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Compounds which exhibit liquid crystal phases have been widely used in display technology. The majority of display applications utilize the nematic liquid crystal phase, which is a liquid-like phase which has partial orientational order at the molecular level. The nematic phase exhibits birifringence which can be manipulated through the application of an external field. Subsequently, all liquid crystal-based display technology utilizes the application of an external field to ???switch??? or tune the optical properties of a nematic domain into a desired optical state. In addition to an external field, the geometry and surface interactions of the liquid crystal domain must be precisely controlled in order for the display to operate properly. Liquid crystal displays (LCDs) utilize a rectangular domain, or pixel, within which the nematic domain is exposed to surface anchoring conditions that result in a twist of the nematic alignment through the thickness of the domain. In this work, a different type of liquid crystal domain that is elliptic is studied which is formed through ???bottom-up??? techniques, such as phase separation of a liquid crystal/polymer mixture to form a polymer-dispersed liquid crystal (PDLC) composite. Nematic domains within PDLCs are spheroidal, as opposed to rectangular for a pixel, and thus exhibit substantially different behaviour in the presence of an external field. The fundamental difference between spheroidal and rectangular nematic domains is that the former requires the presence of defects in nematic order while the latter does not. The overall objective of this work is to study, for a simplified elliptic cylinder domain, the formation of the nematic domain, the resulting domain texture in the presence of an external field, and the domain texture following release of the external field. These three states are directly related to applications of PDLC films as optical functional materials, where an external (electric) field is used to manipulate the optical properties of the film. The effects of geometry (aspect ratio), surface anchoring, and external field strength are studied through a simulation-based approach using the Landau-de Gennes theory of the nematic phase.

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本論文致力於棒狀液晶體動力學系統的幾個方面的研究，其中包括:空間週期問題及初邊值問題弱解的整體存在性，初值問題及初邊值問題強解的局部存在唯一性及相應的爆破準則。 / 首先，借助於一類新的逼近系統，我們證明了非等溫液晶體系統的二維空間週期問題存在整體弱解。與經典的Ginzburg Landau逼近系統不同的是，我們所採用的系統不是奇異型逼近。由於無法從基本能量等式中獲得方向場的二階導數的估計，我們採用局部能量不等式，於局部時間內，重新獲得了這些估計。與等溫系統不同，我們所獲得的弱解保持整體能量不變。由此可以看出，系統所損失的內能和動能全部轉化為熱能。此外，我們所獲得的弱解至多含有限個奇異時間點。而且，在每個奇異時間點上，由於能量集中現象的發生，系統的溫度必然會在某些區域內突然升高。 / 其次，通過Ginzburg- Landau逼近，我們證明了三維Ericksen-Leslie系統的初值問題存在唯一的局部強解，並建立了相應的爆破準則。我們共建立了如下四種爆破準則: (i) Serrin型準則; (ii) Beale-Kato-Majda (縮寫為BKM) 型準則; (iii) 混合型準則，即對速度場和方向場之一提Serrin型條件，而對另一場提BMK型條件;(iv) 一個新型準則，即用Ginzburg-Landau逼近系統的強解的Serrin型範數來刻畫Ericksen- Leslie 系統的強解的最大存在區間。其中，借助於一新的對數型Sobolev不等式，我們對速度場所建立的BKM型條件是經典條件的一個BMO型弱化。此外，我們還證明了，于強解的存在區間內， Ginzburg-Landau系統強收斂至Ericksen- Leslie 系統。 / 然後，我們將我們的關於強解的局部存在唯一性的結果推廣至有界域的情形。我們採用的依舊是Ginzburg-Landau逼近。為克服由於缺乏方向場的法向三階導教估計而帶來的困難，我們採用了一新的Sobolev型嵌入不等式。該嵌入不等式將Sobolev函數嵌入至一適當的混合範教LP 空間。此外，我們亦證明了局部強解的相應爆破準則 。 / 最後，通過利用前述提及的技街，我們證明了二維有界域上Ericksen-Leslie 系統的初邊值問題存在唯一的局部強解並存在一整體弱解。特別地，通過利用局部能量不等式，我們證明了強解的存在區間以及局部時間內的估計只依賴於初值的基本能量及其L²積分連續性。借助於此，通過對強解取極限，我們獲得了Ericksen-Leslie 系統的弱解。 / This thesis is devoted to some studies on the hydrodynamical model of nematic liquid crystals, including: the global existence of weak solutions to the spacial periodic and the initial-boundary value problems, the local well-posedness and blow-up criteria of strong solutions to the Cauchy and the initial-boundary value problems. / First, we prove the global existence of weak solutions to the non-isothermal nematic liquid crystal system in T², based on a new approximate system. Different from the classic Ginzburg-Landau approximation, this new system is not a singular type one. Local energy inequalities are employed to recover the estimates on the second order spacial derivatives of the director fields locally in time, which cannot be derived from the basic energy balance. Different from the isothermal case, the weak solutions we obtained conserve the total energy, and thus the kinetic and potential energies transfer to nothing but the heat energy. Also our weak solutions have at most finite many singular times at which the energy concentration occurs, and as a result, the temperature must increase suddenly at each singular time on some part of T². / Next, we prove the local well-posedness and blow-up criteria of strong solutions to the Ericksen-Leslie system in R³, based on the Ginzburg-Landau approximation. Four kinds of blow-up criteria are established, including: (i) the Serrin type; (ii) the Beale-Kato-Majda (BKM for short) type; (iii) a mixed type, i.e., a Serrin type condition for one field and a BKM type condition on the other one; (iv) a new one, which characterizes the maximal existence time of a strong solution to the Ericksen-Leslie system in terms of the Serrin type norms of the strong solutions to the Ginzburg-Landau approximate system. Besides, thanks to a new logarithmic Sobolev type inequality, our BKM type condition for the velocity is a BMO type improvement version. We also show that the strong solutions of the Ginzburg-Landau approximate system converge to a strong solution of the Ericksen-Leslie system up to the maximal existence time of this solution. / Then, we generalize our results on the local well-posedness of strong solutions to the Ericksen-Leslie system for the whole space to bounded domains of R³, still by the Ginzburg-Landau approximation method. A new Sobolev embedding inequality into mixed-norm L{U+1D3E} space is exploited to overcome the difficulty caused by the lack of the uniform estimates on the third order normal derivative of the director field to the Ginzburg-Landau approximate system. We also establish a blow-up criterion of the local strong solutions to the Ericksen-Leslie system. / Finally, using the technics exploited in the previous results, we obtain the local existence of strong solutions and the global existence of weak solutions to the Ericksen-Leslie system in bounded domains of R². In particular, by employing the local energy inequality, we prove that the lower bound of the existence time and the local in time estimates of a strong solution depend only on the basic energy and the L² integral continuity of the initial data. Thanks to these properties, by taking the limit of a sequence of strong solutions, we obtain a weak solution to the Ericksen-Leslie system. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Li, Jinkai. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 179-185). / Abstracts also in Chinese. / Introduction --- p.1 / Chapter 1 --- Global existence of weak solutions to the nematic liquid crystals in T² --- p.14 / Chapter 1.1 --- Introduction and main results --- p.14 / Chapter 1.2 --- Preliminaries --- p.21 / Chapter 1.3 --- Faedo-Galerkin scheme --- p.24 / Chapter 1.4 --- The limit n → ∞ --- p.27 / Chapter 1.5 --- The limit M → ∞ --- p.33 / Chapter 1.6 --- The limit N → ∞ and the local existence --- p.47 / Chapter 1.7 --- The global existence --- p.51 / Chapter 2 --- Local well-posedness and blow-up criteria of strong solutions to the Ericksen-Leslie system in R³ --- p.60 / Chapter 2.1 --- Introduction and main results --- p.60 / Chapter 2.2 --- Local existence --- p.67 / Chapter 2.3 --- Blow-up criteria --- p.83 / Chapter 2.4 --- Convergence of Ginzburg-Landau to Ericksen-Leslie --- p.99 / Chapter 3 --- Local well-posedness and blow-up criteria of strong solutions to the Ericksen-Leslie system in bounded domains of R³ --- p.112 / Chapter 3.1 --- Introduction and main results --- p.112 / Chapter 3.2 --- Local well-posedness --- p.119 / Chapter 3.3 --- Blow-up criteria --- p.141 / Chapter 4 --- Global existence of weak solutions to the Ericksen-Leslie system in bounded domains of R² --- p.156 / Chapter 4.1 --- Introduction and main results --- p.156 / Chapter 4.2 --- Strong solutions and blow-up criteria --- p.161 / Chapter 4.3 --- Global weak solutions --- p.172 / Chapter 5 --- Discuss on future works --- p.177 / Bibliography --- p.178

