 0.9 1.0 1.1 1.2 MC T (a) 1.3 1.4 1.05 1.00 MC T1 MC T NI • MC T2 4 6 8 10 12 14 16 18 20 N (b) Slika 17 (a) Energija (kvadratki) in specifiˇcna toplota (polni krogci) v 14 plastni hibridni celici. Vrh pri T2M ~ 1,115 ustreza strukturnemu prehodu med dvoosno in upgnjeno strukturo, manjˇsi vrh pri T1MC ~ 1,165 pa ustreza ureditvi tekoˇcega kristala v sredini celice. (b) Temperaturi TM C in T2M C kot funkciji ˇstevila plasti v celici. Crtkana ˇcrta oznaˇcuje temperraturo prehoda v neograjenem sistemu. Po priˇcakovanjih so rezultati Monte Carlo simulacije pokazali obstoj tako dvoosne kot upognjene strukture. Homogene strukture v sistemu ni, ker sta bili moˇci sidranja na obeh ograjujoˇcih stenah enaki. Simulacija je bila izvedena na razliˇcno velikih sistemih (30 x 30 x N', kjer je N' = N + 2 = 6,8,10,12,16,22 in predstavlja N “plasti” molekul ali skupkov molekul tekoˇcega kristala in po 1 krajno “plast” na vsaki strani celice, ki ustreza molekulam ograjujoˇce stene) in pri razliˇcnih vrednostih Monte Carlo temperature (TMC = kBT/e, kjer je e energija interakcije med sosednjima molekulama), predvsem v okolici prehodov. Na sliki 17 (a) je predstavljena tipiˇcna odvisnost energije in specifiˇcne toplote hibridne celice v odvisnosti od temperature. Dobro razviden visok vrh specifiˇcne toplote ustreza strukturnemu prehodu iz dvoosne strukture v upognjeno. Temperaturo prehoda smo oznaˇcili s T2MC. V celici z N = 14 plastmi je T2MC f« 1.12. Temperatua strukturnega prehoda je niˇzja od temperature faznega prehoda v neograjenem sistemu, ki ustreza T MC NI = 1.1232 [152]. Nekoliko skrit v glavni vrh in precej manj jˇsi je vrh, ki us- treza ureditvenemu prehodu v sredini celice. Pri visokih temperaturah so urejene le molekule blizu sten, ko pa se omoˇcitveni plasti stakneta, dobimo znano dvoosno strukturo. Temperaturo ureditvenega prehoda smo oznaˇcili z T1 , v 14 plastni celici je njena vrednost T1M C ~ 1.18. Zaradi urejujoˇce narave sten nastopi ureditev v celici pri temperaturi viˇsji od prehoda v neograjenem sistemu, o ˇcemer smo govorili v poglavju o heterofaznih sistemih. S poveˇcevanjem debeline celice se T1M C pribliˇzuje vrednosti TNMIC, prav tako se proti tej vrednosti pribliˇzuje temperatura strukturnega prehoda T2MC. xxviii Razˇsirjeni povzetek 1.0 0.5 0.0 -0.5 -1.0 '--__ ' ,..-- ¦f1/ / N.0 ,'' \x --.-.. / v/ 2.0 1.6 1.2 0.8 0.4 0.0 0.2 0.4 0.6 0.8 1.0 z/d (a) 0.0 '?3 7 ¦ 2 ' i 1 0-^~^^~ 0.00 0.02 0.04 0.06 0.08 0.10 ? 2 (b) Slika 18 (a) Relaksacijska hitrost nekaj osnovnih direktorskih fluktuacijskih naˇcinov v odvisnosti od debeline filma. Ker direktorske fluktuacije vodijo strukturni prehod, pade njihova relaksacijska hitrost na 0 ob prehodu v upognjeno strukturo, ˇce je prehod zvezen, oziroma pri debelini, nad katero dvoosna struktura ni veˇc niti metastabilna, v primeru nezveznega prehoda. Pikˇcasta in ˇcrtkana ˇcrta ustrezata mejni debelini za stabilnost oziroma debelini prehoda. (? = 0,9, aS = 1,1 in g1 = g2 —> oo) Fluktuacije ureditvenega parametra Pri ˇstudiju fluktuacij se zanimamo za fluktuacije v dvoosni strukturi v bliˇzini prehoda v upognjeno strukturo. Loˇcimo ˇstiri neodvisne fluktuacijske naˇcine. Direktorske fluktuacije, povezane s fluktuacijami v smeri baznega tenzorja T-1, predstavljajo deformacije direktorskega polja v ravnini (x,z). Njihov osnovni naˇcin povzroˇca prav tako deformacijo direktorskega polja, kot je znaˇcilna za upognjeno strukturo. Zato direktorske fluktuacije vodijo strukturni prehod. Njihova relaksacijska hitrost pade ob debelini (temperaturi) prehoda na 0, ˇce je prehod zvezen. V primeru nezveznega prehoda je relaksacijska hitrost direktorskih fluktuacij ob prehodu konˇcna, a pade na 0 pri debelini (temperaturi), ki predstavlja mejo stabilnosti dvoosne strukture. Te lastnosti direktorskih fluktuacij so znak, da prav te fluktuacije predstavljajo mehanizem prehoda. Odvisnost relaksacijske hitrosti direktorskih fluktuacij v primeru nezveznega strukturnega prehoda kaˇze slika 18 (a). Fluktuacije dvoosnosti (fluktuacije amplitud ßi v smeri tenzorjev T±2) merijo porazdelitev molekul v ravnini pravokotni na ravnino obeh znaˇcilnih smeri v hibridni celici, torej ureditev v ravnini (y, z) in v ravnini (x, y). Prve predstavljajo fluktuacije dvoosnosti v delu celice, kjer je direktor v smeri osi x, medtem ko predstavljajo v delu filma z direktorjem v smeri osi z direktorske fluktuacije. Za drugi tip fluktuacij velja ravno nasprotno. Ker so v urejeni nematski fazi fluktuacije dvoosnosti precej trˇse od direktorskih fluktuacij, je nekaj najniˇzjih fluktuacijskih naˇcinov ß±2 lokaliziranih le v n = oziroma n = delu filma, kjer predstavljajo direktorske fluktuacije. Viˇsji Razˇsirjeni povzetek xxix 1.0 0.5 0.0 -0.5 -1.0 2.0 1.6 1.2 0.8 0.4 0.0 0.2 0.4 0.6 0.8 1.0 z/d (a) 0.0 / /najnižji simetrični / način ^^najnižji r-^ antisimetrični način 0.00 0.02 0.04 0.06 0.08 0.10 ? 2 (b) Slika 19 (a) Profila osnovnega naˇcina fluktuacij stopnje urejenosti. Polni ˇcrti predstavljata sklopnjena fluktuacijska naˇcina stopnje urejenosti glede na direktor v smeri osi x oziroma z, ˇcrtkani ˇcrti pa ustrezata pripadajoˇcima ravnovesnima profiloma. (b) Relaksacijska hitrost nekaj najniˇzjih fluktuaci-jskih naˇcinov. Pikˇcasta in ˇcrtkana vertikala ustrezata debelini filma ob meji stabilnosti dvoosne strukture in ob strukturnem prehodu. (? = 0,9, ?2 = 0,01258, aS = 1,1 in g1 = g2 —> oo) naˇcini deformirajo red v celotnem filmu. Fluktuacij ski naˇcini ±2 so degenerirani, njihovi profili pa so zrcalno simetriˇcni glede na sredino filma. Degeneracija se zgubi v filmih, kjer moˇci sidranja na obeh stenah nista enaki. Fluktuacije stopnje urejenosti predstavljajo sklopljene fluktuacije obeh neniˇcelnih ravnovesnih ureditvenih parametrov. Upodobimo jih lahko kot fluktuacije stopnje urejenosti glede na oba direktorja v filmu. Naˇcini so sodi ali lihi glede na sredino celice, kar je posledica sodosti poteniciala za fluktuacije [130]. Najniˇzji fluktuacijski naˇcin je najniˇzji lihi oziroma antisimetriˇcni naˇcin. Ta ustreza fluktuacijam debeline omoˇcitvene plasti ob steni in smo ga spoznali ˇze v poglavju o heterofaznem sistemu. Maksimum naˇcina ustreza mestom z najveˇcjim naklonom v spreminjanju stopnje urejenosti, kot je razvidno tudi s slike 19 (a). Oba naˇcina sklopljena skupaj predstavljata fluktuacije poloˇzaja osrednjega dela, kjer pride do zamenjave lastnih osi ureditvenega tenzorja. Viˇsji naˇcini spreminjajo obliko profilov in s tem izmenjal-nega podroˇcja. Relaksacijska hitrost osnovnega naˇcina fluktuacij stopnje urejenosti se ob prehodu oziroma ob meji stabilnosti moˇcno zmanjˇsa, vendar ostane konˇcna, saj strukturni prehod ni povezan z mehanizmom urejanja, ki ga te fluktuacije predstavljajo [glej sliko 19 (b)]. Relaksacijska hitrost doseˇze ob prehodu niˇzjo vrednost, ˇce gre za prehod v debelejˇsih filmih (prehod doseˇzemo z niˇzanjem temperature) oziroma pri temperaturah blizu temperature faznega prehoda v neograjenih sistemih (prehod doseˇzemo z veˇcevanjem debeline), ko dvoosna struktura izgine, namesto nje pa je pred upognjeno strukturo izotropna faza in je urejen le majhen del molekul tik ob steni. xxx Razˇsirjeni povzetek Slika 20 Fotografija tankega nematskega filma po razpadu zaradi spinodal-nega razomoˇcenja. Fotografijo so posneli F. Vandenbrouck in sodelavci objavljena pa je v reviji Phys. Rev. Lett. 82, 2693 (1999). Stabilnost tankih hibridnih nematskih filmov Zaradi vpliva povrˇsin se povpreˇcni red in fluktuacije v bliˇzini prehoda moˇcno spre-ˇ menijo, zaradi ˇcesar deluje med stenama dodatni tlak. Ce je tekoˇcekristalni film ograjen le z ene strani, ostale povrˇsine pa so proste, je v odvisnosti od dodatnega tlaka film lahko stabilen ali pa zaradi ojaˇcenja kapilarnih valov razpade v kapljice nematika in vmesna suha podroˇcja [104,158]. Primer nematskega filma po spinodal-nem razomoˇcenju kaˇze slika 20. K tlaku v hibridnem nematskem filmu prispevajo strukturni tlak, psevdo Casimir-jev tlak in van der Waalsov tlak [28,30]. Vsi so odvisni od nematske strukture v filmu. Na sliki 21 lahko vidimo odvisnost strukturnega tlaka od ravnovesne strukture in debeline filma. V okviru direktorskega opisa v homogeni strukturi ni strukturnega tlaka, strukturni tlak v upognjeni strukturi pa je moˇcno odbojen in pada z razdaljo kot 1/d2. Upoˇstevanje ostalih prostnostnih stopenj nematske ureditve privede do neniˇcelnega tlaka tudi v homogeni strukturi. Ta je kratkega dosega in zelo ˇsibek, saj je posledica moˇcno lokalizirane in majhne deformacije stopje urejenosti. Popravki k strukturnemu tlaku v upognjeni strukturi so zanemarljivo majhni, saj so nekaj velikostnih redov manjˇsi od glavnega prispevka zaradi elastiˇcnih deformacij direktorskega polja. Tudi strukturni tlak v dvoosni strukturi je odbojen, kar je posledica nasprotujoˇcih si robnih pogojev, ki povzroˇcajo deformacije, tem veˇcje ˇcim manjˇsa je debelina filma [29]. Pri majhnih debelinah je deformacija ureditvenega parametra vezana na celoten film, zato je tam tlak pribliˇzno enak tlaku v upognjeni strukturi. Z veˇcanjem debeline je deformacija vedno bolj vezana na osrednji izmenjalni del filma, tako da velikost tlaka pade moˇcneje kot v upognjeni strukturi. Primerjani sili ne ustrezata upognjeni in dvoosni strukturi pri istih parametrih, ampak gledamo upognjeno strukturo globoko v nematski fazi, kjer zadoˇsˇca opis z direktorskim poljem, dvoosno strukturo pa v temperaturnem obmoˇcju, kjer je ta stabilna, torej v bliˇzini temperature TNI. Razˇsirjeni povzetek xxxi 3.0 2.5 2.0 ¦V 100 10 ........: 1.5 - \ 1 ^^^^ ""•-- ] 1.0 0.5 . \ 0.1 r \ i 0.0 0 v~_ 0.01 01 ,\ 10 20 30 40 5 d [nm] 0 10 d [nm] (a) (b) Slika 21 Strukturna sila na enoto povrˇsine v hibridni celici (a) s homogeno strukturo ter (b) z upognjeno (ˇcrtkana ˇcrta) in dvoosno (polna ˇcrta) strukturo. Strukturna sila v hibridni celici z upognjeno ali dvoosno strukturo je nekaj velikostnih redov veˇcja, saj je posledica moˇcnih elastiˇcnih deformacij. Pri debelinah nad nekaj 10 nm pada velikost sile v dvoosni strukturi hitreje kot v upognjeni strukturi, saj postaja deformacija vedno bolj lokalizirana. (G1,G2 › ?, upognjena struktura globoko v nematski fazi in dvoosna struktura blizu temperature TNI.) V podrobnosti izraˇcuna in lastnosti psevdo Casimirjeve sile se tu ne bomo spuˇsˇcali. V sploˇsnem je njen izraˇcun v hibridni celici zaradi nehomogenosti ureditvenega parametra zelo oteˇzen. Rezultate, ki jih bomo uporabili pri ˇstudiju stabilnosti in interpretaciji eksperimentalnih rezultatov [158], bomo povzeli po ˇstudiji Ziherla in sodelavcev [65]. Izraˇcun velja za najpreprostejˇso, homogeno, strukturo in to dovolj globoko v nematski fazi, da lahko zanemarimo deformacije stopnje urejenosti. Poleg tega je ˇstudirana hibridna celica taka, v kateri stena, ki vsiljuje red v ravnini stene, na makroskopskem skali ne preferira nobene smeri. Na mikroskopskem in mezoskopskem nivoju to seveda nikoli ni res in je rotacijska simetrija zlomljena, ravnovesne strukture pa take, kot smo jih opisali v tem poglavlju. Do pomembne razlike pride pri opisu fluktuacij in fluktuacijske sile, saj opisana stena ne poruˇsi zvezne rotacijske simetrije okrog normale na ograjujoˇci steni. Psevdo Casimirjeva sila v takem sistemu je potem odvisna od dveh parametrov, razmerja ekstrapolacijskih dolˇzin ? = ?n/?p ter razmerja med debelino celice in kritiˇcno debelino za prehod v upognjeno strukturo, d/dc. Odvisnost psevdo Casimirjevega tlaka od debeline filma je upodobljena na sliki 22. Poleg opisanih strukturnega in psevdo Casimirjevega tlaka prispeva k skupnemu tlaku ˇse van der Waalsov tlak [31]. V eksperimentalnem sistemu, ki ga ˇzelimo obravnavati, je bila na trdni steni ˇse dodatna tanka oksidna plast, tako da sistem sestavljajo ˇstiri plasti. Njihove dielektriˇcne in optiˇcne lastnosti so zbrane v tabeli 1. Ker imata silicij in njegov oksid razliˇcen vpliv na van der Waalsovo interakcijo v xxxii Razˇsirjeni povzetek 0 -2 -4 -6 -8 -10 0 5 d * 10 15 d [nm] 20 25 Slika 22 Dodatni tlak v hibridnem nematskem filmu v stiku s trdno steno in s prosto povrˇsino globoko v nematski fazi (polna ˇcrta). Crtkana ˇcrta ustreza van der Waalsovemu tlaku, pikˇcasta in ˇcrtkano pikˇcasta ˇcrta pa psevdo Casimirjevi in strukturni sili. Poveˇcan je del v bliˇzini prehoda tlaka iz naraˇsˇcajoˇcega z debelino filma v padajoˇcega. povezavi z danim tekoˇcim kristalom in zrakom, je van der Waalsova sila v sestavljenem sistemu nemonotona. V grobem lahko reˇcemo, da je privlak pri majhnih debelinah filma posledica interakcije z dodatno oksidno plastjo, medtem ko je pri d > t, kjer je t debelina oksidne plasti, njen vpliv zanemarljiv in je odbojni tlak posledica interakcije s silicijem. Pri debelinah d ~ t sta pomembna oba vpliva. Odvisnost van der Waalsovega tlaka od debeline nematskega filma je predstavljen na sliki 22. Skupni tlak v hibridnem nematskem filmu ima nemonotono odvisnost od debeline. Za debeline manjˇse od marginalne debeline d* tlak naraˇsˇca z debelino, za d > d* pa pada. V prvem primeru je film nestabilen napram spinodalnemu razomoˇcenju, v drugem je stabilen. Pri sobni temperaturi dobimo za vrednosti parametrov ?# = 33 nm in ?p = 133 nm (? = 0,25 in dc = 100 nm) za marginalno debelino 18 nm, Tabela 1 Dielektriˇcne in optiˇcne lastnosti snovi, ki sestavljajo ˇstiriplastni sistem, pri sobni temperaturi. e je statiˇcna dielektriˇcna konstanta in n lomni koliˇcnik v obmoˇcju vidne svetlobe. Vsi parametri so podani pri sobni temperaturi. material e n silicij 12 3,5 silicijev oksid 14 1,5 5CB 18,5 7 n\\ 1,71 n± 1,53 4 2 Razˇsirjeni povzetek xxxiii 120 100 80 60 40 20---------— 20 22 24 26 28 30 32 34 36 T [°C] Slika 23 Temperaturna odvisnost marginalne debeline filma za razliˇcne vrednosti razmerja ekstrapolacijskih dolˇzin ?. Kvadratki oznaˇcujejo eksperimentalne rezultate, ki so jih izmerili Valignat s sodelavci [154]. kar ustreza rezultatom eksperimenta. Parametra ?# in ?p sta primerljiva ˇze znanim vrednostim [159,43,28]. S pribliˇzevanjem prehodu v izotropno fazo izmerjene vrednosti marginalne debeline moˇcno narastejo. Naraˇsˇcanja ne opraviˇci temperaturno spreminjanje dielektirˇcnih in optiˇcnih lastnosti tekoˇcega kristala, ki so povezane s stopnjo urejenosti. Zaradi spreminjanja le-teh se van der Waalsova in strukturna sila le malo spreminjajo. Tudi eksplicitna temperaturna odvisnost tako van der Waalsove kot tudi psevdo Casimirjeve sile ne da opaˇzenega, skoraj divergentnega obnaˇsanja. Tako obnaˇsanje pojasni temperaturno spreminjanje ekstrapolacijskih dolˇzin. V okviru direktorskega opisa in z razvojem povrˇsinske proste energije do ˇclenov drugega reda so ekstrapolacijske dolˇzine konstantne. Vendar pa rezultati meritev kaˇzejo, da postanejo ob pribliˇzevanju prehodu pomembni tudi ˇcleni viˇsjega reda, tako da se ekstrapolacij ske dolˇzine s temperatuo spreminjajo nekako kot ? oc S~2(T) [17,46-48,16]. Pri interpretaciji spinodalnega razomoˇcenja smo privzeli najpreprostejˇsi model: ekstrapolacijski dolˇzini se spreminjata na enak naˇcin, tako da ostaja njuno razmerje stalno ,ob pribliˇzevanju temperaturi prehoda pa se poveˇcuje kritiˇcna debelina dc. Ekstrapolacijski dolˇzni se ujemata z izmerjenimi vrednostmi globoko v nematski fazi, kjer je opis z direktorskim poljem dobro definiran. Na sliki 23 vidimo primerjavo temperaturne odvisnosti izmerjenih vrednosti za marginalno debelino z rezultati naˇsega modela za razliˇcne vrednosti parametra ?. Vrednosti ?# in ?p sta izbrani tako, da se ujemata izraˇcunana in izmerjena vrednost marginalne debeline globoko v nematski fazi. Najboljˇse ujemanje dobimo pri prej omenjenih parametrih. Spinodalno razomoˇcenje je eden od mehanizmov za opazovanje strukturnih in psevdo Casimirjevih sil v tanki plasti tekoˇcega kristala. xxxiv Razˇsirjeni povzetek Zakljuˇcki V doktorskem delu sem predstavila razliˇcne vidike vpliva ograjujoˇcih sten na ravnovesno ureditev in fluktuacije ureditvenega parametra v bliˇzini faznih in strukturnih prehodov. Pokazala sem, pri katerih pogojih privede (raz)urejevalna moˇc sten do zmanjˇsnja nezveznosti prehoda oziroma ga celo odpravi in postane prehod zvezen. V vseh ˇstudiranih sistemih sem poiskala mehanizme, ki vodijo prehod, poleg teh pa so me zanimali tudi lokalizirani fluktuacijski naˇcini, ki so povezani z obstojem faznih ali strukturnih mej. Fazni prehod v heterofaznem prehodu vodijo fluktuacije stopnje urejenosti, ki so odgovorne za rast omoˇcitvenih plasti, ko se pribliˇzujemo prehodu. Na omoˇcitveno plast so vezani niˇzji direktorski naˇcini v paranematskem sistemu in niˇzji fluktuacijski naˇcini dvoosnosti v nematskem sistemu v stiku z razurejujoˇcimi stenami. Strukturni prehod v hibridni celici vodijo direktorske fluktuacije, saj pri prehodu ne gre za urejanje nematika, ampak spreminjanje ureditve. Strukturni prehod je lahko zvezen ali nezvezen, vmes pa je trikritiˇcna toˇcka, katere zgornjo mejo sem doloˇcila. Fluktuacije stopnje urejenosti v hibridni celici predstavljajo povezano spreminjanje dveh parametrov, ki opiˇsejo povpreˇcno ravnovesno ureditev. Te prispevajo k urejevanju nematika v celici. Niˇzji fluktuacijski naˇcini dvoosnosti so zanimivi, ker so lokalizirani v le eni omoˇcitveni plasti, tisti, kjer te fluktuacije predstaljajo direktorske fluktuacije. Strukture v hibridni celici in prehode med njimi sem ˇstudirala tudi ob primerjavi rezultatov, ki jih da Monte Carlo simulacija nematskega tekoˇcega kristala v hibridni celici. Pokazala sem kvalitativno ujemanje rezultatov, za kvantitativno primerjavo pa moramo vzpostaviti ˇse nekaj povezav med temperaturami v okviru Monte Carlo simulacije in fenomenoloˇskega opisa ter med moˇcmi sidranja in vsiljevano stopnjo ureditve. Ograjujoˇce stene vplivajo posredno preko vpliva na ureditev tekoˇcega kristala tudi na van der Waalsov tlak v plasti ograjenega tekoˇcega kristala. Ker do zdaj ni bilo znano, kakˇsen vpliv ima anizotropija snovi na van der Waalsovo interakcijo, sem jo za primer enoosne simetrije sistema doloˇcila v tem delu. Pokazala sem, da se v enoosni snovi dielekriˇcne in optiˇcne lastnosti renormalizirajo tako, da so pomembni parametri za opis sistema n^)ij — (I)ij] aiaj = 1 + 5(Q)ij aiaj (2.3) 2 and Q = —S (3n ® n^ — I) (2.4) 1 — 2 is the tensorial order parameter of the nematic liquid crystal which represents the quadrupolar moment of the distribution, i.e., deviations from the perfect sphere. Here, I is the second rank unit tensor. If there are external fields acting on the nematic liquid crystal the cylindrical symmetry of the order can be lost. In that case, the tensorial order parameter is somewhat more complicated. With the procedure similar to the one in Eq. (2.3) one ends up with 1 1 —S (3n ® n^ — I) + 2 2 Q = —S (3n ® n^ — I) +P (e^1 0 e^1 — e^2 e^2) (2.5) 2 Phenomenological description 21 where P = (3/2) (sin ? cos 2?) denotes parameter of biaxiality of order, e\ represents a secondary director, and n, e\, and e^2 form the orthonormal triad. The full tensorial order parameter has five independent degrees of freedom. This can be easily recognized if one considers, that the order parameter is a second rank symmetrical traceless tensor: The second rank tensor has 9 degrees of freedom. Because it is symmetric, (Q)ij = (Q)ji, there are 3 degrees of freedom less, and, finally, the constraint tr Q = 0 reduces the number of independent degrees of freedom for one which leads to only 5 independent degrees of freedom. In the parametrization considered in Eqs. (2.4) and (2.5) these degrees of freedom are two angles determining the orientation of the director, the scalar order parameter, the angle specifying the orientation of the secondary director, and the parameter of biaxiality. In most of the thesis the order parameter will be parameterized with respect to the 5 base tensors of the symmetrical traceless tensor [32,33] 3n n^ - I To = ----------i=------, v6 e^\ ® e^\ - e^ 2 e2 e^\ ® e^2 + e^2 e^\ T\ =--------------j=---------- , T-i =--------------j=----------, (2.6) v2 v2 ei®n^ + n^®ei e^2 n^ + n ® e^2 T2 = -------------7=---------- , T-2 = --------------j=----------. v2 v2 All the above tensors are traceless and they are orthogonal with respect to the metric Tn : Tm = tr (TnTm) = ?n,m. (With this the magnitude of a tensor can be defined as || A|| = v tr A2.) In this parametrization the order parameter reads 2 Q = ^ qmTm, (2.7) m=-2 where qm = tr(QTm). The multiplicative constants are set so that the amplitude qo represents the scalar order parameter, parameters q±i are nonzero if the order is biaxial, and parameters q±2 represent deviations in the orientation of the director with respect to the assumed director n. If we interpret the nematic order in the eigenframe it can be schematically visualized by a block with the length of sides equal to ?i + ?>, where ?i’s are the eigenvalues of the tensor Q and ?> is the largest of them. In the case of uniaxial order the ratio between the sides is 1 : 1 : 4. Schematic representation of the nematic order is plotted in Fig. 2.1 together with representation of amplitudes along different base tensors Ti. The order that is represented by a tensorial order parameter is reflected in its influence on the macroscopic tensorial physical quantities, such as magnetic susceptibility. In the eigenframe, the magnetic susceptibility can be written as ' ?i 0 0 \ ?= 0 ? 2 0 , (2.8) 0 0 ?s J 22 Phenomenological description (c) (b) Figure 2.1 Schematic representation of (a) uniaxial and (b) biaxial nematic order. In (c) deviations of amplitudes from the uniaxial nematic order are visualized along base tensors. Variations of the degree of order correspond to a breathing mode (T0); variations of the degree of biaxiality simultaneously decrease one of the short sides and increase the other by the same amount (T1), whereas the deviations of the biaxial and nematic director are identified by rotations of the block about the long (T-1) and the two short axes (T±2), respectively. Phenomenological description 23 where ?'i > ?i, ?2 are susceptibilities in the direction of the director and in two perpendicular directions e^\ and e^2, respectively. By use of the derived order parameter [Eq. (2.5)] and the scalar product A : B = tr (AB) it can be rewritten as r- r- 2 r1 n^n^ i r 1 1 ? = ~?a ~S (3 - I) + ?b _P (ei ? ei - e2 ? e^2) + ?iI. (2.9) — 3 2 2 Here, ?a = ?~a - (?~i + ?~2)/2 is the biggest anisotropy of the magnetic susceptibility and ?~? = ??/S are susceptibilities of the perfectly ordered nematic, ?b = ?~i - ?~2 is nonzero in biaxially ordered nematic (?~? = ??/P), and ?i = (?i + ?2 + ?s)/3 is the average magnetic susceptibility, i.e., its isotropic part. 2.2 Phenomenological Landau–de Gennes theory Before we proceed with the Landau-de Gennes theory of phase transitions in liquid crystals let us first recall the general properties of phase transitions and the basic assumptions within the Landau description of phase transitions. 2.2.1 Landau theory of phase transitions The term phase transition denotes a change in the medium which is accompanied by discontinuity of some of the thermodynamic potentials and by a change of a certain physical quantity, e.g., density of a medium, macroscopic magnetization or polarization of the material, magnetic, electric and optical properties of the medium, etc. If entropy of the system is a continuous function of thermodynamic variables at the transition then the transition is of second order or continuous whereas it is of the first order or discontinuous if the entropy is discontinuous. In the first order transitions, heat is absorbed by the system in going from the low temperature to the high temperature phase. The absorbed heat or the latent heat of the transition is Ql = Tc?S, where ?S is the difference of the entropies of the two coexisting phases at the transition and Tc is the phase transtition temperature. The two phases of the medium can either possess the same symmetry or the symmetry is changed at the transition. The Landau theory pertains to the latter category of transitions. In the following, the Landau theory will be reviewed [34]. Although it was first established for continuous phase transitions it is used also for describing the discontinuous transitions, however, in this case special care should be taken because the order parameter exhibits a jump at the transition and is, thus, not necessarily small. In Landau theory, the information about the change of physical quantities as well as of the change of the symmetry of the system is gathered in the order parameter 24 Phenomenological description ?0 which is a macroscopic quantity, ?0 = V~1 ddr?(f), and neglects spatial and temporal fluctuations. The basic concept of the Landau description of phase transitions is the introduction of Landau free energy. The Landau free energy takes into account the symmetry of the system through the power series expansion in terms of the order parameter in the vicinity of the transition. This originates in the Legendre transformation of the grand potential ?(T, hi) of the system with which the external fields hi related to physical quantities that change with order parameter(s) ?i are replaced by the order parameter(s), F(T, ?i) = ?(T, hi) + / d drhi?i. From this, the equation of state of the system reads ?f — = hi, (2.10) ??i where F = J d drf. When there are no external fields, the equilibrium state is the one that minimizes f. If a solution to Eq. (2.10) with hi = 0 exists for a nonzero ?0 there can be spontaneous order provided that the free energy of the state with nonzero ?0 is lower than the free energy of the state with ?0 = 0. The Landau theory assumes that the order parameter is small in the vicinity of the transition so that only the lowest terms required by symmetry and preventing the free energy from diverging are kept in the expansion, 1 2 1 3 1 45 f = f0 +A?0-----B?0 + -C?0 + O(?0), (2.11) 2 3 4 where f0 is the free energy density of the disordered phase. Coefficients A, B, and C are determined phenomenologically and are temperature dependent. Coefficient C > 0 in order for f to have a local minimum for ?0 < ? whereas the parameter A must change the sign at the temperature below which the solution ?0 = 0 is not possible anymore so that the solution ?0 > 0 becomes stable. In the expansion A = a(T - T*) + ... only the first term is taken into account. The transition is discontinuous. Usually, parameters B and C can be regarded as being constants. If at some temperature B = 0 the temperature is referred to as a critical temperature. In systems with the center of inversion the odd-power terms are not allowed by the symmetry and the phase transition is continuous. If in addition the parameter C is negative in some temperature interval next even-power term has to be added to the expansion and the transition becomes discontinuous. The temperature at which the line of continuous transitions changes to a line of discontinuous transitions between the two phases is denoted as a tricritical point. In the equilibrium, the free energy of the system has its minimum, ?f — = 0 ??0 ?2 f ? 2 > 0. (2.12) m Phenomenological description 25 w T > TC T = TC \ J J T < TC \ / 1 1 (a) (b) Figure 2.2 (a) Free energy of a system with a continuous phase transition as a function of the order parameter for various temperatures. (b) Temperature dependence of the order parameter. If the transition is continuous the order parameter ?0 changes continuously to dis ?0T = V a = 0 in the disordered phase (2.13) ± — (T - Tc) C in the ordered phase which occurs below Tc = T*. Free energy of a system with center of inversion and a continuous phase transition is plotted in Fig. 2.2. In Eq. (2.13) we recognize the well-known critical exponent for the order parameter within the mean-field theories, ß = 1/2. At the phase transition, the first term in Eq. (2.11) vanishes and the free energy of a system is a function of the fourth power of ?0. This means that the free energy of the system is undistinguishable from the equilibrium value even if the order parameter deviates significantly from its equilibrium value. Thus, the continuous phase transition is characterized by increased fluctuations of the order parameter. At the discontinuous phase transition the temperature of the transition is shifted from T* which is by definition the temperature below which the disordered phase can not exist. Therefore, in the interval T* < T < Tc the disordered phase with ?0 dis = 0 is metastable whereas it is stable above Tc. Here, 2B Tc = T +------. 9aC (2.14) The ordered phase with ?0 ord = B 2C 4aC 1 + \ 1 -( -T*) - = ?c 1 +1 ____________________X Y 1 - 8?/9 ) - (2.15) 26 Phenomenological description T* T T T** 1 lCl (a) (b) Figure 2.3 (a) Free energy of a system with a discontinuous phase transition as a function of the order parameter for various temperatures. Dashed and dotted lines correspond to Tc < T < T?? and T? < T < Tc, respectively. (b) Temperature dependence of the order parameter in stable (solid line) and metastable state of the appropriate phase (dotted lines). is stable below Tc and it is metastable in the temperature interval Tc < T < T** = T* + B2/4aC. Here, ?c = ?0Td(Tc) and ? = (T — T*)/(Tc — T*). Temperature dependence of the order parameter in (meta)stable (dis)ordered phases is plotted in Fig. 2.3(b) and the free energy of the system as a function of ?0 is plotted in Fig. 2.3(a). The latent heat associated with the transition is ^ TcaV 2 Qi = Tc?o =-------?c (2.16) At the phase transition, the free energy has a parabolic shape and the pretransitional fluctuations are much smaller than the pretransitional fluctuations at the continuous transition. Fluctuations of the order parameter within the mean-field theory Although in bulk the order is a macroscopic quantity and, on the average, depends only on the external parameters such as temperature, pressure, and external fields, due to finite temperature it is characterized by spatial and temporal deviations from the average value, caused by thermal fluctuations. What is important when con-sidering fluctuations is their correlation length which determines how big are the islands characterized by different order. When the correlation length of the fluctu-ations is small compared to the typical dimensions of the system the fluctuations do not mask the average behavior of the system. On the other hand, highly cor-related fluctuations change the order on large scales and perturb the macroscopic appearance of the system. Since we are interested in spatial deviations of the order Phenomenological description 27 parameter with respect to the mean-field value ?0 defined either in Eq. (2.13) or in Eq. (2.15) ?(r) = ?0 + ?(r*), (2.17) where {?(f)) = ?0 and ?(r*) are spatially-dependent fluctuations, (?(r*)) = 0. To describe spatial variations of the order parameter a gradient term has to be added in the free energy density which reads in the first nontrivial order L(??)2/2. Here, LV~2/3 is the measure for free energy density associated with deformation of the order parameter and V is the volume of the system. The correlation function of fluctuations reads ?(r) = (?(r)?(0)) - (?(0)) , (2.18) where we have used the equality {?(r)) = (?(0)) which is due to transitional invari-ance of the system. Taking into account Eq. (2.17) the correlation function reduces to ?(r) = (?(r*)?(0)). (2.19) Because of the continuous translational invariance of the system the functions of r can be expanded in a Fourier series, f(r) = J2q f(o) e~iq'r, so that ?(q) = (|?(q)|2). The amplitudes ?(q) are derived from the free energy of the system which can be by applying the Fourier series expansion rewritten in a form F = drf (2.20) V(f0 +A?0-----B?30 + —C?0) + VzJ(A - 2B?0 + 3C?0 + Lq )|?(q)| . 2 3 4 2 ^ Since the fluctuations are assumed to be small the free energy is written out only up to the quadratic terms in fluctuations. From the equipartition theorem we know that each degree of freedom which enters into the energy with a quadratic term holds the internal energy kBT/2. Taking into account this and the Eq. (2.20) the Fourier transform of a correlation function reads ~ (q) = kBT/V(A - 2B?0 + 3C?02 + Lq2) which gives in the direct space ?(r*) =e~r/. (2.21) Here, ? = L/(A - 2B?0 + 3C?02) is the correlation length of fluctuations. For finite ?, correlations are weak and decrease exponentially whereas for infinite correlation length the correlations are long-range and decrease inversely with distance. From the result for ?0 in Eq. (2.13) the correlation lengths in the vicinity of a continuous transition (B = 0) have the following temperature dependence, L a(T - Tc) 28 Phenomenological description ? = V (2.22) L ; T < Tc. 2a(Tc — T) On approaching the phase transition the correlation length in both, high temperature disordered phase and low temperature ordered phase, diverges. The critical exponent is in both cases ? = 1/2 which is again the common feature of mean-field theories. Due to fluctuations the islands of ordered (disordered) phase occur in the disordered (ordered) phase. Typical dimension of these islands is the correlation length. Far from the transition the correlation length is small and the islands of (dis)ordered phase do not change the macroscopic behavior of the system. Correlations are smaller in the ordered phase. Getting closer to the transition the dimension of islands grows and, finally, at the transition domains of different order extend over the whole system. In systems with a discontinuous phase transition the correlation lengths in the two phases read ? = L a(T — T?) ? = L —-— ; T > Tc, ; T < Tc. (2.23) 2a(T? — T) + B2(1 +1 + 4aC(T? — T)/B2)/2C Neither of them diverges at the phase transition, however, they diverge at the temperature where the given phase ceases to exist even as a metastable phase. Nondi-vergent behavior of the correlation length of fluctuations at the transition is a characteristics of a discontinuous phase transition and leads to occurrence of metastable phases and a jump of the order parameter at the transition. At the transition, both of the phases are energetically equally favored, thus, the correlation lengths of fluctuations in both phases are equal there. Goldstone mode and soft mode As it was derived in Eq. (2.20) the free energy associated with a certain fluctuation mode is (2.24) •Fq =----(? + q ) . In the following Sections this free energy will be related to the relaxation rate µ^ for the decrease of the excited mode, µ^ oc (?-2 + q2) and consequently, T$ oc µ^. There are two specific types of fluctuation modes in which we will be interested in the study of thermal fluctuations in confined nematic liquid-crystalline systems: Goldstone modes and soft modes. Phenomenological description 29 When the correlation length of fluctuations diverges the relaxation rate of a long wavelength mode (q —> 0) drops down to 0 — the free energy associated with a deformation caused by the mode is very small. Soft modes are modes whose relaxation rate drops to zero at the transition whereas it is nonzero elsewhere. From the consideration of correlation lengths we can deduce that the q = 0 fluctuation mode of the order parameter characteristic for a given continuous transition belongs to the category of soft modes. On the other hand, the relaxation rate of a Goldstone mode drops to zero at the transition and stays critical within the entire range of a given ordered phase. At the phase transition from the disordered to the ordered phase the symmetry of the system is typically lowered. This spontaneous breaking of the symmetry is accompanied by a multiple degeneration of the ground state. The system is brought from one ground state to the other by symmetry operations of the high-symmetry phase. If the broken symmetry is continuous a fluctuation mode occurs whose deformation of the system represents a continuous change from one ground state to another and so on. In the thermodynamic limit with q —> 0 this excitation does not increase the free energy of the system. The mode is called a Goldstone mode and the condition for its existence is stated in a Goldstone theorem [35-37]. Ginzburg criterion The measure of the importance of fluctuations of the order parameter is the difference between the order parameter and its mean-field value averaged over the correlation volume V? ~ ?3, where d is the dimensionality of the system. Fluctuations are negligible if in the ordered phase ((??cor)2) = V?