Elastic constants of nematic solutions of rod-like and semi-flexible polymers

Abstract

Analytical expressions for the elastic constants of nematic polymer solutions are derived when the macromolecules are monodisperse rod-like, bidisperse rodlike or semi-flexible. These expressions are derived in the gaussian approximation so that they constitute exact leading terms of a general asymptotic expansion for high degree of nematic ordering although the results are justifiable only within the range of validity of the second virial approximation. The limiting forms are in complete agreement with recent numerical work for monodisperse rods.     

Chiral nematic liquid crystals in cylindrical cavities. A classification of planar structures and models of non-singular disclination lines

A part of homotopy theory is applied to classify planar structures in chiral nematic liquid crystals confined to cylindrical cavities. The resulting classification is exact in the approximation of undeformed chiral nematic surfaces. Within this approach the relative stability of possible planar structures with surface and bulk disclination lines is discussed. The number and the shape of these disclinations, which in some cases form spiral structures, are predicted. Further approximate analytical expressions for non-singular director fields close to disclination lines with integral strength are introduced. Our predictions, which are also in agreement with some previously suggested pictures of such director fields, are used to improve stability considerations of the confined planar chiral nematic structures in tubes and droplets.     

Phase relationships in a ternary solution composed of a solvent and two monodisperse homologous polymers

Exact expressions are derived for the spinodal, critical conditions, and separation factor of a ternary solution consisting of a pure solvent and two monodisperse homologous polymers in which the Flory-Huggins interaction parameter χ depends separately on the concentrations of the polymer components. The results allow one to see the difference from previous expressions obtained with χ depending on the total concentration of the polymer, and are expected to be useful for experimental determination of the Gibbs free energy of ternary solutions.     

Closed-form expressions of quantum electron transfer rate based on the stationary-phase approximation

Closed-form rate expressions are derived on the basis of the stationary-phase approximation for the Fermi golden rule expression of the quantum electron-transfer (ET) rate. First, on the basis of approximate solutions of the stationary-phase points near DeltaG = 0, -lambda, and lambda, where DeltaG is the reaction free energy and lambda is the reorganization energy, three closed-form rate expressions are derived, which are respectively valid near each value of DeltaG. Numerical tests for a model Ohmic spectral density with an exponential cutoff demonstrate good performance of the derived expressions in the respective regions of their validity. In particular, the expression near DeltaG = -lambda, which differs from the semiclassical approximation only by a prefactor quadratic in DeltaG, works substantially better than the latter. Then, a unified formula is suggested, which interpolates the three approximate expressions and serves as a good approximation in all three regions. We have also demonstrated that the interpolation formula can serve as a good quantitative means for understanding the temperature dependence of the quantum ET rate.     

On the Viscosity of Dilute Emulsions

The slow motion of a liquid droplet in a shear flow in the presence of surfactants is studied. The effects of the interfacial viscosity, Gibbs elasticity, surface diffusion and bulk diffusion of surfactants in both phases are taken into account. The analytical solution of the problem for small Reynolds and Peclet numbers gives a simple criterion for estimation of the tangential mobility of the droplet interface. By applying the standard procedure for averaging of the stress tensor flux at an arbitrary surface of the dilute emulsion, an analytical formula for the viscosity of emulsions in the presence of surfactants is derived. The result is a natural generalization of the well-known formula of Einstein for the viscosity of monodisperse dilute suspensions and of the expressions derived by Taylor and Oldroyd for the viscosity of monodisperse dilute emulsions taking into account the Marangoni effect. Copyright 2001 Academic Press.     

Light scattering from acrylate-based polymer dispersed liquid crystals: theoretical considerations and experimental examples

A light scattering (LS) study made using acrylate-based polymer dispersed liquid crystals (PDLCs) is presented. The polarized component IVV is measured for blends of a polyacrylate and the liquid crystal (LC) E7 at several compositions. Only the off-state configuration of the droplets with no external fields is considered here. These composites consist commonly of micron-sized nematic LC droplets dispersed in a solid polymer matrix. Theoretical expressions for the scattered intensities in the case of isotropic and anisotropic spherical droplets are given both in the Rayleigh-Gans approximation (RGA) and in the anomalous diffraction approximation (ADA). Series of VV and VH components of the scattering intensities are calculated using the models of Meeten, Stein and coworkers. The model calculations are compared with the light scattering data. This comparison enables us to extract information on the size and the shape of droplets assuming that the size distribution is uniform and that the scattering is due to single droplets, neglecting inter-particle correlation and multiple scattering effects. This paper demonstrates that the LS technique is a useful tool for studying the morphology of PDLC samples and estimating the average size of nematic droplets.     

