Univerza Ljubljani Fakulteta za matematiko in fiziko Gregor Skačej odeliranje močno ograjenih ekočekrisalnih sisemov disertaija Ljubljana 2002 University of Ljubljana Faculty of athematics and Physics Gregor Skačej odeling of strongly confined liquid-crystalline system thesi Ljubljana 2002 Za pomoč in številne nsvete na raziskovalni poti ter pri nast tega dela se zahljuje svojemu mentorju, Sobodanu Žumru. • I am grateful to Paoo Pasini, Claudo Zannoni, and Cesare Chiccli (all from Bologna), as well as to Nikolai V. Priezjev d Robert elcovits (Providence, RI), for sharing teir experience in Monte Carlo methods and lattice models. I also wis thank Giovanni Barbro (Torino) Caudio Frrero and Thomas Ginzel both from renoble for joint forts in t phenomenoogy wor Morover, I wish to acknowledge the suppor y CINECA through t EC Access to Researh Infrastructures acti f the Improving Huma Potential Programme. Computatiol work in support of tis resear wa pformd also at the Toretic Physics Computing Facility Brown University * Za dobro delovn vzdušje in številna razčiščen vprašanja hvala vse tekočekristlce z petega adstropja zsti Andr, Danielu, Piki in Prmožu. Z vsestransko podporo e velika zah ud moj domači in prijatelm. Povzetek To delo se ukvarja z izbranimi problemi modeliranja močno ograjenih tekočekristalni sistemov: z urejanjem molekul in površinskim sidranjem v plasti nematika, z molekul sko dinamiko in učinki zunanjih polj v nematskih kapljicah, ter z urejanjem, ki ga vsiljujejo v nematiku dispergirane polimerne mreže pravilne ali nepravilne oblike. Modeliranje teh sistemov sloni na fenomenološkem Landau-de Gennesovem opisu in na simulacijah vrste Monte Carlo v okviru mikroskopskih mrežnih modelov. Posebna pozornost je posvečena vzpostavitvi povezave med rezultati simulacij in eksperimentalnimi opazljivkami, kot so spektri 2H NMR, električna kapacitivnost in intenziteta skozi vzorec prepuščene svetlobe. Rezultati študija urejanja v plasti nematika kažejo, da spreminjanje stopnje nematske ureditve ob površini privede do pojava elastične deformacije, ki jo spremlja prispevek k notranjemu sidranju. Pri študiju nematskih kaplji je bila razvita nova metodologija za napoved spektrov 2H NMR iz rezultatov simulacij Monte Carlo, ki pravilno opiše gibanja molekul v vzorcu. Metodologija je bila preizkušena pri obravnavi fluktuacij dolgih osi molekul, translacijske difuzije in učinkov zunanjega električnega ali magnetnega polja. Na koncu so predstavlje ne simulacije nematikov z dispergiranimi polimernimi mrežami. Te kažejo, da so vlakna mreže sposobna urediti nematik, ki jih obdaja, četudi je njihova površina dokaj groba. Poleg tega lahko zunanje polje povzroči strukturne prehode, prikazane v preprostem faznem diagramu, pri čemer igra pomembno vlogo sama topografija mreže. Nad temperaturo prehoda v izotropno fazo je v sistemu moč opaziti tudi paranematsko urejanje. Rezultati simulacij so omogočili tudi vpogled v notranjo strukturo topoloških defektov v nematski ureditvi, do katerih lahko pride v primerno ograjenih sistemih. Ključne besede: nematski tekoči kristal, ograditev, polimerne disperzije, elastič nost, sidranje, topološki defekti, zunanje polje, mrežni modeli, simulaije Monte Carlo, 2H NMR, kapaitivnost, prepuščanje svetlobe PACS: 61.30.Cz, 61.30.Gd Abstract This thesis addresses selected topics in the field of strongly confined nematic liquid crystals: molecular ordering and surface anchoring in a nematic slab, dynamical and external field effects in nematic droplets, and the orienting ability of regular or irregular polymer networks dispersed in liquid crystals. The modeling of these composite systems is based both on phenomenological (Landau-de Gennes) and mi croscopic simulation (lattice model Monte Carlo) approaches. A special attention is paid to establishing a relation between the simulation output and experimental observables, in particular 2H NMR line shapes, electric capacitance, and transmit ted light intensity. In the simple nematic slab geometry, a subsurface variation in the degree of nematic order is shown to result in a subsurface elastic deformation, accompanied by an intrinsic contribution to surface anchoring. The section on ne matic droplets presents a novel methodology for the calculation of 2H NMR line shapes from the output of Monte Carlo simulations, pointing out the importance of translational diffusion, molecular fluctuations, and external magnetic or electric field effects. Finally, simulations of nematics with dispersed polymer networks indi cate that the networks are capable of aligning the surrounding liquid crystal even if their surface is fairly rough. In a regular fiber array external field-induced struc tural transitions are studied in detail, presenting a stability phase diagram for the observed structures. The topography of the network itself is seen to be intimately related to the characteristics of the external field-driven molecular switching process Moreover, above the nematicisotropic transition temperature paranematic ordering is detected in the system. Eventually, suitable topological constraints can lead to the formation of defects in nematic ordering. The inner defect structure could then be resolved from molecular simulations Keywords: nematic liquid crystal, confinement, polymer dispersions, elasticity, anchoring, topological defects, external field, lattice models, Monte Carlo simulations 2H NMR, capacitance, light transmission PACS: 61.30.Cz, 61.30.Gd ontents ntroduction 13 Theoretical background 23 2.1 Phenomenological Landau-de Gennes theory.............. 23 2.1.1 Nematic order parameter..................... 23 2.1.2 Nematic-isotropic phase transition................ 25 2.1.3 Inhomogeneous phases and curvature elasticity......... 27 2.1.4 External anchoring........................ 30 2.1.5 External fields........................... 32 2.1.6 Topological defects........................ 35 2.2 Molecular simulations and lattice models................ 37 2.2.1 LebwohlLasher model...................... 37 2.2.2 Hexagonal lattice model..................... 40 2.2.3 Monte Carlo simulation technique................ 42 2.3 Experimental observables......................... 46 2.3.1 2H NMR.............................. 46 2.3.2 Electric capacitance........................ 49 2.3.3 Polarized light transmission................... 50 Nematic slab 53 3.1 Planar cell: Landau-de Gennes analysis................. 53 3.1.1 Quasi splay-bend elastic constant and intrinsic anchoring energy 54 3.1.2 Numerical solution of the variational problem......... 56 3.2 Twisted cell: Landau-de Gennes analysis................ 63 3.3 Planar cell: molecular approach..................... 68 Polymer-dispersed liquid crystals 77 4.1 Dynamical effects and H NMR line shapes............... 79 4.1.1 Molecular fluctuations...................... 79 4.1.2 Homogeneous translational diffusion............... 82 4.1.3 Inhomogeneous translational diffusion.............. 84 4.1.4 "Powder" sample: bipolar droplets............... 85 4.2 External field effects ........................... 87 4.2.1 Nematic phase .......................... 88 4.2.2 Isotropic and field-induced nematic phase............ 94 11 12 CONTENTS Nematics with dispersed polymer networks 99 5.1 Aligning ability of the network......................100 5.1.1 Planar anchoring.........................102 5.1.2 Homeotropic anchoring: topological defects...........104 5.1.3 2H NMR spectra.........................109 5.2 External field-induced switching.....................Ill 5.2.1 Regular fiber array........................Ill 5.2.2 Irregular fiber array........................117 5.2.3 Experimental observables and network irregularity.......121 5.3 Pretransitional ordering in the isotropic phase.............126 Conclusion 133 References 137 azširjen ovetek Uvod Na prvi pogled se morda zdi, da izraz tekoči kristali vsebuje neskladje, saj se hkrati sklicuje na tekočnost in kristaliničnost snovi. V resnici gre pri tekočih kristalih za organske snovi, kijih sestavljajo molekule močno anizotropne oblike, in čeprav so ob primernih pogojih tekoči, oblika molekul botruje anizotropiji njihovih makroskopskih lastnosti: lomnega količnika, dielektrične konstante, magnetne susceptibilnosti in drugih. Prav zato so tekočekristalni materiali danes izjemno razširjeni v indus triji optičnih naprav, kot so hitre zaslonke ali tekočekristalni zasloni, pogosto pa jih srečujemo tudi v vsakdanjem življenju. Prve tekočekristalne snovi so bile odkrite že pred več kot sto leti [1,2] in kaj kmalu se je izkazalo, da gre za novo agre gatno stanje med navadno (izotropno) tekočino in trdnino. Danes je znanih mnogo tekočekristalnih faz, od katerih se bomo tukaj omejili na najpreprostejšo izmed njih nematske ("nitaste") tekoče kristale [3]. Sestavljajo jih podolgovate molekule, ki se v velikem vzorcu v povprečju uredijo v določeni smeri. To smer označimo z enotskim vektorjem n in ga imenujemo direktor [4]. Pri tem sta smeri +n in —n enakovredni težišča molekul pa so razporejena naključno po prostoru kot pri navadni tekočini (slika 1). Ponavadi je porazdelitev molekul okoli n osno simetrična: nematik je tedaj enoosen. Če to ne drži, govorimo o dvoosnem nematiku. Z višanjem temperature stopnja ureditve okoli n postopoma pojema, končno pa pride do faznega prehoda v navadno (izotropno) tekočinsko fazo, ki je šibko nezvezen. Obnašanje nematikov postane veliko bolj zanimivo, če jih ogradimo — z njimi zapolnimo drobne (mikroskopske) kapljice ali pore (slika 1) [5]. Pojem močne ogra-ditve se pri tem nanaša na sisteme z visokim razmerjem površine in prostornine. Pomemben je pojav sidranja tekočekristalnih molekul na ograjujoči površini [6], pri katerem se molekule lahko urejajo vzdolž površine (planarno sidranje [7-10]) pravokotno nanjo (homeotropno sidranje [11-13]), ali pa poševno [14]. Red, ki ga vsiljuje površina, lahko ob stenah obstaja tudi nad temperaturo faznega prehoda v izotropno fazo (paranematski red). Močno ograditev lahko zagotavljajo tudi v tekočem kristalu razpršene (dispergirane) polimerne mreže [15]. Podobno je moč (raz) urejuj oče učinke pričakovati tudi na prosti površini nematika [16,17], čemur pravimo otranj sidranje (za razliko od zunjega, ki zadeva urejanje ob trdni površini). V ograjenih sistemih postane direktor krajevno odvisen, n = n(r). Vsako odstopanje od homogeega direktorskega profila z n ^ n(r) predstavlja elačno deforma- zširjeni povzetek cijo, ki jo spremlja prispevek k prosti energiji [18-20]. Tudi vsakršno odstopanje od smeri, ki jo predpisuje sidranje, poviša prosto energijo [21,22]. Zaradi anizotropije v električni in magnetni susceptibilnosti je na orientacijo molekul (in n) mogoče vpli vati tudi z zunanjim poljem [4,20]. Ravnovesni n(r) potem najdemo z minimizaijo elotne proste energije. V določenih primerih ograditve naletimo na mesta v vzorcu kjer n iz topoloških razlogov ni definiran. To so lahko toče, črte ali stene, ki jim pravimo topološki defekti [4,23,24]. Danes so na voljo najrazličnejše vrste ograditev: nematske plasti (celice), membrane z mikronsimi valjaimi porami [27], nematske kapljice v polimerni matri ki (PDLC) [28], v tekočem kristalu dispergirane polimerne mreže nanometrsih vlaken [26,30,31] in podobno. Te in podobne sisteme so v preteklosti raziskovali z različnimi pristopi. S teoretične plati prednjačijo fenomenološki pristopi temelječi na teoriji Landaua in de Gennesa [16,32], ki jim sledijo študije z gos-totnim funkcionlom [33-35]. Z razvojem hitrih računalnikov je doživelo razcvet področje simulacij molekularne dinmike [36] in simulacij vrste Monte Carlo [37]) Oba pristopa sta zasnovana na parskih interacijah med molekulami in omogočata povezavo med mikroskopskimi in makroskopskimi lastnostmi sistema. Izmed eks perimentalnih metod je bila za študij tekočih kristalov prva uporabljena polari-cijsk mikroskopija [1,2,26,38]. Sledile so ji ostale metode: meritve električne kapaciivnosti [39], devterijeva jedrska magnetna resonanca (2H NMR) [5,4041] podvajanje frekence slobe (SHG) [42,43], elipsometria [44], dinmično sipnje lobe [4546], klorimerija [47] in še številne druge. Va^0^ (a) Slika 1 Ureditev molekul v nematski fazi, n označuje direktor (a). Primer ograjenega nematika v kapljii (b) V tem doktorskem delu se bomo dotaknili modeliranja izbranih problemov s področja močno ograjenih tekočih kristalov. Pri tem se bomo posvetili dvema ci ljema: (i) boljšemu razumevanju urejanja molekul v bližini ograjujočih površin in (ii) opisu nematskega urejanja v primerih kompleksne (tudi nepravilne) ograditve. Temu poglavju bo sledil kratek pregled nekaterih pojmov, potrebnih za opis nematskega urejanja in za razumevanje modeliranja, ki bo predstavljeno v nadalje vanju. Prvi izmed problemov, ki se ga bomo lotili, bo nematsko urejanje v tanki plasti tekočega kristala. Posebna pozornost bo posvečena spremembam stopnje ureditve ob površini, ki jih spremljajo notranje sidranje in elastične deformacije, opažene tudi eksperimentalno [43,48-51]. Hrati so bile podobne deformacije napovedane tudi teoretično v okviru fenomenološkega Landau-de Gennesovega opisa z elastično konstanto K13 [52,53]. Deformacije, do katerih tak opis privede, so močne [54-58] in zato v nesladju z uporabo kontinuumske teorije [59-61]. Hrati novejše analize Razširjeni povzetek (tudi z uporabo gostotnega funkconala) kažejo, da za stopnčast profil gostote i stopnje ureditve ob ravni površini velja Ki3 = 0 [62-65]. To sicer navide sicer reš omenjen problem, a kljub temu pušča odprto vprašanje obstoja eksperimentalno opaženih deformacij. Elastične deformacije je možno zasledit, kadar profil gostote ni stopničast in lahko variira tudi stopnja nematske urejenosti [66,67]. V tem delu se bomo najprej lotili planarne ureditve z namenom podrobno raziskati posledice variacij stopnje reda ob površini [68], pri čemer v deformacij ski prosti energ člena s konstanto Ki3 ne bomo izrecno upošteval tudijo bomo na koncu ra še na neravninske zvojne deformacije [71]. Pojav notranjega sidranja, ki pomembno vpliva na ureditev blizu površine ne matskega vzorca [17,72] bomo v plasti nematika obdelali tudi z mikroskopskega stališča. Fenomenološki opis bo nadomestl preprost model s šestkotnško mrežo, temelječ na prostorsko anizotropni interakcij med induciranimi dipoli. Problema se bomo lotili s smulacijami Monte Carlo pri končnih temperaturah. Glavni namen te študije je izmeriti jakost notranjega sidranja in njeno temperaturno odvs nost. V preteklosti je blo nanizanih že več mikroskopskih študij sidranja ob prostih površinah nematka ali za nematik v stku s trdno steno: psevdomolekulsk konti-nuumsk pristop elipsastimi molekulami [73], simulacijske študije v sistemih Gay-erneovih delcev [74-78] ter študije sistemov trdih elpsoidov [79,80]. Nekatere simulacije molekularne dinamke Gay-Berneovimi delci kažejo tudi na plastovito urejanje v bližini površine, kar vodi do znatne modulacije v profilu molekulske gos tote [84-87] in je bilo opaženo tudi v eksperimentih [88,89]. onavadi so v simulac ja določene energije sidranja velko všje od ekspermentalih vrednosti [6]. Sledila bo obravnava krogelih (PDLC) nematskih kapljc premera pod 1/iin, k bo omejena na radialne [38,40] n bipolarne [38,40,90] kapljice, prve s homeotrop- dranjem, druge s planarnim (slika 1). V aplikativne namene so posebej za nimivi primeri, ko kapljice postavmo v unanje polje [40,90]. Eksperimentalno so kapljice preučevali z 2H NMR [40,90] in polarizacijsko mikroskopijo [38] teo retično pa fenomenološko (Landau-de Gennesov opis) [5,25] in s simulacijami Monte Carlo [91,92]. Do sedaj predstavljene simulacije praviloma temeljijo na mrežnem modelu Lebwohla in Laserja [93] in so se izkazale kot korstne tudi pri napovedovanju eksperimentalnih opazljivk: statičnih spektrov 2H NMR in slik polariza cijske mikroskopije [91,94]. Tukaj se bomo posvetil razvoju nove metodologije za računanje dnamčnih spektrov 2H NMR v prisotnosti molekulskega gbanja — fluk-tuacij dolgih osi molekul in translacijske difuzije [95] — ter jo preizkusili tudi v prmeru nehomogene difuzije [96,97]. Vhodne podatke a izračun spektrov bodo priskrbele simulacije Monte Carlo, izvedene v modelskem sistemu Lebwola in Lash erja. Obravnavo kapljic bomo zaključili s podrobnm študijem učinkov unanjega polja [98-100], ki jih bomo spet spremljal predvsem skozi ravoj spektrov 2H NMR Naslednje poglavje bo zadevalo nematike z dispergiranimi polimern mrežami kar bo tudi najkompleksnejša od vseh obravnavanih ograditev. Tudi t sistemi so obetavni za aplikacije, ki v glavnem temeljijo na preklapljanju orientacije molekul iz smeri, ki jo določajo polimerna vlakna, v smer zunanjega polja. Nenaden preklop molekulskih orentacij je moč zaznati preko sprememb v kapacitivnosti, optičn prepustnost ali z MR spektroskopijo [31,39] ogajanje v procesu preklaplja Razširjeni povzetek nja je močno odvisno tako od pogojev sdranja na poršin, kot tudi od topografije same mreže, kar lahko oboje reguliramo med njenim natankom [15101]. Obstoječ eksperimentalne študije ponavadi spremljajo fenomenološke analize [5,26,31,39 medtem ko je bilo za te sisteme torjenega zjemno malo na področju molekulskih si mulacij. Zaradi vsega navedenega bo najprej predstavljena temeljita mikrokopka študija orientaijke sklopitve med polimernimi vlakni in tekočim kristalom, obdaja. Kot primeru kapljic bomo uporabli mreži model Lebwohla in Lah-erja, in sicer vlakna z dobro definirano povprečno smerjo. Obravnavali bomo prmere različno grobostjo površine laken [102], pa tud take pri katerh pride do tvorbe topoloških defektov [10]. Pri slednjh bomo notranjo zgradbo defektov primerjali s fenomenološim napovedmi [104,105]. Simuliral bomo še proces preklapljanja v unanjem polju, pri tem pa bomo posebej pozorn na vplive nepravilnost topografiji polmerne mreže. Stabilnost opaženih direktorkih struktur v prmeru pravilne polmerne mreže bo prikazana v preprostem fanem dagramu. Na koncu bo preučena še možnost paranematskega urejanja nad temperaturo prehoda v izotropno fao. Rezultati simulacij bodo predstavljen tudi v obliki ibranih eksperimentalnih opazljivk: pektrov 2H NMR kapaivnoi in intenztete skoi vorec prepušene svetlobe. V zaključku bodo predstaljen gla doeži i so b predtaljeni v tem delu pa tud še odprta vprašanja n možne umeritv a prihodnot. Teoretičn ozadje Fenomenološki opis nematske uredtve Fenomenološk Landau-de Genneov opis nematika temelj na uvedbi simetrskm lastnostim nematika primernega parametra urejenoti, ga uporabimo apis proste energije sistema. Ravnovesno tanje nato najdemo iskanjem njenega m muma. Orientacijsko urejanje dolgega dosega v enoosnem nematku opemo s tenzor skim parametrom urejenosti (zaradi enakorednoi ±n vektorski ne b bil primeren) Q = i53n®n-I], (1) kjer I pomeni identeto [4]. Vpeljali smo S = (|(2 8 1)}, skalami parameter urejenosti, kjer je co = n • u, enotski vektor u določa trenutno orentaijo dolge si posamezne molekule n (.) pomeni ansambelsko povprečje čez več molekul V prmeru idealne nematske ureditve imamo S = 1 v neurejeni iotropni fazi pa S = 0. Enakoreden opis daje tudi redtvena matrika iu®u-I V njen latni vrednost največjo abolutno rednostjo preponamo S, v prpadajočem lastnem vektorju pa direktor n. Iz raike preostalh lastnih rednosi je mogoč klepat na stopnjo dvoonost orientake poradeltve molekul P Razširjeni povzetek Ureditev, ki jo opisuje Q, se odraža ti na makrkopskih opazljivkh. Ta ahko tenzor magnetne susceptibilnosti % v enoosnem primeru zapišemo v obik X = §XaQ + Xi') kjer pomeni %a največjo anizotropijo susceptibinosti ideano urejenem nematiku Xi Pa njeno rednost iotropni fai Slika 2 Prehod med nematsko in izotropno fazo: (a) prosta energija F ko funkcija parametra urejenosti S in (b) ravnovesna vrednost S pri različnih temperaturah. Nezvzni fazni phod med izotropno (S = 0) in nmasko (S ^ 0) fazo v okviru Landauove teorije v ližini faznega prehoda opišemo z analitičnim razvojem gostote proste energije /0 po tistih invariantah parametra urejenosti Q, ki so sogasju s simetrijo manj urejene fae [32] /o (S, T) = L (T) + \a{T - T*)S - ±B + \C\ (3) Tukaj predstavljajo a > 0, B > 0 ter C > 0 fenomenološke snovne paramet re, fi (T) gostoto proste energije izotropne faze in T* najnižjo možno temperaturo njene podhladitve. Potek proste energije F = J f0(S,T)dV je prikazan na si 2 prav tako pa tudi temperaturna odvisnost ravnovesnega parametra urejenosti, ki jo dobimo pri minimizaciji F: S (T) = ^[1 + Ji — ^§r(T — T*)] pri T < Tc (ne matska faza) in S (T) = 0 pri T > Tc (izotropna faza). Temperaturo fanega prehoda Tc in najnižjo temperaturo podhladitve izotropne faze T* povezuje ve Tc = T* + 2B2/9aC. Možna so tudi metastabina stanja s pregretim nematikom vendar le v zkem območju pod T** = Tif + /4C. Ponavadi vlja Tc —T* ~ 1 K in T** — Tc ~ 01 K. Ker je prehod šibko nezen, ga pri Tc spremlja gostota utajene toplote Q = ^aTcS(Tc)2, ki je ponavadi velikostnega reda 106 J/m3. Kadarkoli se otimo obranave ograjenih sistemov, lahko postane stopnja uredit ve S krajevno odvisna. V takem nehomogenem primeru je potrebno gostoti proste energije (3) dodati člen obke |L(VS) (tudi L je snovna konstanta) in variacij ko najti ravnovesni profi = S (t). Značino dožino sprememb S — korelacijsko olžino — dooča ve = V (T - T*) - Sb + 6 Razširjeni povzetek kjer pomni Sf, ravnov ost dvenega parametra v velikem homom vzorcu ki jo določimo z minimizacijo izraza (3). Dolžina L narašča, ko se pribujemo faznemu prehodu, a tam ne divergira: L(TC) ~ 10 nm. V ograjenem sstemu je ponavadi poleg stopnje ureditve S (t) tudi poprečna smer ureditve — direktor n(r) — od kraja odvisna. V tem prmeru je prikladno gostoto proste energije izrazii z od kraja odvisnm tenorsk parametrom ure jenosti Q(r) (1) (r) = fi{T) + \a(T - T,)trQ2 - §BtrQ3 + ±C(trQ2 + 2 2^2 Qik,k 2^ kj (5) kjer smo vpeljali krajevne odvode Q^-^ = dQij/dxk [4,107], Li, L2 n L3 pa so elastični snovn parametr prmeru, ko S / (t) = S t, lahko prepšemo neho- mogen del izraa (5) Mr) = \ {#11 [V • n] #22 V x n)]2 + #33 [ x (V n)]2} - V • [n (V • n) + n V x n)] (6) menovan tudi Frankova gostota proste energije 18] s tremi elastčnimi konstantam predstavljajo razlčne deformacijske načine: #n pahljačno, K22 zvojno in #33 upogibno deformacijo (slika 3). Zadnji, divergenčn člen prpada sedlasto-ahljani deformaciji. Pdemo lahko do naslednj zvez: #n = K33 |5,2(2Li + L ) #22 \SjL #24 = fSf^Li + L3). V prmeru, ko je + L3 = 0 velja #n #22 = #33 = #, čemur pravimo približek z eno konstanto. Ponavadi velja # 5 x 10-12 kar je mogoče izmeriti na prmer s preučevanjem elastčn deformac v zunanjem polju 108109] Velja tud 110] (a) (b) (c) Slika 3 Frankovi načini elastične deformaije: pahljača (a), zvoj (b) in upoib () Divergenčnemu členu s #24, ki pri iskanju ravnovesnega direktorskega profila pliva le na robne pogoje, je podoen tudi pahljačno-upogibniprispevek oblike #3 V- (V • n)] [5253]. Od člena s #24 se ralikuje po tem, da vsebuje tudi druge odvode n in kot tak povroča težve pri matematični formulaciji variacijskega problema. Tudi po njihovi odpravi napoveduje (pre)močne elastične deformacije ob površi 55-58] vendar novejše analize kažejo da je prašanje pravzapra brezpredmetno in da za dealno ravno površino velja #i3 = 0 6265111]. Zarad navedenh nejasnost se omo v nadaljevanju členu s konstanto #i izognili. V okviru fenomenološkega opisa interakcije nematika z ograjujočo trdno površino — menovane tudi znanje sidranje — poršna siljuje določeno smer (n0) n stopnjo uredve (o). Ooje lahko ajamemo tenorskem parametru 0 poršnsko Razširjeni povzetek Vi gostoto interakcijske proste energije pa modeliramo z [22] U = ±wMQs - Qo)2 (7) kjer Qs opisuje dejansko ureditev ob površini (n, in Ss). Kadar vzamemo S = konst se zgornji izraz poenostavi v Rapini-Papoularjevo formulo f^p = ^W cos2 a [21] ki pogosto predstavlja dovolj dober približek. V njej a pomeni kot odstopanja ns od predpisane smeri n0 (cos a = ns • n0) energija W = \WeSl pa jakost sidranja. Značilne vrednosti zanjo so od 10~6 J/m do 10~4 J/m2 [6]. Bolj nazorna je izražava s Kleman-de Gennesovo ekstrapolacijsko dolžino K/W (slika 4) [4], ki pri navedenih vrednostih K in W niha med 50 nm and 5 /im. Jakost sidranja merijo na mnogo načinov: s povzročitvijo elastične deformacije z zunanjim poljem [6], z analizo direktorskih struktur v kapljicah in porah [112113] ali pa s študijem fluktuacij direktorja v ograjenih sistemih [446] S ^ KIW 0 z Slika 4 Direktorski profil cf)(z) v deformirani plasti nematika in definicija ekstrapolacijske dolžine K/W 4]. Ograjujoča površina se nahaja pri z = 0 Če nematik izpostavimo zunanjemu polju (na primer magnetnemu jakosti H) to nanj deluje z navorom, do česar pride zaradi anizotropije tenzorja magnetne susceptibilnosti x (an dielektrične konstante e v električnem primeru). Pripadajoči prispevek k prosti energiji je f \^SH cos2/3, (8) kjer / pomeni kot med in H (cos/ = n • H/H), Xa anizotropijo molekulske susceptibilnosti jj,0 pa indukcijsko konstanto (analogen izraz je moč zapisati tudi za urejanje v električnem polju). V primeru, ko je Xa > 0, se molekule nematika uredijo v smeri zunanjega polja, ki ga na ta način lahko izkoristimo za krmiljenje povprečne smeri molekul v vzorcu. Ta pojav za svoje delovanje izkorišča velik del tekočekristalnih optičnih naprav. V ograjenem nematiku neizbežno pride do nesoglasij med ureditvenimi težnjami zunanjega polja in ograjujočih sten. Pričakovati je, da se bodo molekule nematika dovolj daleč od sten usmerile vzdolž smeri polja, blizu sten pa bo nematik zaradi površinskega sidranja deformiran (slika 5). Debelino te deformirane plasti v grobem določa koherentna dolžina ki jo v primeru magnetnega polja zapišemo takole: K oXSH Razširjeni povzetek Če imamo potemtakm opravka z ograjenimi sistemi, katerih značilna dimezija j majša od Lm, zunje polje na ureditv sploh ne bo vplivalo. Za značilen teko kristal v magnetnem polju gostote 1 T znaša Lm ~ 10 /im. Če zvišamo jakost zunega polja tako daleč, da zane veljati Lm < K/W', se zunaje sidranje ne more več upirati premočnemu zunjmu polju in dopusti, da s molekule nematika obrnejo v smer polja tudi tik ob steni. Pri tem gre za zvezi strukturi prhod, ki ga imenujmo tudi saturacijski prehod (slika 5) [6] (a) H=0 (b) H . (c) H , l t z=0 l ^ ^ lika 5 Polneskončni vzorec nematika s plarnim sidranjem unjem polju H, usmerjenem pravokotno na steno vzora: nedeformirana (a) deformirana (b) in zasičena struktura () Nkoliko drugače se obnša lia debeline d, v katri je nematik med dvma vzporednima plošama s planarnim sidranjem i je zunanje polje — kot prj usmerjeno pravokotno nanju. Za razliko od polneskončnega vzora dobimo elastično deformiran direktorski profil e nad doloeno vrostjo poljske jakosti, pod njo pa urditev določa planaro zunanje sidranje. V primru neskoo močnega sidranja ta zvni prhod imenovan Freederickszov 108,109 — dogodi, ko LTO = /w [4], medtm ko pri šibkem sidranju in K/W 0, r pa predstavlja enotski vektor ki povezuje oba delca. Parameter v nadzira prostorsko anizotropijo interakcije: za = 0 dobimo izotropno interakcijo ki je neodvisna od r (kot v modelu LL), za v = 1 pa anizotropno interakcijo med induciranimi dipoli. Če je interakcija anizotropna {v ^ 0), se na površini vzorca pojavijo ureditveni učinki ali notranje sidranje [17,81]. Poleg tega za 0 model ustreza približku z eno elastično konstanto, sicer pa ne [129]. Uporaba prostorsko anizotropnih potencialov skupaj s kubično mrežo privede do pojava preferenčnih smeri urejanja tudi globoko v vzorcu, zaradi česar postane model neuporaben za študij sicer tekočinskih nematikov [81]. Izkaže pa se, da šestkotniška mreža (slika 8) te hibe nima, če orientacije delcev Uj omejimo na ravnino šestkotnikov [17]. Namesto šestih imamo zdaj osem sosedov, enotski vektorji u^ pa so le dvodimenzijski. Da se v celoti izognemo obstoju periodičnih rešitev, ki so pos ledica mrežnega približka in anizotropnega potenciala (12) je potrebno zahtevati še Razširjeni povzetek v < 0.3 [17]. Ker imamo s t v ^ 0, je notanje sidanje ob površii še prisotno, sicer pa se delci najraje urejo tako, da so njihov dolge osi vzporene lednje rečamo tudi v sistmih Gay-Berneovih delc [130]). ^ ..S-. S y^vf V; lika 8 Mo s šiško mr reži modeli so zčilni pimeri v katerih je imern poabiti simulacije st Mon Crlo M) za iskanje ravovesja v sist za izčun ermodiamiih povprečij. Damika mtode Monte Carlo ne elj na reševanju enačb ibanja ampak a Mopoisovm postopku, ki si ga bomo ogledai v adaljevanju. Navne dinamike v mrežih sistemih pravzaprav sploh nima smisa simulati saj jim po efinicii manjka del sicer naravih postostih stopenj. Termodiamičo povprečje spremenljivk A v sist el je efino z Jduu)e-^u^ 13) kjer je /3 — ksT (ks Botzmannova kostanta absolutna tmpaatur), u pa dolg vektor, ki doča orientacije Uj vse N delcv i tako opisuje stanje -delčnega sistma v faznem postoru. Poleg tega predstavlja % hamiltonko sistema ki je sestavljena iz vsote vsh ierakciskih energij md pari sosedov pa tudi iz pispevkov njihove sklopitv z zunnjim polje. Mtoda Moe Crlo omogoča izračun sam- elskega povprečja A), ki je v ergodiih sistmih enakovreo časovnemu povprečju A, dobljenemu iz simulaci moleularne diamike, kjer sist ledimo v času. Pri mtodi Me Carlo gre za primern uežen naključi sprehod po faznem ostoru, pri katerem v vsake korak vzorčimo koči A. Naključni sprehod ledi Metropolisovemu postopku [37], ki ahko strnemo v naslednje redpise: 1. Vzmi stao konfiguracij (s); izunaj njeno energo ^(s). . Naključo izberi enega od delev iz stare konfiguracije zasuči naključ da dobi novo poskno konfiguracij u' ); izunaj energ ov konf guracije u'N n). rejmi premik u' (s z verjeost mi 1, e m(n)-U(s)] Vrni s koak ega postopka oge je pokazati, da pived rnji postopek do kaoične porazdelitv z dob- o doočeno tmperaturo ki smo o predpostavili pri definicii povpreč ) (13). Razširjeni povzetek Zanljivost ocene a povpreče (.4.) je tem večja, čim č delcev sodeluje v simulacij n čim lje je simulacija tekla. Če velja A oc N, se relativna napaka povprečja manjša kot l/y/NM, kjer M označuje števil korakov mulacije. Naslednje vprašanje zava tvorbo oskusnih konfiguracij u'N. V modelu LL navdi uprabljamo tehko Barkerja in Wattsa [131]: najprej naključno izbere eno izmed tre kartezih osi labratorijskega koordinatnega sistema in nat oli nje zavrtimo naključno izbran mlekulo za naključo izbran kot. Največj možn kot zasuka (ampltudo) lahk med smulacijo spreminjamo. Če so namreč zasuki preveliki, je možno, da nova knfiguracija sploh ne bo sprejeta. Če pa so, po drug strani, zasuki premajhni, Metrpolsov postopek bišče premajhen del faznega prostra, da bi ocena za (A) bla kokaj dobra. Pnadi se amplitudo zasuka pr lagaja dinamino, in sicer tako, daje selej sprejeta oko polovica vseh nameravanih zasukov. V prmeru modela s šestkotniško mrež, kjer so vektorji u, le dvodimenzij sk, se zgornji postopek poenostavi v preprosto vrtenje znotraj ravn šestkotnkov Včasih je pomembna tudi izbira začetne konfiguracije. V ergodičnih sistemih b morali rezultati simulacije načeloma biti od nje neodvisni, vendar je treba bti pre viden zlasti v blžin metastablnih stanj, še posebej, če smulacija n tekla dovolj dolgo. Na koncu je treba določii še količino A, ki jo želimo ovprečiti. To o lahko komponente uredivene matrke Q, ureditven parametri PJ = |(3(v • u^)2 — l) kjer prestavlja v neko fiksno smer, al pa orientacijske korelacijske funkcije S bolj zanive so ekspermentalne opazljivke, na primer spektri 2H NMR, električna kapaciivst ali intenzteta prepuščene svetlbe. Tem se b vetili v naalje vanju Eksperimentalne opazljivke Prv izmed obravnavanih opazljivk bodo spektri devterijeve jerske magnetne reso nane (2H NMR. Ta tehnika je primerna zlasti za raziskave devteriranih nematkov v mkroskopskih votlinah razsežnosti pod 1 ^m, ko optične metode odovedo. 2H NMR daje infrmacije o orientacijskem urejanju v vzorcu, pa tudi o dinam lekul — fluktuacijah dolgih osi in translacijski difuziji [2540132,133] V izotropni fazi v spektru devteriranega nematika vidimo eno sam črto pr Zeemanovi frekvenci uoz- Ko vrec ohladimo v nematsko fazo se pojavi drupoln razcep coq, zaradi česar vidio v spektru dve črt. Razep uq znaša prmeru enoosne uredive [5134135] ojq = ±SojqIS[3co291] (14 ter je odvisen od kota 9 med zunanjim magnetnim poljem NMR spektrometra direktorjem n, pa tudi od stopnje ureditve . Značilna velikost razcepa je Sloq ~ 2n x 40 kHz. V ograjenem sistemu sta n in S od kraja odvisna, posledičn pa tudi uoq = u>q(t). Na ta način je mogoče iz spektrov razbrati, za katero izme možnih direktrskih struktur v vzorcu gre. Razpoznavanje pstane teža zlasti v drobnih votlna k translacijska difuzija bl spektralnih rt zazna pači [136] zširjeni povzetek Za prailno obavna molekulskih gibanj z značilno časovno sklo krašo od značilne skale spektroskoije NMR (i0 ~ 2tt/Suq ~ 2.5 x 10~5 s) je potrebno uporabiti polklasični pristop asono odvisno hamiltonko deteijevih spinov [135]. V oku tega pristopa je nprej otebno generirati elkscijsko funkcijo (t) = ex{icozt)(ex(i|*f[ri(i/),*/] dO) (15) kje je fžg[rj (L),L] = ±5uq | [3(u, • B/i?)2 — 1] "trenutni" kvadrupolni razcep (povprečje po fluktuacijah u tu še ni opravljeno), oklepaji {...)j pa pomenijo ansmbelsko ovprečje o delcih. Gibanje molekul (fluktuacije ali difuzij) stopa v (t) skozi asovno odvisne u». Spekter I(oj) izračunmo potem s Fourierovo transformacijo I(u) = J eluj G (t) dt. Značilna časovna ska fluktuacij molekulskih dolgih osi je okrog tp ~ 10~8 s in je kot taka znatno krajša od i0 [41]. Tudi značilni čas, ki je potreben za difuzijski premik molekule približno za svojo dolžino (1 nm), je v tem območju: to ~ 10~8 s [41]. Vsekakor pa je pri difuzijskem procesu bolj merodajen s t, ki ga molekula potrebuje za premik v področje z znatno drugačnim loq. Sektri so tko vedno izpovprečeni preko fluktuacij dolgih osi U preko difuzijskeg gibnja pa le v primeru močne ograditve. Naslednja izmed eksprimentalnih metod za študij urejanja nematikov je merite elektčne kapacitinosti [39]. Temelji n dejstvu, da je dielektrična konstanta e v nemtikih nizotropna. Zamislimo si plast nematika v ploščatem kondenztorju, čigar normalo označimo za os z (xy potem predstavlja ravnino plošč). V poeno stavljeni sliki si plast zamislimo razdrobljeno na množico drobnih kondenzatorjev kateih kapacitivnosti so sorazmerne z lokalno dielektrično konstanto. Ta je na mestu itega delca z orientacijo Uj enka e(x,y, z) = e_i_ + (e| — ej_)(uj • z), kjer pomeni enotski vektor v smei osi z, e in e|| pa lstni vrednosti molekulskega dielektričneg tenzoj. Kondenztorje si zd predstavljamo vezane zaporedno vzdolž osi z, tako astale vige pa vzporedno po vsej ravnini xy. Celotno kapacitinost sti velikosti x b x d otem izračunmo iz zveze dx dy[---------- (16) Jo Jo Jo (x,y,z)J kjer pomeni 0 influenčno konstanto. Kapacitivnost otemtakem predstavlja nazor no merilo za oientacijo molekul nematika in je kot t prabn tudi z pazovanje ojava preklapljanja v zunnjem polju [39]. retjo in zgodovinsko najstarejšo skupino metod za raziskave nematikov predstavljajo optične metode 1,2]. Podobno kot statična dielektrična konstanta e je tudi visokofrevenčna e°° v nematikih nizotop, z njo pa tudi lomni količnik. Nematiki so otično anizotrpna enoosna edstva katerih loklna smer optične osi sovpa z loklnim direktorjem n. Pr razšinju elektromgnetnega valovanja skozi nematik imamo tako dva žark različnima polarizacijama in hitrostjo razširjanj (lomnim količnikom): rednega in izrednega. Polarizacija prvega je hkrati pravokotn na optično os in na smer razširjanja valovanja k, lomni količnik pa je enak n0 = \/^ Polarizcij ugeg je pravokotn a k, hkrati pa leži v ravnini, ki jo doloa k in Razširjeni povzetek optična os. Ustrezni lomi kličnik ne j odvisen od kota ed k optično oso, pri čemer lja [137] 1 cos 9 , -* =------------------• 17) ne e± 6| sod v zgornjih izrazih označujeta ^ in e lastni vrednost matrike e Pri preučevanu razšrjanja polarizirane svetlobe skozi nehomogen optično ani-zotropen nematik si lhko pomagamo s formalizmom Jonesovih vektorv. Zamis limo s enako geometrijo kot pri meritvah kapacitivnosti, svetlobni žarek pa naj se razrja v smeri osi z. Vzorec naprej razdelimo na majhna območja z dobro določeno lokalno optično osjo — ta na sovpada kar z orientacijo posameznega delca Uj. Nato razcepimo vpadno poarizacijo na redno n izredno komponento, ki se potem vsaka zase razata skozi plast, in ta postopek ponavljamo, dokler ne pridemo skozi ves zorec. Pri tem zanemarimo lom, uklon in spane. Na drugi strani vzorca lahko me rimo intenziteto svetlobe, prepuščene skozi analizator, ki ga postavimo pravokotno na smer poarizacije vhodne svetlobe. V primeru, da je kot med optično oso ceot nega zorca vhodnega poarzatora enak (p0 mamo za ntenzteto [138] = /(2^)sin2(A$/2), 18) kjer je I0 intenzteta vhodne svetlobe, A$ (2tt/X) /0 [ne(z) — n0]dz pa razlika v faz med rednim in zrednm žarkom po potovanju skozi vzorec (A pomeni vlovno dolžino svetlobe). Otno je, da bo najveja za (p0 7r/4. Takšna ekspermen talna postavitev je bila uporabljena za meritev elastičnih konstant [116] kost dranja [6] in za opazovane paranematskega urejanja [26]. o tod smo s ogledali lavna orodja in prstope, potrebne za zpolitev zadanih si ciljev: preuitve urejanja molul v bliž poršine tanke plast nematika nematskega urejanja zapetenejših ograditvah — kapljicah in sstemih polimernih mrež. V pogavjih ki edijo, bodo predstavljen napomembnejš rezutat omen ih študij Plast nematika Fenomenološki opis take plasti tem pogavju se bomo lotili obravnave plast nematika, še zlasti v bliž sten. Radi bi razkali vpliv varacij stopnje nematske ureditve — neizbežnih v bliž vsake ograjujoče povine — na smer nematskega direktorja. Sprva se bomo posvetili ravnnskim deformacijam, pri katerih lhko drektor parametrziramo z n (z) = (sin<^), 0, cosc/)(z)), stopno urejenosti pa opisuje profl S (z) (pri tem os z sovpada s ploskovno normao sten zorca). Ko nato tenzorski parameter urejenosti Q 1) zrazmo z (z n 4>(z), zapišemo Landaude Gennesovo gostoto proste energije (5 v obliki / /o(5, T) + /!