Maier-Saupe Model Research Articles

Phase behavior of a lattice-gas model for biaxial nematics

We employ a lattice-gas extension of the Maier-Saupe model with discrete orientation states to study the phase behavior of a statistical model for biaxial nematogenic units in mean-field theory. The phase behavior of the system is investigated in terms of the strength of isotropic interaction between anisotropic objects, as well as the degree of biaxiality and the concentration of those units. We obtain phase diagrams with isotropic phases and stable biaxial and uniaxial nematic structures, various phase coexistences, many types of critical and multicritical behaviors, such as ordinary vapor-liquid critical points, critical end points, and tricritical points, and distinct Landau-like multicritical points. Our results widen the possibilities of relating the phenomenological coefficients of the Landau-de Gennes expansion to microscopic parameters, allowing an improved interpretation of theoretical fittings to experimental data.

Computational molecular field theory for nematic liquid crystals.

Nematic liquid crystals exhibit configurations in which the underlying ordering changes markedly on macroscopic length scales. Such structures include topological defects in the nematic phase and tactoids within nematic-isotropic coexistence. We discuss a computational study of inhomogeneous configurations that is based on a field theory extension of the Maier-Saupe molecular model of a uniaxial, nematic liquid crystal. A tensor order parameter is defined as the second moment of an orientational probability distribution, leading to a free energy that is not convex within the isotropic-nematic coexistence region, and that goes to infinity if the eigenvalues of the order parameter become nonphysical. Computations of the spatial profile of the order parameter are presented for an isotropic-nematic interface in one dimension, a tactoid in two dimensions, and a nematic disclination in two dimensions. We compare our results to those given by the Landau-de Gennes free energy for the same configurations and discuss the advantages of such a model over the latter.

The formation of biaxial nematic phases in binary mixtures of thermotropic liquid-crystals composed of uniaxial molecules

ABSTRACTMonte Carlo simulations in the isothermal-isobaric ensemble are used to investigate the formation of an ordered, biaxial nematic phase in a binary mixture of thermotropic liquid crystals. The orientational dependence of the interaction between molecules of each pure component is the same as in the well-known Maier-Saupe model; each pure component of the mixture is therefore capable of forming a uniaxial nematic phase. For the interaction between molecules of different components, we use the same Maier-Saupe model but change the sign of the coupling constant. As a consequence a T-shaped arrangement of these molecules is energetically favoured. The formation of the biaxial phase occurs in two steps. At higher temperatures T, one of the components forms a uniaxial nematic phase whereas the other is in a quasi two-dimensional restricted isotropic liquid state. We develop a simple theoretical model to understand the high degree of (ostensible) nematic order in the latter. At lower T, the second component becomes nematic and then the entire mixture of the two compounds has biaxial symmetry. The biaxial nematic phase does not demix into domains rich in molecules of one or the other species.

Temperature-Dependence of Persistence Length Affects Phenomenological Descriptions of Aligning Interactions in Nematic Semiconducting Polymers

Electronic and optical properties of conjugated polymers are strongly affected by their solid-state microstructure. In nematic polymers, mesoscopic order and structure can be theoretically understood using Maier-Saupe (MS) models, motivating us to apply them to conjugated macromolecular systems and consider the problem of their material-specific parameterization. MS models represent polymers by worm-like chains (WLC) and can describe collective polymer alignment through anisotropic MS interactions. Their strength is controlled by a phenomenological temperature-dependent parameter, υ(T). We undertake the challenging task of estimating material-specific υ(T), combining experiments and Self-Consistent Field theory (SCFT). Considering three different materials and a spectrum of molecular weights, we cover the cases of rod-like, semi-flexible, and flexible conjugated polymers. The temperature of the isotropic-nematic transition, TIN, is identified via polarized optical microscopy and spectroscopy. The polymers...

Uniaxial and biaxial structures in the elastic Maier-Saupe model.

We perform statistical mechanics calculations to analyze the global phase diagram of a fully connected version of a Maier-Saupe-Zwanzig lattice model with the inclusion of couplings to an elastic strain field. We point out the presence of uniaxial and biaxial nematic structures, depending on temperature T and on the applied stress σ. Under uniaxial extensive tension, applied stress favors uniaxial orientation, and we obtain a first-order boundary along which there is a coexistence of two uniaxial paranematic phases, and which ends at a simple critical point. Under uniaxial compressive tension, stress favors biaxial orientation; for small values of the coupling parameters, the first-order boundary ends at a tricritical point, beyond which there is a continuous transition between a paranematic and a biaxially ordered structure. For some representative choices of the model parameters, we obtain a number of analytic results, including the location of critical and tricritical points and the line of stability of the biaxial phase.