Micro-actuators: when artificial muscles made of nematic liquid crystal elastomers meet soft lithography

The production, via soft lithography, of micron-sized thermoresponsive pillars made of side-on nematic LCE leads to the formation of surface-responsive structures. The individual, monodisperse pillars, when cut out from the surface, behave as micro-actuators, showing contraction of around 35% at the nematic to isotropic phase transition.     

Effect of persistent flexibility on the isotropic, nematic and columnar ordering in a self-assembling system

Concentrated solutions of reversibly assembling amphiphilic molecules often exhibit a variety of liquid-crystalline mesophases due to the asymmetry of their aggregates. For strongly elongated rod-like micelles flexibility may significantly influence the relative stability of these phases. This question is addressed for a simple model for a self-assembling system of monodisperse linear rod-like aggregates which are considered to be persistent flexible. Analogously to a recent calculation for a self-assembling system in which the rod-like aggregates are assumed to be perfectly rigid, persistent flexible aggregates may exhibit the same isotropic-nematic-columnar progression of phases. However, increasing the flexibility reduces the concentration range over which the nematic phase is stable. For sufficiently flexible aggregates the nematic phase is abandoned altogether and a direct isotropic-columnar transition occurs.     

Interaction of electromagnetic waves in planar waveguide with a nematic layer

This work is a theoretical study of energy exchange between two coupled TE-wave modes on director diffraction grating in a planar waveguide containing a layer of nematic liquid crystal. The diffraction grating is produced by an external electric field in the nematic layer with spatial periodic anchoring energy between director and waveguide surface. The intensity of a signal mode at the output of the nematic layer has been calculated in dependence of anchoring energy amplitude and modulation period, the size of nematic layer and electrical field value. The cases of co-propagating and oppositely propagating modes have been analysed. The analytical expressions that describe the maximum values of signal mode intensity have been derived. The maximum intensity value output from the nematic has been shown to depend monotonously on the anchoring energy parameters in the case of oppositely propagating wave modes and non-monotonously in the case of co-propagating wave modes. In both cases, the maximum value of signal mode intensity grows with the increase in electric field.     

On the existence of the thermomechanical terms of Akopyan and Zel’dovich in cholesteric liquid crystals

We revisit a theoretical paper of Akopyan and Zel’dovich about the thermomechanical coupling terms in nematic liquid crystals. We show that the expressions of these terms given by these authors must be corrected to satisfy the Onsager reciprocity relations, a point already stressed by Pleiner and Brand in 1987. We then extend this calculation to the cholesteric phase and show that there are no additional terms in the uniaxial approximation of this phase. Finally, we give the correspondence between the Akopyan and Zel’dovich terms and those calculated by Pleiner and Brand in 1996 by making a different choice for the forces and the fluxes in the theory.     

Thermodynamically consistent closure approximation for hard spheres systems

We present a new closure relation that is an extension of the Percus-Yevick approximation. In the proposed closure, we introduce an additional term and a mixing coefficient that can be determined by imposing a condition of thermodynamic self-consistency. Moreover, the mixing coefficient is calculated analytically within a linear approximation. In the case of a monodisperse system of hard spheres, we compare the results of our model to well-established thermodynamic expressions and also to the structural properties of fairly known closure approximations. In the second case, and using an equivalent scheme, the new closure relation is extended to the depletion potential between two large hard spheres immersed in a liquid of small hard spheres. In both cases, the results of our model are in good agreement with numerical simulations performed at intermediate concentrations.     

Critical point in a random side-chain nematic copolymer: Mean-field theory

A model for anisotropic phases in a random side-chain nematic copolymer is proposed and solved in the mean-field approximation. The main results of the calculation are the appearance of paranematic and nematic phases and a critical point, without the application of external electric or magnetic fields.     

Theoretical direct correlation function for two-dimensional fluids of monodisperse hard spheres

The direct correlation function plays an important role in describing the effects of the structure of particle systems with respect to light diffraction, x-ray diffraction as well as transmission and transmission fluctuations of radiation through a dense suspension. In this paper, the direct correlation function for a monolayer of monodisperse hard spheres or disks is derived theoretically. Based on the approximation of Baus and Colot [Phys. Rev. A 36, 3912 (1987)] and the equation of state for a fluid of hard disks by Santos et al. [J. Chem. Phys. 103, 4622 (1995)], we propose a new direct correlation function, which compares well to the approximate analytical expressions and gives a good prediction of the structure factor in a wide range of monolayer density or suspension concentration. The resulting radial distribution function also agrees well with Monte Carlo computer simulation data. The corresponding contact values of the radial distribution function compare well with the results of analytic approximations, numerical solutions, and computer simulations. Our proposed direct correlation function is applied to the transmission fluctuation spectrometric study. Experimental results show good agreement with the theory.     

