On the theory of liquid-crystalline ordering of polymer chains with limited flexibility

We propose a universal approximate method which enables us to study the statistical mechanics of the liquid-crystalline ordering in the solutions of stiffchain macromolecules with limited flexibility of arbitrary nature at arbitrary solution concentrations and arbitrary temperatures (solvent quality). In the variables temperature and concentration we construct the full phase diagrams for the nematic transition for the cases of long rigid rods and semiflexible macromolecules with freely jointed, persistent, and rotational-isomeric flexibility mechanisms. The method proposed can be easily generalized for the study of thermotropic polymer liquid crystals, as well as more complex polymer chain models (e.g., copolymers containing stiff and flexible fragments in the chain).     

Dipole-induced, first-order phase transitions of nano-rod monolayers

The isotropic-nematic transition of nano-rod monolayers with fore-aft symmetry is second order, in stark contrast to the first-order phase transition explained by Onsager [L. Onsager, Ann. (N.Y.) Acad. Sci. 51 (1949) 627] for rods in three dimensions. Here we show that the coupling of a dipole potential to excluded volume is sufficient to re-instate a first-order phase transition of rods confined to two dimensions.     

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.     

Phase diagram and effective shape of semiflexible colloidal rods and biopolymers

We study suspensions of semiflexible colloidal rods and biopolymers using an Onsager-type second-virial functional for a segmented-chain model. For mixtures of thin and thick fd virus particles, we calculate full phase diagrams, finding quantitative agreement with experimental observations. We show that flexibility, which renders the rods effectively shorter and thicker depending on the state point, is crucial to understanding the topologies of the phase diagrams. We also calculate the stretching of wormlike micelles in a host fd virus suspension, finding agreement with experiments.     

On the statistics of a three-dimensional gas of long thin rods

A gas of long thin rods undergoes an order-disorder phase transition as a function of rod concentration. We have evaluated the critical concentration at which this first-order transition occurs using Onsager's hard-core interaction model. We obtain the nematic angular distribution function of rods in the ordered phase expanded in a series of Legendre polynomials.This work was partially supported by the National Science Foundation through Grant No. GP-10536.     

Rigid rod adsorption including long-range dispersion interactions

The Caillé-Ågren analysis of rigid rod adsorption is extended by applying the van Kampen theory of condensation to include long-range dispersion interactions between the adsorbed rods. We discuss in greater detail the characteristics of the phase transition predicted by Caillé and Ågren between states having an isotropic and anisotropic distribution of rods adsorbed parallel to the surface. The maximum density range over which the anisotropic adsorbed phase is stable is determined as a function of the length-to-breadth ratio x of the rigid rods and the strength of the anisotropic dispersion energy. Critical surface adhesion and anisotropic dispersion energies necessary for anisotropic adsorption are also obtained as a function of x. In agreement with Caillé and Ågren the isotropic-anisotropic transition for rigid rod adsorption with attractive forces present is found to be second order. We also discuss the spreading pressure-density or area per molecule isotherms obtained for adsorbed rods having various values of x and surface adhesion and dispersion energies. Whenever feasible we compare our results with the spreading pressure isotherms obtained for monolayers of lyotropic molecules on either aqueous or mercury subphases and obtain qualitative agreement. In particular, the critical density and pressure associated with the two dimensional adsorbed gas-liquid condensation for rigid rods with x = 10 gives reasonable agreement with the critical constants observed by Hawkins and Benedek and Kim and Cannell for the corresponding condensation of pentadecanoic acid monolayers on neutral and acidified aqueous subphases. This agreement suggests that considerable dimerization of the pentadecanoic acid molecules may occur on aqueous surfaces.     

Disorder induced transitions in layered coulomb gases and superconductors

A 3D layered system of charges with logarithmic interaction parallel to the layers and random dipoles is studied via a novel variational method and an energy rationale which reproduces the known phase diagram for a single layer. Increasing interlayer coupling leads to successive transitions in which charge rods correlated in N>1 neighboring layers are nucleated by weaker disorder. For layered superconductors in the limit of only magnetic interlayer coupling, the method predicts and locates a disorder induced defect-unbinding transition in the flux lattice. While N = 1 charges dominate there, N>1 disorder induced defect rods are predicted for multilayer superconductors.     