- V ddr ?(r) —d2T2. (2.26) Here, ?cv = a2Tc/2C denotes the difference between heat capacity per unit volume - ??cor = V?- d r (?(r) - ?0), (2.25) in ordered and disordered phase, and ?0 = L/2aTc is the bare correlation length. Parameter Id = f ddx x-(d-2)Y(x) depends solely on the dimensionality of the system, whereas the integral Y(x) = /0°° dz zd-1 J(d?d/(2?)d) eizcos? /(z2 + x2) comes from the correlation function, ?(r) = (kBT/Lrd-2)Y(r/?). The criterion was first derived by Ginzburg [39] which yields the name Ginzburg criterion. Although the 30 Phenomenological description criterion was derived by considering fluctuations in the ordered phase it also applies in the disordered phase [38]. For d > 4 the left hand side of Eq. (2.26) diverges as T approaches Tc and the Ginzburg criterion is always satisfied near the critical point. On the contrary, for d < 4, \Tc — T\4~d tends to zero as T —> Tc and the Eq. (2.26) is never satisfied near the critical point. Thus, the mean-field theory provides an adequate description of continuous transitions for d > 4 and it breaks down for d < 4. The dimension dc = 4 below which the mean-field theory fails to describe the continuous transition adequately is called the critical dimension. For d < dc, the mean-field theory is valid far from the phase transition and up to the Ginzburg temperature TG, \T T l / \ 2/(4—d) \C — G\ / kBId \ -----------~-----d . (2.27) Tc 2?cv$j In systems with d < dc , however, with ?0 —> oo the mean-field theory is valid even close to the transition; ?0 —> oo in systems with long-range forces. 2.2.2 Phase transition in a nematic liquid crystal Isotropic-nematic phase transitions are typically only weakly discontinuous which is reflected in the narrow interval of metastable phases. Typically, the isotropic phase can be undercooled for ~ 1 K and the nematic phase can be overheated for ~ 0.1 K. Therefore, the isotropic-nematic transition can be described within the Landau theory of phase transitions. Here, we will use the extended theory which permits local variations of order parameter, however, the variations are on the lengthscales much larger than the length of a molecule (typically 1 nm) which preserve continuum description of the system. To do this, the free energy is expanded in a power series of scalar invariants of the order parameter and the gradient terms are considered only up to the quadratic term. The order parameter for the isotropic-nematic transition was derived in Section 2.1. Here, we will use the parametrization in terms of five base tensors in Eqs. (2.6) and (2.7), so that in the uniaxial nematic trQ2 = S2. The free energy of the system must be invariant to all symmetry operations that preserve the system in the high symmetry phase. The isotropic phase has full symmetry. Scalar invariants of a tensor are its trace and determinant. Both invariants are preserved with rotations, however, the determinant changes sign with reflections. Thus, the free energy is expanded in a power series of traces of powers of the order parameter. As defined in Section 2.1, tr Q = 0, so the first nontrivial term is tr Q2. The absence of the first order term is in accordance with the existence of the stable high temperature phase with Q = 0. There are no other second order terms than tr Q2 and there is also only one third order invariant, tr Q3. It is included in the Phenomenological description 31 n 1 2 1 3 1 22 fhom = fiso +A(T - T ) tr Q-----Btr Q + —C( tr Q ) , (2.28) a) b) c) Figure 2.4 Schematic representation of three basic elastic deformations of the director field in an uniaxial nematic liquid crystal: (a) splay with (V-n)2 = 0, (b) twist with [n^ ¦ (V x n^)]2 = 0, and (c) bend with nonzero [n^ x (V x n)]2. expansion because ±Q have different physical meaning; from Eq. (2.9) we can see that for materials with positive magnetic anisotropy, which are studied in this thesis, Q > 0 corresponds to nematic phase in which molecules tend to orient along the director whereas Q < 0 is associated with a nematic phase in which molecules orient in the plane perpendicular to the director. There are two fourth order invariants, trQ4 and (trQ2)2, however, for symmetric tensors trQ4 = (trQ2)2/2 and we will use only (trQ2)2 in the expansion. Higher order invariants are not considered here so that the homogeneous part of the free energy density reads 1 2 1 3 1 —A(T - T ) tr Q-----B tr Q + — 2 3 4 where A, B, and C are temperature independent, and T? is the supercooling temperature. The magnitude of A is typically 10-5 J/m3K whereas B ~ C ~ 10-6 J/m3. Since spatial variations of order parameter are allowed gradient terms have to be added to the free energy. There are many symmetry allowed invariants related to gradients of the tensorial order parameter. Up to second order derivatives: L(1)Qij,ij; L(2)Qij, Qij, L(2)Qij,iQ j, L(2)Qij, Q j,i L(2)Qij,iQj L(2)Qij, Qij; L(3)QijQij, Q L2 QijQik,jQkl,l, L3 QijQik,kQjl,l, L4 QijQik,lQjk,l, L5 QijQik,lQjl,k, L6 QijQik,lQkl,j, etc., where Qij,k = ?Qij/?xk [40,41]. In the vicinity of the phase transition, one is not interested in elastic deformations of nematic director but rather in spatial variations of the degree of nematic order. Therefore, the pretransitional nematic system is described adequately within the usual one-elastic-constant approximation, 1 f el = -LvQ.vQ. (2.29) 2 Here, L = L(12 is typically in order of 10-11 N to 10-10 N. Deep in the nematic phase, variations of the scalar order parameter do not play significant role and the main contribution to the elasticity of the system is due to elastic deformations of the director field which contribute to the free energy density Frank 1 ( „ n^2 ^ „ 2 ^ „ ^ f el = ~ K11 (v • ) + K22[n ¦ (V x n^)] + K33[n x (V x n^)] \ (2.30) 2 32 Phenomenological description +K13V • [n(V • n^)] — K24SJ ¦ [n(V • n^) + n x (V x n^)]. The terms apply to invariants with respect to rotations of the system as a whole, space inversion, and the transformation n —> —n^. Here, the first three terms correspond to three basic deformations of the director field, splay, twist, and bend deformations, respectively. These deformations are schematically represented in Fig. 2.4. The description of the nematic liquid crystal within the frame of elastic deformations of the director whereas the other degrees of freedom of the nematic order are considered to be constant is known as the Frank elastic description [42]. The stiffness of the nematic with respect to a given deformation is determined by parameters Kij; typical magnitude being 10 pN [4]. The other two terms in Eq. (2.30) correspond to splay-twist and saddle-splay deformations, respectively. From their form it can be seen that unlike the three bulklike terms they represent a kind of surface terms, therefore, they are often neglected in studies. In the last decade the interest in these terms was renewed in the studies of spontaneously deformed nematic liquid crystals characterized by stripe-domain structure [43,44]. However, there are still arguments in favor or against these terms. The main reason for that is that unlike the first three terms which are obviously positive definite for the K13 and K24 terms the existence of the lower limit is not that evident. In fact, it can be shown that the saddle-splay term has the lower limit [45] whereas the K13 term does not. The one-elastic-constant approximation used in Eq. (2.29) corresponds to the case K ii = K, K13 = 0, and K24 = K in the Frank elastic theory and does not allow any spontaneous deformation of the director field. In the uniaxial nematic liquid crystal parameters L and K relate through K = 9LS2/2. Within the mean-field theory there are no spontaneous elastic deformations since any deformation increases the free energy. As discussed before, if we take into account thermal fluctuations of the director even in bulk the director field is not uniform but bent due to Goldstone director mode. However, when the nematic liquid crystal is subject to interactions with the confining walls the equilibrium order can be also perturbed. In this thesis we are interested in highly constrained nematic liquid crystals where the interaction with the walls and their effect can not be neglected. On the microscopic level, the molecules of the walls and of the liquid crystal attract each other via van der Waals interaction. This interaction is rather short-range since it decreases with distance as 1/r6. Therefore, it is assumed that only the molecules that are in a contact with the wall interact with it — contact interaction. On the macroscopic level, this interaction reduces to the quadruple-quadruple interaction and can be, as the interaction between molecules of the liquid crystals, expanded in a power series over scalar invariants. Usually, only the first Phenomenological description 33 order term is taken into account although some experimental evidence suggest that higher orders could be important as well [17,46-48,16]. (Temperature dependence of extrapolation length of the substrate-liquid crystal interaction can not be modelled by taking into account only the quadratic term. This is discussed in more detail in Section 5.4 on page 140.) The contribution from this interaction to the free energy density of a nematic liquid crystal is modeled by an improved expression of the Rapini-Papoular expression [49], first suggested by Nobili and Durand [17], 1 2 fS =—Gtr (Q — QSi ) ?(z — zS), (2.31) 2 where G is the strength of the interaction, QS is the preferred value of the tensor order parameter at the substrate, and the wall is located at z = zS. In the case of uniaxial nematic order the anchoring strength G can be related to anchoring strength W for the bare director description as W = 3GS2, where S is the scalar order parameter. Often, the anchoring strength is measured in terms of the extrapolation length which denotes the length on which the director field would relax to the one preferred by the substrate. It is the measure for the relevance of the competing elastic distortions vs. violating substrate induced order, ? = K/W or, similarly, ? = 3L/2G. The criterion whether the anchoring is strong or weak is the ratio between the extrapolation length and typical dimension of the system; ?/d —> 0 corresponds to strong anchoring whereas ?/d 3> 1 is associated with weak anchoring. It is useful to rewrite the quantities into a dimensionless form. Thus in the following, all coordinates will be measured in terms of the film thickness d and the correlation length ?NI = ?d = 27CL/B2 f« 10 nm. The order parameter will be rescaled in units of the scalar order parameter of the nematic phase at the phase transition temperature, Sc = 2B/3y6C which is typically between 0.2 and 0.6, and the temperature will be controlled by ? = (T — T*)/(TNI — T*), where TNI = T* + B2/27AC is the bulk nematic-isotropic phase transition temperature; the reduced temperatures ?=1,0, and 9/8 correspond to the bulk phase transition temperature, and to the supercooling and superheating limits, respectively. Thus, the dimensionless free energy density reads 1 2 r 3 1 22 1 2^Q .^ f = fiso + ?trQ — V6tr Q +(tr Q ) + ? VVQ, (2.32) 2 2 2 where f is measured in units of f = L?N~2S2 and V corresponds to derivatives with respect to dimensionless coordinates xi <— xi/d. The substrate induced contribution to the free energy density is rescaled in 1 2 fS = —g tr (Q — QS) ?(z — zS), (2.33) 2 — QS) ?(z — 34 Phenomenological description where g = (?2NI/Ld)G or g = (3?2NI/2d)?-1 in terms of the extrapolation length. Unless stated otherwise, the calculations will be performed for a nematic liquid crystal 5CB with A = 0.13 x 106 J/m3K, B = 3.89 x 106 J/m3, C = 3.92 x 106 J/m3, L = 9 x 10-12 N, and T? = 307.1 K [50,51]. Here, Sc ~ 0.27, TNI — T? f« 1.1 K, and the latent heat ql = B4/729C3[TNI/(TNI — T?)] ~ 1.5 • 106 J/m3 which is rather small comparing to the latent heat of the typical first order transition — for melting of ice ql = 3.36 • 108 J/m3. 2.2.3 Correlation lengths of the nematic order parameter Again, we will discuss the validity of the mean-field description of the isotropic-nematic phase transition in terms of correlations of fluctuations of the nematic order parameter. In the bulk, the average equilibrium order is uniaxial with the order parameter Qmf = A = a0T0 and a0 = S. Due to thermal fluctuations, the local order can deviate from the average mean-field value, Q(r) = A + B(r*), where (B(r)) = 0 and ||B|| 0) is the Goldstone mode. Fluctuations of other degrees of freedom of the nematic order are much more energy consuming. The correlation length of fluctuations of the scalar order parameter diverges at the superheating temperature ?** = 9/8. In the vicinity of the transition to the isotropic phase ?n o/?ni ~ 6 — 5?, ?n±i/?ni ~ 18 — 9?. (2.38) Now we can set the meaning to the correlation length ?NI — it is the correlation length of fluctuations of the scalar order parameter at the bulk isotropic-nematic phase transition. The hardest type of fluctuations in the uniaxial nematic phase are fluctuations of biaxiality since they oppose to the established symmetry of the phase. Temperature dependence of correlation lengths of different types of fluctuations are depicted in Fig. 2.5. 2.3 Dynamics of the ordered fluid At a phase transition or on applying external fields the order of the system changes. To describe the changes on the macroscopic level we have to restrict ourselves to phenomena which happen on timescales much larger than the characteristic times for microscopic phenomena — the time between two distinct collisions. The dynamics on the macroscopic level is related to acquiring the equilibrium; the process associated with that is dissipative. In this Section the equations describing the dissipative dynamics will be derived [52,53]. 36 Phenomenological description 14 12 10 8 6 4 2 0 -0.5 >\' fluctuations of biaxiality nematic phase \ isotropic phase \. fluctuations of the scalar order parameter director ^\ fluctuations "\, all types of fluctuations _———---- 0.0 0.5 1.0 ? 1.5 2.0 2.5 Figure 2.5 Temperature dependence of correlation lengths in the isotropic and nematic phase. Continuations of the lines across the dotted vertical cor-respond to the correlation lengths in the corresponding metastable phase. First, we will introduce the basic terminology and physical quantities on the example of a homogeneous body moving in an isotropic liquid. The state and motion of such system is described by a set of macroscopic observables xi, where i = 1,.., s and s is the total number of degrees of freedom of the system. In general, these are the 3 spatial coordinates, 3 components of the macroscopic velocity of the system, Euler’s angles, etc., and components of the order parameter. The entropy of a system is a function of the thermodynamic state of the system, thus, it depends on the just mentioned macroscopic observables, S = S(xi). Let us define thermodynamically conjugate quantities to observables xi, Xi = - S . ?xi With these, the production of the entropy can be written as S = S = - ^ Xix˙i. (2.39) (2.40) In the equilibrium, the entropy reaches its maximum, dS = J2?S/?xi dxi = 0, which is realized if Xi = 0 for each i. Simultaneously, the production of entropy is equal to 0. The production of entropy accompanies irreversible processes, i.e., the approach of the equilibrium. Thus, the system which approaches the thermal equilibrium acts like an origin of the entropy. So far as the state of the system is not too far from the equilibrium the linear relations between “flows” xi’s and “forces” Xi’s can be assumed [54], s (2.41) = - y^?ijXj, j =1 Phenomenological description 37 with kinetic coefficients ?ij. (The relaxation of slightly non-equilibrium system is described by x˙i = - J2j ?ijxj. From the definition of Xi in Eq. (2.39) and, again, not to far from the equilibrium, the thermodynamic conjugate Xi’s can be expanded in Xi = J2j ßijxi, where ßij = -?2S/?xi?xj and, thus, ßij = ßji. Then, by expressing xi from the latter expansion and by using this in the expression for x˙i the Eq. (2.41) follows directly [52].) In 1931 Onsager showed that from the reversibility of dynamical laws governing the microscopic processes behind observable macroscopic phenomena ?ij = ?ji, (2.42) which is known as the Onsager’s reciprocity principle [55,54]. The state of a rigid body under a dissipative motion in a viscous medium is described by generalized coordinates Qi and impulses Pi, where i = 1,.., s. If H = H(Qi,Pi) is the energy of a system then the non-dissipative motion is determined by ?H — where Pi = ?H/?Qi. When the processes become dissipative the dissipative terms introduced in Eq. (2.41) have to be added to equations of motion [Eq. (2.43)], Pi = - H y^?ijXP.. (2.44) The deviation of the entropy of a closed system, which is not in its equilibrium, from its maximum value equals ?S = - Amin/T, where Amin is the minimum work required to bring the system (body in a system) in the given non-equilibrium state and T is the temperature of the system [52]. In the case discussed here Amin = ?H, where P i = - H , (2.43) thus, XP. = -?S/?Pi = 1/T(?H/?Pi) = Qi/T and P˙i = - J2s=1?ijQ j dis j ?ij ‹ ?ij /T. Since the matrix ? is fully symmetric the latter eqution can be reduced to (Pi)dis = -?D/?Qi, where 1 2 i,j=1 D = — ^ ?ijQiQj (2.45) is a bilinear form of velocities, which is called the dissipation function. Now, the equations of motion can be rewritten in ?H ?D ?Q --------- P i = -?Q - D . (2.46) Temporal evolution of the system is determined by a reactive part and a dissipative part describing the part of the energy of the system which converts into heat. The 38 Phenomenological description rate of the decrease of the energy of the system is thus, H˙ = ^i[Qi(?H /?Qi) + P˙i(?H /? Pi)] = J2iQi[(?H/?Qi) + Pi] = - J2iQi(?D/?Qi) = -2D. The above derived equations of a dissipative motion can be generalized for macroscopic motion of continuum media such as liquids and liquid crystals. Here, we have to take into account the spatial dependence of macroscopic velocity as well as of macroscopic observables. In that case, the entropy of the system is a function of macroscopic observables and their gradients, S = /d 3 r s(xi, ?xi). The production of entropy is then ?s ?s \ —x˙i + ?—?x˙i ?xi vXi ?s ?s \ / 3 ^-^ / ?s \ ------?— x˙i + d r /_, ? · ? x˙i , (2.47) ?xi ??xi oVXi d r s = ?xi d r TJ-----? x˙i + d r TJ ? · ?xi ??xi where the first term can be rewritten in terms of products of flows x˙i and forces Xi, ?s ?s Xi = ------- ?—, (2.48) ?xi ??xi which determine the dissipative nature of the motion. The second term in production of entropy, /d 3 r$^i? · [xi(?s/??xi)] = J2iS dS · [xi(?s/??xi)], corresponds to dissipation of energy on the surfaces. This can be neglected in bulk systems, however, we will neglect it even in the case of systems with high surface-to-volume ratio. The deviation of the entropy density from its maximum value is again related to the minimum work required to bring the system in a given non-equilibrium state, ?s = - amin/T. Due to inhomogeneity of the system we do not now the exact amount of this work, however, we know that it depends on variations of macroscopic observables and their gradients which cause internal friction in the system. The change of the macroscopic velocity for a constant value or the rotation of a system as a whole do not cause any relative internal motion, thus, they do not contribute to the dissipation. Therefore, the minimum work and consequently, the dissipation function do not depend on velocity but rather on its gradient. The work related to changes of the order of the system is phenomenologically determined to be equal to f(?,??) as it was written out in Eqs. (2.11) and (2.32). In addition to these contributions there is also a contribution which couples the velocity field and the order of a system. The equations of motion of a continuum system are written in the same way as for the rigid body [Eqs. (2.44) and (2.46)], however, the energy H has to be replaced by the energy density h = H/V. The dissipative part of equation is described by a dissipation function which is constructed to be a bilinear form of invariants of macroscopic observables and their gradients allowed by symmetry. Coefficients Phenomenological description 39 ?˙ = - D . (2.49) ?v = -Vh + V • D , (2.51) before the terms are determined phenomenologically. The state of the system and its dynamic behavior is further determined by the dynamics of the order parameter(s), ?D ?? The evolution of the order is purely dissipative. In general, it is coupled to gradients of the velocity field which makes the description even harder. The energy density of an isotropic fluid reads h=-?v + p, (2.50) 2 where ? is the density of the fluid, v is the spatially dependent macroscopic flow, and p is the pressure in a given point. The state of the system in a point r is described by the impulse p = ?h/?v = ?v. The reactive part of the equation of motion then reads ?v = - Vp, where v is the material time derivative of the velocity, v = ?v/?t + (v ¦ V)v. Taking into account the dissipative nature of the motion, X^= - (1/T)V • (?amin/?Vp), where we have taken into account that amin depends solely on gradient of the velocity. From this, the equation of motion reads ?D ?Vv where (?D/?^v) = ?(?amin/?\Jv) and D is the dissipation function which is in the case of isotropic fluid D = [?'(Vv) : (Vv) + ?" tr (Vv)2 + ?'"( tr Vv)2]/2. The viscous coefficients ?', ?", and ?" are determined phenomenologically. In incompressible fluids ?'" = 0, whereas ?1 = ?" because D must be zero for pure rotations, v = ? x f. Taking into account all these, the Eq. (2.51) reduces to the well-known Navier-Stokes equation, ?v ?t In the anisotropic liquid, the dissipation is also due to variations of the order parameter(s). Therefore, first, terms bilinear in invariants of the order parameter and, secondly, terms coupling the order and the motion of the system, have to be added in the dissipation function. The latter terms, denoted as Da, depend on the character of the order parameter and will be written out at the end of this Section. The dissipative terms related solely to the variations of order can be written out in terms of linear laws, ?D ?? where by definition X? = V • [?s/?(V?)] - (?s/??). The change of the entropy density due to variations of the order parameter, s = -f(?, V?)/T, thus, ?f ?f\ da V • t=-------—-----D . (2.54) ? \/? ?? ?—+?( v ¦ V)v = - Vp + ?V v. (2.52) ?˙ = -?'X?-----D , (2.53) ?? ?? 40 Phenomenological description The left-hand side of Eq. (2.54) and the second term of its right-hand side are associated with the friction whereas the force Xv corresponds to the generalized elastic force which is due to the macroscopic order. In equilibrium, both forces are equal to 0. Due to the macroscopic order the equation of motion changes as well, ?v , ?Da ?+ ?(v ¦ V)v = — Vp + ? V v + V • -r-,^ . (2.55) ?t ?vv The part of the dissipation function which couples the macroscopic flow and the order in the system is a bilinear form of gradients of velocity and the order parameter. In the case of a scalar order parameter bilinear invariants coincide with scalar invariants of gradients of velocity, thus, 1 2 „v2 1 2 T-r^2 1 2t-t^ t-t - D = -?\? (tr Vi>) + ?2? tr (v v) +?s? v v : v v. (2.56) 2 2 2 For the vectorial order parameter, ? —> f?, D = -?i(? ¦ vv) +?2(vv ¦ r?) + ?^(? ¦ vv) ¦ (vv ¦ r?), (2.57) 2 2 2 whereas in the case of a tensorial order parameter, ? —> ?, 1 n^ 2 1 t-7-> r7^ 1 T-,-> D = —?i(? : vv) +?2(vv?) : (vv?) +?s(vv?) : (?vv) 2 — 2 — — 2 — — +?i(?vv) : (vv?) +?^(?fvv) : (?vv) +?6(vv : ?)(? : Vv) 2 — — 2 — — 2 — — 1 T-7-M2 1 _^^ 1 T-,^2 +?7[tr (?vv )] + ?s tr (? vvvv ?) +?gtr(?Vv) . (2.58) 2 — 2 — — 2 — In the case of a symmetric, traceless, second rank tensor the dissipation function Da is less complicated since (? : Vv)2 = (Vv?) : (?Vv) = (?f\7v) : (Vv?) = (Vv : ?)(? : Vv) = [tr (?Vv)]2 = tr (?Vv)2 = [n • Vv • n^]2. In the case of anisotropic liquids, due to the coupling between the macroscopic motion and the order of the system the description of dissipative processes is rather complex. One has to deal with (1 + 3 + the number of components of the order parameter) coupled partial differential equations: V • v = 0 for the incompressible fluid, and Eqs. (2.55) and (2.54), for the same number of observables: pressure, 3 components of the velocity field, and independent degrees of freedom of the order parameter. However, in nematic liquid crystals the typical time for equilibration of the velocity field, ?v oc 1/?, is several orders of magnitude smaller than the typical time ?n for the relaxation of the nematic director, ?^/?u ~ 10-6 [4,56]. Thus, the adiabatic approximation which assumes the velocity field to immediately follow the orientational changes of the system is not far from reality. In the following, we will be interested in pretransitional dynamics of the liquid crystal without any macroscopic Phenomenological description 41 = —?—, (2.59) motion. If there is no macroscopic velocity when the system is far from the phase transition and if the transition is approached with small and slow changes, within the adiabatic approximation, no motion will be induced. In that case the dissipative equations derived in this Section reduce to ?? ?f — = — ?— ?t ?? where ?/?? = ?/?? — V- (?/?V?) denotes functional derivative with respect to ? and ?_1 is a generalized viscosity associated with the relaxation of the order parameter. The same result could be obtained also directly from the Landau-Khalatnikov equation [7]. In 1954 Landau and Khalatnikov proposed that in the case of a non-equilibrium configuration of the order parameter the latter relaxes to the equilibrium value as d?/dt = —?~1??/??, where ? is the appropriate thermodynamic potential, and ? is a generalized viscosity coefficient. Now, the equation is known also as a time-dependent Ginzburg-Landau model (TDGL) [38]. The Landau-Khalatnikov equation obtained in Eq. (2.59) can be understood as follows: The equilibrium configuration of the order parameter is determined by the minimum of the free energy, ?T = JdV?f(?, V?) = / dV(?f /??)??, which is satisfied for ?f/?? = 0. If the system is out of the equilibrium ?f/?? acts like a generalized elastic force which is balanced by viscous forces. When the macroscopic velocity can be neglected the viscosity is related solely to the rate of change of the order parameter ?˙. 2.3.1 Pretransitional collective dynamics in a nematic liquid crystal In the equilibrium, the macroscopic order corresponds to the minimum of the free energy. Due to T > 0 there are motions on the microscopic level which can result in a collective motion — thermal excitations with characteristic energy ksT. When the system is excited out of the equilibrium it relaxes back following the relaxation equation [Eq. (2.59)]. For the nematic liquid crystal with a tensorial order parameter the generalized viscosity is in general a tensor. In the ordered phase, one can expect that due to the anisotropy of the order the viscosities differ with respect to directions along different eigenvectors. In nonhomogeneously ordered nematic the generalized viscosity can be also spatially dependent. However, in order to simplify the description the generalized viscosity will be assumed to be isotropic with the effective value equal to the average viscosity, ?~1 = ?_1I. In the case of a nematic liquid crystal with the free energy density written out in Eqs. (2.32) and (2.33) the relaxation equation reads ?Q ?t 3 V = —?Q + 3v6 Q —2QtrQ +? V Q, (2.60) —?Q + 3v6 Q2 —2QtrQ2 + ?2V2Q, 42 Phenomenological description where time is measured in units of time ?a = 27C?-1/B2, which is related to the relaxation of the alignment, t <— t/?a. The value of ?a ~ 10-8 s is determined phenomenologically [32]; typical values of the effective generalized viscosity are ?-1 ~ 0.01 kg/ms [4]. The operator ... denotes the traceless part of the tensor in question. The surface part of the free energy is non-negligible only in the immediate proximity of the confining wall, thus, it reduces to boundary conditions for the order parameter, (nS • V)Q = ±—2 (Q - QS) ? S , (2.61) ? on surface where nS is the surface normal and ± correspond to the wall placed at fS where rS+t is either within the liquid crystal or outside, respectively. As already mentioned, the dissipation of energy on the confining substrates is neglected in this study. Usually, the motion of molecules close to the substrates is hindered due to the anchoring and the energy dissipated at the substrate can be assumed very small. If this would not be the case, the second term in Eq. (2.47) would contribute to the boundary conditions as well. In the equilibrium, Q = J2i=-2 ai(f)Ti is not time-dependent and the Eq. (2.60) reduces to five coupled, scalar, differential equations for five scalar amplitudes ai(f), ( \ r 2 2 0 = - [? - ? V ) ai + 3v6 J^ ajak tr (TiTjTk) - 2ai ^ aj, (2.62) j,k=-2 j=-2 with boundary conditions g (n^S ¦ V)ai = ± —(ai - aiS ) ?2 (2.63) on surface Here, it was assumed that the system can be described by a constant set of base tensors T i which is usually true in planar geometry and in some special cases in curved geometries. In a particular geometry and with given anchoring conditions the problem is generally simplified, since due to the symmetry reasons some of the amplitudes are equal to 0. In addition to that, only few of the products TiTjTk have a nonzero trace; the combinations of base tensors which have a nonzero trace are listed in Table 2.1. In planar geometry, in the case of a uniaxial nematic order (ai=0 = 0) one is left with only one scalar equation. When considering pretransitional collective dynamics let us repeat that the derived equations hold for states not far from the equilibrium. Thus, the order can be assumed to be only slightly perturbed with respect to the equilibrium value, Q(f,t) = A(r) + B(r*, t), where A is the equilibrium order parameter determined by the mean-field theory [Eq. (2.62)] and B is a small fluctuating part, Phenomenological description 43 Table 2.1 List of products of up to three base tensors T? with nonzero trace: trTi = 0, tr(TiTj) = Sij, and tr(TiTjTk) is as follows in the Table. Base tensors are defined in Eq. (2.6). i j k tr (TiTjTk) 0 0 0 1/v6 0 ±1 ±1 -1/76 0 ±2 ±2 1/2 v6 1 ±2 ±2 ±1/23/2 2 -1 -2 1/23/2 llBll , (2.69) 2 I i=12 I where b stands for each of the fluctuating degrees of freedom of the order parameter and ? is a generalized correlation length characteristic of a particular type of fluctuations. From the discussion of fluctuations in previous Sections, ?-2 =1 ?2f L ??2 , (2.70) ?=?o where ? corresponds to fluctuations of the order parameter ? whose equilibrium mean-field value is ?0. In the case of a uniform nematic order parameter tensor the relevant temperature-dependent correlation lengths are the ones introduced in Eqs. (2.36) and (2.37) for the isotropic and nematic phase, respectively. We have divided the free energy of the system associated to the equilibrium order into the mean-field part corresponding to the order calculated by means of a phenomenological mean-field theory and to the part associated with collective thermal fluctuations, Tcas. The latter is given by a partition function 0 ' Vbe-H[b]/kBT) , (2.71) where kB is the Boltzmann constant, T is the temperature, and the integral is over all configurations of a fluctuating field b which satisfy boundary conditions [54,38]. The name pseudo-Casimir force for the fluctuation-induced interaction is due to the analogy with a Casimir effect which was first recognized by Casimir in 1948 [68]. In his study he found out that at T = 0 quantum fluctuations of the electromagnetic field in a cavity yield a weak yet measurable attraction between the walls of the cavity. Because the force between the walls is determined by a derivative of the free energy of a system rather than by a derivative of its energy similar effect is expected above absolute zero where the interaction is not just due to quantum but also due to thermal fluctuations. In liquid crystals, the fluctuation-induced interaction is due to thermal fluctuations of order parameter field instead of the electromagnetic field. The main problem when calculating the pseudo-Casmir force is in fact that the total free energy of collective excitations diverges within the continuum description. Within the theory of the Casimir effect there are many methods for determining the finite interacting part from the divergent total free energy, such as dimensional regularization, the introduction of the lower limit of the wavelength of excitations, Phenomenological description 47 Zeta regularization, methods based on Green’s function of fluctuations, etc. Describing techniques of calculation of the pseudo-Casmir force is beyond the scope of this thesis. When needed, we will just quote the results already obtained by others. To get the feeling about the pseudo-Casimir force few basic characteristics should be known. Depending on the boundary conditions the fluctuation-induced interaction can be either repulsive or attractive [27,63,64]. Its magnitude depends strongly on the surface interaction. In general, the sign of the pseudo-Casimir interaction is determined by the type of the boundary conditions, provided that the system is not subject to electric or magnetic field; b(z = 0,d) = (?/d) b'(z = 0,d), where b' = db/dz and d is the separation between the walls [see Section 2.3.1 and Eq. (2.66)]. Fluctuation modes constrained by strong, ?\t2 d, anchoring at both substrates lead to an attractive force. In a mathematical language this corresponds to Dirichlet, b(z = 0) = b(z = d) = 0, or Neumann, b'(z = 0) = b'(z = d) = 0, boundary conditions at both substrates. In contrast the asymmetric situation with one surface enforcing a strong anchoring and the other a weak anchoring yields a repulsive force (mixed boundary conditions). This is the universal property of Casimir forces [69-71]. 