We mathematically model Smectic-A (SmA) phases with a modified Landau-de Gennes (mLdG) model. The orientational order of the SmA phase is described by a tensor-order parameter $\mathbf{Q}$, and the positional order is described by a real scalar $u$, which models the deviation from the average density of liquid crystal molecules. Firstly, we prove the existence and regularity of global minimisers of the mLdG free energy in three-dimensional settings. Then, we analytically prove that the mLdG model can capture the Isotropic-Nematic-Smectic phase transition as a function of temperature, under some assumptions. Further, we explore stable smectic phases on a square domain, with edge length $λ$, and tangent boundary conditions. We use heuristic arguments to show that defects repel smectic layers and strong nematic ordering promotes layer formation. We use asymptotic arguments in the $λ\to 0$ and $λ\to\infty$ limits which reveal the correlation between the number and thickness of smectic layers, the amplitude of density fluctuations with the phenomenological parameters in the mLdG energy. For finite values of $λ$, we numerically recover BD-like and D-like stable smectic states observed in experiments. We also study the frustrated mLdG energy landscape and give numerical examples of transition pathways between distinct mLdG energy minimisers.
We investigate stability properties of the radially symmetric solution corresponding to the vortex defect (so called "melting hedgehog") in the framework of the Landau - de Gennes model of nematic liquid crystals. We prove local stability of the melting hedgehog under arbitrary $Q$-tensor valued perturbations in the temperature regime near the critical supercooling temperature. As a consequence of our method, we also rediscover the loss of stability of the vortex defect in the deep nematic regime.
We explain theoretically peculiarities of the smectic A -- hexatic B equilibrium phase coexistence in a finite temperature range recently observed experimentally in free standing smectic films [I.A.Zaluzhnyy et al., Physical Review E, volume 98, 052703 (2018)]. We quantitatively describe this unexpected phenomenon within Landau phase transitions theory assuming that the film state is close to a tricritical point. We found that the surface hexatic order leads to diminishing the phase coexistence range as the film thickness decreases shrinking it at some minimal film thickness, of the order of t…
Ginzburg-Landau-de Gennes -type covariant theories are extensively used in connection with twist grain boundary (TGB) phases of chiral smectogens. We analyze the stability conditions for the linear, covariant elasticity theory of smectic-C liquid crystals in the context of achiral materials, and predict an equilibrium modulated structure with an oblique wavevector. We suggest that a previous experimental observation of stripes in smectic-C is consistent with the predicted structure.
Frustration is a powerful mechanism in condensed matter systems, driving both order and co plexity. In smectics, the frustration between macroscopic chirality and equally spaced layers generates textures characterised by a proliferation of defects. In this article, we study several different ground states of the chiral Landau-de Gennes free energy for a smectic liquid crystal. The standard theory finds the twist grain boundary (TGB) phase to be the ground state for chiral type II smectics. However, for very highly chiral systems, the hierarchical helical nanofilament (HN) phase can form and is…
We study nematic liquid crystal configurations in confined geometries within the continuum Landau--De Gennes theory. These nematic configurations are mathematically described by symmetric, traceless two-tensor fields, known as $\Qvec$-tensor order parameter fields. We obtain explicit upper bounds for the order parameters of equilibrium liquid crystal configurations in terms of the temperature, material constants, boundary conditions and the domain geometry. These bounds are compared with the bounds predicted by the statistical mechanics definition of the $\Qvec$-tensor order parameter. They gi…
We investigate the solution landscape of a reduced Landau--de Gennes model for nematic liquid crystals on a two-dimensional hexagon at a fixed temperature, as a function of $λ$---the edge length. This is a generic example for reduced approaches on regular polygons. We apply the high-index optimization-based shrinking dimer method to systematically construct the solution landscape consisting of multiple defect solutions and relationships between them. We report a new stable T state with index-$0$ that has an interior $-1/2$ defect; new classes of high-index saddle points with multiple interior …
The mistakes of the classical solution of screw dislocations in smectic-A liquid crystals are pointed out. This reveals a serious problem of the well-known theory, which may be named de Gennes-Kleman-Pershan paradox, for many decades in the scientific community of liquid crystal study. The correct solution is given in this letter in terms of simplest, elementary and straightforward solution method. Furthermore, the stress filed and energy of dislocation are discussed in detail. The present article gives a corrected stress field and dislocation energy as well.