Sedimentation equilibrium in nonideal heterogeneous systems. I. Fundamental equations for heterogeneous solute systems and some preliminary results

The sedimentation-equilibrium method is extended to treat nonideal solutions of heterogeneous macromolecules. The solute is assumed to be heterogeneous not only in molecular weight but also in other quantities such as partial specific volume, second virial coefficient and specific refractive increment. General expressions for various observable molecular weights, especially for weight-average, z-average, and number-average molecular weights, are derived. Their dependences on sedimentation parameter and solute concentration are discussed in detail. For the extrapolation of observable molecular weights, giving a type of weight-average, and z-average, to infinite dilution to estimate the molecular weight and the second virial coefficient, average concentration is superior as a concentration variable to original concentration. The plots of observable molecular weight versus average concentration are usually less influenced by the choice of the sedimentation parameter, especially of rotor speed. The general expressions are applied to a few special cases; monodisperse polymer, polydisperse homologous polymer, and polymer blend. The results are compared with experiments on a monodisperse, polystyrene, a polydisperse poly(methyl methacrylate), and a mixture of the two polymers, all in 2-butanone at 25°C. The agreement between the theory and experiments is satisfactory.     

Comments on "a study of the collisional fragmentation problem using the gamma distribution approximation"

M. Kostoglou and A.J. Karabelas [J. Colloid Interface Sci. 303 (2006) 419-429] proposed using a gamma distribution approximation to study a collisional fragmentation problem. This approximation involved two types of integrals and the use of continued fraction expansions for their computation. In this Comment, explicit expressions are derived for computing the integrals.     

From the double-stranded helix to the chiral nematic phase of B-DNA: a molecular model

B-DNA solutions of suitable concentration form left-handed chiral nematic phases (cholesterics). Such phases have also been observed in solutions of other stiff or semiflexible chiral polymers; magnitude and handedness of the cholesteric pitch are uniquely related to the molecular features. In this work we present a theoretical method and a numerical procedure which, starting from the structure of polyelectrolytes, lead to the prediction of the cholesteric pitch. Molecular expressions for the free energy of the system are obtained on the basis of steric and electrostatic interactions between polymers; the former are described in terms of excluded volume, while a mean field approximation is used for the latter. Calculations have been performed for 130 base pair fragments of B-DNA. The theoretical predictions provide an explanation for the experimental behavior, by showing the counteracting role played by shape and charge chirality of the molecule.     

Many-body interactions between particles in a polydisperse polymer fluid

We use a continuum chain model and develop an analytical theory for the interaction between many spheres immersed in a fluid of ideal polydisperse polymers. Assuming local spherical symmetry of the polymer field about each particle, combined with a local approximation, compact expressions are derived for the many-body interaction between the spheres. We use a mean-field approximation to investigate the fluid-fluid phase diagram for the mixture.     

Critical point in a random side-chain nematic copolymer: Mean-field theory

Abstract

A model for anisotropic phases in a random side-chain nematic copolymer is proposed and solved in the mean-field approximation. The main results of the calculation are the appearance of paranematic and nematic phases and a critical point, without the application of external electric or magnetic fields.     

Solution of a molecular statistical model for chiral nematic liquid crystals

A molecular statistical model for the chiral nematic phase of liquid crystals is investigated. The model is treated in the molecular field approximation. The resulting set of coupled integral equations for the order parameters and the pitch determining equation are solved numerically. The model hamiltonian consists of a nematic and a twist producing term. If only the nematic term is present, the model is known to have a first order phase transition. The model containing only the twist producing term is shown to exhibit a second order phase transition. The order parameters and wave length are presented for three ratios of the coupling constants in the case that both interactions are present.     

The Effective Hamiltonian and Electron Correlation Energy

An effective,hermitian hamiltonian is derived in amodel space. Its perturbation expressions to third order approximation are given.The correlation energy is also given to the third order approximation.The effective hamiltonian deviates form the actual one by the presence of acorrelation operator.The cprrelation operatop is given in an explicit form.