STABILNOST IN METASTABILNOST NEMATIČNIH STEKEL

Structures exhibiting continuous symmetry breaking are extremely susceptible to various perturbations. The reason behind is the existence of Goldstone modes in the gauge component of the order parameter describing broken symmetry. The so-called Larkin-Imry–Ma argument claims that even infinitesimally weak random field-type disorder destroys long range order (LRO) which would otherwise be present in the absence of random disorder. Furthermore, it claims that the system breaks into domain type configuration having short range order (SRO), where the characteristic domain size scales as ksi= W^-2/(4-d). Here W measures the strength of random field interaction and d is the dimensionality of space. However, some studies claim that structures with quasi long range order (QLRO) are established instead of SRO. The main focus of this doctor thesis is the character of nematic structures in the random field. I studied theoretically and numerically nematic structures that are obtained by continuous symmetry breaking in orientational degrees of freedom on decreasing the temperature T, starting from the ordinary liquid, the so called isotropic phase. In particular, I investigated conditions for which the Larkin-Imry-Ma theorem holds true. So far statistical interpretations of such systems have typically used two different semi- microscopic type models: i) the Random Anisotropic Nematic (RAN) and ii) the Sprinkled Silica Spin (SSS) model. The RAN model is a Lebwohl-Lasher (LL) model with nematic molecules locally coupled with uncorrelated random anisotropic field at each site, while the SSS model has a finite concentration of impurities frozen in random directions. I used a three dimensional (d = 3) model intermediate between SSS and RAN models, with finite concentration p of frozen impurities, where p < pc (pc stands for the percolation threshold). The simulations were performed at different temperatures for temperature-quenched (TQH) and field-quenched histories (FQH), as well as for temperature-annealed histories (AH). The first two of these limits represent extreme histories encountered in typical experimental studies. Numerically, I studied the impact of control parameters (T, p, W) and history of samples (TQH, FQH, AH) on structural properties of the system. Within the model I was varying p, temperature T, interaction strength W and also sample histories. From final configurations, I calculated orientational order parameters and two-point correlation functions. Next, I estimated the size of the Larkin-Imry-Ma domains d. Finite size-scaling was also used to determine the range of the orientational ordering, as a function of W, p, T and sample history. The main results of my study are the following. In general, the system exhibited strong memory effects, indicating important role of history of samples. Furthermore, obtained results were relatively robust (from macroscopic point of view), indicating substantial energy barriers among competing states. On increasing the strength W, I typically obtained the following sequence of orders: LRO, QLRO, and SRO. For some concentrations p,however, SRO was absent. The crossover anchoring strength between QLRO and SRO strongly depends on history of samples, and it has the lowest values for TQH. From my simulations it follows that for the model used the Larkin-Imry-Ma argument holds only in limited range of model parameters. In most cases I obtain QLRO instead of SRO. However, in all structures there is imprint of Larkin-Imry-Ma domains, exhibiting scaling d 1/ (W2p) in the weak anchoring regime. This suggests that we do not have a “classical ” QLRO with algebraic decay with distance. Similar results were obtained in the studies of magnetic systems.Strukture, pri katerih prihaja do zloma zvezne simetrije, so zelo občutljive na različne motnje. Razlog za to je obstoj Goldstonovih nihajnih načinov v umeritveni komponenti ureditvenega parametra, ki podaja zlomljeno simetrijo. Tako imenovani Larkin-Imry–Ma-jev teorem pravi, da celo infinitezimalno šibek nered tipa naključnega polja uniči red dolgega dosega (LRO), ki bi sicer bil prisoten. Dalje trdi, da sistem razpade v konfiguracijo domen z redom kratkega dosega (SRO), z značilno velikostjo domen ksi= W^-2/(4-d). kjer je W jakost sklopitve med sistemom in naključnim poljem, d pa je dimenzija prostora. Vendar pa nekatere študije kažejo, da namesto tega nastanejo strukture z redom kvazi-dolgega dosega (QLRO). Glavni poudarek te doktorske teze je na značaju nematičnih struktur v prisotnosti naključnega polja. Teoretično in numerično sem študiral nematične strukture, ki jih dobimo z zlomom zvezne simetrije v orientacijskih prostostnih stopnjah pri znižanju temperature T od stanja običajne kapljevine ali izotropne faze navzdol. Posebej sem raziskal pogoje, pri katerih zares velja Larkin-Imry-Ma-jev teorem. Pri statistični interpretaciji teh struktur so doslej uporabljali dva različna semi-mikroskopska modela: i) naključni anizotropni nematični (RAN) in ii) SSS model. RAN model je Lebwohl-Lasher-jev (LL) model, kjer so nematične molekule lokalno sklopljene z nekoreliranim naključnim poljem v vsaki celici simulacijske mreže, medtem ko ima SSS model končno koncentracijo nečistoč z zamrznjenimi naključnimi smermi. Uporabil sem tridimenzionalni (d = 3) model, ki je vmes med modeloma SSS in RAN in ima končno koncentracijo p zamrznjenih nečistoč, tako da velja p < pc (pc označuje perkolacijski prag). V okviru tega modela sem spreminjal parametre p, T, W kot tudi zgodovino sistema. Simulacije so bile narejene pri različnih temperaturah za zgodovino temperaturne zamrznitve (TQH), zgodovino zamrznitve v zunanjem polju (FQH) in zgodovino s postopnim spreminjanjem temperature (AH). Prvi dve zgodovini sta limiti, ki ustrezata značilnim eksperimentalnim študijam. Numerično sem raziskoval vpliv parametrov (T, p, W) in zgodovine sistema (TQH, FQH, AH) na njegove strukturne lastnosti. Iz končnih konfiguracij sem izračunal orientacijske ureditvene parametre in dvotočkovne korelacijske funkcije. Iz slednjih sem ocenil velikost Larkin-Imry-Ma-jevih domen d. Za ugotovitev dosega orientacijskega reda v odvisnosti od omenjenih parametrov in zgodovine vzorca je bila narejena tudi analiza končne velikosti mreže. Glavni rezultati moje študije so naslednji. Na splošno je sistem izkazal močne spominske učinke, to je bistveno vlogo zgodovine vzorca. Dobljeni rezultati so relativno robustni (kar se tiče makroskopskih parametrov), kar kaže na velike energijske bariere, ki ločujejo tekmujoča stanja. S povečevanjem jakosti interakcije W sem večinoma dobil zaporedje ureditev: LRO, QLRO in SRO, za določene koncentracije p pa je manjkal red SRO. Kritična jakost W za prehod med redoma QLRO in SRO je zelo odvisna od zgodovine sistema in je najnižja za TQH. Iz simulacij sklepam, da Larkin-Imry-Ma-jev teorem dobro velja v opisanem modelu samo v omejenem območju parametrov, drugod pa imamo QLRO namesto SRO, kot predvideva teorem. Vendar pa so v vseh strukturah prisotni \u27\u27odtisi\u27\u27 Larkin-Imry-Ma-jevih domen, za katere velja d 1/(W2p) v območju majhne sklopitve W. To nam daje sklepati, da nimamo klasičnega QLRO z algebrajskim pojemanjem reda z razdaljo. Podobni rezultati so bili dobljeni v študijah magnetnih sistemov