An extension to a high-order model for the direct isotropic/smectic-A liquid crystalline phase transition was derived to take into account thermal effects including anisotropic thermal diffusion and latent heat of phase-ordering. Multi-scale multi-transport simulations of the non-isothermal model were compared to isothermal simulation, showing that the presented model extension corrects the standard Landau-de Gennes prediction from constant growth to diffusion-limited growth, under shallow quench/undercooling conditions. Non-isothermal simulations, where meta-stable nematic pre-ordering preced…
Composed of microscopic layers that stack along one direction while maintaining fluid-like positional disorder within layers, smectics are excellent systems for exploring topology, defects and geometric memory in complex confining geometries. However, the coexistence of crystalline-like characteristics in one direction and fluid-like disorder within layers makes lamellar liquid crystals notoriously difficult to model - especially in the presence of defects and large distortions. Nematic properties of smectics can be comprehensively described by the Q-tensor but to capture the features of the s…
Optical birefringence measurements on a rod-like liquid crystal (8OCB), imbibed in silica channels (7 nm diameter), are presented and compared to the thermotropic bulk behavior. The orientational and positional order of the confined liquid evolves continuously at the paranematic-to-nematic and sizeably broadened at the nematic-to-smectic order transition, resp., in contrast to the discontinuous and well-defined second-order character of the bulk transitions. A Landau-de-Gennes analysis reveals identical strengths of the nematic and smectic ordering fields (imposed by the walls) and indicates t…
In the Landau-de Gennes theory on nematic liquid crystals, the well-known Landau-de Gennes energy depends on four elastic constants; $L_1$, $L_2$, $L_3$, $L_4$. For the general case of $L_4\neq 0$, Ball-Majumdar \cite {BM} found an example that the Landau-de Gennes energy functional from physics literature \cite{MN} does not satisfy a coercivity condition, which causes a problem in mathematics to establish existence of energy minimizers. In order to solve this problem, we observe that the original third order term on $L_4$, proposed by Schiele and Trimper \cite{ST} in physics, is a linear comb…
In our previous work,, we studied asymptotic behavior of minimizers of the Landau-de Gennes energy functional on planar domains as the nematic correlation length converges to zero. Here we improve upon those results, in particular by sharpening the description of the limiting map of the minimizers. We also provide an expression for the energy valid for a small, but fixed value of the nematic correlation length.
We study transformations of self-organized defect arrays at the nematic-smectic A liquid crystal phase transition, and show that these defect configurations are correlated, or "remembered", across the phase transition. A thin film of thermotropic liquid crystal is subjected to hybrid anchoring by an air interface and a water substrate, and viewed under polarized optical microscopy. Upon heating from smectic-A to nematic, a packing of focal conic domains melts into a dense array of boojums---nematic surface defects---which then coarsens by pair-annihilation. With the aid of Landau-de Gennes num…
Defects in liquid crystals are of great practical importance and theoretical interest. Despite tremendous efforts, predicting the location and transition of defects under various topological constraint and external field remains to be a challenge. We investigate defect patterns of nematic liquid crystals confined in three-dimensional spherical droplet and two-dimensional disk under different boundary conditions, within the Landau-de Gennes model. We implement a spectral method that numerically solves the Landau-de Gennes model with high accuracy, which allows us to study the detailed static st…
Uniaxial nematic liquid crystals whose molecular orientation is subjected to a tangential anchoring on a curved surface offer a non trivial interplay between the geometry and the topology of the surface and the orientational degree of freedom. We consider a general thin film limit of a Landau-de Gennes Q-tensor model which retains the characteristics of the 3D model. From this, previously proposed surface models follow as special cases. We compare fundamental properties, such as alignment of the orientational degrees of freedom with principle curvature lines, order parameter symmetry and phase…
Recent experiments on mixed liquid crystals have highlighted the hugely significant role of ferromagnetic nanoparticle impurities in defining the nematic-smectic-A phase transition point. Structured around a Flory-Huggins free energy of isotropic mixing and Landau-de Gennes free energy, this article presents a phenomenological mean-field model that quantifies the role of such impurities in analyzing thermodynamic phases, in a mixture of thermotropic smectic liquid crystal and ferromagnetic nanoparticles. First we discuss the impact of ferromagnetic nanoparticles on the isotropic-ferronematic a…
