Modulated nematic structures and chiral symmetry breaking in 2D

ABSTRACT

We have studied the properties of biaxial particles interacting via an anisotropic pair potential, involving second-rank quadrupolar and third-rank octupolar coupling terms, using Monte Carlo simulation. The particles occupy the sites of a 2D square lattice and the interactions are restricted to nearest neighbours. The system exhibits spontaneous chiral symmetry breaking from an isotropic phase to a chiral modulated nematic phase, composed of ambidextrous chiral domains. When twofold axes of quadrupolar and octupolar tensors coincide this modulated phase appears to be the ambidextrous cholesteric phase with pitch comparable to a few lattice spacings. The associated phase transition is first order.     

Pair correlation functions and a free energy functional for the nematic phase

In this paper we have presented the calculation of pair correlation functions in a nematic phase for a model of spherical particles with the long-range anisotropic interaction from the mean spherical approximation (MSA) and the Percus-Yevick (PY) integral equation theories. The results found from the MSA theory have been compared with those found analytically by Holovko and Sokolovska [J. Mol. Liq. 82, 161 (1999)]. A free energy functional which involves both the symmetry conserving and symmetry broken parts of the direct pair correlation function has been used to study the properties of the nematic phase. We have also examined the possibility of constructing a free energy functional with the direct pair correlation function which includes only the principal order parameter of the ordered phase and found that the resulting functional gives results that are in good agreement with the original functional. The isotropic-nematic transition has been located using the grand thermodynamic potential. The PY theory has been found to give a nematic phase with pair correlation function harmonic coefficients having all the desired features. In a nematic phase the harmonic coefficient of the total pair correlation function h(x1,x2) connected with the correlations of the director transverse fluctuations should develop a long-range tail. This feature has been found in both the MSA and PY theories.     

Theory of solvation in polar nematics

We develop a linear response theory of solvation of ionic and dipolar solutes in anisotropic, axially symmetric polar solvents. The theory is applied to solvation in polar nematic liquid crystals. The formal theory constructs the solvation response function from projections of the solvent dipolar susceptibility on rotational invariants. These projections are obtained from Monte Carlo simulations of a fluid of dipolar spherocylinders which can exist both in the isotropic and nematic phases. Based on the properties of the solvent susceptibility from simulations and the formal solution, we have obtained a formula for the solvation free energy which incorporates the experimentally available properties of nematics and the length of correlation between the dipoles in the liquid crystal. The theory provides a quantitative framework for analyzing the steady-state and time-resolved optical spectra and makes several experimentally testable predictions. The equilibrium free energy of solvation, anisotropic in the nematic phase, is given by a quadratic function of cosine of the angle between the solute dipole and the solvent nematic director. The sign of solvation anisotropy is determined by the sign of dielectric anisotropy of the solvent: solvation anisotropy is negative in solvents with positive dielectric anisotropy and vice versa. The solvation free energy is discontinuous at the point of isotropic-nematic phase transition. The amplitude of this discontinuity is strongly affected by the size of the solute becoming less pronounced for larger solutes. The discontinuity itself and the magnitude of the splitting of the solvation free energy in the nematic phase are mostly affected by microscopic dipolar correlations in the nematic solvent. Illustrative calculations are presented for the equilibrium Stokes shift and the Stokes shift time correlation function of coumarin-153 in 4-n-pentyl-4'-cyanobiphenyl and 4,4-n-heptyl-cyanopiphenyl solvents as a function of temperature in both the nematic and isotropic phases.     

Perturbation theory for nematic liquid crystals of axially symmetric molecules: Numerical results at high pressures including second- and fourth rank orientational order parameters

Abstract

A statistical mechanical perturbation theory is applied to study the thermo-dynamic properties of nematic liquid crystals at the nematic-isotropic (N-I) phase transition under pressure, retaining second- and fourth-rank long-range orientational order parameters. We report calculations for a hard ellipsoidal system superposed with an attractive interaction and subjected to different external pressures. The repulsive interaction is represented by a repulsion between hard ellipsoids characterized by a length-to-width ratio and the interaction arising from the dispersion interaction between two asymmetric molecules represents the attractive interaction. The inclusion of the fourth-rank order parameter in the effective one-body potential ψ(Ω) does not lead to a significant overall improvement. The influence of pressure on the stability, ordering and thermodynamic functions at the N-I transition is analysed. We find that our theoretical predictions are in accordance with experimental observations.     

