Effect of Disclination Lines on Free Energy of Nematic Liquid Crystals

In the light of φ-mapping method and topological current theory, the effect of disclination lines on the free energy density of nematic liquid crystals is studied. It is pointed out that the total Frank free energy density can be divided into two parts. One is the distorted energy density of director field around the disclination lines. The other is the saddle-splay energy density, which is shown to be centralized at the disclination lines and to be topologically quantized in the unit of kπ /2 when the Jacobian determinant of the director field does not vanish at the singularities of the director field. The topological quantum numbers are determined by the Hopf indices and Brouwer degrees of the director field at the disclination lines, i.e., the disclination strengthes. When the Jacobian determinant vanishes, the generation, annihilation, intersection, splitting and merging processes of the saddle-splay energy density are detailed in the neighborhoods of the limit points and bifurcation points, respectively. It is shown that the disclination line with high topological quantum number is unstable and will evolve to the low topological quantum number states through the splitting process.     

Contribution of Disclination Lines to Free Energy of Liquid Crystals in Single-Elastic Constant Approximation

In the light of φ-mapping method and topological current theory, the contribution of disclination lines to free energy density of liquid crystals is studied in the single-elastic constant approximation. It is pointed out that the total free energy density can be divided into two parts. One is the usual distorted energy density of director field around the disclination lines. The other is the free energy density of disclination lines themselves, which is shown to be centralized at the disclination lines and to be topologically quantized in the unit of kπ /2. The topological quantum numbers are determined by the Hopf indices and Brouwer degrees of the director field at the disclination lines, i.e. the disclination strengths. From the Lagrange‘s method of multipliers, the equilibrium equation and the molecular field of liquid crystals are also obtained. The physical meaning of the Lagrangian multiplier is just the distorted energy density.     

Contribution of Disclination Lines to Free Energy of Liquid Crystals in Single-Elastic Constant Approximation

In the light of C-mapping method and topological current theory, the contribution of disclination lines to free energy density of liquid crystals is studied in the single-elastic constant approximation. It is pointed out that the total free energy density can be divided into two parts. One is the usual distorted energy density of director field around the disclination lines. The other is the free energy density of disclination lines themselves, which is shown to be centralized at the disclination lines and to be topologically quantized in the unit of kn/2. The topological quantum numbers are determined by the Hopf indices and Brouwer degrees of the director l~eld at the disclination lines, i.e. the disclination strengths. From the Lagrange‘s method of multipliers, the equilibrium equation and the molecular field ofliquid crystals are also obtained. The physical meaning of the Lagrangian multiplier is just the distorted energy density.     

Effect of Saddle-Splay Elasticity on Stability of Disclination Rings in Nematic Liquid Crystals

In this paper, the stability of disclination ring in nematie liquid crystals is studied. In the presence of saddle-splay elasticity （characterized by k24） the disclination ring has a universal equilibrium radius. Depending on the values of the saddle-splay constant k24, the universal equilibrium radius is altered. When k24 〉 0.92k （m = 1/2） and k24 〉 0.88k （m =-1/2）, the disclination will be a point rather than a ring, where k is the Frank elastic constant in the one-constant approximation.     

Effect of Saddle-Splay Elasticity on Stability of Disclination Rings in Nematic Liquid Crystals

In this paper, the stability of disclination ring in nematic liquid crystals is studied. In the presence of saddle-splay elasticity (characterized by k₂₄) the disclination ring has a universal equilibrium radius. Depending on the values of the saddle-splay constant k₂₄, the universal equilibrium radius is altered. When k₂₄>0.92k (m=1/2) and k₂₄>0.88k (m=-1/2), the disclination will be a point rather than a ring, where $k$ is the Frank elastic constant in the one-constant approximation.     

Topological Aspect and Bifurcation of Disclination Lines in Two—Dimensional Liquid Crystals

Using φ-mapping method and topological current theory,the topological structure and bifurcation of disclination lines in two-dimensional liquid crystals are studied.By introducing the strength density and the topological current of many disclination lines,the total disclination strength is topologically quantized by the Hopf indices and Brouwer degrees at the singularities of the director field when the Jacobian determinant of director field does not vanish.When the Jacobian determinant vanishes,the origin,annihilation and bifurcation processes of disclination lines are studied in the neighborhoods of the limit points and bifurcation points,respectively.The branch solutions at the limit point and the different directions of all branch curves at the bifurcation point are calculated with the conservation law of the topological quantum numbers.It is pointed out that a disclination line with a higher strength is unstable and it will evolve to the lower strength state through the bifurcation process.     

