Various results concerning homogenisation of nematic liquid crystals

In this thesis we present various results concerning homogenisation of nematic liquid crystals: two of them are in perforated domains, while the other one concerns rates of convergence for boundary homogenisation. The first work described in this thesis is a $\Gamma$-convergence result for the Landau-de Gennes model in $3D$ domains with connected perforations. The goal of the analysis is to find new terms in the energy functional that are independent of the gradient. The second result is an error estimate for a $2$D toy model used to describe rugosity effects in nematic liquid crystals via homogenisation problems, using once again the Landau-de Gennes model. The last problem is a local $L^2$-convergence result for a homogenisation problem in $\mathbb{R}^2$ with isolated perforations. Here we use the Oseen-Frank model, with the goal of finding new gradient-dependent terms in the energy functional. We start, in Chapter 1, with a brief introduction to nematic liquid crystals. We introduce two major variational models used to describe nematic liquid crystals: Landau-de Gennes (LdG) and Oseen-Frank (OF). For LdG theory, which uses $Q$-tensors as the order parameter, we present typical choices for each type of energy contribution (bulk, elastic and surface). For OF theory based on the order parameter $\textbf{n}\in\mathbb{S}^2$ we discuss the elastic energy, that depends on the director and its gradient. We then present a short summary of the main mathematical results obtained for LdG and OF. In Chapter 2, we analyse a homogenisation problem in $\mathbb{R}^3$ using the Landau-de Gennes model in which the perforations form a cubic microlattice. We assume a dillute regime, that is the volume of the cubic microlattice tends to $0$ as its characteristic length scale tends to $0$. The goal of this problem is to show that, given this geometrical setting, by choosing various types of surface energies one can obtain a new material in the limit of vanishing characteristic size of the microlattice. This material also behaves like a nematic liquid crystal, but now with different bulk coefficients. At the end of this chapter, we discuss a rate of convergence of the approximating surface energies to a homogenised term. In Chapter 3, we concentrate on achieving and improving error estimates in homogenisation problems, since they can give us crucial information for manufacturing processes. Here, we consider a simplified $2$D model in which we highlight how one could replace a rugose boundary with the imposed homeotropic alignment by a flat boundary with an effective alignment depending on the initial geometry of the rugosity. We are able to improve an $L^2$ error estimate for a class of linear nonhomogeneous Robin problems. In Chapter 4, we consider a general elastic energy in a $2$D case for the Oseen-Frank model in domains with isolated perforations. The goal of this study is to analyse how one could obtain a nematic liquid crystal with novel elastic properties via homogenisation procedures. Under suitable conditions we can analyse the $\mathbb{S}^1$-valued homogenisation problem via a scalar problem obtained through the lifting procedure. We also prove a local $L^2$ convergence result

Zaharia Bârsan and The Establishment of the National Theatre in Cluj

The article tackles the circumstances in which the National Theatre in Cluj came into being and its relevance to the Romanian cultural scene. It was inaugurated on the 1st of December 1919 after the Unification of Transylvania and Romania on the 1st of December 1918. To have professional theatre performed in the mother tongue of the majority was the embodiment  of an age-old yearning of the Romanians living in this region, and was supported by the entirety of the theatre movement. In this respect, the most important aspects are mentioned. Before the Unification, Transylvania was part of the Austro-Hungarian Empire, and Romanians, despite forming the largest segment of the population, were completely stripped of rights. Therefore, the new-born National Theatre in Cluj was invested with a plurality of missions, the main one being to stand as an expression and a stimulus of Romanian creativity in all Transylvania, which was materialized through tours and series of performances in the region. The paper also attempts to capture the complexity of Zaharia Bârsan’s personality, a reputable theatrical animator, actor, director, teacher, dramatist, poet, prose writer, who was the first to be invested as director and founder of the institution. The difficulties he had to overcome, which revolved around founding a troupe of talented actors, finding a repertoire and bestowing the theatre with an aura of prestige are analysed. Initially, the repertoire included classical pieces, which were thought to carry a clearer,  more accessible message, but he introduced modern and Romanian pieces as well. Zaharia Bârsan managed to produce extraordinary performances right from the first theatrical seasons: Vlaicu Vodă by Al. Davila, Oedipus Rex by Sophocles, Shakespeare’s Hamlet and Macbeth

Cubic microlattices embedded in nematic liquid crystals: A Landau-de Gennes study

We consider a Landau-de Gennes model for a connected cubic lattice scaffold in a nematic host, in a dilute regime. We analyse the homogenised limit for both cases in which the lattice of embedded particles presents or not cubic symmetry and then we compute the free effective energy of the composite material. In the cubic symmetry case, we impose different types of surface anchoring energy densities, such as quartic, Rapini-Papoular or more general versions, and, in this case, we show that we can tune any coefficient from the corresponding bulk potential, especially the phase transition temperature. In the case with loss of cubic symmetry, we prove similar results in which the effective free energy functional has now an additional term, which describes a change in the preferred alignment of the liquid crystal particles inside the domain. Moreover, we compute the rate of convergence for how fast the surface energies converge to the homogenised one and also for how fast the minimisers of the free energies tend to the minimiser of the homogenised free energy.SEV-2013-0323-17-1 (BES-2017-080630

Various results concerning homogenisation of nematic liquid crystals

164 p.In this tesis we present various results concerning homogenisation ofnematic liquid crystals: two of them are in perforated domains, while the other oneconcerns rates of convergence for boundary homogenisation. The first workdescribed in this thesis is a Γ-convergence result for the Landau-de Gennes model in 3D domains with connected perforations. The goal of the analysis is to find newterms in the energy functional that are independent of the gradient. The secondresult is an error estimate for a 2D toy model used to describe rugosity effects innematic liquid crystals via homogenisation problems, using once again the LandaudeGennes model. The last problem is a local L2-convergence result for ahomogenisation problem in R2 with isolated perforations. Here we use the Oseen-Frank model, with the goal of finding new gradient-dependent terms in the energyfunctionalThis research is supported by the Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2013-0323-17-1 (BES-2017-080630) and through project MTM2017-82184-R funded by (AEI/FEDER, UE) and acronym “DESFLU”

Effective surface energies in nematic liquid crystals as homogenised rugosity effects

We study the effect of boundary rugosity in nematic liquid crystalline systems. We consider a highly general formulation of the problem, able to simultaneously deal with several liquid crystal theories. We use techniques of Gamma convergence and demonstrate that the effect of fine-scale surface oscillations may be replaced by an effective homogenised surface energy on a simpler domain. The homogenisation limit is then quantitatively studied in a simplified setting, obtaining convergence rates