«>, S') + /(^ S) + /3& < S'), kjer omogeni člen fo,T) dočen z izrazom 3), ostali trije pa z f^ fLij ^^cos i)}^ 19) zširjeni povzetek f2(tf>',S) = |L1 ^}^ (20) /3(0 , S, S) = -§ ) sm(2) Drugi čn predstavlja Franko prispevek (6) ( ^ je v zbani gemetriji enak nič). Tretji len je d ni azličen l a L2 L3 ^ 0 — kadar ne upoštevamo pblka z eo ksto — plja stp jnosti in kota (f). Dnimo, da imamo pneskončno plast nemaka in da stena predpisuje stopnjo ureditve So te kot 4>q. Dovolj daleč o stene je vednost S ena Sb in jo določimo z minimizacijo fo(S, T) (3). Varicija v profilu S () se dogdi na razda, ki je bl koeaci dni L (4) jo moemo tak smatrati za povrinski pov. Izkže se da sta prav zato tudi W = J0°° fi)dz in Q = /0°° f3(z)dz egijska prispka ki ju lahko pripimo povšini. Prvi zradi odvisnosti d k sta pk k tranmu sidra st mo z Wi « f \ \(Sb - S0)(A*)_ \Kxl faaKA*)-^ - f f (22) preferenčna mer ja pa je lhko bodisi plana (za + L3 > 0) bodi homeotropna (za L2 + 3 < 0). Pri tem pmeni A* dožino kostneg reda L Podobno je moč uvideti, da ima dgi člen (Q) enako ktno odvost, ko bi jo v obravaani geometri imel člen s K13 [52,53], ki pa ga v azvoju proste nergij (5) nismo izrecno upoštevali, aj smo povsod zapisai le prve krajevne odvode Q. I dobsti lahk dj imo kvi ahljčno-uogino ečno konnto K* = f 2 3)(S ) = \{K22 - Kn)[l (|)] (23) ima drugačn izvor 3 in kot tka ne poča tež ri e iacijsa problema. K ^ 0 dobimo l v prime ko Sb ^ So in L2 + 3 ^ 0. e vedno pa prek sklopive S" in ' čle /3 rivde do estčne dr mae A(j ob povrini, e oba pdbo k v meu prave 13: Af -{13/2K)sm2cj)o [149]. Da bi podrobno preučili ojav notranjega sidanja in deformacij ob povši, zdaj oglejmo popolno numerio minimizacijo poste enegije v ončni plst ne matika. Steni zorca nj bosta vzporedni in naj se nahajata pri = ±|. V prost enegiji upoštvajmo pstorninske člene (5), prispevek zunanjega sidranja pa mo-delirajmo z izrazom (7) Mimia privede do Eler-Lrageih enačb in us trez bnih pj h j a do reš (j)( S (z), przanih n k 9. Opazimo lahk, d so variae S (z) dejansko omeje na tanko površinko plst (debline oki 10 nm) in da jih v plsti iste dbeline spremlja deformaca v odvis nost 4>{z). Predznak defomacje A se obnaa skladno z zornjo napoedjo in j odven od predakov +L3 ter So — Sb- b primerjai dejanskih površinkih vred st (z) in S(z) za (f>0 = 0.1(180°/w « 5.73° (preferenčna smer zunanjega sidranja), Sf, ~ 0.3747 in L? + L3 = +Li, 0, —Zq [primeri (a), (b) in (c)]. Razurejujoča (Sq = 0.35, /ew) in urejujoča površina (So = 0.4, desno). Debelina vzora je enaka d = 1 /jm, o = 0.13 x 106 J/m3 B = 1.6xl06 J/m3 C = 39xl0 J/m3 (podatki za 5CB [148]) T-T* = 04 in Li 10"11 N notranjega sidranja. Značilna razdalja vseh variacij je nekoliko odvisna tudi predznaka L2 + L3, saj se s tem spreminja tudi vloga stabilizrajočih prispevkov prosti energiji (5). Jakost celotnega (notranjega n zunanjega) idranja je mogoče ocen s poskusom zunanjem magnetnem polju, kjer molekule v zorcu poravnamo v smeri polja, hkrati pa opazujemo odstopanja od te smeri v bližini stene zaradi sidranja. Ta so potem merilo za njegovo jakost in s tem za ekstrapolacijsko dlžino K/W (slika 4). Iz tako določenih K/W ter z analitične oene (22) lahko skepamo, da A* «6 — 7 nm, kar je pri izbranih podatkih dejansko blizu korelacijske olžine L. Pri isth izbrah jakost skopve s poršno We v enačbi (7) ki še ajo realistične vrednosti za K/W (100 nm n več) so variacije profilih S (z) in (z) šbke: za we = Wed/Li = = 0.5, S& = 0.3747 in imamo zdaj še zvojn kot to, merjen gede na os x: n(z) = (sin (z) cos u (z), sin if>(z) si to (z), c j)(z)). V stti proste eergie edina sprememba nastopi v Frankovem nu f2 ', t, S) = \LX& { (l + Ž) ^ si2 i1 + #T 2} (24) kje smo veljal L + L3. Ž njegova zgradba saa napoveduje, da v zvoj- em profilu to (z) ne smemo pričakovati deforci kakrše povzroči prisevek dvisnosti (z), saj nikjr ne najdemo členov obke S00 Oglm si rešitev problema imiti močega sidranja a (±|) = 7r/2, ko n prčakovati nikakršnih deformcij v profilu 4(z). Hkrati zahtevajmo lo(—|) = 0 to(^) = 7r/4, kar privede do zvojne deformcije, ki se razširja po vsem vzorcu. Č predpostavim, da 5o = Sb in tako S ^ S (z), j profil zvojnega kota to (z) preprosta linearna funkcija . V nasprotnem primeru — če im variacijo AS = Sb — So 7^ 0 v plasti debelie A* (kot zgoraj za (j>) — dobim za A* (±f) = tt/2, S6 w 0.375, AS « 0 0, -0.025 (zgor- nja, srednja in spodnja črta), w(f) = 0, w(|) = 7r/4; L . Vrednosti a C, T — T* in L so enake tistim pri sliki 9 Na koncu je poebno povedti, d datna zvojna deformcija, ki se pojavi zaradi sprememb S ob površii ima istega izvora kot tista v kou pri planarnih deformcijah: namesto člena /3 je njen izvor navadni Frankov elastičn člen /2 Deformacijo s tem povzročajo spremembe elastičnih konstant, pa še to le v primeru ko je zvojna deformacija v vzorcu že prisotna. Razširjeni povzetek kroskopski opis tanke plasti Zdaj želimo obravnvati ureja v plasti nematika še n molekulki ravni. Poebno pozornost bomo posvetili pojavu notrega sdnja in ga modelirali v okviru modela s šestkotniško mrežo in anizotropno interkcio med inucirnimi dipoli (12) [17] Še zlasti se bomo zanimli za temperaturno odvisnost ekstrapolcijske dolžine K/W', ki so j že poskušali določiti za zunnje side [152]. Eksperimentaln opažanja mreč kažejo, da jakost sranja W s temperaturo pada, vendar ni vi splošneg pravila z ovinost W(S) [46,112,143,153,154]. Tako lahko poskui kažeo celo n W (x SA [46], medtem ko nekateri modelski sistemi napovedujejo le W oc S [157,158] V prvem primeru ob bližanju faznemu predu z upoštevem K on S2 zaznamo povečanje K/W, v drugem pa zmanjšanje. Se pred samo meritvijo temperaturne odvinosti K/W bo potrebno preučii sam fzni preh med nematsko in izotropno zo (prehod NI). Simulcie, predstavlene v tem razelku, ledile meodi Mone Carlo (gle prejšnje poglavje). Rzdelitev stkotniške mreže na tri podmreže je omogočila vek-torizcijo in s tem popešev računalniškega algoritma. Prehod NI je bil preučen v vzorcu velikosti 30 x 30 x 30 delcev s periodičnimi robnimi pogoji v vseh treh smere, K/W pa je bila merjen v plasti velikosti 48 x 48 x 46 delcev s periodičnimi robnimi pogoi v mereh x in y, prosto površino pri z = 0 in dno steno pri z = d. Za uravnovešan je bilo izvedenih po 2 x 105 MC korakov (v enem koraku pride na vrsto za morebino reorientacio vsak izmed delcev), še nadaljnjih 105 a za izračun povprečij Za ačetno konfigurcijo je bila izbrana tista z naklčnimi orientacijami delcev u,, lahko pa tudi že uravnovešen konfigurcija pri emperaturi ne predaleč d obravnavane. Poleg notraje energie predstavlja pomembno opazljivko dvoi-menzijska ureditvena matrik, ki jo po analogii z izrazom (2) lahko vpeljemo ko rpo 1NI rpo (a) 0.05 1.375 ± 0.025 1.3 15.5 ±1 1.2 13 ±1 1.1 13 ±1 0.1 1.225 ± 0.025 1.175 4±2 1.1 5±1 1.0 5±1 0.2 0.950 ± 0.025 0.9 -1±0.5 0.8 +1±0.5 0.7 +1±0.5 0.3 0.675 ± 0.025 0.6 -4± 1 0.5 -4± 1 0.4 -4± 1 Tabela 1 Tempeaturna odvisnost ekstrapolacke doline notanga sidranja L (merjena v enotah mreže a) a ralične v in pripadajoče reducirane temperature fanega prehoda T Razširjeni povzetek XI q = 2(u, ® Ui)i — I in iz nje po diagonalizaciji določimo direktor n ter dvodimen-zijski skalami parameter urejenosti s = (2(iij • n)2 — l)j. Direktor lahko ponovno parametriziramo z n n(z) = (sm (j)(z), 0, cos (f>(z)) Temperaturo faznega prehoda T^7 je mogoče določiti iz temperaturnih odvisnosti notranje energije. Rezultati v reducirani skali T° = ksT/e' so za različne vrednosti parametra anizotropije v zbrani v tabeli 1. Opazimo, da T^7 z rastočim v pada, kar je v skladu z upadanjem Frankove elastične konstante pri povečevanju v [129]. Meritev ekstrapolacijske dolžine je bila izvedena v plasti, podobni hibridni celici a prosti površini [z = 0) deluje notranje sidranje, ki je za v < 0.3 homeotropno ri z = d imamo močno planarno sidranje, ki v vzorcu zadostne debeline privede do elastične deformacije [114] To lahko vidimo tudi iz odvisnosti (f)(z), prikazanih na sliki 11 za v 0.05, razen v primeru, ko se je zaradi bližine prehoda v izotropno fazo nematik v celici stalil — slednje je razvidno tudi iz profilov s(z). Z ekstrapolacijo direktorskega profila, kot je nakazana na sliki 4, dobimo dolžino L, ki je ekvivalentna K/W le v primeru dovolj šibkega sidranja, ko L ^ L [L označuje ko-relacijsko dolžino (4)]. Iz tabele 1 je razvidno, daje razen za(/ = 0.05 L mikroskopska dolžina (reda nekaj a) in da se veča z manjšanjem parametra anizotropije v Poleg tega vidimo, da je za v > 0.2 sidranje tako močno, da postane ekstrapolirana l zaradi zanemarjenih sprememb stopnje urejenosti s negativna in zato ne predstavlja K/W [143]. Zdi se tudi, da je temperaturna odvisnost l le šibka in slabo izražena, razen za v = 0.05, kjer L ob bližanju prehodu v izotropno fazo blago naraste. V grobem torej kaže, da imamo v pričujočem sistemu W oc S6 z 8 ~ 2. 0 10 20 30 40 . 50 0 10 20 30 40 50 z/a z/a Slika 11 Temperaturni odvisnosti (z) in s (z) za v = 0.05. Iz ektrapolacije profilov 4>{z) proti levi lahko določimo t notranjega sidranja. Pri temperaturah blizu faznega prehoda se nematik v vzorcu stali, zmanjša stopnjo urejenosti s in se s tem izogne upogibni deformaciji. Hkrati v bližini trdne stene vedno opazimo povišanje s Kaže, da je ujemanje L z eksperimentalnimi vrednostmi (100 nm in več) mogoče doseči le za dovolj majhne v. V takih primerih medmolekulski potencial (12) vzpodbuja pretežno vzporedno ureditev molekul, kot jo, na primer, tudi sterične odbojne interakcije med podolgovatimi delci, ki jih nismo nikjer izrecno upoštevali. Ostali neupoštevani pojavi zajemajo še dipolne in kvadrupolne interakcije, prisotnost nečistoč [161], krajevne spremembe v gostoti nematika [6667], elektrostatske in- XX Razširjeni povzetek terakcije zaradi selektivne adsorpcije ionov [160] in grobost trdne površine (zadnja pojava le v primeru sidranja na trdni podlagi) Nematske kapljice V tem poglavju se bomo ukvarjali z nematskim urejanjem v nekoliko zapletenejši geometriji: v podmikronskih kapljicah. Najdemo jih v sistemih PDLC, kjer so ujete v polimerno matriko. Obdelali bomo primer s homeotropnim sidranjem — radialno kapljico — ter takšnega s planarnim — bipolarno (slika 12). Številne fenomenološke, simulacijske in eksperimentalne študije so v preteklosti privedle do precej dobrega poznavanja teh sistemov [5,91]. Tukaj se bomo lotili še ne v celoti obdelanega vprašanja razlage rezultatov simulacij Monte Carlo skozi spektre 2H NMR, vključno z dinamiko molekul in urejujočimi učinki zunanjih polj Slka 12 Površinska plast delcev "duhov", ki pipadaj polimeni matiki in dločajo robne pogoje: (a) radialna in (b) bipolarna kapljica (pučica označuje iplarn simetijsko os). Ueditev molekul v vsaki imed kapljic je pikaana tudi shematičn imulacije urejanja v kapljicah (vrste Monte Carlo) so bile izvedene v okviru LebwohlLasherjevega modela (glej poglavje o simulacijah). Kapljice s polmerom R = 12a in zahtevanimi robnimi pogoji so bile izžagane iz kubične mreže, kot kaže slika 12. Jakost interakcije e med nematskimi delci — pri izbranem R jih je 5832 — je bila enaka jakosti interakcij delcev nematika z delci, ki pripadajo polimerni matriki (teh je skupaj 1352). Za začetek vsake simulacije je bila privzeta konfiguracija, ki ustreza idealni ureditvi v skladu z izbranimi robnimi pogoji, ali pa že uravnovešena konfiguracija pri bližnji temperaturi, čemur je sledila uporaba Metropolisovega postopka z BarkerWattsovo tehniko. Sistem delcev je bil urav-novešan skozi vsaj 5 x 104 korakov, potem pa je bilo zajetih 1024 konfiguracij, iz katerih so bila izračunana vsa povprečja in spektri 2H NMR. V nadaljevanju bodo Razširjeni povzetek XXI vse temperature izražene v brezdimenzijsk skali, doeni s T* = ksT/e. Smulacij v nematski fazi s ble tako izvedene pr T* = 0.8 v izotropni pa pri T* = 1.2. Za izračun spektrov 2H NMR je bil uporabljen prstp s časvno odvisno spin-sko hamiltonko (glej poglavje o eksperimentalnih opaljivkah), i omogoa pravilno upoštevane proces kot sta translacijska difzija in fluktuacij dogih osi mlekul Slika 13 Simuliram spektri 2H NMR za eno bipolarno kapljico v nematski faz pri T* = 0.8 in za različne jakosti zunanjega polja (oc y/rj); brez difuije (a), s hitro difuzijo (b). Kvarupolni racep narašča r\. ačetek si ogle kapljice v odtnsti zunanjih olj {r = 0), najpre bpo larno. Od zunanjih lj imejm edio pe NMR spektrometra potrebno za mertev spektrov, vendar na bo tak šibko, da ne vpliva na mlekulsk ureditev v kapljica [R J§Dto meni fektivno difuzijsko nstanto, t pa načiln čas spektroskopij NMR) — orej doolj velikih kapljic. Edina gibanja s katerimi imamo zda opravka v vzou, so fluktuacije dolgih o molekul u, ki doočajo S in n. Izkaže se da tovrstne fluktuacije sicer nenaravna dinamka metode Mnte Carlo opiše dovol dobro. Ureditev v bipolarni kapljici je takšna, da je vel del molekul usmerjen priblžno v smeri biplarne i Ce usmerimo magnetno plje spektrometra vzdolž te smeri (tega se bo držali do nadaljnjg), naletim v spektru na dobro izražena vrhova pr približo ujz ± S5uq (slka 13. Iz njune lege je tako mogoče prblno določiti S ~ 0.73 ± 0.01. Do ocene za lah pridem tudi nepsreno iz smu-lacijskih podatko z diagonalizacijo kalnih ureditvenih matrk Q ), i jih tem ovprečimo vsej kaplji: S ~ 0.76 ±0.04. Priomniti je treba še, d mam v realnem vzo mnogo kapljicami prazdelitev p smere bolarnih o. Prmer ko XX1 Razširjeni povzetek ga obravnavamo tukaj, srečamo le, če že sam proces nastajanja kapljic (na prmer fazna separacija [28]) poteka v zunanjem polju, k usmeri vse bipolarne osi v isto smer [40]. Poleg tega je bilo a glajenje spektrov potrebno izvesti konvolucijo jedrom šrine 0.04&jq. Kadar velja R z ± \5uq n rameni, k segata do ujz ± SSloq. CO 0.84 17=1 •......•......».....¦*-.....*¦.....•......•¦.....* ~~A 0.8 0.8 . J$ 0.7 0.76 Ju ^ 0.6 0.5 0.72 77=0 Razširjeni povzetek xxiii Simulacije v primeru brez translacijske difuzije dejansko dajo opisanemu podoben spekter (slika 15), ki spet omogoča določitev S iz položaja vrhov in ramen (S ~ 0.72 ± 0.02), pa tudi iz podatkov simulacije (S ~ 0.73 ± 0.10). Kadar je difuzija hitra, dobimo v spektru zaradi izotropnosti orientacijske porazdelitve en sam vrh pri uq « 0, iz podatkov Monte Carlo pa (coq) « 0.03 5uq. Kot kaže, je v obeh limitah (z in brez difuzije) s primerjavo spektrov vedno mogoče razlikovati med radialno in bipolarno kapljico. Razlika v obnašanju spektrov je še posebej nazorna, če vzorec obračamo v polju spektrometra: za razliko od spektrov bipolarne kapljice se spektri radialne pri tem ne spreminjajo. Slika 16 prikazuje radialno odvisnost parametrov S in (P2JB- Za r\ = opazimo, da je v središču radialne kapljice — kjer pričakujemo topološki defekt — stopnja ureditve S dejansko znatno nižja kot v zunanjih delih kapljice, kjer je radialno -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 {0i-(O^ld0iQ {0)-(O^ld0iQ Slika 15 Simulirani spektri MR radialne kapljice v nematski fazi za T* = 0.8 in različne rj; (a) brez difuzije, (b) s hitro difuzijo. S povečevanjem rj pride do prehoda med radialno strukturo in strukturo z večino molekul usmerjenih vzdolž zunanjega polja. xxiv Razširjeni povzetek urejanje jasno izraženo. Namignili smo že, da je difzija pomembna zlasti v majhnh kapljah. Če vzamemo 5ujq f 2tt x 40 kHz, D ps 4 x 10-11 m2/s n S m 0.8 (v poprečju), lahko ocenimo, da so spektri kap s polmeri pod R ~ 60 nm znatno difuzijsko izpovprečeni. Po drugi strani simucije sprektrov kažejo, da pri isth podatkh v kapicah polmerov nad R ~ 330 nm inki ifuzije niso več zaznavn. Zaradi omejitv z velikostjo vzorca (R = 12a n a ;$ 5 nm) so predstavljenih simuacijah realističi predsem primeri z R ^ 60 nm, ko smo že v limti tre difuzije. o daj smo predpostavljali, da je difuzija prostorsko homogena. Hkrat ekspermenti kažejo, da je lahko v bližini ograjujočih sten znatno upočasnjena, to pa je posebej opano v močno orajenh sstemih [96,166]. Tedaj so izračunani spektr sestavljeni iz superpozicije difuzijsko izpoprečenega prispevka notranjosti kapljic prspeka poršinske pasti, kjer difuzija ne igra pomembne vloge [97] Če pa s poeg tega ogledamo še vzorec mnogi polarnmi kapljami, katerih smetrijske osi so usmerjene po prostoru naključno, ponovno naletimo na spekter Pakeo vrste [97] Vzore je namreč makroskopsko izotropen, čeprav posamezne kaplje nso. Tud tra difuzija Pakeov spekter ohrani le da je ta nekoliko ožji kot primeru bre ifuzije. ,0 4 8 12 shell 0 4 8 12 shell 0 0.5 1 rlR o 0.5 1 rlR Slika 16 Parametri urejenosti za radialno kapljico pri T* = 0.8: (a) S in (b) (P2)b kot funkcija razdalje od središča r. Krivulje so narisane za r\ = 1, r\ = 05, t] = 0.2, r/ = 0.05, rj = 0.03, r\ = 002 in rj = 0 (od zgoraj navzdol) Jedro defekta se z višanjem poljske jakosti pretvori v strukturo, urejeno v smeri polja Prljumo sedaj unanje polje, k naj bo dovolj močno, da dodatno uredi ne-matik kapljicah (R >> Lm). V simuacji skloptev nematskih delcev z zunanjim poljem opisuje izraz (11), ki za 77 > 0 privede do urejanja delcev v smeri polja f primeru bpoarnih kapljic je prvi učinek zunanjega polja reorientacija bipoarnh osi, ki se obrnejo v smer polja [2590]. Tukaj bomo obravnavali primer, ko je že vzporeden bipoarnm osem. Tako spektri (slika 13) kot profili parametrov urejenosti (slika 14) kažejo, da zunanje polje povzroči dodatno urejanje moeku v smeri polja, hkrati pa tudi poviša stopnjo urejenost S [117]. Oboje se odraž povečanju kvadrupolnega razcepa, kar opazimo ne glede na stopnjo difuzijske izpoprečenost spektrov. Zavedati pa se moramo, da so obravnavana zunanja polja izjemno močna, saj so tudi mulirane kapljice relativno majhne. Tako imamo 1 7/=l 0.8 i1 ! 11!;::; • • ^0.6 ^ 0.4 • * * 0.2 Vo ¦ . ¦ : : * • 0 * •------.:.....i.J Razširjeni povzetek magnetnem primeru pri rj = 0.2 za značilen nematik (e = 0.023 eV, \aS ~ 10~6 in Vo ~ 100 nm3) B ps 150 T, kar je precej nad trenutnimi eksperimentalnimi zmožnostmi. Na podobne težave naletimo tudi pri uporabi električnih polj. V primeru radialnih kaplji so spremembe, ki jih povzroči zunanje polje, bolj dramatične: defektno strukturo v središču kaplje polje razbije in delce uredi v smeri f. Urejeno jedro obkroža defektna linija moči \1 ki se z višanjem rj razteza, medtem ko je v plasteh ob površini za ne prevelik r\ radialno urejanje še vedno prisotno (slika 16). Takšno obnašanje lahko razberemo tudi iz izračunanih spektrov (slika 15), ki jih odsotnosti difuzije uteženo sestavljata prispevek urejene sredie in Pakeo zorec. V primeru hitre difuzije se spekter z enim rhom pri ujq 0 v dovolj močnem polju prelev v dvovrhega. e dvignemo temperaturo do T* = 1.2 nad temperaturo prehoda izotropno fazo, opazimo že odsotnosti zunanjega polja (77 = 0) ob steni kapljie urejeno paranematsko fazo. Ce poleg tega vključimo še močno zunanje polje, se nematsko urejanje razširi po vsej kapljici [118], to pa ponovno privede do znatnega kvadrupol-nega razcepa. Ta pojav je lahko opazen zlasti zato, ker je — kot že rečeno —jakost zunanjega polja izjemno visoka. Kljub temu, da predstavljena metodologija za izračun spektro dobro opiše dinamiko molekul in vplive zunanjih polj, pa sedanji rezultati za bolj natančno kvan titatino primerjao z eksperimentalnimi še niso zreli: zaradi relatino majhnega števila delcv v simulaciji so spektri precej zašumljeni, konvolucije pa jih sicer zgladijo, a s tem tudi nekoliko prizadenejo njihovo obliko. Nematiki z dispergiranimi polimernimi mrežam Zadnje poglavje bo obravnvalo simulaije nematskega urejanja v sistemih z dispergiranimi polimernimi mrežami. Te sestaljajo tanka (nanometrska) polimerna vlakna ali nekoliko debelejši snopiči vlaken. Topografija mrež je lahko precej nepravilna, zato predstavljajo najzapletenejšo vrsto v tem delu obravnavanih ogradite [15,101] Slika 17 Shematski prikaz polimerne mreže (desno) in simulacijska škatla z valjastim vlaknom in eno izmed valjastih plasti (levo). Označena je tudi smer zunanjega polja E. Razširjeni povzetek Najpomembnejša značilnost mrež j a, da ahko že pr niz konceaciji u dijo okoliški tekoči kristal [5,26,39] so kot take zanimive za uporabo v različnih optičh eementih. V nadaljevanj si bomo ogledali učinke grobosti površine na uredtveno spoobnost mreže [102] opološke defekte [103], pojav preklapljanja orienacije molekul v zunanjem poju pri različnih mrežnih topografijah n parane matsko urejanje nad prehodom v izotropno fazo. Navedene pojave bomo spremljali tu koz opazljivke, kot so 2H NMR, elektna kapacitivnost n optična prepst nost Ko prvi korak k modeliranju polimerne mreže si ogejmo valjasto vlakno v smu lacijski škatli s periodinimi robnim pogoji, ko kaže lika 17. Na a način dobmo pravilno mrežo ranih in zporednih vlaken. Simulacije o bile — podobno kot v prejšnjem poglav za kapjice — izpeljane v okviru modela Lebwohla in Lasherja. Površino vlakna tudi tukaj modeliramo z zamrznjenimi delci "duhovi" robni pogoji pa zajemajo planarno sdranje (vzdolž smeri vlaken, z), homeotropno (pravokotno na lokalno površino) in delno ali v celot neurejeno (groba površna). Stopnjo grobost površine lahko kvantittivno podamo s parametrom urejenosti (P2)g, ki predstavlja po absoluni vrednost največjo lastno vrednost uredvene make Q, izračnane za sne duhoe". (Pzjg = 1 tako streza idealnemu planarnemu urejanju, (/2)9 = —0.5 idealnemu homeotropnemu, (^2)9 ~ 0 pa popolnoma naključni orienacijsk porazdelivi (merjeno akič ede na z). Vmesne vrednosti ustrezajo deno rejen grobi površni Pri izbranem polmeru vlakna R 5a i likost katle 30 x 0 x 30 dev imamo v sistemu 24600 nematskih delcev i 840 površnh delcev "duhov" Simulacije so bile zagnane iz popolnoma naključnih konfigracij in uravnovešane saj 6 x 10 korakov na enak način kot v primeru PDLC kapic. Nato je bilo 6.6 x 10 zaporednih konfiguracij uporabljenh za izračn povrečij. Ta zajemajo komponente reditvene matrike Q n parameer (P22) = \ 3{(u» • z) 1] k opiuje stopnjo redve gede 0.8 0.6 0.4 0.2 <\ (?&*) (a) 6 8 10 12 14 16 18 20 r/a 0.6 0.4 0.2 1 0>) ,<^2>g=l v^ 6 8 10 12 14 16 18 20 r/a Slika 18 Pravilna mrea raih in vzporednih vlaken debeline R = 5o odvisnost {P2) od r (razdalje do središča simulacijske škatle); (a) nematska (T* = 1.0) in (b) izotropna (T* = 1.2) faza. Krivulje zgoraj navol: (P2)g ~ 10 (idealno planarno urejanje), 075, 050, 025 in 0 (naključna ri entacijska poradelitev) Razširjeni povzetek XXV1 na smer vlaken z. Slika 18 prikazuje radalne odvisnost parametra urejenosti (P|) za različne stopnje grobosti povrne. V nematski faz pri T* = 1.0 za (p2)g = 1 je direktor usmerjen vzdolž z, vrednost (P2) pa sovpada s skalarnm parametrom urejenosti, S (pr zbrani T* = 1.0 enak S ~ 0.6). Vidimo lahko, da je S ob vlaknu rahlo povišan. V primeru denega nereda {P2)g < 1 stopnja uredive ob vlaknu upade, vendar je vlakno še vedno sposobno urediti nematk zdolž z. Zd se, da to zmožnost izgubi šel za (P)g ~ 0, saj direktor takrat več i vzporeden z. Pri višji temperaturi T* = 1.2 (nad Ttfj) tudi v sstemu vlaken opazmo paranematsko urejenje, ki je tem ibkejše, čim bolj grobo je vlakno. Vse spremembe v stopnji urejenost ob vlaknu se dogajajo na razdali korelacijske dolžine L (4), ki je reda vlikost nekaj a. Zgornje ugotoviv potrjujejo tu spektri 2H NMR, k ahko izraunamo po metodoogiji, razvi za nematske kapice. Do podobnh izsledkov ede grobosti vlakna n sposobnosti urejenja pridemo tud za {P-iig < 0, homeotropno sidranje. V dealno homeotropnem primeru postane nematska ureditev še posebej zanimiva: v bližii vlakna pride v sistemu do tvorbe topoloških defektov (pravzaprav linij, k teejo vzdolž vlakna). Lvi del slike 19 prikazuje direktorsko polje v na vlakno pravokotni ravnin, skupaj s prpadajoč stopnjo urejenost S (oboje dobmo z diagonalizacijo okanih ureditvenh matr 30 i x/a 30 y/a 10 15 20 25 30 0.6 ft, co" 0.2 :i.......jvC S -0.2 v P 0.8 0.4 O) O) 0 ca -0.4 ~~...-...*::....-...^ A-----^fe*.-««, *«*-'*» 10 15 20 25 30 yla Slika 19 Nematsko urejanje v ob ravnem homeotropnem vlaknu. Levo: prerez direktorskega polja n(x, y) (n predstavljen s tokovnicami). Stopnja osenčenos ti predstavlja skalami parameter urejenosti S(x,y) (temna območja ustrezajo nizkemu S). Par defektov moči — ^ se pojavi blizu diagonale prikazanega pre reza. Desno: lastne vrednosti ureditvene matrike Q_ (Qi, Qi in Q3), narisane skozi levega izmed obeh defektov vzdolž osi y. Zgornji del slike prikazuje pri padajoča poteka parametra urejenosti S in stopnje dvoosnosti P. Nesimetrijo glede na jedro defekta (to leži pri y/a = 20) povzroča prisotnost vlakna. Ostali parametri: T* = 10 in R/a = 5. XXV1 Razširjeni povzetek ). Dei dl slike 19 kaže obnašnje lastnih vrednosti Q, ko pčkam efekt Vidimo, da je daleč jedra defekta urejnje enoosno. Ko se bližamo jedru začne stopnja urejenosti S padati hkrai pa araste dvoosnost P. samm jedru je urditev ponovno enoosna vendar S < 0, pri čemer je dirktor usmrjen vzdol smeri vak, z. To obnašnje je v kladu s fenomenološkimi npovedmi [04105] d drugim opazimo tudi, da do tvobe defektov vselej pride blizu ene od diagon imulacijsk škatle. To lahko pripišemo st odboju med defetoma, deloma tudi koletivnim fluktuacijam v kkast imuijski škatli [168. Topološko je obravnavane sistemu dopustna tudi en ama defektna nija mči —1, vendar stlna [24]. Oglejmo si še pojav ekapljnja orientcije molekul z zunanjim polje. Sidranje aj bo idealno planarno vzdolž vlaken (osi z), zunje polje E p aj bo merjeno ravkotno nanje (vzdolž osi y) — slika 17. Obnašnje pravilnega sistema vzporednih vaken v zuanje polju je podobno obnašanju nematsk elice (lej poglavje o fenoenološke op nematika): pri višanju poljske jakosti do Freederickga prehoda opazimo heno strukturo (h) z n||z, nad njim deformirano strukuro (d), nad turacijskim phodom pa zaičeno strukturo (s) z molekuami usmrje imi v smri polja n||E||y. Ke sta kriiči poljski jakosti za oba pehoda odvisn d jakost idranja, si boo oledali prim z različnim w = eg/e. Tuka eg pomen jakos ntrakije d molekuami nemaika in polimnega vak terak d molekulami nematika amimi [enačb (1] akos (b) 3 s .* 2 O 1 d ^^ 0 -1 h -1 0 1 lognK/Wd Slika 20 Pravilna mreža vzporednih vlaken in preklapljanje v zunanjem pou: (a) strukturni fani diagram za T* = 1.0 in R/a = 5; 770 pomeni točko Freederickszovega rehoda pri w = 5. (b) Primerljiv fazni diagram a navadn last nematika z [115]); h je sramerna jaksti polja. imue so bile izvdene pri T 1.0 v mu 30 x 30 x 30 dlce s pol akna R 5a. Pri danem je blo pri vaki vrednosti r\ potebnih 1.2 x 105 korakov za uravnovšanje in nadaljnjih 1.2 x 105 za izračn opazljivk. Simulacije so potekale tako, da je vrnost r\ naraščala postopoma d r\ = 0 do navišje vrnost (onstra sauracijskega pda), nto pa postopoma padala azaj k -n = , kar aj b golo razkritje orebitnih pojavov histere. Pričakovana strukturna prhda ahk išmo z opazovanjem arametra Pf |[3((uj • )2) če te porečje ( sega vzor, je Pf obtljiv Freedrik če p Razširjeni povzetek XXI le po tanki plasti tk ob površini vlakna, pa na saturacijskega. Še bolj zanesljivo j opazovanje pripadajočega efektvnega odmika ay, saj fuktuacije P% ob strukturnih preodili opazno narastejo. S slike 20 lahko raberemo območja stabilnosti treh napovedanih struktur za primeravo pa je prikazan enakovreden diagram a plast nematika [115]. Kvalitativno ujemane med obema je dobro. V šibkem polju vselej opazimo homogeno strukturo h, v elo močnem pa zasičeno strukturo s. Med njuni območji stabilnost se vriva območje deformirane strukture d, ki postane pri šibkem sidranju (majhn w) zelo ozka. Kljub temu kaže, da neposrednega prehoda h - s ni, s tem pa tud ne trojne točke v diagramu. Poleg tega n opaziti histeree, iz česar sklepamo, da sta oba prehoda zvezna. Kot kažejo slike elektronske mkroskopije [26,39], je topografija polimernih mrež precej bolj nepravilna, kot smo jo bili zmoži opisati s preprostim modelom s slike 17 (imenujmo ga "vzorec A"). Vlakna so lahko zavita in naključno razporejena po prostoru, vendar s še vedno dovolj dobro določeno povprečno smerjo. Prv korak k takšni topografiji mreže predstavljjo ravna in vzporedna vlakna, ki so razpore jena naključno po nekoliko povečani simulacijski škatli ("vzorec B"). Tudi v takem primeru pride v zunanjem polju do Freederickszovega prehoda, vendar pri nižji poljski jakosti kot pri pravilni mreži z enako debelino in koncentracijo vlaken. Vzrok za tak premik je najti v porazdelitvi razdalj med vlakni, saj do Freedericksovega prehoda pride, ko postane koherenčna dolžina zunanjega polja Lm primerljiva z naj daljšo izmed njih. Zaradi tega prihaja do preklaplanja molekulskih orentacij v vzorcu postopoma, in sicer najkasnee v področjih, ker so vlakna raporeena nadpovprečno gosto. Slika 21 Polimerna mreža z neravnimi vlakni (vzorec C): delci "duhovi" predstavljajo ogo mrežo. Sidranje na površini vlaken je planarno. XXX Razširjeni povzetek Najzapletenejši od vorcev s pimimi mrežami — "vzorc C" — vključuje nevna lana in je prikazan na iki 21. Kot pri vorcu B so bile simulacije ene v vzorcu 50 x 50 x 50 delcev osmimi vlakni pa R = 3a (usmerjenimi v povpečju vdolž z) ter s planaim sidranjem jakosti w 1 vzdolž lokalne tangentne smeri vlaken. Zdaj si bomo ponovno ogleali pjav klapljanja zunanje plju, pri te pa primerjali obnašanje tgafsko različih vorcv A, B C s ribžno enakimi konentrcijami pimnih vlaken (9%). V vzorcih B in C je bilo naejenih 8 104 simulacijskih korkov za uovešanje in 6.6 x 04 za računanje povprečij, v manjšm vzorc vrste A (velikosti 18 x 18 18 delce) pa je bi avovešanje skrajšan na 6 x 10 korkov. Rezultate bomo dstavii v obliki ibranih ksperimentalnih opazljivk. 20 16 tu 12 : /s -6 log 77 log 77 Slika 22 Preklapljanje orientacije molekul v zunanjem polju: meritve ka-pacitivnosti (levo) in intenzitete skozi vzorec prepuščene svetlobe (desno, Iq označuje intenziteto vpadne svetlobe); vzorec A (majhne pike), vzorec B (večje pike) in vzorec C (največje pike). V desni sliki puščice označujejo kritične vrednosti 77, kot jih odberemo z leve. Začimo z električno kapacitivnostjo Cy opazovano vdolž smeri zunanjega plja (E||y) glej pglavje eksperimentalnih opazljivkh. lik 22 prikazuje odvisost Cy(\ogrj). V rimeru idealne ditve delv bi dobili Cy/eo = e± za ureditev vzdolž smeri vlaken (z) in Cy/eo e\\ za uditev vdolž zunanjega plja, iz čes ledi a lahko i edenja količine Cy izluščimo informcijo o orientaciji molekul v orcu. Tukaj pomenita e± i e|| latni vrdosti moleulskega dielektričnega ten- orja (v priujoči simulaciji e± = 6.1 i || 9.8) S ike 22 lahko razeremo leg Freericksovega phoda ta zajam veliko tevio delv — za vsakega i reh vorcev, medte ko satuaciskega phoda Cy ne mor azkriti. Freerickszov pehod na prikazanih grafh sovpada točko, kjer Cy občutn naaste [39]. Ocenjene kritične vrnosi so tja = 0.022 ±0.01 za vzorec A, t\b = 0.013±0.01 za vzorc r\c = 0.01 ± 0.01 za orc C. Kot že omenjen, velja t\b < Va zadi orazel efeivih razdalj md vlakni. Naalje immo rje < Vb, kar ivira iz dejstva, d v vorcu C delci pri nikih tj niso smrjeni natanko vzdolž z in deluje magnetn navor nanje že za poljuno majen r. Zdi tega v vzorcu C przaprav ne smemo ovorii o osem Freedrickszovem prehodu. Prlapljanje je najpnejš v vzorcu B (krivulja Cy najpožnejša), kjer se neatik v delih vorca z visoko osto vlaken Razširjeni povzetek XXXI uredi v smeri polja š ri zlo visokih rj. Povedati j reba, aj udi tkaj zunj polje zelo močno (glej oceno v prejšnjem poglavju) n da gre naaščanje Cy pri najvišjih rj na račun povečevanja stope ureditve S [117] ne pa račun reoentacij molekul Vzorec lahko postavimo tudi med reižaa polarizato nalizato i merimo inenzitto prepuščene svetlobe (/, kot je bilo nakazano v poglavju o ksperimen talnih opazljivkah. Inenzitta / je sorazmerna sin2(A$/2) (18), kjer A$ pomen azliko med fazama rednega in izrednega žarka, ki se nabere, ko prepotujeta vzorec. Če v izbrani geomeiji svetimo vzdolž smeri zunanjega polja, se količia A$ spre eni s kočne vrednosti pi rj = 0 do zlo majhne vrednosti za velik r), kar vod do nihajočga obnašanja v odvisnosti I(rj) takoj, ko v vzorcu pride do preklapljanja moleklskih orientacij [116]. Tako obnašanje zares opazimo tudi na sliki 22, ki nudi enake zakljčke glede kritičnih poljskih jakosti in hitrosti preklapljanja kot opazovae Cy. Kivlje so il simulirane za svetlobo valovne dolžine 632 nm, efektivn debelino vzoca 10 //m in loma količnika nematika 15270 za redno polaizacijo ter 17445 za izredno (največja moža vrednost). Omenimo l še to, da tudi izračunani spekt 2H NMR bro ujemajo z rezltati obeh predstavljenih ksperimentov. w=l.O w=0.5 w=0.l -1 -0.5 0 0.5 1 {(o-oi^ld(oQ 1 k -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 r*=i.i25 r*=i.i3 r*=i.i35 r*=i.i4 T*= 1.145 r=i.i5 r*=i.i55 r=i.i6 r*=i.i65 r*=i.i7 T*= 1.175 r*=i.i8 r=i.i85 r*=i.i9 r=i.i95 r*=i.2 r*= 1.205 r*=i.2i r*=i.2i5 Slika 23 Paanematsko urjanje v vzoru C: difuzijsko izpovprni spektri 2H NMR v odvisnosti od T* za različne jakosti sidranja: w = 0.1 (levo) w = 0.5 (sredina) in w = 0.1 (desno) XXX1 Razširjeni povzetek Zadnji od predsavljih pojov bo paranematsko urane nad tempeat prehoda v izotrpno fzo, i ga pooča površina. Opazili smo ga že v vzorcu A [slka 18 (b)] u pa si omo ogledali simulacije v vzorcu C (slka 21). Parametri simulacije so enak kot v primer preklapljanja le daje bila izvedena a planarn sidranje z različimi vrednostmi w, brez zunanjega plja (rj = 0) in za različne tem perature nad Ttfj = 1.1232. Kot smo že opazili na sli 18 (b), imam blizu vlaken nezanemarljivo stpnj nematske uredtve ki pade prti nič na korelacijski razdalj L ~ 5a. Ta red lahko zaznamo z optičnimi metodam (med prekrižanima polariz trjem in analizatrjem kakr zgoraj) [26] ali pa z 2H NMR (slika 23). Spektr ili raunani v limi hitre difuzije, polje spektrometra pa je bilo usmerjeno vzdo z. Dvvrhi spektri so posledica urejanja b vlaknih, kvadrlni razcep pa je tem večji, čim višja je stopnja uredtve. Vidimo, da razep pada z išanjem temperature i nižanjem jakosti sidranja saj se s tem zmanjšuje tudi stopnja urejenosti. D bnih opažanj prdem pri ilaijah intenzitete prepušene svetlbe. Zaključki V zaključku prelejmo glavne rezultate pričjočega dela skpaj s še odprti vpra šanji in nekaterimi smernicami za prihodnost. V delu smo se dotaknili izbranih prblemov mdelranja ograjenih tekočekrstalnih sistemov, v ta namen pa pora bii fenomenološki Landau-de Genneso i še večjem merilu, ikrskpsk mulacijsk metodo Monte Carlo zzvani z eksperimenti, k kažejo na znatne elastične deformacije ob površini ne matika, smo najprej bdelali urejanje ob stenah nematske plasti. V okviu Landau-de Gennesovea prstopa smo pokazali da spreminjanje stpnje urejenosti ob površini lahko vodi do pojava lkaliziranih elastičih deformacij, hkrati pa tudi do notranjega prspevka k površinskemu sidranju (oboje le takrat, če Frankove elastične onstante no vse enake). Omenjene lkalizirane deformaije smo opazili le pri ome jitvi na ravninske deformaije ne pa tudi pr neravninskih zvojnih, in ne izvirajo iz sporneg člena s pahljačo-uogibn elastično knstanto. Če sklopitev nematika ograjujočo površio prlagoimo tako, da ekstrapolacijska dlžina zunanje in no ranjega sdranja dosež znalne eksperimentalne vrednsti okoli 100 nm, postanej deformaije b povr tako šibke da nji ne mrem pojasni ekspermentaln paenih gornjo fenmenoško štu smo nato polnili z mikroskopsko, temelječ na metodi Monte Carlo in model šestkotnišk mrežo ter prstorsko anizotrpnim interakcijami med induciranimi d. Glavn namen te študije je bil pdrobneje raziskat notranje sdranje, kje pri dolj nizkih stpnjah interakcijske anizotropije homeotropno in tudi takrat še vedno preej močo: pripadajoče ekstraplacijske dolžine i so mikroskopske — reda nekaj mlekulskih rasežnosti. Poleg tega a razko od eksperimentv v realstičih sistemih i ne kaže znatne temperaturne dvi osti ko se bližam prehodu v izotrpno fzo. avedena pažanja je moč pripisat preproststi dela nadaljnji koraki bi lahko zajemali ostitev mrežne prbližka kar bi oogoč vključtev spremnjanja gostte blizu ovrine ter Razširjeni povzetek XXX1 grobih površin. Od obojega je pričakovati znižanje jakosti sidranja. Zatem so bile obravnavane kompleksnejše vrste ograditev: nematske kapljice in nematiki z dispergiranimi polimernimi mrežami. Obeh smo se lotili s preprostim mikroskopskim mrežnim modelom Lebwohla in Lasherja, spet z uporabo metode Monte Carlo. V primeru kapljic — najdemo jih v sistemih PDLC — smo obravnavali radialne in bipolarne robne pogoje ter se posvetili zvezi med numeričnimi rezultati simulacij in spektri 2H NMR kot eni izmed možnih eksperimentalnih opazljivk. Razvita je bila metodologija za napoved spektrov v prisotnosti znatnega gibanja molekul — fluk-tuacij molekulskih dolgih osi ter translacijske difuzije (homogene in nehomogene), kvalitativno ujemanje z eksperimenti pa je dobro. Tudi urejujoči učinki zunanjega polja se jasno odražajo v računanih spektrih: zunanje polje molekule reorientira, a tudi zviša stopnjo urejenosti nematika. V poljih visoke jakosti pride do nematskega urejanja celo daleč nad temperaturo prehoda v izotropno fazo. Zadnjo in najkompleksnejšo izmed obravnavanih ograditev so predstavljali nematiki z vključki polimernih mrež. Sprva smo se posvetili pravilni mreži ravnih in med seboj vzporednih vlaken ter vplivu grobosti njihove površine na ureditvene sposobnosti take mreže. Rezultati kažejo, da mreža ohrani svojo sposobnost urejanja nematika tako dolgo, dokler površina vlaken ni popolnoma neurejena. primeru gladkih vlaken in homeotropnega sidranja v sistemu opazimo topološke defekte, katerih notranjo strukturo smo lahko razbrali iz rezultatov simulacij: jedro defekta obdaja prstan dvoosne ureditve, v kateri je stopnja urejenosti S zmanjšana, v samem jedru pa je ureditev enoosna, toda z negativnim S. Nadaljevali smo s študijami preklapljanja molekulskih orientacij z zunanjim poljem. Obnašanje pravilnega sistema vzporednih vlaken v zunanjem polju smo predstavili v faznem diagramu, iz katerega lahko razberemo premikanje Freederickszovega in saturacijskega prehoda s spreminjajočo se jakostjo zunanjega sidranja. Obnašanje je podobno kot v navadni nematski plasti. Ogledali smo si tudi preklapljanje v vzorcih z bolj zapleteno topografijo mreže, kjer so bila ravna ali neravna vlakna nameščena po prostoru naključno. Preklapljanje smo spremljali z opazovanjem simuliranih spektrov 2H NMR, električne kapacitivnosti in intenzitete prepuščene svetlobe. Vse navedene opazljivke kažejo, da igra topografija mreže polimernih vlaken pri procesu preklapljanja izjemno pomembno vlogo. a konec smo si ogledali še paranematsko urejanje nad temperaturo prehoda v izotropno fazo. Nekatere od zgornjih zaključkov lahko prenesemo tudi v sorodne kompleksne sisteme: nematske gele in koloide. Na koncu lahko rečemo, da je simulacijski del postregel s številnimi sicer že znanimi pojavi, vendar je pomembnost predstavljenih rezultatov in metod zlasti v tem, da so omogočili vzpostavitev povezave med podrobnostmi na mikroskopski ravni in makroskopskim obnašanjem vzorca. Zaradi majhnosti obravnavanih sistemov iskanje kvantitativnega ujemanja z eksperimenti za zdaj ni prav uspešno, kar pa se bo z naraščanjem razpoložljive računalniške moči s časom gotovo spremenilo. xxxiv Razširjeni povzetek Introduction To those not familiar with the subject, the expression liquid crystals may at first seem somewhat selfcontradictory. Indeed, the term refers both to liquid and solid-like properties of matter, and appears to suggest that in a liquid-crystalline substance both are manifested simultaneously. Without usually being aware of the very subtle nature of these complex materials, most people regularly use liquid crystal-based devices in everyday life. These devices mainly comprise various types of displays ranging from simple seven-segment watch displays that became operational already in the early 1970s, to sophisticated color displays built into up-to-date laptop computers. The smallness and low energy consumption were the features that made such displays suitable for mass production, resulting in a subsequent boom in elec tronic industry. Other related applications are switchable windows and rapid light shutters, the latter being very promising in the field of optical telecommunications Despite this vast development over the last decades, a conventional cathoderay tube display (still cheaper than a state-oftheart liquid-crystal display of comparable performance) was used to display results of the research presented in this thesis and, eventually, also for typing it Figure 1.1 Structural formula of 4'' pentyl-4-cyanobiphenyl (5CB), an elonated molecule showin liquid-crystalline behavior Roughly speaking, liquid crystals are anisotropic fluids made up of rodlike or dis cotic organic molecules. While the main liquid-like property is the ability to flow, the solid-like character comprises anisotropies of dielectric, magnetic, and optical properties, which all originate in the strong anisotropy of the effective molecular shape. The first liquid-crystalline compound — cholesteryl benzoate — was studied by the botanist Reinitzer [1] as early as in 1888, followed by Lehmann a year later [2]. Performing polarizing microscopy experiments, Reinitzer observed two firstorder phase transitions upon heating the solid compound: first it melted into an unusual turbid fluid state, which at a higher temperature transformed into a clear liquid. Studying the intermediate turbid phase, Lehmann realized that it is bire fringent, ie., optically anisotropic, and called it liquid crystal. Today a rather wide 13 Introduction collection of different liquid-crystalline phases is known and, as they are dwelling in between the ordinary solid and liquid phases, the preferred common description for them is mesophases. The mesophases can be divided into two broad categories, thermotropics and lyoropics. Thermotropics are one-component systems and have temperature-dependent phase behavior, while lyotropics are solutions of a liquid-crystalline solute in a solvent (e.g., water) and exhibit concentration-dependent phase behavior. The simplest of all thermotropics are the nemaic liquid crystals (nematics), owing their name to a translation from the Greek word urjfia ("thread") by Friedel who, in the early 1900s, observed the threadlike appearance of topological defects under a microscope between crossed polarizers [3]. Nematics consist of rodlike molecules whose elongated and rigid torso is typically formed by two or more aromatic rings, while the flexible limbs are usually made of alkyl chains (Fig. 11). Macroscopically, a nematic sample is fluid and, like in an ordinary liquid, there is no positional order in center of mass positions. There is, however, significant long-range orientational order in orientations of molecular long axes, and in a large (bulk) sample nematic molecules on the average align along a well-defined (but arbitrary) direction, denoted by a unit vector, the director n (see Fig. 1.2) [4]. Note that in ordinary nematics states described by n and —n cannot be distinguished. Moreover, at finite temperatures the ordering along n is not perfect. The corresponding degree of order decreases with increasing temperature. In absence of severe external constraints, such as confinement and electric or magnetic fields, the orientational distribution of molecules around n is uniform, corresponding to uniaxial symmetry. The optical axis in such uniaxial nematics then also coincides with n. Under strong external constraints, however, the uniaxiality in ordering may be lost and a second characteristic direction occurs in the mesophase, now called biaxial >,t-V^ (a) ^o^a ^ ^ (b) n u 0^n (c) t- Figure 1.2 Molecular ordering in the isotropic (a), nematic (b), and one of the possible smectic phases (c). n denotes the nematic director When in a nematic sample temperature is increased far enough, a first-order phase transition takes place. The long-range orientational order is now lost and the resulting isotropic phase behaves as an ordinary liquid. If, on the other hand, temperature is decreased, either a direct transition to a solid or to one of the smectic mesophases can follow, depending on the exact nature of the liquid-crystalline species. On the average, in smectics molecular centers of mass are arranged into parallel layers (see Fig. 12), thereby exhibiting one-dimensional positional order, in addition to orientational order present already in nematics. The term "smectic" is common to a rich variety of mesophases and comes from the Greek expression for Introducton 15 soap (afiejfjia), a substance also capable of forming thin molecular layers. In general, smectic phases possess two characteristic directions and are therefore optically biaxial. In this thesis, however, the emphasis will be given to thermotropic nemat-ics. One of the reasons for this lies in the fact that it is mostly nematic materials that are currently being used for applicative purposes, while other mesophases are primarily a matter of fundamental research. The behavior of nematics (as well as other mesophases) becomes even more interesting if they are confined to microcavities like droplets and pores [5], or simply in between two parallel plates. In such systems, the aligning tendency of solid substrates significantly affects the equilibrium molecular ordering, especially in strongly confined systems with a high surface-to-volume ratio. Solid substrates can impose different types of alignment