2.4.1 Stability of thin liquid films Imagine a thin liquid film deposited on a solid substrate so that it has a free liquid-air interface. Due to thermal fluctuations the free interface of the film is not flat but rather wrinkled. The fluctuation waves of the interface are known as capillary waves. Because of the interfacial interactions and long-range interactions, such as the van der Waals interaction, and structural and pseudo-Casimir interaction in an ordered liquid, the total pressure in the film depends on its thickness, being either higher or lower than the external pressure. This difference in pressures, ? = po — p where po is the external pressure, is denoted as a disjoining pressure. The term disjoining pressure was first introduced by Derjaguin as a pressure due to the van der Waals interaction [72]. Later, its use was extended also to structural interaction [59]. Here, it will denote the total pressure which is due to interactions in the film. The disjoining pressure can be either repulsive, ? > 0, or attractive, ? < 0, and it has to vanish in the limit of infinitely thick film. Thus, if the disjoining pressure is characterized by a monotonic behavior the magnitude of the disjoining pressure is larger for smaller film thicknesses. The two possible situations for the monotonic disjoining pressure are depicted in Fig. 2.7 (top). Imagine now a film whose interface is perturbed at a given moment. If the disjoining pressure is repulsive the repulsion is stronger in a thinner region of film. Thus, the tendency to thicken the thinner part of the film will be stronger than to thicken the thicker part and the differences 48 Phenomenological description d d : : ¦ ifnT"MMI (a) (b) Figure 2.7 Schematic representation of conditions for the stability of thin liquid films. Thermal fluctuations of the free liquid–air interface are (a) di-minished and (b) amplified, resulting in stable film or decomposition of the film in liquid drops and dry patches, respectively. Top: schematic representa-tion of the disjoining pressure; middle: uniform film with a thickness equal to its average value and a sketch of a capillary wave with a disjoining pressure indicating the sign and the magnitude; bottom: resulting film profile. Phenomenological description 49 Figure 2.8 A liquid droplet on a solid substrate in the equilibrium; 9 is the contact angle. in the film thickness will be smeared. On the other hand, if the disjoining pressure is attractive the stronger attraction in thinner regions yields amplified differences in the film thickness when finally the thickness of thinner regions drops to zero and dry patches occur. These situations are schematically presented in Fig. 2.7 (a) and Fig. 2.7 (b), respectively. The described mechanism of a decomposition of a film is called the spinodal dewetting. Once we are familiar with basic relations between the disjoining pressure and the stability of a thin film the stability conditions will be derived quantitatively. In the derivation we will follow the calculations made by de Gennes [73] and Brochard-Wyart and Daillant [74]. The spreading of the liquid on the solid substrate is determined by a spreading coefficient S = ?SO — (?SL + ?) (2.72) where ?SOA is the interfacial energy of the bare solid, ?slA is the interfacial energy of the solid-liquid interface, and ?A is the interfacial energy of the liquid-air interface; A is the surface area. S > 0 leads to a complete wetting of the solid by liquid whereas for S < 0 the wetting is only partial. On the other hand, the wettability of the liquid is described by a contact angle ?e as it is plotted in Fig. 2.8. The subscript e denotes that ? refers to contact angle in the equilibrium with the vapor. The contact angle and the interfacial energies are related through the Young’s relation, ?SV — ?sl — ?cos?e = 0, where ?SVA is the interfacial energy of the solid-vapor interface [75]. For organic liquids the difference ?SO — ?SV > 0 is usually very small. The energy of a liquid film is T = Tq — S A + V(d)A, (2.73) where V(d) represents the energy of interactions which contribute to the disjoining pressure, ? = —?V/?d. The equilibrium thickness of a film, de, is defined by the minimum of the energy while the volume V = Ad of the liquid is preserved, S = V(de) + ?(de)de. Imagine now a film with the average thickness do, however, with small deviations of the thickness with respect to the coordinate x parallel to the solid substrate, ?(x). 50 Phenomenological description q (a) qM q/qc (b) Figure 2.9 The inverse relaxation rate of capillary waves as a function of a wavevector for (a) repulsive disjoining pressure and (b) attractive disjoining pressure. In the latter, for q < qc the fluctuations are amplified. The pressure distribution in a film is then ?2 ? p(x) = p0 — ?^ — ?(?), ?x2 (2.74) where the term with ? corresponds to the pressure due to the curved interface (1/R = ?2?/?2x, where R is the radius of the curved path). In the lubrication approximation (flat liquid film and Poiseuille type of flow), the horizontal current in the liquid is given by ?3 ?p j = ?v = ( ?p\ —— ?x , (2.75) — 3? \ ?x where v is the average horizontal velocity. In addition, the mass conservation law gives ?j/?x + ??/?t = 0. Consider now the expansion of the deviations of the film thickness from its average value ?(x) = d0 + ^ uq eiqx e t/?q. (2.76) Here, ?q is the relaxation time of the capillary wave with wavevector q and uq <^i d0 is the amplitude of a given capillary wave. Using the ansatz defined in Eq. (2.76) in Eq. (2.75), considering the mass conservation, and linearizing the obtained equations one is able to determine the relaxation times, — = 0 ?q 3? ?q - q ?'(d0) . (2.77) For ?/(d0) < 0, which corresponds to repulsive disjoining pressure, ?q > 0 for every q and fluctuations decrease exponentially after they are excited. On the other hand, if ?'(d0) > 0 a wavevector qc can be found so that ?q• r d (a) (b) Figure 3.2 Schematic representation of van der Waals interaction between (a) two neutral molecules and (b) two macroscopic bodies. 3.1 Van der Waals interaction Van der Waals interaction is a common name for the dispersion interaction, which originates in the pair-wise interaction of fluctuating dipoles arising from dynamic redistribution of electrons in molecules, and the orientational interaction which results from the interaction of permanent yet fluctuating electric dipoles. The value of the interaction is the quantum mechanic expectation value of the corresponding interaction term in the Hamiltonian. For small separations, the interaction energy is proportional to 1/d6, whereas for large separations the dispersion interaction falls off faster and approaches 1/d7 behavior. The decrease of the dispersion interaction is a consequence of finite velocity of light; when the time it takes for the electromagnetic field of one molecule to reach the second one and to return becomes comparable to the period of the fluctuating dipole (usually, d > 100 nm) the phase coherence between the two interacting molecules is getting lost. The effect is known as retardation. The 1/d7 behavior of the dispersion interaction in the limit of high retardation which was first calculated by quantum electrodynamic approach developed by Casimir and Polder [91,68]. The orientational interaction is proportional to 1/d6 for all separations. For materials with small permanent electric dipoles and for distances up to 100 nm the dispersion interaction dominates over the orientational one. Several approaches have been employed to calculate the van der Waals interaction between macroscopic bodies. The simplest way, known as the Hamaker approach [92], is to sum all pair-wise interactions between constituent molecules. In condensed media these are not independent but rather strongly influenced by the surrounding medium, therefore, taking into account many-body interactions becomes essential. On the mesoscopic and macroscopic scale the many-body system can be regarded as a continuum and can be described by macroscopic quantities, such as permittivity, which take into account the screening of the surrounding molecules. Van der Waals force 57 The continuum approach is known as the Lifshitz approach. Lifshitz was the first who calculated the van der Waals interaction energy between two semi-infinite dielectric bodies separated by a gap of vacuum [93]. Later, the interaction energy for the system with a gap filled with an isotropic dielectric medium was calculated by Dzyaloshinskii et. al [94,95]. The basic idea of the continuum theory is that the interaction between the bodies is considered to take place through a fluctuating electromagnetic field. The interaction arises from the change in the zero-point energy of the electromagnetic field modes when the latter are perturbed by the coupling of the field with the polarization currents induced on the molecules. The electromagnetic field modes are obtained by solving Maxwell’s equations. These findings were obtained by a quantum field theory approach based on determination of the Green’s function for the electromagnetic field in the presence of molecules in a dielectric media characterized by isotropic permittivities. The poles of the Green’s function represent the electromagnetic field resonances of the system. There are two sources of poles: the poles of the permittivity, t(?) = 0, and the poles of the denominator of the Green’s function, T)(?) = 0. The former modes refer to bulk modes and they do not depend on the separation between the macroscopic bodies. The poles arising from the secular equation T)(?) = 0 refer to the surface modes and are responsible for the change of the zero-point energy of the electromagnetic field modes. Once it is recognized that the van der Waals interaction arises from the change of the electromagnetic field surface modes, the latter can be determined directly by solving Maxwell’s equations in a classical way. The equivalence of the van der Waals force between macroscopic bodies and the surface modes interaction was first shown by van Kampen et. al [96] for the non-retarded interaction, and later it was extended by Gerlach [97] and Schram [98] for the retarded interaction. The change of the free energy of a perturbed system for temperatures T > 0 instead of the change of the energy of a perturbed system at T = 0 was first introduced by Ninham et. al [99]. In the following Sections the continuum procedure will be used to derive the van der Waals interaction between uniaxial macroscopic bodies. Before we start with the derivation of the van der Waals force for uniaxial media we shell first quote the known expression for a three-layer system of isotropic media (see e.g. [100]), ? k B T /? 2 ? 12 ? 23 e-x = ------ dx x ———— (3.1) 16?d3 0 1 + ?12?23 e-x k B T ? ~3 r? I ? Ro? 23 e 2pd ? R 2 ? R 3 e 2pd 1 +------d3 / d dp p-------------------------~ +---------------------------~ ^ n=1 1 ' RR -2pd —R—R -2pd ' 12?23 Here, d is a thickness of a gap between two semi-infinite media 1 and 3, and the gap is filled with material 2; d = d-^ft2?n/c and ?ij’s are functions of frequency dependent 58 Van der Waals force permittivities of the media. They are defined in Section 3.2 [Eqs. (3.24) and (3.36)]. It should be noted, that in general there is no explicit separation dependence of the van der Waals force. Except for very large separations the second term in Eq. (3.1) is dominant. At small d’s it reduces to an expression similar to the first term, and the force has a 1/d3 separation dependence. For intermediate separations the force falls off quicker, and would obtain a 1/d4 dependence. However, for very large separations the second term becomes negligible comparing to the first one, and the 1/d3 separation dependence is recovered. In practice, the van der Waals force is calculated by using the well-known approximate expression ? = — A/6?d3 (see e.g. [89]), which is in a good agreement with Eq. (3.1) for separations of order of few nanometers and provides satisfactory qualitative insight in the van der Waals force for larger separations. Here A is a Hamaker constant, 3 t1 — t2 t3 — L2 3h?e (n21 — n22)(n23 — n22) A = —k b T--------------------------:= , ,— —, ------, , (3.2) 4 e1 + e2e3 + e2 8V2 n21 + n22n23 + n22(n21 + n22 + yjn23 + n22) ti is the static dielectric constant of a medium i, ni is its refractive index in visible, and ?e is the plasma frequency of a medium, taken to be equal for all media. 3.2 Electromagnetic field surface modes In this Section the spectrum of electromagnetic field modes in the gap between two dielectric macroscopic bodies will be calculated for the system with no external electromagnetic field and no macroscopic electric charges and dipoles. The fluctuations of the electromagnetic field are a consequence of a dynamic redistributions of electrons in molecules due to thermal fluctuations. However, the interest is not in fluctuations of a single molecule, but in collective behavior on the macroscopic scale. Therefore, the electromagnetic field in the gap is to be determined. The field obeys Maxwell’s equations Vx E = —— , V • D = 0, (3.3) ?t ?D Vx H = — , V • B = 0, ?t and satisfies corresponding boundary conditions ?B = 0 ?EH boundary = 0 ?H11 boundary = 0 (3.4) boundary = 0. boundary Here D = t0tE and B = µ0H; e is permittivity tensor. It is not the subject of this thesis to calculate the electromagnetic field modes for general geometry of the Van der Waals force 59 cavity and for general types of the wall. Therefore, as already stated in previous Section the electromagnetic field modes are calculated for the planar geometry. The coordinate system is set so that the z axis is perpendicular to the gap between the macroscopic bodies and axes x and y lie in the plane of one of the interfaces. In the lateral directions, there are no constraints for the electromagnetic field, therefore, the x and y dependencies of the electromagnetic field are the one of the plane wave. The dependence on the z coordinate is still to be determined. The ansatz function for the electromagnetic field is A = A 0 (z) ei?rfe~i?t, (3.5) where A stands for either electric field E or magnetic induction B and ? = (?x, ?y, 0) is a wavevector in the plane of the boundary. In this study, the permittivity tensors of the macroscopic bodies as well as the permittivity of the interposed layer are uniaxial with eigenframes coinciding with the coordinate system, (ti± 0 0 \ 0 i± 0 . (3.6) in Here, index i = 1,2, or 3, represents different media, and e is frequency dependent. As already stated before, the uniaxial symmetry with perpendicular optical axis with respect to the gap is the only non-isotropic geometry, which allows one to calculate electromagnetic field modes quite easily. Even if the eigenframe of the permittivity tensor remains the same, but the optical axis lies in one of the lateral directions, the eigenmodes can not be calculated analytically, except in the nonretarded limit, which is equivalent to describing static fields. In the case of uniaxial symmetry and optical axis parallel to the gap normal, the full rotational symmetry around the z axis is preserved. This allows us to divide the eigenmodes in two groups: transverse magnetic modes (TM) and transverse electric (TE), as defined by Rayleigh already in 1897 [101]. Transverse magnetic modes The TM waves are defined as waves with Bz = 0. By using this and the ansatz function for the electromagnetic field introduced in Eq. (3.5) to solve the Maxwell’s equations the wave equation is obtained, ?------en ? = 0, (3.7) c2 with boundary conditions ?? 1 . = 0 and ?—? boundary Cj_ = 0. (3.8) boundary 60 Van der Waals force Here, ?' = d?/dz and ?(z) defines the z dependent amplitudes of corresponding components of the electromagnetic field, Bx(f,t) = B0?(z) ei?'r -i? , By(r,t) = -B0 — ?(z)e ' , ?y c2 ? x , i?r f-i?t Ex(r,t) = -B0---------? (z) e ' , (3.9) i?t± ?y c2 , i?-i Ey(f,t) = -B0------?/(z)em'r ^ , Ez(f,t) = B0----------?(z) ei?'r-i?t. With respect to the coordinate normal to the gap, the electromagnetic field decays/grows exponentially with the inverse penetrations depth [ 1/2 e_i_ / 2 ? W — ?-----eii . (3.10) en c2 " In a composed system of three dielectric media the amplitude ? has the following z dependence {Ae?—1z ; z < 0 B e?—2z + Ce- ?—2z ; 0 < z < d . (3.11) D e- ?—3z ; z > d Taking into account the boundary conditions in Eq. (3.8) one ends up with a secular equation, which determines the dispersion relation for the TM surface modes DTM(?) = 1 + ?12(?)?23(?)e- 2?—2(?) = 0. (3.12) Here, ?—ij(?) = ——----------—. (3.13) ei±(?)?—j(?) + ej±(?)?—i(?) Similarly, for the four-layer system one obtains the secular equation m(?) = 1 + R — R -2?—2(?)d-2?—3(?)t DTRM(?) = 1 + ?12(?)?23(?)e 2?—2(?) + ?23(?)?34(?)e 2?—3(?) (3.14) +?12(?)?34(?)e 2?—2(?) e 2?—3(?) = 0, where d and t are thicknesses of the layers of media 2 and 3, interposed between semi-infinite surrounding media. Transverse electric modes The TE waves are defined as waves with Ez = 0. Using the same procedure as when calculating TM modes the wave equation for the z dependent amplitude ? is Van der Waals force 61 obtained, - ?2------ej_ c2 with boundary conditions ?? ? 2 ? \ -----e_i_ ? = 0, (3.15) c2 = 0 and ?? boundary = 0. (3.16) boundary The components of the electromagnetic field depend on the function ?(z) as Ex(r,t) = E0?(z) ei?'r-i? , Ey(r,t) = -E0 — ?(z)e ' , ?y ? x 1 . i?p-i?t Bx(r,t) = E0-----— ?(z)e ' , (3.17) ?y i? By(f,t) = E0 i? ?(z) ei?'r-i? , i?2 1 i? r-i?t - E0-------_ ?y i? Bz(r,t) = -E0------—?(z)e The characteristic length for the penetration of the TE waves into the dielectric walls is then / 2 \ 1/2 ?(?) = ?-----e± . (3.18) In a composed system of three dielectric media the amplitude ? has the same z dependence as defined in Eq. (3.11), with ?i instead of ?—i. Taking into account the boundary conditions in Eq. (3.16) one ends up with a secular equation, which determines the dispersion relation for the TE surface modes DTRE(?) = 1 + ?R12(?)?R23(?)e-2?2(?)d = 0. (3.19) Here, R ?i(?) - ?j(?) ?i j(?) =----------. (3.20) ?i(?) + ?j(?) Similarly, for the four-layer system one obtains the secular equation DTRE(?) = 1 + ?R12(?)?R23(?)e-2?2(?)d +?R23(?)?R34(?)e-2?3(?)t (3.21) +?12(?)?34(?)e- 2?2(?) e- 2?3(?) = 0, where d and t are the thicknesses of the layers of media 2 and 3 interposed between semi-infinite surrounding media. If all the media are isotropic, the electromagnetic field surface modes can still be divided into TM and TE but they are described by the same wave equation ?2------e ? = 0 (3.22) c2 J 62 Van der Waals force with boundary conditions Eqs. (3.8) and (3.16) for TM and TE modes, respectively. Here, the transformation e_i_,e|| —> e has to be made. The penetration depth of the modes is then / 2 \ 1/2 / 2 ? \ ?(?) = ?-----e , (3.23) c2 J and the secular equations for the three-layer system read D (?) = 1 + ?R12(?)?R23(?)e 2?2(?)d = 0, (3.24) TM dte(?) = 1 + ?R12(?)?R2 3(?)e~2?2(?)d = 0, where ?Ri j(?) = ---()----()---------------, (3.25) R ?i(?) - ?j(?) ?i j(?) =----------. 3.3 The zero-point energy of surface modes The roots of secular equations for TM and TE modes give frequencies of the corresponding modes, ?, as a function of a wave vector ?, i.e., the dispersion relation for the surface modes. At the absolute zero, the interaction energy is then given by the difference of sums over the zero-point energies of each mode, E = —h?/2, for each value of the wave vector ?, for the composed system of dielectric media with either finite thickness of the interposed medium or with d —> oo, y^?i - y^?i/ . (3.26) i i Here, ?i’s are the zero-point frequencies of the perturbed system and ?i/’s are the zero-point energies of unperturbed system (only the medium which is inbetween the walls). Solving the secular equations for the electromagnetic field surface modes is far from being easy, however, in the standard contour integral representation [102] the expression in Eq. (3.26) reduces to 1 ,—^ / —h ?? d D(??) ?E = — 2, d??-------—m (3.27) 2?i ^ 2 d?? D0(??) = -—i \] d?? ln ? , h— D ( ) — 2_] d?? ln ? 4?i ^ D0(??) where D(?) = 0 and D0(?) = 0 are the secular equations for the perturbed and unperturbed systems, respectively. The secular equations derived in previous Sections are already the corresponding ratios, which can be easily tested by performing Van der Waals force 63 the limit d —> oo. The planar system is not bounded in the lateral directions, thus, the spectrum of wave vectors ? is continuous, and the sum over the wave vectors can be replaced by an integral, J2g ~^ (2 A 2 J d 2 ?. Here, A is the surface area of the interfaces. In the case of optically isotropic media and uniaxial media with the optical axis parallel to the interface normal the frequencies ? depend only on the magnitude of the wave vector ?, therefore, the zero-point energy reads ?E = — At the absolute zero, there are no excited states and the interaction is due to the change of the zero-point energy as derived above. At higher temperatures, not only the zero-point energy of the electromagnetic field is changed but also the occupancy of states. The whole information is gathered in the free energy of the system rather than in its energy. The free energy of the electromagnetic field with modes characterized by frequencies ?i is then d? ?d? ln. o Dq(?) (3.28) F = kBT 2_j ln 2 sinh 2 ßh—?i\ ------- 2 . (3.29) To calculate the interaction free energy of the perturbed system one has to substitute the energy of the system with the free energy, hL¦ —> kBTln(2sinh^^) = 2 h2 — kBT V\?=i -e-fihujn. If a further substitution ? = i? is made we get the final expression for the change of the free energy of the electromagnetic field, ?F = = = = hA z? z? — / d?? j d? 8? O-? hA z? ? —2 / d?? / d? 4? O O hA z? z? / d?? J d? o o 1 + 2 ^ n=l i/3h^n \ ln 1 + 2 2_s cos(ß—h?n) ln Do(i?) n=\ 4? o kBTA 2? 1+2 ? 53 ? o ?(ßh—? — 2?n) — d?? ^ ln Do(i?n) ln . Do(i?) (3.30) In Eq. (3.30) the second line is obtained by recognizing that D(?) is an even function of ?, therefore, the odd, sine, part of the function under the integral gives 0 when integrating over the symmetric interval. The third line is obtained by using the equality J2?n=i cos(nx) = ? J2?n=-? ?(x ~ 2?n) — 2, which is derived from the Poisson sum formula [102]: J2?n=-? f(n) = 2? J2m=-? F(2?m), where F is Fourier transform of f. The last line is obtained by integrating over the frequencies ?. In the final expression, ?n = 2?kBTn/h. To relate the free energy of the van der Waals interaction to the van der Waals force we should use the thermodynamic definition of force written out in Eq. (2.67). 64 Van der Waals force In the planar parallel system consisting of two infinitely large surfaces, the force is always perpendicular to the surfaces and, consequently, F F = - F , (3.31) V,.4. where d is the separation between the two objects. The positive value of F corresponds to a repulsive force and the negative value corresponds to an attractive force. 3.4 Van der Waals force in a multi-layer system As explained in previous Sections, the van der Waals force between macroscopic objects is due to the change of the free energy of electromagnetic field in a system perturbed by interfaces. By the definition of the force in Eq. (3.31) and the free energy of electromagnetic field surface modes obtained in Eq. (3.30) the van der Waals force per surface unit area, ? = F/A, reads kbT f°° ^ 1 ?D?(i?n) ? =-------- d?? y y. (3.32) 2? 0 n^oo ?= D?(i?n) ?d TM,TR In order to evaluate the sum in Eq. (3.32) the frequency dependence of the permittivity tensor has to be known. The permittivity varies with frequency in much the same way as does the atomic polarizability of an atom, e = 1+n?/t0, where n is the density of molecules and ? is their polarizability [101]. Basic polarization processes are reorientation of permanent dipoles and deformation of electronic configuration due to excited electrons. These yield e(?) = 1 + (e-n2)/(1 - i?/?r)+(n2 -1)/(1-?2/?e2) [100]. Here, e and n stand for each of the components of the corresponding tensors, e = e(0) is the static dielectric constant, n is the refractive index of the medium in the visible, ?r is the molecular rotational frequency, and ?e is the plasma frequency. Usually, ?r < 1013 s_1 ?r, the dispersion relation is determined solely by the electronic absorption, n2 - 1 e(i?n) ~ 1 +-------, (3.33) 1 + ?n2/?e In this case, D(i?n) = D(-i?n) and kBT f°° ^.f 1 ?D(i?n) ? =-------- d?? 2,, (3.34) ? 0 n=0 D(i? n ) ?d where prime over the sum denotes, that the term with n = 0 should be multiplied by 1/2. From the secular equations determined in Section 3.2 the van der Waals Van der Waals force 65 force per unit area between the two interfaces in the three-layer system reads kBT L2|i (0) °° 2 ?—12?—23e_x ?(d,T) = 16?d3 L2„ (0) II e2±(0) r°° ?12?2 dx x------- 0 1 + ?12?23 e~x + y. ?d n= d 3 11 dp p R R -2pd ?? e ~ ? 12 23 1 + ? RR ? 12?23 ~ -2pd ~ + II 2_L ?—R ?—R e~2pd ? 12 23 1 + ? RR ? 12?23 ~ 2pd (3.35) . The first term in the expression corresponds to the static response of the medium and the second term corresponds to the dynamic response. Functions ?ij = ?ij(0), ?i j = ?i j(i?n) and —j- = —j-(i?n) are determined in Eqs. (3.13) and (3.20), and d = dJ~t2?n/c. In Eq. (3.35) the integration over the wavevector ? was substituted by the integration over a dimensionless parameter p = ?2 (?)c/(?nv/e2±), so that ? ij = ?i j = si — sj si + s j —„ = —is—j ij —is—j —jsi , , = t-i\\1^ , si = p2 - 1+ ^i±_ / ^±_, (3.36) = p2 - 1+ t-iii / 2ii , and ti = ti(i?n), unless stated otherwise. By following the procedure of calculating the van der Waals force and by comparing the Eq. (3.35) to the Eq. (3.1) (or similarly, by comparing it to the limiting case of the Eq. (3.35) where e^ei,, —> q), it can be seen that the influence of the anisotropy of the permittivity tensor affects the van der Waals interaction through the change of the penetration depths of the surface fluctuation modes and via changing the boundary conditions for the electromagnetic field at the interface between two different media. For the uniaxial permittivity the anisotropy of the interacting media 1 and 3 changes the interaction via renormalized effective permittivity, či —> \/ei\\ei±. On the other hand, the uniaxiality of medium 2, which mediates the interaction between the other two media, does not only renormalize the effective permittivity but also explicitly affects the magnitude of the interaction. Beside these differences, the structure of van der Waals force for uniaxial media is similar to the one for the isotropic media. The properties of the force and the differences between the isotropic and uniaxial expressions will be discussed in more detail after the analytic expression for the force, i.e., the Hamaker constant, will be derived. Often, in experimental set-up the solid substrate is covered by a natural oxide layer, a layer of condensed water, etc., and the system in question is at least four-layered (see Fig. 3.3). For such cases, combining relations for Hamaker constants are known [89] and are frequently used in practice. These relations are by itself approximate and beside that, they combine Hamaker constants, which are already , 66 Van der Waals force Figure 3.3 Schematic representation of the four-layer system: two semi-infinite macroscopic bodies separated by the third medium. One of the substrate is covered by an additional layer, e.g., a natural oxide layer or a layer of condensed water. The thickness of the additional layer is t and the separation between the macroscopic bodies, which is subject to change, is d. obtained by some other approximations that will be discussed in the following Section. Therefore, the force calculated from a combined Hamaker constant can only serve as a qualitative measure of the force rather than a quantitative. From the secular equations obtained for the four-layer system [Eqs. (3.14) and (3.21)] and the definition of the van der Waals force [Eq. (3.34)], the expression for the force in a four-layer system can be easily determined. Both, the expression in Eq. (3.34) and the van der Waals force for the four-layer system can only be calculated numerically. Since the results of such numerical calculations will be used in the study of stability of thin nematic deposition, the expression for the van der Waals force acting on a layer of media 2 with thickness d, and surrounded by semi-infinite medium 1, a layer of medium 3 (thickness t), and semi-infinite medium 4, are written out as well, ? = kBT 16?d3 + y ?d3 n= d 3 1 1 e2ii (0) e2±(0) ?— ?— e —x +?— ?— e—x(1+a) dx x 2-------------------------------------------------------------(1 0 1 -I- — 12 — 23 e—x -4- — 23 — 34 e—xa -4- —t->— 34 e—x~ra) dp p ?R ?R e-12 23 2p + ?12?34 e ~ 2pd(1+a) 1 + ? R1 2 ?23 e ~ 2pd + ?23 ?34 e + 1 ? 12 ? 23 e 2pd + ?12? R 34 ?23?R3 2pd(1+a) 2pda + ? R ? R e 2pd~(1+a) 12?34 L2± 1 + ?R12 ?23 e 2pd _|_ — 23 ?R34 e 2pda _|_ — R — 34 e ~ 2pd(1+a) (3.37) Here, d dw^2±?n/c, a = t/d ?Ri j(i?n),? = —. ( 2i /^2j_ )( 3± /^3ii ), and functions ?ij = ?ij(0), AR = (i?n) are defined in Eq. (3.36). In a four-layer system, for small thicknesses d the van der Waals interaction is mostly due to the interaction between media 1 and 3 over a layer of medium 2, whereas for large separations d the effect of the additional layer 3 is negligible and the interaction is due to the interaction of media 1 and 4 over a medium 2. For intermediate separations the interaction is due to the non-trivial composition of layers and is not just a superposition of the two interactions described above. From the arguments just discussed the force can have . Van der Waals force 67 a non-monotonic behavior and the turn-over of the behavior of the force occurs at d ~ t. 3.5 Uniaxial Hamaker constant Frequently, one is not interested in a precise separation dependence of the van der Waals force, but rather in its attractive/repulsive character and its order of magnitude. In such case, one is satisfied with the approximate analytic expression which is far easier to analyze and calculate with. For that reason the approximate analytic expression for the uniaxial van der Waals force is determined. In the nonretarded limit the dependence on the separation between the two bodies can be extracted from the material properties and the van der Waals interaction can be written in a form of an explicit separation dependence and the “Hamaker” constant. In this limit the velocity of light is assumed to be infinitely large, so that the phase coherence between interacting fluctuating dipoles is not lost for any separation. This is always true for static electromagnetic fields, whereas for the dynamic electromagnetic fields it only holds approximately if the distance between the two interacting molecules is small comparing to the wavelength of the radiation of the fluctuating dipole. For most materials, ?e = c/?e ~ 100 nm. Neglecting the retardation, the expression in Eq. (3.32) reduces to ? = -A/6?d3, where A is a Hamaker constant calculated from the Lifshitz theory, 3kBT ™ . L2,, (i?n) f°°, 2 ?—12(i?n)?—23(i?n)e~x A =------ y — dx x ------ r -----. (3.38) 4 ^ 0 t2±(i?n) 0 1 + —.12(i?n) —.23(i?n)e~x Here, the prime over the sum denotes that the term with n = 0 should be multiplied by 1/2 and the function ?ij = ?ij(i?n) is defined in Eq. (3.36). After performing the elementary integral in Eq. (3.38), 3kBT ^ e2|| (i?n) L ^ (-?—12?—23) 1 A =-----------/ — ?—12?—23 1 + y(-------3 . (3.39) 2 ~=0 e2± (i?n) k=1 k + 1) Usually, I ?ij I <^i 1, except if —i <^i — j or vice versa, which is usually not true. Therefore, the terms of higher orders of products ?12?23 can be neglected — neglecting “many-body” interactions. Here, by analogy with the microscopic description, the contributions from different orders in ?12?23 are termed as: “pair-wise” interaction, if only the first order is taken into account, and as “many-body” interaction, if the complete series of orders is considered. If the temperatures are not very high, the sum over n can be replaced by integral with respect to the frequency ?. (For room temperatures, ?1/?e = (kBT/hc)?e ~ 1/80 <^i 1, and the sum over discrete spectrum of frequencies ?n can be safely replaced by an integral.) In the case of 68 Van der Waals force isotropic media, these approximations allow one to calculate the Hamaker constant analytically. Here, it should be taken into account that from Eqs. (3.36) and (3.33) —j7-(i?) =2------n2 ------nj . (3.40) n + j + 2?2/?e For uniaxial media the latter expression, with the additional transformation ni —> n—i, is only approximately valid, since terms [(?ni/ni||)2±(?nj/nj )2](?/?e)2 and higher orders in both, (?ni/ni,,) and (?/?e), are neglected; ?ni = ni,, -ni?, and is in liquid-crystalline systems up to 10% of ni,,, thus, neglecting the higher orders contributes the error not larger than few percents. With this additional approximation, the Hamaker constant for uniaxial media can be determined: A = A?= + A?> = —kBT—--------------- (3.41) 4 L2? —1 + —2 —3 + —2 + -----7=(n—21 - n— 2)(n3 - n—22) 8\/2 V2(n22-n22) n2?(2n22 - n—21 - n—22)(2n22? - n—23 - n—22) ? 2n2,, - n—21 - n—22 2n22 - n—23 - n—22 -------------/---------------------------------------------------------------------------------------i -------------------------------------------------------------------------------- \ln—21 + n—2(2n22 - n—21 - n—22)(n—21 - n—23) Jn—23 + n—2(2n22 - n—23 - n—22)(n—21 - n—23 ) , —-------------------------------------------------- \/n—21 + n—22 (2n22? - n—21 - n—22)(n—21 - n—23) \/ni-{-n2(2n2,, - n—23 - n—22)(n—21 - y y3 2 aj_ where ai = -/a^ai? and a stands for either static dielectric constant e, or the refractive index in visible n. The first term in Eq. (3.41), A?=0, corresponds to the static response of the media, and the second term, A?>0, corresponds to dynamic response. If the interacting macroscopic bodies are isotropic, the effective parameters — and n are replaced by isotropic parameters e and n, respectively. In the case all three media are isotropic, the expression reduces to the well known formula written out in Eq. (3.2). Frequently, the two semi-infinite media are the same, e.g., freestanding liquid-crystalline film surrounded by air, membrane in a solution, etc., and the Hamaker constant reduces to A 3 k t2|i (^1 —2) 3h—?e (n— 2 n— 2 ) 2 (3 42) 4 L2? (—1 + ^2) 8v2 X y2(n22 - n22 ) (n—21 + n22)2 + 4n42 - 2(n—21 + n—22)(3n22 - n2 ) + n2?