Within the framework of the generalized Landau-de Gennes theory, we identify a $Q$-tensor-based energy that reduces to the four-constant Oseen-Frank energy when it is considered over orientable uniaxial nematic states. Although the commonly considered version of the Landau-de Gennes theory has an elastic contribution that is at most cubic in components of the $Q$-tensor and their derivatives, the alternative offered here is quartic in these variables. One clear advantage of our approach over the cubic theory is that the associated minimization problem is well-posed for a significantly wider ch…
General symmetry arguments, dating back to de Gennes dictate that at scales longer than the pitch, the low-energy elasticity of a chiral nematic liquid crystal (cholesteric) and of a Dzyaloshinskii-Morya (DM) spiral state in a helimagnet with negligible crystal symmetry fields (e.g., MnSi, FeGe) is identical to that of a smectic liquid crystal, thereby inheriting its rich phenomenology. Starting with a chiral Frank free-energy (exchange and DM interactions of a helimagnet) we present a transparent derivation of the fully nonlinear Goldstone mode elasticity, which involves an analog of the Ande…
We study uniaxial solutions of the Euler-Lagrange equations for a Landau-de Gennes free energy for nematic liquid crystals, with a fourth order bulk potential, with and without elastic anisotropy. In the elastic isotropic case, we show that (i) all uniaxial solutions of the Euler-Lagrange equations, with a director field of certain symmetry, necessarily have the radial-hedgehog structure modulo an orthogonal transformation, (ii) the "escape into third dimension" director cannot correspond to a purely uniaxial solution of the Landau-de Gennes Euler-Lagrange equations and we do not use artificia…
Landau-de Gennes Modelling of Confinement Effects and Cybotactic Clusters in Bent-Core Nematic Liquid Crystals
Center for Applied Mathematics, Renmin University of China（中国人民大学应用数学中心）；Department of Mathematics and Statistics, University of Strathclyde（斯特拉斯克莱德大学数学与统计系）；College of Science, Princess Nourah bint Abdulrahman University（公主诺拉·宾特·阿卜杜勒拉赫曼大学理学院）；University of Manchester（曼彻斯特大学）
We construct a phenomenological Landau-de Gennes theory for hard colloidal rods by performing an order parameter expansion of the chemical-potential dependent grand potential. By fitting the coefficients to known results of Onsager theory, we are not only able to describe the isotropic-nematic phase transition as function of density, including the well-known density jump, but also the isotropic-nematic planar interface. The resulting theory is applied in calculations of the isotropic core size in a radial hedgehog defect, the density dependence of linear defects of hard rods in square confinem…
We investigate the solution landscapes of a simplified Ericksen--Leslie (sEL) vector model for nematic liquid crystals, confined in a two-dimensional square domain with tangent boundary conditions. An efficient numerical algorithm is developed to construct the solution landscapes by utilizing the symmetry properties of the model and the domain. Since the sEL model and the reduced Landau--de Gennes (rLdG) models can be viewed as Ginzburg--Landau functionals, we systematically compute the solution landscapes of the sEL model, for different domain sizes, and compare with the solution landscapes o…
Recent experiments probing a new ferroelectric liquid crystal (CLF08) confined in cells with planar alignment have shown dielectric and optic anomalies suggesting the onset of ferrielectric ordering within the surface lattice of dechiralisation lines. We present a phenomenological theory describing the corresponding phase transitions sequence SmA->SmC*->Ferri. Phase diagrams, thermodynamic, dielectric and optic properties are worked out and compared with experiments. The anomalies are related to the predicted tristability of the experimental cells under applied electric field. The order parame…
We consider a theoretical model for the chiral smectic A twisted ribbons observed in assemblies of fd viruses condensed by depletion forces. The depletion interaction is modeled by an edge energy assumed to be proportional to the depletant polymer in solution. Our model is based on the Helfrich energy for surface bending and the de Gennes model of chiral smectic A liquid crystals with twist penetration at the edge. We consider two variants of this model, one with the conventional Helfrich Gaussian curvature term, and a second with saddle-splay energy. A mean field analysis of both models yield…
Experimental and theoretical studies of a smectic-hexatic transition in freely suspended films of 54COOBC compound are presented. X-ray investigations revealed a discontinuous first-order transition into the hexatic phase. Moreover, the temperature region of two phase coexistence near the phase transition point diminishes with film thickness. The coexistence width dependence on film thickness was derived on the basis of the Landau mean-field theory in the vicinity of the tricritical point (TCP). Close to TCP the surface hexatic ordering penetrates anomalously deep into the film interior.