The phase transition in a one-dimensional lattice of axisymmetric bodies

The partition function is studied for an array of axisymmetric, hard bodies (capped cylinders, etc.) with each fixed at the base on a regular one-dimensional lattice. It is shown that if a phase transition occurs in a system ofn molecules, then it also occurs in a system of two molecules for the same value of the spacing parameter. With certain additional technical assumptions the converse is also true. Results are reported specifically for a system of thin, hard rods. Necessary and sufficient conditions are shown for a first-order transition to occur in the thermodynamic limit; there is only one transition and that happens when the spacing parameter is equal to the length of the rod. As expected, there is no phase transition when the rod is contracted to a point.     

Attractive colloidal rods in shear flow

The effect of shear flow on the isotropic-nematic phase transition of attractive colloidal rods is investigated by a combination of simulations and experiments. The isotropic phase aligns with the flow, while the nematic phase undergoes a collective rotational motion which frustrates the merging of the coexisting regions. The location of binodals, spinodals, and the tumbling-to-aligning transition line in the shear-rate versus concentration plane are investigated. The phase diagrams in the shear-concentration plane for the various strengths of attractions can be mapped onto a master curve by appropriate scaling.     

Coulomb-frustrated phase separation phase diagram in systems with short-range negative compressibility

Using numerical techniques and asymptotic expansions we obtain the phase diagram of a paradigmatic model of Coulomb-frustrated phase separation in systems with negative short-range compressibility. The transition from the homogeneous phase to the inhomogeneous phase is generically first order in isotropic three-dimensional systems except for a critical point. Close to the critical point, inhomogeneities are predicted to form a bcc lattice with subsequent transitions to a triangular lattice of rods and a layered structure. Inclusion of a strong anisotropy allows for second- and first-order transition lines joined by a tricritical point.     

Mesomorphic polymer solutions

Summary A mean-field theory for the nematic-isotropic phase transition in semi-rigid polymer solutions is proposed. Phase diagrams are calculated. New effects due to chain flexibility are found for the transition temperature and the nematic order parameter of the polymer. An abrupt increase of chain extension in the nematic phase is demonstrated. Paper presented at the ?Meeting on Lyotropics and Related Fields?, held in Rende, Cosenza, September 13–18, 1982.     

Phase transitions in binary mixtures of rodlike colloids and stiff polymers

A suspension of long rodlike colloids and long stiff polymers is modelled as a mixture of hard rods. The diameter of the colloid particle is finite, while the polymer is considered in the limit of zero diameter. Two types of first-order phase transition may occur in such mixtures: an isotropic-nematic phase transition if the density (or the pressure) is high enough, and a demixing transition involving two isotropic phases. The demixing transition has a critical point, and a triple point with one nematic and two isotropic phases may also exist. Phase diagrams are calculated. For the demixing isotropic-isotropic transition to be observed the ratio between the polymer length and the colloid length must exceed 0.36.     

External field induced tricritical phenomenon in the isotropicnematic phase transition of hard non-spherical particle systems

The effect of static external field is studied on the isotropic–nematic phase transition of a system of hard non-spherical particles (rods or platelets) with negative anisotropic polarizability (susceptabilities). On the basis of Onsager theory, the phase coexistence curve is calculated numerically without any approximation. It is found that a weakly ordered nematic phase (uniaxial planar) is in coexistence with a highly ordered biaxial nematic phase which ends at a tricritical point. In the limit of infinite field strength, the orientations of the particles are confined in a plane perpendicular to the field and continuous isotropic–nematic phase transition takes place.     

Untersuchungen am Kubischen Bariumtitanat Nach der Methode des elektrostriktiven Resonators

The temperature dependence of the constant of elasticity could be measured in the cubic phase on [100] rods in the absence of a d.c. field using the electrostriction resonator method. The temperature range reached from the transition point up to 300°C. A steep decrease in mechanical quality could be observed several degrees above the transition point.The influence of the dielectric non-linearity on the observed electrostriction resonance is discussed. The piezoelectrical resonances found additionally are explained as flexural vibrations excited by a tetragonal boundary layer.     