(2n22? - n—21 - n—22)2 2(n—21 + n—22)3/2(2n22 - n—21 - n—22) In the following, the derived Hamaker constant for uniaxial media [Eq. (3.41)] will be first compared to the isotropic Hamaker constant [Eq. (3.2)] and, secondly, its validity with respect to the full Lifshitz theory [Eq. (3.35)] will be discussed. . Van der Waals force 69 Uniaxial vs. isotropic In the derived expressions, the difference between isotropic and uniaxial media is explicitly manifested: The relevant parameters, which determine the character of the interaction, are in the case of isotropic interacting media e = čso and n = niso. In the case of uniaxial media the relevant parameters are not the traces of the corresponding tensors, čso = (ey + 2ej_)/3 and (niso)2 = (n2< + 2n2L)/3, but rather products of their eigenvalues, — = v/ejjeX and n = y/n\\n±. The sign of the static part of the Hamaker constant depends on the relative sequence of the introduced renormalized static dielectric constants: for —2 < —1,—3 or —2 > —1,—3 the static Hamaker constant is positive and static part of the van der Waals interaction is attractive, whereas for —1 < —2 < —3 or —1 > —2 > —3 the static Hamaker constant is negative and the corresponding interaction is repulsive. Similar conditions can be determined for the dynamic part of the Hamaker constant. It can be shown that the part in the square brackets in Eq. (3.41) is positive definite, thus, the sign of the dynamic part of the Hamaker constant depends solely on the sign of the product (n21 — n—22)(n23 — n—22). For n2 < n1, n3 or n2 > n1, n3 the dynamic Hamaker constant is positive and the dynamic part of the van der Waals interaction is attractive, whereas for n—1 < n—2 < n3 or n—1 > n—2 > n3 the dynamic Hamaker constant is negative and the corresponding interaction is repulsive. The sign of the isotropic Hamaker constant has the same sequence dependence as the uniaxial Hamaker constant, however, with isotropic parameters čso and niso instead of effective — and n, respectively. As already noted, until now in studies concerning uniaxial layers the van der Waals interaction has been calculated by use of the isotropic Hamaker constant and isotropic parameters. There are two sources of mistakes when doing this. First, even if the effective parameters are very close to the isotropic parameters, the magnitude of obtained Hamaker constant differs from the uniaxial one because the anisotropies ?L2 = L2M — e2± and ?n2 = n2|| ~n2± are neglected. (The static part is always smaller whereas the dynamic part can be either smaller or larger.) In the static part, the neglected dependence is easily recognized in the ratio t2,,/t2±, which is by definition always larger than 1, whereas in the difference of the dynamic parts the dependence on the anisotropy is not that clear. Secondly, the difference between the isotropic and uniaxial Hamaker constants can be even more profound if the sequences of isotropic and effective parameters are different. In that case, the isotropic expression yields wrong sign of the interaction. The change of the sequences is very easily obtained with static dielectric constants, whereas the optical anisotropies are usually small and the sequences of isotropic and effective refractive indices change only in a very narrow range of possible combinations. Since the static Hamaker constant is about an order of magnitude smaller than its dynamic part the effect is not very common 70 Van der Waals force 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 0.0 0.5 1.0 ß 1.5 2.0 Figure 3.4 Dynamic part of Hamaker constant as a function of ß = n1/n2±. Solid line corresponds to reduced uniaxial and the dashed line to reduced isotropic Hamaker constant; A0 = 3—h?en2±/8\^2, n2,,/n2± = 1.2, and n3/n2± = 0.67. in experimental set-ups. The narrowness of possible combination of materials that satisfy the described conditions (see Fig. 3.4) could be one of the reasons why the problem of wrong sign of the Hamaker constant in systems consisting of uniaxial media has not been recognized before. However, it can be expected that the effect has already been observed but not recognized and/or understood. The explained change of the attractive/repulsive character of the van der Waals interaction to the repulsive/attractive character due to the increased optical anisotropy can be also one of the reasons for change in the stability of thin soft layers when crossing the (dis)ordering transition. As an example, the Hamaker constant is calculated for a system, which is often a part of the experimental set-up: a thin liquid-crystalline film deposited on a solid substrate and in contact with air on the other side. For liquid crystal 5CB (ey = 18.5, e± = 7, n\\ = 1.702, and n± = 1.539) on silica (e = 14 and n = 1.5) the van der Waals interaction is attractive in both, uniaxial and isotropic phase; for the same liquid crystal on a material with e = 11 and n = 1.6, which can be found among glassy materials, the interaction is repulsive in the isotropic and attractive in the uniaxial phase (see Fig. 3.5). Hamaker vs. Lifshitz After the comparison between the isotropic and uniaxial Hamaker constants the validity of the approximations that lead from the full Lifshitz theory to the derived Hamaker constant should be discussed. By comparing the Hamaker constants in which either “many-body” interactions [Eq. (3.35)] or just “pair-wise” interactions [Eq. (3.41)] are taken into account, it can be seen that in the nonretarded limit neglecting the higher orders does not considerably alter the magnitude of the inter- Van der Waals force 71 40 20 0 -20 -40 0 20 40 60 80 100 d [nm] 0 20 40 60 80 100 d [nm] Figure 3.5 Van der Waals force per unit area in the layer of nematic liquid crystal (en = 18.5, e± = 7, ni = 1.702, and n± = 1.539) in a contact with a solid substrate — (a) silica (e = 14, n = 1.5), (b) some glassy material (e = 11, n = 1.6) — and air. Solid curves correspond to full Lifshitz theory for uniaxial media, dashed curves are the van der Waals force per unit area calculated with uniaxial Hamaker constant, and dotted lines are calculated with Hamaker constant for isotropic media. action. Nevertheless, one should bear in mind that the screening of the surrounding molecules decreases the interaction, however, for realistic parameters ~ 5% at the most. As already known, the main defect of the introduced “Hamaker” procedure is not in neglecting “many-body” interactions but in neglecting the retardation. The latter becomes important when the time it takes for the electromagnetic field of one molecule to reach the second one and to return becomes comparable to the period of the fluctuating dipole. Usually, this happens when the interacting molecules are about ten nanometers apart. In Figs. 3.5 and 3.6 the van der Waals force as calculated from the full Lifshitz theory for uniaxial media is compared to the forces in the nonretarded limit, either taking into account or neglecting the anisotropy. Although, strictly speaking, approximation of no retardation is valid only when the interacting bodies are in close proximity to each other one should keep in mind that the retardation does not change the character of the interaction but only decreases its magnitude when the separation between the bodies is increased. If there is another, stronger interaction, which acts in the system, and the van der Waals interaction contributes only a correction to the primary interaction or if the van der Waals interaction is itself the primary interaction one can be satisfied with the approximate analytic expression which is far easier to calculate with and gives better insight in the effect of dielectric and optical properties of constituent media on the van der Waals interaction. Here is the opportunity for the analytic expression derived here. Especially lately, in studies, which aim to explain the experiments of 72 Van der Waals force 106 105 104 103 102 101 100 10-1 10-2 1 10 d [nm] 100 Figure 3.6 Van der Waals force per unit area in the layer of nematic liquid crystal in a contact with silica and air. Solid curve corresponds to the full Lifshitz theory for uniaxial media and dashed curve to the van der Waals force per unit area calculated with uniaxial Hamaker constant. Parameters used are the same as in Fig. 3.5. spinodal dewetting of thin soft organic materials usually characterized by uniaxial dielectric permittivity and refractive index, the correct determination of at least the character of van der Waals interaction is very important [103,104]. 4 Heterophase nematogenic system In this study of influence of the confining substrates onto the equilibrium order and pretransitional dynamics of the nematic liquid crystal we will first stop at substrate-induced effects which are localized in the vicinity of confining walls. They are char-acteristic for systems in which there is no competition between antagonistic fields inducing preferred direction of the nematic order in different directions, such as the surface, magnetic, etc., fields. Here, we are interested in systems that are subject to surface-induced nematic order. Especially interesting is the case, in which the surface potential is such as to induce a sufficiently large orientational ordering as compared with the bulk phase — a paranematic system. Then, a mesoscopic layer of nematic phase intervenes at the substrate–isotropic phase interface as the isotropic–nematic transition is approached from above [see Fig. 4.1 (a)]. The described situation is known as (orientational) wetting; the wetting can be either partial or complete. In the case of complete wetting, a surface transition occurs prior to the isotropic– nematic phase transition; the former being associated with the occurrence of the well defined layer of a nematic phase. The thickness of the surface-induced ordered layer diverges at the isotropic–nematic phase transition which, thus, becomes contin-uous. Above the surface critical point (GSC, TSC) the surface-induced layer grows continuously. In the case of partial wetting, the thickness of the surface-induced layer is saturated before the isotropic–nematic phase transition and the transition remains discontinuous. By changing the aligning power of the substrate, i.e., chang-ing the anchoring strength and/or changing the value of the induced nematic order, the complete wetting can change to partial wetting (or vice versa) between which the wetting transition occurs. Temperature dependence of surface-induced wetting layers in the case of complete and partial wetting is plotted in Fig. 4.2. The surface interaction may also have a disordering effect if, for example, the inner surface of the host material is rough [105–107]. In this case a reduction of the degree of order in the boundary layer is expected below the phase transition tem-73 74 Heterophase nematogenic system z=0 z=0 (a) (b) Figure 4.1 Schematic representation of the order of molecules close to the substrate in the system with surface-induced (a) nematic order and (b) disorder. perature, and the substrate induces wetting by the isotropic phase [see Fig. 4.1 (b)]. Again, the complete wetting refers to the case where the thickness of the surface-induced isotropic layer diverges on approaching the isotropic-nematic phase transition from below whereas the wetting is partial if the thickness of the wetting layer remains finite at the transition. The divergent nature of the thickness of the wetting layer, either isotropic in a nematic phase or nematic in isotropic phase, is associated with the surface aligned semi-infinite nematic systems. In the case the liquid crystal is bounded from more than one side, the thickness of the wetting layer can not diverge because the wetting layers get in contact before that. Still, the discontinuity of the isotropic-nematic phase transition can be significantly reduced. In highly constrained surface-aligned nematic systems the transition between the isotropic and nematic phase can be lost whereas the nematic order grows gradually on lowering the temperature. The interest in wetting transitions was initiated by study of Sheng in 1976 in which it was found that above some critical film thickness the isotropic-nematic phase transition in a surface aligned nematic can become continuous [18]. The original study of a nematic in contact with two substrates with infinitely strong anchoring was extended to cases of oblique anchoring strength [50]. Allender et. al studied extensively a semi-infinite nematic wetting system for which they determined in an analytical manner the scalar order parameter profiles in stable and metastable phases above and below the isotropic-nematic phase transition [108]. The isotropic-nematic phase transition in restricted geometries was further studied by Sluckin and Poniewierski [109,59], both, for the case of homeotropic and planar anchoring. In the latter, the nematic order in the wetting layer is biaxial due to the broken symmetry in the plane of the confining substrate. Telo da Gama et. al studied the wetting and interfacial phenomena by use of the density functional method [110,111]. In highly restricted geometry, the pretransitional nematic order and the structural force among the confining substrates was studied by Borˇstnik and Zumer [60]. Studies of equilibrium order were followed by studies Heterophase nematogenic system 75 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 10 20 30 z [nm] 40 50 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 10 20 30 z [nm] 40 50 (a) (b) Figure 4.2 Portrait of profiles of the scalar order parameter for various tem-peratures for the case of (a) complete wetting and (b) partial wetting (dashed line corresponds to the metastable paranematic solution). The scalar order parameter S is measured in units of bulk value of the order parameter at the phase transition, SNI. of pretransitional collective dynamics in wetting systems. First, pretransitional ex-citations were determined for the paranematic slab above the phase transition by Ziherl et. al [14] which was followed by an extended study in the inverse case of a nematic sandwiched between order melting substrates [15]. The analytical solutions in a semi-infinite system were derived by Ivanov [112]. Beside theoretical studies, the wetting and wetting transitions were subject of series of experimental studies since they reveal anchoring and interfacial properties of the liquid crystal in contact with substrates. The studies performed by means of optical ellipsometry methods are due to Miyano [113], Yokoyama [114], Moses [106], and many others, and lately, the experiments involving the force microscopy were performed by Moreau et. al [22] and Koˇcevar et. al [115,67]. Similar effect as in the case of surface-aligned nematic order is observed also for the case of smectic order — wetting by a smectic phase — which is associated with both, the vicinity of the transition to the smectic A phase and with the fact that the presence of the wall breaks the continuous translational symmetry resulting in a smectic-like order close to the confining substrate. The former effect was studied mostly by Moses et. al [116], and the latter effect was studied by Rosenblatt [117], Ocko et. al [118], etc. As it can be noted from the above review of work done in the field of wetting by isotropic or nematic phase there was not much known about the pretransitional collective dynamics in wetting systems before our study was performed. Though, it can be expected that the pretransitional dynamics of wetting systems should differ significantly from pretransitional dynamics in bulk systems, especially, when the conditions for the complete wetting are fulfilled. If the average equilibrium order in the wetting systems reveals the anchoring properties of the confining substrates and 76 Heterophase nematogenic system the nature of the liquid crystal interface the study of pretransitional dynamics offers insight into the evolution of the order characteristic for the low-temperature phase in the case of paranematic system and the evolution of the disorder in the case of a nematic confined by “order-melting” substrates. Regardless of the geometry of the host medium, these systems are characterized by high surface-to-volume ratio and, thus, they are very susceptible to any interaction between the constituent molecules and the surrounding walls. The surface coupling encountered in actual confined systems is usually not strong enough to be describable by a fixed value of the degree of order at the wall which was assumed in the study of pretransitional collective dynamics in the paranematic system [14], however, it reveals the basic physics behind the phenomena studied. Secondly, as it has been already discussed, the surface interaction may also have a disordering effect which is realized in the case of silica and some other materials for various kinds of the surface treatment [107]. It is evident, that there are a number of parameters of wetting in liquid-crystalline systems that seem to be pertinent to the behavior of collective excitations of the ordering in the vicinity of the isotropic–nematic phase transition. In order to provide a complete account of the phenomenon first discussed in a preliminary study of ˇ Ziherl and Zumer [14], I elaborate some of them theoretically by (i) comparing the spectra of fluctuations in geometries with surface-induced order and disorder, and (ii) by extending the analysis to substrates with finite strength of the surface interaction. For the case of “order-melting” substrates I present a detailed study of both, equilibrium order determined within the phenomenological mean-field theory and collective pretransitional dynamics. One of the difficulties encountered in any theoretical description of confined liq-uid crystals is the curved or even irregular and random internal geometry of the host material, which is often not easy to model. However, in the case of wetting, the anchoring effect of the confining surface is either partly or completely screened, and thus the actual topology of walls is not really important: it can be expected that the basic physics of these systems can be captured by a model planar geometry consist-ing of a nematic liquid crystal sandwiched between two parallel substrates, which is adopted in the present analysis. Schematic representation of the modeled system is visualized in Fig. 4.3. Two types of walls are considered: The disordering substrate gives rise to an isotropic boundary layer below the isotropic–nematic phase transi-tion temperature, where the largest part of the sample is nematic, and to perfectly isotropic phase above the transition. The system will be termed as the surface-molten nematic system. Second type of walls pertains to the ordering substrate and the corresponding liquid-crystalline system is paranematic. In the paranematic sys-tem, the equilibrium configuration is nematic below TNI, and above TNI it remains Heterophase nematogenic system 77 a a as <—> <—> as y ©----------------------------------------------1-------> O d Z Figure 4.3 Schematic representation of the studied system. Nematic liquid crystal is sandwiched between two parallel planar substrates with equal anchoring properties. They either induce uniaxial nematic order in the homeotropic direction (paranematic system) or they decrease the nematic order (surface-molten nematic system). nematic within the boundary layer whereas the core melts into isotropic phase. The forthcoming analysis is based on the Landau-de Gennes model of the phase transition and the nematic is described by a tensorial order parameter. In the study, the surface-induced smectic layering is not considered since in the case of liquid crystals which can be found in nematic phase but do not form smectic layers in the bulk (which is the case for 5CB I refer to all through the thesis) the smectic layering is not very pronounced. Further, it is assumed that the average nematic director is homogeneous in the system and only the scalar order parameter varies as a function of a coordinate z perpendicular to the confining substrates. Skaˇcej et. al showed that when the elastic constants associated to splay and bend deformations of the nematic director are different form the twist elastic constant (which is usual in liquid-crystalline materials) a spatial variation of the scalar order parameter induces variation of the director [119]. However, the coupling and the resulting variation of the director is very small and it is safe to adopt the one-elastic-constant approximation in which this effect can not arise. In 1996 Braun et. al studied wetting in the system where there was a competition between the surface-induced direction for the director and the preferred orientation of the director at the nematic-isotropic interface [120]. There are many experimental [121,122] and theoretical [123-125] studies considering the orientation of the director at the nematic-isotropic interface and also at the free nematic or isotropic surface. In most of them, the reported measured directions are in the range of 40° to 60°, however, from the Landau-de Gennes theory with assuming uniaxial nematic order only homeotropic and planar molecular arrangements can be obtained [123]. Marcus removed the assumption of uniaxial nematic order and showed that the order at the nematic-isotropic interface is highly biaxial and that the angle between the interface normal and the nematic director can be oblique [124]. However, in all cases the anchoring strength at the 78 Heterophase nematogenic system nematic-isotropic interface is rather weak, ? 10-5 J/m2, [107] so that it can be assumed that the orientation of the director is determined by the stronger anchoring of the substrate in the case of a paranematic system whereas in the case of a nematic in contact with disordering substrates the homeotropic direction is maintained by the aligning action of the magnetic field. In order to compete with the anchoring at the nematic-isotropic interface, B ? 0.01 T, which is very weak and, thus, its contribution to the free energy of the system need not to be considered explicitly. The case where the competing antagonistic preferred orientations of the director are comparable will be studied in the following Chapter which deals with the hybrid nematic cell. In the following, the equilibrium profiles of the nematic order are calculated within the phenomenological mean-field theory (Section 4.1). The pretransitional dynamics of all five degrees of freedom of the order is determined in Section 4.2. Structural forces which arise among the confining substrates because of their (dis)-ordering action are discussed in Section 4.3. Last Section in the Chapter deals with the van der Waals force acting on a wetting layer due to the inhomogeneity of the order. 4.1 Equilibrium profiles For a uniform director field with n = ez, the base tensors T i [Eq. (2.6)] are uniform themselves provided that the orientation of the two arbitrary vectors e^1 and e^2 is also position-independent, e.g., identified by ex and ey. Being uniaxial, wetting structures are characterized by an inhomogeneous profile of the degree of order, S/Sni = a0. The lateral dimensions of the system are much larger than its thickness, Lx, Ly ^ d, thus, a0 depends only on the distance from (one of) the substrates. Since the confining substrates are equal the profile of the degree of order is symmetric with respect to the plane in the middle of the liquid-crystalline cell. Thus, it will only be calculated in one half of the cell. The other four coefficients in the expansion A = J2i=-2 ai(f)Ti are all equal to 0. The Euler-Lagrange equation [Eq. (2.62)], which determines the profile of a0, reduces to ? !a0 - ?a0 + 3a0 - 2a0 = 0, (4.1) where prime denotes d/dz. Since QS is assumed uniaxial and homeotropic, QS = aST0, the boundary conditions at z = 0 and z = 1/2 read a0 = ?2 (a0 - aS), (4.2) where aS is the preferred degree of order at the substrate, and a'0 = 0, (4.3) Heterophase nematogenic system 79 1.2 1.2 1.0 0.8 1.0 0.8 .0.9 0.6 0.4 0.999/ /0.99999 0.6 0.4 0.2 0.2 0.0 0. 0.0 20 0 0.1 z/d 0. 0.0 0.1 z/d 0.2 (a) (b) Figure 4.4 Equilibrium profiles of the degree of order in the vicinity of (a) a disordering substrate at ? = 1 - 10-1,1 - 10-3, and 1 - 10-5, and (b) an ordering substrate at ? = 1 + 10-1,1 + 10-3, and 1 + 10-5. In both cases, the surface interaction is modeled by a prescribed degree of order equal to 0 and 1.1, respectively. respectively. This and all further differential equations are solved numerically using the relaxation method [126]. In both wetting geometries the equilibrium profile of the degree of order exhibits a substrate-induced variation in the boundary layer and levels off at the bulk value in the center of the sample (Fig. 4.4). In the case the wetting is partial, the thickness of the wetting layer dW has a very moderate temperature dependence whereas in the case of a complete wetting it exhibits a pronounced pretransitional increase. Since the thickness of the liquid crystal is finite the increase can not diverge like it diverges in the semi-infinite sample [108], however, as presented in Fig. 4.5 the actual temperature dependence is not far from the logarithmic behavior typical for semi-infinite samples. In the case of partial wetting, the thickness of the wetting layer levels off on approaching the bulk phase transition temperature. Temperature dependence of the thickness of the isotropic and nematic wetting layer in the case of complete and partial wetting is presented in Fig. 4.5. Although the wetting behavior of liquid crystals can be quite complex as it was discussed in the beginning of this Chapter, complete wetting is generally related to substrates with large (dis)ordering power, whereas otherwise partial wetting is to be expected. For the quadratic surface interaction used in present study, complete S > wetting of the disordering wall occurs only if a = 0 and G 0.0023 J/m2. In ? case of an order-inducing substrate the critical value of G depends on the preferred degree of order, which must exceed 1: for example, Gc(aS = 1.1) = 0.0006 J/m2. The obtained results are consistent with the results of the earlier study performed by Sheng [50], based on a somewhat different type of surface interaction. To illustrate the role of the strength of the surface interaction in the wetting behavior, some 80 Heterophase nematogenic system 60 50 40 I 30 -^20 10 60 50 40 30 20 10 0.80 100 80 60 40 20 0 0.85 0.90 ? (a) 0.95 1.00 1.0 1.2 1.4 6 1.6 (b) 1.8 2.0 1E-5 1E-4 1E-3 0.01 |? - ?NI | (c) 0.1 0 1E-5 1E-4 1E-3 0.01 |? - ?NI | (d) 0.1 Figure 4.5 Temperature dependence of the thickness of the wetting layer dW in (a) surface-molten nematic system and in (b) paranematic system in the case of complete (solid lines) and partial (dashed line) wetting. Dotted lines correspond to the thickness in the semi-infinite sample. Parameters used in the calculation of finite samples are for the surface-molten nematic system: aS = 0, d = 792 nm, and G › ? in the case of complete wetting and G = 0.001 J/m2 in the case of partial wetting, and in the case of a paranematic system: aS = 1.1, d = 792 nm, and G › ? in the case of complete wetting and G = 0.0006 J/m2 in the case of partial wetting. In the semi-infinite sample: G › ? and aS = 0 and aS = 1.1, respectively. In (c) and (d), the curves are plotted in logarithmic scale. 1 1 Heterophase nematogenic system 81 1.2 1.2 1.0 0.8 1.0 0.8 /""/"' r~ \\ 0.6 ¦ / / 0.6 \ \\ 0.4 / 0.4 \\ 0.2 ;/' / 0.2 - \ \\ 0.0 0. ___^y, 0.0 20 i s~> 0 0.1 0. .0 0.1 0. z/d z/d (a) (b) Figure 4.6 Some profiles of the degree of order in the finite anchoring model: (a) a disordering substrate at ? = 1 -10-5 with G —> oo (solid line), 0.001 J/m2 (dashed line), and 0.0003 J/m2 (dotted line), (b) an ordering substrate at ? = 1 + 10-5 with G —> oo (solid line), 0.0007 J/m2 (dashed line), and 0.0006 J/m2 (dotted line). equilibrium profiles of the degree of order in the two wetting systems are shown in Fig. 4.6. As the thickness of the liquid-crystalline sample is varied the thickness of the wetting layer changes even though it looks like the two wetting layers do not interact with each other. In an earlier study within the Landau-Ginzburg theory, Gompper et. al studied a heterophase system of water confined with hydrophobic substrates. They derived the asymptotic behavior for the thickness dependence of the wetting layer thickness, ddW/dd ~ -const. x exp(-d/L) [127], where L is the correlation length of the degree of order in the bulk phase. In Fig. 4.7 the temperature dependence is presented for the case of complete and partial wetting in surface-molten nematic and paranematic systems. In both, dW is a decreasing function of d and approaches the thickness corresponding to the semi-infinite sample. The behavior differs from the predicted asymptotic behavior, however, the latter is valid in the limit of large d’s and small dW’s. On decreasing the sample thickness, the bulk-like core of the sample is more and more affected by the presence of the wall and finally the two wetting layers merge. In a confined geometry, the transition between a surface-induced heterophase ordering and a homophase structure occurs at a temperature somewhat different from the clearing point. In case of disordering walls, the transition from the low-temperature phase characterized by molten boundary layer to the high-temperature isotropic phase is shifted below the nematic-isotropic phase transition temperature. Conversely, in the order-inducing geometry the transition from nematic to paranematic phase takes place above 9ni. The actual magnitude of the shift depends on the size of the sample and on the parameters of the surface interaction, and is prac- 82 Heterophase nematogenic system 51.1 51.0 50.9 16.74 16.72 16.70 16.68 16.66 300 400 500 600 700 800 900 1000 d [nm] 84.5 84.0 83.5 83.0 9.75 9.70 9.65 9.60 300 400 500 600 700 800 900 1000 d [nm] (a) (b) Figure 4.7 Thickness of the wetting layer as a function of the thickness of the (a) nematic and (b) paranematic sample. Solid lines correspond to the regime of the complete wetting and dashed lines correspond to partial wetting. In semi-infinite sample, the corresponding thicknesses are in the nematic case 50.6 nm and 16.3 nm, and in the paranematic case 81.1 nm and 9.4 nm, respectively. Parameters used in calculation are in the surface-molten nematic case: aS = 0, ? = 1 — 10-5, and G —> oo (complete wetting) and G = 0.001 J/m2 (partial wetting), and in the paranematic case: aS = 1.1, ? = 1 + 10-5, and G —> oo (complete wetting) and G = 0.0006 J/m2 (partial wetting). tically negligible in micron-size cavities: For example, for ? = 0.01 (d = 792 nm) and perfectly disordering wall with G —> oo and aS = 0, the transition occurs at ? = 0.99274. In the paranematic system with the same thickness and anchoring strength but with aS = 1.1, the transition occurs at ? = 1.0073. Both examples indicate that the shifts do not exceed 0.01 K. However, in smaller cavities the effect can be far more prominent. Below some cell thickness, which depends on the anchoring properties of the confinement, the phase transition is lost and the order grows continuously which is in agreement with experimental studies [128,129]. 4.2 Pretransitional dynamics Once the relevant equilibrium structures in the two wetting geometries have been described, the scene is set for the analysis of fluctuations. The dynamics of the five scalar components of collective excitations — introduced by the expansion B(r*, t) = J2i=-2 bi(f, t)Ti — is derived by projecting the equation of motion [Eq. (2.64)] onto the base tensors. Because the equilibrium order is characterized by only one nonzero amplitude, a0, the five fluctuation modes are uncoupled. In addition, due to the uniaxial symmetry of the system the two biaxial modes are degenerate and so are the two director modes. Since the mean-field equilibrium profiles depend on the Heterophase nematogenic system 83 z-coordinate only, the normal modes can be factorized as follows: where //i’s are the dimensionless relaxation rates of the eigenmodes measured in units of r~1 ~ 108 s_1 [Eq. (2.60)]. Their normal components $ are uncoupled and determined by C P0 ~ P ~ 6a0 + 6a0 — A0 P0 = 0, C P'±1 ~ \P + 6a0 + 2a0 — A±1] (3±1 = 0, (4.5) C P±2 — 0 — 3a0 + 2a0 — A±2j P±2 = 0, where (3[ = d$/dz and \ = \ii — Cp(k2x + ky) are the reduced relaxation rates of the modes. The in-plane components of the wave vector, kx and ky, are assumed to be subject to periodic boundary conditions. In terms of correlation lengths Ai = Ci~ /CjvI + C2qz2, where Li’s are the corresponding correlation lengths introduced in Eq. (2.35) and qz is the wavevector of the deformation parallel to the substrate normal. In case of finite anchoring strength, the corresponding boundary condition at the substrate is given by . g pi(0) = —2pi(0), (4.6) and otherwise /3i(0) = 0. Due to symmetry arguments, the normal modes must be either even or odd with respect to the center of the sample, thus, $(1/2) = 0 in the former case and $(1/2) = 0 in the latter case. (Since the equilibrium mean-field profile a0 is even with respect to the plane in the middle of the cell and parallel to the confining substrates the corresponding potentials for the fluctuation modes are even as well. In the case of even potential the modes are know to be of the two types — either even or odd with respect to the same symmetry plane [130].) Homophase ordering If a nematic layer is bounded by the walls characterized by strong surface interaction and a bulk-like value of the preferred degree of order, $’s reduce to sine waves, and their relaxation rates may be cast into ^i = Cn,i/Cni + C [(n+1)71"] , (4.7) which is the same as in bulk except that due to the finite dimension in the z direction the wavevector qz can only have discrete values, qz,n = ((n + 1)tt, where n is the number of nodes of the sine function between the two substrates. In confined sample, the minimum wavevector is ^7r = 0, thus, even the relaxation rate of the Goldstone 84 Heterophase nematogenic system director mode is finite, though, it is very small. LN,i’s are defined in Eq. (2.37) and their temperature dependence is presented in Fig. 2.5. Above the clearing point, a disordering wall produces a perfectly isotropic phase. In this case, all five types of fluctuations are degenerate, and their relaxation rate is determined by T ^—2 n—2 s2 2 Ai = 4/ /4Ni + C [(^+1)7r] • (4.8) LI is introduced in Eq. (2.36) and its temperature dependence is plotted in Fig. 2.5. It should be stressed again that the hardness of a given type of fluctuations can be characterized by its correlation length: the shorter the correlation length, the higher the energy of fluctuations. To understand the pretransitional behavior of the system, it is important to know how the energy levels of excitations in nematic phase compare with those in isotropic phase. In the vicinity of the phase transition, fluctuations of the degree of order are equally hard in both phases. In nematic phase, the biaxial modes are energetically far more costly than in isotropic phase — as opposed to the director modes, which are characterized by an infinite correlation length in nematic phase, whereas LI is finite at the phase transition temperature. At the phase transition, the correlation lengths of the scalar order parameter are equally hard in both, nematic and isotropic phase, because at the transition both phases are in equilibrium. In general, the surface-induced degree of order differs form the bulk value and the profile of the degree of order is inhomogeneous. Thus, the generalized correlation lengths of the fluctuation modes are spatially dependent and the eigenmodes of fluctuations in the two wetting geometries can only be determined numerically. In the following, the spectra of collective excitations in nematic phase with molten boundary layers and in paranematic phase are interpreted simultaneously. Fluctuations of Degree of Order In both systems, the primary effect of wetting is related to the existence of a slow mode characterized by soft dispersion of its relaxation rate, whereas the upper part of the spectrum remains more or less the same as in homophase system (Figs. 4.8 and 4.9). The elementary mode of fluctuations of the degree of order is localized at the phase boundary between the wetting layer and the bulk phase, and since it is even with respect to the center of the sample, it corresponds to fluctuations of thickness of the central part of the slab. Similarly, the lowest odd mode — also localized at the nematic-isotropic interface — represents fluctuations of position of the core. However, the relaxation rates of these two modes are the same within numerical accuracy, indicating that the two wetting layers are effectively uncoupled. Heterophase nematogenic system 85 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.5 0.0 -0.5 0.90 0.92 0.94 0.96 0.98 1.00 ? (a) -1.0 0.0 0.1 0.2 0.3 z/d (b) 0.4 0.5 Figure 4.8 Disordering substrate: (a) spectrum of fluctuations of the degree of order, illustrated by the portraits of a few typical modes labeled by the number of nodes (b); ? = 1 — 10-5. The lowest mode characterized by soft dispersion of the relaxation rate corresponds to fluctuations of the thickness of the wetting layer, whereas the upper part of the spectrum is basically the same as in purely nematic sample and the corresponding fluctuations disturb the whole sample. Red line in (a) represents the lower limit of the spectrum of a homophase system and the dashed line in (b) corresponds to the equilibrium profile of the degree of order; G —> oo and aS = 0. This is directly related to the thickness of the sample, which is much larger than LN,0, the typical length scale of the variation of the degree of order. Were the system thinner, the correlation between the (dis)ordered regions induced by the two substrates would be stronger and the degeneracy of the lowest two normal modes would be removed. In the complete wetting regime, the relaxation rate of the elementary excitations of the degree of order exhibits a linear critical temperature dependence typical for soft modes: /^0,0 ±C±(0 — 1), (4.9) where ’—’ and ’+’ correspond to nematic phase with molten boundary layer and paranematic phase, respectively. The difference between the coefficients C- and C+, which are approximately equal to 5.6 and 3.0, can be attributed to the fact that the thickness of the isotropic wetting layer at the disordering wall at 9 = 1 — 8 is half of the thickness of the nematic wetting layer at the ordering substrate at 9 = 1 + 8; cf. Fig. 4.4 (a) and Fig. 4.4 (b). The slowdown of the relaxation rates of the surface-induced soft modes, i.e., the divergence of their relaxation times, at the phase transition temperature is a well-known and clear signature of the continuity of the transition, which is actually just another face of the advancing phase boundary in any complete wetting geometry [18,113]. In a finite system, however, a wetting-driven phase transition can never be truly continuous, because the heterophase 86 Heterophase nematogenic system 1.2 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.5 «f 0.0 0.5 1.00 1.02 1.04 1.06 1.08 1.10 ? 