Orders Exhibited by Ensemble of Headless Spins Preferring Twisted Alignment: Phase Diagram of Extended Maier–Saupe Model on Simple Cubic Lattice

The effect of the preference for nonparallel alignment without a preferred twist sense of neighboring headless spins is examined through computer simulations on extended Maier–Saupe models on the simple cubic lattice with/without next-nearest-neighbor interactions. Phase diagrams containing four phases are constructed for a varying degree of the preference for the local twist and for a strength of the next-nearest-neighbor interaction. It is shown that the preference for the local twist brings about not only the instability for the nematic (uniaxial) order but also that for a spatial order to reduce the frustration arising from the local twist. The next-nearest-neighbor interaction can produce even a weak spatial order of local chirality. The results are discussed in relation to the formation of a liquid crystalline gyroid phase and chiral phases in ensembles consisting of antispindle molecules that are axially symmetric and achiral.

Modulated phases in a three-dimensional Maier-Saupe model with competing interactions.

This work is dedicated to the study of the discrete version of the Maier-Saupe model in the presence of competing interactions. The competition between interactions favoring different orientational ordering produces a rich phase diagram including modulated phases. Using a mean-field approach and Monte Carlo simulations, we show that the proposed model exhibits isotropic and nematic phases and also a series of modulated phases that meet at a multicritical point, a Lifshitz point. Though the Monte Carlo and mean-field phase diagrams show some quantitative disagreements, the Monte Carlo simulations corroborate the general behavior found within the mean-field approximation.

Thermodynamics of a Compressible Maier-Saupe Model Based on the Self-Consistent Field Theory of Wormlike Polymer.

This paper presents a theoretical formalism for describing systems of semiflexible polymers, which can have density variations due to finite compressibility and exhibit an isotropic-nematic transition. The molecular architecture of the semiflexible polymers is described by a continuum wormlike-chain model. The non-bonded interactions are described through a functional of two collective variables, the local density and local segmental orientation tensor. In particular, the functional depends quadratically on local density-variations and includes a Maier–Saupe-type term to deal with the orientational ordering. The specified density-dependence stems from a free energy expansion, where the free energy of an isotropic and homogeneous homopolymer melt at some fixed density serves as a reference state. Using this framework, a self-consistent field theory is developed, which produces a Helmholtz free energy that can be used for the calculation of the thermodynamics of the system. The thermodynamic properties are analysed as functions of the compressibility of the model, for values of the compressibility realizable in mesoscopic simulations with soft interactions and in actual polymeric materials.

Maier-Saupe model of polymer nematics: Comparing free energies calculated with Self Consistent Field theory and Monte Carlo simulations

Self Consistent Field (SCF) theory serves as an efficient tool for studying mesoscale structure and thermodynamics of polymeric liquid crystals (LC). We investigate how some of the intrinsic approximations of SCF affect the description of the thermodynamics of polymeric LC, using a coarse-grained model. Polymer nematics are represented as discrete worm-like chains (WLC) where non-bonded interactions are defined combining an isotropic repulsive and an anisotropic attractive Maier-Saupe (MS) potential. The range of the potentials, σ, controls the strength of correlations due to non-bonded interactions. Increasing σ (which can be seen as an increase of coarse-graining) while preserving the integrated strength of the potentials reduces correlations. The model is studied with particle-based Monte Carlo (MC) simulations and SCF theory which uses partial enumeration to describe discrete WLC. In MC simulations the Helmholtz free energy is calculated as a function of strength of MS interactions to obtain reference thermodynamic data. To calculate the free energy of the nematic branch with respect to the disordered melt, we employ a special thermodynamic integration (TI) scheme invoking an external field to bypass the first-order isotropic-nematic transition. Methodological aspects which have not been discussed in earlier implementations of the TI to LC are considered. Special attention is given to the rotational Goldstone mode. The free-energy landscape in MC and SCF is directly compared. For moderate σ the differences highlight the importance of local non-bonded orientation correlations between segments, which SCF neglects. Simple renormalization of parameters in SCF cannot compensate the missing correlations. Increasing σ reduces correlations and SCF reproduces well the free energy in MC simulations.