Perturbation theory for nematic liquid crystals of axially symmetric molecules: Numerical results at high pressures including second- and fourth rank orientational order parameters

A statistical mechanical perturbation theory is applied to study the thermo-dynamic properties of nematic liquid crystals at the nematic-isotropic (N-I) phase transition under pressure, retaining second- and fourth-rank long-range orientational order parameters. We report calculations for a hard ellipsoidal system superposed with an attractive interaction and subjected to different external pressures. The repulsive interaction is represented by a repulsion between hard ellipsoids characterized by a length-to-width ratio and the interaction arising from the dispersion interaction between two asymmetric molecules represents the attractive interaction. The inclusion of the fourth-rank order parameter in the effective one-body potential ψ(Ω) does not lead to a significant overall improvement. The influence of pressure on the stability, ordering and thermodynamic functions at the N-I transition is analysed. We find that our theoretical predictions are in accordance with experimental observations.     

Computer simulation study of a nematogenic lattice model based on an elastic energy mapping of the pair potential

Director configurations in a nematic liquid crystal can be determined by minimizing its total elastic free energy, for given elastic constants and specific boundary conditions. In some cases, these configurations have been obtained by numerical procedures where the elastic free energy density plays the same role as the overall potential energy in a standard Metropolis Monte Carlo simulation. The interaction energies or potentials used in these studies are short ranged but, in general, not pairwise additive, unless the three elastic constants are set to a common value, thus reducing the potential to that in the well-known Lebwohl-Lasher lattice model. On the other hand, we can construct, in different ways, a lattice model with pairwise additive interactions, which approximately reproduces the elastic free energy density, where the parameters defining the pair potential are expressed as linear combinations of elastic constants. An anisotropic nematogenic pair interaction of this kind, originally proposed by Gruhn and Hess (T. Gruhn and S. Hess, Z. Naturforsch. A51, 1 (1996)), has recently been investigated by one of us, using a Monte Carlo simulation (S. Romano, Int. J. Mod. Phys. B 12, 2305 (1998)). Here we propose another approximate procedure for the mapping, and study the resulting pair potential model with the aid of Monte Carlo simulations. The behaviour of the nematic phases formed by the two models is compared together with the predictions of molecular field theory and the properties of the Lebwohl-Lasher model.     

Glass Formation of n-Butanol: Coarse-grained Molecular Dynamics Simulations Using Gay-Berne Potential Model

Using coarse-grained molecular dynamics simulations based on Gay-Berne potential model, we have simulated the cooling process of liquid n-butanol. A new set of GB parameters are obtained by fitting the results of density functional theory calculations. The simulations are carried out in the range of 290-50 K with temperature decrements of 10 K. The cooling char-acteristics are determined on the basis of the variations of the density, the potential energy and orientational order parameter with temperature, whose slopes all show discontinuity. Both the radial distribution function curves and the second-rank orientational correlationfunction curves exhibit splitting in the second peak. Using the discontinuous change of these thermodynamic and structure properties, we obtain the glass transition at an estimate of temperature T_g=120±10 K, which is in good agreement with experimental results 110±1 K.     