1.0 0.0 0.1 0.2 0.3 0.4 0.5 z/d (b) (a) Figure 4.9 Ordering substrate: (a) relaxation rates of fluctuations of the degree of order in paranematic phase, and (b) some typical modes for ? = 1 + 10-5. As in Fig. 4.8, the soft mode represents fluctuations of the thickness of the wetting layer, and the upper part of the spectrum is more or less the same as in perfectly isotropic sample, which is also reflected in the sinusoidal behavior of ßo,n>\(z). Again, the red line is (a) is associated to the lower limit of the spectrum in a homophase system and in (b) the equilibrium profile of the degree of order is plotted with the dashed line; G —> oo and aS = 1.1. configuration eventually becomes unstable in the immediate vicinity of the clearing point — but in samples of thickness > 100 nm this effect is detectable only if the temperature resolution of the experimental method is better than ~ 0.01 K. In both wetting geometries, the upper part of the spectrum is more or less the same as its homophase (i.e., nematic and isotropic) counterpart, which is reflected in its regularity as well as in the sinusoidal profiles of the normal modes (Figs. 4.8 and 4.9). This also means that the upper, quasi-homophase modes are more or less independent on the strength of the surface interaction, which has been verified numerically. On the other hand, the behavior of the wetting-induced elementary mode does depend strongly on the magnitude of the anchoring strength, G: if the wetting is partial instead of complete, the pretransitional decrease of the localized modes’ relaxation rates is less pronounced. They do not drop to 0 but remain finite at ?ni = 1, so that the transition from surface-molten nematic to isotropic phase or from nematic to paranematic phase is discontinuous even in semi-infinite systems. Though, the corresponding latent heat may be reduced considerably compared to the bulk isotropic-nematic transition. The temperature variation of the relaxation rates of the lowest modes remains linear, implying that the underlying mechanism is basically the same as in the complete wetting geometry. These findings are quantitatively summarized in Fig. 4.10, where the lowest mode’s relaxation rate at the isotropic-nematic phase transition temperature is plotted as a function of the anchoring strength. In the partial wetting regime, which corresponds to small Heterophase nematogenic system 87 1.0 0.8 0.6 0.4 0.2 0.0 10-5 10-4 10-3 G [J/m2] 10-2 Figure 4.10 Relaxation rate of fluctuations of the thickness of the boundary layer in disordering (solid line) and ordering wetting geometry (dashed line) as a function of the strength of the surface interaction. In both cases, ?0,0 is finite for G < Gc (partial wetting) and 0 otherwise (complete wetting), the critical values of G being equal to 0.0023 J/m2 for disordering substrates and 0.0006 J/m2 for order-inducing walls. Note that the two geometries differ in the type of the behavior of ?0,0 in the vicinity of Gc. G’s, ?0,0 is finite; in the complete wetting regime, on the other hand, it is (within numerical accuracy) equal to 0. The two geometries give rise to slightly different behavior of ?0,0 in the vicinity of the critical strength of the surface interaction: in case of disordering wall, ?0,0 approaches 0 somewhat more slowly than in case of an order-inducing wall which is due to the fact that in the former case the regime of the complete wetting is bounded to the very edge of the phase diagram (G, aS), i.e., to aS = 0. In addition to the two elementary modes corresponding to fluctuations of thickness of the boundary layers there are actually two more localized modes with relaxation rates that do depart from the quasi-homophase spectrum although not as distinctly as the soft dispersion of ?0,0. These modes represent fluctuations of the shape of the phase boundaries: the even one is related to simultaneous sharpening/flattening of the phase boundaries, whereas the odd one describes out-of-phase fluctuations of their slope. The relaxation rates of these two modes are degenerate, which is, as it has already been established, related to the fact that the system considered is rather thick, so that the correlation between the two wetting layers is very weak. These modes are depicted in Fig. 4.11. There are, therefore, two localized modes associated to each interface between nematic and isotropic phase: one of them corresponds to fluctuations of the position of the phase boundary, and the other one changes its profile. Since the theoretical approach used in this analysis is quite universal in its very nature, it seems that the 88 Heterophase nematogenic system 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 0.0 0.1 0.2 0.3 0.4 0.5 z/d -1.0 0.0 0.1 0.2 0.3 0.4 0.5 z/d (a) (b) Figure 4.11 Portrait of the second-lowest order parameter modes in the case of (a) disordering and (b) ordering substrates. The even mode is responsible for simultaneous sharpening/flattening of the phase boundaries and the odd one describes out-of-phase fluctuations of their shape. The modes are degenerate within the numerical accuracy. same should hold true for any interface that can be described by a scalar variable. However, in case of a phase boundary with a more complex structure, additional and more sophisticated localized modes are expected. As the mean-field structures discussed here are characterized by inhomogeneous profiles of the degree of order and homogeneous profiles of the degree of biaxiality and director fields, the wetting-specific dynamics is primarily related to fluctuations of the degree of order. On the other hand, any critical behavior of the biaxial and director modes is merely an indirect effect of the surface-induced heterophase ordering. Biaxial fluctuations Biaxial modes are the hardest type of fluctuations in uniaxial nematic phase, which is related to the fact that thermal excitations of transverse molecular order have to compete with the existing uniaxial alignment. In systems with intrinsic biaxiality, biaxial fluctuations are much softer which will be evident in the hybrid nematic system studied in the following Chapter. At the phase transition temperature, the lower limit of the relaxation rates of biaxial fluctuations in nematic phase is 9 times larger than in isotropic phase [cf. Eqs. (4.7), (4.8) and Eqs. (2.37), (2.36)]. This considerable difference in the energy levels of biaxial modes in the two phases is reflected in their spectra in the two wetting geometries. In case of nematic phase confined by a disordering wall, the lowest modes are bounded to the isotropic wetting layer. A strong elastic deformation of the modes in the thin isotropic region of the sample is energetically more favorable than a moder- Heterophase nematogenic system 89 10 8 6 4 1.0 0.5 0.0 -0.5 0 0.90 0.92 0.94 0.96 0.98 1.00 ? -1.0 0.0 0.1 0.2 0.3 0.4 0.5 z/d (a) (b) Figure 4.12 Biaxial modes in a nematic sample bounded by disordering substrates: (a) the lowest modes exhibit pretransitional slowdown on approaching the clearing point and are confined to the isotropic wetting layer (b); ? = 1 - 10-5, aS = 0, G —> oo. The upper part of the spectrum is more or less nematic-like, the modes being spread over the whole slab. Red line represents the lower limit of the spectrum in a homophase system and the dashed line corresponds to the equilibrium profile. ate deformation in the thick nematic core (see Fig. 4.12). The number of bounded modes depends on the thickness of the wetting layer and, thus, on temperature: as the sample is heated towards the clearing point, more and more levels depart from the upper, nematic-like part of the spectrum, which corresponds to modes that disturb the whole sample. In paranematic phase induced by the ordering substrate, biaxial fluctuations are, conversely, expelled from the ordered boundary layer (see Fig. 4.13), so that the allowed wavelengths of the normal modes are determined by the thickness of the central isotropic part, not by the actual thickness of the sample. The difference between these two is not significant except in the vicinity of the phase transition temperature, where the nematic wetting layers squeeze the isotropic core and speed up the relaxation rates of the biaxial modes. Director fluctuations Director modes are, as opposed to biaxial fluctuations, excited very easily in nematic phase, where their Hamiltonian is purely elastic, whereas in isotropic phase they are characterized by finite correlation length [Eqs. (2.37) and (2.36)]. This implies that their wetting-induced behavior should be quite the inverse of what is predicted for the biaxial modes. In the disordering geometry, the director modes are forced out of the substrate-induced isotropic boundary layer into the nematic core (see Fig. 4.14) just like the biaxial modes are expelled from the nematic boundary layer into the isotropic core 2 90 Heterophase nematogenic system 1.2 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.5 0.0 -0.5 1.00 1.02 1.04 1.06 1.08 1.10 ? -1.0 0.0 0.1 0.2 0.3 0.4 0.5 z/d (a) (b) Figure 4.13 Biaxial modes in paranematic phase are all expelled from the quasi-nematic boundary layer (b); ? = 1 + 10-5, aS = 1.1, and G —> 0; and on approaching ?ni their relaxation rates must therefore increase along with the thickness of the boundary layer (a). Red line represents the lower limit of the spectrum of a homophase system and the dashed line depicts the equilibrium profile. 0.10 ^_^^J 1.0 0.5 0.0 -0.5 -1.0 00 0. 0.08 _______^ " /''' 1 M \ / \ r 0.06 ____^ 0.04 ___. "Aw w \ 0.00 0. ----------------------- 90 0.92 0.94 0.96 0.98 1. ? (a) 0 0.1 0.2 0.3 z/d (b) 0.4 0.5 Figure 4.14 Director modes in a nematic sample bounded by disordering substrates: (b) director fluctuations are forced out of the boundary layer; ? = 1 — 10-5, aS = 0, and G —> oo; and must speed up on approaching the clearing point (a), just like the biaxial modes in paranematic phase (Fig. 4.13). Red line represents the lower limit of the corresponding spectrum in a homophase system and the dashed line depicts the equilibrium profile. Heterophase nematogenic system 91 1.2 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.5 0.0 -0.5 1.00 1.02 1.04 1.06 1.08 1.10 ? -1.0 0.0 0.1 0.2 0.3 0.4 0.5 z/d (a) (b) Figure 4.15 Director modes in paranematic phase: (a) the relaxation rates of the lowest modes, which are restricted to the nematic wetting layer (b); ? = 1 +10-5, aS = 1.1, and G —> oo; decrease to 0 as ? —> 1 due to the growth of the wetting layer (cf. Fig. 4.13). Red line represents the lower limit of the corresponding spectrum in a homophase system and the dashed line depicts the equilibrium profile. of the paranematic phase induced by the ordering substrate. Far from the phase transition temperature, their relaxation rates are temperature-independent, whereas in the vicinity of the clearing point they all increase because of rapid growth of the wetting layer. In paranematic phase a few lowest director modes are confined to the nematic boundary layer, whereas the upper ones extend over the whole sample and are more or less the same as in perfectly isotropic phase (see Fig. 4.15). The relaxation rates of the lowest modes exhibit a cusplike slowdown similar to that observed in biaxial modes in a disordering geometry (Fig. 4.12). Moreover, their pretransitional slowdown resulting from the increase of the thickness of the wetting layer is actually critical, since in this case the fluctuations confined to the wetting layer are Goldstone modes. The discussed results correspond to infinitely strong surface interaction. For G’s which are large enough to induce complete wetting, the spectra of fluctuations re-main qualitatively the same, whereas otherwise the slow modes are no longer critical. Eventually, if the strength of the surface interaction is very weak, all fluctuations become cosine-like and their spectrum is described by Eqs. (4.7) and (4.8). The analysis has revealed a close relationship between the wetting regime induced by (dis)ordering substrates and the pretransitional behavior of thermal fluctuations of the ordering in confined liquid crystals. Both geometries are characterized by a wetting-induced interface between nematic and isotropic phase, which gives rise to two localized normal modes: the first one represents fluctuations of the position 92 Heterophase nematogenic system of the phase boundary and is characterized by a soft dispersion of its relaxation rate (provided that the wetting is complete), and the second one corresponds to fluctuations of the shape of the interface. Moreover, there are a few additional slow modes, which are restricted to the wetting layer and whose pretransitional behavior is related to its growth. If the wetting is partial, the slowdown of the localized modes is not as pronounced as in complete wetting regime, but the underlying physics remains the same. The wetting-induced pretransitional behavior of the fluctuations of the liquid-crystalline ordering is certainly not limited to geometries discussed in this study. A similar phenomenon is expected in nematic and isotropic samples with substrate-stabilized smectic boundary layer, which should exhibit critical slowdown in the vicinity of the nematic–smectic and isotropic–smectic phase transition, respectively. But the analogy with the substrate-stabilized nematic layer above the isotropic– nematic phase transition is not complete due to layered structure of the smectic ordering, which presumably gives rise to nontrivial features of the wetting-induced fluctuations in this system. It seems possible that the effect described here has already been detected exper-imentally in some microconfined liquid-crystalline systems, where a huge increase of the decay time of fluctuations has been observed in the vicinity of isotropic– nematic [9] and nematic–smectic A phase transition [131]. However, conclusive evidence could only be provided by a detailed and comprehensive analysis of the existing data or by an experiment designed to probe the dynamics within the bound-ary layer. The latter could be based on, for example, the evanescent light scattering technique [132,24]. The results of the study can be extrapolated beyond the geometries discussed once it has been realized that the slow dynamics of the localized modes is actually directly related to the existence of the phase boundary and that the wetting itself is merely a mechanism which introduces a heterophase structure — and, therefore, a phase boundary — into the system. In confined liquid crystals, heterophase order-ing is very often induced by topological constraints imposed by curved walls, which result in singularities of the director field, where very strong elastic deformation of the nematic phase is avoided by reducing the degree of order. Since the dis-ordered regions called defects are more complex than the planar nematic-isotropic interface [133–135], they should be accompanied by several localized modes related to fluctuations of their structure as well as those corresponding to fluctuations of their size, position, and shape. This indicates that the defects should be considered as possible generators of slow nondirector fluctuations in confined liquid crystals. Another aspect of structure and pretransitional dynamics applicable to defects in Heterophase nematogenic system 93 0 -200 -400 -600 -800 -1000 0 20 40 60 d [nm] 80 100 0 -200 -400 -600 -800 -1000 0 20 40 60 d [nm] 80 100 (a) (b) Figure 4.16 Structural force in a nematic system confined with (a) disordering substrates (? = 0.9, aS = 0, and G —> oo) and (b) ordering substrates (? = 1.1, aS = 1.1, and G —>¦ oo). Solid lines correspond to the force in the nematic phase and dashed lines correspond to the force in the isotropic phase. Dotted verticals denote the thickness at which the isotropic phase becomes metastable in the surface-molten nematic system (94.6 nm) and the nematic phase becomes metastable in the paranematic system (91.3 nm), respectively. nematic liquid crystals is given in Chapter 5. 4.3 Structural and pseudo-Casimir forces Structural force The structural force in nematogenic heterophase systems studied in this thesis arises from the deformation of the scalar order parameter. Thus, the force is short-range and attractive. A detailed study of structural forces in a paranematic system was ˇ performed by Borˇstnik and Zumer [60]. Similarly, the structural force can be cal-culated also for the nematic system with molten wetting layers. Typical thickness dependence of the structural forces in both nematogenic systems is presented in Fig. 4.16. The order in thin cells is determined by the surface interaction. If the cell thickness is increased the corresponding amount of the liquid-crystalline material enters the cell and its order is changed from the bulk order to the one induced by the substrates. Thus, the free energy of the system changes for (f - fbulk)V which produces a constant force between the confining walls. The force is attractive, because in the given systems the surface-induced order corresponds to higher free energy. In thick enough cells, the order parameter profile is characterized by a bulk-like core and surface-induced wetting layers. Now the change of the free energy of the system when changing the cell thickness is due to the minor changes in the order parameter profile while most of the material enters the core of the cell, thus, its free 94 Heterophase nematogenic system energy is not changed. In this regime, the structural force decays exponentially with respect to the cell thickness, ? oc exp(—d/?), (4.10) where ? = ?n,o in the case of a nematic with molten wetting layers and ? = ?I in the case of a paranematic cell. In Fig. 4.16 hysteresis loops can be clearly seen — they are the evidence of the metastable phases. Details of the profile of structural force depend on details of the surface-induced interaction, whereas the existence of the hysteresis depends on the temperature. Pseudo-Casimir force For the sake of completeness, in this Section, I will quote the main results of fluctuation-induced force in the paranematic phase which were obtained by Zi-herl et. al [28]. As we have seen, there is a strong correspondence between the paranematic system and the surface-molten nematic system. Thus, the main features of the pseudo-Casimir force in the surface-molten nematic system, as they can be predicted from the anchoring properties and fluctuation modes, will be discussed at the end of this Section. The pseudo-Casimir force in a heterophase system can be interpreted in terms of two contributions: (i) the interaction between the substrates and the phase boundaries and (ii) the interaction between the two phase boundaries. Interaction between solid substrate and phase boundary consists of three contributions corresponding to three non-degenerated fluctuation modes. The fluctuations of the degree of order and biaxial fluctuations give rise to a short-range repulsion between the substrate and the phase boundary proportional to exp(—2dW/?), where dW is the thickness of the wetting layer and ? = ?n,o or ?n,±i for order parameter and biaxial fluctuations, respectively. The short-range of the interaction is a consequence of finite correlations in both nematic and isotropic phase. The repulsion between the substrate and the phase boundary can be understood in terms of boundary conditions. The anchoring at the substrates is strong, whereas the maxima of the order fluctuation modes at the phase boundaries (see Fig. 4.9) indicate that these modes experience effectively weak “anchoring” conditions. Because of the dissimilarity of the boundary conditions the interaction due to fluctuation of the degree of order is repulsive. A similar argument applies to biaxial fluctuations. Here weak “anchoring” condition at the phase boundary can be understood as a consequence of the fact that biaxial fluctuations are much more favorable in the isotropic phase than in the nematic phase. The main contribution to the interaction between the solid substrate and the phase boundary is induced by the director fluctuations, Heterophase nematogenic system 95 which are characterized by an infinite correlation length in the nematic phase. The leading term of this interaction reads kBTS?(3) ---------7>----, (4.11) where ? is the Riemann zeta function. This long-range interaction is attractive which can again be interpreted in terms of (dis)similarity of boundary conditions. In the isotropic phase, the director fluctuations are very “hard” compared to the ones in the nematic phase, therefore, as we have seen (Fig. 4.15), the lowest normal modes are actually confined to the nematic surface layer. The effective boundary condition at the phase boundary is very similar to strong anchoring at the solid substrate, and the force induced by director fluctuations is attractive. Interaction between phase boundaries gives rise to an attractive fluctuation-induced force which is proportional to exp(-2(d-2dW)/?I). The attraction is due to the identical boundary conditions. Except in the vicinity of the metasta-bility limit of the paranematic phase, the distance between the substrate and the phase boundary is much smaller than the distance between the two boundaries, and the interaction between the phase boundaries is very weak. The range of the pseudo-Casimir interaction between the phase boundaries is half of the range of the structural interaction [Eq. (4.10)]. Thus, for dW 1, dW ^ d) the fluctuation-induced force between the two substrates is governed by the interaction between the solid substrate and the phase boundary. It is mediated by the structural interaction which determines functional dependence of the wetting layer thickness on the sample thickness which has already been discussed to be ?dW/?d ~ -const. x exp(-d/?I) [127]. Therefore, the leading term in the substrate-to-substrate fluctuation-induced force kBTS?(3) ?dw _d/? Fcas ~---------^------ oc e (4.12) 2?dW ? d is repulsive and short-range. Its range, ?I, is identical to the range of the structural force, whereas its magnitude is smaller but comparable to the magnitude of the structural attraction. Since the two forces have the same range, the structural force is proportionally diminished by fluctuation-induced contribution. 96 Heterophase nematogenic system The pseudo-Casimir force in a surface-molten nematic system: From the knowledge of general properties of the pseudo-Casimir force and by being familiar with fluctuations in the system the force can be predicted also for the case opposite to the paranematic, for the nematic in contact with disordering walls. Here, the wetting layers are in the isotropic phase where all fluctuations modes are characterized by the same, finite correlation length, thus, all the corresponding contributions to the fluctuation-induced force are short-range and proportional to exp(—2dW/?I). Fluctuations of the degree of order and director fluctuations yield a repulsive interaction because they both experience effectively weak “anchoring” at the phase boundary. In the case of former fluctuations, this is indicated by the position of their maxima whereas in the case of the latter, the “weak” anchoring is due to the fact that the director fluctuations are much more favorable in the nematic core. Biaxial fluctuations give rise to a short-range attraction because they experience effectively strong “anchoring” at the phase boundary which results from the fact that biaxial fluctuations are extremely “hard” in the nematic phase and, thus, they are localized to the wetting layers. On the other hand, the fluctuation-induced interaction between the two phase boundaries is dominated by Goldstone director fluctuations which give rise to a long-range attractive interaction proportional to (d — 2dW)~2. Fluctuations of the degree of order and biaxial fluctuations yield short-range attraction. From this, it can be expected that the pseudo-Casimir force in the nematic system with molten wetting layers should be long-range and attractive and it should dominate the structural force since the range of the latter is finite, i.e., ?n,0. 4.4 Van der Waals force in heterophase liquid-crystalline systems As it has been derived in previous Chapter dealing with the van der Waals force, two media with the same “isotropic” dielectric and optical properties (here, isotropic quantities are aiso oc tra and a is the corresponding tensor), however, with different symmetry of relevant tensors, have different effect on the electromagnetic field modes. In a heterophase system, the two phases — the nematic and isotropic phase — have the described properties. Therefore, the isotropic-nematic phase boundary acts as an additional wall in the system and it forms, together with the solid-liquid-crystalline interface, a cavity which perturbs the electromagnetic field modes. The van der Waals interaction originating in different symmetries of the phases together with the pseudo-Casimir interaction, thus, yields a correction to the equilibrium thickness of the (dis)ordered wetting layer. In Chapter 3, the van der Waals force was determined for a system characterized Heterophase nematogenic system 97 (a) (b) Figure 4.17 Cartoon of the nematic–isotropic heterophase system: semi-infinite nematic liquid crystal in a contact with a solid substrate which (a) induces high uniaxial homeotropic nematic order or (b) melts the nematic order. In a graph, the solid line corresponds to the spatial dependence of the scalar order parameter S and the dotted line corresponds to the modeled function. dW is the thickness of the (a) disordered isotropic wetting layer in the bulk uniaxial nematic liquid crystal or (b) ordered uniaxial wetting layer in the bulk isotropic liquid. by sharp, discontinuous changes in the dielectric (and optical) properties at the walls whereas these properties were constant between two walls. The obtained results are applicable to the systems where the thickness of the interface between two distinct media is finite, however, much smaller than typical dimensions in the system, i.e., the gap between the walls, and the wavelength ?e. In wetting system close to the isotropic–nematic phase transition, the interface between the two phases is well defined and the order parameter profile can be approximated by a step-like function, as is schematically represented in Fig. 4.17. The approximation is the better the closer is the temperature to the phase transition. (In the immediate proximity of the phase transition, the thickness of the region where the nematic order falls off to the isotropic phase or, similarly, where the isotropic order grows to the nematic phase, is well below the thickness of the wetting layer as it is shown in Figs. 4.2 and 4.4.) In the case of a two-phase system the separation dW between the two “semi-infinite” media — solid substrate and the bulk liquid-crystalline phase — is not a directly controllable parameter, instead, it corresponds to the minimum of the free energy of the system; it can be changed by changing the temperature of the system. In Section 4.1 only the mean-field part of the free energy was taken into account to determine the equilibrium state of the heterophase system. However, there are at least two sources of correction to that description. The fluctuations of 98 Heterophase nematogenic system 10 0 -5 -10 0 20 40 60 dW [nm] 80 100 Figure 4.18 Van der Waals force per unit area acting on a wetting layer in a surface-molten nematic system (solid line) or in a paranematic system (dashed line). the order parameter tensor contribute to the pseudo-Casimir interaction between the interfaces; the leading term of the interaction energy in the paranematic system is written out in Eq. (4.11) [28]. Another contribution to the free energy of a heterophase system is due to the spatial dependence of the permittivity tensor which yields nonzero van der Waals interaction. The interaction is very weak, because the difference in the permittivities of two of the media — the isotropic and nematic part of the system — is only due to different eigenvalues of the permittivity tensors, whereas the traces of the tensors are the same. In Fig. 4.18 the van der Waals force acting on a wetting layer is plotted for a typical liquid-crystalline heterophase system. Although the thickness of the wetting layer is not subject to direct change, the force has been calculated as it has been defined in Eq. (3.31), i.e., as a derivative of the corresponding free energy with respect to the gap between the walls. One can imagine that due to thermal order parameter fluctuations the thickness of the wetting layer can change. The van der Waals force acting on such wetting layer is then the change of the interaction free energy of the system because of the changed separation between the walls. In a system with silica as a solid substrate the van der Waals force between solid-liquid-crystalline and nematic-isotropic interface is attractive in the surface-molten nematic system and it is repulsive in the paranematic system. In the surface-molten nematic system the thickness of the equilibrium isotropic wetting layer is decreased due to the nonzero van der Waals interaction; in the paranematic system, the thickness of the ordered wetting layer is increased. To determine quantitatively the change of the equilibrium wetting layer thickness the van der Waals free energy should be included in the total free energy which is subject to minimization. However, the van der Waals interaction is very weak. 5 Heterophase nematogenic system 99 3 2 1 0 -1 -2 -3 20 30 40 50 60 70 80 dW [nm] (a) 1E-5 1E-4 1E-3 0.01 0.1 ? - ?NI (b) Figure 4.19 (a) Van der Waals force (solid line) and structural force per unit area in the wetting layer of a nematic liquid crystal at the order-inducing solid substrate for various temperatures close to the isotropic-nematic phase transition: T — TNI = 10-2 K (dashed line), T — TNI = 10-3 K (dash-dotted line), and T — TNI = 10-4 K (dotted line). The nonzero van der Waals force yields the increase of the equilibrium wetting layer thickness. (b) Close to the phase transition, the temperature dependence of the correction of the equilibrium wetting layer thickness exhibits power-law behavior, ?dW oc (T — Tni)-? , where ? = 0.42 is obtained from fitting the corresponding data (red line). In the limit T — TNI —> 0, ? = 0.5. Therefore, the forces acting on the wetting layer will be compared rather than the corresponding free energies. Here again, one should imagine that due to thermal fluctuations a heterophase structure with different wetting layer thickness is estab-lished. Since that structure is not the equilibrium one its mean-field counterpart of the free energy is raised with respect to the equilibrium value and an effective structural force starts acting in the system in order to reestablish the equilibrium state. In Fig. 4.19 (a) the structural force acting on the ordered wetting layer in the paranematic system is plotted together with the van der Waals force. The effec-tive structural force is repulsive if the layer thickness is smaller than the equilibrium thickness, and it is attractive if the thickness is larger than the equilibrium thickness. In the equilibrium, there is no force acting on the layer. If the van der Waals force is taken into account the point of zero force is moved towards larger layer thicknesses. The increase of the equilibrium layer thickness is in the range of few nanometers. In 5CB in contact with order-inducing silica, at T - TNI = 10-2 K the equilib-rium layer thickness determined from the phenomenological mean-field approach, dMF = 28.13 nm, is increased by 0.34 nm, at T - TNI = 10-3 K the equilibrium layer thickness dMF = 45.28 nm is increased by 0.51 nm, and at T-TNI = 10-4 K the equilibrium layer thickness dMF = 63.39 nm is increased by 1.29 nm. Temperature dependence of the van der Waals induced correction is presented in Fig. 4.19 (b). On approaching the bulk phase transition temperature ?dW exhibits critical power-law 1 100 Heterophase nematogenic system behavior, ?dW oc (T — Tni)~?, where ? = 0.5. Due to the van der Waals interaction between the solid-liquid-crystalline and two phase interface the critical behavior of the growth of the wetting layer changes — critical behavior determined from the mean-field phenomenological approach is characterized by a critical exponent 0 (logarithmic divergence). Except in immediate proximity of the phase transition where ?dW diverges the correction due to the van der Waals force is rather small and, thus, experimentally undetectable. However, it should be noted that for the sake of simplicity the structural force has been determined in the limit of infinitely strong surface anchoring which yields larger deformations and therefore larger forces and stronger thickness dependence as compared to more realistic values of the surface coupling. On the other hand, in calculation of the van der Waals force the usual experimentally determined values of dielectric permittivity and refractive indices have been used [136]. Therefore, in real wetting systems the effect of the van der Waals force is stronger and can be estimated to contribute up to 10% to the structural force and, therefore, also to the equilibrium wetting layer thickness. In the immediate vicinity of the phase transition, the order parameter fluctuations become critical and the fluctuation-induced interaction exceeds the structural