or anchoring, the possibilities being planar, homeotropic, and tilted [6]. Close enough to the substrates surface-induced paranematic phase can persist even above the nematic-isotropic transition temperature. Planar alignment in the plane of the confining substrate can be achieved by unidirectional rubbing of the clean or coated substrate [7-10]. Homeotropic alignment (normal to the substrate), on the other hand, can be provided by treating the substrate with a surfactant [1113], while tilted anchoring conditions can be achieved through oblique evaporation of silicon oxide onto glass plates [14]. Alternatively, directionless evaporation can result in a substrate with irregular surface topography, acting disorderly rather than orderly. Similar completely or partially random topography can be encountered on polymer surfaces, and depends sensitively on the polymerization parameters [15]. Analogous (dis)ordering effects can be expected also at a free nematic surface, or at the nematic-isotropic interface [16]. Indeed, in a liquid crystal intermolecular interactions are highly anisotropic, and missing-neighbor effects can account for the aligning tendencies at such interfaces [1 The actual preferred orientation (the easy axis) is intimately related to individual properties and molecular structure of the given nematic species. The free-surface orienting effects are an intrinsic property of the nematic and are therefore often referred to as intrinsic anchoring (as opposed to the solid substrate-imposed external anchoring discussed above). Under strong confinement, different easy axes at the opposing substrates (as in a slab) or more complex geometric constraints (in droplets and pores) give rise to a conflict in determining the average molecular orientation. In other words, the director becomes spatially dependent, n = n(r), in order to meet all boundary conditions imposed by the confining substrates. Any deviation from the homogeneous bulk director profile with n ^ n(r) represents an elastic deformation, accompanied by an increase in the deformation free energy [18-20]. In addition, if the actual surface molecular orientation is to deviate from the local anchoring easy axis, this is also penalized by an increase in free energy [21,22] Then, the equilibrium director profile is the one minimizing the total free energy. Unlike in solids, the elasticity encountered in nematics is referred to as curvature elasticity. While still being able to flow (and thus unable to resist shear deformations), nematics are able to transmit substrate-imposed torques through the curvature of the director field. There is no characteristic length scale associated with such elasticity: the elastic distor- Introduction tion spreads smoothly throughout the sample. In crtain geometries, howeve, the director field may contain places where the director — for topological reasons — cannot be defined. Such point, line, or wall-like singularities, accompanied by a strong variation in the degree of ordering, are called topological defects [4,23,24] A further consequence of the non-spherical molecular shape is the anisotropy in the dielectric and magnetic susceptibility tensor of nematics. Consequently, in an external (electric or magnetic) field an orienting torque acts on nematic molecules thereby allowing for a controllable variation of the director orientation [4,20] A the same time, the orientation of the optical axis (n) is varied as well, which is the main effect liquid-crystalline optical devices are based on. Inevitably, external fields conflict with confining substrates, and — unless extremely strong — they are unable to reorient the nematic in the very vicinity of the confining wall. The thickness of this region is given by the electric or magnetic coherence length and decreases with increasing field strength [4] Figure 1.3 Scanning eleon microscopy picures takn after rmoal of the nematic. (a) PDLC sample with nematic droplets [25], (b) polymer networks dispersed in nematics [26] The simplest type of confinement can be encountered in every nematic cell including displays, where the liquid crystal is sandwiched between two flat (usually parallel) plates, or in any thin (also free-standing) liquid crystal film. However, more recent developments in material science have provided novel confining matrices, which are attractive both for basic research, as well as for applicative purposes. Nucleopore and Anopore membranes are thin membranes penetrated by channels providing cylindrical confinement, the channel radii being of the order of 1 ju [27]. Polymer-dispersed liquid crystals (PDLCs) consist of roughly spherical nematic droplets, with typical radii from a few hundred Angstrom to well above a micron, embedded in a polymer matrix [see Fig. 13 (a)] [28]. They can be used to design switchable windows, projection devices, reflective holographic displays [29] etc. Low concentration polymer networks consist of thin polymer fibers (even nano-metric in diameter), or of somewhat thicker fiber bundles [see Fig. 13 (b)]. They are capable of orienting the surrounding liquid crystal [30], being useful in building bulk-stabilized optical switches [31], and even allow for the formation of parane-matic order [26]. Other types of complex confinement include also filled nematics Introducton 17 and aerogels, where agglomerates of nanometric silica particles are dispersed in a nematic host, dividing it into several domains, each with its own local order [5] Until now, confined systems have been studied extensively both theoretically and experimentally. Theoretically, most widespread are the analyses performed within the phenomenological Landau-de Gennes theory, which is based on describing the rather complex molecular ordering by introducing a macroscopic order parameter. Then a free energy functional is constructed [16,32] and minimized with respect to order parameter profiles for the given boundary conditions. The second class of approaches stems from the statistical mechanics-based density functional theory. Here the molecular ordering is described in terms of the density and orientational distribution function profiles, the equilibrium being found by minimizing the grand potential with respect to these profiles [33-35] Unlike the phenomenological Landau-de Gennes theory, this approach includes microscopic intermolecular interactions and is therefore somewhat more fundamental With the advent of fast computing machines in the late 1940s, the field of molecular simulations began to burgeon. Indeed, from then on it was possible to simulate an ensemble of particles with microscopic degrees of freedom, such as particle position and orientation, and express the simulation output in terms of macroscopic thermodynamic averages and experimental observables. In such studies, the system evolution can be provided either from integrating Newton's equations of motion (molecular dynamics simulations [36]), or performing an appropriately weighted random walk in phase space (Monte Carlo simulations [37]). Both approaches are based on intermolecular interactions and are in ergodic systems supposed to yield equivalent results for thermodynamic averages. Experimentally, polarizing microscopy was one of the first tools used for the study of liquid crystals [1,2], whereby the sample is put between two crossed polarizers and the transmitted light pattern (or just the intensity) is analyzed. The method relies on the birefringence of the liquid-crystalline material, in particular on its molecular orientation-dependent index of refraction. It has been successfully applied, e.g., to distinguish between different director structures in supramicron confined systems [38], to detect paranematic order in a system with dispersed polymer networks [26], as well as for investigations of defects [23], to name just a few examples. Similarly, capacitance measurements reckon on the anisotropy of the static dielectric constant of the material and can therefore give information on significant molecular reorientation in the sample, such as director switching in an external field [39]. Yet another experimental technique is the deuterium nuclear magnetic resonance (2H NMR) [5]. It is extremely powerful for the recognition of nematic structures, especially in submicron cavities not accessible to optical techniques due to limitations set by the light wavelength [40]. The method can provide information on molecular ordering either through spectra governed by the molecular orientation and order-dependent quadrupolar splitting, or through nuclear spin relaxation phenomena [5,41]. A further optical technique is the second harmonic generation (SHG) which is a non-linear optical phenomenon and — contrary to the previous "bulk" techniques — gives information on molecular ordering in a thin subsurface layer [42]. Therefore, combined with conventional "bulk" methods, it is suitable for Introductio investigations of possible subsurface variations in molecular orientation [43]. Similar information on molecular alignment in thin films can be extracted also from the characteristics of reflected or transmitted polarized light in ellipsometry measurements [44]. Information on surface anchoring strength can be deduced from dynamic light scattering experiments, analyzing the light scattered by collective di rector fluctuations in a nematic confined to a cell [45] or to cylindrical pores [46] Further, calorimetry (i.e., measurements of the heat capacity) has been applied to study phase transitions, also in confined systems [47] This thesis aims at covering some selected modeling problems in the physics of strongly confined liquid crystals. Two main objectives will be followed: (i) a better understanding of mechanisms driving the nematic ordering close to an interface (still not fully understood as of now) and (ii) gaining more insight into nematic ordering in complex (even irregular) types of confinement. The geometries treated here will include a simple nematic slab [to cover objective (i)], nematic droplets (as encountered in ordinary and holographic PDLCs, covering objective (ii)], and nematics with dispersed polymer networks [objective (ii); related systems comprising colloids and gels]. Especially the two latter confinement types are particularly interesting also for applicative purposes, as already mentioned. In the analysis, phenomenolog-ical Landau-de Gennes-type modeling and simple large-scale molecular simulations will be combined. Therefore, the next Chapter will be devoted to these approaches as well as to methods used for calculating selected experimental observables from the simulation data. Starting with the simplest type of confinement, Chapter 3 will be dealing with molecular ordering in a thin nematic slab, giving particular emphasis to variations of the degree of nematic order, as well as to the resulting intrinsic anchoring and subsurface deformations. In fact, a number of experimental investigations [43,48-51 shows that liquid crystal molecules in the surface layer can have an orientation dif ferent from that in the bulk material. In some cases this can be attributed to the variation of nematic order [50,51] and biaxiality [43]. On the other hand, theoretical predictions about subsurface deformations have been published by different groups mainly in connection with the splay-bend (.K13) elastic constant introduced long ago in the Landau-de Gennes phenomenological description [52,53]. As the splay-bend contribution to the elastic free energy can be shown not to be bounded from be low [54], an additional higherorder term is to be included in the free energy density for stabilization [55-58], producing a strongly localized (but finite) subsurface defor mation. Based on elastic theory and yielding strong deformations, this description raised many questions and was followed by alternatives avoiding the strong de formations [59-61]. Possible variations of the nematic order were not taken into account, and tacitly the scalar order parameter was assumed constant. More recent macroscopic considerations indicate that for an ideal flat nematic interface with a step-like density profile and no order variation K13 = 0 holds [6265]. This then apparently solves the problem of strong subsurface deformations in the macroscopic description, but does not specify the microscopic source of deformations observed experimentally. However, a nonzero K13 is recovered if one allows either for a vari ation of density close to the nematic interface, or for a variation in the degree of Introduc nematic order [66,67]. The analysis presented in this Chapter will first focus on planar deformations, allowing for order (but not density) variations [68], aiming to explore the coupling between subsurface order variations and elastic deformations similar to those induced by the splay-bend elastic contribution in case of constant nematic order. For this purpose, the free energy will be expanded only up to first spatial derivatives of the order parameter, thereby explicitly avoiding the controversial splay-bend constant. Complementary analyses in a similar system have been performed also by other authors [50,51,6970]. In addition, the present analysis will be extended to twist deformations [7] The second part of Chapter 3 will still be concerned with the nematic slab, however, omitting the phenomenological description used in the first part and replacing it with a simple microscopic hexagonal lattice model presented in Ref. [17]. According to Ref. [17], the essential mechanism in aligning nematic molecules close to an interface seems to be the competition between intrinsic and external anchoring, the same conclusion coming also from the density functional approach [72]. Therefore, in this Chapter intrinsic anchoring will be investigated in some more detail. Like in Ref. [17], the simple hexagonal lattice model will be considered, which — combined with the spatially anisotropic induced dipole-induced dipole interaction — can reproduce the intrinsic anchoring at a nematic interface. The current analysis will be extended to nonzero temperatures, performing Monte Carlo simulations in the abovementioned model system. In particular, the intrinsic anchoring strength, including its temperature dependence, will be estimated by imposing a bend deformation in a hybrid cell-like system. In the past, intrinsic anchoring has been studied, e.g., in a pseudomolecular continuum approach with ellipsoidal molecules [73], analyzing a nematic-vapor or nematic-isotropic interface for Gay-Berne particles [74-78] and considering a system of hard ellipsoids in contact with a hard wall [780]. The anchoring energy (if reported) mostly shows that anchoring reproduced by these model systems is rather strong, exceeding experimental values [6] by one or two orders of magnitude. Moreover, simulations in similar systems yield different anchoring easy axes for intrinsic anchoring [75,76], depending sensitively on parameters entering the intermolecular potential [78]. A number of other (zero-temperature) lattice or continuum approaches has also been devised, mainly to study subsurface deformations in nematics [81-83]. Further, molecular dynamics simulations of particles interacting via the Gay-Berne potential show a substrate-induced spatial variation of the nematic scalar order parameter, accompanied by density modulations and smectic ordering [84-87], seen also experimentally [88,89] Moving now to more complex confining geometries, Chapter 4 will cover nematic ordering in spherical PDLC nematic droplets. The molecular ordering in such confined systems is affected by the competition between ordering effects of the polymer matrix (anchoring) and of external fields, plus by disordering temperature effects. Depending on the polymer matrix and on the surface treatment methods, different director structures — including topological defects — can be encountered inside droplets. The discussion in this Chapter will be limited to radial [40,38] and bipolar droplets [38,40,90] with homeotropic and planar anchoring conditions, respectively (see Fig. 14). Applying an external electric or magnetic field, additional ordering ef- 20 Introduction fects can be observed [40,90], and turn out to be of great importance for technical applications. Experimentally, deuterium nuclear magnetic resonance (2H NMR) [40,90] and polarizing microscopy [38] are the usual techniques to study PDLCs. From the theoretical point of view, however, Landau-de Gennestype elastic continuum approaches [5] and Monte Carlo (MC) simulations [91] have been widely used to study PDLC in a variety of physical situations, including ellipsoidal droplets [25,92]. The simulations were mostly based on the simple LebwohlLasher lattice model [93]. In particular, the MC technique has proved to be a powerful method not only for investigating the thermodynamical behavior of confined nematics, but also for the prediction of quantities directly observable in experiments. The observables simulated to date include static H NMR spectra and polarizing microscopy patterns thereby bridging the gap between simulation and experiment [91,94]. Chapter 4 will present a novel and widely applicable methodology for the calculation of 2H NMR spectra that also starts from MC configurations, but takes into account dynamical effects as well [95]. These include fluctuations of molecular long axes and transla-tional molecular diffusion (also spatially inhomogeneous [96,97]). A many-bipolar droplet sample will also be considered [97]. To conclude this Chapter, external field-induced changes in spectra will be studied in great detail [98]. Note that analyses presented in Refs. [99,100] already deliver thorough studies of external field effects in radial and bipolar droplets, containing also the corresponding 2H NMR spectra, but none of these studies presents a line shape calculation in the presence of significant molecular motion. ^^ (a) ^^ (b) (P/Jh mm Figure 1.4 Director configurations in a PDLC droplet: (a) radial, (b) bipolar The following Chapter 5 will be concerned with nematic samples containing dispersed polymer networks. These can be topographically quite irregular and, consequently, this is the most complex confinement type presented in the thesis. Having a rather high surfaceto-volume ratio, even at low network concentrations these composite systems are similar to "ordinary" confined systems with "closed" cavities [5] Again, apart from exhibiting a variety of interesting ordering and confinement-related phenomena, composite materials like liquid crystal-dispersed polymer net works are promising also for the construction of novel bulk-stabilized electrooptical devices, which are usually based on the switching phenomenon. This phenomenon consists of changing the molecular orientation — initially imposed by the polymer network — by applying an aligning external electric field. Above a certain threshold the net molecular orientation changes, which can be observed through changes in optical transmission, electric capacitance, or by 2H NMR spectroscopy [31,39]. Both the nature of the switching process and the performance of electrooptical devices are intimately related to the anchoring and ordering conditions at the fiber surface Introducton 21 as well as to the topography of the network. These properties can all be regulated during the network formation process [15,101], and by surface treatment after the network has been formed. The existing experimental studies were usually accompanied by phenomenological (Landau-de Gennes-type) analyses [5,26,31,39], but so far almost nothing has been done for such network-like confinement at the microscopic level. For all these reasons here a thorough microscopic simulation study of the ori-entational coupling between polymer fibers and the surrounding liquid crystal will be presented. Again, the analysis will be based on the Lebwohl-Lasher model [93] focusing on polymer networks with a well-defined fiber net direction (as in Fig. 1) and on effects of roughness at the fiber surface [102]. Furthermore, for homeotropic anchoring, due to topological constraints defects in nematic ordering can be expected to form [103]. Their inner structure will then be explored and compared with the existing phenomenological studies [104,105]. Moreover, the switching process in an external field will be simulated and the role of network irregularities in switching characteristics (threshold etc.) will be examined. For a regular polymer fiber array (also applicable to model colloidal crystals), a simple stability phase diagram will be derived for the director structures observed in an external field. Finally paranematic surface-induced ordering above the bulk nematic-isotropic transition temperature will be explored in the system. For practical purposes, the simulation output will be expressed in terms of selected experimental observables, that is 2H NMR spectra, electric capacitance, and transmitted intensity of polarized light. The final Chapter — the Conclusion — will summarize the main achievements and findings presented in the thesis together with open problems and possible future directions. Introdutio heoretical background In this Chapter a brief review of theoretical and simulation approaches for treating confined nematics will be given, together with selected experimental methodologies applicable to these systems 2.1 Phenomenological Landau-de Gennes theory Unlike in rare gases or idealized solids, in liquids (including liquid crystals) a full statistical mechanics treatment is highly nontrivial. On the other hand, with present computing power also molecular simulation studies of macroscopic liquid samples are still not within reach. In addition, very often the detailed microscopic description of the system is not even necessary to predict its macroscopic properties sufficiently well. Therefore, in such cases a phenomenological Landau-type description can be suitable. Within this description type, an appropriate macroscopic order parameter must be introduced first. The order parameter is supposed to reflect the microscopic state of the system and should be in agreement with its symmetry. The equilibrium state of the system is then searched for by minimizing the appropriate thermodynamic potential with respect to the order parameter 2.1.1 Nematic order parameter An isotropic liquid possesses neither positional nor orientational order and has thereby the highest possible symmetry. The nematic phase, on the other hand, is orientationally ordered, while positional ordering is still absent. A smectic phase however, possesses both orientational and positional long-range order. In the following we are going to omit smectic phases and treat exclusively nematics. Conse quently, an order parameter containing information on solely orientational ordering of molecules will be sufficient to characterize nematics and distinguish them from isotropic liquids. Nematics consist of elongated molecules whose orientations can be characterized by unit vectors u along the molecular long axes. Due to rapid tumbling, the effective molecular shape is cylindrically symmetric. For the same reason, the molecular tic "head" and "ail" also cannot be distinguished on the average, although the head tail symmetry is not an inherent property of such molecules. Consequently, nematic molecules can be regarded as directionless objects with a well-defined orientation (note that terms direction and orientation are not used as synonyms here). In a reference frame where the z-axis is chosen to coincide with the director n, molecular orientations u for each molecule can be defined by the polar and azimuthal angles, 9 and (j), respectively, with cos 9 = n • u. Then the state of molecular alignment can be described by an orientational distribution function f(9, )dVt = 1, if integrated over the whole solid angle. In an unconstrained sample f(9: ) is uniaxial— axially symmetric with respect to n — and hence ^»-independent. In such uniaxial nematics / = f9) can be expanded as 9) = J2fnPncOs9) 1) n=0 where Pn(x) is the nth Legendre polynomial and fn = (2n + l)fž~x 9)Pn(cos9)dL are the expansion coefficients. The monopole term is a constant, /0 = 1, and therefore not sensitive to any type of orientational molecular ordering. Hence, higher-order terms are to be considered in the quest for an appropriate order parameter Given the head-tail symmetry of the nematic phase, f(9) = f(ir — 9) holds, which results in fn = 0 if n is odd and in /„ ^ 0 otherwise. The first nonzero contribution thus comes from the quadrupolar term /2 = 5Q-1 9)cos 9)dQ = 5(cos 9)) = 55, defining the scalar order parameter S = (|(3cos2L? — 1)), where the average (..) is to be performed over an ensemble of molecules. The order parameter defined in this way is able to quantify the degree of nematic ordering with respect to n: in a perfectly aligned nematic with u||n one has S = 1, while in the isotropic phase with 9) uniform also in 9, (cos2 9) = | holds and the order parameter vanishes, i.e. S = 0. A meaning can be assigned also to negative values of S; S = — \ corresponds to perfect ordering in the plane perpendicular to n. Up to the first nontrivial term the orientational distribution can now be rewritten as 9) = l + 5S'Pcos0) = l + — [3(n-u)l] 55 1 + — [fiiUj - S] UiUj = 1 + 5 QiUjUj assuming summation over repeated indices) where Q given by Q = |[n®n-I is the tensorial order parameter [4] — the quadrupolar moment of the orientational distribution f(9) — and I the identity matrix. The Q-tensor contains the inforation both on the average molecular orientation and on the degree of order S. Alternatively, nematic orderi n b d a n tms mme traceless ordering matrix [4] = i[3(u®u)-I] 25) where the eigenvalue with the largest absolute value can be identified as S and the corresponding eigenvector as n. In the uniaxial case, the other two eigenvalues are then both equal to — ^S and the eigenvectors perpendicular to n. Under external constraints such as electric and magnetic fields, or severe confinement, elastic deformations can break the cylindrical symmetry of the f(0,(j)) distribution function assumed so far. In addition to the director n, two additional characteristic directions appear in the system, ei and e2, forming an orthonormal triad together with n. In such biaxial nematics the tensorial order parameter be comes somehat more complicated and reads P - [3 n n - I] + — [ <8> e - e <8> e] 26) introducing P = |(sin2 6cos2(j)) as the biaxiality. Diagonalizing the ordering ma trix (2.5) now gives three eigenvalues that are different, S, — |(n-l) / — (e <8>e -e(g>eJ + X\ 27) Here Xa = X3 ~ 2(xl + X2) is the anisotropy of the molecular susceptibility, i.e. the anisotropy for a perfectly ordered nematic (S = 1), where x'kS = Xk (recall that Xk are macroscopic parameters obtained at finite temperatures and S < 1) Further, Xb = x'[ ~ X2 is nonzero in case of biaxial ordering {xtP = Xk) and Xi |(X + X + X3) denotes the average susceptibility i its isotropic part 2.1.2 Nematic-isotropic phase transition In phase transitions matter undergoes changes in microstructure and symmetry. A phase transition is accompanied by an abrupt change of a acroscopic observable tic like density, electric or magnetic polarization, or simiar. While the relevant ther modynamic potential — determining the equilibrium state in a system for given parameters (e.g., free energy F for well-defined temperature, volume, and particle number) — remains continuous across the transition, its derivatives may not. Phase transitions characterized by a discontinuity in, e.g., entropy S = —(dF/dT)v, are referred to as discontinuous (or first-order), while the ones with no discontinuity in S are called continuous (or second-order). As opposed to continuous phase transitions discontinuous transitions present also a discontinuity in the aforementioned macroscopic observable playing the role of the order parameter. In addition, the possible discontinuity in entropy, AS, results in an exchange of latent heat Q = TCAS, where Tc stands for the transition temperature. Moreover, metastable states can be found in the vicinity of discontinuous transitions Figure 2.1 Nemaic-isotrpic tnsition: fre enegy v. order paramer at different temperatures (a), equilibrium order parameter vs. temperature (b) The macroscopic observable sufficent o chaterize uniaial ordering in a bulk nematic sample is the scalar order parameter S introduced in the previous Sec tion. Experimental studies have shown that the nematic-to-isotropic transition is accompanied both by a discontinuous drop in S and by absorption of latent heat [4] Moreover, metastable supercooled isotropic and superheated nematic phase could be observed as well. These findings all indicate that the nematic-isotropic (NI) tran sition is discontinuous. Following the Landau description of phase transitions the free energy density of the system is to be written in terms of the order parameter in form of a simple polynomial expansion [32]. Close enough to the transition this expansion converges and may hence contain a small number of lowestorder terms that is /o(5, T) = fi(T) + \a{T - T*)S2 - ±BS3 + ^CSA 8) Here a > 0, B > 0, and C > 0 are positive phenomenological material constants while T* is the lowest temperature still allowing for the existence of the supercooled isotropic phase. Typically a is of the order of 105 J/m3K, while B and C range around few times 106 J/m. Further, /j denotes the free energy density of the isotropic phase. he expression (28) ensures with C > that F is bounded from Thel background 27 below and ages with the symmetry of the order parameter S. In particular, odd order S^-term is allowed because of the non-equivalence of S and — S, while the linear S-term is prohibited so as to obtain S = 0 in the isotropic phase if external fields are absent. Unlike constants B and C, the proportionality constant of the 5'2-term a(T — T*) is taken to be temperature-dependent and therefore plays an important role in driving the phase transition. The equilibrium in the system is then found by minimizing the free energy F(S,T) = J fo(S,T)dV [see Fig 1 (a)] at xed temperature with respect to S, yielding 0, ______________ T > Tc, S{T) = ž (l + y/l-mT-T-)) > T Tc) and Sb ^ 0 following Eq. (2.9) in the nematic phase (T < Tc). The correlation length L increases on approaching the transition temperature Tc from either side, but does not diverge. For a typical nematic L can be evaluated to be of the order of ~ 10 nm at Tc. A similar characteristic length can be derived for biaxiality (P) variations as well. As soon as the average molecular orientation is subject to spatial variations too, it is more convenient to write the free energy density in terms of the tensor order parameter Q and its gradients. For simplicity, only uniaxial cases will be treated here, and Q will be given by Eq. (2.4). As free energy is a scalar quantity scalar invariants have to be constructed from the Q tensor, the candidates being the trace and the determinant. At the same time, the free energy expression must be invariant to all symmetry operations allowed in the high-symmetry (isotropic) phase possessing full symmetry, including invariance to rotations and reflections The tensor trace is then the only scalar meeting these invariance requirements. For this reason, the homogeneous part of the free energy density (2.8) can be written in terms of traces of powers of Q, i.e., trQ2 = |52, trQ3 = l^3, and (trQ2)2 = l^4 Note that the first-order term trQ is absent because it is zero by definition, and that another fourth-order term is possible, trQ4, but is just proportional to (trQ2)2. In absence of external fields, the free energy density for a homogeneous (undeformed) sample /0 ) in tensor notation then reads /o(Q, T) = fi(T) + \{T T,)trQ ^trQ ^C(trQ2 1 In inhomogeneous phses, the Q-tensor components become spatiallydepndent. In the weak deformation (continuum) limit the first-order derivatives Qij,k = dQij/dxk are small quantities. They can be used to construct additional scalar invariants with proper symmetry, representing free energy contributions associated with elastic distortions. The free energy density in a deformed sample can now be written as [4107] (r) = /0(Q, T) + -LxQ^Q^ + -L2QlJQlkk + -L3QlhkQlk (213) where Li, L2 and L3 are the "elastic parameters" entering this phenomenological Landau-de Gennes model. For a deformed state / > fQ(Q,T) is expected, therefore in Eq. (2.13) there are no linear terms in the first-order derivative of Q^, and, further, all terms quadratic in Q^-^ are assumed to be positive definite. This yields the following restrictions: L\ > 0 and L2 + L3 > — §Z>i [107]. Using the definition of Q (2.4) and taking into account that n is normalized (n • n = 1 and hence r^njj = 0), it is possible to rewrite f(r) given by Eq. (2.13) in terms of the director n and the scalar order parameter S. In the resulting expression the variations of S and n are highly coupled. Neglecting for the moment variations of S to treat only curvature elasticity (setting VS1 = 0 and S = Sj) one is left with (r) = f0(Sb T) + -Sl[(22,! U L3){(V • n)2 + (n x (V x n))2} (214) +2L1(n • (V x n)) (2Li + L3)V • (n(V • n) + n x (V x n))] or equivalents (r) = f0(SbT) + fF{*)- Here fF(v) = i {Ku [V • n] K22 [n ¦ (V x n)] K33 [n x (V x n)]2} -K24V-[n(V-n) + nx (V xn)] (215) is the standard Frank elastic free energy density [18], and Kn, K22, and K33 are the Frank elastic constants for the corresponding deformation modes: Kn for splay with (V • n)2 7^ 0, K22 for twist with (n • (V x n))2 ^ 0, and K33 for bend with (n x (V x n))2 jL 0; see Fig. 2.2. The last divergence term belongs to the saddle-splay deformation and is nonzero if n depends on more than one Cartesian coordinate introducing the K2i elastic constant (a) (b) (c) Figure 22 Frank elastic deformation modes: splay (a) twist (b) and bend (c). Comparing Eqs. (214) and (2.15) following relations are found: Kn = K33 = l(2LiL2+L3) K22 fLi and K24 |(2LiL3) showing that within this tic approach the splay and bend constants are equal and different from the twist elastic constant. Only in the special case with L2 + L3 = 0, all three Frank elastic constants have the same value Ku = K22 = -^33 = K = §5f Li (one-constant approximation) while the value of the saddle-splay K2^ elastic constant is still different. The typical magnitude of Frank elastic constants is K ~ 5 x 10~12 N. They can be measured reliably, e.g., by studying elastic distortions in an external field (described in more detail later) [108,109]. The K2± constant, on the other hand, can be deduced, e.g., by investigating three-dimensional structures in capillaries [110] its magnitude being similar to that of Frank elastic constants. Mathematically, the saddle-splay contribution is a divergence term and therefore does not contribute to the bulk Euler-Lagrange equations while it modifies the surface boundary conditions. Another divergence term derived in the past together with the K24 term was the splay-bend term /13 (r) = K13V • [n (V • n)] [52,53]. Unlike the K24 term, it explicitly contains second-order derivatives of n and could therefore not be recovered from Eq. (2.13) containing first derivatives only. It is unbounded from below unless an additional higher-order stabilizing contribution is added to the bulk free energy density. In this case it induces a strong subsurface variation of n [55-58]. More recent studies show that for a step-like nematic density profile at a flat interface it is canceled by the spontaneous splay contribution Ki(V • n) [111] which is prohibited in the bulk due to inversion symmetry, but allowed in a thin subsurface layer whose thickness is related to the range of intermolecular forces [62,65] Due to its controversy coming from the strength of the subsurface deformation it produces the Ki3 contribution is usually excluded from considerations of confined systems. 2.1.4 External anchoring Phenomenologically, the effect of the bounding interface is to impose a certain degree of ordering and a preferred average molecular orientation, the easy axis (or, alternatively, the "easy" Q-tensor denoted by Q0). Any deviations from the interface-imposed ordering are penalized by a surface contribution to free energy, which can — similarly as in the bulk case — be written in terms of appropriate scalar invariants. These are now constructed from the tensor Qs — Q0 measuring the deviation of the actual surface ordering Qs from Q0. The corresponding bulk free energy density reads /a(r) = ^ [tr(Qs - Qo) + /i{tr(Qs - Q0)2}2] (r - r0). (2.16) Often it is enough to consider the first contribution only, setting jj, = 0 [22]. Since the close-packing and van der Waals forces (mainly) responsible for anchoring effects are relatively short-ranged, fa is regarded as a purely contact term, r0 representing the locus of points defining the interface. The quantity We has units of energy and is related to the strength of anchoring. For uniaxial nematics with Q0 = ^So(3no <8 n0 —I) and Qs = ^Ss(ns = 0 and a finite anchoring strength and that the anchoring at the opposite (z = d) surface is infinitely strong with a different easy axis. In a thick enough sample such a set-up induces an elastic deformation [11 and the deviation of the actual surface tilt (j)(0) from the easy axis (/>0 = 0 at z 0 will yield an information on the anchoring strength. Then, in terms of the (f(z) profile the total [bulk (2. plus surface (2. free energy per unit surface is given by F = f{j fF(z)dz + fs = ^ Jq (f)l2dz \W sin 0) in the one-constant approximation and assuming the tic degree of order no to vary, i.e., S = Sf,), the prime ' denoting derivation with respec to z. Minimizing F with respect to " = 0 is found to be a linear director profile (z) = C\Z + Co, the constants Ci to be determined from the boundary conditions at both sample walls At the z = 0 wall the boundary condition reads (-^-)z=0 = -^fc- or (d(f)/dz)z=0 = \iyVjK) sin2(0) and represents the surface torque balance condition. This relation enables one to deduce K/W from estimated (d(f)/dz)z=0 and (f>(0). Note that for small (f)(0) (i.e., strong enough anchoring) the above condition simplifies to (d/dz)z=0 = (W/K)(f)(0) and makes it possible to determine K/W simply by extrapolating the profile (f)(z) graphically across the sample boundary to (f> = 0 corresponding to the homeotropic easy axis (see Fig. 2.3). Note also that if the degree of nematic order is allowed to vary — which usually is the case near confining boundaries — the profile (f)(z) may deviate from the linear behavior predicted above. In this case the extrapolation of the profile towards the surface must be performed from far enough in the bulk where the order parameter profile is constant 2.1.5 External fields If nematic molecules are exposed to an external field, an additional contribution appears in the bulk free energy density. In the following, the derivation will be per formed for magnetic fields, but the procedure for an electric field would be identical Denoting the external field strength by H, the resulting magnetization can be written as M = %H, with the magnetic susceptibility tensor x given by Eq. (27). In case of uniaxial ordering (P = 0) % has only two different eigenvalues, %3 = X3S = X\\ and Xi,2 = x'12^ = X-L> corresponding to directions parallel to the director n and perpendicular to it. Splitting the field strength into components parallel and perpendicular to n, one has H|| = n(H • n) and Hi = H — n(H • n). Consequently the magnetization reads = X±H + (x|| - x±)(n • H)n. (19) In an experiment performed in an external field with strength fixed, the additonal free energy density contribution reads fm = —/j,0 J^1 M • dH (jU0 standing for the permittivity of the vacuum) and, taking into account (19) becomes /m(r) = -f [xx# + (X| " X±)(n • H)2] 0) The second term is orientation-dependent and shows that for positive XaS = X\\ ~Xi the field free energy is smallest when the director is aligned along H, the field direction. Similarly, Xa < 0 would have resulted in an alignment perpendicular to H. This orienting mechanism stems exclusively from the nonzero anisotropy Xa and is not present in media where the molecules (and the tensor %) are isotropic Denoting the angle between n and H with f3 (cos/3 = n • H/H) Eq. (.20) can be simplified to /m(r) = f(H) - \n,XaSH cos/3, 1) where the orientation-independent term has been denoted by f^(H). Analogously, in an external electric field E directed at an angle /3 with respect to n, the field free energy contribution is given by /e(r) = (E) - heaSE2 cos /3, 22 where eQ = (e|| t±)/S is the dielectric constant anisotropy and e0 the dielectric constant of the vacuum. Note that both f and /e are position-dependent through the j = ?