Field induced memory effects in random nematics

We studied numerically external field induced memory effects in randomly perturbed nematic liquid crystals. Random anisotropy nematic-type lattice model was used. The impurities imposing orientational disorder were randomly spatially distributed with the concentration p below the percolation threshold. Simulations were carried for finite temperatures, where we varied p, interaction strength between LC molecules, and impurities and external field B. In the {B, T} plane we determined lines separating short range—quasi long range and quasi long range—long range order. Furthermore, crossover regime separating external field and random field dominated regime was estimated. We calculated remanent nematic ordering in samples at B = 0 as a function of the previously experienced external field strength B

History-dependent patterns in randomly perturbed nematic liquid crystals

We study the characteristics of nematic structures in a randomly perturbed nematic liquid crystal (LC) phase. We focus on the impact of the samples history on the universal behavior. The obtained results are of interest for every randomly perturbed system exhibiting a continuous symmetry-breaking phase transition. A semimicroscopic lattice simulation is used where the LC molecules are treated as cylindrically symmetric, rod-like objects interacting via a Lebwohl-Lasher (LL) interaction. Pure LC systems exhibit a first order phase transition into the orientationally ordered nematic phase at T = Tc on lowering the temperature T. The orientational ordering of LC molecules is perturbed by the quenched, randomly distributed rod-like impurities of concentration p. Their orientation is randomly distributed, and they are coupled with the LC molecules via an LL-type interaction. Only concentrations below the percolation threshold are considered. The key macroscopic characteristics of perturbed LC structures in the symmetry-broken nematic phase are analyzed for two qualitatively different histories at T << Tc. We demonstrate that, for a weak enough interaction among the LC molecules and impurities, qualitatively different history-dependent states could be obtained. These states could exhibit either short-range, quasi-long-range, or even long-range order