interaction. Thus, to probe the critical exponent due to the effect of the van der Waals interaction the mean-field theory should be renormalized also by the fluctuationg-induced contribution. Further from the phase transition, the effect of the van der Waals interaction should be more prominent in systems with substrate-induced smectic layering. In such systems, the layer of the surface-induced order and the corresponding phase interface are better defined even further from the phase transition. In the case of presmectic layering in the isotropic phase the two phases — smectic A and isotropic phase — differ more than do the smectic A and nematic phase in the case of presmectic layering in the nematic phase. Thus, in the former case the van der Waals interaction between the solid substrate and the phase boundary is stronger than it is in the latter case. 5 Hybrid nematic cell In hybrid nematic cells the liquid crystal is confined by substrates inducing nematic order in different directions, often close to being perpendicular. In such systems the nematic order can not be unperturbed. Due to the frustrating boundary conditions the equilibrium order usually results in a nonlocally perturbed, both, director field and degree of nematic order. Hybrid nematic cells are mainly used for studying anchoring properties of the confining substrates but also as optical switches. In practice the hybrid frustration can be achieved by several ways: (i) If the nematic liquid crystal is confined with substrates that have been prepared in a way to induce nematic order one in homeotropic direction and the other in one of the lateral directions. (ii) If the nematic liquid crystal is deposited on the solid substrate whereas on the other sides it has free liquid-crystalline–air interface. In many liquid crystals the free liquid-crystalline–air interface induces rather strong homeotropic anchoring of the nematic liquid crystal. Thus, in combination with a solid substrate which induces nematic order in a direction in the plane of the substrate such liquid crystal is subject to frustrating hybrid conditions. (iii) The hybrid situation can also be a result of geometrical constraints such as in cylindrical cavity with homeotropic anchoring at the wall whereas the geometry of the cavity prefers orientation along the symmetry axis. Different possibilities of achieving hybrid nematic cells are schematically represented in Fig. 5.1. Possible technological applications have stimulated the increase of interest in hybrid nematic geometries [137]. Using a quasi-elastic light scattering method Wit-tebrood et. al experimentally studied thickness dependence of the nematic-isotropic phase transition temperature and stability of ordered structures in a hybrid nematic film obtained after a spread of a liquid crystal droplet on a solid substrate [138]. In their experimental setup with unequal anchoring strengths of the confining sub-strates (solid substrate and a free liquid crystal interface) they were able to determine the critical cell thickness for the hybridly aligned order which was in good agreement 101 102 Hybrid nematic cell (a) (b) (c) Figure 5.1 Schematic representation of different systems characterized by hybrid frustration: (a) Confining substrates prepared in a different way, so that one induces homeotropic anchoring and the other homogeneous planar anchoring. (b) Hybrid frustration due to opposing geometry induced direction of order and the direction induced by anchoring. (c) Liquid crystal in a contact with a solid substrate and with a free liquid-crystalline-air interface. with the theoretical expression obtained long ago by Barbero and Barberi [19]. In their study an approximate director picture omitting positional dependence of the scalar order parameter and biaxiality was used. In the framework of Frank elastic theory an extensive study of pretransitional director dynamics in a hybrid cell was done by Stallinga et. al [12]. Using the director description of the nematic liquid-crystalline ordering they calculated relaxation times for tilt and twist fluctuations in hybridly aligned structure and director fluctuations in uniform director field structure. However, in their study they neglected spatial dependence of the uniaxial and biaxial degrees of nematic order, which are quite important in the case of strong anchoring and thin cells. A couple of years ago, Palffy-Muhoray et. al [20] showed that in highly constrained hybrid cells the nematic order can be either biaxial with the steplike profile of the director’s tilt angle or the director field can be bent continuously. They predicted a structural transition between the two possible ordered configurations but did not probe the stability of both configurations. A more detailed description of the nematic order in planar hybrid geometry has been provided by Galabova et. al [139]. They calculated the phase diagrams for a hybrid cell in relation to film thickness and anchoring strength of one of the surfaces. Another aspect of a nematic liquid crystal in a planar hybrid geometry are stripe domains studied by Pergamenshchik [140]. In his study, using Frank elastic theory with surface terms, it was shown that equilibrium modulated structures can appear. However, in that study only spatial dependence of the nematic director was taken into account whereas other degrees of freedom of the nematic order have been neglected. In a cylindrical geometry Ziherl and Zumer [13] studied director fluctuations in the vicinity of a disclination line of strength 1 whose structure is similar Hybrid nematic cell 103 to structures in hybrid cells. They extended the approach based on Frank elastic theory by introducing spatially dependent rotational viscosity and elastic constants. More extensive study of structure of point and line defects in a capillary was carried out by Kralj et. al [25]. They described the nematic order in an infinite cylindrical cavity with three parameters, scalar order parameter, parameter of biaxiality, and the angle between the local director and capillary long axis. However, they assumed tr Q2 to be a constant, which leaves only one, either the scalar order parameter or the parameter of biaxiality, independent. Furthermore, they neglected the third order term in the expansion of the free energy density. The obtained results have the expected resemblance to results of our study [21]. Lately, an increased interest was focused in the field of fluctuation-induced forces. Ziherl et. al determined the pseudo-Casimir force in the hybrid cell with degenerate planar anchoring in the most simple case, where the equilibrium nematic director is not perturbed. The force was calculated within the bare director picture and the Frank elastic theory. This brief review shows that there was a lack of information on the dynamics re-lated to the structural transition between different nematic configurations in highly constrained systems when nondirector degrees of freedom are crucial. This moti-vated us to start our analysis. In order to provide a simple but detailed description of a highly frustrated system a thin planar film with hybrid surface conditions was examined. In contrast to previous studies [19,12,13] we have focused our attention on highly constrained films where biaxiality and nonhomogeneous degree of nematic order play an important role. Although the origins of the high frustration in a system can be different, i.e., specific confining substrates in planar geometry or ge-ometry induced hybrid properties (see Fig. 5.1), its effects on the liquid-crystalline order and pretransitional dynamics are similar [25,21,134,135,141]. Therefore one can study the basic effects of high frustration within the analysis of a planar system. Our model system is a very thin hybrid film consisting of a nematic liquid crys-tal confined by two parallel substrates inducing uniaxial nematic order in mutu-ally perpendicular directions. The hybrid frustration can be also achieved if both substrates induce order with the nematic director perpendicular to the substrates, however, with positive scalar order parameter at one of the substrates and with a negative scalar parameter at the other. The latter case — the degenerate planar anchoring — corresponds to molecules lying in the plane of the substrate, but being randomly distributed around the substrate normal. A system with such boundary conditions possesses full rotational symmetry around the substrate normal, whereas the equilibrium bent director structure breaks this symmetry. Breaking of the sym-metry results in an infinite number of equilibrium states with the same energy and 104 Hybrid nematic cell *1 — &x G2 "S2 k2= ez y 6----------------------------------------------1-------> O d Z Figure 5.2 Schematic representation of the model system. The substrate at z = 0 induces uniaxial nematic order in a particular direction in the plane of the confining substrate, say parallel to the x axis. The other substrate, in the plane z = d, is then characterized by a homeotropic anchoring. k^i is the easy axes of the i-th substrate, aS i is the induced degree of uniaxial nematic order, and Gi is the free energy associated with the anchoring of the i-th substrate. The lateral dimensions of the cell are much larger than its thickness, Lx,Ly ^ d. characterized by a mirror plane, which contains the previous axis of full rotational symmetry. Such system is easily described within a director picture and Frank elastic theory. In that way Lavrentovich et. al have studied point defects which are due to infinite degeneracy of equilibrium states [142,143]. On the other hand, the same system is hard to handle within the phenomenological description and the tensorial order parameter which favors the nonfrustrated solution because it preserves the original symmetry on expense of varying the scalar order parameter. In practice the full rotational symmetry at the planar substrate is never ideally realized and there exists a locally preferred direction. Therefore, if one is interested in defect free structures even the hybrid situation with degenerate planar anchoring can be described by a hybrid cell with two well defined preferred directions. In order to simplify the description it is assumed that there is no surface induced smectic order although at least partial formation of smectic layers is often observed [117,144]. Suppose that the first substrate (z = 0) induces uniaxial nematic order in a particular direction in the plane of the confining substrate (say parallel to the x axis); the other substrate (in the plane z = d) is then characterized by a homeotropic anchoring. Here, d is the thickness of the nematic film which is much smaller than the lateral dimensions of the system, d W2 ~ 1.1 x 10-5 J/m2) the critical film thickness is found to be approximately 0.4 µm [138]. However, in the case of two confining substrates (without free surface) with very different surface anchorings (such as substrates modified with different aliphatic acids with, e.g., W1 ~ 10-3 J/m2 and W2 ~ 10-4 J/m2 [51]) the critical value would be as small as dc ~ 40 nm. As implied by the above expression for the critical film thickness this value would be even smaller if the anchoring strengths of the confining substrates would be comparable, therefore, in such hybrid films the order would always be distorted. However, in the case of hybrid yet equally strong anchoring conditions the director field is uniform below a finite critical film thickness, whereas the boundary conditions are fulfilled with the eigenvalue or director exchange [20,21]. Here the term “uniform director field” refers to the corresponding uniform orthonormal triad, whereas the director’s tilt angle exhibits a steplike change [see Fig. 5.3(b)]. The other interesting consequence of equivalent confining substrates is a geometry induced biaxial ordering of a uniaxial nematic liquid crystal. The effect is interesting because in thermotropic nematic liquid crystals biaxiality cannot be observed very often. In the following Sections first different possible equilibrium structures in a hybrid film are described in detail, and the structural transition toward the bent director structure is discussed. In Section 5.1.1 the comparison between the phe-nomenological description and the results from the Monte Carlo simulations is provided [145,146]. Next, in Section 5.2 the pretransitional orientational dynamics is determined for the biaxial structure, which is characteristic for highly constrained hybrid systems. The perturbed equilibrium nematic order and the changed spectrum of thermal fluctuations give rise to structural forces discussed in Section 5.3. Once acquainted with the forces acting on a slab of a nematic liquid crystal the 106 Hybrid nematic cell stability of thin liquid-crystalline depositions are discussed in Section 5.4. 5.1 Equilibrium structures The free energy of liquid-crystalline system consists of terms of three different types. The bulk-like terms control the melting/growing of the nematic order. If the order is smaller or larger than its bulk value these terms contribute to the increase of the free energy. If the nematic order is spatially dependent the free energy of the system in-creases due to the elasticity related to deformations of the tensorial order parameter. The elastic part of the free energy corresponds either to variations of the uniaxial and biaxial parameters of the order around the director or to elastic deformations of the director field. The larger the nematic order the bigger the resistance of the sys-tem toward the elastic deformations of the director field; the elastic energy related to deformations of the director scales as S2, if S is the degree of the nematic order. The third contribution to the free energy of a confined nematic liquid-crystalline system is the free energy associated with (dis)obeying the substrate-induced order. The system reaches its thermal equilibrium when its free energy is minimal. In a hybrid cell the confinement induces deformation of the director field. If other de-grees of freedom of the tensorial order parameter are neglected, the minimum of the free energy is reached by the interplay of energetic penalty for violating the surface-induced order and the energy due to elastic distortions. In a one-elastic-constant approximation, for d > dc elastic deformations are more favorable than severe vio-lation of the surface anchoring, whereas for d < dc it is energetically more favorable to disobey surface anchoring than to be submitted to elastic deformation. However, the nematic order has also other degrees of freedom. If these are taken into account the increase of the free energy due to elastic deformations can be compensated by localization of elastic deformation together with melting the nematic order in the region of high deformations. The violation of the surface-induced order can be also balanced by decreased nematic order at the surface with the mismatched director and the surface easy axis. In addition to melting of the nematic order the uniaxial distribution of molecules around the director can become biaxial. The equilibrium ordering of a hybrid nematic film can exhibit either distorted (hybridly bent) or undistorted director structure (see Fig. 5.3). The undistorted structure is characterized by either biaxial director exchange configuration in the case of equally strong but hybrid surface anchorings or uniform director field in the case of hybrid confining substrates characterized one by a strong anchoring and other by a weak anchoring. Which of the two possible configurations — distorted or undistorted — will actually occur depends on the temperature and film thickness. Hybrid nematic cell 107 z=0 (a) z=d z=0 z=d (b) A 2" A z=0 z=J (c) Figure 5.3 Schematic representation of three possible ordered configurations in a hybrid film: (a) the bent-director structure, (b) biaxial structure with director exchange, and (c) uniform director structure. However, the existence of either of the two undistorted structures depends on the strength of the surface coupling. Both distorted and undistorted structures are studied using the same free en-ergy density expansion [Eqs. (2.32) and (2.33)]. By comparing the total free energy dependencies on temperature and film thickness the structural transition is deter-mined. The nematic order is described by a tensorial order parameter, however, only the nontrivial degrees of freedom are actually taken into account. Each partic-ular parametrization of the order parameter tensor is described when describing the corresponding structure. The obtained differential equations are solved numerically using the relaxation method [126]. Uniform director field If the antagonistic anchorings are very different in magnitude it is energetically more favorable to disobey one of the surface anchorings than to be subject to elastic deformation. In order to minimize the surface contribution to the free energy the nematic director will lie in the direction of the easy axis of the substrate with stronger anchoring, say the substrate at z = d, i.e., G2 > G1. Thus the nematic order can be described with a scalar order parameter S and, in general, with the additional parameter P measuring biaxiality of the order; Q = ST0 + PT1, with the orthonormal triad n^ = e^z, e^1 = e^x, and e^2 = e^y. Usually, the biaxiality of the nematic order is neglected in calculations but in highly frustrated systems such an approximation is not justified. The positional dependence of the two chosen 108 Hybrid nematic cell ^^ 1.0 S ~~~~^ 0.8 \ 0.6 v 0.4 0.2 P ; 0.0 0.2 0.4 0.6 z/d 0.8 1.0 Figure 5.4 Uniform director structure in a hybrid film with unequal anchoring strengths. Equilibrium profile is characterized by a spatially dependent degree of nematic order and increasing biaxiality profile when approaching the substrate with weaker anchoring. Parameters used in calculation were ? = 0.9, ?2 = 0.03, aS = 1.1, G1 = 1.2 x 10-3 J/m2, and G2 = 1.2 x 10-4 J/m2 (solid line), G2 = 4 x 10-4 J/m2 (dashed line). parameters can be obtained by solving Euler-Lagrange equations 2 = 0, = 0, ? S" — ?S + 3S — 2S — 3P — 2SP ? P" — ?P — 2P — 6SP — 2S P where the prime denotes d/dz, S = S (z), and P = P (z). boundary conditions are determined by , z=0 (5.1) The corresponding . g 1 S a S\ S = ——2 [ -\------- ? 2 z=0 . g 1 ( a SV P = — jt: P +-------- ? 2 3\ (5.2) . g 2 ? P = —P , z=1 where aS is the preferred value of the scalar order parameter which is taken to be equal at both substrates and gi = (?27/Ld)Gi is the dimensionless strength of the surface interaction. The dimensionless parameters used in this calculation are the same as introduced in Eqs. (2.32) and (2.33) and discussed on page 33. Since the confining substrates induce uniaxial order the biaxiality is small, especially near the substrate whose easy axis is parallel to the nematic director (Fig. 5.4). By increasing the weaker anchoring the biaxiality increases as well. Progressively increasing biaxiality and simultaneously decreasing degree of nematic order along , Hybrid nematic cell 109 the director lead to transformation of the uniform structure to the biaxial structure. The transformation does not correspond to a structural transition. If one of the confining substrates is characterized by very weak anchoring (g —> 0) the parameter of biaxial order can be omitted and the equations reduce to ?2S" - ?S + 3S2 - 2S3 = 0, i.e., describing Sheng’s surface aligned nematic ordered structures [18]. Bent-director structure The discussion of the bent-director configuration is only slightly simplified by the assumption that the order is uniaxial but it allows both positionally dependent scalar order parameter and director field, whereas in most previous studies only the variation of the director field has been taken into account. The effect of biaxiality can be neglected since it is very small comparing to the scalar order parameter, especially, deep in the nematic phase. If one minimizes the free energy of the hybrid system with constant parameters S, P, and ? ~ ?/2, deep in the nematic phase, the value of biaxiality can be estimated to be no larger than P = (?2/4\/3)?2 CS^1. The biaxiality is negligible for hybrid cells with d > 100 nm (P < 0.01). Using the dimensionless form of the free energy density expansion [Eq. (2.32)] and the ansatz Q(z) = S(z)(3n n^ - I)/v6, where the nematic director has the form n = (sin?, 0, cos?) and ? = ?(z), the two independent parameters S — the scalar order parameter — and ? — the angle between the nematic director and the substrate normal — are determined by the equations ? S" - ?S + 3S - 2S - 3? S(?') = 0, (S ?')' = 0. (5.3) In the case of a very strong surface anchoring the boundary values of S and ? are set to the values preferred by the confining substrates [S = aS and ?(z = 0, d) = ?S12, where ?S1 = ?/2 and ?S2 = 0] otherwise they are determined with boundary conditions z=0,1 S = i?2 2S + aS - 3aS cos (? - ?S) , g i S ? =----— aS sin2? , (5.4) 2?2 z=0,1 where the signs + and -, and the subscripts i = 1, 2 correspond to z = 0 and z = 1, respectively. If spatial variations of the scalar order parameter are neglected the Eq. (5.3) reduces to the well known equation for the director’s tilt angle which can be derived from the Frank elastic description within the director picture, ?" = 0. Resulting director field is characterized by a linearly varying director’s tilt angle with boundary conditions ?'(z = 0,1) = - (g1,2/2?2) sin2?(z = 0,1). Meanwhile, 110 Hybrid nematic cell 1.0 ^^^^ S ^^^^^ 0.8 -^ 0.6 0.4 " 'a 0.2 - /'' "v 0.0 0.2 0.4 0.6 z/d 0.8 1.0 Figure 5.5 Equilibrium bent-director structure in a hybrid film. The solid line corresponds to the scalar order parameter and the dashed line represents the a_1 = (\/3/2)S sin 2? amplitude of the tensor order parameter, which describes the bending of the director field in the plane (x, z). Parameters used in calculation are ? = 0.9, ?2 = 0.01258, g1 = g2 —> oo, and aS = 1.1the scalar order parameter is constant through the liquid-crystalline film and its temperature dependence is described by the equation ?effS — 3S2 + 2S3 = 0, where ?eff = ? + (3? /4)? . (5.5) Thus, the scalar order parameter corresponds to its bulk value at the increased temperature [cf. Eqs. (2.15) and (5.5)]. Due to elastic deformation of the director field the temperature of the hybrid system effectively increases. The increase of the effective temperature results in a smaller degree of order along the director with respect to the degree of order in bulk where there are no elastic deformations. The latter fact is often forgotten in studies within the bare director description. However, these difference is in micron-size cells, which are usually studied within the bare director description, negligible; for a typical liquid crystal Teff — T f« 0.5 mK. As suggested, above the critical film thickness or below the critical temperature (with constant temperature or film thickness, respectively) the order in a hybrid nematic film can be described by a bent-director field. Since we allow the scalar order parameter to vary with the distance from one of the substrates the director tilt angle is not changing linearly as it would in the case of the uniform scalar order parameter (see Figs. 5.5 and 5.6). However, the difference is very small and, as expected, decreases further with the increasing film thickness and when the boundary value of the scalar order parameter is getting closer to the value Sb(?eff). Here, Sb(?) = 0.75(1 + J1 — 8?/9) is the bulk degree of the nematic order and ?eff is the renormalized dimensionless temperature defined in Eq. (5.5). In the case of strong surface anchoring the discrepancies from the case of uniform scalar order parameter Hybrid nematic cell 111 7T/2F; ?/4 0 1.6 1.4 1.2 1.0 0.0 0.2 0.8 1.0 0.4 0.6 z/d Figure 5.6 Spatial dependence of the tilt angle (top) and its derivative (bottom) for the bent-director structure. Dashed lines correspond to the appropriate parameters in the case of the uniform scalar order parameter. Parameters used in calculation are ? = 0.9, ?2 = 0.0126, aS = 1.1, and g1 = g2 —> oo. are ?S = S(0) — S(1/2) f« aS — Sb and ??' = ?'(0) — ?'(1/2) ~ ?(a2S — Sb2 )/(a2S + Sb2), where Sb = Sb(?eff). On approaching the transition to either of the uniform director structures, i.e., by increasing the temperature or decreasing the film thickness, the discrepancies become larger. Simultaneously, the biaxiality becomes larger and the assumption of its negligibility is not justified anymore. Biaxial configuration As already mentioned in the introduction of the Chapter, the biaxial configuration was introduced by Palffy-Muhoray et. al [20] and later discussed by Galabova et. al [139]. However, their studies were made for a special case where the temperature corresponded to the bulk supercooling limit (T = T*), whereas some other choices of temperature give rise to different physical phenomena. In order to better understand the pretransitional dynamics in such biaxially ordered structure, a detailed description of the biaxial configuration is presented. In general, in the case of a hybrid film the director field is not uniform. However, the easy axes of the confining substrates are one in the direction of the x axis and the other parallel to the z axis; therefore it can be assumed that the director will lie in the plane (x,z), i.e., perpendicular to the y axis. Thus n = ey, e^1 = ez, and e^2 = ex can form a suitable uniform orthonormal triad. The biaxial configuration is determined by using the expansion of the tensor order parameter in terms of the base tensors, A(r*) = J2i=-2ai(z)Ti. Due to the symmetry reasons and boundary conditions, the configuration can be described by two amplitudes, a0 and a1. The former refers to the scalar order parameter with 112 Hybrid nematic cell respect to the y axis, whereas the latter denotes biaxiality of the order in perpendicular directions. In our case, the negative/positive sign of the amplitude a1 tells whether the actual director is in the direction of the xorz axis, respectively. Both nonzero amplitudes are the solutions of two coupled equations arising from minimization of the free energy [Eqs. (2.28) and (2.62)] and the expansion of the tensor order parameter in terms of the base tensors, ? a0 — ?a0 + 3(a0 — a1) — 2a0(a0 + a1) = 0, ? a![ — ?a1 — 6a0a1 — 2a1(a0 + a1) = 0, (5.6) where the prime denotes d/dz. The boundary conditions are determined by the surface interaction. In our case, the induced order is assumed to be uniaxial at both substrates but in mutually perpendicular directions, therefore QS(0) = aS(3ex(dex — I)/v6 and QS(1) = aS(3ez ® ez — I)/v6, where aS > Sb(?) is the preferred degree of order at the substrates (and is assumed to be equal at both substrates). Thus, the boundary conditions read a'0 = ±^ a0 + aS/2 , ?2 z=0,1 g r ^ i I 1,2 /7 a1 = ±— a1 ± aSV3/2 , (5.7) ?2 2=0,1 where the signs + and —, and the subscripts 1 and 2 correspond to z = 0 and z = 1, respectively. If the anchoring is very strong (g —> oo), the order at the surface is the same as the one preferred by the confining substrate, otherwise, the parameters can differ from the preferred ones. The actual significance of the two nonzero amplitudes is obvious when they are rewritten into ax0 = — (a0 + y3a1)/2 and az0 = (—a0 + y3a1)/2, where the former sum refers to the scalar order parameter with respect to the director n = ex and the latter sum denotes the scalar order parameter with respect to the director n = ez. As shown in Fig. 5.7, on the average, near the first surface (z = 0) the liquid-crystal molecules are oriented parallel to the x axis while they are parallel to the z axis close to the other substrate (z = 1). In the vicinity of the surfaces the order is uniaxial, however, with increasing distance from the substrates it becomes slightly biaxial. In the case of equal anchoring strengths, both, biaxiality and order parameter profiles are symmetric with respect to the middle of the film (plane z = 1/2). The biaxiality profile has two maxima near the symmetry plane. In between them the molecular ordering can be described with a director perpendicular to the plane of the molecules (n = ey), yet the scalar order parameter is negative. In the region of negative scalar order parameter the director or eigenvalue exchange occurs. The maximum biaxiality and the thickness of the exchange region depend on the film thickness, the temperature, and the anchoring strength. The biaxiality is more Hybrid nematic cell 113 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 a x\ \S P a z A-—-. ^. _^^ ^-^ 0.0 0.2 0.4 0.6 0.8 1.0 z/d 0.55 (a) (b) Figure 5.7 (a) Equilibrium profiles of the nonzero degrees of freedom. Thin solid line refers to the scalar order parameter with respect to the uniform direc- tor y whereas thicker lines represent scalar order parameters with respect to the easy axes x and z, ax0 = — (a0 + v3a1)/2 and az0 = (—a0 + v3a1)/2, respectively. Dashed lines correspond to scalar order parameter [S = (V6/2)|Qii|, where Qii has a sign opposite to that of the two other eigenvalues of Q] and biaxiality of the order (P = |Qjj — Qkk|/2, where j, k = i), respectively. (b) Magnified detail of the profiles in the exchange region. Parameters used in calculation are ? = 0.9, ?2 = 0.01258, aS = 1.1, and g1 = g2 —> oo. pronounced in thinner films and when the temperature is closer to the phase transition temperature. From the point where the surface wetting layers are in contact the exchange region thickness is — within the numerical accuracy — independent of temperature. On the other hand, the relative exchange region increases with decreasing film thickness, however, the absolute exchange region thickness decreases. The biaxially ordered configuration is typical for highly constrained nematic liquid crystals, i.e., systems with high surface-to-volume ratio and strong surface anchoring (G > 10-3 J/m2). In such systems the surface wetting layers may be in contact with each other, thus, the structure they form becomes progressively ordered on approaching the phase transition temperature. Because of the continuous growth of the ordered biaxial structure there is no nematic-isotropic phase transition. However, there is the transition to the low-temperature bent-director field configuration. Because the initial structure is ordered too, the transition between the two phases is structural rather than the phase one. In the case of unequal but strong surface anchorings the high-temperature phase is biaxial as well, but the exchange region is located closer to the surface with weaker anchoring (Fig. 5.8). As already discussed before, biaxial structure reduces to the uniform director field state with spatially dependent degree of nematic order and negligible biaxiality (except at the substrate where the nematic director is perpen- 114 Hybrid nematic cell 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 0.0 0.2 0.4 0.6 z/d 0.8 1.0 Figure 5.8 The degree of nematic order with respect to mutually perpendicular directors n^ = ex and n^ = ez. Different lines correspond to different anchoring strengths: G1,G2 —>¦ oo (solid line), G1 = G2 = 1.2 x 10-3 J/m2 (dashed line), G1 = 1.2 x 10-3 J/m2 and G2 = 1.1 x 10-3 J/m2 (dotted line), and G1 = 1.2 x 10-3 J/m2 and G2 = 0.6 x 10-3 J/m2 (dash-dotted line). Parameters used in calculation are ? = 0.9, ?2 = 0.03, and aS = 1.1. dicular to the easy axis) if one of the confining substrates is characterized by weaker anchoring (G <• 10-4 J/m2). Structural transition between bent-director structure and biaxial structure By comparing the total free energies of the two ordered configurations we determine the structural transition film thickness. However, the bent-director structure was determined approximately, therefore, the total free energy of the actual configuration is lower than the one obtained in our calculations. Since the neglected biaxiality is of order of P ~ c?2, where 0 < c < 1, the difference between the actual and approximated free energy should be very small, i.e., T-^"approx ~ - ?4[c(?2v3/4)S-c2(?/2 + 3S + S2)], where S = Sb(?) is the bulk degree of nematic order parallel to the director. As expected, the correction is getting smaller as the film thickness is increased. Near the isotropic-nematic phase transition temperature (TNI - T = 0.1 K) and in a hybrid film of a typical liquid-crystalline material (such as 5CB) the nematic order is distorted if the film is thicker than dt ~ 47 nm, whereas the metastable biaxial structure ceases to exist if the film thickness is larger than ds ~ 71 nm. The latter critical thickness is determined by pretransitional dynamics, which will be discussed in detail in Section 5.2. As the temperature is decreased both values are decreased too and so is the difference between them. The same structural transition can be realized if the film thickness is held constant and the temperature is varied. Hybrid nematic cell 115 0.10 0.08 0.06 0.04 0.02 0.00 -0.02 -0.04 t s biaxial^ /bent 0.8 0.9 1.0 1.1 6 Figure 5.9 Temperature dependence of the total free energy of biaxial and bent-director structure. The structural transition occurs at the point where the free energies of the two configurations are equal (?t = 0.951), whereas ?s = 0.869 represents the “supercooling” limit of the biaxial structure. In a typical liquid-crystalline material, the difference between the two temperatures is very small, ?T = Ts — Tt ~ 0.09 K, and the corresponding latent heat is by an order of magnitude smaller than the latent heat of the nematic-isotropic phase transition. The dashed continuation of the total free energy represents the region where the given structure is metastable. Parameters used in calculation are ?2 = 0.02, aS = 1.1, and g1 = g2 —> oo. A typical temperature dependence of the free energies of both ordered phases at constant film thickness (?2 = 0.02) is shown in Fig. 5.9. It is obvious that the slopes of the functions are not equal at the transition point, so that the structural transition is discontinuous. However, the corresponding latent heat, ql = ?(?T/?T)Tt = ?(?Jr/??)[?t + T*/(Tni — T*)] ~ 8 x 104 J/m3, is even smaller than the isotropic-nematic phase transition latent heat (~ 1.5 x 106 J/m3) [147], therefore the structural transition is only weakly discontinuous. The (dis)continuity of the structural transition can be changed if the temperature and film thickness are low enough. Within our approximation, in such case the free energies of bent and biaxial structures do not intersect and the free energy of the bent structure exceeds the biaxial one at the “supercooling” limit. This can be understood if one considers the approximate determination of the bent-director structure in which the biaxiality was omitted. However, even a rough calculation such as the one introduced at the beginning of this section shows that taking into account the estimated biaxiality lowers the free energy of the bent structure so that the transition becomes continuous. Such nature of the structural transition was found also in studies of Palffy-Muhoray et. al [20] and Galabova et. al [139]. The two different regimes are separated with a tricritical point (tricritical temperature and film thickness) below which the transition is continuous. Because of the approximate 116 Hybrid nematic cell 0.4 0.3 0.2 0.1 0.0 0.0 0.2 0.4 0.6 0.8 0 Figure 5.10 Temperature dependence of the total free energy of biaxial and bent-director structure at the “supercooling” film thickness [(s = (s(d)]. Because of the approximate determination of bent director structure the corresponding free energy is too high but even rough calculation shows that the correction (dotted line) would cause the transition to become continuous below some critical temperature and film thickness (tricritical point). Above the tricritical point (the upper limit 6tp = 0.746 and LTP = 0.054) the transition becomes progressively discontinuous. description of the bent-director structure only its upper limit has been determined: ?tp = 0.746 and ?TP = 0.054, which corresponds to T/v/ — TTP = 0.28 K and dtp = 34 nm for a typical nematic liquid crystal, such as 5CB. In Fig. 5.10 the temperature dependence of total free energies of both ordered structures at the “supercooling” film thickness is shown. One should notice that the dimensionless total free energies are decreasing functions of temperature. That indicates that in the range of film thicknesses where biaxial structure can be realized the elastic part of the free energy is dominant over the ordering terms. The elastic term whose magnitude is determined by ?2 oc 1/d2 is decreasing with temperature because the “supercooling” film thickness is an increasing function of temperature. 