(r) dependence Liquid crystals are — like most organic molecules — diamagnetic: nematic molecules usually contain aromatic rings capable of carrying magnetic field-induced currents (see Fig. 1). Consequently, both %|| and X-L are negative and small with typical absolute values around 10~7 10~6, and the macroscopic anisotropy aS = X|| X_L is also m this range [4]. In the magnetic case one usually has Xa > 0. On the other hand, in the electric field case the anisotropy of the dielectric constant comes from the anisotropic molecular polarizability, as well as from permanent dipoles associated with strongly polar (e.g., cyano) groups — Fig. 11. The typical values for the dielectric constant are of the order of ~ 5, with ea ~ 0.2 (magnitude and sign strongly dependent on the chemical structure of the given nematic species [4]. (a) H=0 (b) H . (c) H , X X z=0 I ^ ^ Figure 2.4 Semi-infinite nematic sample with planar anchoring in an external field H applied along the surface normal: undeformed zero-field structure (a) deformed structure (b), and saturated structure (c) Imagine now a semi-infinite nematic sample (as shown in Fig 2.) whose substrates promote planar molecular alignment, the z-axis being the substrate normal. Let further (f> denote the angle between n and the z-axis. In absence of external fields, one has 4> = 7r/2 throughout the sample [Fig. 2.4 (a)]. However, once an external field is applied along z, the conflict between the field and the confining substrate produces an in-plane elastic deformation [Fig . The minimization of the Frank (15 plus field (221) free energy F = /0° f{z)dz = \ J™[K(f)'2 + foXaSbsm ] (written per unit surface in one-constant approximation, again assuming that S = St, is constant) yields the Euler-Lagrange equation ¦^{§tf) = {%,)• This equation, if solved for infinitely strong surface anchoring (i.e., no deviations from the surface easy axis tt/ are allowed, gives the director profile Lm nematic molecules are aligned along the field and Lm can be regarded as the thickness of the deformed layer. Note that the thickness Lm is inversely proportional to the field strength H: the stronger the field, the smaller Lm. In a 1 T magnetic field for typical nematic Lm can be estimated to be of the order of 10 yum, which means that in order to align the nematic in submicron cavities external fields of extreme strength are required. Similarly, the electric conterpart of Lm — the electric coherence length is given by & = {K/^eaShE2)ll. In cases where the anchoring strength W is finite [and modeled, e.g., by the Rapini-Papoular form (2.18)], the qualitative character of L, the core energy can be ignored in comparison with the elastic contribution. Note that in this case and for low (and relevant) values of m the total energy Fd + Fc is an increasing function of and that, consequently, high strengthdefects are less likely to form. If there are two (or more) defects positioned at different locations in the sample, the overall director field is a linear combination of director fields obtained for sepa rate defects (this follows from the linearity of the Euler-Lagrange equation). Then the free energy for the director field deformation for, e.g., two defects of strength and separated by b is given by [24] Fd = + 27Km log p (2.27) where Fj + F] is the distortion energy of the isolated defects and he last term is the "interction" term. Frm (2.27) it follos tat defects of opposit stre sin att (and, eventually, annihilate), whil thos of equal streng si repel his attration/repulsion is mediated by he long-range crvature elastiit of the rector field. Moreover, defects of given strength can split into several lower-strength ones hereby maintainin the total defect strength m = X^rv For example, a —1 streh defec an split into two —| strength on. Cmbining Eq. (2.25-(2.27) ve = -KKim log------2-KKm log — (2.28 ro r0 for he deformation free energy of two interting defects and implies tat a pair of, e.g., — \ strength defects indeed is re stable an a singe —1 streg one [24] ere, however, both he defec re free energy and differece in re sizes were nored, but ore detailed analys still suggest that it is fa for defect of higher stren to split into several lower strength-one. Very often a simple "rul of tmb" for the defect stabilit analsis is used, accrding to which the leain ontution in the crresponin free eergy is sipl opotional to m2. 2.2 olecular simulations and lattice odels he econd lass of approaches addressed in this thesis will be based on sipl microsopic models of the nematic, stating fro give interparticle pairwis in eration law. Sch approaches become inevitae if one is interested, e.g., in performing detailed stus of phase ansitions accmpanied by sinificant fluctuations of the phenomenologcal order pameter, studying cases where he rder parameer hos spatial viations too stron to be ccmmated within continuum thees or, simply, if one is mere interested in relatin certain pameer enterin he intermolecula potntial ith mcrosopi erv 2.2.1 LebwohlLasher model On the molecular evel wo main interction types are responsib for the formation of the nemati phase. he first t is the pure steric intermolecular repulsion it no long-range attrction. In fa if the partice density in a system of strongy longated moleculs is increased, following Onsager a nematic-toisotropi tansition taks place [119] n the other hand, a temperature-drve firstrder ematic-to isotopic transition was swn to happ (by aier and Saup also in a sstem of olecules it a large anisotop in the molecula polaizaility resultin in anisotropi atttive dispersion (van der Waals) forces [120,1] To overcome the mean-fiel approximation used in Rfs. [120,121] a simple microscopic lattice mde was put forward by Lebwohl and Lasher, evolvin into a coputer siulation study of the ematic-isotopic (NI) transition 93,122] he lattice approxation was partiula suita fo he pioneering cputer stu becaus of its putational tic /C. f^............ 1 /? a ¦-**:.... -K u, Vh -^ Figure 2.6 Lebwohl-Lasher lattice odel of ematic: articles attached to a cubic lattice of spacing a. cheapness in comparison to the off-lattice models. With the enormous increase of the computing power in the past decades, lattice models are still appealing because they facilitate the performance of large-scale simulations, as well as simulations in medium-size systems for a broad range of physical parameters. In the framework of the Lebwohl-Lasher (LL) lattice model uniaxial nematic molecules are represented by particles that can rotate freely, but are arranged into a simple cubic lattice with spacing a (Fig. 2.6). Thereby there are no translational degrees of freedom in the system, but despite this the model can be seen to reproduce the orientational behavior well enough. Equivalently, particles may also be regarded as close-packed molecular clusters (consisting of up to 102 molecules), maintaining their short-range orientational order in the relatively narrow temperature existence range of the nematic, as well as across the NI phase transition [94,100]). Supposing that the effective volume occupied by one molecule is ~ 1 nm3, the lattice spacing a can be estimated by a ^ 5 nm. The orientation of the particle located at the «th lattice site is denoted by a three-dimensional unit vector Uj. The particles i and j then interact through a pair potential given by Uij = -eijP(cos^ = -dj [-(ui ¦ - (2.29) where e^- = e is a positive constant (of the order of ^0.02 eV) if i and j are nearest neighbors, and e^ = 0 otherwise. Further, is the second-rank Legendre polynomial and P^ the angle between Uj and Uj In this sense the LL model is similar to the Heisenberg model used for modeling magnetic systems, yet accounting also for the head-tail symmetry encountered in nematics. The effect of the interaction (2.29) is to align neighboring particles parallel to each other and is at low enough temperatures sufficient to induce nematic ordering. Simulating bulk systems, standard periodic boundary conditions are usually applied [93,122,123]. An improved "cluster" boundary condition has also been devised where particle orientations at the simulation box surface are sampled from a dis- tribution with the same value of the order parameter S as calculated for particles inside the box [124]. On the other hand, in confined systems a certain fraction of nematic particles (the "ghost" particles) is used to fix the boundary conditions. For example, to model a PDLC droplet, a jagged sphere is carved from the cubic lattice by considering all molecules lying closer than R (the droplet radius) to the chosen center [1]. Then the interaction with the confining polymer substrate is mimicked by assuming an additional layer of ghost particles whose orientations are chosen in accordance with the desired boundary conditions and are kept fixed during the simulation [125]. Both nematic-nematic and nematic-ghost interactions are modeled by the interaction law (2.29), but the interaction strengths e^ are not necessarily equal for both interaction types -0.6 -0.8 * -i b -1.2 -1.4 -1.61___: 1 Figure 2.7 Bulk Lewohl-Lasher system: temperature depndence of reduced internal energy U* = U/e (a) and scalar order parameter S (b). The NI phase transition occurs at T* = fcgT/e = 11232 n reduced units, as ndcated by the arrows [123] Performing Monte Carlo simulations (described in more detail later in this Chapter), it was shown that the LL model reproduces a weakly first-order NI transition in a bulk system — a large sample with periodic boundary conditions [93,123] — while in confined systems (e.g., droplets) of sufficiently small size the phase transition is suppressed [126,127], which is in agreement with experimental data. The temperature dependences of the total internal energy U = J2(i 0 (xa > ) nematic particles are aligned along f. Matching aligning effects in the electric and the magnetic field, for a given nematic species one can translate any magnetic field strength (or B) into an equivalent strength of the electric field (E) [99]. In the electric case one then has er\ = eaeoVoE2/3, where ea = (e\\ — e±)/S. 2.2.2 Hexagonal lattice model The Maier-Saupe theory [120,121] has been devised from anisotropic van der Waals forces in the mean-field approximation that can be believed to work well deep enough in the bulk. he Lebwohl-Lasher model [93] goes beyond the mean-field approximation, but still assumes the interparticle interaction to promote strictly parallel alignment, regardless of the relative position of particles. While certainly reasonable in the bulk, close to a confining substrate such a simplified picture is not necessarily appropriate. Therefore, in what follows, anisotropic van der Waals forces will be reconsidered, leading to a generalization of the interaction law (2.29): the induced dipole-induced dipole interaction. In a simplified picture, a nematic molecule can be thought of as an elongated and electrically neutral cloud of positive and negative charge. Due to thermal fluctuations, centers of mass for the positive and negative charge do not overlap. The characteristic time for these charge fluctuations is much shorter than the times associated with the reorientation of the molecule as a whole. Then, in a given moment, in the molecule an instantaneous electric dipole pe is created whose time-averaged value equals zero. However, during its existence pe gives rise to a dipolar electric field (r)«i[^p.] The electric field (2.31) polarizes neighboring molecules, thereby inducing electric dipoles p'e(u', r), where r and u' denote molecular positions and orientations, respectively, and r = |r|. Assuming that nematic molecules are on the average uniaxial, the molecular polarizability tensor a has only two different eigenvalues, u\\ and a±, referring to an electric field aligned along the molecular long axis u' and perpendicular to it, respectively. he dipole moment Pg induced by the electric field (r) can now be written as pe(u, r) = a(r) = a±(r) + (a,, - a±)((r) • u)u (2.32) and the corresponding interaction energy as U -p(i,r) • (r) oc -a±E - (a\\ - a±)((r) • u)2. (2.33) Note that the molecular orientation-dependent term E(r) • u enters Eq. (2.33) as a square and therefore always yields a nonzero average if averaged over rapid theral fluctuations of the molecular charge (and pe), regardless of the fact that the (2.31) instantaneous pe is actually aveaged to zero. Furher, given a high anisotropy o the molecular polarizability with a\\ >> a±, one can assume that the most probable orientation of the instantaneous fluctuation-induced pe is along the molecular long axis u. Then, taking the expression (2.31) for the dipolar field E(r the orientation-dependent part of the interaction energy is given by (u,u'r) = -^[u.u'3,(u.^(u'^)] (234) Here v — the interaction anisotropy parameter — has been introduced: for v = 1 the expression (2.34) represents the induced dipole-induced dipole interaction, while for v = 0 it reduces to the MaierSaupe interaction used in the LL model. The magnitude of interaction strength e is similar to that of e introduced in Eq. (2.29 for the LL model. Having assumed a\\ >> a±, e' is positive and the interaction is attractive. It is also decreasing rapidly with the interparticle distance r. Unlike the MaierSaupe interaction law used in the LL model (2.29), the interaction energy U is a function not only of the particle orientations u and u', but also of their relative position r, and is therefore spatially anisotropic. The anisotropy parameter v allows one to continuously vary the importance of the anisotropic (r-dependent contribution in the interaction law (2.34). Then, a nonzero v results in orienting effects at the sample surface [17,81] and in an anisotropy of Frank elastic constants. In fact, the Maier-Saupe interaction corresponds to the one-constant approximation while for v ^ the twist elastic constant is different from the splay and bend ones [129]. y ,S-, Sf' V; igre 28 Simle hexagoal lattice odel Having a spatially anisotropic pairwise potential, a question should be posed as to whether discrete lattice models can still be used together with such potentials. In principle, in this case artificial lattice approximation-induced bulk easy axes could appear, depending on the lattice geometry. Indeed, the effect of the LL potential is merely to impose parallel alignment of neighboring particles, regardless of their relative position, and for this potential none of the lattices imposes any bulk easy axes. Using the induced dipoleinduced dipole interaction together with, e.g., the simple cubic lattice (as in Ref. [81]) is inappropriate because easy axes appear along the three main lattice directions. Then the model is not capable of reproducing the soft curvature elasticity in the bulk and is hence useless for studying nematics. Alternatively, one can use the simple hexagonal lattice — see Fig 2.8. In this case for in-hexagonalplane molecular alignment, on the average bulk easy axes can be seen to vanish [17], and planar elastic deformations in nematics can be treated appropriately with this model system. The most striking leftover from the lattice approximation is the existence of periodic solid-like solutions for the director profile, which — at low enough temperatures — are more stable than the smooth nematic ones [17]. One can avoid, however, these difficulties by reducing the anisotropy parameter v to below v « 0.3. Then solid-like solutions disappear and the smooth nematic ones become stable, while the spatially anisotropic interaction character is still retained since v ^ 0. In fact, for v ^ 0.3 side-to-side molecular alignment is favored (similar to that of the Gay-Berne interaction [130]), promoting parallel ordering. This result is compatible with the one obtained in the mean-field analysis by Maier and Saupe, as well as with the behavior of the LL model. However, while disguised in the bulk, the anisotropic interaction character reappears in form of orienting effects close to the sample surface (absent in the LL model), which can be interpreted as intrinsic anchoring. To cover these phenomena qualitatively, it is sufficient to consider nearest-neighbor interactions. 2.2.3 onte Carlo simulation technique This Section will be devoted to Monte Carlo simulations that can be applied to the above model systems for finding equilibrium molecular configurations, as well as for the calculation of macroscopic observables. While molecular dynamics simulations aim at mimicking the "natural" dynam ics of the simulated system, the system evolution provided by Monte Carlo-type simulations is different. Molecular dynamics consists of the brute-force step-by-step solution of Newton's equations of motion, thereby tracing the system in time, which corresponds to a realistic experimental situation. In this case it is possible to calculate time averages of thermodynamic quantities (A). On the other hand, the Monte Carlo method can be interpreted as a procedure for sampling many-particle configurations from the statistical ensemble. This method then enables one to calculate ensemble averages (A) Then, assuming that the ergodic hypothesis for a given system holds (A = (A)), both molecular dynamics and Monte Carlo techniques should yield equivalent results for the averages of interest. However, the advantage of Monte Carlo methods is that — in contrast to molecular dynamics simulations — they do not require any calculations of forces and torques. Moreover, in highly simplified model systems like the lattice ones ("non-natural" by construction), Monte arlo simulations seem to be the method of choice. In lattice models the microscopic state of the system consisting of N particles is univocally defined by specifying all particle orientations u,. In the iV-particle configuration space the set of all orientations Uj can be given by a single long vector uN. In the LL model system (Sect. 1), for example, the Hamiltonian %{x\.N) in presence of an external field reads U{u = -e J2 P2(ui ¦ uj) - er^P^m • f), 35) the first sum Si*4l4(u) (2.37 In this way one chooses (samples) only important states (i.e., those having a high Boltzmann weight). The sampling scheme is hence called importance sampling. Now an algorithm is needed to generate the L sample points u^ according to the scheme just introduced. Instead of generating all of them at once, one can simulate a random walk in configuration space, producing a trajectory of states that obeys the probability distribution N(uN). Let the random walk start with the configuration u, referred to as "o" (old). Now generate a new trial configuration u', labeled as "n" (new), by adding a random displacement to u. The probability for such a move to occur is denoted by tt(o —>• n). It is a product of the probability to generate such a trial move a(o —> n), and the probability to accept it acc(o —>¦ n). The sampling scheme described by the probability matrix it must necessarily be ergodic ie., the scheme must be able to reach every accessible point in configuration space in a finite number of steps from any other point of that space. Here it has already been assumed that the simulated system is ergodic in itself. The old and new states have a different Boltzmann weight factor and thus differ ent probabilities N{6) and J\f{n) of finding the equilibrated system in one of these states. If J\f(n) > A/"(o), i.e., the new state is more likely, the move should automatically be accepted. Otherwise, it should be accepted with a certain probability depending on both J\f{n) and J(o). To find this acceptance criterion, one has to notice that the transitions o —)¦ n and n —>• o should not destroy the equilibrium probability distribution Jf(uN) once it has been reached. Thus the number of moves o —>• n must be exactly canceled by moves n — o, which gives rise to the detailed balance condition: Af(o)no^n) = J\f(n)in-K)). (238 Assume that a(o —> n) = a(n —> o), i.e, the moves o — n and the reverse move are attempted with equal probabilities. Then one can rewrite the detailed balance condition as acco -)• n) Af(n) = e_m{n)_n{o)] accn —> o) A/"(o) Consequently, the above procedure is in agreement with the detailed balance condition if the acceptance probability in trial moves with J\f(n) < Af(o) is set to acc(o —> n) = Af(n)/Af(0) = e~^n^~n^. At the same time, for trial moves with M{n) > A/"(o), acc(o —> n) = 1 must hold. Summarizing, a basic Monte Carlo algorithm for a A^-particle system has the form: 1. Take the configuration uN; calculate its energy ^u*) =%o) 2. Select one particle from the old configuration at random; move the selected particle randomly to obtain the new trial configuration u'N; calculate the energy of the new configuration %{xi!N) = %{n). 3 ccept the move uN — u' (o —> n) with a probability acco - n) = min [l, e'^^-71^] . Return to step of this algorithm. At each step of the algorithm the quantity A, whose average is to be evaluated, has to be sampled. It must be stressed that a sample must be taken also if the trial move has been rejected since 7r(o —> o) = 1 — En^o7r(° -^- n) ^ 0. Not sampling A in such cases can result in non-negligible systematic errors Moreover, the instantaneous value of the quantity A is subject to fluctuations Also the fluctuations of A can, beside the average (A), yield some interesting information. For example, in the canonical ensemble considered above, fluctuations of the total energy % are related to the specific heat Cy at constant volume <4 = (H) - (H) = kBTCv The ensemble average (.4) is due to fluctuations accompanied by an error AA Suppose that one has performed M simulation steps and sampled A M times. As the period of a fluctuation is usually several simulation steps (typically of the order 10), all M samples of A are not uncorrelated. Let r be the corresponding correlation time (measured in units of steps). Since M/t is then approximately the number of independent sample points taken, the error of (A) can be written as o A standing for {{A) — (A))1/. If (A) oc iV, o a oc \/N holds, and the relative error A.A/(A) decreases with the increasing number of particles as 1/y/ Moreover, it decreases also with the number of Monte Carlo steps as 1/y/M There is also the issue of how to generate trial moves — the Uj reorientations For instance, the Lebwohl-Lasher model simulations presented in this thesis have followed the Barker-Watts technique [131]. It was first applied to perform simulations of water structure [131], but was then used extensively together with the LL model. In all cases it proved to ensure appropriate and ergodic sampling in configuration space [91]. The technique consists of selecting an individual particle at random, then selecting one of the three fixed coordinate axes in the Cartesian laboratory reference frame (again at random), and performing a rotation of the unit vector Uj around the chosen axis. The rotation angle and direction are also chosen at random, the maximum rotation magnitude being an important simulation parameter. Indeed, if the moves are too small, the random walk will visit only a small fraction of the configuration space and the estimate for (A) will be poor. On the other hand, if the trial move is too large, it is very likely to be rejected due to the low probability J\f(n) of the new state. Although more sophisticated criteria for the selection of the trial move magnitude exist, here a "rule of thumb" will be adopted, stating that on the average 50% of attempted trial moves be accepted. Instead of performing individual trial moves, collective moves involving several particles can be performed as well. Efficient collective moves can, however, be constructed only if the displacements of individual particles are not chosen independently, but follow, e.g., Newton's equations of motion as in molecular dynamics simulations leading to the hybrid Monte Carlo technique Performing Monte Carlo simulations with the hexagonal lattice model (Sect. 2.2.2) a procedure similar to that used for the LL model can be adopted. The main differ ence between the two models comes from the fact that here nematic particles must be prevented from moving out of hexagon planes, thereby avoiding bulk easy axes. Again the Metropolis algorithm can be applied, however, the generation of trial moves has to be reduced to a simple particle rotation, so as to meet the in-plane constraint for particle orientations. Another important topic are the initial conditions. In an ergodic system, after equilibration the simulation results are supposed not to be initial configuration-dependent. Yet some caution is necessary whenever the system can be trapped in metastable states (especially if the simulation run is too short). Finally, it is necessary to specify the quantity A to be sampled during the MC evolution provided by the Metropolis algorithm. In analyses of confined nematics, it is convenient to accumulate the ordering matrix Q components (2.5) to obtain the order parameter, biaxiality, and director maps. Further, one can define and accumulate PI = |(3(v • u,)2 — l)j, where v is a fixed unit vector (also spatially-dependent if needed), quantifying the degree of order with respect to a given direction (v) or a given pattern v = v(r). Moreover, various types of positional and orienta-tional correlation functions can be sampled (only the latter being relevant in case of lattice models). Another important class of observables are the ones detectable experimentally. Some of the examples are 2H NMR line shapes, electric capacitance and transmitted polarized light intensity. They will be discussed in more detail in Sect. 23 and later on. 46 Theretcal background 2. Experiental observables An essential part of any theoretical or simulation work is to bridge the gap between these approaches and experimental observables. Attempting to do this, from the broad variety of experimental techniques listed in the introductory Chapter, here three will be treated in more detail: deuterium nuclear magnetic resonance, measurements of electric capacitance, and measurements of light intensity transmitted through a sample between two crossed polarizers 2.3.1 2H NM Deuterium nuclear magnetic resonance (2H NMR) is a powerful technique, frequently applied to investigate confined nematics such as PDLC and systems with dispersed polymer networks [25,40,132,133]. The technique is very convenient for the study of such heterogeneous systems since, if applied to deuterated nematics, it only gives direct information on the behavior of the confined liquid crystal — rather than on the non-deuterated confining matrix. Further, it is applicable also for small, i.e. submicron cavities, where optical methods fail to yield useful information because the light wavelength is too large in comparison to the cavity diameter. Moreover, 2H NMR spectra can give a complete idea about orientational molecular ordering, that is both about director configurations and dynamic processes like fluctuations of molecular long axes and translational diffusion. Figure 2.9 2H NMR and geometry: molecular long axis, n nematic director and B spectrometer magnetic field. A 2H NMR spectrum of a selectively deuterated nematic in the bulk isotropic phase consists of a single line whose position in the spectrum is determined by the Zeeman splitting of deuteron energy levels in the spectrometer magnetic field. The corresponding line width is well below 2tt x 100 Hz. Since deuterons possess a nonzero quadrupolar moment and interact with the electric field gradient (EFG) of C-H bonds in nematic molecules, there is an additional quadrupolar perturbative contribution to their energy levels. This contribution is averaged out by molecular motions in the isotropic, but not in the nematic phase. Indeed, once in an undis torted nematic, the single narrow line splits into a doublet, the frequency splitting now being typically of the order of ujq ~ 2tt x 40 kHz. In general, the splitting ujq depends on the relative orientation of the EFG tensor and the direction of the NMR spectrometer magnetic field B. In deriving the expression for uq, one usually assumes that nematic molecule exhibit fast rotations, mainly around the long molecular axis. These occur on a time scale much faster than any other motion type and lead to an averaged EFG tensor that is uniaxial, with the principal axis along the long molecular axis u. The instantaneous quadrupolar frequency splitting so depends only on the relative orientation of u and B, and is given by Qq = ±5uq | [3(u • B/_B)2 — 1], where 5u>q represents the maximum effective splitting (averaged over fast molecular rotations). In a ne matic, however, molecular long axes u are not fixed and fluctuate also themselves around the director n. An average over these fluctuations can be easily performed only if they are also rapid enough on the NMR time scale given by t0 ~ 27t/5o;q and results in a reduction of the quadrupolar splitting. Neglecting biaxiality in molecular ordering this effective splitting can be written as see for instance Refs [54135 uQ = ±5uQS[C°*2°-\ 24 where S = (|[3(n • u)2 — 1]) is the usual uniaxial nematic order parameter (2.2) and 6 the angle between the local director n and the spectrometer magnetic field B see Fig. 2.9). In a confined system, both n and S become position-dependent. Since the NMR spectrum corresponds to the overall response of all molecules in the sample, the n = n(r) and S = S(r) dependences reflect in NMR spectra via the spatial dependence of the local value of the quadrupolar splitting ujq = ojq{y). The resulting characteristic spectral patterns make it then possible, e.g., to distinguish between different structures inside PDLC droplets, or to monitor the switching of the molec ular orientation in a nematic with dispersed polymer fibers. The identification of di rector configurations can, however, become very problematic in small cavities where translational self-diffusion significantly affects the 2H NMR line shape. One of the ways to calculate spectra is a simple superimposition of individual molecular static Lorentzian lines from all over the sample positioning them into the spectrum either according to n(r) and S(r) profiles [136], or according to molec ular configurations obtained from MC simulations [94,99], in both cases following Eq. (2.41). Note that such a procedure is correct only in absence of significant dif fusive motion or, alternatively, in systems where the confinement is less severe and the spatial dependence ujq = ojq(v) is weak. To properly take into account any kind of motional effects, it is convenient to use a semiclassical approach with the time-dependent deuteron spin amiltonian 35 This approach consists of generating the relaxation function t) = exp(zo;zt)(exp(z|iQQ[r,(t')t']^')) 242) with fžg[rj(i), t] = ±5luq | [3(u, • B/.B)2 — 1], the "instantaneous" quadrupolar split ting not to be confused with uq from Eq. (2.4) where an average over fluctuations of Uj has already been performed. Further, in Eq. (2.42) ujz denotes the Zeeman frequency, while the brackets {...)j stand for the ensemble average over all molecules in the sample. The resonance frequency of the th molecule located at r, is given by coz + fŽQ[rj(L),*] and depends on time (t) through Uj = iij[rj(L),L], i.e., the in stantaneous orientation of the molecular long axis. Generating G(t) is equivalent to mimicking an actual pulsed-NMR experiment. In fact, placing the nematic sample into the spectrometer magnetic field B gives rise to a net magnetization M along B, coming from 2H spins. Then, applying a so-called tt/2 magnetic field pulse with a coil mounted perpendicular to B, M is switched into a plane perpendicular to B. Now, to withstand the torque exerted by B, M starts to precess (still in the plane perpendicular to B if spin-lattice relaxation phenomena are neglected). Then, in the coil (same as used to apply the 7r/2-pulse) a periodic induced voltage is detected and coincides with the relaxation function G(t) introduced above. Then the H NMR line shape I(co) is calculated as the Fourier transform of G(t) namely I(u) = f et G(t) dt (243 This procedure for calculating I (to) can establish a correct link between any type of molecular motion and the spectra. The information on the motion enters G(t) via the Uj = Uj[r,(L),*] dependence. The motion is dominated by two processess: fluctuations of molecular long axes and translational self-diffusion. Neglecting for the moment the diffusion, the coordinates of a given molecule rj can be taken as fixed and time-independent during the G(t) acquisition. The long molecular axis u,, however still fluctuates around the local average, i.e., the director n(r,), and thus still depends on t. Hence, the time dependence in Qq is caused solely by Uj fluctuations, while the average Uj (the director) remains unchanged. The characteristic time scale for such fluctuations tF in a typical liquid crystal is of the order of ~ 10~8 s and is much shorter than the typical NMR time scale t0 ~ 2.5 x 10~5 s [41]. If now translational diffusion is considered as well, also the molecular coordinates change during the NMR experiment: r, = i"j(i). Thereby the average u, for a given molecule changes with time since — in a confined system — n = n(r,). Then the instantaneous Qq depends on t also indirectly through r, = r,(t). The typical time scale tD for diffusion to yield a molecular displacement of approximately one molecular length (~ 1 nm) is, like tF, also of the order of ~ 10~8 s [41]. It is, however, more relevant to know the time t'D needed for molecular diffusion to yield a displacement over which the average molecular orientation n(r) changes noticeably. The time scale t'D obviously varies with system size and can become much larger than tD, especiall in systems where the confinement is weak. In order to estimate how dynamic processes influence the spectra, it is necessary to compare their characteristic time scales to the characteristic NMR time scale t0 If the molecular motion is sufficiently slow on the t0 time scale, the spectra can be calculated as I(uS) = (S[co — uz ± |^q(i"j)|])j [40], i.e., it is possible to use the static approach used in Refs. [94,99]. If, at the other extreme, molecular motion is very fast on the t0 scale, the spectrum is completely motionally averaged and consists of a sharp doublet I(co) = 5[u — coz± \{^Q(Ti))\] whose peaks are positioned at average frequencies loz ± \(uQ(ri))i\ [0] 32 Elecic cpaitnc Alternatively, confined nematics can be studied also by electric capacitance mea surements. As already discussed, the anisotropic shape of nematic molecules gives rise to an anisotropy in the dielectric constant. In an external electric field E the dielectric displacement D can be written as D = eof E, where e is the dielectric constant tensor. Therefore, for a given E the displacement D and the effective value of the dielectric constant depend on the relative orientation of the principal axes of e (being in an intimate relation with the director n) and E. As a consequence, a mea surement of electric capacitance can yield information on the orientation of nematic molecules, i.e., n. Inversely, capacitance can also be predicted from known direc tor profiles derived phenomenologically [39], or directly from molecular simulation providing configurations of molecular orientations u(r). Consider now a nematic slab of thickness d between two capacitor plates where an external electric field E is applied along z, the unit vector along the sample normal while the plates are parallel to the rcy-plane. For simplicity, it will be assumed that the local electric field is everywhere directed along z. Although in an inhomogeneous dielectric sample this is not strictly true, a full solution of electrostatic Maxwell equations to determine the correct E(r) dependence is beyond the scope of this analysis. Given the above assumption, one can write for the normal component of the dielectric displacement at the site of each molecule Dzx, y) = e0ex, y, z)Ezx, y z) where the molecular ex, y z) is defined as x, y z) = e'± + (ej, - e'±)u ¦ z)2 (2.44 Here eji and e'± denote the eigenvalues of the molecular dielectric tensor, and u stands for the orientation of the long molecular axis. Following the Gaufi theorem §s D • dS = 0 in absence of free (e.g., ionic impurity) charges, and provided that spatial variations of u within the xy-plane are weak, for given x and y Dz can be taken to be ^-independent. Then Dz has to be same both for z = 0 (at the sample surface) and for any other value of z, resulting in e(x,y z)Ez(x,y,z) = e(x,y,0)Ez(x,y,0). Moreover, the voltage V across the sample is independent of where at the plates it is measured, i.e., V = /0 Ez(x,y, z)dz = J0 Ez(0, 0, z)dz. At the same time, the electric charge loaded onto a a x b portion of the capacitor plates is given by e = /0a dx /0& dy Dz(x,y). Assembling all the above relations the capacitance of the sample can be identified as y = e0dx dy(-^-] (245 J x,yz)J This formula can be interpreted as follows. The aforementioned assumptions are equivalent to replacing the sample with a circuit of small capacitors, each with its own capacitance proportional to e(x,y,z). These capacitors are connected in series along the z-axis and, after this, these capacitor series are connected in parallel within the xy-plane. The expression (245) then represents the effective capacitance of such circuit tic 33 lad li anmision Historically, optical methods were the first to reveal the unusual phase behavior of liquid-crystalline materials [1,2]. Like the static dielectric constant, its high-frequency counterpart — the refractive index — is anisotropic, too. Solving the electromagnetic wave equation in an anisotropic dielectric medium, one finds two principal ray types with different polarization, ordinary and extraordinary, characterized by different values of the refractive index and hence also by different propagation speeds. In the following, exclusively polarized and monochromatic light beams will be considered. Note that the polarization in an anisotropic medium is given by the dielectric displacement vector D that is perpendicular to the direction of beam propagation, k (the wave vector). As nematic molecules are anisotropic in shape, it is important to know how the polarization D of incident light is oriented with respect to the principal axes of the dielectric tensor. For uniaxial molecules only two out of three eigenvalues are different, en° and 6j°, symbols || and _L again referring to the molecular long axis u. The eigenvector corresponding to the non-degenerate eigenvalue en0 is identified as the optical axis (here denoted by the unit vector a) and in a macroscopic sample coincides with the director n, while at the molecular level it can be taken along the long molecular axis u. If now the incident light polarization D — always perpendicular to k — is perpendicular also to the optical axis a, one is dealing with the ordinary ray. The ordinary refractive index is given by n0 = y^T and does not depend on the relative orientation of k and a, i.e., on cos6 = a • k/fe. On the other hand, if D is lying in the plane defined by a and k, the corresponding ray is referred to as extraordinary For the extraordinary ray, however the (extraordinary) refractive index ne becomes ^dependent [137] and satisfies 1 cos sin . „ . -;--------------------• 246 „2 oo n Consequently, any arbitrary polarization vector D can be decomposed into ordinary and extraordinary polarization components, each of them propagating through the sample with the corresponding index of refraction. Note that if the light beam is directed along the optical axis (k||a) the refractive indices of both components are equal, n0 = ne = v/e±J- The polarization vector D(r) at a given point r in the sample can be conveniently described in terms of the Jones vector with components in the plane perpendicular to k. These can also be complex, which corresponds to a general elliptical polarization of the light. Assume again that the nematic is confined to slab geometry and that z is the slab normal, coinciding with the incident light direction, k. Assume further that diffractive and interference phenomena can be neglected and let the sample be split into thin layers normal to k, of thickness 5z each. Considering a single layer at z, the polarizations of incident and propagated light, D(z) and T>(z + 5z) respectively, and are related via T)(z Sz) = A(z)T>(z), where the matrix A(z) is given by i2n[ne(z)-n0]Sz/\ _ (247) i = (z) are the components of a, and A is the light wavelength in vcuum Propting through the whole sample, the incoming and outcoming polarizations D0 = (z = 0) and D' = D (2 = d) are related by D' = B D0, where B is given by the mix poduct B = Yin=i &{nSz) and i = d/Sz is the numbe of layes in the mple. A typical expeimentl set-up consists of putting the sample between two polarizes cossed at ight ngle and mesuing the intensity of transmitted light I oc Da{Da)* = [$t(D'a)]2 + [^(Da)]2. He Da is the component of the Jones vector D' along the diection of the polarizer analyzing the outcoming light and (D'a its complex conjugate Assuming that the director is homogeneous (and a constant nd directed at ngle ( with espect to the polarize, the transmitted intensity n be witten 138 7q sin2 (2(p0 sin2 (A$/2) (2.48 with Iq denoting the intensity of the incoming polarized light. Mreover, A$ is the phse difference between the ordinary and extraordinary ray accumulated upon passing through the sample (also called birefringence) and is given by A$ = (2ir/X) /0 [(z) — n0]dz. According to Eq. (2), the maximum output intensity is obtained for cp = tt/4. The set-up described above has been used to monito the switching of the molecular orientation in the Freedeicksz transition [116] to measur the temperature dependence of Frank elastic constants, to measure the nchoing strength [6], s well s for the detection of suce-induced paranemtic oder [2 to nme just few exmples Figure 2.10 Defect structure in a nematic with polymer inclusions as viewed between crossed polarizers (photo by L Drevenšek) In the above analysis it was tacitly ssumed that the optical axis (and the nematic diector) depends only on the norm ^-coordinate, but not on the in-plane ones, x and y, s, for example, in PDLC droplets o nematics with dispersed polyme networks f the diecto modultions in the nd y diections are occurring on 52 Theretical background length scales much shorter than the light wavelength, in an experiment one measure the light intensity averaged over the part of the xy-pl&ne occupied by the probing light beam [26]. If, on the order hand, n variations are smoother, one ends up with a rc-and-y-coordinate-resolved intensity pattern I(x, y), which is the underlying mechanism, e.g, for the recognition of nematic structures in polarizing microscopy experiments [38] — see Fig. 2.10. ematic slab This Chapter will be concerned with the simplest confining geometry, a nematic slab. The motivation for the following study comes from experimentally detected subsurface variations of the average molecular orientation, as discussed in the Introduction. In the first part of the Chapter (Sects. 