Demixed and ordered phases in hard-rod mixtures

We analyse demixing and ordering transitions in systems of hard cylindrical particles. The second virial approximation of Onsager and a bifurcation analysis, as introduced by Koda and Kimura, are used to evaluate the free energies, pressures, and density distribution functions in mixtures of equally long but differently wide cylinders. The spatial density distribution along the one relevant coordinate is of particular importance as it provides more detailed information concerning the nature of the phase transition than the bare bifurcation diagnosis. Detailed results are given for the nematic–nematic spinodal and the nematic–smectic transitions. Allowing for the absence of an isotropic phase, our results are in good qualitative agreement with those for freely orienting rods reported previously, and indicate a complex sequence of phase diagrams as the diameter dissimilarity of the two components is increased, with upper and lower critical points bounding nematic and smectic demixing regions. However, experimental results on colloidal rods show that nematic demixing occurs at a diameter ratio much smaller than ours or those for freely rotating fluids, indicating that Onsager-type theories may be insufficient to reproduce this phenomenon in a quantitative manner and, consequently, that more sophisticated approaches, presumably incorporating particle flexibility and additional interactions, are required.     

Transverse alignment of fibers in a periodically sheared suspension: an absorbing phase transition with a slowly varying control parameter

Shearing solutions of fibers or polymers tends to align fiber or polymers in the flow direction. Here, non-Brownian rods subjected to oscillatory shear align perpendicular to the flow while the system undergoes a nonequilibrium absorbing phase transition. The slow alignment of the fibers can drive the system through the critical point and thus promote the transition to an absorbing state. This picture is confirmed by a universal scaling relation that collapses the data with critical exponents that are consistent with conserved directed percolation.     

Phase transitions in a solution of rodlike macromolecules of two different sizes

Phase equilibria for a solution of long rigid rods of diameters D_i and lengths L_i (i = 1,2) are studied analytically using Gaussian trial functions. The conditions on the molecular size parameters for a nematic—nematic phase transition to be possible are determined, and found to be satisfied everywhere except in a closed domain in the diameter ratio—length ratio plane. Information on the relative ordering of the two components is also provided.     

二维声子晶体中简单旋转操作导致的拓扑相变

构建了一种简单的二维声子晶体:由两个横截面为三角形的钢柱所组成的复式元胞按三角点阵的形式排列在空气中,等效地形成了一个蜂巢点阵结构.当三角形钢柱的取向与三角点阵的高对称方向一致时,整个体系具有C_(6v)对称性.研究发现:在保持钢柱填充率不变的条件下,只需要将所有三角柱绕着自己的中心旋转180°,就可实现二重简并的p态和d态在布里渊区中心Γ点处的频率反转,且该能带反转过程实质上是一个拓扑相变过程.通过利用Γ点的P态和d态的空间旋转对称性,构造了一个赝时反演对称性,并在声学系统中实现了类似于电子系统中量子自旋霍尔效应的赝自旋态.随后通过k·p微扰法导出了Γ点附近的有效哈密顿量,并分别计算了拓扑平庸和非平庸系统的自旋陈数,揭示了能带反转和拓扑相变的内在联系.最后通过数值模拟演示了受到拓扑不变量保护的声波边界态的单向传输行为和对缺陷的背向散射抑制.文中所研究的声波体系,尽管材料普通常见,但其拓扑带隙的相对宽度超过21%,比已报道的类似体系的带隙都要宽,且工作原理涵盖从次声波到超声波的很大频率范围,从而在实际应用上具有较大的优势和潜力.     

On the stueckelberg formula for non-adiabatic transitions

The phase corrected Stueckelberg and Landau-Zener formulae for the non-adiabatic transition probability in a two state problem are compared. It is shown that validity criteria on the derivation of the former imply precise equivalence between the two formulations for the linear curve crossing problem; but the Stueckelberg approach is seen to have greater flexibility. A proposed uniformization of the Stueckelberg form for use at low velocities is shown to be incorrect.     

Mixtures of anisotropic and spherical colloids: Phase behavior, confinement, percolation phenomena and kinetics

Purely entropic systems such as suspensions of hard rods, platelets and spheres show rich phase behavior. Rods and platelets have successfully been used as models to predict the equilibrium properties of liquid crystals for several decades. Over the past years hard particle models have also been studied in the context of non-equilibrium statistical mechanics, in particular regarding the glass transition, jamming, sedimentation and crystallization. Recently suspensions of hard anisotropic particles also moved into the focus of materials scientists who work on conducting soft matter composites. An insulating polymer resin that is mixed with conductive filler particles becomes conductive when the filler percolates. In this context the mathematical topic of connectivity percolation finds an application in modern nano-technology. In this article, we briefly review recent work on the phase behavior, confinement effects, percolation transition and phase transition kinetics in hard particle models. In the first part, we discuss the effects that particle anisotropy and depletion have on the percolation transition. In the second part, we present results on the kinetics of the liquid-to-crystal transition in suspensions of spheres and of ellipsoids.