5.1.1 Monte Carlo simulations of a hybrid cell Theoretical description of physical phenomena has its purpose when the results can be related to the observed phenomena. In the case of hybrid nematic cell, the distorted bent-director structure and undistorted uniform structure have been probed by many experimental methods taking advantage of the effect of certain ordered structure on physical observables, such as dielectric and optical properties, etc. On the other hand, the experimental evidence of biaxial structure is rather limited since the existence of this structure is very delicately tuned by the anchoring properties of the confining substrates and since it is only realized in a very narrow temperature in- Hybrid nematic cell 117 terval. However, the results can also be compared to the ones obtained by computer simulations which act not only as a bridge between microscopic and macroscopic length and time scales but also as a bridge between a theory and experiment. In computer simulations we provide a guess for the interactions between molecules and probe it by comparing predicted macroscopic physical properties of a system with its actual properties. Once the model interactions between molecules are set the computer simulations can be used as an “experimental” method which may also reveal hidden details behind macroscopic observables. There are two basic types of computer simulations, molecular dynamics and Monte Carlo simulations. The former method consists of a brute-force solution of Newton’s equations of motion, therefore, it corresponds to what happens in “real life” — it generates configurations time step after time step in their natural time sequence. On the other hand, the latter method can be thought of as a prescription for sampling configurations from a statistical ensemble; on achieving the equilibrium the system goes from one state to the next, not necessarily in a proper order [148,149]. Here, I present results obtained from the Monte Carlo simulation of the hybrid nematic film which was performed by the group of Pasini and prof. Zannoni in Bologna in collaboration with me and prof. Zumer. Further, I suggest some preliminary correspondence between outcome of the phenomenological theory and computer simulation. The hybrid nematic film is simulated using a well known cubic-lattice spin system put forward by Lebwohl and Lasher [150]. It is based on modeling the interactions between the molecules through a second rank Lebwohl-Lasher potential Uij = tij P2(cos ßij), (5.8) where tij equals e > 0 for nearest-neighbor particles i and j, and it is zero elsewhere. P2 is the second rank Legendre polynomial and ßij is the angle between the long axes of the corresponding spins; spin denotes either a single molecule or a cluster of molecules whose short-range order is maintained in the examined temperature interval [151]. The only free parameter of the model is the temperature of the system, TMC = kBT/e. The model reproduces well the behavior of bulk nematic liquid crystals, particularly, it reproduces a weakly discontinuous phase transition and the correct temperature dependence of the order parameter [150,152]. The bulk isotropic-nematic phase transition occurs at TNMIC = 1.1232 [152], which yields for 5CB with Tni = 308.3 K the inter-spin interaction strength e = 0.0237 eV (comparing this to the phenomenological estimate A(TNI — T*)V0 one would get for the volume of correlated molecules V0 ~ 26 nm3 with the corresponding correlation length v V ~ 3 nm which is comparable to the phenomenological value ?NI ~ 8 nm). 118 Hybrid nematic cell Q -e- N "layers' -p----------<$----------e- -e- / ?NI) the order in the middle of the film is subject to a continuous evolution and the ordering transition is lost. On lowering the temperature, first the order in the middle of the film grows and then, below some temperature the director field starts to bend. The corresponding temperature of the structural transition between undistorted biaxial and distorted bent-director structure is denoted as T2MC; in a 14-layer hybrid film T2MC ~ 1.12. Temperature dependence of the bending parameter a1 as a function of temperature and distance from (one of) the substrates is presented in Fig. 5.14. Structural transition temperature is lower than the bulk transition temperature. This difference with respect to the TNMIC is a consequence of the elastic deformations which are due to the frustrating boundary conditions. The effect has already been discussed within the phenomenological theory. Elastic deformations are stronger in thinner films, thus, the temperature T2MC is an increasing function of the film thickness which is presented in Fig. 5.13 (b). Increasing the film thickness the temperature interval of the stable biaxial structure decreases and in very thick films the two transitions would merge; the remaining transition would correspond to a direct change from the disordered to ordered bent-director structure. The existing transitions can also be recognized as peaks in the heat capacity of the film. In Fig. 5.15, the temperature dependence of the internal energy of the 14-layer film and the corresponding heat capacity are shown. The well developed peak corresponds to the transition to the bent-director structure. Smaller peak, which is to some extent hidden by the higher one, is associated with the change of the ordering in the core of the film. The results obtained from the Monte Carlo simulation confirm the existence of the biaxial structure and the corresponding transitions between (dis)ordered configurations. In some other simulation study of a hybrid film [153], the biaxial structure was not found which was due to the choice of the surface potential which yielded extremely unsymmetric properties for the homeotropic and planar anchoring. Still, some further work is needed to establish the closer quantitative correspondence between the results obtained from Monte Carlo simulations and from the phenomenological description. 122 Hybrid nematic cell -2.0 -2.2 -2.4 -2.6 -2.8 -3.0 -3.2 -3.4 /•••• nnnnnnD / • \ nnnDD / \ • \n ,pDDDDDD • tf ••• n« • ••• • p P *». 3 :\ • ^»m' D-D' \ • ?D •v.-, D ••• • •••• ?D •• T2 MC T 1 MC ^ 5 4 3 2 1 0 0.9 1.0 1.1 1.2 1.3 1.4 Figure 5.15 Energy of the 14-layer hybrid nematic film (squares) and the corresponding specific heat (full circles). The highest peak located at T2M ~ 1.12 corresponds to the structural transition between bent-director and biaxial structures and the lower peak which is masked by the higher one corresponds to the ordering transition and is associated to the middle of the film, T± ~ 1.17. Lines are guides to the eye. 5.2 Pretransitional dynamics Once we have calculated and tested the equilibrium profiles we can begin with the analysis of fluctuations. Here, the analysis is restricted to fluctuations in the biaxial (director exchange) structure with equal anchoring strengths. We study their temperature/film thickness dependence when approaching the structural transition to the bent-director configuration and the “supercooling” limit. The same approach can be used also with the uniform director structure. This analysis is not performed here because due to very small biaxiality the fluctuations do not differ much from fluctuations in homogeneous systems which are discussed in Section 2.2.2 and Section 4.2 on page 83. The detailed analysis of the pretransitional dynamics of all five degrees of freedom around the bent-director configuration is somewhat more complicated because of the nonuniformity of the base tensors. The simplified description within the director picture was studied in detail by Stallinga et. al [12], however, the results are not quoted here. The Gaussian dynamics of five scalar components of collective excitations — introduced by the expansion B(r*, t) = J2i=-2bi(r,t)Ti — is derived by projecting the linearized form of the relaxation equation [Eq. (2.64)] onto the base tensors. Since the equilibrium profiles depend on the z coordinate only, the normal modes can be factorized as follows, b (r t) = ei\kxx\kyyj ßi ( z) e -µit (5.12) where kx and ky are the in-plane components of dimensionless wave vector of fluc- 6 Hybrid nematic cell 123 tuations which are assumed to be subject to periodic boundary conditions, µi’s are the dimensionless relaxation rates of the eigenmodes, and time is measured in units of ?a = (27C/B2)? ~ 10-8 s [32]. Considering the introduced ansatz [Eq. (5.12)] and the equilibrium profiles of the system the amplitudes ßi(z) are determined by the equations ? ß0 — (? — ?0,1 — 6a0 + 6a0 + 2a1)ß0 — 2a1(3 + 2a0)ß1 = 0, ? ß1 — (? — ?0,1 + 6a0 + 2a0 + 6a1)ß1 — 2a1(3 + 2a0)ß0 = 0, ? ß-'!_1 — (? — ?-1 + 6a0 + 2a0 + 2a1)ß-1 = 0, (5.13) ? ß±2 — (? — ?±2 ~~ 3a0 -F 3v3a1 + 2a0 + 2a1)ß±2 = 0, where ß^ = dßi/dz and ?i = µi — ?2(k2 + ky2) are the reduced relaxation rates of the modes. When deriving these equations, one must consider that the modes which are coupled relax with the same relaxation rate, therefore ?0 = ?1 = ?0,1. In the case of a very strong surface anchoring (g —> oo) no fluctuations are allowed at the substrate, thus ßi(z = 0,1) = 0, otherwise the boundary conditions read nl g 1,2 n ßi = ±pPi , (5.14) ?2 z =0,1 where the signs + and —, and the subscripts 1 and 2 refer to z = 0 and z = 1, respectively. In the case of a purely uniaxial nematic ordering and uniform director field (ai = 0, i = 0) the five fluctuating modes are independent, therefore the two equations for the amplitudes ß0(z) and ß1(z) are uncoupled. Furthermore, due to the symmetry reasons the two biaxial modes (bi with indices i = ±1) are degenerate and so are the two director modes (indices i = ±2). The system reduces to the one studied in Chapter 4. As implied by equations in Eq. (5.13) this is not the case when dealing with fluctuations in a biaxially ordered hybrid film. Since the equilibrium profiles are described by two nonzero amplitudes, a0 and a1, the corresponding fluctuation modes, ß0 and ß1, are coupled. The significance of these modes is transparent when considering their linear combinations ß0x 1 = —(ß0 + v3ß1)/2 and ß0z,1 = —(ß0 — v3ß1)/2, which denote the order parameter fluctuations with respect to the nematic director parallel to the x and z axis, respectively. The other three fluctuation modes are uncoupled and represent either director fluctuations (ß-1 modes and low ß±2 modes) or biaxial fluctuations, high ß±2 modes. Due to the inhomogeneous equilibrium profiles the eigenmodes of fluctuations can only be determined numerically. In the following, the spectra of collective excitations and the eigenamplitudes for different fluctuating modes will be interpreted. 124 Hybrid nematic cell z/d Figure 5.16 Spatial dependence of the lowest order parameter mode. Solid lines correspond to coupled fluctuations of the degree of order with respect to the two mutually perpendicular easy axes. Dashed lines correspond to the equilibrium profiles (Fig. 5.7). Parameters used in calculation are ? = 0.9, ?2 = 0.01258, aS = 1.1, and g1 = g2 —> oo. Order parameter fluctuations The term order parameter fluctuations denotes coupled fluctuations of the two nonzero equilibrium amplitudes. As it is well known the eigenfunctions of an operator invariant to the space reflection are either symmetric or antisymmetric with respect to the same transformation [130]. Since the operator which governs the order parameter fluctuations [see Eq. (5.13) and the results for the equilibrium profiles a0 and a1] is symmetric with respect to the plane z = 1/2 the eigenfunctions of the system can be divided into two classes, i.e., the symmetric and antisymmetric functions with respect to the symmetry plane. The lowest symmetric mode is associated with fluctuations of the thickness of the central director exchange region and therefore also with the fluctuations of the magnitude of biaxiality of the nematic order. However, the lowest antisymmetric mode corresponds to fluctuations of the position of the boundary between the two parts of the film which are determined by mutually perpendicular nematic directors. The portrait of the lowest antisymmetric order parameter mode is plotted in Fig. 5.16. It can be noticed that the two corresponding profiles, ß0x 1 and ß0z, 1, are “localized” at the part of the film with director ex and ez, respectively. The positions of their maxima coincide with the position of maximum slope of the scalar order parameter. Thus, the lowest antisymmetric order parameter mode is responsible for the growth of the surface wetting layers. The same fluctuation mode was found also in heterophase systems (see Section 4.2). Higher symmetric and antisymmetric modes change the shape of the exchange region in a symmetric or an antisymmetric manner, respectively. The lowest relaxation rate corresponds to the lowest antisymmetric mode. When Hybrid nematic cell 125 2.0 1.6 1.2 0.8 0.4 0.0 0.00 0.02 0.04 0.06 0.08 ? 2 0.10 0.0126 0.02 rf[nm] (a) (b) Figure 5.17 (a) Lower part of the spectrum of collective order parameter excitations with respect to the film thickness. Dotted and dashed verticals denote the “supercooling” and the structural transition film thickness, respectively. (b) Magnified detail of the lowest order parameter relaxation rate. Notice that the relaxation rate remains finite even at the “supercooling” limit and approaches the limit with the zero slope. Parameters used in calculation are ? = 0.9, aS = 1.1, and g1 = g2 —> oo. increasing the film thickness toward the structural transition thickness (decreasing the parameter ?2) all the relaxation rates are decreased, especially the lowest one (see Fig. 5.17). However, it stays finite (?0,1;n=0 > 0) even at the “supercooling” limit/transition point above/below the tricritical point, respectively. Similarly, the relaxation rates decrease with decreasing temperature. At this point it should be emphasized again that the film thickness turns out to be a parameter that rescales the temperature [Eq. (5.5)]. Director fluctuations Director fluctuations ß-1 represent changes of the orientation of the nematic director in the plane of the two easy axes. They bend the nematic director in the n = ex half of the film toward the direction ez and the n = ez director in the other half toward the perpendicular x direction. The corresponding eigenmodes are spread over the whole sample and are similar to the sine functions as it is shown in Fig. 5.18. The lowest director mode represents the change of the tensor order parameter that is similar to the one characteristic for the bent-director configuration plotted in Fig. 5.5. Its relaxation rate exhibits a critical slowdown when the film thickness approaches the “supercooling” limit/transition point above/below the tricritical point, respectively. In the case of discontinuous structural transition the lowest director mode is almost critical even at the structural transition, which is in agreement with our previous 126 Hybrid nematic cell 1.0 0.5 0.0 -0.5 -1.0 '---_._ _..--""- y<>c / /"- \ ' \s0 / / 7\1 / / \\ / \/ ¦y' Yx / _,---'' \ ^--.. / 0.0 0.2 0.4 0.6 z/d 0.8 1.0 Figure 5.18 Spatial dependence of the lowest two director modes labeled by the number of nodes. Dashed lines correspond to the equilibrium profiles plotted in Fig. 5.7. Parameters used in calculation are ? = 0.9, ?2 = 0.01258, aS = 1.1, and g1 = g2 —> oo. 2.0 13 7 1.6 /2 1.2 0.8 / 1 0.4 0^^^^ 0.00 0.02 0.04 0.06 0.08 0.10 ? 2 Figure 5.19 A few lowest relaxation rates of director modes versus film thickness. The rates are decreasing with increasing film thickness, especially the lowest mode’s rate which drops to zero as the film thickness approaches the “supercooling” limit. Dotted and dashed verticals denote the “supercooling” and the structural transition film thickness, respectively. Parameters used in calculation are ? = 0.9, aS = 1.1, and g1 = g2 —> oo. Hybrid nematic cell 127 1.0 0.5 0.0 -0.5 -1.0 0.0 0.2 0.4 0.6 z/d 0.8 1.0 Figure 5.20 Portrait of two typical biaxial fluctuation modes ß2 (the ß-2 modes are just their mirror images with respect to the symmetry plane z = 1 /2). The lowest modes are expelled from the part of the film where these fluctuations represent biaxial fluctuations. Higher modes are spread over the whole sample. Labels denote the number of nodes of the mode and the dashed lines correspond to the equilibrium profiles of biaxial structure. Parameters used in calculation are ? = 0.9, ?2 = 0.01258, aS = 1.1, and g1 = g2 —> oo. conclusion that the transition is only weakly discontinuous. Therefore, in both regimes the soft director mode can be assumed to govern the structural transition between the two ordered configurations. As shown in Fig. 5.19, higher modes relax faster and do not contribute essentially to the pretransitional change in the director field. Biaxial fluctuations Biaxial fluctuations ß±2 are described by the last two equations in Eq. (5.13). If these equations are rewritten in more appropriate form ?2ß±'l ±2 - [? - ?±2 + 6a0,z + 2(a0 + a1)]ß±2 = 0, (5.15) and the symmetry relations between the equilibrium amplitudes ax0 and a0 are considered (see Fig. 5.7) it can be easily seen that the spectra for the two biaxial modes are degenerated, whereas the eigenfunctions are just mirror images with respect to the plane z = 1/2. As shown in Fig. 5.20 the few lowest modes of fluctuations ß2(z) and ß-2(z) are expelled from the part of the film which is characterized by directors ex and ez, respectively. This can be easily understood if we consider that ß±2 represent amplitudes of the projection of the tensor order parameter along the base tensors T±2 which couple directions y and z or y and x, respectively. That means that ß2 refers to director fluctuations in the n = ez part of the film but to biaxial 128 Hybrid nematic cell 0.00 0.02 0.04 0.06 0.08 0.10 Č Figure 5.21 The lowest part of the spectrum of relaxation rates of biaxial fluctuations. Note that the relaxation rates are higher than the relaxation rates of other modes (cf. Figs. 5.17 and 5.19). Since the biaxial fluctuations represent deformations of the order parameter in the y direction they do not play any important role at the structural transition. Dotted and dashed verticals denote the “supercooling” and the structural transition film thickness, respectively. Parameters used in calculation are ? = 0.9, aS = 1.1, and g1 = g2 -*¦ oo. fluctuations in the other part, and vice versa for the (3-2 fluctuation modes. Since in the uniaxial nematic phase director fluctuations are much more favorable than biaxial fluctuations [15], (3±2 fluctuations tend to be localized at the appropriate half of the film only. Higher modes are spread over the whole film whereas the unfavorable manner of biaxial fluctuations is compensated by the shorter wave vector of a deformation. In addition, it is well known that the higher the modes, the smaller the effect of the shape of the potential on them. The biaxial relaxation rates are higher than the rates of other fluctuation modes, therefore the biaxial fluctuations do not play any important role in the structural transition discussed. The spectrum of biaxial relaxation rates is plotted in Fig. 5.21. 5.3 Structural and pseudo-Casimir forces Structural force The common feature of all ordered structures in hybrid nematic cell is the repul-sive character of the structural force for separations above a few nanometers. The repulsion is due to the antagonistic boundary conditions, which always lead to at least small deformation, and to the fact that within certain ordered structure the frustration is stronger if the confining substrates are brought closer to each other. The magnitude of the force is tuned by the anchoring strength at both substrates Hybrid nematic cell 129 whereas its functional form depends on the structure in question. The uniform director structure is characterized by a weak localized deformation of the nematic scalar order parameter whereas the director field is undeformed. The corresponding structural force is short-range and very weak comparing to the structural forces characteristic for other configurations in a hybrid film. Depending on the details of the induced order the force can be either monotonically repulsive or can exhibit a nonmonotonic behavior characterized by attraction at very small film thicknesses and a weak repulsion for larger separations (see Fig. 5.22). The latter case corresponds to the substrates that induce strong order, higher than characteristic for given temperature. (E.g., the excess order at the free surface of some nematic liquid crystals observed by Kasten et. al [122] and studied theoretically by means of density functional approach by Martin del Rio et. al [125].) Here, the turn-over between the regime of the increasing/decreasing force with the increasing film thickness is related to the fact that above certain film thickness the order at the side with stronger anchoring is above its bulk value whereas it is below it on the other side. If the spatial variation of the scalar order parameter is neglected the free energy of the uniform director structure corresponds only to the penalty for violating the induced order at the substrate with weaker anchoring, F = —GS A, (5.16) where G is related to the substrate with weaker anchoring, S is the uniform degree of the nematic order, and A is the surface area of the confining substrate. This free energy does not depend on the separation between the two confining substrates and, thus, does not give rise to the structural force. Within the bare director picture there is no structural force in the uniform director structure, however, the elastic deformation of the director field in the bent-director structure gives rise to strong structural repulsion even within the simplified description. In this structure, in the limit of infinitely strong surface anchorings the elastic contribution to the free energy reads 32 A1 A2 3?2LS2A F = -LS Ad dz(?) =-------------, (5.17) 2 0 8d which is only slightly perturbed by the coupling with the deformed field of the scalar order parameter. From Eq. (5.17) in the bent director structure the structural force per unit area is roughly given by 3?2LS2 ? =-------. (5.18) 8 d2 For finite anchorings the structural pressure becomes ? ? (3/2)LS2(?1 - ?2)2d~2, where ?1 and ?2 are the director’s tilt angles at the substrates located at z = 0 and 130 Hybrid nematic cell ¦ A\(c) ¦ 'v^- ^_(d) ¦ -(b) 0 10 20 30 40 50 d [nm] (a) 1.52 1.50 --------0.8 nm --------------------___2.8 nm ..........4.8 nm --------6.7 nm - -------8.7 nm s ..........10.7 nm ^lf --------25 nm / J /" 0.5 1.0 0.0 0.5 1.0 0.0 0.5 zld zld zld (b) (c) (d) Figure 5.22 (a) Structural force per unit area in a hybrid cell characterized by the uniform director structure. Solid line corresponds to confining substrates which induce high nematic order [aS = 1.01Sb(?)] and the dashed line corresponds to the hybrid cell with substrates with no excess order [aS = Sb(?)]. Labels (b) - (d) refer to the corresponding equilibrium profiles of the scalar order parameter in a hybrid cell confined by substrates characterized by the induced excess order. Parameters used in calculation are ? = 0, G1 = 2 • 10-5 J/m2, and G2 —>¦ 00. 0 Hybrid nematic cell 131 100 10 1 0.1 0 10 20 30 d [nm] 40 50 0.01 10 d [nm] 100 (a) (b) Figure 5.23 (a) Structural force per unit area in the bent-director structure (dashed line) and in the biaxial structure (solid line). In (b) the same plot in the logarithmic scale where it can be clearly seen that for large cell thicknesses the localization of the deformation in the biaxial structure results in shortened range of the structural force. z = d, respectively. They are determined by Eq. (5.4) and depend on the separation between the two confining substrates. The structural force in the bent-director structure is plotted in Figs. 5.23 (a) and 5.23 (b). In biaxial structure, the force is repulsive and exhibits 1/d2 behavior at small cell thicknesses whereas at large d’s the exponent of the power law is smaller than -2 which is clearly evident from Fig. 5.23 (b). For small cell thicknesses, the elastic deformation, although of the scalar fields rather than the director field, is spread over the whole cell and the force exhibits typical elastic dependence. The decrease of the range of the force for larger cell thicknesses is a consequence of the localization of the deformation when approaching the stability limit of this structure. Pseudo-Casimir force As we have seen, depending on the temperature, surface anchorings, and the thick-ness of the cell the nematic liquid crystal in a hybrid cell can be found in one of the three ordered structures described in previous Sections. Unlike in the case of the heterophase nematic system, where a systematic study of the pseudo-Casimir force was performed by Ziherl et. al [154,28], in the hybrid cell the pseudo-Casimir force has only been calculated for the simplest ordered configuration, i.e., the uniform director structure (Ziherl et. al [65]). The studied hybrid cell was characterized by stronger homeotropic and weaker degenerate planar boundary conditions yielding homeotropic director field below the critical thickness dc which preserves the full ro-tational symmetry with respect to the normal to the planar parallel cell. The study was performed within the bare director description. In the cell with the uniform nematic order the effective correlation length in the Hamiltonian of the correspond- 5 132 Hybrid nematic cell ing fluctuating mode [Eq. (2.69)] is a constant and the partition function of the fluctuation modes can be derived analytically. In this thesis, we do not intend to present any new results for the pseudo-Casimir force in the hybrid cell although to complete the description of the nematic liquid crystal in a hybrid geometry pseudo-Casimir force should be determined also in other possible structures, i.e., the bent-director structure and biaxial structure. Determination of the force in the latter structure is more complex due to deformed equilibrium order. In the following Section the stability of thin nematic depositions subject to hybrid boundary conditions will be studied. In the system, the pseudo-Casimir force plays crucial role for the mechanical stability of the film and, thus, has to be taken into account. In the study, we will use the results obtained by Ziherl et. al [65], however, in this Section, we present in short the calculation of the pseudo-Casimir force in the given system and we discuss the obtained results. In the uniform director structure with homeotropic and planar boundary conditions the Hamiltonian of the liquid-crystalline system consists of elastic and surface terms. As already done in previous studies of order parameter fluctuations, the calculation is restricted to harmonic approximation, therefore, the director is expanded around the equilibrium configuration and only the lowest-order terms are kept. The corresponding Hamiltonian of fluctuations is diagonal 7i[nx, ny] = 7i[nx]+7i[nx], (5.19) where in the one-elastic-constant approximation K \ f(n 2 -1 f 2 -1 f 2 S 1 TC[n] = — \n) dV — ?P n dSP + ?H n (ih . (5.20) 2 Here, n is either of the two fluctuating scalar director fields, K = 3LS2, and ?P,H are the extrapolation lengths of the degenerate planar and homeotropic substrate, respectively. The negative sign of the planar surface term indicates the frustrating role of the two competing substrates, which eventually destabilizes the uniform structure. The interaction free energy of the two degenerate director modes is determined by the partition function written out in Eq. (2.71). Due to the in-plane trans-lational invariance of the system the modes can be Fourier decomposed, n(r) = X^exp (iq- f?)n^(z) — q is the in-plane wavevector and f? is the projection of the vector r onto the plane of the confining substrate —, in to the ensemble of independent one-dimensional harmonic oscillators. The corresponding partition function is Hybrid nematic cell 133 factorized, ? = ?q?q^, where and H[n~q>] = KA ^q = / d , 0 Dn~q?e-m[n~qi ?) q2n~2q > dz - ?-n~ P + ?-1n~ =,+ . (5.21) (5.22) 2 0 Here, A is the substrate surface area, the prime denotes d/dz, and n± = n(z = 0, d). Further calculation of the partition function is based on the analogy with the calculation of the probability for a particle with the Hamiltonian H to remain at a given point within a certain time interval [155]. According to this, °° / K A dnq - CO 00 ?qf ? / dn~q,- - CO - exp fA ( ? -P n~2q, - - ?-H 1n~2q, +) )?qf;n--,n~q- (5.23) and q;n~q,-, n~q,+ ? 1 sinh(qd) -------k—~T [n~q + n~2 +) coth(qd) - sinh(qd) (5.24) where we have disposed of irrelevant multipliers. Here, ?qf;n~q--,n~q-+ is the partition function associated with the fluctuation modes with boundary conditions n^- and n~q^+, and ?q*in Eq. (5.23) is obtained by an extension of a point-to-point Green function to a region-to-region Green function, where the width of each region is defined by a characteristic length scale analogous to the extrapolation length [155]. In practice this means, that a finite surface interaction with a given easy axis is represented by a superposition of strong surface interactions each characterized by some easy axis and multiplied by a statistical weight corresponding to the energetic penalty for the deviation from the given easy orientation. The integrals in Eq. (5.23) can be calculated analytically and by omitting further irrelevant multipliers we obtain the final expression for the interaction free energy, ? - kBT ^ ?-P ?-H q ?-P + ? H y sinh(qd) + cosh(qd) 1/2 . (5.25) Here, we stress that by replacing - ?P with ?H/ the obtained result corresponds also to the free energy of fluctuations between two substrates with homeotropic anchoring and with different anchoring strengths characterized by extrapolation lengths ?h> and ?h. Rather than in the free energy of the interaction we are interested in the force acting between the two confining substrates. The force is, like in the case of the 134 Hybrid nematic cell 10 8 6 4 2 0 -2 -4 -6 -8 -10 0.0 1.0 0.4 0.6 d/dc Figure 5.24 Pseudo-Casimir force for different values of ? = ?h/?P in unit of ~ = (kBT/dc). Dotted lines correspond to the pseudo-Casimir force in the two analytical limits, ? —> 0 (only the zeroth-order) and ? —> 1. weak anchoring weak anchoring strong anchoring' H weak anchoring d ^ Kp^Kjj "kiilX XH< d 0.5 attraction (a) (b) Figure 5.25 Relation between the cell thickness and the effective anchoring strength in hybrid cell below the critical thickness dc. (a) For d < ?h, ?P both anchorings are effectively weak whereas for ?h < d < ?P the homeotropic anchoring becomes effectively strong. (b) For ?h/?P > 0.5 both anchorings are effectively weak for all thicknesses d < dc. Hybrid nematic cell 135 structural force, obtained by the derivative of the corresponding part of the free energy of the system with respect to the distance between the substrates. According to this and by replacing the sum over q by an integral, J2q ~^ (2?) j d q, the pseudo-Casimir force per unit area reads kBT f? p2dp =------ -1------------, (5.26) - ?d d3 0 p-----1111111111111----- p+1-?p-?-1 + 1e2px - 1 where ? = ?h/?p and x = d/dc. The shape of the pseudo-Casimir force depends on the ratio between the two extrapolation lengths whereas its magnitude is tuned by the difference between them, i.e., the critical thickness dc for the existence of the uniform director configuration. The above expression can not be calculated analytically except in few limiting cases. For the finite but similar extrapolation lengths (? —> 1) the force is attractive and decreases as d-3 which is a typical behavior for equal boundary conditions [27,63], kBT ?(3) 2? 2d3 where ?(r) = YZm=1 m-r is the Riemann zeta function. In the case of infinitely strong homeotropic anchoring (? —> 0) the system reduces to mixed boundary conditions for the fluctuation modes and the zeroth-order of the pseudo-Casimir force has a typical monotonic repulsive behavior with the characteristic separation dependence 1/d3, ? (? —> 1) f«------------—, (5.27) [ 3?(3) ln 2 1 —n—^—2 . (5.28) 8d ?P d -^ 0) ~ 2? 8d ?Pd The second term in Eq. (5.28) is related to the fact that the substrate with weak planar anchoring promotes fluctuations and therefore enhances the discrepancy between the two effective boundary conditions which gives rise to an additional repulsion. In real hybrid systems with both ?P and ?H > 0, the fluctuation-induced force is attractive at small d/dc’s, then it becomes repulsive and may reach a local maximum before the pretransitional logarithmic singularity, which is common for all combinations of extrapolation lengths in the hybrid geometry [65]. However, it should be stressed that in the vicinity of the structural transition to the bent-director structure the anharmonic fluctuations may also play an important role. Nevertheless, the higher-order corrections are expected not to modify the divergent pretransitional behavior qualitatively [65]. The portrait of the pseudo-Casimir force for few values of the ratio ?H/?P is plotted in Fig. 5.24. The nonmonotonic behavior of the pseudo-Casimir force can be simply understood by means of the influence of the type of the boundary conditions for the fluctuating modes on to 136 Hybrid nematic cell the fluctuation-induced force which we have already met in the introductory Section 2.4. At very small thicknesses (d/dc 0.5 both extrapolation lengths are larger than the cell thickness for all thicknesses yielding the uniform director configuration and the nonmonotonic behavior of the pseudo-Casimir force is lost. The numerical calculation gives for this critical ratio of the two extrapolation lengths the value 0.7; above this, the pseudo-Casimir force is attractive almost right up to the structural transition to the bent-director configuration. The described relations between extrapolation lengths and the effective strength of the anchorings are schematically represented in Fig. 5.25. 