3.1 and 3.2), the nematic orientation close to a flat solid substrate will be investigated by means of a Landau-de Gennes phenomenological model. In particular, it will be shown that when the splay and bend elastic constants are different from the twist elastic constant, a spatial variation of S near the bounding walls yields a subsurface deformation in the nematic director (and vice versa), as well as an intrinsic contribution to the anchoring strength. An expression for the quasi splay-bend elastic constant will be derived and the effective anchoring — a combination of the external and the intrinsic contribution — will be analyzed. Matching elastic and magnetic effects, the corresponding effective extrapolation length will be estimated. In conclusion, a non-planar twisted sample will be considered, analyzing the possibility of finding a localized subsurface deformation also in the twist angle profile. Such twisted-cell geometry is particularly interesting because it is widely used for display applications. In the second part (Sect. 3.3), a microscopic Monte Carlo study of the intrinsic anchoring at a planar free nematic surface will be presented. To estimate the corre sponding extrapolation length and its temperature dependence, a bend deformation in a hybrid celllike nematic slab will be imposed (Sect. 2.1.4), while the simulation will be based on the simple hexagonal lattice model with nematic particles interact ing via the spatially anisotropic modified induced dipoleinduced dipole potential (see Sect. 2.2.2) 3.1 Planar cell: Landau-de Gennes analysis Consider now a nematic sample sandwiched between two parallel plates. Let the sample normal be the z-axis, with the plates positioned at z = ±|. Further, consider a uniaxial nematic and allow for planar deformations only. Then one has S = S (z) and n = n(z) = (sin (f)(z), 0, cos (z)), (f>(z) being the angle between n and the sample normal. Parametrizing the tensorial order parameter Q (2.4) with S(z) and (z) the Landau-de Gennes free energy density (2.13) in the oneconstant approximation tic sla f0(ST) + ^L1S' + -4L1Scj> 3.1 where the prime denotes a derivative with respect to z. Note that in the planar geometry discussed here the contribution giving the K2± term vanishes identically Expression (3.1) has been considered by different authors, mainly to describe the influence of the spatial variation of the elastic constant on the nematic tilt angle profile (f)(z) [70,139-143]. A simple variational analysis shows that in a symmetric sample in the strong anchoring case, in which values at both walls of the nemati slab are equal and fixed, a possible S = S (z) dependence does not induce any subsurface deformation. In fact, a minimum of / (3.1) corresponds to = 0. The spatial variation of S can induce an additional (^)-variation only if the deformation is already present. Consider now a more general case, where L2 + L3 ^ 0 and Ku = K33 ^ K2. I the planar eometry the free energy density iven by (2.13) has four terms = fQ(S T) + Af S) + /( S) + f3( S, S) (3.2) introducing three elastic contributions (/ f /). The f\ term quadratic in S depends also on /(^ = ^{ ^^cos i)} 3.3 he Frank elastic term /2(\S) = ?LlS ^±i2 3. has similar structure as the correspondi term i 3.1. The third part of the free ener h{ S, S) = ~(L2 + L3) sin(2 SS 3. is absent in the equal elastic constant case, coupling variations in (f> and S. If the substrates impose a scalar order parameter different from the bulk one, the free energy / is no longer minimized by a solution with ' = 0. Hence a scalar order parameter spatial dependence — usually localized near the substrates — induces a spatial variation of the tilt angle [69]. The influence of /3 on structural transitions in nematic liquid crystals has been partially analyzed by Jerome [01]. 3.1.1 Quasi splay-bend elastic constant and intrinsic anchoring energy Assume for the moment that the nematic sample occupies the z > 0 halfspace and that in the planar onedimensional case S(0) = S and (0) = (f>0 are fed by short-range forces. In the bulk the value of S depends only on temperature and is denoted by Sf, [found by minimizing (2.8)]. Assuming further that 5& ^ So, S(z) relaxes to S& over a length which is of the order of the correlation length L (2.11). This characteristic length, in general, does not apply to 0-variations, since in absence of external fields there is no preferred orientation of nematic molecules 0j in the bulk of the sample, which would be analogous to Sf,. Bulk ^-variations, i.e., bulk elastic deformations, caused by external fields or confinements, occur usually over a scale considerably larger than L. In the following, it will be shown that in a nematic layer thick compared to L it is possible to include a spatial variation of S in two additional surface energy terms, one corresponding to intrinsic anchoring and one to a quasi splay-bend elastic term. To show this, one has to consider the second and the fourth term of / defined in Eq. (3.2). The total energies per unit surface coming from these contributions are given by W = J0° fi(z)dz and Q = /0° f3(z)dz. hen for W one obtains ^{l + ^±^cos^) + \)}\*(S'2) (3.6) where (S2) (A)" /0A* S'2dz, A being a length of the order of a few L. In Eq. (3.6) z* is an effective distance in the range (0, A). Since A* is a mesoscopic length, W can be considered as an additional surface energy, whose anisotropic part \Wi cos 4>{z can be interpreted as intrinsic anchoring with a strength defined by Wt-\L2 {b )~-|L2------------- -----—------^IfJ 03-7) assuming that (S) ~ (St, — So)2/(X) and taking into account expressions for the elastic constants derived in Sect. 2.1.3. The sign of L2 + L3 determines the direction of the easy axis, while the anchoring strength Wi is proportional merely to the modulus of L2 3. he Kleman-de Gennes extrapolation length / Ku/Wi is then given by |An — A22| Ob — boy and depends strongly on the difference Sf, — So. Also the integral of the term coupling the order and angle variations Q can be rewritten in an effective form as G ~(L2 L3) [sin(20) of the splay-bend elastic term introduced by Nehring and Saupe [52,53]. he corresponding quasi splay-bend elastic constant can then be identified as a |i, 3)(^ - S ^-^11 - |)]. (3.10) tic sla It should be stressed that Q is only effectively a surface term and cannot produce any divergent subsurface deformation, as it is the case with the ordinary K13 term. Essentially, it is a bulk term effective only in a thin layer of thickness A*. There fore the Q term-induced subsurface deformations are stabilized by the bulk elastic terms f\ and /2. The detailed director profile, requiring a complete free energy minimization, will be discussed in the following. The intrinsic anchoring strength Wi given by Eq. (3.7) and a quasi Ki3 given by Eq. (3.10) are both temperature-dependent because both the bulk value of the scalar order parameter S& and the length A* oc L exhibit a significant temperature dependence on approaching the nematic-isotropic (NI) phase transition. According to the pseudomolecular model proposed by Vertogen [144,145], it is possible to evaluate elastic constants if the interparticle interaction U responsible for the formation of the nematic phase is known. In the framework of Vertogen's model it can be shown that if U is spatially isotropic (i.e., molecular orientations and their relative position are not coupled), the relations Kn = K22 = -K33 and Ki3 = 0 hold [146]. This is in accordance with the above result that the quasi K± vanishes in the oneconstant approximation ie for L2 + L3 = 0 3.1.2 Numerical solution of the variational problem To complete the previous approximate analysis, in the following a full minimization of the total free energy will be carried out by solving numerically the corresponding variational problem. Again a nematic slab of thickness d is considered, with the confining surfaces at z = ±|, and again the deformation is assumed to be planar. To solve the minimization problem, Euler-Lagrange equations and the corre sponding boundary conditions must be derived first. The total free energy to be minimized can be written as F = J fB[{±i) fa S(±l), S0dS, (311) where fB = /0 + /1 + /2 + /3 and fs (specified later) are the bulk and the surface free energy density, respectively, while 0O and So denote the substrate-induced values of

o and S(±|) = S0 while in the weak anchoring case they become B\ dfs _ b\ dfs The above system (3.12-3.13) was solvd numerically by means of the relaxation method for boundary value problems [147]. First the case with infinitely strong anchoring will be considered. After this, it will be upgraded to a more general situation with an arbitrary strength of anchoring 5.86 5.76 60 •8 5.68 5.6 -0.50 0.37 CO 0.36 0.35 -0.50 -0.48 -0.46 -0.50 -0.48 -0.46 z/d z/d Figure 3.1 Nematic slab: (±f) is fixed to 0 ((f>b being the bulk tilt angle), is found to be proportional to the quasi Kis elastic constant introduced above, similarly as in the case of normal Ki3 elastic constant. If the amplitude of the S-variation (ie, S& — So) is small enough to neglect the variation of the Frank elastic constant K o Sf close to the interface, the same relation as for the 58 Nematic slab ordinary K\3 may be used to approximately predict the deformation amplitude [149]: A^«-^sin^(±f) (314 ote that whereas the deformation stabilization in Ref [149] is governed by second-order elasticity, it is here by the positive definite first-order terms /i oc S" and f oc 4> introduced in Sect. 31.1 The quasi Ki3 elastic constant given by Eq. (3.14) depends on both L2 + L% and the difference between the bulk and the surface scalar order parameter The numerical solutions confirm that if L2 + L3 changes sign, the deformation amplitude A(j) oc K13 changes sign as well. If L + L3 = 0, the subsurface deformation vanishes and A = 0. Further, the change in sign of A occurs also if the sign of So S& is changed. From Fig 31 it can be deduced that also the characteristic length of the subsurface distortion depends on L + L3. In comparison to cases with negative L+L3, positive L2+L3 yield larger proportionality constants in the stabilizing terms /1 and /2 [see Eqs (33-3)], indicating that stabilization effects for L2 + L3 > 0 are stronger than for L2+L3 < 0. Hence the corresponding deformations are weaker, ie. occurring over a larger distance and having smaller amplitude, the former holding for both (f)(z) and S(z) profiles, while the latter is true for (f)(z) profiles only, since Si So is fixed if anchoring is strong onsider now a more realistic nematic-surface coupling, ie., an anchoring situation where actual surface values S*(±|) and (±f) are allowed to vary. Any deviation of these values from the surface-induced ones (So and 0, respectively) is penalized with an increase of the anchoring energy This short-range interaction free energy is usually modeled by (217) [22], ie fs = lWe[l{{±% S2} (±lSo cos (±|) = 4>o] its form is /5 o [S(±|) — So]2. In the most general case, however, S and (/»-variations are coupled in the anchoring energy (3.15) The weak anchoring case has been considered previously in Ref. [141] in the one-constant approximation, in which the quasi splay-bend elastic constant is identically zero Now this analysis shall be generalized by allowing L2 + L3 ^ 0 The same Euler-Lagrange equations as in the strong anchoring case were solved, however, with modified boundary conditions. As the actual surface tilt angle is not fixed any more, effects of S'-variation-induced intrinsic anchoring can now be revealed. The easy axis for this intrinsic anchoring contribution can be either planar (for L + L3 > 0) or homeotropic (for L2 + L3 < 0), as it follows from Eq (3.6) The calculated (f)(z) profiles confirm this prediction, which is evident from Fig 3: for L + L3 < 0 0 ^(±|) > The subsurface deformation is still present and behaves in the same manner as in the strong anchoring case However, suppsed the same S& and S n in te sg a case since |S& — S(±|)| < |S& — S0|. It should be stressed that molecular models where, for instance, the intermolec ular interaction is described as a superposition of the Maier-Saupe and the induced dipole-induced dipole coupling (2.34), yield L2 + L3 < 0, i.e., Ku = K33 < K22 [146] which corresponds to a homeotropic easy axis in our study. Thus further discussion will be restricted only to cases with L2 + L3 < 0. 5.64 -0.50 -0.48 -0.46 z/d 5.72 -0.50 -0.48 -0.46 z/d 0.372 ^0.368 0.364 -0.50 -0.48 -0.46 z/d 0.382 to 0.378 0.374 -0.50 -0.48 -0.46 z/d igre 3.2 Nematic slab: (j){z) and S{z) profles in the weak anchoring case (we = 10); Disordering (So = 0.2, left) and ordering substrate (S = 0.5, right). All other parameters and labels are same as those in Fig. 3.1. A suitable method to estimate the strength of the effective anchoring is to investigate its competition with a distortion-imposing magnetic field H [17] tilted at an angle a with respect to the surface normal. Therefore the magnetic energy term (221 fm -/XaS(n • H -jXaSH cos ((z) - a) (36) should be added to the bulk free energy density (3.2). Then 4>{z) and S(z) rofiles can again be calculated by solving the Euler-Lagrange equations, which now differ from those derived in absence of the additional magnetic free energy contribution The influence of subsurface deformations on the large scale director profile enters only via the effective intrinsic anchoring contribution. Therefore, ignoring the thin subsurface layer in which the subsurface deformation occurs one can for small fit 60 Nematic slab the calculated 4>(z) profiles by the ansatz (see Fig. 3.3) (j)(z) =a + A cosh(z/gm) cosh(d/2Lm) (3.17) following from the standard Frank elasticity, the parameter A being related to the amplitude of the deformation, d denoting the sample thickness, and Lm the characteristic length of this field-induced deformation — the magnetic coherence length (2.23) [4]. Fig. 3.3 (b) shows the enlarged subsurface region of Fig. 3.3 (a), in which the ansatz (3.17) describing the macroscopic director profile fails to match with the calculated profile. Since this region is of microscopic — nanometric — thickness, it will be neglected in the determination of the anchoring strength, as already stated above. If (j)0 is the direction favored by the effective anchoring, the parameters o the fit (b) -0.5 -0.4 -0.3 -0.2 -0.1 0 z/d -0.50 -0.48 (c) -0.46 z/d -0.5 -0.4 Figure 3.3 Calculated director profiles (dots) in magnetic field compared to the hyperbolic cosine fit (solid line): we — 5, L2 + L3 — —-^l? Sf, ~ 0.3747, and So = 0 [cases (a) and (b); no external anchoring] or So = 05 [case (c); homeotropic external anchoring with 0O = 0°]. The sample thickness is equal to d = l/zrn, the magnetic field direction a = 01(180°/it) ~ 573. The magnetic field strengths expressed in terms of the coherence length Lm amount to ~ 65 nm, 90 nm, 205 nm, 290 nm, 650 nm, and 920 nm; the first value corresponding to the top and the last to the bottom curves of Figs, (a) and (c) Comparing cases (a) and (c), it is evident that the external anchoring is considerably stronger than the intrinsic one. Fig. (b) presents the enlarged section of Fig. (a) marked with a dashed line. All other parameters are equal to those of Fig. 31 A, ma — a yd te effecve en lh [ This anchoring is a superposition of the intrinsic and external contribution. From the analysis performed above, it is possible to estimate extrapolation lengths for both sources of anchoring separately. For intrinsic anchoring one can rewrite (3 as [S ~ S(±l 2 + (319 Provided that A* is known, the approximate value /j (3.19) can be compared with the "measured" one, /e// (3.18). Similarly it is possible to derive an estimate for the external anchoring extrapolation length le = Ku/We The external anchoring strength WE can from Eq. (3.15) be identified as WE = §WeS(±|)S0) while the elastic constant K1X is still given by K = |Sf (2Li + L2 + L3). In terms of the dimensionless anchoring strength we = ed/Li the length l can be expressed (for So + 0) as le =-----^—r- h±±± (3 Assuming that both intrinsic and external anchoring have the same easy axis (e.g homeotropic), the effective anchoring strength can be written as Weff = W + We hen for the corresponding extrapolation lengths the relation 1 . r (3 holds. If, e.g., le (z) and S(z) profiles in magnetic field were calculated for different values of the field strength H, the surface-imposed order parameter So, and anchoring strength we (the example we = 5 is given in Fig. 3.3). In all cases L2 + L3 = —L\ was assumed, resulting in Ktl < K22 and yielding a homeotropic easy axis for intrinsi anchoring he Landau parameters a, B, C, and the temperature were chosen such that S 3747 he estimates for the "measured" effective extrapolation length 62 Nematic slab We S0 s(±i) k le (nm) leff (nm) 0.3720 58000 A* 400000 0.5 0.3756 500000 A* 374 375 0.3613 2350 A 16700 0.5 0.3791 22000 A* 74 75 10 0.3484 610 A* 4300 10 0.5 0.3832 5800 A* 37 37 50 0.2606 33 A* 205 50 0.5 0.4072 400 A 100 0.1861 12 A* 63 100 0.5 0.4255 160 A* Table 3.1 Effeive anchoring extapolaion lengths le/f cmpared to the values li and le, predicted for intrinsic and external anchoring, respectively All estimates for leff with So = 0 refer to pure intrinsic anchoring, while the ones ith So = 0.5 refer to a superposition of intrinsic and external anchoring, where the latter prevails. Easy axes for both kinds of anchoring are homeotropic. The angle between the magnetic field direction and the sur face normal is equal to a = 0.1 (180° /tt) « 5.73°, the bulk value of the order parameter to Sb f» 0.3747, and the sample thickness to d = 1/m. leff are listed in Table 3.1. The results for So = show that if the coupling with the surface has a strength we < 50, the intrinsic anchoring is rather weak (Ii > 100 nm). Its strength increases with increasing we as St, — S(±|) increases which is in agreement with formula (3.19). However if So ^ 0, the external contribution to the anchoring is nonzero as well and is for, g So = 0.5 considerably stronger than the weak intrinsic part [compare Figs. 3.3 (a) and (c)]. Consequently, leaving other parameters unchanged the effective extrapolation length decreases significantly in comparison to the So = 0 case and now only we < 5 yields extrapolation lengths of the order of those observed experimentally (> 100 nm) [6]. Since the external contribution to the effective anchoring seems to completely overwhelm the intrinsic one, one cannot expect to observe any temperature-driven anchoring transitions due to their competition Comparing the predicted values for le in cases with So = 0.5 [Eq (3.20)] and the "measured" effective ones [Eq (3.18)] a very good agreement is observed (see Table 3.1), which again shows that in these cases the intrinsic anchoring is negligible with respect to the external one Further, setting So = 0 and considering intrinsic anchoring alone, the agreement of predicted [Eq (3.19)] and "measured" values of /, can be achieved by setting A* ~ 6 7 nm which is comparable to the thickness of the layer in which the S and (j> variations occur. Note also that in all cases the deformation strength of the subsurface deformation is rather small For instance, for we = 5 and So = 0.5, yielding a still reasonable extrapolation length and close to Ne 0o = 7!"/4 one finds d/dz w3x 10_4//?0 '(z),u(z),S(z),S(z)] dz. (3.2 J-2 Minimizing (3.24) with respect to S(z), (z), and u(z) yields the following Euler-agrange equations (EE) L d___ L d_L_ _ _ d _ dŠ~dz"&~0, djd~z~0, ^~Tzd^~°- (3-25) Since in / there is no explicit dependence on u (also for chiral nematics), the last of the above EE can be rewritten as — — = « = const. (3.2 Taking into account Eq. (3.23), one can find that a = - sin ^{U + ^ cos } (3.27) which, once integrated with respect to z over the whole slab, results in coil u($) = a [d/2 -------------^--------------- (3.28) J-d/ ILi+ ^ cos ) sin If w(|) = u(|) is chosen, it follows that a = 0 must hold and from (3.27) also that a/ = 0 everywhere in the sample. Hence in this case the twist deformation is absent and the director is lying in a plane — the problem is degenerate with respect to u; see previous Section). If, however, w(|) 7^ u(—|), a = 0 no longer holds and the twist deformation is present. For such a twisted nematic slab the ELE (3.26) must be solved numerically. As weak anchoring would only reduce the deformation, here we will concentrate on strong anchoring boundary conditions, where the actual surface values of S, (j), and u cannot deviate from those imposed by the substrate. First consider 5-profiles in a twisted nematic sample. The scalar order parameter always relaxes monotonously from the surface value Sq to the bulk value S& given by Eq. (2.9). Like in the planar case, the variation here occurs over a distance characterized by the nematic correlation length L (2.11) (see Fig. 3.4, bottom) and is hence again localized to a thin subsurface layer. For the present choice of a, Ll5 and T T*, its thickness is about 20 nm. S'-profiles in presence of the twist deformation are very similar to those reported in the previous Section. In general, any variation of (f> or u affects S'-profiles only very weakly since the free energy contributions associated with the elastic deformation (/l /2, and /3) are considerably smaller than the homogeneous one (/0). On the other hand, in a twisted nematic slab there is quite a significant change in tilt angle profiles (z) in comparison to the non-twisted case. hese profiles show -0.46 -0.45 z/d 0.49 -0.48 -0.47 -0.46 -0.45 z/d Figure choring. or So = tively]. = 16 3.4 (z) and »S^-profles in a twisted nematic slab with strong an <^(±f) = 45°, w(-|) = 0°, w(|) = 45°, Sb » 0.375, S0 = 0.35 (left) 0.4 (right), and L2 = Li, 0, — Li [curves (a), (b), and (c), respec-The sample thickness is equal to d = 1 //m, a = 0.13 x 106 J/m3K, x 106 J/m3, C 39 x 106 J/m3 T-T* = 04 K, and Lx = 10"11 N considerable variation of 4> that spreads over the whole slab (Fig. 3.4, top) — even in the symmetric anchoring case in which 4>(—|) = <^(f). The source of this delocalized deformation is the coupling between a/ and in the Frank elastic term /2 (3.23). Namely, since the a/-term — nonzero whenever the twist deformation is present — always gives a positive free energy contribution, the proportionality factor appearing in front of co' must be as low as possible, i.e., for L2 = 0, ±Li (as chosen for profiles plotted in Fig. 3.4) \(j>\ must decrease. However, this decrease is compensated by the (j)2 term which is also present in /2 and gives a positive free energy contribution as soon as (j> varies. ote that because the ratio of the proportionality constants belonging to a/ and 0). Further, it should be noticed that for L'2 ^ 0 the coupling between $ and S' described by the /3-term yields a localized Svariation-induced subsurface deformation in to be constant throughout the sample. The resulting profile is then a linear function of z, ie. u(z)=U}(-i) + Au)(^ + ± (3.29) where Ao "(f) ui(—|) is proportional to the constant a introduced in (3.2). tic sla If, however, either 0 (with S=const.; Frank solution) or both (j) and S are allowed to vary with z, deviations 5u(z) from the linear profile given by (3.29) may occur. For example, global variations of appearing, e.g., when {—^) ^ 0(f), or even in a symmetric case where varies due to the twist deformation, give rise to deviations 5u(z) that are global as well [see Fig. 3.5 (a)]. On the contrary, a localized variation of S(z) induces a variation of u that is localized as well. Explore now this "subsurface deformation" in u in more detail and compare it to the one appearing in the 0(z)-profile. Let, for sake of clarity the (/»(^-dependence be omitted by setting 0(f) = 0(—f) = 90°. In this case there is no source of the subsurface deformation in since the /3-term (3.5) vanishes. Numerical solutions of the ELE (3.25) confirm indeed that then cf)(z) = 90° = const holds throughout the whole sample. Putting now (j)(z) = 90 into (327) yields ; 2t . W=5I^M <330 ssume for the moment that the S'(^)profile is modeled by *M-*-^aS- <331 representing the localized variation of S close to the confining walls with an am plitude AS = Sf, — Sq and a characteristic length A* that is to be characterized by the nematic correlation length L. The integration of (3.30) is particularly simplified if | AS"| {±%) = 90°, Sb » 0.375, AS « 0.025, 0, - 0.025 (top, center, and the bottom curve, respectively), co(|) = 0, co(^) = 45° L'2 = 0 [for 0(±|) = 90 the problem is degenerate with respect to the value of L'2] . The profiles cross in the middle of the sample and exhibit a similar behavior at the opposite side, the top curve, e.g., now corresponding to the negative value of AS. The values for a, B, C, T — T, and L\ are same as in Fig. 3.4. Note again that the source of the subsurface variation of the twist angle Sco(z) is the Frank elastic term (/2) and that the origin of this w-variation is different from that responsible for the in-plane variation of (f) examined in the previous Section — in that case the source was the /3 term. Both subsurface (f) and a;-variations follow from a localized variation of S, but the latter can exist only if a global distortion in co is already present (since Sco ex Aw) while for the former no deformation in {—tj) ^ . Then the additional variation has to be added to the one already analyzed in Sect. 3.1, where this kind of deformation was not discussed since only symmetric planar samples were considered 68 Nematic slab 3. Planar cell: molecular approach In this Section the properties of intrinsic anchoring (analyzed to some extent already earlier in the Chapter) will be addressed from the molecular point of view, especially its temperature dependence. The analysis will be based on the hexagonal lattice model [17], along with the spatially anisotropic modified induced dipole-induced dipole pairwise potential (2.34) — see Sect. 2.2.2. Before proceeding to anchoring studies, however, the nematic-isotropic transition in this model system will be briefly investigated. A similar molecular study of intrinsic anchoring was already given in Ref. [17], however, in the zero-temperature approximation. More recently, a finite-temperature Monte Carlo analysis of external anchoring was performed in Ref. [152] for the LL model, attempting to present a full temperature dependence of the external anchoring strength (expressed through the extrapolation length K/W) Phenomenologically, the anchoring energy W decreases with increasing temper ature, being roughly proportional to S2 if modeled by (2.17), and is unable to yield any significant temperature dependence in K/W (recall that K oc S2 as well). To get rid of such behavior — in contrast also to most experiments — the anchoring energy can be modeled directly by (2.16) with fi ^ 0. In fact, experimental analyses typically show a decrease of W — or an accompanying increase of the extrapolation length K/W when the NI transition is approached [46,112,143,153,154] In addition, thermal director fluctuations can substantially renormalize W [155,156]. While it is well known that K oc S, there seems to be no general rule for the functional form of the VF(S)-dependence [and, consequently, the temperature dependence of W]. In fact, W seems to be strongly related to specific properties of a given confining surface. For example, theoretical considerations in a system of hard rods confined between parallel hard walls show that W oc [157], while in experiments measuring anchoring at rough pore surfaces even W oc S4 could be observed [46]. Given K oc S"2, in the former case K/W decreases when the NI transition is approached (agreeing with simulation results in athermic systems of hard elongated particles [158]), but increases in the latter (as found experimentally for most thermotropics). Motivated by these developments, here the analysis per formed in Ref. [17] will be upgraded to nonzero temperatures and K/W for intrinsic anchoring will be reestimated, attempting to extract its temperature dependence from the data. However, first a set of bulk sample simulations will be presented to determine the NI transition temperature. Consider again a nematic sandwiched between two flat and parallel substrates as shown in Fig. 2.8, the z-axis being the sample normal and d its thickness. To avoid bulk easy axes, nematic particles are allowed to rotate only in parallel xz-planes (coinciding with hexagon planes). Following Eq. (2.34), for nearest-neighbor particles i and j with orientations given by unit vectors u^ and Uj, the interaction energy is given by Ui = -e' [Ui • Uj 3i/(u, • r*)(Ui • r*)]2 , (3.34) where e' > 0 and r is the interparticle unit vector. Although the range of the van der Waals potential (2.34) is proportional to r~6, for computational reasons in Nematc sab 69 the present simulation only interactions between nearest neighbors were taken into account. Thereby the error in the estimation of the intrinsic anchoring energy is expected not to exceed 15%. There are eight nearest neighbors (in contrast to the LL model with six neighbors only): six in the xz hexagon plane, plus two outofplane ones (along the ?/-axis). It must be stressed again that while the lattice itself is three-dimensional, the particles Uj are two-dimensional objects. As already suggested in Sect. 2.2.2, the analysis must be restricted to low values of the anisotropy parameter to avoid solid-like periodic director solutions [17] (a) ˇ¦..................¦..................•..................*¦...................¦..................• #..................ˇ...................¦..................•..................ˇ...................¦ ¦..................¦..................ˇ"..................¦..................¦..................T A (b) a a Figure 3.6 Tripartite hexagonal lattice and the three sublattices: squares circles, and triangles, (a) rcz-plane, (b) ay-plane Then Monte Carlo (MC) simulations were performed as follows. For the measurement of the extrapolation length the simulation box size was chosen to be 48 x 48 x 48 particles, which amounts to 105984 particles taking part in the simulation (the boundary layers at z = 0 and z = d being already excluded in this figure), while for the bulk simulation a smaller 30 x 30 x 30 lattice with 27000 particles was considered. The hexagonal lattice was divided into three sublattices, as shown in Fig. 3.6 Thereby the bonds between neighboring particles on the lattice never connect two particles from the same sublattice. Considering the simple hexagonal lattice as tri partite made it possible to vectorize the simulation algorithm, which provided a significant speed-up in calculations. Moreover, in the bulk calculation in all directions periodic boundary conditions were assumed, while for the K/W measurement this was the case only along the x and y-axes. Then the simulation was started either from a random configuration in two dimensions (recall that u, are restricted to hexagonal planes), or from an equilibrated configuration at a temperature slightly higher than the simulated one, if this was available. Then the standard Metropolis algorithm was applied [37]. For the vectorized algorithm to work correctly, in each MC cycle trial moves (in-plane rotations) involving particles in the first sublattice were attempted (and accepted/rejected) first, only then proceeding to the second one, and after this to the third one. In generating a new trial configuration for acceptance/rejection, each time only a single particle was involved. Typically 2 x 105 MC cycles for equilibration and after that another 105 production cycles (to accumulate averages of interest) were performed. 70 Nematc sab In a bulk sample, temperature scans were carried out to determine the NI transition temperature, calculating the average internal energy U by summing up pair wise contributions (3.34). The second relevant quantity for the problem is the order parameter, which, however, for two-dimensional in-plane ordering differs from the standard S introduced in the usual threedimensional space (2.2). Further, attempt ing to measure the extrapolation length, the third important average is the director profile n(z). Both the order parameter and n can be extracted from the ordering matrix, which in two dimensions can be defined by 2(uj ® Uj) - I (3.35) differing from its three-dimensional counterpart Q given by Eq. (2.5). Further, the (...) average goes both over particles Uj and over MC production cycles. Diagonal ization of q leads to two eigenvalues, ±s. The positive eigenvalue s can be identified as the two-dimensional nematic order parameter, while its eigenvector corresponds to the director n. The order parameter is now given by s (2(u, • n)2 — 1), giving s 1 for a nematic aligned perfectly along n and s = 0 in the two-dimensional isotropic phase. Note that in the two-dimensional case there is no biaxiality. Finally, to deduce the z coordinate-resolved order parameter and the director profiles s(z) and n(z), the ordering matrix q(z) was averaged separately for each layer with fixed z and then diagonalized. Alternatively, the director profile can also be expressed in terms of the tilt angle profile (/>(z), with cj) again measured from the sample normal. In this case cf> = 0 corresponds to homeotropic, while cf> = n/2 to planar alignment. Note that in Ref. [152] a different and less reliable method was used to obtain 4>(z) 0.4 y y x v=0.3 ^ t--~ 0.6 v=0.2 y" 4>~—* v=0.1 S >-v=0.05 / y 0.8 v=0 (a) 0.4 0.8 1.2 1.6 rpo (b) Figure 3.7 Temperature dependence of the reduced internal energy U/e (a) and the order parameter s (b) in a bulk hexagonal lattice sample. The NI transition positions (indicated by the arrows) depend strongly on the v parameter Turning now to a bulk sample, the NI transition temperatures TNj can be approximately determined from the kinks observed in temperature scans of the internal energy U(T) and of the order parameter s(T), shown in Fig. 3.7. More reliably TNI can be found by monitoring the variance of internal energy fluctuations, i.e., by determining the maximum of the heat capacity [123]. Here all temperatures are reported in a reduced scale defined by T° = ksT/e'. Note that even for v 0 — Nematc sab 71 1NI rpo L(o) 0.0 35 ± 0.0 3 5.5 ± 2 3 ± 3 ± 0. 25 ± 0.0 4±2 5± 0 5± 0.2 0.90 ± 0.0 0.9 ±0.5 0.8 ±0.5 0. ±0.5 0.3 0.65 ± 0.0 0.6 -4 0.5 -4± 0.4 -4± Table 3.2 Tempeure dpendence of the intinsic anchoring erapolation length L (measured in units of lattice spacing a) for different u7 and the corresponding reduced bulk NI phase transition temperatures Tfrj. corresponding to the spatially isotropic Maier-Saupe interaction — the transition temperature is higher than in the LL model (having already taken into account the difference in energy and temperature scale definitions in both models). This difference can be attributed to a different coordination number (8 vs. 6 nearest neighbors) and nematic particle dimensionality (two vs three). Both distinctions lead to a different balance between the decrease of internal energy and loss of ori-entational entropy when going from the isotropic into the nematic phase. Further, from ig. 3.7 it is evident that the position of the NI transition strongly depends on the anisotropy parameter v In particular, in the range 0 < v < 0.3 T^7 decreases with increasing is, which indicates that the parallel aligning tendency giving rise to the nematic phase weakens with increasing v. This seems to be in agreement with a decrease of the Frank elastic constant and an increasing softness of the nematic if v is increased [129]. A full summary of reduced transition temperatures T^7 can be found in Table 3.2. A standard method to measure the strength of any anchoring is to impose an elastic distortion so that the average surface molecular orientation (f)(0) deviates from the easy axis 4>o defined by the anchoring. The magnitude of this deviation can then be used to estimate the anchoring strength and the corresponding extrapolation length [4] (see also Sect. 2.1.4 Note, however, that the length L obtained by simple geometrical extrapolation (as in Fig. 2.3) can be straightforwardly interpreted as K/W only when anchoring is sufficiently weak, so that L ^ L. Here L is the correlation length associated with interface-induced S-variations (2.11), matching also with the length scale over which subsurface (^-variations can occur, as shown in the first part of this Chapter. The elastic distortion in a nematic slab can now be imposed either by applying a magnetic field whose orientation must not coincide with the direction of the easy axis [17, or by antagonistic anchoring conditions Nematc sab at the opposing surfaces [152]. In the zero-temperature analysis of Ref. [17] the magnetic field method was used. On the other hand, for nonzero temperatures a strong enough magnetic field can enhance the degree of nematic order and even shift the NI phase transition, which can present additional difficulties in interpreting the results. Therefore, it is more appropriate to avoid any external field and consider a hybrid cell-like sample with antagonistic boundary conditions instead In the present simulation, the left (z = 0) surface was chosen to represent the free nematic surface (where the intrinsic anchoring is to act), while the right (z = d) surface corresponds to a solid wall represented by a layer of fixed particles with planar alignment. Since intrinsic anchoring was seen to promote homeotropic alignment for v < 0.3 [17], a bend deformation is expected to appear in the sample. The deformation should be present as long as the sample thickness d exceeds dc = \{K/W)o - {K/W)d\, where (K/W)0 and {K/W)d refer to anchoring at the left and the right interface, respectively [114] 0 10 20 30 40 50 0 10 20 30 40 50 z/a z/a Figure 3.8 Temperature dependence of director profiles (f>{z) for different v. An extrapolation from the bulk to the left provides an estimate for I at the free surface. At temperatures closest to the NI transition (for bulk Tjy-j see Table 3.2) in the bulk the nematic melts down to avoid elastic distortions [compare with s[z) profiles given in Fig. 3.9] igs. (38) and (39) show the order parameter and the director profiles for v = 0.05, 0.1, 02, and 03. To reliably treat cases with even lower v (or cases with extremely weak anchoring), a thicker nematic slab should have been considered, ensuring d > dc, so that the bent director structure becomes stable. In all cases displayed in Figs (38) and (39), at the right surface the nematic was strongly Nematc sab 73 1 0.8 0.6 ® "0.4 0.2 0 v=0.05 f i i i i r=i.i ___k J T°=l.3 r=1.35 0 10 20 30 40 50 z/a 1 0.8 v=o.i r r°=i.o J 0.6 ^«"-"""""""" * ----- A "0.4 /' r=i.'i 4 0.2 0 ¦L T°= 1.175 ^*-'-------r°=i.2 10 20 30 40 50 z/a 1 0.8 v=0.2 f r=0.7 .J 06 -___.............. ..............