Bimodal form distribution from modelling biaxial lyotropic liquid crystal solutions through a polydisperse Maier–Saupe model

ABSTRACTBiaxial nematic phases have been the subject of a long list of studies. In particular, they were found for a few lyotropic micellar solutions. There is a debate in the literature on whether the micellar aggregates should be biaxial, or if biaxiality could be the result of perpendicular alignment of uniaxial particles of cylinder-like and disc-like geometry in a mixture. Based on recent studies on the phase stability of such mixtures, we have investigated a polydisperse distribution of uniaxial particles interacting through a Maier–Saupe potential. Our calculations were developed for a general distribution of micellar anisometries. The distribution was obtained from the fitting of our results to the experimental data of Yu and Saupe’s well-known 1980 paper, near the Landau point, yielding a bimodal distribution, with the presence of two quadrupoles referred to objects of opposite symmetry, that is to prolate and oblate micelles. This result lends support to the rationalization of the biaxial phase for lyotropic systems in terms of a polydisperse mixture of rod-like and disc-like micelles.

Lattice Statistical Models for the Nematic Transitions in Liquid-Crystalline Systems

We investigate the connections between some simple Maier-Saupe lattice models, with a discrete choice of orientations of the microscopic directors, and a recent proposal of a two-tensor formalism to describe the phase diagrams of nematic liquid-crystalline systems. This two-tensor proposal is used to formulate the statistical problem in terms of fully-connected lattice Hamiltonians, with the local nematic directors restricted to the Cartesian axes. Depending on the choice of interaction parameters, we regain all of the main features of the original mean-field two-tensor calculations. With a standard choice of parameters, we obtain the well-known sequence of isotropic, uniaxial, and biaxial nematic structures, with a Landau multicritical point. With another suitably chosen set of parameters, we obtain two tricritical points, according to some recent predictions of the two-tensor calculations. The simple statistical lattice models are quite easy to work with, for all values of parameters, and the present calculations can be carried out beyond the mean-field level.

Recent theoretical development in confined liquid-crystal polymers

Liquid-crystal polymers in confined system is a fundamental issue in soft matter. Theoretical method plays animportant role in studying these systems. The intention of this work is to give a thorough reviewof the theoretical methodologies used in tackling confined liquid crystals. At first, some basic concept of liquid crystal, such as a vital order parameter for orientation, phases of liquid crystal, the uniaxial and biaxial of liquid crystal, are presented. After that, a brief review of the development of liquid-crystal theories, which include the Onsager model, the Maier-Saupe model, the McMillanmodel, the Landau-de Gennes expansion, the Frank elastic model and the self-consistent field model for liquid-crystal polymers, are given. All these theories havetheir own advantages and disadvantages. For example, the phenomenological Frank elastic model is the most widely used model due to its simplicity. In contrast, parameters in the self-consistent field model are physically meaningful, however, it is rather complicated. During recent decades, with these theories and suitable boundary treatment, plenty confined liquid crystal systems are investigated. In this review, we focus on three kinds of confined systems: 1) the surface wetting behavior in slits; 2) the two-dimensional liquid crystals confined by a boundary line and 3) defects in the orientational field of rigid rods on spherical surface. Results arrived from different At the end of this review, we give a list of frontier issues and an outlook for thecoming ten years.

Maier-Saupe model for a mixture of uniaxial and biaxial molecules

We introduce shape variations in a liquid-crystalline system by considering an elementary Maier-Saupe lattice model for a mixture of uniaxial and biaxial molecules. Shape variables are treated in the annealed (thermalized) limit. We analyze the thermodynamic properties of this system in terms of temperature T, concentration c of intrinsically biaxial molecules, and a parameter Δ associated with the degree of biaxiality of the molecules. At the mean-field level, we use standard techniques of statistical mechanics to draw global phase diagrams, which are shown to display a rich structure, including uniaxial and biaxial nematic phases, a reentrant ordered region, and many distinct multicritical points. Also, we use the formalism to write an expansion of the free energy in order to make contact with the Landau-de Gennes theory of nematic phase transitions.

Theoretical Studies of the Effects of Magnetic Field on the Phase Transition of Swollen Liquid Crystal Elastomers

The effect of magnetic fields on the swelling of liquid crystal elastomers (LCE) dissolved in liquid crystal (LC) solvent have been studied. The Flory-Huggins model used to calculate the free energy of an isotropic mixing and the Maier-Saupe model used to calculate the free energy of a nematic mixing. Numerical integration method used to calculate the orientational order parameter and the total free energy of system (consists of : nematic free energy, elastic free energy, isotropic mixing free energy and magnetic free energy) and the calculation results graphed as a function of temperatures for various magnetic fields and as function of magnetic fields for various of temperatures. We find that the magnetic field shifts the transition points towards higher temperatures, increases the energy transition, and induces an isotropic phase to paranematic phase.