Stability of Disclinations in Nematic Liquid Crystals

In the light of φ-mapping method and topological current theory, the stability of disclinations around a spherical particle in nematic liquid crystals is studied. We consider two different defect structures around a spherical particle: disclination ring and point defect at the north or south pole of the particle. We calculate the free energy of these different defects in the elastic theory. It is pointed out that the total Frank free energy density can be divided into two parts. One is the distorted energy density of director field around the disclinations. The other is the free energy density of disclinations themselves, which is shown to be concentrated at the defect and to be topologically quantized in the unit of (k-k₂₄)π/2. It is shown that in the presence of saddle-splay elasticity a dipole (radial and hyperbolic hedgehog) configuration that accompanies a particle with strong homeotropic anchoring takes the structure of a small disclination ring, not a point defect.     

Topological Aspects of Liquid Crystals

Using -mapping method and topological current theory, the properties and behaviors of disclination points in three-dimensional liquid crystals are studied. By introducing the strength density and the topological current of many disclination points, the total disclination strength is topologically quantized by the Hopf indices and Brouwer degrees at the singularities of the general director field when the Jacobian determinant of the general director field does not vanish. When the Jacobian determinant vanishes, the origin, annihilation, and bifurcation of disclination points are detailed in the neighborhoods of the limit point and bifurcation point, respectively. The branch solutions at the limit point and the different directions of all branch curves at the first- and second-order degenerated points are calculated. It is pointed out that a disclination point with a higher strength is unstable and will evolve to the lower strength state through the bifurcation process. An original disclination point can split into at most four disclination points at one time.     

扭曲向列相液晶电光效应的微观机理研究

为了探究液晶电光效应的微观机理,以扭曲向列相液晶为研究对象,对液晶分子的变化情况进行了理论分析。运用连续体弹性理论分析了扭曲向列相液晶分子的指向矢取向变化,并对液晶电光效应曲线不同区间范围液晶微观结构的变化进行了系统分析。结果表明:液晶分子在零电场条件下的扭曲是均匀的;当外加电场大于某一值时,液晶分子开始转向,液晶产生电光效应。研究结果对液晶显示器件设计和深入理解液晶电光效应机理具有一定的指导意义。     

双轴向列相液晶的Monte Carlo模拟

利用Monte Carlo方法模拟棒状液晶分子随温度变化的相变行为,采用热力学统计方法得到能量与温度的关系,得出比热随温度的变化规律,通过图像的峰值确定相变点;重点模拟计算序参数矩阵元与温度的关系,得到不同分子结构参数下液晶系统的温度相图,模拟结果显示,当两个短棒长度相等时,不存在双轴相.     

Topological quantization of disclination points in three—dimensional liquid crystals

Using the φ-mapping method and topological current theory, we study the inner structure of disclination points in three-dimensional liquid crystals. By introducing the strength density and the topological current of many disclination points, it is pointed out that the disclination points are determined by the singularities of the general director field and they are topologically quantized by the Hopf indices and Brouwer degrees.     

Remarks on Dynamics of the Freedericksz Transition in Nematic Liquid Crystals

An improved discussion on the dynamics of the Freedericksz transition in nematics is proposed. It is pointed out that the relaxation time γ^-1 of the directors follows, an a²(a - 1) law in the twist deformation case. The possible chaotic motion of the directors in an external field ia briefly discussed.     

Some Interesting Observations on the Free Energy Principle

Biehl et al. (2021) present some interesting observations on an early formulation of the free energy principle. We use these observations to scaffold a discussion of the technical arguments that underwrite the free energy principle. This discussion focuses on solenoidal coupling between various (subsets of) states in sparsely coupled systems that possess a Markov blanket—and the distinction between exact and approximate Bayesian inference, implied by the ensuing Bayesian mechanics.     

Multiscale Free Energy Analysis of Human Ecosystem Engineering

Unlike ecosystem engineering by other living things, which brings a relatively limited range of sensations that are connected to a few enduring survival preferences, human ecosystem engineering brings an increasing variety and frequency of novel sensations. Many of these novel sensations can quickly become preferences as they indicate that human life will be less strenuous and more stimulating. Furthermore, they can soon become addictive. By contrast, unwanted surprise from these novel sensations may become apparent decades later. This recognition can come after the survival of millions of humans and other species has been undermined. In this paper, it is explained that, while multiscale free energy provides a useful hypothesis for framing human ecosystem engineering, disconnects between preferences and survival from human ecosystem engineering limit the application of current assumptions that underlie continuous state-space and discrete state-space modelling of active inference.     