—*i 0.4 r°=o.8 n K A 0.2 0 V r=o.9 \/ r°=0.925 10 20 30 40 50 z/a 1 t 0.8 T°=OA J >------------ —A 0.6 r=6.5 J. \ T°=0.6 / 0.4 *¦——. ~__ . „~~------- r=0.65 0.2 0 v=0.3 10 20 30 40 50 z/a Figure 3.9 Temperature dependence of order parameter profiles s(z) for dif ferent v (see also the caption of Fig. 3.8) anchored at (f> = tt/2 = 90° (planar orientation), while the intrinsic anchoring at the left surface tends to impose homeotropic alignment with 0 = 0°. Wherever there is no significant variation in s(z) — like deep enough in the bulk — the ^(,z)-profiles roughly exhibit the linear behavior predicted by the Frank theory. On the other hand, if close to the sample surface the degree of order exceeds its bulk value, the nematic will reduce the deformation in this region because it is (free)energetically more expensive due to a larger elastic constant (and vice versa). As it is evident from order parameter plots, in the vicinity of the right solid wall with strong anchoring there is always an increase in s(z), accompanied by a reduction in slope (d(f)/dz) in the director profile /dz. As also seen in the first part of this Chapter, all inhomogeneities in s(z) and, consequently, in (0) decreases with increasing v. In other words, the corresponding intrinsic anchoring strength is nonzero for v ^ 0 and increases with i/, as already seen in Ref. [17], along with the decrease of the elastic constant K [129]. Microscopic values of the extrapolation length disagree with the typical experimental ones, that is 100 nm and above [6]. To reach a quantitative agreement of present results and experiments, a significant decrease of the v parameter seems to be inevitable, as also suggested in Ref. [17]. A small v in Eq. (3.34) promotes parallel molecular alignment, as it is also favored, e.g., by steric repulsions in a system of hard rods. A decrease in v might therefore be regarded as an effective inclusion of steric repulsions excluded in the present model At the same time it should be borne in mind that at this stage looking for a strict quantitative agreement with experimental results is somewhat far-fetched. In fact, a number of other phenomena may also affect nematic ordering in the vicinity of an interface [159], such as dipolar and quadrupolar interactions, electrostatic interactions due to selective ion adsorption [160], presence of impurities [161], density variations [66,67], or roughness of a solid surface. All these phenomena can alter the local elastic properties of the liquid crystal, which affects the elastic torque transmitted from the surface to the sample bulk, and can thereby affect the determination oil. Further note that close to the NI transition the nematic can melt in the center of the sample if a strong enough elastic deformation is imposed (see igs. 3.8 and 3.9). Then molecular alignment becomes homeotropic in the vicinity of the left surface, followed by a region of (nearly) isotropic liquid in the slab center, and by a region of planar alignment close to the right surface (see the snapshots shown in Fig. 3.10), as predicted also phenomenologically [162]. Thereby the elastic distortion vanishes, which — in a sample of current thickness — makes it difficult to measure K/W in the vicinity of TNI. Moreover, the variations in the 0(z)-profile additional to the linear profile predicted by Frank elasticity seem to be a purely elastic effect driven by subsurface variations of the elastic constants. In this sense they are similar to the localized variations of the twist angle u explored in Sect. 3.2. In the strongly bent nematic studied here, localized quasi-splay-bend-like subsurface deformations could not be identified, although — in principle — they might exist since for v ^ 0 one has L2 + L3 ^ 0. Nematc sab 75 Figure 3.10 Snapshots of molecular configurations for u = 0.05 in a hybrid cell: right surface with strong planar anchoring, left surface is free (homeotrop-ically anchored through intrinsic anchoring). Left: T° = 1.1 (bent profile), right: T° = 1.35; just below bulk T^j (homeotropic against planar alignment: the nematic melts in the middle of the cell) The final note in this Section goes to possible residual bulk easy axes connected with the use of the spatially anisotropic pair potential (2.34) in a lattice model Strictly speaking, in the present model there is no bulk easy axis for in-plane alignment only provided that the nearestneighbor particles are all aligned strictly along the same direction. At any finite temperature, however, this never is the case, and therefore some lattice-induced orientational correlations may be expected for molec ular ordering on short length scales comparable to the lattice spacing. However, for the applicability of any lattice model it is important that it appropriately reproduce large-scale elastic deformations and that the corresponding (f)(z) director profiles be smooth and follow qualitatively the phenomenological picture. By all means, this is the case for the model presented here Nematc sab olymer-dispersed liquid crystals The following Chapter will be devoted to nematic ordering in microscopic spherica cavities, as encountered in polymerdispersed liquid crystal (PDLC) systems. Spherical nematic droplets can be obtained by dispersing the liquid crystal in a polymer binder through either phase separation [28] or emulsification [163,164]. Reviewing PDLC samples, one encounters a rather broad distribution of droplet sizes, the lower bound for their diameter being several tens of nanometers (e.g., 20 nm [132]), and the upper well above 1 /im. In the past, nematic ordering in PDLCs has been subject to extensive studies (phenomenological, experimental, and simulation) and is well understood by now [5,91]. For this reason, here another question will be addressed, namely, of how to establish a correspondence between the information on nematic ordering obtained from Monte Carlo (MC) simulations and one of the possible experimental observables, deuterium (2H) NMR spectra. A novel methodology for the calculation of 2H NMR line shapes will be presented that is — in contrast to the existing MC approaches — applicable in presence of significant molecular motion. In particular, radial and bipolar droplets will be considered (see Fig. 4) The MC simulations reported in this Chapter were performed in the framework of the LebwohlLasher (LL) lattice model (Sect. 2.2.1). To model a PDLC droplet, a jagged sphere was carved from the cubic lattice, considering all the molecules lying closer than R = 12a (the droplet radius, a denoting the lattice spacing) to the droplet center. The interaction with the polymer matrix was mimicked by assuming an additional layer of ghost particles, with orientations chosen in accordance with desired boundary conditions. The radial boundary conditions were obtained by orienting the ghost particles normal to the local droplet surface. Similarly, in case of bipolar droplets the ghosts were fixed tangentially to the local surface and in planes containing the symmetry axis of the droplet (see Fig. 4.1). Further, the nematicnematic and nematicghost interaction strengths were assumed equal, which corresponds to a rather strong anchoring with a microscopic extrapolation length of the order of a the lattice spacing. The number of particles inside the droplet was set to N = 5832 whereas in the additional surface layer fixing the boundary conditions to Ng = 1352 For radial and bipolar boundary conditions the calculations at the lowest temperature were started from perfectly ordered (zero-temperature) configurations. These are, in accordance with the given boundary conditions, a perfect hedgehog and a 77 78 Polymedisped liquid crysals perfect bipolar structure in the radial and bipolar cases, respectively. At higher temperatures the simulations started from an already equilibrated configuration at the nearest lower temperature, when this was available. The standard Metropolis algorithm (Sect. 2.2.3) was then applied to update nematic particle configurations. Typically, 5 x 104 MC cycles (sets of N attempted trial moves) were performed for equilibration, followed by a set of further 1024 production cycles used to calculate averages of interest. Within one MC cycle, each of the nematic particles was selected at random for a trial move, using a random shuffling algorithm [123]. Then a new trial orientation for the chosen particle was generated by a controlled variation from the previous one, applying the Barker-Watts technique [131] (Sect. 2.2.3). The trial move amplitude was adjusted dynamically so as to ensure a rejection ratio not too far from 50%. The reduced temperature scale used in this Chapter is defined as T* = ksT/e, with e introduced just below Eq. (2.29). The simulation runs corresponding to the nematic phase were all performed at T* = 0.8. Similarly, cases in the isotropic phase were run at T* = 1.2. Note again that in this temperature scale the bulk nematic-isotropic (NI) transition takes place at T^T = 1.1232 [123] and is reduced to around 1, e.g., for a bipolar droplet in absence of external fields [100]). Figure 4.1 Surface layer of ghost particles defining the boundary conditions (a) radial (b) bipolar droplet (the arrow indicates the bipolar axis) Recall that the dynamics of MC simulations is determined by the "non-natural" molecular evolution process driven by the Metropolis procedure (in contrast to molecular dynamics simulations following the equations of motion), and hence the time scale assigned to fluctuations generated by this technique does not necessarily have to match with the natural time scale indicated in Sect. 2.3.1. However, the update process adopted here rotates one molecule (particle) at a time for a certain angular step, which is a plausible physical evolution process. In this sense, one can map the MC dynamics onto a plausible real one, apart from an arbitrary time unit. For the calculation of 2H NMR line shapes the approach with the time-dependent nuclear spin Hamiltonian (Sect. 2.3.1) was used, allowing for a full treatment of dynamical effects in calculating the line shapes. Combining this approach with MC Polymesped lqud crys simulations, in this Chapter it will be possible to link 2H NMR line shapes to fluctuations of molecular long axes, translational self-diffusion of molecules (homogeneous and inhomogeneous), as well as to aligning effects of external electric/magnetic ields. In the analysis of the spectra we will proceed from the static limit to the limit of completely motionally averaged spectra, in order to find out to which extent diffusive processes smear the spectra and thus make the identiication of director configurations impossible. 4.1 Dynamical effects and 2H line shapes To begin with, droplets in absence of external fields will be considered, assuming also that there is no coupling between the spectrometer magnetic field B and the molecular orientation. Indeed, in microconfined liquid crystals the aligning effects of a magnetic ield can be ignored if the magnetic coherence length Lm oc \jB (223 is much larger than the characteristic dimension of the confned system, i.e., if the magnetic eld is weak enough not to overwhelm the aligning effect of the confining walls. In the droplet case, the condition Lm ^> R must be fulfilled to justify omitting the molecular coupling with the spectrometer field. 4.1.1 olecular fluctuations First consider spectra in absence of translational diffusion, or, equivalently, spectra of large enough nematic droplets in which this kind of molecular motion can be considered as rather unimportant. The droplet size R in this case must satisfy R 3 6-Dio) where is the diffusion constant, to the characteristic NMR time scale, and y/6Dto the root-mean-square diffusive displacement covered by a molecule during to Now the only relevant molecular dynamics is caused by fluctuations of long molecular axes Uj. In order to obtain a spectrum with a sufficient resolution, it is necessary to simulate a relaxation signal G(t) that is long enough, i.e., lasting for several NMR cycles of duration to each. Comparing the time scales of molecular fluctuations tp and the characteristic NMR "time-window" to (see Sect. 2.31), it is evident that there should be approximately 103 molecular fluctuations per each NMR cycle, to. This relation between to and t p did not allow for generating a sufficiently long G(t), as, for technical reasons, at this point only data for 1024 MC particle configurations were available. Therefore, generating Git), the nematic particle confguration was updated from the MC data less frequently than required by the natural time scale tp so as to cover a long enough period in time. This approximation is not of essential importance, given the already mentioned arbitrariness in the Monte Carlo time scale. In the case without diffusion, particle congurations inside the droplet were updated 8 times per NMR cycle, this being much less than the natural scale tp However, this made it possible to generate a G(t) signal whose length is 128 to yielding spectra with a resolution of 256 points in the relevant (nonzero) part of the spectrum. By comparing the order parameter S deduced from NMR spectra with that calculated directly from MC data, it is possible to check whether the chosen Polymedispeed liquid crystals frequency of configuration sampling is sufficient or not. Further, the resulting spectra show some "noise" because the number of particles inside the droplet is still relatively small. To smoothen these spectra, a convolution with a Gaussian kernel of width 0.045coq was performed. For Sluq ~ 2tt x 40 kHz this width equals ~ 2-7T x 1600 Hz, which is well above the natural line width, typically given by ~ 2tt x 100 Hz. Note that neither the width of the kernel nor its shape (Gaussian instead of Lorentzian) match with the features of natural single-spin NMR lines and that the purpose of performing such a convolution is merely to smoothen the spectra and not to simulate the natural linewidth. According to Eq. (2.41), the maximum width of a doublet in the spectrum equals 2SloqS. In the perfectly aligned nematic phase with 5 = 1 the spectral width amounts to 2Scoq, but as soon as dynamic effects are taken into account, the spectrum is narrowed, and molecular fluctuations effectively yield S < 1. The line shape obtained, for example, from the radial droplet, is somewhat more complex because now one also has n = n(r). In the radial droplet most molecules are directed radially from the droplet center, while in the center there is a fairly small defect core [165] (the core structure will be discussed in more detail later). Consequently, molecular orientations are distributed almost isotropically over the whole solid angle Q, which is equivalent to considering a polycrystalline sample with dN/dLt = const., where dN is the fraction of molecules whose orientations fall into dLl. Then, taking into account (2.41), the spectrum is given by I(co) dN/dcu oc [to — (uiz ± ^coqS)]-1^ {ujz denoting the Zeeman frequency) and is called Pake-type powder pattern [135]. It consists of two sharp asymmetric peaks positioned at ujz ± \&wqS, with two "shoul ders" reaching up to uoz ± SujqS. In fact, similar features can be found also in the spectrum calculated from the MC simulation data for the radial droplet at T* 0. see the top curve in Fig. 4.2, left. Now it is possible to deduce the value of S from the actual position of the peaks (or shoulders): S ?» 0.72 ± 0.02. To check this result, one can calculate 5 also directly from the MC data by diagonalizing the MC timeaveraged ordering matrix Q (2.5) for each particle and then averaging the largest eigenvalues obtained in this way over the whole droplet. Such a calculation then gives S ?» 0.73 ±0.10. Here the variance comes from the ensemble average over particles. This rather good agreement also indicates that even sampling MC structures rather infrequently (i.e., only 8 times per NMR cycle) still reproduces the effect of molecular fluctuations sufficiently well In the bipolar case, surface anchoring is planar and in the droplet core most particles are aligned along a given direction, determined by the bipolar symmetry axis [Fig. 4.1 (b)]. Where the axis intersects the droplet surface, there are two topological defects at the "poles" of the droplet. The spectrum of the bipolar droplet in the no-diffusion limit differs considerably from that of the radial droplet. If the NMR magnetic field B is applied along the bipolar axis, it still has two asymmetric peaks, which, however, are now located approximately at u>z ± SujqS. This reveals that indeed most of the molecules are aligned parallel to B — see the top curve in Fig. 4.2, right, calculated for T* = 0.8. Evaluating S from the peak positions, S ps 0.73 ± 0.01 is obtained, while calculating 5 directly from the MC data yields S m 0.76 ± 0.04 gain the agreement of the two estimates is good. In general Polymedispersed lquid crystal 81 bipolar symmetry axes in droplets of a real PDLC sample can have arbitrary spatial orientations. Summing up contributions originating from droplets all over the sample then yields a spectrum similar to the Paketype powder spectrum even for bipolar droplets [132] (this topic will be discussed later in this Chapter). If, however, the process of bipolar droplet formation in a polymer matrix is occurring in a sufficiently strong external magnetic field, the bipolar axes align along the field direction. This frozen-in alignment ("memory-effect") can be retained also after the field is switched off [40], which then corresponds to the case considered here. In the following it will be assumed that the NMR spectrometer magnetic field is directed along the droplet symmetry axes. -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 (a)-(i)z)/douQ (co-cd^/ccDq Figure 4.2 Left: 2H NMR spectra of the radial droplet for different values of the diffusion parameter A: A — 0 corresponds to the no-diffusion limit while A = 32 corresponds to the fast diffusion limit. The Pake-type powder spectrum obtained for A — 0 collapses into a single line centered at zero quadrupolar splitting for A — 32. Right: Same as left, however, for the bipolar droplet. The spectrometer magnetic field was aligned along the bipolar symmetry axis, which results in a spectrum consisting of two lines both in absence of diffusion and in the fast diffusion limit. In all cases T* — 0.8, ensuring the existence of the nematic phase. Here and elsewhere in the thesis all spectra were normalized so as to obtain the same peak height 82 Polymedispersed liquid crystals 4.1.2 Homogeneous translational diffusion In addition to fluctuations of the molecular long axes, now also translational molec ular diffusion will be included into the analysis. Note that since nematics are anisotropic liquids, the process of translational diffusion is anisotropic, characterized by a tensor D_. In a bulk unconstrained nematic phase the ratio of its eigenvalues the diffusion constants D\\ and D^ measured along the director n and perpendicular to it, respectively — can typically range around ~ 2 [4,40]. However, since here we are primarily interested in qualitative features of spectra, in the following analysis the diffusive process will be assumed isotropic and characterized by a single motional constant D = D\\ = D±. Some preliminary tests with anisotropic diffusion indeed showed that qualitatively current results do not change. Translational diffusion was simulated by a simple random-walk process in which the particles representing one or several nematic molecules jump between lattice sites. In every simulation step each of the particles is allowed to move to its near est neighbor site with equal probability in the present isotropic case, while in the anisotropic case this probability should be biased so as to increase the diffusion probability along the director [40]. After the diffusion jump has been performed, the particle acquires a new orientation, whose average (i.e., the local n) in a distorted sample is different from the average calculated at the old coordinates. Calculating G(t), like in the diffusionless case nematic particle configurations were updated from the MC data 8 times per NMR cycle. Now additional diffusion steps were added in between these structural updates, with their number A ranging from 1 to 32 In this last case the spectra are completely motionally averaged due to diffusion effects since for A = 32 each of the particles exhibits a total of 256 jumps within the duration of one NMR cycle, this already corresponding to the fast diffusion limit with ^6Dt0 ~ 16a ^ R. It should be stressed that the diffusive random walk does not interfere with the MC evolution process; in fact, it only uses the particle configurations generated by the Metropolis algorithm, forwarding this information to the line shape calculation. Starting by considering the radial droplet at T* = 0.8, Fig. 4.2 (left) shows a sequence of spectra for this type of boundary conditions, ranging from the no-diffusion limit (A = 0) to the limit of fast diffusion (A = 32). In general, for any type of boundary conditions the fast diffusion spectrum consists of two lines centered at uz±(oJq), where the average frequency is given by (uq) = 5uq ( o (y/6Dto 2), equal to e ^$ 0.5. Here it was assumed that most molecules are aligned along the NMR field direction and that the nematic order is homogeneous with S = 0.8 throughout the droplet. Note that although e 0 show that with the increasing field strength more and more molecules orient along z (i.e., along f) thereby increasing the size of the droplet core where the nematic liquid crystal is almost undistorted and n\\z. The thickness of the distorted region is related to the external field coherence length Lm and is obviously microscopic because the applied field is extremely strong. According to the above observations and to snapshots shown in Fig. 4.7, the increase of the quadrupolar splitting ujq in strong fields can be attributed both to the overall increase in the local degree of ordering, i.e., to an increase of S — as observed also experimentally [117] — and to the additional molecular alignment along f resulting in an increase of (P2)b = {^(cosfl)); see formula (2.41). Also the narrowing of the spectral lines is related to the increase of (i^s since in the droplet -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 {0i-(O^ld0iQ {0)-(O^ld0iQ Figure 4.8 2H NMR spectra of a sngle radial droplet in the nematic phase at T* = 0.8 for dfferent values of 77; no-diffusion limit (a), fast diffusion lmit () A hdd s n os wih ig 7 Polymedisped liqid cals core the bipolar configuration is replaced by the "aligned" one. The spectral line narrowing further follows from changes in the distribution of local S(r). In fact, in strong fields the field-enhanced "bulk" value of S approaches the surface-induced value and thus the distribution of S becomes narrower. Considering now Fig. 4.5 (b) and the spectra of bipolar droplets in the fast translational diffusion limit (or, equivalently, in small enough nematic droplets), the spectra for all r\ still consist of two well-defined lines now positioned, however, at an average quadrupolar frequency ±{coq), where coq is spatially-dependent and given by Eq. (2.41), and the averaging is to be performed over diffusive motions of all molecules inside the droplet [40,136]. The quantity (uq) can be calculated also directly from MC data and, according to values presented in Table 4.1, the agreement with actual peak positions is very good. 0 CO 12 shell 1 7,= 1 0.8 0.6 ,/ 0.4 / 71=6 l<=> /?;?"=»; ws^ 0.2 0 0.5 1 rlR D 4 8 12 1 77=1 0.8 ! 1 \ I I ; "*"*¦** •¦¦ » ! t • . ¦» • •n 0.6 • * * • * • • 0.4 4 * * * 0.2 7/=0 • 0 • • -........... » j 0 0.5 shell 1 rlR Figure 4.9 Order parameters calculated for the radial droplet at T* = 0.8 (nematic phase): (a) standard nematic (S) and (b) external field ((P2)b) order parameter as a fuction of the distance from the droplet center. Curves are plotted for (top to bottom): rj = 1, r\ = 0.5, rj = 02, r\ = 005, r\ = 003, 7] = 0.02, and r\ = 0, respectively. The defect core transforms into an aligned structure. The molecular aligment for rj = 0 is depicted schematically as set We now turn to radial droplets. In absence of external fields the nematic director is directed radially from the droplet center, giving rise to a "hedgehog"-like structure. Then in the very vicinity of the center the elastic deformation (splay) becomes rather strong and therefore it is convenient in this tiny region for the nematic to melt, ie. to decrease the degree of nematic order (S) so as to reduce the elastic deformation free energy. In the rest of the droplet the radial alignment still exists and therefore in each of the intermediate and outer shells particle orientations are distributed evenly through the whole solid angle. As far as NMR is concerned, this situation is equivalent to having a polycrystalline powder sample and, indeed, as already discussed, the spectrum of the radial droplet for r\ = 0 is the Pake-type powder pattern shown in Fig. 4.8 (a) for the diffusionless case. A calculation of the nematic order parameter S [Fig. 4.9 (a)] for r\ = 0 shows that the value of S in the center of the droplet is nonzero, yet considerably smaller (« 0.32) than the value obtained in the intermediate and surface layers (^ 0.75). This confirms the existence of a small ( 4a in diameter) and fairly disordered defect core. Looking at snapshots Polymedisped lquid crysal 93 y f O Figure 4.10 Radial droplet at T* — 0.8 (nematic phase): snapshots of molec ular configurations for different jj; rcy-cross sections (left colmn), rr^-cross sections (right colmn). The external field f is applied along z. of molecular configurations (Fig. 4.10), it can be readily observed that here one is not dealing with a point defect: the defect core actually consists of a weakly aligned nematic, encircled by a ring disclination line of strength +|. The plane of the ring takes an arbitrary orientation and can even rotate during the MC evolution. The characteristic time scale associated with reorientations of the ring is expected to be much larger than the time span covered by the 1024 MC cycles employed in the line 94 Polymedisped liquid crysals shape calculation. Applying an external field, with increasing r\ the disclination ring first orients normal to f, the field direction, and after that increases also in size (see Fig. 4.10). Thereby the radial "hedgehog" structure is transformed into an axially symmetric structure with a well-pronounced ring defect. The degree of ordering in the center therefore increases significantly and the molecules of the core align along the field direction (compare with order parameters S and (PzJb plotted in Fig. 49). There is no critical field characterizing the transition between the "hedgehog" and the aligned structure: the size of the aligned core increases gradually with the increasing field strength [99] This can be confirmed also by inspecting the resulting NMR spectra shown in Fig. 8 (a). For strong external fields with r\ > 0.2 the Pake-type pattern transforms into a spectrum with two narrow peaks, similar to those observed for bipolar boundary conditions. Again, this indicates that for r/ > 0.2 most of the molecules are aligned along f, except for those lying close enough to the polymer substrate (compare with Fig. 4.10). In fact, surface-induced radial order persists in the outermost molecular layers, which results in a strong decrease of the order parameter (i^B [Fig. 49 (b)] in the surface region. The thickness of this region is again roughly equal to the field coherence length Lm. In the intermediate regime with 0 < T] < 0.2 the spectra are composed both of the Pake type contribution originating from the surface layers and of two narrow peaks being a signature of the field-ordered core. With increasing r\ the latter contribution prevails, as it is clearly evident from Fig. 48 (a). Again it is possible to check the agreement of values for S deduced from peak positions and from MC data (Table 41). As seen in previous Sections, in the fast diffusion regime the spectrum of the radial droplet for r\ = 0 consists of a single line located at (ljq) = 0 [0,136] A soon as there is a preferred direction (like in the bipolar case with all droplets aligned or in a strong enough field), (uq) = 0 no longer holds and the spectrum splits into two narrow and symmetric peaks. Fig. 48 (b) shows the fast diffusion spectra for radial boundary conditions. As expected, one observes a single line in the spectrum only if the external field is of or relatively weak with r\ < 0.02. Increasing the field strength, the spectrum gradually transforms into the two-peak pattern described above. For a comparison of peak positions and (ujq) calculated from MC data see Table 41; the agreement of the two estimates is fairly good. 4.2.2 Isotropic and field-induced nematic phase In the following we will consider both types of boundary conditions at T* = 1.2, a higher temperature already above the nematic-isotropic transition. In absence of external fields molecular motion in the bulk isotropic phase is spatially isotropic (ie., there is no long-range orientational order), hence one has S = 0. Consequently, the quadrupolar perturbative contribution to the deuteron energy levels is averaged out and the frequency splitting ujq oc S vanishes [see Eq. (2.41)]. In this case a single line positioned at ujq = 0 appears in the NMR spectrum. Consider the bipolar droplet for r\ = 0 first: the corresponding spectrum consists of a single line at zero-splitting, as expected [Fig. 11 (a)] A calculation of Polymedisped lquid crys 95 the nematic scalar order parameter 5 [Fig. 4.12 (a)] reveals some residual bipolar ordering with S > 0 in the outer molecular layers that is responsible for a rather large line width. Applying the external field, the two peaks characteristic for or dered nematic phases reappear. Also the nonzero values of both order parameters 1 3 (a) r, = l j?=0.5 j?=0.2 rj=0 -1 -0.5 0 0.5 1 (a)-coz)/do)Q 1 3 J J ¦ft lU 1 1 I u 0>) -1 -0.5 0 0.5 1 (w-co^/do)Q (c) J I A r L j?=0.5 1 |l »7=0.2 r/=0 (d) -0.5 0 0.5 (co-o)^/dcoQ -1 -0.5 0 0.5 (o)-co^/dcoQ Figure 4.11 2H NMR line shapes of droplets at T* = 12 (above Tfa). Top bipolar droplet in the static (a) and fast diffision limit (b). Bottom: same as top, but for the radial droplet: static (c) and fast diffision limit (d). For strong enough fields a nonzero quadrupolar splitting is restored Polymedisped liquid crysals 0 12 shell co 0.8 0.6 0.4 0.2 0 0.8 0.6 *a 0.4 0.2 0 0 i,=l f ,/' / / 77=0 JT ..... ...... ...-- »7=1 •-. r ""••-. ¦¦¦.. -... ../ »7=0 ...... ..... •¦*'** _>'' 0.8 0.6 Q^0.4 0.2 0 0.5 1 rlR 0 0 4 8 12 0.8 !J=l.......................................... . 0.6 *......«..., ¦••¦.., B 'v. 0.4 ....................p 0.2 ....•••-"'" n nT?....,.....-.-...................."" 77=1 ^. *¦.....¦»... \ N \ ¦¦¦•.......... \ \ ">. \ \ \> ..1=9... ¦-¦. 0.5 1 rlR Figure 4.12 Bipolar (top) and radial droplet (bottom); same as Figs. 4.6 and 4.9, but at T* = 1.2 (above T^j). Even for tj = 0 there is some residual surface-induced nematic order in the outer layers of the droplet, while in a strong enough external field nematic order is restored throughout the droplet (S and (^2)5) given in Fig. 4.12 are rstoed again despite the fact that the at T* = 1.2 one is already quite far from the NI transition. All these observations show that a strong enough external field can overwhelm temperature effects and induce nematiclike molecular ordering also above the nematic-isotropic transition temperature [99,100,167], which has been observed also experimentally [118], however, not that far from the NI transition in the absolute temperature scale. Like in the nematic phase, the quadrupolar splitting increases with increasing r/. Again peak positions and MC-data values for S match (Table 4.2). One can also compare the peak positions (and, simultaneously, the estimates for S) with those obtained for the lower temperature T* = 0.8. As expected, for T* = 1.2 the estimated values for S are lower than those for T* = 0.8 (see above). This suggests that although unrealistically strong, the external field is still not the only important element in de termining the structure inside the droplet and that disordering temperature-induced effects are still well-pronounced. In the fast diffusion limit all spectra, including that for 77 = 0, consist of two peaks since (coq) ^ 0 [Fig. 4.11 (b)]. This result is expected for r) ^ 0 and also for r\ = 0, where the ordering effect of the external field is absent, but there is residual surface-induced ordering [for the behavior of S see Fig. 4.12 (a)]. Checking the matching of peak positions and the MC-deduced values of (coq), the agreement is still very good (Table 4.2). Repeating the analysis for radial boundary conditions and T* = 12 gives similar results: in the no-diffusion limit the single broad peak centered at uq = 0 for r/ = splits into a doublet for r\ ^ 0 and the splitting increases with r\ [Fig. 4.11 (c) Polymedisped lquid crys 97 i^peak Smc ((WQ) /^Q)peak {{uq)/5u)Q)mc bipolar 0.0 0.1 0.1 0.2 0.51 ±0.01 0.53 ±0.08 0.46 0.46 0.5 0.63 ±0.01 0.63 ±0.05 0.5 0.60 1.0 0.2 ±0.01 0.1 ±0.03 0.6 0.6 adial 0.0 0.001 0.2 0.46 ±0.01 0.48 ±0.11 0.33 0.34 0.5 0.62 ±0.01 0.5 ±0.08 0.52 0.52 1.0 0.1 ±0.01 0.68 ±0.08 0.65 0.65 Table 4.2 Bipolar and radial droplet at T* = 1.2: comparison of S and (ojq) deduced from NMR spectra with those calculated directly from Monte Carlo data. Smc for r\ = 0 is not given since in the isotropic phase the method employed to calculate S results in an overestimate because in a disordered phase the nematic director is not welldefined [124] If the field is absnt, the order parameters S and {PzJb plotted in Fig. 4.12 sow the existence of residual radial order in the surace layrs of the droplet, while n the core the liquid rystal is isoropic. For nnzero \ both order parameters acquire a nonzero value d hence again confirm field-iduced nemati orderi above TNI. A comparison of pak positiondetermined S and MC-alculated S given in Table 4.2 again shows tht the agreement f the two values is fairly good. Like for the bipolar droplet one can compare these valus with those obained at T* = 0.8 in the nemati phas d again find tht the degree of nemati ordrin is lower at higher tempraur inally, Fig. 4.11 (d sows the corrsponding fas diffusion limi spectra. For r\ 0 the line is single-peaked i spite f residual orde with nonzero close to the surface (Fig. 4.12). Note, how, tha n this layer one is daling with radial ordr yildin cuq) 0 already in itself In other ases with external field two paks reappear and the corresponding splitting inceass with r\ The agreement of (luq) evaluated from ak posiios ad from MC daa is vry good again; see Table 4.2. For a mor srious quantitative comparison of all calculated spectra with the experimental ones, usually recorded for largr droplets than the ones studid hee, would be necessary to simulate droplets cntainig a significantly larger numb o partiles so as to access radii of the order of ~ 1 /im. Morover it turns ou that the distribution in orientatios of droplet symmetry axes is closely related the shape of the droplets, which is ot necessarily spherical — as assumd so far — bu rather somewhat ellipsoidal, whch may reflect in NMR spectra [25], as well as i ther observables. Som simulatio work with ellipsoidal droplets has also been formd in the past prenti, how, n exrimental obsrvabls [92] 98 Polymedisped liquid crysals ematics ith dispersed polymer networks In this Chapter the complex-most of all confinement types in this thesis will be treated — low-concentration polymer networks dispersed in a nematogenic liquid. The networks typically consist of thin fibers — few nanometers thick — or of some what thicker bundles of such fibers. As already mentioned in the introductory Chapter, polymer fibers can play an important role in aligning the surrounding liquid crystal even at low polymer concentrations [5,26,39]. As such, they are important for the construction of electrooptical devices based on the external field-induced switching process. The detailed characteristics of this process are closely linked to the anchoring and ordering conditions at the fiber surface, as well as to the shape and regularity of the network. Experimentally, these network properties can be regulated during the network formation (photopolymerization from the monomerliquid crystal mixture) through various parameters: monomer solubility, curing temperature, ultra-violet (UV) light curing intensity, and the degree of orientational ordering in the liquid-crystalline component [5]. In particular, poorly soluble monomers re sult in polymer fibers with a grainy and coarse surface morphology, while highly soluble monomers can form smooth fiber surfaces [15]. Further, high curing temperatures, as well as high UV light intensities, result in larger voids between polymer fibers [101]. If the liquid-crystalline component of the mixture is isotropic during the polymerization process, polymer fibers form directionless strands. On the other hand, performing the polymerization in the nematic phase, or applying an external aligning magnetic field, fibers can form bundles with a welldefined average direc tion [5]. Similar types of network-like confinement can be achieved also in silica aerogel systems, where irregular chains of silica particles play the aligning role of polymer fibers [5]. While thin (nanometric) polymer fibers typically promote planar surface anchoring along the fiber direction, thicker fibers or fiber bundles (several 10 nm in diameter) can be treated with surfactants to yield homeotropic anchoring conditions Like for PDLC droplets (Chapter 4), Monte Carlo simulations have been performed in the LebwohlLasher lattice model system. In this Chapter following phe nomena will be investigated in more detail: (i) a relation between the polymer fiber surface roughness and the long-range orienting ability of the network [102], (ii) the switching behavior in network systems with regular and irregular surface topog- 99 00 Nematics with disped polymer neworks raphy, (iii) pretransitional paranematic ordering, and (iv) topological defect lines (disclinations) [103]. The first two phenomena are of great interest for applicative purposes. Issues (i), (ii), and (iv) will be explored in a system of parallel and straight fibers. Such a regular fiber array is similar also to colloidal crystal systems. In addition, the analysis of issue (ii), the switching behavior, will be extended also to irregular fiber networks, dropping the initial requirement that the fibers be straight and regular in position. Pretransitional ordering, issue (iii), will also be addressed in an irregular sample. The simulation output will be expressed in form of 2H NMR spectra (all issues), electric capacitance (the switching behavior), and transmitted polarized light intensity (the switching behavior and pretransitional ordering) 5.1 Aligning ability of the network n this ection a thorough microscopic simulation study of the orientational coupling between polymer fibers and the surrounding liquid crystal will be presented, focusing on polymer networks with a well-defined net fiber direction (as shown in Fig. 13), and on effects of roughness at the fiber surface. In simulations the same model as for studying nematic droplets was used (see Chapter 4 and the references therein), however, only after appropriately adapting the simulation box and the boundary conditions to mimic the geometry of the current network sample As a first step towards modeling the complex topology of the polymer network, a single straight cylindrical fiber (oriented along the z-axis) was considered. The shape of the fiber was defined by carving a jagged cylinder from the cubic lattice and taking all particles that are lying closer than R — the fiber radius — from the center of the xy-plane (Fig. 5.1). The particle orientations in the surface layer of the fiber (ghost particles) were chosen in agreement with the desired boundary conditions and were kept fixed during the simulation. The strengths of nematic-nematic and nematic-ghost interactions were set equal, which corresponds to the strong anchoring limit Further, periodic boundary conditions at the simulation box boundaries were assumed. Such a set-up in fact corresponds to a regular array of straight and parallel fibers. It is believed that such simplified topography of the fiber network does not qualitatively affect the effect of surface roughness studied here, at least for low-density polymer networks More realistic network models will be presented later, including curved fibers positioned randomly inside the simulation box, allowing for cross-linking between them at somewhat higher polymer concentrations n the case of "perfect" anchoring ghost particle orientations were chosen either along z (a unit vector along the z-axis) for planar anchoring, or along the local radial unit vector for homeotropic anchoring For cases with partially disordered anchoring simulating the still cylindrical but rough fiber surface — the perfect planar or homeotropic ghost orientations were perturbed by performing an additional rotation for each of the ghost particles, characterized by a set of polar (9) and azimuthal ((f)) angles. While the (j> angle was sampled from a uniform distribution within [0,27r] the sampling of 9 (or, alternatively, cos 9) was biased so as to regulate the degree of randomness in ghost particle orientations. The biasing distribution was chosen to Nematcs with dsped polymer neworks 01 Figure 5.1 Schematic depiction of the polymer network (right) and the simlation box with the cylindrical fiber and one of the cylindrical shells (left) be dp/d cos 9 oc exp(P cos2 9) (with p denoting the probability and cos 6? G [1,1]) where for small V the resulting orientational distribution of ghosts becomes almost isotropic, while for large values of V it becomes strongly peaked at cos# = ± (i.e., 9 = 0,7r) and therefore approaches that of the perfect anchoring cases. In the case with completely disordering anchoring ghost orientations were sampled from a fully random orientational distribution. The degree of randomness can be given quantitatively by diagonalizing the ordering matrix Q = |(3(uj Uj)9 — I) (the average {...)g taken over ghosts), which gives the ghost director and the corresponding order parameter {Pzig. In all cases the (P2)g order parameter is referred to the z-axis, the fiber direction. Therefore, cases with {P2)g = 1 and (P)g = —0.5 stand for perfect planar and homeotropic alignment, respectively, and (P2)g ~ 0 for a random orientational distribution. Intermediate values of {P2)g then correspond to partial planar ({^2)9 > 0) or partial homeotropic order ((P2)g < 0 in ghost orientations. In principle, the phenomenological analogue to such partially disordered interfaces (characterized by a surface-imposed degree of order, {P2)g, and by a given easy axis, as here z) can be found in modeling