Computing stable configurations of confined smectic liquid crystals with a deep variational framework
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Columbia University（哥伦比亚大学）；Center for Applied Mathematics, Renmin University of China（中国人民大学应用数学中心）；School of Mathematical Sciences, Beijing International Center for Mathematical Research, Center for Quantitative Biology, Center for Machine Learning Research, Peking University（北京大学数学科学学院、北京国际数学研究中心、定量生物学中心、机器学习研究中心）
Experimental and modeling/simulation studies of phase equilibrium and growth morphologies of novel polymer-dispersed liquid crystal (PDLC) mixtures of PS (polystyrene) and liquid crystals that exhibit a direct isotropic/smectic-A (lamellar) mesophase transition were performed for PS/10CB (decyl- cyanobiphenyl) and PS/12CB (dodecyl-cyanobiphenyl). Partial phase diagrams were determined using polarized optical microscopy (POM) and differential scanning calorimetry (DSC) for different compositions of both materials, determining both phase separation (liquid/liquid demixing) and phase ordering (is…
A mesh adaptivity scheme on the Landau-de Gennes functional minimization case in 3D, and its driving efficiency
Fakulteta za matematiko in fiziko, Univerza v Ljubljani（卢布尔雅那大学数学与物理学院）；Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions（索邦大学，巴黎第六大学，UMR 7598，雅克-路易·利翁斯实验室）；CNRS, UMR 7598, Laboratoire Jacques-Louis Lions（法国国家科学研究中心，UMR 7598，雅克-路易·利翁斯实验室）；INRIA-Paris-Rocquencourt（法国国家信息与自动化研究所巴黎罗昆库尔中心）；Inštitut Jožef Stefan（约瑟夫斯蒂芬研究所）
We have structurally characterized the liquid crystal phase that appears as an intermediate state when a dielectric nematic, having polar disorder of its molecular dipoles, transitions to the almost perfectly polar-ordered ferroelectric nematic. This intermediate phase, which fills a 100-year-old void in the taxonomy of smectics and which we term the "smectic $Z_A$", is antiferroelectric, with the nematic director and polarization oriented parallel to smectic layer planes, and the polarization alternating in sign from layer to layer. The period of this polarization wave ($\sim$180 A) is mesosc…
Thermotropic biaxial nematic phases seem to be rare, but biaxial smectic A phases less so. Here we use molecular field theory to study a simple two-parameter model, with one parameter promoting a biaxial phase and the second promoting smecticity. The theory combines the biaxial Maier-Saupe and McMillan models. We use alternatively the Sonnet-Virga-Durand (SVD) and geometric mean approximations (GMA) to characterize molecular biaxiality by a single parameter. For non-zero smecticity and biaxiality, the model always predicts a ground state biaxial smectic A phase. For a low degree of smectic ord…
We study nematic equilibria on three-dimensional square wells, with emphasis on Well Order Reconstruction Solutions (WORS) as a function of the well size, characterized by $λ$, and the well height denoted by $ε$. The WORS are distinctive equilibria reported in [10] for square domains, without taking the third dimension into account, which have two mutually perpendicular defect lines running along the square diagonals, intersecting at the square centre. We prove the existence of WORS on three-dimensional wells for arbitrary well heights, with (i) natural boundary conditions and (ii) realistic s…
Structures and orientational transitions in thin films of tilted hexatic smectics
我们提出了详细的系统的研究在薄液晶薄膜的近晶-C的六角相变的结构转变。对于第一次所有可能的结构在文献中报道的一种材料（5 O. 6）在温度和厚度的变化观察。在不寻常的调制结构中，条纹的平衡周期是磁畴尺寸的两倍。我们解释这些模式的现象学朗道型理论的框架工作，作为一个自然的几何挫折在一个系统中产生的平衡现象，具有自发的展曲变形。