Space Dependent Mean Field Approximation Modelling

It is shown that the self-consistency condition which is the basic equation for calculating the mean-field order parameter of any mean-field model Hamiltonian can be replaced by the standard Metropolis Monte Carlo scheme. The advantage of this method is its ease of implementation for both the homogeneous mean-field order parameter and the heterogeneous one. To be specific, the mean-field version of the Ising model spin system is discussed in detail and the resulting magnetization is the same as in the case of solving the respective mean-field self-consistency equation. In addition, it is shown that if a high temperature phase of such system is quenched below critical temperature then the mean field experienced by spins develops into a network of domains in analogous way as it happens with the spins in the case of the exact many-body Hamiltonian system and the coarsening processes start to take place. To show that the introduced Metropolis Monte Carlo method works also in case of the continuous variables the order parameter for the Maier-Saupe model for nematic liquid crystals has been calculated.

Calculation of the nematic entropy using digital images

In this work we will use digital images to compute the entropy dependence on temperature of a nematic lyotropic sample. The set of images comprehend the entire temperature range between a reentrant nematic isotropic phase transition, at a low temperature, and a usual nematic isotropic phase transition at a higher temperature. We will show that, inside the nematic phase, the image entropy profile agrees accurately with the entropy given by the Maier-Saupe model. As far as we know, this is the first time that the entropy of a lyotropic nematic phase is evaluated by this method, which introduces a way to measure their macroscopic variables. Namely, being that the entropy is a thermodynamical potential, this result implies that digital images can be used to compute mean values of nematic random variables.

Molecular field theory for biaxial smectic A liquid crystals

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 order, the phase diagram is very rich, predicting uniaxial and biaxial nematic and smectic phases, with the addition of a variety of tricritical and tetracritical points. For higher degrees of smecticity, the region of stability of the biaxial nematic phase is restricted and eventually disappears, yielding to the biaxial smectic phase. Phase diagrams from the two alternative approximations for molecular biaxiality are similar, except inasmuch that SVD allows for a first-order isotropic-biaxial nematic transition, whereas GMA predicts a Landau point separating isotropic and biaxial nematic phases. We speculate that the rarity of thermotropic biaxial nematic phases is partly a consequence of the presence of stabler analogous smectic phases.

Enhancement of Nematic Order and Global Phase Diagram of a Lattice Model for Coupled Nematic Systems

We use an infinite-range Maier–Saupe model, with two sets of local quadrupolar variables and restricted orientations, to investigate the global phase diagram of a coupled system of two nematic subsystems. The free energy and the equations of state are exactly calculated by standard techniques of statistical mechanics. The nematic–isotropic transition temperature of system A increases with both the interaction energy among mesogens of system B, and the two-subsystem coupling J. This enhancement of the nematic phase is manifested in a global phase diagram in terms of the interaction parameters and the temperature T. We make some comments on the connections of these results with experimental findings for a system of diluted ferroelectric nanoparticles embedded in a nematic liquid-crystalline environment.

Alignment of Small Organic Solutes in a Nematic Solvent: The Effect of Electrostatic Interactions

The origin of the alignment with respect to the director observed for solutes in a nematic host remains unclear, and various mechanisms ranging from steric repulsions to dispersive or electrostatic interactions have been invoked. Here we present atomistic molecular dynamics (MD) computer simulations of rigid solutes of small dimensions dissolved in a nematic liquid crystal solvent, 4-n-pentyl-4'cyanobiphenyl (5CB), that aim to quantitatively predict the orientational order. We have validated the results comparing the dipolar couplings obtained by atomistic simulation with their experimental NMR counterparts. To help assess the separate effect of the various types of anisotropic interactions on the orientational order of solutes, we have modeled solute molecules with their partial atomic charges present or absent (switching them to zero), finding that, at least for the cases studied, the alignment mechanism is largely dominated by steric and van der Waals dispersive forces rather than Coulomb ones. We have compared the anisotropic aligning potential with the predictions of the Maier-Saupe and surface tensor models and discussed their performance.

Biaxial nematic phase in the Maier-Saupe model for a mixture of discs and cylinders

We analyze the global phase diagram of a Maier-Saupe lattice model with the inclusion of shape-disordered degrees of freedom to mimic a mixture of oblate and prolate molecules (discs and cylinders). In the neighborhood of a Landau multicritical point, solutions of the statistical problem can be written as a Landau-de Gennes expansion for the free energy. If the shape-disordered degrees of freedom are quenched, we confirm the existence of a biaxial nematic structure. If orientational and disorder degrees of freedom are allowed to thermalize, this biaxial solution becomes thermodynamically unstable. Also, we use a two-temperature formalism to mimic the presence of two distinct relaxation times, and show that a slight departure from complete thermalization is enough to stabilize a biaxial nematic phase.