the surface free energy by (2.17), also used in Chapter 3. To study the radial dependence of order parameters, it is convenient to split the cubic simulation box into cylindrical layers surrounding the fiber (see Fig 5.1) The observables accumulated during the production run were (Pf)? quantifying the degree of ordering with respect to the z-axis, (P2C), indicating how the order deviates from perfect radial ordering in the xy-plane, and the standard ne matic order parameter S. Then, for example, (P Uj)r — I) averaged over sites in the nematic layer with radius r, and over MC cycles. The eigenvalue with the largest absolute value can then be identified as S and the difference between the remaining 102 Nematics with disped polymer networks two eigenvalues corresponds to biaxiality. n absence of significant collective molecular reorientation during the MC evolution, it is instructive to calculate also spatially-resolved director and order parameter maps n(i-j) and 5(rj), respectively, where r, denotes the position of the ith lattice site. For this purpose the local ordering matrix Q(r^) = |(3{uj (g> Uj)ri — I) was averaged over MC cycles and then diagonalized, yielding the local value of the order parameter S fa), as discussed above, and the corresponding eigenvector, ie., the local director n(i-j). Similarly, the biaxiality map can also be deduced from the data. In simulations presented in this Section, the simulation box size was set to 30a x 30a x 30a, which for the chosen fiber radius (R = 5, denoting the lat tice spacing) amounts to 24600 nematic and 840 ghost particles in total. Simulation runs were started from a completely random (disordered or isotropic) orientational configuration not to impose any preferred orientation in the system. n general, simulation results were rather insensitive to changing the initial particle configuration. Then the standard Metropolis scheme [37] was employed to perform updates in par ticle orientations [9,13]. Once the system was equilibrated (after at least 6 x 104 MC cycles), a sequence of 6.6 x 104 (or more) successive particle configurations was used to calculate relevant observables. Results from MC simulations were expressed using selected order parameters and 2H NMR spectra following the methodology presented in Chapter 4 he following were obtained at two different reduced temperatures T* = kT/e, T* = 1.0 and T* = 12, deep enough in the nematic and isotropic phases, respectively (recall again that the nematic-isotropic transition in the bulk takes place at T^j 1.123 [123]). Other calculations, not reported here for reasons of space, were performed at T* = 11 with results qualitatively similar to those obtained for T* = 10 The correlation length for orientational ordering at these temperatures was found not to exceed « 5a, which with the present choice for the simulation box size is expected to be enough to avoid spurious correlations originating from periodic boundary conditions. n this study the fiber radius was fixed to R 5a Another set of runs for a thinner fiber with R = 3a has also been performed, but has shown no major difference in comparison with the R 5 case and is therefore not reported here. 5.1.1 Planar anchoring First consider a nematic sample at T* = 10, with planar anchoring along the z di rection and with possible deviations from this perfect alignment, as described above. This situation corresponds to a series of polymer fibers whose surface morphology varies from smooth to rough and disordered. Fig. 5. (a) shows how the (P%) order parameter changes across the simulation box from the fiber surface to the outer sample boundaries. Different curves shown in the plot correspond to different degrees of order in the ghost particle system, (i^g- For perfect planar anchoring ||z with (P2)g = 1 the nematic director n is parallel to z. n this case (P2) becomes a direct measure for S because n and z coincide. Far enough from the fiber the value of {P Nematcs with dsped poymer newo approaches « 0.6, matching with that of S in a bulk saple at T* = 0 [123], while close to the fiber there is an increase in (P|), reflecting the fiber-induced enhanceent of nematic order. The same effect is further confirmed by the behavior of the S(r) profile depicted in Fig. 5.3 (a) as well as by the S(x,y) local order parameter map shown in Fig. 5.3 (b). For all three profiles the characteristic length of the nematic order variation L (211) roughly amounts to « 3a Studying cases with reduced (imperfect) planar anchoring ||z [Figs. 5.2 (a) and 5.3 (a)], one can see that at least down to {P-zjg ~ 0.25 the bulk value of both order paraeters remains essentially unchanged if compared to the perfect {P2)g = 1 case. Note that now for e.g., (i^g ~ 0.75 the increase of order close to the fiber is smaller than for {Pzjg = 1, and that already for {P2)g ~ 050 (as well as for (^2)9 025) the surface degree of order is somewhat lower than its bulk value. Fro these observations one can conclude that the first effect of the partial disorder in surface anchoring is merely a slight decrease in the degree of nematic order in the vicinity of the fiber but that at this point the long-range orienting ability of the polymer network is not lost. This ability, however weakens upon further decreasing (P2)g, but is present at least down to {^2)9 ~ 009 (the corresponding profiles not plotted here). Then only in a sample with a copletely disordering fiber — for (^2)9 0 — the net orientation of the nematic for the intermolecular potential (2.29) is copletely detached from the fiber direction. This follows from the behavior of the (P2} order parameter (calculated with respect to the fixed fiber direction) which now — in principle — can take any arbitrary value, and from the fact that the liquid crystal is still nematic, as suggested by a nonzero value of the S order parameter throughout the saple. Note that the bulk value of S remains almost unaltered in comparison to e.g. the {^)9 = 1 case. The fact that it is actually slightly lower 0.8 0.6 0.4 0.2 <\ (P&*) (a) 6 8 10 12 14 16 18 20 r/a 0.6 0.4 0.2 1 0>) ,<^2>g=l v^ 6 8 10 12 14 16 18 20 r/a Figure 5.2 Order parameter (P|) versus r (the distance from the simuation box center, measured in lattice units a) in a sample containing a singl cyindrical fiber with R = 5a. Planar anchoring along the L-axis; (a) nematic (T = 1.0) and (b) isotropic phase (T* = 1.2). In the plots each of the curves corresponds to a different degree of ordering in the ghost particle system: (^2)9 ~ 1-0, 0.75, 0.50, 0.25, and 0 (top to bottom). The dotted ines serve as a guide to the eye (aso in following Figures) 104 Nematics with disped polymer networks Figure 5.3 Planar anchoring ||z. (a) Order parameter radial profiles S(r) for different values of {^2)9 in the nematic phase (T* = 1.0); curves are labeled as in Fig. 5.2. (b) Perfect planar anchoring ({^2)9 = 1); xy-cvoss section of the local S(ri) order parameter map in the nematic phase (T* = 1.0). (c) Same as (b), but in the isotropic phase (T = 1.2). The distances along x and y are plotted in lattice units than the value obtained for {P2)g = 1 (~ 0.6) can be attributed to slow olletive oleular motion during the prodution run. If teperature in the LL model is inreased to T = 1.2, in a bulk sample the isotropi phase is stable. However, like in PDLC droplets also in a nematic with a dispersed polyer network one should expect surfae-induced paraneati ordering. For the ase of planar anchoring | Figs. 5.2 (b) and 5.3 (c) show the (PL) and the local S(x, y) profiles and in fact onfir the existene of surfae-indued planar ordering. Note that the layer-averaged neati order paraeter profile S(r) would have looked exactly like the (P|)profile and is therefore not shown here. The net oleular orientation is still along z as iposed by the fiber, and the corresponding degree of order given either by {P or S deays to zero over a harateristi length of the order of L 5a (211) 5.1.2 Homeotropic anchoring: topological defects Proeeding now to ases with {^2)3 < 0, ie., to perturbed homeotropi ordering, already for {P2)g —0.08 the polymer fiber is able to align the liquid rystal Moleules are now aligned perpendiular to z the fiber diretion ie. ainly within Nemacs with dsped poymer newo 0.8 0.6 0.4 0.2 0 -0.2 \^.(P2)g=-0.5 \ \ \ X. ****. ***••. "**¦-»... ••»- •¦*¦ '.-: ¦S: - .... .^. .-. - ^ ^(P2)^0 — .... — ¦-¦ ¦-- ¦-¦ ¦-¦ ¦"¦ ¦* (a) 6 8 10 12 14 16 18 20 r/a 6 8 10 12 14 16 18 20 0.6 V<^=-°-5 0.4 * \ 0.2 \ \ 0 \, ''*.. *(P2)^0 o \ s------.~^& -0.1 co -0.2 -0.3 (b) 6 8 10 12 14 16 18 20 r/a /^ „=-0.5 (d) r/a 6 8 10 12 14 16 18 20 r/a Figure 5.4 Order parameter profiles (i^K7*) and S(r) for homeotropic anchoring with {P)g ~ -0.50 -025, and 0 (top to bottom), a) and (): nematic phase (T* = 1. (b and ): isotropic phase (T* = 1. the xy-pl&ne This can be deduced from (P|) profiles, the (P|) values being now negative for all r (not plotted here). Similarly as for planar anchoring there is a decrease in the degree of nematic ordering close to the fiber, e.g., for (P2)g ~ —0.25 (partial disorder) and an enhancement for {P2)g = —0.50 (perfect homeotropic order). Studying cases with homeotropic surface alignment, it is more convenient to plot the (P2C) order parameter profiles. Note now that for (^2)3 = —0.5 the (P|) profile — shown in Fig. 54 (a) — is always positive and that (P2C) values are rather high close to the fiber. This is a signature of strong radial ordering, along with the negative values of S in the first layer next to the fiber surface; see Fig. 5.4 (c) Going away from the fiber, (P|) exhibits a plateau-like behavior, before relaxing to the bulk value close to ~ jS, which is characteristic for homogeneous (undeformed) nematic ordering. Such alignment far from the fiber is compatible with strong radial ordering in the fiber icinity only if topological defects are to form. In fact, as shown in the director map n(i-j) (Fig. 55, left), a pair of — \ strength defect lines forms along the fiber and close to the simulation box diagonal. The plateau in (P^-profiles then corresponds to the distortion of the director field imposed by the defect lines As concluded from topological considerations, either a —1 strength disclination line or a pair of — ^ lines can form in the neighborhood of the particle. This can be inferred from the fact that the insertion of a homeotropic fiber into a uniform nematic (the uniformity provided by periodic boundary conditions) is equivalent to an insertion of a +1 strength defect line. Then, in order to meet the conservation 06 Nematics with disped polymer neworks rule for topological charge (the total defect strength ^wij), defects of opposite sign have to form in the nematic, which is exactly what is seen also in the present simulation. The —1 line, however, does not seem to be stable and splits into a pair of — | lines during the MC evolution, even if it is taken as initial configuration in the simulation run. This behavior agrees with simple estimates of defect line free energies where the defect line free energy scales as m2 (where m is the defect strength; see Sect. 2.16) [4,24]. Moreover, a stable "escaped" structure (bent in the z-direction) could also not be observed in present simulations. Here it should be stressed that the defects are able to form because the anchoring strength w is rather high. Decreasing w or considering a rough fiber surface, both defects vanish hich is because for small w (or for a rough surface) the anchoring penalty for homogeneous alignment also in fiber vicinity is lower. Note also that the following analysis of the defects is relevant not only for fiber network systems but also for elongated colloidal particles embedded in a liquid crystal host [103] The pair of defect lines alays forms close to one of the simulation box diagonals although the cross section of the fiber is axially symmetric (ignoring its jagged shape); see the director fields shown in Fig. 5.5 and 5.6. This symmetry breaking may be attributed to two effects of different origin. The first one (and, according to preliminary tests, the more important one for the 30 x 30 x 30 system size) is the repulsion between defects maximizing the defect-to-defect distance (recall the periodic boundary conditions), while the second one is a finite-size effect originating from collective fluctuations, resulting in a tendency to align the nematic along the simulation box diagonal [168]. These phenomena, as well as the presence of the polymer fiber, are believed not to affect any of the qualitative features characterizing the defect line inner structure studied in the following. Moreover, the presence of the fiber is reflected only in an enhancement of nematic ordering in its immediate surroundings. The inner structure of a defect line is characterized by variations in the three eigenvalues of the ordering matrix Q, Qi, Q2, and Q3. The eigenvalues along with the corresponding eigenvectors, were obtained by diagonalizing Q (2.5) for each of the lattice sites. The scalar order parameter S, biaxiality P, and director n maps can then be derived from these data, as discussed earlier. Fig. 5.5 (left) shows the director field and the order parameter map in the plane perpendicular to the long axis of the fiber. Fig. 5.5 (right) shows the Qi Q, and (^-profiles plotted along the y-axis through the left of the two disclinations. Note that the left-right asymmetry of the profiles with respect to the defect position is caused solely by the presence of the polymer fiber. After passing through the disclination, the Q-component changes from its positive bulk value (~ 0.6) coinciding ith the value of the order parameter S, to some negative value (~ —0.3). At the same time, the Q2-component increases from a negative value (~ —0.3) to a large positive value (~ 0.6) that roughly equals wice the absolute value of the negative one. This behavior is attributed to the director rotation by approximately n/2 when one crosses the defect along the y axis (just recall that the orientation of the eigensystem changes continuously on passing through the defect). On the other hand, the value of the Q3-component does not change too much indicating that the variation in the nematic ordering mostly occurs Nematcs with disped polymer neworks 07 in the xy-plane pependicular to the fiber dirion, z. Alternatively, Q±, Q2, and (^profiles can be interpreted also in terms of order parameters S and P (see Fig. 5.5, right). When the defect line is approached, the uniaxial order parameter S decreases from its temperature-defined bulk value and drops even below zero in the defect center. Note that there the nematic director i.e., the eigenvector corresponding to the negative eigenvalue, is directed along the fiber. On the other hand, the biaxiality — close to zero far enough from the defect increases when the defect line is approached, reaches a maximum and, finally, in the very center of the defect, again drops to a value that is close to zero. The characteristic length scales for these variations are of the order of a few (« 5) lattice spacings a and agree with the estimates for the corresponding correlation lengths in the nematic phase. Qualitatively, molecular ordering close to a disclination line can be summarized as follows. In the very center of the defect molecular ordering is uniaxial with S < 0 and P —> 0. Far enough from the defect line the nematic liquid crystal is uniaxial again, however, with S > 0 and P = 0, as expected in a homogeneous or in a weakly distorted bulk sample. In the intermediate ring-like region, nematic 30 f x/a 30 yla 10 15 20 25 30 0.6 a, .... — .... ... ---- -•* \ Z'" "~'s" &a* 0.2 .... _aK.__ '"p" -0.2 V 0.8 ........... *"" 0.4 01 0 „ oj1 -0.4 *¦¦* *-•*•-* *:* *¦¦* ---- ¦*••• / .*..*. **»» 10 15 20 25 30 yla Figure 5.5 Left: cross section of the director field n(x,y) represented by "streamlines". The shading represents the value of the order parameter S(x,y) (dark values correspond to a low S). A pair of — ^ defects has formed on the diagonal. In the defect core molecules are (on the average) aligned in the xy-pl≠ ordering is uniaxial with S < 0 and the corresponding eigenvector n, is directed out-ofplane (along the z-axis). Right: eigenvalues of the ordering matrix Qi, Q2, and Q3 plotted across the left defect along the y axis. In the upper panel the order parameter S and biaxiality P are plotted. The leftright asymmetry with respect to the defect core positioned at y/a = 20 is due to the presence of the fiber. While S exhibits a simple decrease if the defect is approached from any direction, P shows a more complex craterlike profile 08 Nematics with disped polymer neworks ordering is biaxial with P / 0. These conclusions agree also with results from alignment tensor-based phenomenological analyses of topological defects both of half-integer [104] and integer strength [105] To conclude the discussion about defects, here are some final remarks related to their position. There are two parameters characterizing the position of each defect line: its distance from the fiber surface and the corresponding polar angle in the rry-plane. Increasing the system temperature from T* = 1.0 to T* 11, the pair of defect lines moves away from the fiber which increases the thickness of the deformed region where radial ordering is well-pronounced, as imposed by strong surface anchoring; see Figs. 5.6 (a) and (b). The increase of T* results in an overall decrease of S and, consequently in a decrease of the corresponding elastic constants (proportional to S2). Moreover, when approaching the fiber surface, at higher T* the increase in S is larger and occurs over a somewhat larger characteristic length L (2.11), which makes the defect formation and the accompanying elastic deformation in the immediate fiber vicinity rather unfavorable. Therefore the defects are pushed away from the fiber surface when T* is increased. On the other hand, at fixed T* the defects move away from the fiber also as the fiber radius R is increased In addition for a given R the defect-to-fiber distance seems to be rather insensitive to changing the simulation box size. Indeed for large R (ie., for a low curvature of the fiber surface) the elastic deformation imposed by the defect is more compatible with the radial aligning tendency of the fiber if the defect is located far enough from the fiber surface Finally, for the current system size the main mechanism for the formation of the defects close to the simulation box diagonal seems to be the repulsion between defects corresponding to adjacent fibers as discussed above Also this repulsion is mediated by curvature elasticity and weakens upon increasing temperature. The actual locus of defects is then determined by the subtle interplay between all the effects listed above. Note that for T* ^ 1 the defect position becomes almost temperature-independent. Note also that the defect size increases with temperature which qualitatively agrees with the increase of the characteristic length L on approaching the nematic-isotropic transition Figure 5.6 Dirr fld for perfect homeotroic ain a;y-cross section. (a) T* = 1.0 , (b) T* = 1.1 (both nematic), and (c) T* = 1.2 (isotropic phase) n the nematic phase a pair of — ^ defects has formed close to the diagonal Nematcs with dsped poymer newo Analogous conclusions as for planar anchoring can be drawn in the isotropi phase (T* = 1.2) also for the homeotropic case, now inspecting the decays in (P2C) and S order parameters shown in Figs 5.4 (b) and (d). Note that the two defect lines observed in the nematic phase for T 1.0 and T* = 1.1 have now vanished and that the residual surface-induced ordering is simply radial [Fig. 5.6 (c)]. This is because the current fiberto-fiber distance exceeds 2L and hence the degree of ordering at the simulation box boundaries is already negligibly small. Consequently, no relevance should be attributed to the "randomly" distributed directors plotted in the outer cylindrical layers of the sample, Fig. 56 (c) 5.1.3 2H NM spetra The observations listed so far can be confirmed also by calculating 2H NMR spectra using the numerical output from MC simulations. Fig. 5.7 shows the NMR spectra calculated for the 30 x 30 x 30 particle sample and a single fiber (R = 5a) in the nematic (left, T* = 1.0) and in the isotropic phase (right, T* = 1.2), with the NMR spectrometer field applied along the fiber direction z. The calculation was based on generating the relaxation function G(t) 2.42) from the MC data and calculating its Fourier transform representing the spectrum, as described in detail in Sect. 2.3.1. Generating G(t), effects of homogeneous translational diffusion were included. Following the methodology applied in Chapter 4 to PDLC droplets, the diffusive molecular motion was simulated by a random walk on the cubic lattice performing 1024 diffusion steps per NMR cycle. The effective diffusion constant for such a random-walk process can be measured to be D = 256a25wQ/37r, yielding a rootmean-square molecular displacement of y/QDt0 = 32a in each NMR cycle Here a stands for the particleto-particle spacing on the cubic lattice, while t0 = 2t/5coq denotes the NMR cycle duration. Since this displacement is comparable to the sample size, the calculated NMR spectra are expected to be highly diffusion-averaged. Note that here a sufficient amount of MC data was available to update nematic particle configurations 1024 times per NMR cycle — thereby matching with the natural time scale tp for fluctuations of molecular long axes — and not less frequently as in the PDLC case. For smoothening, a convolution of the spectra with a Lorentzian kernel of width « 0.075o;q was performed. Finally, note that the NMR spectrometer magnetic field is assumed to be weak enough not to align nematic molecules, which, again, is the case only for strongly confined systems. The calculated spectra are shown in Fig. 5.7, left. In the nematic phase with T* = 1.0, for perfect planar anchoring {{Pzjg = 1) in the spectrum one has two peaks positioned at u)q/5cuq ~ ±0.6. In the chosen geometry, \coq/Scoq\ is supposed to be roughly equal to the value of S, the nematic order parameter, since the director and the direction of the NMR spectrometer magnetic field coincide. Indeed, for T* = 1.0 one finds S ~ 0.6, as already seen above from various order parameter profiles. Translational diffusion in this case affects the spectra only negligibly: the nematic director is homogeneous throughout the sample and the degree of order is enhanced only slightly in the vicinity of the fiber. Therefore, the effect of diffusion should be merely a slight increase in quadrupolar splitting, but the resolution of the Nematics with disped polymer newo spectra is not high enough to clearly see this surface ordering-induced shift. Proceeding now to fibers with partially disordered anchoring, in the spectra there is no noticeable change at least down to {P2)g ~ 0.25, reflecting the ability of the polymer network to align the surrounding liquid crystal along z. In the case when anchoring is completely disordered with {Pzjg ~ 0, the spectrum typically still consists of two peaks, however, the corresponding splitting can be arbitrary because there is no preferred direction in the system — note that only one example of the possible spectra is plotted. Note also that sometimes during the acquisition of the G(t) signal slow collective molecular motion can occur, which results in an increase of the spectral line width. On the other hand, in homeotropic cases with (-^2)^ ^5 0, molecular ordering is confined to the rcy-plane. The quadrupolar splitting now decreases by 50% with respect to perfect planar anchoring because the director is perpendicular to the spectrometer field direction (see the two spectra in the bottom of Fig. 57, left). In the bulk isotropic phase, however, quadrupolar interactions giving rise to the uq splitting are averaged out by the rapid molecular motion. Therefore, ignoring translational diffusion, in a confined system for S ~ 0 one should expect a single-peaked spectrum at ujq « 0, as already suggested above, and somewhat broadened U t 3 r*=i.o T*=l.2 -1 -0.5 0 0.5 1 (0)-(Oz)/d(OQ -1 -0.5 0 0.5 {(o-(o^I6(oq Q. O V o Figure 5.7 2H NMR spectra; T* = 1.0 (left) and T* = 1.2 (right). Top to bottom: spectra for {P2)g=1.0 (planar), 0.75, 0.5, 0.25, 0 (random), -0.25, and -0.5 (hometpi). The alning cy of the ntwok is lt only f (P2)g « 0. Nematcs with disped polymer neworks 11 by the surface-induced order. The spectra shown in Fig. 5.7, right, were calculated with fast translational diffusion, and it is evident that some of them are actually double-peaked. This is a clear signature of surface-induced paranematic order. In fact, the peak-to-peak distance decreases with decreasing degree of surface order; compare with Figs. 5.2 (b) and 5.4 (d). For {Pzjg ~ 0 exhibiting no surface order the spectrum is again single-peaked. Finally, note that the splitting observed for perfect planar anchoring ({^2)9 = 1) roughly amounts to twice the splitting seen in the perfectly homeotropic case ((^2)9 = —0.5). This is again because the nematic director close to the fiber is parallel to the NMR spectrometer magnetic field in the first case and perpendicular to it in the second. 5.2 External field-induced switching This Section is going to address the external field-induced switching of the molecular orientation in a nematic with dispersed polymer networks. First a regular array of straight and parallel polymer fibers will be treated, as in the preceding Section, here however, exclusively with perfect planar anchoring. In this simple model system the stability of different director configurations in field presence will be analyzed in de tail. Then we will proceed to more complex network topographies, first simulating irregularities in fiber positions (the fibers still taken as straight and parallel). Af ter this, a sample with a full topographical complexity — a system of curved and potentially cross-linked fibers — will be treated. Also in these latter cases the fiber surface will be assumed smooth, with planar anchoring along the local (tangential) fiber direction. In the end, the switching process will be monitored by inspecting selected experimental observables predicted from the simulation output 5.2.1 Regular fiber array Now a regular array of parallel fibers (Figs. 5.1 and 5.8) will be considered. Again the fiber direction was taken along the z-axis and an external field was applied perpendicular to the fibers along the y-axis. Note that switching experiments are y Figure 5.8 Regular fibr array: the switching geometry depicted schemati cally. Anchoring at fiber surfaces is planar along the fiber direction, z, while the external field E is applied along y Iz. E Nematics wth disd polymer n usually performed in an external electric field and that in an inhomogeneous dielectric (as encountered here) the electric field is not constant and homogeneous throughout the sample. In such situations, Maxwell equations are to be solved in order to obtain the local electric field strength vector E. For simplicity, however, in the following E will be assumed homogeneous. Alternatively, one could perform a switching experiment also in an external magnetic field where the inhomogeneities of the corresponding susceptibility are weaker and the above homogeneity assumption for the external field is more plausible. As discussed in Sect. 2.15, the switching threshold is sensitive to changes in the surface anchoring strength. Therefore, here cases with different anchoring strengths will be examined, with a dimensionless anchoring strength defined as w = eg/e. Here eg denotes the nematic-ghost coupling strength and e the nematic-nematic one; see Eq. (2.29). efore investigating external field-driven orientational transitions in the fiber array model system, it is instructive to draw a rough analogy with a nematic slab (compare also with Sect. 2.1.5). Let the slab of thickness d be confined between two parallel plates imposing strong planar anchoring along the z-axis, and let there be an external field E applied along the slab normal (y-axis). In a weak external field the equilibrium director profile (n) is homogeneous with nlE ("/i-structure"). Increasing the field strength, at first the director profile does not change, but once the Freedericksz threshold EF = JK/eo€aS(Tr/d) is reached, a transition to a deformed structure is observed ("d-structure"). Here K denotes the effective Frank elastic constant and eaS the macroscopic anisotropy of the dielectric constant. For finite anchoring strengths W the Freedericksz threshold EF is reduced, satisfying the relation n(K/Wd)(Ep/EF) = cot [(ir/2)(EF/EF)] [115]. Increasing the field strength E even further, for finite W the orienting effect of the field overwhelms the anchoring and a second, saturation transition takes place. Above this second threshold Es nematic molecules are aligned uniformly along the field, with n||E ("s-structure"). Here Es can be found from ir{K/Wd)(Es/E°F) = coth[(ir/2)(Es/EF)] [115]. Note that in case of weak anchoring or in a very thin nematic slab with K/W >> d, the thresholds EF and Es attain similar values, indicating that the region of stability for the deformed d-structure becomes extremely narrow. This can be inferred also from the phase diagram derived numerically for the h, d, and s structures in Ref. [115]. A similar switching behavior can be observed also in a regular array of polymer fibers favoring planar anchoring along z, the fiber direction, if the field E is applied, e.g., along y, as shown in Fig. 5.8. For comparison, yz-cross sections of director profiles calculated for each of the three structure types are shown in Fig. 5.9. It is interesting to notice that for the (i-structure the nematic is bent in the yz-plane, while it is twisted in the xz-plane, as opposed to the slab case where the twist deformation is absent. Consequently, the transition threshold values may differ from those estimated for the slab geometry. Qualitatively, the switching behavior is also expected not to change if the field is applied in some direction other than y, but still within the xy-plane so that E _L z is fulfilled. The specific in-plane orientation of E may, however, still affect the switching threshold position. In the MC simulation, the sample size was again set to 30a x 30a x 30a and the fiber radius to R = 5a. ll simulations were be performed at T* = 1.0, deep in the Nemacs with dsped poymer nwo E -----> y 77=0.5 Figure 59 Switcing in a rglar fiber ay: examples of director fields for different rj oc E2; T* = 1.0, R = 5a, and w = 1 (yz -cross sections through the fiber center). From left to right: homogeneous (/1), deformed (d), and saturated (s) structure. Anchoring easy axis is planar and ||z, while the external field E is directed along y. nematic phase. To reliably estimate the field thresholds for the Freedericksz and saturation transitions from the MC simulation, for a given anchoring strength w external field strength scans were performed as follows. The first run was performed at f) = 0 in absence of an external field. Then the external field strength ^ff\ was increased stepwise: at each stage, the last molecular configuration from the previous simulation run was used as input for the next run performed at somewhat higher 77, and so on until the maximum field strength was reached. Then the field strength was decreased stepwise (following the same strategy as upon increase) so as to detect a possible hysteresis in the simulated switching process. In the vicinity of structural transitions, orientational fluctuations in the system can increase significantly. Therefore, equilibration runs somewhat longer than in preceding simulations were needed to produce reliable results: approximately 1.2 x 105 cycles for equilibration and further 1.2 x 105 cycles for average production. Transitions between the three director structure types can conveniently be monitored by plotting the radial dependence of a suitable order parameter defined as P^if) = (|[3(uj • y)2 — l])r, where the brackets (...)r represent an average over MC cycles and over nematic particles lying within a cylindrical layer centered at radius r. Moreover, y is a unit vector along the external field direction. In case of perfect nematic alignment along y one has Pf = 1, while for perfect alignment perpendicular to y (e.g., along z) Pf = —| is obtained. For non-perfect alignment at finite temperature with S < 1 the above values reduce to Pf — S* and P% —> — ^S for alignment along y and z, respectively. The Pf (r)-profiles for strong (w = ) and weak (w = 01) anchoring and different field strengths (oc y/fj) are reported in Fig. 5.10. The corresponding anchoring extrapolation lengths can be estimated to be the order of ~ a for w = 1 and ~ 10 a for w = 01, which is still below the fiber-to-fiber distance deff ~ 20 a. For strong anchoring (w = , Fig. 5.0, left) at low r] the homogeneous ^-structure with n _L y is E E -—---~-Os\ \ I vvo— 77=0.005 77=0.05 Nematics with disped polymer newo found and, consequently, P^ir) is negative and almost constant. The slight variaton n P|(r) close to the fiber originates from an enhancement in the degree of ordering (S) close to the fiber, as discussed in Sect. 5.1.1. Above the first (Freedericksz threshold at tjf = 0.0085 ± 0.0015, P% s negative close to the fiber, while it i positive far enough from it. Such behavior is a signature of the deformed d-structure where close to the fiber the alignment is governed by anchoring, while far enough i the bulk it is governed by the external field. The characteristic distortion length — closely related to the external field coherence length Le — can be deduced directl from the P2(r) dependences. After a further increase of r\ beyond tjs = 030 ± 0.015 the second threshold is reached and P% becomes positive for all r, which indicates that now the homogeneous saturated s-structure is stable. The variation of P$ in the vicinity of the fiber can be ascrbed to a reduction of S with respect to its bulk value, occurring due to the confcting algnment tendences of the fiber and of the external field. Decreasing the anchoring strength (w = 0.1, Fig. 5.10, right), one can readily dentfy homogeneous h and s-structures, while the deformed d-structure seems to be missing. Ths is in agreement with the narrowness of the stability region for th (i-structure predicted in the nematc slab [115]. The Freedericksz and the saturation threshold can be estimated to lie within the interval r\ = 0.0034 ± 0.008. Note that now the variations of Pf close to the fiber for all r\ indicate a decrease of S in th region, which s a consequence of having weaker nematicfiber than nematc-nemat nteractions aving found three possble director structures {h, d, and s) for w = 0.1 and = 1, it is nteresting to explore their stability in an external field also at interme iate anchoring strengths and summarize the results in a structural phase diagram. However, the determination of threshold fields (both for the Freedericksz and the saturaton transon) is a rather delicate task. In fact, the recogniton of different 6 8 10 12 14 16 18 20 0.8 0.4 jpxxe*—— 0 ¦0.4 6 8 10 12 14 16 18 20 rla Figure 5.10 Regular fiber array: P|(r)-profiles calculated for T* = 10 Rja = 5, planar anchoring ||z, and an external field ||y. Left: w = 1; to to bottom: »7 = 1, 0.5, 0.1, 0.05, 001, 0.005, and 0.001. Right: w = 0.1; top to bottom: »7 = 1, 0.1, 0.01, 0.005, 0001, 0.0005, and 0.0001. Solid, dashed, and dotted lines represent /i, d7 and s-structures respectively. For w — 01 it is almost imossible to find the rstructure Nematcs with dsped polymer neworks 0.8 0.6 g 0.4 ^ 0.2 5> o -0.2 » -fc * •.*¦•» -7 -5 -6 -4 -1 0 log// 0.6 rT^-------------" " J.4 if ^—~-~~" a? Jj!"/" ^ 0.2 In/ ^-»C IM' &¦ 0 J/ In -0.2 Z^j -3 log 77 log J] Figure 5.11 Order parameters (Pf) (solid line) and (Pf)s (dotted line), plus the corresponding variances ov and of. Top: w = 1, bottom: w 0.1. losed circles correspond to scans upon increasing 7], the open ones to scans upon decreasing. Freedericksz and saturation thresholds can be deduced from the positions of the a and a\ peaks, respectively. For w = 0.1 both transitions virtually overlap structure types from snapshots or director fields turns out to be highly unreliable and subject to human eye-generated bias. Therefore, one can tackle this problem exclusively by finding a more quantitative criterion for structure identification. Hav ing applied an external field along the y-axis, an appropriate order parameter for monitoring is the overall (Pi), where the averaging (..) must be performed over all particles and MC cycles. This order parameter is suitable for the detection of the Freedericksz transition involving a significant portion of nematic particles. Th saturation transition, however, is expected not to reflect that well in the overall (P2) because it only affects a thin surface layer of particles. At this point one can conveniently define (P|)s, calculated exclusively for particles in the surface layer of thickness a. This "surface" order parameter is sufficiently sensitive to particle reorientations that actually contribute to the saturation transition. Furthermore, the sudden rise either of (Pj') or (-ff)s is accompanied by a significant increase of fluctuations of these order parameters. Then the corresponding variances of (P? and (P^Oi °v and olf, can also be used to spot the field strength threshold for both transitions. This is equivalent to studying the NI transition by monitoring fluctuations of the internal energy where the corresponding variance is proportional to the heat capacity [123] Nematics with disped polymer neworks For w = 0.1 and w = 1 the behavior of (Pf) and (Pf}s upon increasing field strength, along with their variances is shown in Fig. 5.11. Note that the reproducibility of the data points is good (comparing scans up and down in field strength) yet it is somewhat poorr in the w = 0.1 case where the surface anchoring is much weaker than for w = 1. The fact that there is no obvious hysteresis suggests that both the Freedericksz and the saturation structural transitions are secndorde (hence cntinuous), in agreement with phenomenological studies. After identifying both thresholds also for anchoring strength values ther than w = 1 and w = 0.1, a stability phase diagram can be plotted. For T* = 10 and R = 5a it is shown in Fig. 5.12, together with the equivalent phase diagram for a nematic slab, derived in Ref. [115]. In weak enough fields one can always find the /structue while in strong enough fields the sstructure is always seen. The dstructure appear at intermediate field strengths, but its stability region gets narrower with decreasing w. As shown in Ref. [115] for the case of a nematic slab, the h-d and ds coexistence lines do not merge upon deceasing the anchoring strength and thee is no triple point where all three structures would coexist. Due to a finite accuracy in field threshold estimates, in the present stability analysis the Freedericksz and saturatio transitions below a certain anchoring strength (w ~ 0.1) coalesce and cannt be distinguished any more. This, however, should not be regarded as an indication for the existence of the triple point Note that the qualitative agreement with results presented in Ref. [115] is athe ood. Further note that here only esults for a single temperature value, T* = 1.0, are resented. The position of the coexistence lines in Fig. 512 may shift with changing T, the temperature dependence entering the field threshld values via the elastic cnstant K and the anchoring extrlation length K/W. (b) 3 s .* 2 o i d ^^ o -1 h 0 1 lognK/Wd Figure 5.12 (a) Structural pase diagram log \Jr)jr\§ vs. logl/w calculated for T* = 1.0 and R/a = 5; r/o corresponds to the Freedericksz threshold at the strongest anchoring considered, w = 5. (b) Phase diagram for a nematic slab (from Ref. [115]) plotted in scales and ranges allowing for a compari son with (a), h denotes field strength values normalized with respect to the Freedericksz threshold found for infinite anchoring Moreover, the position of the hd cexistence line correspnding to the Freede icksz tansition is exected to deend n the effective fibero-fiber distance d// Nematcs with dsped polymer newo 117 (determined also by the polymer concentration in the sample): with decreasing deff at fixed w and T* the line is expected to move towards higher critical field strengths. In other words, the nematic in fiber-rich samples is more difficult to switch. On the other hand, the d-s saturation transition line is expected not to shift significantly. To check these statements, one can explore transitions in a network consisting of e.g., somewhat thinner fibers. Setting R/a = 3 (instead of R/a = 5 above) and considering a 18 x 18 x 18 sample with a single fiber (instead of the 30 x 30 x 30 one) roughly maintains the polymer concentration, but decreases the fiberto-fiber distance. For the thresholds at w = 1 one now finds rfp = 0.027 ± 0.003 and ??18 = 0.30 ± 0.03, which for both lines is in agreement with the behavior predicted above (just recall the corresponding thresholds for the larger 30 x 30 x 30 sample: rjf = 0.0085 ± 0.0015 and r/f = 0.30 ± 0.015). Note that like in slab geometry (and for strong enough anchoring) the Freedericksz threshold field strength scales approximately as l/deff, if deff is taken to be the shortest fiber-to-fiber distance. In fact, comparing deff for the 18 x 18 x 18 and 30 x 30 x 30 sample gives d^f/d^f = 0.6, while the ratio of threshold field strengths yields (^°/^8)1//2 = 0.56. On the other hand, the saturation transition threshold remains almost unaltered. All in all, the switching behavior in a regular fiber array system seems to be very close to that of a simple nematic slab although in the fiber array system the elastic deformation is somewhat different (twist and bend deformation combined). The latter, however, poses no problems because the LL model anyway is equivalent to the one-constant approximation. The difference in the type of geometrical confinement reflects at most in different quantitative behavior. 