Statistical mechanical theory for discotic liquid crystals Discotic nematic-isotropic transition properties

A statistical mechanical theory is applied to study the equilibrium properties of discotic nematic liquid crystals. We report the calculation of thermodynamic properties for a model system composed of molecules interacting through angle-dependent pair potentials which can be broken up into rapidly varying short-ranged repulsions and weak long-range attractions. The repulsive interaction is represented by a repulsion between hard oblate ellipsoids of revolution and is a short-range, rapidly-varying, potential. The influence of attractive potentials, represented by dispersion and quadrupole interactions on a variety of thermodynamic properties is analysed. It is found that the thermodynamic properties for the discotic nematic-isotropic transition are highly sensitive to the form of effective one-body orientational perturbation potential. The discontinuity in the transition properties is more pronounced in the case of quadrupole interaction than for anisotropic dispersion interaction. A remarkable symmetry in the transition properties between prolate ellipsoids (ordinary nematic) and oblate ellipsoids (discotic nematic) is observed.     

On the parallel-perpendicular transition for a nematic phase at a wall

We use an Onsager-level density functional theory to investigate the behaviour of the nematic phase in contact with a solid wall. The nematic consists of hard rigid rods having perfect uniform alignment and uniform spatial density. In the absence of any particle-wall interactions besides excluded-volume forces, we predict a director orientation parallel to the wall. We show that this preference for parallel alignment is due to the entropy associated with the larger volume available to the particles in their parallel orientation. An adsorption energy favouring normal alignment gives rise to a transition from a high temperature parallel orientation to a low temperature normal orientation. We derive expressions for the temperature of this transition, relating it explicitly to the wall adsorption energy, particle axial ratio, and nematic density. Effects such as layering near the wall and imperfect nematic order are argued not to be necessary for the existence of this transition.     

Monte Carlo lattice simulations of the elastic behaviour of nematic liquid crystals

A pairwise additive potential, which approximately reproduces the free energy density for the elastic deformations of a nematic liquid crystal, originally proposed by Gruhn and Hess, has been investigated by simulating the three Freedericksz transitions as well as that of the Schadt-Helfrich cell. The pair potential depends on the three elastic constants K₁, K₂ and K₃ for the splay, twist and bend deformations, respectively. The results of the simulations are compared with the analytical solutions obtained from continuum theory in order to test the accuracy of the model potential at a quantitative level. This comparison is also made for different temperatures to explore the influence of director fluctuations on the elastic behaviour.     

Molecular dynamics investigation of a density-driven glass transition in a liquid crystal system

Molecular dynamics simulations are carried out to address the density-driven glass transition in a system of rodlike particles that interact with the Gay-Berne potential. Since crystallization occurs in this system on the time scale of the simulations, direct simulation of the glass transition is not possible. Instead, glasses with isotropic orientational order are heated to a temperature T, and the relaxation times by which nematic orientational order develops are determined. These relaxation times appear to diverge at a critical density rho(c); i.e., the system can equilibrate at rhorho(c) (at the temperature T). The relaxation times follow a power-law scaling as the critical density is approached, suggesting that this density-driven glass transition concurs with mode coupling theory.     

Density functional theory of freezing of a system of conjugated oligomers parameterised via Gay–Berne potential

Density functional theory (DFT) of freezing is used to study the isotropic–nematic, isotropic–smectic A and nematic–smectic A phase transitions in a system of large, semi-flexible conjugated oligomers parameterised within Gay–Berne (GB) potential. The pair correlation functions of the isotropic fluid, used as structural inputs in the DFT, are calculated by solving the Percus–Yevick integral equation theory. Large number of spherical harmonic coefficients of each orientation-dependent functions has been considered to ensure the numerical accuracy at different densities and temperatures for the system of these model GB ellipsoids having large aspect ratio (length-to-breadth ratio). We found that the system of GB ellipsoids parameterised for conjugated oligomers shows stable isotropic, nematic and smectic A phases. At low temperatures, on increasing the density, isotropic fluid makes a direct transition to smectic A phase. Nematic phase get stabilised in between the isotropic and smectic A phases on increasing the temperature. Using the transition parameter obtained through the DFT, we have plotted the temperature–density and pressure–temperature phase diagrams which are found to be qualitatively similar to the one obtained in simulations for the systems with low aspect ratio GB particles.     