Orientation in Nematic Liquid Crystals Doped with Orange Dyes and Effect of Carbon Nanoparticles

研究了掺杂两种分散染料橙的向列型液晶E7的性质以及碳纳米粒子(单壁碳纳米管或富勒烯C₆₀)的影响. 两种分散染料橙11和13具有较高的溶解度和有序参数,被作为掺杂剂同时使用. 与掺杂单染料相比,同时加入两种染料橙使液晶的有序参数明显提高. 与纯液晶相比,掺杂可引起向列相向各向同性相转变温度的升高.     

Influence of Flexoelectric Effect on Director Alignment of Nematic Liquid Crystals in Axial Arrangement Cylindrical Cells

A positive nematic liquid crystal(5CB) sample is confined in cylindrical cells under strong or weak axial anchoring boundary conditions when a radial nonuniform low-frequency electric field is applied and the flexoelectric effect is taken into account.Based on the Frank elastic free energy,the surface energy of the Rapini–Papoular approximation,the polarization free energy and the flexoelectric free energy caused by electric field,we obtain the free energy density of the nematic and solve the corresponding Euler–Lagrange equation numerically.We investigate the director distribution,the critical voltage and the critical exponent of nematic liquid crystal in cylindrical cells.It follows that the critical exponent is the classical one.It is also shown that the critical voltage in the system is affected by the flexoelectric effect,the geometric effect and radial weak anchoring effect on the cylindrical surfaces.A new type of director transition caused by the flexoelectric effect,the dielectric coupling effect and the radial weak anchoring effect is found.     

Influence of Surface Geometry of Grating Substrate on Director in Nematic Liquid Crystal Cell

The director in nematic liquid crystal cell with a weak anchoring grating substrate and a strong anchoring planar substrate is relative to the coordinates x and z. The influence of the surface geometry of the grating substrate in the cell on the director profile is numerically simulated using the two-dimensional finite-difference iterative method under the condition of one elastic constant approximation and zero driven voltage. The deepness of groove and the cell gap affect the distribution of director. For the relatively shallow groove and the relatively thick cell gap, the director is only dependent on the coordinate z. For the relatively deep groove and the relatively thin cell gap, the director must be dependent on the two coordinates x and z because of the increased elastic strain energy induced by the grating surface.     

C60掺杂向列相液晶薄膜的激光诱导自衍射现象研究

利用Kirchhoff-Fraunhofer衍射积分原理研究了实验中观察到的C60掺杂垂直排列向列相液晶(5CB)液晶薄膜的远场衍射图样.模拟计算结果表明:实验中观察到的衍射圆环是不同实验参数的结果,并给出了确定衍射圆环结构的主要参数C1和C2.当C1和C2的符号相同时,远场会出现中心强度最大,向外逐渐减弱,分布尺度较小的细衍射环;当C1和C2的符号相反时,远场会出现中央较暗,向外逐渐增强,分布尺度较大的粗衍射环.衍射圆环的产生可由高斯光束通过非线性样品是引起附加相位的自相位调制现象来解释.将使液晶材料应用于光限幅等光子学领域.     

On Epistemics in Expected Free Energy for Linear Gaussian State Space Models

Active Inference (AIF) is a framework that can be used both to describe information processing in naturally intelligent systems, such as the human brain, and to design synthetic intelligent systems (agents). In this paper we show that Expected Free Energy (EFE) minimisation, a core feature of the framework, does not lead to purposeful explorative behaviour in linear Gaussian dynamical systems. We provide a simple proof that, due to the specific construction used for the EFE, the terms responsible for the exploratory (epistemic) drive become constant in the case of linear Gaussian systems. This renders AIF equivalent to KL control. From a theoretical point of view this is an interesting result since it is generally assumed that EFE minimisation will always introduce an exploratory drive in AIF agents. While the full EFE objective does not lead to exploration in linear Gaussian dynamical systems, the principles of its construction can still be used to design objectives that include an epistemic drive. We provide an in-depth analysis of the mechanics behind the epistemic drive of AIF agents and show how to design objectives for linear Gaussian dynamical systems that do include an epistemic drive. Concretely, we show that focusing solely on epistemics and dispensing with goal-directed terms leads to a form of maximum entropy exploration that is heavily dependent on the type of control signals driving the system. Additive controls do not permit such exploration. From a practical point of view this is an important result since linear Gaussian dynamical systems with additive controls are an extensively used model class, encompassing for instance Linear Quadratic Gaussian controllers. On the other hand, linear Gaussian dynamical systems driven by multiplicative controls such as switching transition matrices do permit an exploratory drive.