5.4 Stability of thin hybrid nematic films In this Section the stability of a thin liquid-crystalline film is discussed in terms of enhanced/diminished capillary waves which cause the film to decompose through a process known as spinodal dewetting. As discussed in Section 2.4.1 the possible enhancement of thermal fluctuations of a free liquid-air interface is driven by disjoining force of the interactions acting between the two confining surfaces. The sum of all forces per unit area acting on a thin film is therefore denoted as a disjoining pressure. As already noted, the relevant physical quantity is the slope of the disjoining pressure rather than its sign. In the liquid-crystalline film the disjoining pressure consists of van der Waals interaction discussed in Chapter 3, and structural force and pseudo-Casimir force discussed in this Chapter. The electrostatic interactions are neglected here, since we assume that there are no free charges. Lately, there has been an increased interest in the study of stability of thin films, not only liquid-crystalline films but also depositions of liquid metal [156], polymers [103,76], and protein solutions [157]. The reason for that is both, technological interest in stability of thin depositions and the interest in basic physical phenomena involved in the process of spinodal decomposition. Our study was stimulated by the results of the experiment performed by Vandenbrouck et. al [104,158]. In their experimental set-up the system consisted of a nematic liquid crystal 5CB spun cast on a silicon wafer bearing a natural oxide layer. Depending on the initial thickness of the liquid-crystalline film the film either remained stable for days or it dewetted into islands of liquid-crystalline drops and dry patches. The dewetting was moni- Hybrid nematic cell 137 Figure 5.26 Photograph of a film after it has decomposed via spinodal dewet-ting. The picture was taken by F. Vandenbrouck et. al and is published in Phys. Rev. Lett. 82, 2693 (1999). tored using a polarized optical microscopy. A typical picture of a film after it has decomposed via spinodal dewetting is shown in Fig. 5.26. The original experiment was performed at the room temperature and the main conclusion was that the dis-joining pressure exhibits nonmonotonic behavior with the marginal thickness for the dewetting of the film larger than 17 nm and smaller than 20 nm [104]. Later, the experiment was repeated for various temperatures within the nematic phase [158]. As we have seen in Sections 3.4 and 5.3 all forces contributing to the disjoining pressure can exhibit a nonmonotonic behavior. The portrait of the van der Waals force for the system in question is plotted in Fig. 5.27. Since the silicon wafer bears a natural oxide layer of silica the system is at least four-layered. As discussed in Section 3.4 in that case the van der Waals force can exhibit a nonmonotonic behavior if the van der Waals forces in different three-layer systems composed of the given materials have different signs. Here, the interaction between silicon and air across the nematic liquid crystal is repulsive whereas the interaction between silica and air across the same liquid crystal is attractive. Roughly speaking, one would expect that for film thicknesses below the thickness of the oxide layer the interaction would be mostly due to the interaction of silica and air whereas for large film thicknesses the existence of the additional layer would be negligible. Indeed, in Fig. 5.27 we can recognize the explained behavior. The marginal thickness (turn-over thickness) is comparable to the thickness of the oxide layer, which is considered to be approximately 2 nm thick. Actually, the value of the marginal thickness can differ from the thickness of the additional layer if the two competing interactions are very different in magnitude. The dielectric and optical parameters characteristic for the given materials are written out in Table 5.1. 138 Hybrid nematic cell 10 5 0 -5 -10 -15 -20 0 2 4 6 8 10 d [nm] 12 14 Figure 5.27 Van der Waals force per unit area acting on the liquid-crystalline film in contact with a solid substrate bearing an additional layer and a free liquid-air interface. Dashed and dotted lines correspond to the van der Waals force for the partial three-layer systems silicon-5CB-air and silica-5CB-air, respectively. Although the van der Waals force possesses the needed nonmonotonic behavior the marginal thickness is an order of magnitude smaller than the observed value and can not be increased to the appropriate value just by small corrections due to better precision of the input parameters. Similar arguments can be stated also for the structural force which is in addition far too weak to have any significant influence on the disjoining pressure. On the contrary, the pseudo-Casimir force together with the van der Waals force yields suitable set of interactions to describe the process of spinodal decomposition. This was first recognized by Ziherl et. al [28]. In their study they determined the van der Waals and pseudo-Casimir forces acting in the described system and obtained the appropriate marginal thickness within the reasonable set of parameters describing optical and anchoring properties of the materials in question. Here, we present the results of a similar study, however, Table 5.1 Material properties of the media constituting the system for studying spinodal dewetting. e is the static dielectric constant and n is the refractive index of the medium in visible. All parameters are given at the room temperature. mate rial e n silicon 12 3.5 silica 14 1.5 5CB 18.5 7 1.71 n? 1.53 Hybrid nematic cell 139 -2 -4 -6 -8 -10 0 d * 5 10 15 d [nm] 20 25 Figure 5.28 Disjoining pressure within the liquid crystal in contact with a solid substrate and air (solid line) deep in the nematic phase. The dashed line corresponds to van der Waals force, whereas the dotted and dash-dotted lines depict the pseudo-Casimir and structural contributions, respectively. Inset: behavior of the disjoining pressure in the vicinity of the marginal thickness. 100 80 60 40 20 20 22 24 26 28 30 32 34 36 T [°C] dependence of the marginal thickness for the spin- Figure 5.29 Temperature odal dewetting of 5CB obtained by Valignat et. al [158]. with the improved determination of the van der Waals force which in the original study suffered from some defects. Since the van der Waals force is relatively strong in the interval of relevant film thicknesses its best determination is most important. The disjoining pressure on a slab of a liquid-crystalline material subject to spinodal dewetting together with the individual forces taking part in the total force is plotted in Fig. 5.28. In the calculation of the pseudo-Casimir force the extrapolation lengths were taken to be ?# = 33 nm and ?p = 133 nm which yields ? = 0.25. The parameters used are comparable to the ones reported in previous studies [159,43,28]. As the temperature is varied the observed marginal thickness changes [158]: Deep in the nematic phase the marginal thickness only slightly increases with the increased temperature. On approaching the isotropic-nematic phase transition temperature 4 140 Hybrid nematic cell the marginal thickness exhibits a pretransitional singular-like behavior. The experimental values obtained by Valignat et. al are plotted in Fig. 5.29. The pretransitional “singularity” of the marginal thickness for the stability of a film is not just a consequence of the changed occupancy of the fluctuation states, either of thermal fluctuations of order parameter or fluctuations of the instantaneous electromagnetic fields. This explicit temperature dependence induces only weak temperature dependence of the corresponding forces, far from being singular. Within the mean-field theory used in this thesis only the order parameters change with the temperature. They indeed affect the tensorial physical quantities like the permittivity tensor, however, this again only weakly perturbs the van der Waals force through the temperature dependent refractive indices. The extrapolation lengths, being the ratio of the free energy related to elastic deformations and the energetic penalty for violating boundary conditions, are constant within the mean-field description and, thus, do not yield any temperature dependence of the pseudo-Casimir force. However, in experimental studies the extrapolation lengths were found to have a strong temperature dependence characterized by a “critical” increase on approaching the isotropic-nematic phase transition [17,46-48,16]. In their study, Mertelj et. al report this temperature dependence to be approximately ? ? (TNI - T)~1 where TNI corresponds to the bulk isotropic-nematic phase transition [16]. The power law indicates that the extrapolation lengths are inversely proportional to the square of the degree of nematic order. As already noted, within the mean-field theory and with lowest nontrivial terms, both, the elastic free energy and free energy corresponding to the interaction of the nematic liquid-crystalline material with solid substrate, are proportional to S2(T), thus, the corresponding extrapolation length is temperature independent. The observed temperature dependence of the extrapolation length indicates that close to the phase transition higher orders in the interaction between nematic liquid crystal and solid substrate should be taken into account. Considering the quadrupolar symmetry of the constituting molecules up to the fourth order term the surface part of the free energy reads LA { i2] Fsur =-----?~ tr (Q - QS) + ? tr (Q - QS) \ , (5.29) where ? is the ratio of the free energies associated with the second and fourth order terms. For the uniaxial nematic order Q = (S/ ? 6)(3n^ ?n^ - I) and, correspondingly, Q S = (a S / ? 6)(3k ? k - I) if the confining substrate induces uniaxial nematic order with the easy axis k and the preferred degree of nematic order aS. In the case with the uniform degree of nematic order through the film, S = aS = Sb(T), the effective extrapolation length reads approximately ? ?eff(T) =-------- , (5.30) 1 + ? Sb2 (T) Hybrid nematic cell 141 (a) (b) Figure 5.30 (a) Temperature dependence of the extrapolation length for various values of the parameter ol. Large values of ol correspond to dominant fourth-order term which fits well the experimentally obtained behavior. Parameter A is chosen in a way to fit the value of the extrapolation length deep in the nematic phase. (b) Experimental data for extrapolation length in 5CB in polycarbonate Nucleopore membrane reported by Mertelj and Copiˇc [Phys. Rev. Lett. 81, 5844 (1998)]. where ? is related to ?. On approaching the nematic-isotropic phase transition the nematic order decreases and, effectively, the anchoring strength decreases as well. Therefore, the effective extrapolation length increases on approaching the transition to the disordered phase. In Fig. 5.30 the temperature dependence of the extrapolation length is plotted for various values of the parameter ?. There are, however, some other approaches which lead to the increase of the extrapolation length in the vicinity of phase transition. They are based on the renormalization of the anchoring energy due to variations of the degree of nematic order [17] or due to thermal fluctuations [160]. To model the temperature dependence of the marginal thickness for the stability of a thin nematic liquid-crystalline film we have assumed the simplest temperature dependence of the extrapolation lengths: both extrapolation lengths are assumed to change with temperature in the same way, ? oc S~2(T). In that case, the relevant ratio which changes the pseudo-Casimir interaction, ? = ?#/?p, is constant whereas the critical thickness for the structural transition between uniform and bent director configuration changes with temperature. As shown in Fig. 5.31, depending on the value of this ratio, however, with the same extrapolation lengths deep in the nematic phase, temperature dependence can vary from a very weak temperature dependence up to the dependence characterized by a pronounced pretransitional increase. Comparing the results from our model with the experimental results presented in Fig. 5.29 the best fit was obtained for ? = 0.25, and ?# = 33 nm and ?p = 133 nm at T = 293 K. The used parameters fit well with the reported values 142 Hybrid nematic cell 120 100 80 60 40 20---------— 20 22 24 26 28 30 32 34 36 T [°C] Figure 5.31 Temperature dependence of the marginal thickness for different ?’s. Squares correspond to experimental results obtained by Valignat et. al [158]. of extrapolation lengths at free nematic surface of 5CB and with the extrapolation length of 5CB in contact with silica, respectively. Within this, depending on the temperature the marginal thickness of the disjoining pressure is in the interval from 10 nm deep in the nematic phase up to 100 nm close to the transition to the isotropic phase. It should be stressed that the extrapolation lengths and the marginal thick-ness exhibit a pronounced increase in the vicinity of the phase transition, however, they do not diverge at the phase transition. In the literature this increase is often denoted as a pretransitional singularity although there is no reason for that at the discontinuous transition form the nematic to the isotropic phase. 6 Conclusion The primary aim of my thesis was to study the effects of the confinement onto the nematic order parameter fluctuations. These are affected directly by the changed boundary conditions for the fluctuation modes and indirectly through the changed potential arising from the average equilibrium order. The surface-induced effects can be either localized at the confining substrates or the surface-induced deforma-tion can extend over the whole liquid-crystalline sample. On the other hand, the spectrum of fluctuations indicates whether the corresponding transition is contin-uous or discontinuous, which is the mechanism responsible for the change of the order at the transition, etc. The variation of order on the macroscopic scale can be monitored through changed optical properties of the system, by its influence on the NMR response, etc. One of the principal physical observables are, however, the forces. Forces acting among the walls that confine the liquid crystal can be monitored either directly by force spectroscopy methods, such as with surface force apparatus, atomic force microscopy, etc., or indirectly via the effect of the interac-tions in the system onto the mechanical stability of liquid films — via their ability to wet solid substrates. It was found out by other authors that experiments taking advantage of spinodal dewetting of liquid-crystalline depositions are most promising for the observation of fluctuation-induced forces [161]. However, the fluctuation-induced force is just one of the forces among all that are acting on a film and it is far from being the only one relevant. Thus, other forces have to be considered as accurately as possible in order to provide an adequate mechanism for observing the fluctuation-induced forces. There was a lack of knowledge of the influence of the anisotropy of order onto the electromagnetic field fluctuations which give rise to the van der Waals interaction. Although they are not strictly related to the or-der parameter fluctuations they were studied here because of their relevance for the observation of fluctuation-induced forces and because they are too affected by the ordering power of the confining walls. 143 144 Conclusion First, the collective fluctuations were studied in a heterophase system in which the surface-induced change of the order is localized at the confining wall. The analysis has revealed a close relationship between the wetting regime induced by disordering substrates and the pretransitional behavior of thermal fluctuations of the ordering in confined liquid crystals. The disordering action of the confining surface results below the isotropic–nematic phase transition in a heterophase struc-ture consisting of an isotropic wetting layer and a nematic core. The system is characterized by a wetting-induced interface between nematic and isotropic phase, which gives rise to two localized normal modes: the first one represents fluctuations of the position of the phase boundary and is characterized by a soft dispersion of its relaxation rate (provided that the conditions for the complete wetting are ful-filled), and the second one corresponds to fluctuations of the shape of the interface. Moreover, there are a few additional softened biaxial modes, which are restricted to the wetting layer and whose pretransitional behavior is related to its growth. The spectrum of director fluctuations is Goldstone like and the corresponding fluctuation modes are accommodated by the nematic core. If the wetting is partial, the slow-down of the localized modes is not as pronounced as in complete wetting regime, but the underlying physics remains the same. In systems with complete wetting the isotropic–nematic phase transition becomes continuous provided that the thickness of the nematic cell is large compared to the nematic correlation length. The surface-induced influence which perturbs the whole system was studied on the example of the hybrid nematic film. The analysis of nematic liquid crystals confined to highly constrained hybrid films with a biaxial structure has revealed a soft-mode or soft-mode-like dynamics in the vicinity of the structural transition toward hybridly aligned bent-director structure. The soft fluctuation manner is related to the bending director fluctuations which deform the undistorted director profile in biaxial configuration toward the continuously bent-director field in a usual hybridly distorted structure. In addition to this fluctuation mode the lowest order parameter mode exhibits similar slowdown of the relaxation rate, however, it remains finite even at the “supercooling” limit of the biaxial configuration. This mode is related to fluctuations of the position of the interface between the two uniaxial parts of the film. Other fluctuation modes do not contribute to the structural transition. However, low biaxial modes are interesting because they are localized in one half of the film only. The structural transition between the biaxial and bent-director phase can be either continuous or discontinuous — the two regimes are separated by a tricritical point (TTP, dTP). Above the tricritical point, the continuous structural transition becomes progressively discontinuous. The results of the phenomenological description were compared to the ones obtained by the computer simulation of the Conclusion 145 hybrid nematic cell and a good qualitative agreement was found between the two approaches. Another test for the phenomenological description was the study of stability of thin hybrid films. It has revealed that the van der Waals and pseudo-Casimir forces are both important for the stability of the film. We managed to model the temperature dependence for the marginal thickness which is in good agreement with the observed dependence. In order to study the stability of thin hybrid nematic depositions we have de-rived an improved analytic expression for the van der Waals interaction between macroscopic bodies, characterized by uniaxial permittivity tensor. We have shown that neglecting the anisotropy of static dielectric constants and refractive indices can yield wrong character of the interaction leading to incorrect interpretation or prediction of stability of thin uniaxial depositions. The anisotropic van der Waals interaction yields also a correction to the equilibrium order in heterophase nematic and smectic systems. In paranematic systems this changes the critical exponent of the wetting layer thickness. Open problems There are, however, some aspects of the surface-induced influence on the nematic order fluctuations which are not discussed here. Due to the broken translational symmetry of the phases caused by the presence of walls surface-induced layering is expected in the vicinity of walls. The effect is more pronounced in systems with spontaneous smectic layering, however, to some extent it is observed also in nematogenic liquid crystals. The effect of smectic layering onto the fluctuations of the nematic order parameter is still to be investigated. Another influence on the nematic order fluctuations, especially in heterophase systems, is due to the van der Waals interaction which yields a correction to the equilibrium order. This is quite substantial in the vicinity of the phase transition and even changes the critical exponent for the wetting layer thickness. In the complete wetting regime, this behavior is critical even within the phenomenological description used in our study, therefore, the basic properties of the fluctuation spectra are not expected to be changed. It might, however, change the behavior in the regime of partial wetting or at least renormalize the conditions for the complete wetting. In the description of fluctuations it was assumed that the energy is dissipated only in the interior of the system whereas the dissipation at the surfaces was neglected. In order to provide the complete account of the influence of the surfaces the surface viscosity should be discussed in the future. The studied behavior of the nematic ordering and pretransitional dynamics of a liquid crystal is certainly not limited to the simple planar geometry discussed in this 146 Conclusion thesis. Similar phenomena are expected in systems where the heterophase or hybrid order of the nematic liquid crystal is induced by topological constraints imposed by curved walls, such as in cylindrical cavities, in the vicinity of line and point defect, etc. In these, the equilibrium order is already recognized to be characterized by partially molten nematic order characteristic for either of the two systems discussed in the thesis instead of exhibiting deformations of the director field. Thus, it can be expected that the collective modes associated with these geometries are basically the same as described here. However, in curved geometry in both of the discussed systems the director field can not be uniform due to the shape of the confinement. In order to make a quantitative analysis of collective modes in the vicinity of defects and, thus, of their growth, the analysis should be performed from the beginning. -k -k -k The nature of the research work is such as to always open new questions and so the work seems to be never finished and the aim never fulfilled. On my way to prepare this thesis, many new interesting aspects of confined liquid crystals were raised; some of them gave the results presented in this work, some of them led to the dead end, and some of them still have to be investigated. However, this is what makes the research work interesting and what gives the assurance that there will be always something new to work on. Working on this thesis helped in deriving or becoming familiar with certain theories, methods, and models to describe physical phenomena of confined systems and provided some basic concepts characteristic for them. Bibliography [1] G. Friedel, Ann. Physique 18, 273 (1922). [2] R. Reinitzer, Monatsch Chem. 9, 421 (1888). [3] O. Lehman, Z. Physikal Chem. 4, 462 (1889). [4] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1993). [5] S. Chandresekhar, Liquid crystals, 2 ed. (Cambridge University Press, Cam-bridge, 1992). [6] P. Chatelain, Acta Cristallogr. 1, 315 (1948). [7] L. D. Landau and I. M. Khalatnikov, Dokl. Akad. Nauk SSSR 96, 469 (1954). ˇ [8] Liquid Crystals in Complex Geometries, edited by G. P. Crawford and S. Zumer (Taylor & Francis, London, 1996). [9] X. I. Wu, W. I. Goldburg, M. X. Liu, and J. Z. Xue, Phys. Rev. Lett. 69, 470 (1992). [10] G. Schwalb and F. W. Deeg, Phys. Rev. Lett. 74, 1383 (1995). [11] F. M. Aliev, in Liquid crystals in complex geometries, edited by G. P. Crawford ˇ and S. Zumer (Taylor & Francis, London, 1996). [12] S. Stallinga, M. M. Wittebrood, D. H. Luijendijk, and T. Rasing, Phys. Rev. E 53, 6085 (1996). ˇ [13] P. Ziherl and S. Zumer, Liq. Cryst. 21, 871 (1996). ˇ [14] P. Ziherl and S. Zumer, Phys. Rev. Lett. 78, 682 (1997). ˇˇ [15] P. Ziherl, A. Sarlah, and S. Zumer, Phys. Rev. E 58, 602 (1998). ˇ [16] A. Mertelj and M. Copiˇc, Phys. Rev. Lett. 81, 5844 (1998). [17] M. Nobili and G. Durand, Phys. Rev. A 46, R6174 (1992). [18] P. Sheng, Phys. Rev. Lett. 37, 1059 (1976). [19] G. Barbero and R. Barberi, J. Phys. (Paris) 44, 609 (1983). [20] P. Palffy-Muhoray, E. C. Gartland, and J. R. Kelly, Liq. Cryst. 16, 713 (1994). ˇˇ [21] A. Sarlah and S. Zumer, Phys. Rev. E 60, 1821 (1999). [22] L. Moreau, P. Richetti, and P. Barois, Phys. Rev. Lett. 73, 3556 (1994). ˇˇ [23] A. Sarlah, P. Ziherl, and S. Zumer, Mol. Cryst. Liq. Cryst. 329, 413 (1999). ˇˇ [24] A. Sarlah, P. Ziherl, and S. Zumer, Mol. Cryst. Liq. Cryst. 320, 231 (1998). ˇ [25] S. Kralj, E. G. Virga, and S. Zumer, Phys. Rev. E 60, 1858 (1999). ˇˇ [26] A. Sarlah, P. Pasini, C. Chiccoli, C. Zannoni, and S. Zumer (unpublished). [27] A. Ajdari, L. Peliti, and J. Prost, Phys. Rev. Lett. 66, 1481 (1991). 147 148 Bibliography ˇ [28] P. Ziherl, R. Podgornik, and S. Zumer, Phys. Rev. Lett. 82, 1189 (1999). ˇˇ [29] S. Zumer, A. Sarlah, P. Ziherl, and R. Podgornik, Mol. Cryst. Liq. Cryst. 358, 83 (2001). ˇˇ [30] A. Sarlah, P. Ziherl, and S. Zumer, accepted for publication in Mol. Cryst. Liq. Cryst. (unpublished). ˇˇ [31] A. Sarlah and S. Zumer, accepted for publication in Phys. Rev. E (unpub-lished). [32] S. Hess, Z. Naturforsch. Teil A 30, 728 (1975). [33] V. L. Pokrovskii and E. I. Kats, Zh. Eksp. Teor. Fiz. 73, 774 (1977), [Sov. Phys. JETP 46, 405 (1977)]. [34] L. D. Landau, Fiz. Z. Sowejetunion 11, 26 (1937). [35] Y. Nambu, Phys. Rev. Lett. 4, 380 (1960). [36] J. Goldstone, Nuovo Cimento 19, 155 (1961). [37] K. Huang, Statistical Mechanics (Wiley, New York, 1987). [38] P. M. Chaikin and T. C. Lubensky, Principles of condensed matter physics (Cambridge University Press, Cambridge, 1995). [39] V. L. Ginzburg, Fiz. Tverd. Tela. 2, 2031 (1960), [Sov. Phys. Solid State 2, 1824 (1961).]. [40] D. Monselesan and H.-R. Trebin, Phys. Stat. Sol. (b) 155, 349 (1989). [41] A. L. Alexe-Ionescu, G. Barbero, and G. Durand, J. Phys. II (Paris) 3, 1247 (1993). [42] F. C. Frank, Discuss. Faraday Soc. 25, 19 (1958). [43] O. D. Lavrentovich and V. M. Pergamenshchik, Phys. Rev. Lett. 73, 979 (1994). [44] V. M. Pergamenshchik, Phys. Rev. E 61, 3936 (2000). [45] J. L. Ericksen, Phys. Fluids 9, 1205 (1966). [46] K. H. Yang and C. Rosenblatt, Appl. Phys. Lett. 43, 62 (1983). [47] H. Yokoyama, S. Kobayashi, and H. Kamei, J. Appl. Phys. 61, 4501 (1987). [48] L. M. Blinov, A. Y. Kabayenkov, and A. A. Sonin, Liq. Cryst. 5, 645 (1989). [49] A. Rapini and M. Papoular, J. Phys. Colloq. 40, C3 (1969). [50] P. Sheng, Phys. Rev. A 26, 1610 (1982). ˇ [51] G. P. Crawford, R. J. Ondris-Crawford, J. W. Doane, and S. Zumer, Phys. Rev. E 53, 3647 (1996). [52] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 1 (Pergamon Press, Oxford, 1980). [53] L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 3 ed. (Pergamon Press, Oxford, 1986), first published in English by Pergamon Press plc 1959. [54] R. Kubo, Statistical mechanics (North-Holland Publishing Company, Amsterdam, 1965). [55] L. O. Onsager, Phys. Rev. 37, 405 (1931). ˇ [56] D. Svenˇsek and S. Zumer, Liq. Cryst. 28, 1389 (2001). [57] R. G. Horn, J. N. Israelachvili, and E. Perez, J. Phys. (Paris) 42, 39 (1981). [58] S. H. J. Idziak et al., Science 264, 1915 (1994). Bibliography 149 [59] A. Poniewierski and T. J. Sluckin, Liq. Cryst. 2, 281 (1987). ˇ [60] A. Borˇstnik and S. Zumer, Phys. Rev. E 56, 3021 (1997). [61] P. G. de Gennes, Langmuir 6, 1448 (1990). [62] A. Ajdari, B. Duplantier, D. Hone, L. Peliti, and J. Prost, J. Phys. II (Paris) 2, 487 (1992). [63] H. Li and M. Kardar, Phys. Rev. Lett. 67, 3275 (1991). [64] H. Li and M. Kardar, Phys. Rev. A 46, 6490 (1992). ˇ [65] P. Ziherl, F. K. P. Haddadan, R. Podgornik, and S. Zumer, Phys. Rev. E 61, 5361 (2000). [66] P. Ziherl, Phys. Rev. E 61, 4636 (2000). [67] K. Koˇcevar and I. Muˇseviˇc, Liq. Cryst. 28, 599 (2001). [68] H. B. G. Casimir, Proc. Kon. Ned. Akad. Wet. 51, 793 (1948). [69] M. E. Fisher and P. G. de Gennes, C. R. Acad. Ser. B 287, 207 (1978). [70] V. Privman and M. E. Fisher, Phys. Rev. B 30, 322 (1984). [71] H. W. J. Bloete, J. L. Cardy, and M. P. Nightingale, Phys. Rev. Lett. 56, 742 (1986). [72] B. V. Derjaguin, A. S. Titijevskaja, I. I. Abricossova, and A. D. Malkina, Discuss. Faraday Soc. 18, 24 (1954). [73] P. G. de Gennes, Rev. Mod. Phys. 57, 827 (1985). [74] F. Brochard-Wyart and J. Daillant, Can. J. Phys. 68, 1084 (1990). [75] T. Young, Philos. Trans. R. Soc. London 95, 65 (1805). [76] M. Sferrazza et al., Phys. Rev. Lett. 81, 5173 (1998). [77] M. Ibn-Elhaj and M. Schadt, Nature 410, 796 (2001). [78] H. Sirringhaus et al., Nature 401, 685 (1999). [79] S. H. Chen et al., Nature 397, 506 (1999). [80] K. M. Lenahan, Y. Liu, and R. O. Claus, Proceedings of the SPIE—The Interational Society for Optical Engineering 3675, 74 (1999). [81] P. Krecmer et al., Science 277, 1799 (1997). [82] B. A. Grzybowski, H. A. Stone, and G. M. Whitesides, Nature 405, 1033 (2000). [83] H. G. F. Coster, A. J. Phys. 52, 117 (1999). [84] P. Poulin, H. Stark, T. C. Lubensky, and D. A. Weitz, Science 275, 1770 (1997). ˇ [85] A. Borˇstnik, H. Stark, and S. Zumer, Phys. Rev. E 61, 2831 (2000). [86] T. Kihara and N. Honda, J. Phys. Soc. Jap. 20, 15 (1965). [87] K. Okano and J. Murakami, J. Phys. Coll. 40, 525 (1979). [88] R. Podgornik and V. A. Parsegian, Phys. Rev. Lett. 80, 1560 (1998). [89] J. Israelachvili, Intermolecular & Surface Forces (Academic Press, London, 1985). [90] B. V. Derjaguin, Kolloid Z. 69, 155 (1943). [91] H. B. G. Casimir and D. Polder, Nature 158, 787 (1946). [92] H. Hamaker, Physica 4, 1058 (1937). 150 Bibliography [93] E. M. Lifshitz, Zh. Eksp. Teor. Fiz. 29, 94 (1955), [Sov. Phys. JETP 2, 73, (1956)]. [94] I. E. Dzyaloshinskii and L. P. Pitayevskii, Zh. Eksp. Teor. Fiz. 36, 1797 (1959), [Sov. Phys. JETP 9, 1282 (1959)]. [95] I. E. Dzyaloshinskii, E. M. Lifshitz, and L. P. Pitayevskii, Advan. Phys. 10, 165 (1961). [96] N. G. van Kampen, B. R. A. Nijboer, and K. Schram, Phys. Lett. 26A, 307 (1968). [97] E. Gerlach, Phys. Rev. B 4, 393 (1971). [98] K. Schram, Phys. Lett. 43, 282 (1973). [99] B. W. Ninham and V. A. Parsegian, Biophys. J. 10, 646 (1970). [100] J. Mahanty and B. W. Ninham, Dispersion Forces (Academic Press, London, 1976). [101] L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2 ed. (Butterworth-Heinemann, Oxford, 1984), first published in English by Perga-mon Press plc 1960. [102] P. M. Morse and H. Feshbach, Methods of theoretical physics (Mc Graw Hill, Boston, 1953). [103] G. Reiter, Phys. Rev. Lett. 68, 75 (1992). [104] F. Vandenbrouck, M. P. Valignat, and A. M. Cazabat, Phys. Rev. Lett. 82, 2693 (1999). [105] R. Barberi and G. Durand, Phys. Rev. A 41, 2207 (1990). [106] T. Moses and Y. R. Shen, Phys. Rev. Lett. 67, 2033 (1991). [107] H. Yokoyama, in Handbook of Liquid Crystal Research, edited by P. J. Collings and J. S. Patel (Qxford University Press, New York, 1997). [108] D. Allender, G. L. Henderson, and D. L. Johnson, Phys. Rev. A 24, 1086 (1981). [109] T. J. Sluckin and A. Poniewierski, Phys. Rev. Lett. 26, 2907 (1985). [110] M. M. Telo da Gama, Mol. Phys. 52, 611 (1984). [111] M. M. Telo da Gama, Phys. Rev. Lett. 59, 154 (1987). [112] N. B. Ivanov, Phys. Rev. E 60, 7596 (1999). [113] K. Miyano, Phys. Rev. Lett. 43, 51 (1979). [114] H. Yokoyama, S. Kobayashi, and H. Kamei, Mol. Cryst. Liq. Cryst. 99, 39 (1983). [115] K. Koˇcevar, R. Blinc, and I. Muˇseviˇc, Phys. Rev. E 62, R3055 (2000). [116] T. Moses, Y. Ouchi, W. Chen, and Y. R. Shen, Mol. Cryst. Liq. Cryst. 225, 55 (1993). [117] C. Rosenblatt, Phys. Rev. Lett. 53, 791 (1984). [118] B. M. Ocko, A. Braslau, P. S. Pershan, J. Als-Nielsen, and M. Deutsch, Phys. Rev. Lett. 57, 94 (1986). ˇ [119] G. Skaˇcej, A. L. Alexe-Ionescu, G. Barbero, and S. Zumer, Phys. Rev. E 57, 1780 (1998). [120] F. N. Braun, T. J. Sluckin, and E. Velasco, J. Phys. Con. Matt. 8, 2741 (1996). Bibliography 151 [121] S. Faetti and V. Palleschi, Phys. Rev. A 30, 3241 (1984). [122] H. Kasten and J. Strobl, J. Chem. Phys. 103, 6768 (1995). [123] P. G. de Gennes, Mol. Cryst. Liq. Cryst. 12, 193 (1971). [124] M. A. Marcus, Mol. Cryst. Liq. Cryst. 100, 253 (1983). [125] E. Martin del Rio, M. M. Telo da Gama, E. de Miguel, and L. F. Rull, Euro-phys. Lett. 35, 189 (1996). [126] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C: The art of scientific computing, 2 ed. (Cambridge University Press, Cambridge, 1992). [127] G. Gompper, M. Hauser, and A. A. Kornyshev, J. Chem. Phys. 101, 3378 (1994). [128] T. Bellini et al., Phys. Rev. Lett. 69, 788 (1992). ˇ [129] G. S. Iannacchione, G. P. Crawford, S. Zumer, J. W. Doane, and D. Finotello, Phys. Rev. Lett. 71, 2595 (1993). [130] W. Greiner and B. Muller, Quantum Mechanics – Symmetries (Springer– Verlag, Berlin, 1989). [131] T. Bellini, N. A. Clark, and D. W. Schaefer, Phys. Rev. Lett. 74, 2740 (1995). ˇ [132] C. S. Park, M. Copiˇc, R. Mahmood, and N. A. Clark, Liq. Cryst. 16, 135 (1994). [133] I. F. Lyuksyutov, Zh. Eksp. Teor. Fiz. 75, 358 (1978), [Sov. Phys. JETP 48, 178 (1978)]. [134] N. Schopohl and T. J. Sluckin, Phys. Rev. Lett. 59, 2582 (1987). [135] A. Sonnet, A. Kilian, and S. Hess, Phys. Rev. E 52, 718 (1995). [136] Handbook of liquid crystals, edited by D. Demus, J. W. Goodby, G. W. Gray, and H.-W. Speiss (Wiley-VCH, Weinheim, 1998). [137] D. Andrienko, Y. Kurioz, Y. Reznikov, and V. Reshetnyak, Sov. Phys. JETP 85, 1119 (1997). [138] M. M. Wittebrood, T. Rasing, S. Stallinga, and I. Muˇseviˇc, Phys. Rev. Lett. 80, 1232 (1998). [139] H. G. Galabova, N. Kothekar, and D. W. Allender, Liq. Cryst. 23, 803 (1997). [140] V. M. Pergamenshchik, Phys. Rev. E 47, 1881 (1993). [141] E. C. Gartland, P. Palffy-Muhoray, and R. S. Varga, Mol. Cryst. Liq. Cryst. 199, 429 (1991). [142] O. D. Lavrentovich, Phys. Scr. T39, 394 (1991). [143] C. Chiccoli, O. D. Lavrentovich, P. Pasini, , and C. Zannoni, Phys. Rev. Lett. 79, 4401 (1997). [144] J. Stelzer, P. Galatola, G. Barbero, and L. Longa, Phys. Rev. E 55, 477 (1997). [145] P. Pasini, private communication. ˇˇ [146] C. Chiccoli, P. Pasini, A. Sarlah, C. Zannoni, and S. Zumer (unpublished). [147] P. P. Karat and N. V. Madhusudana, Mol. Cryst. Liq. Cryst. 40, 953 (1977). [148] M. P. Allen and D. J. Tildesley, Computer simulation of liquids (Clarendon Press, Oxford, 1989). 152 Bibliography [149] Advances in the Computer Simulations of Liquid Crystals, edited by P. Pasini and C. Zannoni (Kluwer, Dordrecht, 2000). [150] P. A. Lebwohl and G. Lasher, Phys. Rev. A 6, 426 (1972). [151] E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, Phys. Rev. E 50, 2929 (1994). [152] U. Fabbri and C. Zannoni, Mol. Phy. 58, 763 (1986). [153] D. J. Cleaver and P. I. C. Teixeira, Chem. Phys. Lett 338, 1 (2001). [154] P. Ziherl, Ph.D. thesis, University of Ljubljana, Ljubljana, 1998. [155] H. Kleinert and F. Langhammer, Phys. Rev. A 44, 6686 (1991). [156] J. Bischof, D. Scherer, S. Herminghaus, and P. Leiderer, Phys. Rev. Lett. 77, 1536 (1996). [157] U. Thiele, M. Mertig, and W. Pompe, Phys. Rev. Lett. 80, 2869 (1998). [158] M.-P. Valignat and F. Vandenbrouck, private communication. [159] J. E. Proust, Colloid Polym. Sci. 254, 672 (1976). [160] G. Barbero and A. K. Zvezdin, Phys. Rev. E 62, 6711 (2000). [161] P. Ziherl and I. Muˇseviˇc, accepted for publication in Liq. Cryst. (unpublished). Izjava Izjavljam, da sem v disertaciji predstavila rezultate lastnega znanstvenoraziskovalnega dela. Ljubljana, 27. 9. 2001 Andreja Sarlah