A united-residue force field for off-lattice protein-structure simulations. I. Functional forms and parameters of long-range side-chain interaction potentials from protein crystal data

A two-stage procedure for the determination of a united-residue potential designed for protein simulations is outlined. In the first stage, the long-range and local-interaction energy terms of the total energy of a polypeptide chain are determined by analyzing protein-crystal data and averaging the all-atom energy surfaces. In the second stage (described in the accompanying article), the relative weights of the energy terms are optimized so as to locate the native structures of selected test proteins as the lowest energy structures. The goal of the work in the present study is to parameterize physically reasonable functional forms of the potentials of mean force for side-chain interactions. The potentials are of both radial and anisotropic type. Radial potentials include the Lennard-Jones and the shifted Lennard-Jones potential (with the shift parameter independent of orientation). To treat the angular dependence of side-chain interactions, three functional forms of the potential that were designed previously to describe anisotropic systems are evaluated: Berne-Pechukas (dilated Lennard-Jones); Gay-Berne (shifted Lennard-Jones with orientation-dependent shift parameters); and Gay-Berne-Vorobjev (the same as the preceding one, but with one more set of variable parameters). These functional forms were used to parameterize, within a short-distance range, the potentials of mean force for side-chain pair interactions that are related by the Boltzmann principle to the pair correlation functions determined from protein-crystal data. Parameter determination was formulated as a generalized nonlinear least-squares problem with the target function being the weighted sum of squares of the differences between calculated and “experimental” (i.e., estimated from protein-crystal data) angular, radial-angular, and radial pair correlation functions, as well as contact free energies. A set of 195 high-resolution nonhomologous structures from the Protein Data Bank was used to calculate the “experimental” values. The contact free energies were scaled by the slope of the correlation line between side-chain hydrophobicities, calculated from the contact free energies, and those determined by Fauchere and Pliška from the partition coefficients of amino acids between water and n-octanol. The methylene group served to define the reference contact free energy corresponding to that between the glycine methylene groups of backbone residues. Statistical analysis of the goodness of fit revealed that the Gay-Berne-Vorobjev anisotropic potential fits best to the experimental radial and angular correlation functions and contact free energies and therefore represents the free-energy surface of side-chain-side-chain interactions most accurately. Thus, its choice for simulations of protein structure is probably the most appropriate. However, the use of simpler functional forms is recommended, if the speed of computations is an issue. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18: 849–873, 1997     

Monte Carlo lattice simulations of the elastic behaviour of nematic liquid crystals

A pairwise additive potential, which approximately reproduces the free energy density for the elastic deformations of a nematic liquid crystal, originally proposed by Gruhn and Hess, has been investigated by simulating the three Freedericksz transitions as well as that of the Schadt-Helfrich cell. The pair potential depends on the three elastic constants K₁, K₂ and K₃ for the splay, twist and bend deformations, respectively. The results of the simulations are compared with the analytical solutions obtained from continuum theory in order to test the accuracy of the model potential at a quantitative level. This comparison is also made for different temperatures to explore the influence of director fluctuations on the elastic behaviour.     

Comparative study of temperature dependent orientational relaxation in a model thermotropic liquid crystal and in a model supercooled liquid

Recent optical Kerr effect experiments have revealed a power law decay of the measured signal with a temperature independent exponent at short-to-intermediate times for a number of liquid crystals in the isotropic phase near the isotropic-nematic transition and supercooled molecular liquids above the mode coupling theory critical temperature. In this work, the authors investigate the temperature dependence of short-to-intermediate time orientational relaxation in a model thermotropic liquid crystal across the isotropic-nematic transition and in a binary mixture across the supercooled liquid regime in molecular dynamics simulations. The measure of the experimentally observable optical Kerr effect signal is found to follow a power law decay at short-to-intermediate times for both systems in agreement with recent experiments. In addition, the temperature dependence of the power law exponent is found to be rather weak. As the model liquid crystalline system settles into the nematic phase upon cooling, the decay of the single-particle second-rank orientational time correlation function exhibits a pattern that is similar to what has been observed for supercooled liquids.     