Flexoelectricity in liquid crystals is thought to be due to a coupling between dielectric properties and shape anisotropy of the molecules and described by the fiexoelectric coefficients el and e3. Two experiments are needed to measure el and e3 and it is usual to measure the difference (el - e3) and the sum (el + e3) and then calculate el and e3· The first experiment to measure the difference (el - e3) uses a TN structure with an in-plane applied electric field. Due to the dielectric coupling, the director aligns with the electric field and due to the fiexoelectric effect, the director tilts out of plane. This tilt is measured optically using two laser beams at oblique incidence, e.g. 45°. Using a theoretical model the experimental data is fitted and the difference (el - e3) extracted. The second experiment to measure the sum (el + e3) uses a Pi cell. Applying an ac voltage the transmission through the device is a repeating oscillating signal which contains 1st and 2nd harmonics. The 1st harmonic corresponds to the fiexoelectric effect and the 2nd harmonic to the dielectric effect. Using a lock-in amplifier, the harmonics were measured and the sum (el + e3) extracted using a theoretical model to fit the experimental data. Unfortunately, the data proved the experiment to be unreliable and another method was developed, which uses a BAN cell. The third experiment uses simple pulses in a BAN cell and also measures the sum (el + e3). The big disadvantage of the BAN cell is an internal voltage, which is created by the homeotropic alignment layer and the fiexoelectric polarisation. The internal voltage has the same effect on the director profile as the fiexoelectric effect, which is a big problem in measuring fiexoelectricity. Using a material, which is non ionic and has no fiexoelectricity, the internal bias could be measured and taken into account. Applying short de pulses of opposite sign, the fiexoelectric effect can be observed by the optical response and can be measured. Using these experiments, a number of investigation are being carried out such as the correlation between fiexoelectricity and the molecular structure, ions, elastic properties, molecular orientation, dielectric anisotropy 6E, and order parameter S. The results showed that fiexoelectricity only depends on ions and dielectric properties which was very interesting and surprising at the same time.

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This manuscript demonstrates the well-posedness (existence, uniqueness, and regularity of solutions) of the Cauchy problem for simplified equations of nematic liquid crystal hydrodynamic flow in three dimensions for initial data that is uniformly locally L3(R3) integrable (L3U(R3)). The equations examined are a simplified version of the equations derived by Ericksen and Leslie. Background on the continuum theory of nematic liquid crystals and their flow is provided as are explanations of the related mathematical literature for nematic liquid crystals and the Navier–Stokes equations.