We report the observation of a doubly-periodic surface defect-pattern in the liquid crystal 8CB, formed during the nematic--smectic A phase transition. The pattern results from the antagonistic alignment of the 8CB molecules, which is homeotropic at the surface and planar in the bulk of the sample cell. Within the continuum Landau-deGennes theory of smectic liquid crystals, we find that the long period (~10 μm) of the pattern is given by the balance between the surface anchoring and the elastic energy of curvature wall defects. The short period (~1 μm) we attribute to a saddle-splay distortion…
Using a generalized Landau theory involving orientational, layering, tilt, and biaxial order parameters we analyze the smectic-A* and smectic-C* (Sm-A* -- Sm-C*) transition, showing that a combination of small orientational order and large layering order leads to Sm-A* -- Sm-C* transitions that are either continuous and close to tricriticality or first order. The model predicts that in such systems the increase in birefringence upon entry to the Sm-C* phase will be especially rapid. It also predicts that the change in layer spacing at the Sm-A* -- Sm-C* transition will be proportional to the o…
Numerical modeling of nematic liquid crystals using the tensorial Landau-de Gennes (LdG) theory provides detailed insights into the structure and energetics of the enormous variety of possible topological defect configurations that may arise when the liquid crystal is in contact with colloidal inclusions or structured boundaries. However, these methods can be computationally expensive, making it challenging to predict (meta)stable configurations involving several colloidal particles, and they are often restricted to system sizes well below the experimental scale. Here we present an open-source…
We consider a thin film limit of a Landau-de Gennes Q-tensor model. In the limiting process we observe a continuous transition where the normal and tangential parts of the Q-tensor decouple and various intrinsic and extrinsic contributions emerge. Main properties of the thin film model, like uniaxiality and parameter phase space, are preserved in the limiting process. For the derived surface Landau-de Gennes model, we consider an L2-gradient flow. The resulting tensor-valued surface partial differential equation is numerically solved to demonstrate realizations of the tight coupling of elastic…
The biaxial smectic-A* (Sm-A_B*) phase, appearing in the phase sequence Sm-A*--Sm-A*_B--Sm-C*, is analyzed using Landau theory. It is found to possess a helical superstructure with a pitch that is significantly shorter than the pitch of the Sm-C* helical superstructure. The Sm-A_B*--Sm-C* transition can be either 1st or 2nd order, and correspondingly there will be either a jump or continuous variation in the pitch. The behaviors of the birefringence and electroclinic effect are analyzed and found to be similar to those of a Sm-C*_alpha phase. As such, it is possible that the Sm-A_B* phase coul…
Liquid crystal networks combine the orientational order of liquid crystals with the elastic properties of polymer networks, leading to a vast application potential in the field of responsive coatings, e.g., for haptic feedback, self-cleaning surfaces and static and dynamic pattern formation. Recent experimental work has further paved the way toward such applications by realizing the fast and reversible surface modulation of a liquid crystal network coating upon in-plane actuation with an AC electric field. Here, we construct a Landau-type theory for electrically-responsive liquid crystal netwo…