Phase transitions in mixtures of a side-on-side chain liquid crystalline polymer and low molar mass nematic liquid crystals

Miscibility phase diagrams of mixtures of side-on side chain liquid crystalline polymers (s-SCLCP) and low molar mass liquid crystals (E48 and E44) have been established by means of polarized optical microscopy and light scattering. E48 and E44 are cyanobiphenyl-based eutectic nematic liquid crystal (LC) mixtures with nematic-isotropic transition temperatures of 93 and 105 C, respectively. The phase diagram of the s-SCLCP/E48 system reveals the coexistence of an isotropic nematic region and a single nematic phase in order of descending temperature. The single nematic phase suggests that the pair is miscible in the nematic region. On the other hand, the s-SCLCP/E44 mixture shows liquid liquid and nematic nematic coexistence phases, suggestive of the immiscibility character of the pair. These nematic phase diagrams of the s-SCLCP/E48 and s-SCLCP/E44 have been analysed in the context of the combined Flory-Huggins (FH) free energy for isotropic mixing and the Maier-Saupe (MS) free energy for nematic ordering of the mesogens. This combined FH/MS theory is capable of predicting the observed nematic phase diagrams consisting of liquid liquid, liquid nematic, nematic nematic, and the pure nematic regions. The change of colour accompanying the appearance and disappearance of the inversion walls may be attributed to the temperature dependence of birefringence.     

Phase diagrams of binary mixtures of biaxial nematogens

A mean field theory is used to describe nematic phases of binary mixtures of biaxial molecules. Using a general pseudopotential consistent with the D_2h symmetry of the constituent particles, the theory is used to calculate the elements of the order tensors necessary to describe the orientational order in binary mixtures in both uniaxial and biaxial nematic phases. For a single component, the model only requires one parameter, r₂, a ratio of anisotropic interaction strengths, to predict the temperature dependence of the four order parameters. The temperature dependence of the orientational distribution functions is illustrated for both rod-like and plate-like molecules. For binary mixtures, three anisotropic interaction strengths, r₁, r₂, and r₃, are needed to calculate the order parameters of both components as a function of concentration and temperature. The free energy is evaluated to predict the phase stability of the mixture. By systematically varying the anisotropic interaction strengths, temperature-concentration phase diagrams for a variety of molecular shapes are presented. The theoretical predictions suggest that binary mixtures of molecules with highly asymmetric shapes will display stable biaxial nematic phases.     

Coarse grained models for flexible liquid crystals: parameterization of the bond fluctuation model

We extend the bond fluctuation model, originally devised to investigate polymer systems, to contain anisotropic interactions suitable for the simulation of large flexible molecules such as liquid crystalline polymers and dendrimers. This extended model coarse grains the interaction between the flexible chains at a similar level of detail to the mesogenic units. Suitable interaction parameters are obtained by performing trial simulations on a low molar mass liquid crystalline system. The phase diagram of this system is determined as a function of the molecular stiffness. The nematic to isotropic transition temperature is found to increase with increasing stiffness.     

N.M.R. study of segmental molecular interactions in liquid crystals

A simple statistical model of interacting non-rigid molecules, based on a perturbation expansion of the pair correlation function and the additivity of segmental interactions, is applied to the study of orientational order as measured by N.M.R. in the nematic and S_A phases of 4-n-alkyl-4'-cyanobiphenyls (N-CB, N = 5 to 8), the nematic and S_C phases of 4-n-alkyloxybenzoic acids (N-OBA, N = 7, 8) and the D_h0 columnar discotic phase of hexa-alkyloxytriphenylenes (N-THE, N = 5 to 8). The order parameters of each homologous series are correctly described in terms of two isotropic and two anisotropic segmental coupling constants. The model predicts certain relations among coupling constants pertaining to different homogous series. These predictions are supported by the results obtained for the three types of compounds studied.