5.2.2 Irregular fiber array The actual topology of a polymer network shown in typical SEM pictures [26,39] (Fig. 1.3) is much more complex than the regular array of straight and parallel fibers considered so far. In particular, the effective interfiber distance distribution is expected to play an important role in switching studies. Therefore, as a first step towards a more complex network topography an irregular array of straight fibers will be studied, increasing the lattice size to 50 x 50 x 50 particles and including 8 fibers of equal thickness, with R = 3a. The fibers were modeled as straight and parallel (oriented along the z-axis), but distributed randomly within the xy-plane. The polymer concentration and the fiber radii were kept unchanged in comparison to the previous 18 x 18 x 18 case. This should provide the same average fiber-to-fiber distance and thus facilitate the comparison of results. Again, the external field was applied along the y-axis and the surface anchoring was assumed planar (with w = 1) along the z-axis. The concentration of the polymer can be estimated by dividing the number of frozen (ghost) particles by the number of all particles, yielding approximately 9% polymer. In the following, the regular array sample (previous Section) will be referred to as "sample A", while the sample with the fiber position irregularity as "sample B. Moreover, "sample C containing curved fibers will be considered later in this Section. Simulation runs involving samples of B and C-type consisted of 8 x 104 equilibration and 6.6 x 104 production cycles, while for 118 Nematics with dispersed polymer networks Figure 5.13 Array of several (8) straight fibers (sample B): P^i^i) order pa rameter map (xy cross section) for different rj. The switching process starts approximately at r\B = 0.013. The "columns" represent parallel straight polymer fibers. Calculated for T* = 10 w = 1, and Rja = 3 the smaller A-type sample the equilibration was shortened to 6 x 104 cyles Fig. 5.13 shows the evolution of Pf averaged over MC ycles for every particle within a given xy-cvoss setion for sample B. For r\ <¦ 0.012 one has -Ff(r) ~ —03 Nematcs with dsped poymer newo constant and negative, showing that particles are still aligned along z (recall that at T* = 1.0 one has S ~ 0.6) and that the Freedericksz threshold has not been reached yet. Reaching, however, r\ = 0.014, in regions where the fiber density is below average (the interfiber distance above average, the particles start to reorient along the field direction and Pf increases, becoming even positive in some areas upon a further increase in r\. It is important to notice that now the Freedericksz threshold is significantly lower than in the regular array case with the same polymer concentration — sample A. This can be attributed to the fact that the external field always destabilizes the longest-wavelength distortion first. Then, unlike in sample A, in the irreglar sample B there is a distribution of effective fiber-to-fiber distances, allowing also for deformations whose wavelength is larger than the average interfiber distance. As the field strength is increased even further, the parallel-to-fiber alignment persists only in the very vicinity of fibers and P% becomes positive almost everywhere. Finally, for extremely strong fields the saturation threshold is reached as well, and then all molecules are aligned along the external field direction. In addition, the strong field enhances the degree of nematic order [117]. Note also that in extremely strong fields close to each fiber there is a decrease of P% that can be attributed to a decrease in S. Indeed, in the vicinity of each fiber there is a conflict between the strong aligning tendencies of anchoring and of the external field. Note that the switching of the molecular orientation has not occurred in all parts of the sample simultaneously. Indeed, very strong fields are required to switch the particle orientation, e.g. between two (or more) fibers positioned close to each other (see Fig. 5.13, for r\ ;$ 03, in the left corner). Consequently, the Freedericksz transition is not as sudden as in a regular array-system. The saturation transition, on the other hand, is driven merely by a competition between surface anchoring and the external field. Therefore, wherever the fiber density is not too high, the positional irregularity of the network is expected not to affect the satration transition very significantly. The last step in modeling the polymer network topography consisted of dropping the assumption that the fibers be straight and parallel and considering a system of curved fibers instead. Again, the general (average) fiber direction was taken along the z-axis. Then each of the fibers (whose thickness was assumed constant along z) was generated by performing a biased random walk: while progressing along the indirection, random deviations from this direction within the xy-plane were performed with a given probability regulating the crvedness of the fiber. Doing this, care was taken to meet the periodic boundary conditions along the z-axis. Following uch a procedure, at high enogh polymer concentrations the neighboring fibers can meet and cross-link, thereby providing a complexity in the network topography that is already close to that present in a real sample (Fig. 13). The surface anchoring was assumed planar, here, however, along the local fiber direction (yet not necessarily matching with the ^-direction) with w = 1. The fiber network used in switching simulations presented here — also referred to as "sample C" — is shown in Fig. 5.14 Because the orientations of ghost particles everywhere agree with the local fiber directions, the ghost order parameter (i^g introduced in Sect. 5. can be sed also Nematics with disped polymer neworks Figure 5.14 Array of sevral curved fibers (sample C): st particles representing the fixed polymer fiber network; (i^)^ ~ 028 to characterize the curvedness of the fiber system. For sample C shown in Fig. 5.14 (P2)g « 0.28. The switching process was simulated also for the sample C. Fig. 5.5 shows director field yz-cross sections for different r/. Despite fiber curvature, in absence of external fields (r) = 0) the net molecular orientation seems to be still welldefined (along the average fiber direction, z-axis), except for the fiber vicinity where it is affected by the local anchoring easy axis. In Freedericksz geometry where the magnetic field is strictly perpendicular to the easy axis (samples A and B), the ori-entational transition happens abruptly at a welldefined threshold. Here, however due to network irregularity, the magnetic field and the anchoring easy axis are never perpendicular to each other, therefore — unlike in Freedericksz geometry — a magnetic torque acts on nematic particles already at arbitrarily low field strengths. A we shall see later, this results in a decrease of the field threshold value in comparison to more regular samples A and B, while the switching itself is still relatively sudden. At the same time it should be borne in mind that the sample C studied here covers length scales still far below the macroscopic ones The linear dimension of the sample box 50a ranges from 50 nm to nm, depending on which value of is taken into account (1 nm ;$ a ^ 5 nm). As a consequence, the detailed switching behavior depends sensitively on the specific irregularities of the network generated Nematcs with dispeed polymer networks 121 ?7=0 77=0.015 77=0.05 77=1 Figure 5.15 Sample C: yz-cross sections of the director field map at x/a = 39 for different r\. Note that in absence of the field r\ = 0 most molecules are aligned along z (the average fiber orientation), except for those close enough to fibers where they align along local fiber orientations. Applying an external field (rj / 0), the major particle reorientation is initiated at r\c ~ 0.010. for sample C. To provide more general results, either a much larger sample should e considered, or a set of smaller samples, yet with different network topographies. Finally, in any symmetry-lacking sample it is less instructive to show order parameter or director map cross sections because they may not provide the information on the general behavior such complex systems. On the contrary, it is much more relevant to explore simulation-predicted experimental observables, which will be done in the next Section. A detailed comparison of samples A, B, and C will e given as well 5.2.3 Experimental observables and network irregularity This Section will be devoted to the analysis of experimental observables (electric capacitance, intensity of transmitted polarized light, and 2H NMR spectra). As these 122 Nematics with dispersed polymer networks methods all monitor the bulk response of the system, they are suitable for monitoring the Freedericksz transition. For the same reason, they are rather insensitive to the saturation transition, affecting a relatively small number of molecules in a thin subsurface layer. The Freedericksz transition can be spotted most easily by performing an electric capacitance measurement, as described in Sect. 23.2. In the simulation, the experiment was set up as follows. The net fiber direction was chosen along z and, as above, the external field was applied along y. Then, following Eq. (245), normalized capacitance Cy was measured along the y direction to monitor the switching process. Below the Freedericksz threshold, in a nematic aligned perfectly along z (S = 1) one would expect to have Cy/eo = e'±, while for a perfect alignment along the external field (far beyond the saturation threshold) one should expect Cy/eo = e|i (e'± and eji standing for the eigenvalues of the molecular dielectric tensor). On the other hand, for imperfect (but homogeneous) nematic alignment at finite temperatures with S < 1, one has Cy/eo = |(1 — S)e'u + |(2 + S')e'± > ' for alignment along z and Cy/eo = |( + S)e' +1(1 — S)e'± < eji for alignment along y, as it follows from (27). Here ej > e'± has been assumed. Therefore, the measured capacitance depends both on the orientation of the nematic director and on the degree of nematic ordering given by S. Consequently, in sample geometry chosen here, the capacitance Cy is expected to exhibit a more or less sudden increase whenever a major reorientation of nematic molecules takes place in the sample. Fig. 5.16 shows the capacitance versus r\ (the field strength squared) for samples , B, and C in a semi-logarithmic scale. Recall that all three samples are characterized by different network topographies, yet having roughly the same polymer concentration. Like in experiments with a dielectrically highly anisotropic nematic reported in Ref. [39], e| = 29.8 and e'± = 6 were chosen in the simulation. Following Ref. [39], for simplicity same values of e| and e'± were assumed also for the polymer network. Inspecting now Fig. 5.16, one can observe that the switching threshold is lowest in the irregular sample C and highest in the regular sample A: f]A = 0.022 ± 0.01, 7]B = 0.013 ± 0.01, and r)C = 0.010 ± 0.01 Again, r)B is lower than t]a because in sample B there is a distribution of interfiber distances. Moreover, f]C is lower than r\B because the absence of perpendicularity between fibers and the external field direction allows for a field torque to act on molecules already at arbitrarily low field strengths. The second feature of curves depicted in Fig. 5.16 is their slope after exceeding the switching threshold. The curve corresponding to sample A is steeper than that of sample B because — as already seen — in the latter case the switching is a gradual process and in the high-fiber-density areas molecules refuse to switch unless the field is very strong. On the other hand, the curve pertaining to sample C is steeper than the curve B (and approximately as steep as that of the regular sample A), which is because now even molecules in the polymer-rich areas are not able to resist switching since the corresponding fibers are not perpendicular to the external field. Note also that in curve C there are jumps that may be attributed to the irregularities in the fiber network. In a macroscopic sample these would have been averaged out, yielding smooth capacitance curves comparable to those measured in Nematcs with disped polymer neworks 23 cJ^ • ¦ 20 ..»¦•"• • 18 . •*• • 16 mM 14 12 • * 10 •V? 8 ! • • • • ••••••J^,",¦" -7 -6 -5 -4 -3 -2 -1 0 log 77 Figure 5.16 External field-induced switching as monitored by capacitance (Cy) measurements: sample A (small dots), sample B (medium dots), and sample C (large dots). For explanations regarding the sample topographies see text Ref. [39] experimentally. The third feature of capacitance curves is the prethreshold value of Cy. In fact curves A and B in this region almost overlap, which is mainly because of the well-defined molecular alignment along z. For r\ = 0 in sample C the overall molecular alignment still matches with z, especially far enough from the fibers, but this is not quite true in the fiber vicinity (see the snapshots in Fig. 5.15). This is also why already below the threshold there are significant local deviations from z, resulting in an enhancement of Cy, if compared to the A and B cases Finally, note that for all samples at extremely high field strengths (above the saturation transition) the increase of Cy is attributed to an increase in the degree of nematic order (S) rather than to molecular reorientation. Further, as discussed in Sect. 4.2.1, the external fields applied in simulations to the rather small samples are typically unrealistically strong. Another convenient method for the determination of the Freedericksz threshold are the measurements of the intensity of polarized light transmitted through the sample (Sect. 2.3.3). The experimental set-up considered here is identical to that described at the end of Sect. 2.3.3. Let again the average fiber direction coincide with the z-axis and let the external field be applied along the y-axis, with the light beam also propagating in this direction. In a nematic slab with planar anchoring along z (and with no polymer networks) there is no x or z-dependence in the direc tor field. The intensity of light transmitted through such a sample (if put between two polarizers crossed at a right angle) is given by the formula (2.48). As already mentioned in Sect. 2.3.3, the output signal is maximized when the angle between the projection of the director onto the a;z-plane and the incident light polarization plane is eual to tpo = n/4. Having fixed (p0, the output signal I depends only on Nematics with disped polymer newo the birefringence A$ = (2-k/X) Jo[ne(z) - n0]dz, ie, I oc sin2(A$/2). This implies that below the Freedericksz threshold / will be a constant depending on the sample thickness d and on the effective refractive indices of the nematic. Exceeding the threshold, the director reorients, hich is accompanied by a change in the extraordinary index ne [see Eq. (2.46)]. Consequenty, A$ changes continuously as the director field deforms and, accordingly, the utput signal / oc sin2(A$/2) oscillates. The total number of oscillations seen upon increasing the field strength (oc ^ffj can be estimated by comparing A«l> for the homogeneus alignment along z (A$2 ith that for the alignment along the externa field (A$y). In the atter case n0 ~ ne because the optical axis (n) matches with the light beam directin, yielding A«^ « 0. In the former case, however, the refractive index difference is nonzer and is, fr nematics with weak optical anisotropy roportional to the degree order S [4]. Then, A$2/27 nS/X is the approximate number of oscillatins ob- served in the I(rj) graph, where An stands for the maximum difference n nQ0 in a erfectly ordered nematic (the superscript "0" referring to perfect nematic ordering ith S 1). From such a graph, the Freedericksz threshold can be estimated by identifying the point where the I(rj) curve begins to oscillate (as, eg., in Ref [116]) Turning now a sample with dispersed polymer networks, the switching behavior is simiar to that of a nematic sab, as seen in Sect. 5.2. In the simulation, the net fiber direction was chosen along z, while all other geometrical parameters were chosen as described in the previous paragraph, including the incoming light polarization plane fixed at cp0 = 7r/4. The sample thickness in this optical part o the simulation was set to 10/m. This can be done although the linear sample size in samples B and C is just 50a — where a ~ nm is the lattice sacing (or ne 0.8 0.6 ,o0.4 0.2 0 -7 -6 -5 -4 -3 -2 -1 0 log 7/ Figure 5.17 External field-induced switching as monitored by transmitted polarized light intensity I measurements: sample A (small dots), sample B (medium dots), and sample C (large dots). The arrows indicate the switching threshold as determined from the capacitance measurement (Fig. 5.16). 7 denotes the intensity of the incoming light Nematcs with disped polymer neworks 25 molecular dimension) — by letting the polarized light pass several (~ 200) times through the sample so as to accumulate a sufficient birefringence A$. In the smaller sample A 550 light passes were necessary to yield the same optical thickness. On each pass (ie., MC cycle), the molecular configuration was refreshed from the MC data. However, unlike in the nematic slab discussed above, now the local director depends also on the x and z coordinates, and so does the outcoming light intensity / = I(x,z). Because I = I(x,z) is modulated on length scales much shorter than the light wavelength A, I(x, z) is to be averaged across the light beam cross section in the xz-plane to yield an average light intensity observable in a real experiment. In the following it will be assumed that the polymer network is optically isotropic, with a refractive index np = 1.499. It will be further assumed that for the perfectly ordered nematic vPe 1.7445 (maximum value) and n° = 1.520, yielding An0 = n° — nQ0 ~ 0.2175. Probing such a sample with He-Ne laser light with A = 632 nm, at T* = 1.0 with S ps 0.6 A$z can be estimated close to A$z f 27rx2.1 Consequently, two oscillations can be anticipated in the /(^-dependence. Inspecting the simulated /(77) curves shown in Fig. 517, one immediately recognizes the oscillatory behavior predicted above. Again, curves for the A, B, and C sample are characterized by different Freedericksz thresholds, matching with those observed already in the capacitance measurement. Moreover, the oscillations of I upon increasing f) are slowest for sample B, which is in agreement with the lowest slope of the Cy{rj) capacitance curve (Fig. 516). At extremely strong fields I drops to zero which is because now the effective birefringence approaches zero, as discussed above. Note also that the value of below the threshold is different for each sample, originating in different AZ, as also already discussed. Finally, let us inspect 2H NMR spectra calculated for each sample type to monitor the switching process. Fig. 518 comparatively shows the spectra calculated for the samples A, B, and C, with the spectrometer field applied along the (average) fiber direction, z. Again, a smoothening convolution with a Lorentzian kernel of width ~ 0.07Scoq was performed. Like in Sect. 513, translational diffusion was simulated, taking the same rate of diffusion. Consider sample A first. In absence of the external field, as well as for low 77, in the spectrum there are two well-defined lines positioned approximately at a maximum frequency splitting reduced by a factor of S ~ 0.6, as it follows from Eq. (2.41). This is a signature of fiber-imposed molecular ordering along z below the Freedericksz threshold. Once the threshold at tja ~ 0.022 is reached, molecular orientations start to switch along y, the external field direction, and the splitting is reduced. At very high fields already above the saturation transition the quadrupolar splitting should equal half the splitting seen at zero field, provided that the degree of ordering remains unchanged. In a strong external field, however, this never is the case and therefore the splitting keeps on increasing as the field is getting stronger. Moving to sample B, one can observe that the low-field splitting starts to decrease already at t)b ~ 0.013. Moreover, the switching process is also more gradual, which is all in agreement the other two experiments. In sample C at low fields one also finds a double-peaked line shape, yet at a splitting slightly lower than in the previous two cases. This indicates that despite the network is highly irregular, the z-orientation of molecules is still maintained in most of the sample, 26 Nematics with disped polymer networks sample A sample B sample C -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 (co — co^/dau (co—o)z)/<5ft>g (co—co^)/dcoQ Figure 5.18 External field-induced switching as monitored by 2H NMR spec tra: sample A (left), sample B (center), and sample C (right). In all cases T* = 1 and w = 1. The spectra of sample A show more noise because this sample contains 5256 particles only — as opposed to 112200 particles in samples B and C. which agrees with snapshots plotted in Fig. 5.5. The fact that the splitting is somewhat lower than that observed for samples A and B, however, implies that in the vicinity of fibers nematic molecules do follow the local orientation of the network and thus deviate from the z-direction. Note also that the sample C switches already at r\c ~ 0.010, which, again, is the lowest threshold of all. In addition, the switching again occurs more suddenly than in sample B. 5.3 Pretransitional ordering in the isotropic phase The last issue considered in this Chapter will be pretransitional ordering in a nematic sample with dispersed polymer networks above the NI transition (T^7 = 11232) As seen already in Sect. 5.1 studying a regular fiber array, some surface-induced Nematcs with disped polymer neworks 27 Figure 5.19 Pransitional ordering in sample C at T* = 12, fr plana anchoring with w = 1: S(x, y) order parameter map cross section at z/a = 25. The paranematic order decays to zero over the correlation length L « 5a, except in polymerrich areas where nematic "bridges" can form in between fibers (as here in the left lower corner with S < 0.2). Note that while the "columns" denoting fiber positions are vertical, the bers themselves can be tilted with respect to the rry-plane paranematic ordering may persist in the vicinity of fibers. This ordering can be detected by optical means [26], as described in Sects. 23.3 and 5.23. Alternatively it can be detected also by 2H NMR [2] The simulations presented in this Section were performed with sample C presented in Sect. 5.23 (see Fig. 5.14). Again, planar anchoring along the local fiber direction was assumed and there was no external field applied. Paranematic ordering can then be characterized by a director parallel to the local fiber direction, and by a degree of order S decaying from a nonzero surface value to S ~ 0 over L ps 5a (at T* = 12), the corresponding correlation length. Therefore, in a low-polymer-density sample areas with S ^ 0 are limited only to the very vicinity of fibers. In high-polymer-density samples, on the other hand, fibers can approach each other (or even cross-link) and thereby create "bridges" of nematic order with S ^ 0 wherever the lowest fiber-to-fiber distance becomes comparable to ~ 2L — see Fig. 5.19 This effect is similar to the capillary condensation observed experimentally in thin nematic films by force spectroscopy [169] potentially allowing for a self-assembly of colloidal particles. The actual degree of paranematic order depends significantly also on temperature (T*) and on the strength of the orientational coupling of the liquid crystal with the fiber surface (w). Therefore, in this Section simulation results for different T* and w will be presented, focusing on the experimental output: transmitted light intensity and 2H NMR line shapes. In the simulation, the sample was equilibrated during 8 x 104 MC cycles and afterwards, 7 x 104 cycles were used to accumulate the relevant observables. The set up of the optical experiment was identical to that used in Sect. 5.3: Nematics with disped polymer nwo net fiber direction along the z-axis, light beam along the y-axis, and the polarizer and analyzer crossed in the xz-plane, each of them at an angle of 7r/4 with respect to the z-axis. Further, same sample thickness of d = 10 /im and light wavelength A = 632 nm were considered. Again, the polymer was assumed optically isotropic and the refractive indices of the liquid crystal equal to those used in ect. 5.3. As already discussed, the intensity of outcoming light / is proportional to / oc sin2(A$/2) (2.48), where the effective birefringence A$ is proportional to An, the difference between the extraordinary (ne) and ordinary (n0) refraction index averaged along the path of the light beam. Assuming that the net direction of paranematic ordering — averaged over the whole sample — is still along the z-axis, as imposed by the polymer network, and that An is small in comparison to ne and n0, An is simply proportional to the overall degree of order, in this case given by (P2Z). This relation can be obtained from the expression for the eigenvalues of the high-frequency dielectric tensor (2.7), assuming that (P|)Ae/ei < does not always hold (e.g., for w = 1) — and a possible significant variation of A$ is saturated in the i~ oc sin2(A$/2) dependence. For the same reason, the curve for w = 1 is not monotonous close to the NI transition. Moreover, with decreasing anchoring strength w the degree of ordering decreases, which then translates into a lower I/I0 signal. One can also try to estimate I/Io from the formula (2.48) and compare it with the simulated one. Taking An = 0.2175 and (^2)3 ~ 0.28, as above, and estimating (S) from the simulation for a single fiber ((S) ~ 0.13 at T = . and w = 1) yields A$ = An{S)(P2)g(27rd/\) « 0.8. This then gives I/I0 = sin2(A$/2) « 0.15 while the simulated value for T 1.2 and w = 1 is I/I0 0.23. Recall now that in the simulation I/I0 is deduced by averaging transmitted light intensities within the xz-plane — across the light beam cross section. In a real (macroscopic) sample, the outcoming light at each of the points in the xz-plane has accumulated roughly the same phase shift A$ when traveling through the sample, i.e., the intensity of the outcoming light is uniform within the probing light beam cross section. Therefore, in a real experiment the quantities (sin(A$/2))a;2 (calculated in the simulation and sin2((A$)a;ž/2) (estimated above and measured in Ref. [26]) are within reason equal. Here {.)xz stands for a spatial average in the xz-plane. The degree of network irregularity in the simulated 50 x 0 x 0 particle simulation box is, however, still rather far from that of irregularities in macroscopic sample. A a consequence, one Nematics with disped polymer nwor should not be surprised by the disagreement of the estimated and simulated I/Iq, and better focus on the qualitative behavior of I/I0 instead. Alternatively, pretransitional ordering can be detected also via 2H NMR. discussed in great detail in Sect. 2.3.1 and elsewhere, the quadrupolar line splitting cjg(r) (2.41) depends both on the local orientation of the nematic director (matching with the local orientation of the fiber network) and on the local degree of order. The spectra presented in Fig. 5.21 were calculated for sample C in the fast diffusion limit (the rate of diffusion being 1024 diffusive steps per NMR cycle, as earlier in this Section), with the spectrometer magnetic field directed along the z-axis. In this case the spectrum consists of one or two well-defined lines positioned at an average frequency ujz ± {loq(t)), where (...) stands for the spatial average. As above, in case of predominantly uniaxial molecular ordering the spherical harmonics addition theorem can be applied to Eq. (2.4 when (ujq(t)) is calculated, resulting m(uQ(v)) = 5uJQ(S)(P2)9 Fig. 5.21 shows the evolution of H NMR spectra upon increasing T* for different values of the anchoring strength w. A convolution of the spectra with a Lorentzian kernel of width ?» 007Sojq was performed in order to smoothen the spectra. The w=1.0 w=0.5 w=0.1 i -1 -0.5 0 0.5 1 (w—(o^/dcifo -1 -0.5 0 0.5 1 ((o-wz)/do)Q -1 -0.5 0 0.5 1 r*=i.i25 r=i.i3 r*=i.i35 r*=i.i4 T*= 1.145 r=i.i5 r*=i.i55 r=i.i6 r*=i.i65 r*=i.i7 7*= 1.175 r*=i.i8 r=i.i85 r*=i.i9 r=i.i95 r=i.2 J*= 1.205 r=i.2i r=i.2i5 Figure 5.21 Pretransitional ordering: diffusion-averaged 2H NMR spectra as a function of T for different anchoring strengths: w = 0.1 (left), w = 0.5 (center), and w — 0.1 (right). Double-peaked line shapes are a signature of surfaceinduced paranematic ordering. Nematcs with disped polymer networks 131 left sequence of specta in Fig. 5.21 is plotted for w = 1 where the surface degre of order is high enough to yield a doublepeaked spectrum. For T* = 1.2 the peak is estimated to be located at \uq/8coq\ fa 0.04 ± 001. On the other hand, (S)(P2)g 0.036, and hence the agreement of the two estimates is reasonably good. The agreement indicates that in case of paranematic molecular ordering in sample C biaxiality effects are not extremely important, supporting also the simple estimates in the above optical experiment. When decreasing the temperature towards T^T, the effective quadrupolar splitting increases which results from an increase of (S) For weaker anchoring (w = 0.5 and w = 0.1) the overall (5") is smaller and t calculated spectra are only singlepeaked — except rather close to T^T for w = 0. because their resolution is not high enough to detect (S) that approaches zero. In conclusion, the simulations presented in this Section clearly confirm the existence of paranematic ordering above T^T. This ordering weakens with increasing temperature and with the decreasing strength of the orientational coupling with th polymer surface. These effects were both observed also through simulated 2H NMR line shapes and transmitted ligt intensity measurements 32 Nematics with disped polymer networks onclusion In this concluding Chapter we will review the new achievements presented in the thesis, along with the still open questions and possible future directions. The thesis was mainly concerned with modeling confined nematics, applying well-established phenomenological (Landau-de Gennes) and, more extensively, simulation (Monte Carlo-type) approaches to selected open problems in the field. Motivated by experiments indicating significant subsurface elastic deformations in nematics, the first part (Chapter 3) focused on molecular ordering in a nematic slab, with a special emphasis to the vicinity of its interfaces. In the framework of the phenomenological Landau-de Gennes theory it was shown that unless the one-elastic constant approximation is applied, any variation of the degree of nematic order S — as occurring typically close to any interface — yields a subsurface elastic deformation whose characteristic length matches with that of the S-variation, the corresponding correlation length. This subsurface deformation exists only for planar distortions (in the tilt angle cf> profile), while it is absent in case of pure twist distortions. Note that the free energy functional was expanded only up to first spatial derivatives of S and (f), and that the observed subsurface deformation stems exclusively from the S-variation, having thus an origin completely different from that of deformations predicted in connection with the controversial splay-bend elastic constant. In addi tion, the S-variation yields an intrinsic contribution to the surface anchoring energy, with an easy axis either parallel or perpendicular to the sample normal. If the effec tive (intrinsic plus external) anchoring strength is adjusted to yield an extrapolation length approaching those observable experimentally (L ~ 100 nm) — here estimated by matching external field and anchoring effects — the subsurface deformation and the ^-variation are both small. This agrees with the use of a truncated first-order free energy functional. Nevertheless, it should be clearly stressed that the described phenomenological continuous approach cannot explain short-scale peculiarities in the orientation of molecules observed experimentally in the first molecular layer in direct contact with the substrate. The phenomenological study of ordering in a nematic slab was complemented by a molecular Monte Carlo (MC) simulation analysis based on the novel simple hexagonal lattice model. This model allows for the use of spatially anisotropic pairwise potentials (like the induced dipoleinduced dipole one considered here) without cre 133 34 Concusion ating any artificial bulk easy axes. In a deformed hybrid cell-like slab with one free surface, intrinsic anchoring was explored in detail. Its microscopic origin comes from incomplete spatially anisotropic intermolecular interactions close to the free surface. For the induced dipole-induced dipole interaction the corresponding extrapolation length L is microscopic, as estimated from simple profile extrapolation. It approaches experimental values only if the interaction anisotropy is significantly reduced. Moreover, L exhibits almost no temperature dependence, even if the nematic-isotropic transition is approached. This obvious disagreement with experimental findings might be attributed to the great simplicity of the present model. Possible extensions of the model comprise simulations of density variations close to the surface (inserting vacancies into the lattice) and simulations of rough or modulated solid substrates (both expected to reduce the anchoring strength). Nonetheless, a full and also computationally more demanding) description of nematic interfaces can be provided only by dropping the lattice restriction, allowing thereby directly for density variations and the formation smectic phases. This remains one of the goals of the future research. In the following Chapters, a rather simplified picture of interfaces was adopted: anchoring (exclusively external) characterized with a well-defined easy axis and strength. Instead, more complex confining geometries were treated: spherical PDLC droplets (Chapter 4) and polymer networks (Chapter 5). Chapter 4 was concerned with molecular ordering in radial and bipolar PDLC droplets. Recently, PDLCs have gained renewed interest with the advent of holographic PDLC materials. The ordering in spherical PDLC droplets was explored by performing MC simulations in the Lebwohl-Lasher model system. This model is also based on a lattice (however, simple cubic), with particles interacting via a spatially isotropic pair potential giving rise to no intrinsic anchoring. Following extensive simulation studies of nematic droplets in the past, the focal point of the present PDLC study was the development of a novel methodology for calculating 2H NMR spectra — one of the important experimental observables for these systems in presence of significant molecular motion, such as translational diffusion both spatially homogeneous and inhomogeneous) and fluctuations of molecular long axes. The methodology was tested successfully for bipolar and radial droplets, allowing for an identification of both director structures by inspecting the calculated line shapes. The results indicate that molecular fluctuations lead to a narrowing of the NMR spectrum, while its shape remains similar to that obtained in the static case. From this narrowing it was possible to deduce the values of S, the order parameter, which turned out to be in an excellent agreement with values deduced directly from the MC data. Further, molecular diffusive motions result in an averaged spectrum, which in the fast diffusion limit consists of one or two rather narrow lines, depending on the type of boundary conditions. The positions of these lines can also be predicted from MC data, and again the agreement is fairly good. If the diffusive motion close to the confining substrate is significantly hindered, this results in perceptible changes in NMR spectra. These changes should be detectable also experimentally in droplets with a high enough surface-to-volume ratio. Moreover, "powder" spectra for an array of many bipolar droplets with randomly oriented symmetry axes were Concuson 135 calculated as well. Finally, droplets in an external electric or magnetic field were studied, together with the evolution of the corresponding spectra upon increasing the field strength. The field-induced effects seen to reflect in the spectra are: molecular reorientation along the field direction, enhancement of the degree of nematic order, and the formation of the nematic phase at temperatures even higher than the nematic-isotropic transition. The two latter effects can be detected only in external fields that are unrealistically strong. In the radial droplet the external field induces a continuous structural transition from the radial "hedgehog" structure to an aligned one whose spectrum becomes similar to that of the bipolar droplet. Note that currently it is still impossible to fit experimental spectra with the simulated ones because of the smallness of the simulation box. To reduce the noise present in the spectra, smoothening convolutions must be applied, distorting the spectra to some degree. Qualitatively, however, the agreement of experimental and simulated spectra is very good. nother type of confinement — polymer networks dispersed in a nematic — was investigated in Chapter 5. Like PDLCs, these systems are becoming important for applicative purposes, yet to date lacking molecular simulation studies. Here as a first step towards a realistic fiber network a regular array of straight and parallel fibers was studied, focusing on effects of surface roughness. Different types of anchoring conditions at the fiber surface were considered: planar along the fiber direction, homeotropic, and partially or completely random. In cases with perfect planar or homeotropic anchoring in the very vicinity of the fiber nematic order is enhanced. However, once the anchoring conditions are partially distorted, the surface degree of nematic order may drop below its bulk value, but the long-range orienting capability of the fiber network is still retained. This ability seems to be lost only for completely random anchoring imposing no well-defined direction in the system, as confirmed also by the 2H NMR spectra calculated from the simulation output. Further, above the nematic-isotropic transition surface-induced paranematic ordering could be detected, reflecting clearly in the translational diffusion-averaged 2H NMR spectra. In addition, in the nematic phase two — \ strength disclination lines parallel to the fiber were observed for perfect homeotropic anchoring, and were seen to move away from the fiber upon increasing temperature. Plotting order parameter S and biaxiality P maps made it possible to resolve the inner structure of the defect core: in the very center nematic ordering is uniaxial with S < 0, surrounded by a ring where ordering is strongly biaxial with P^O and S > 0. These findings agree very well also with those coming from phenomenological defect studies. Then, in the same model system, an external field was applied in the direction perpendicular to the fibers to investigate the switching behavior of nematic molecules. Monitoring fluctuations of relevant order parameters, both the Freedericskz and the saturation transition were identified for different anchoring strengths. A stability phase diagram for possible nematic structures was plotted and was found to be in good qualitative agreement with the one calculated from the Frank elastic theory for the simple slab geometry. As a second step in increasing the complexity of the fiber network, a system of several straight and parallel, yet randomly positioned fibers was considered. At unchanged polymer concentration, the Freedericskz switching 136 Concusion threshold in such a sample shifts towards lower field strengths, and the switching process is more gradual than in a system of regular fibers. An even more realistic network model includes randomly positioned fibers that are curved and allows also for cross-linking between them. This model represented the most complex type of confinement considered in the thesis. The Freedericskz threshold in such a sample is not welldefined any more, but the switching is still relatively sudden. These conclusions were all demonstrated also by the simulated experimental observables namely 2H NMR line shapes, electric capacitance, and transmitted polarized light intensity. Finally, paranematic ordering was inspected also in the irregular fiber system. 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