Maier-Saupe Theory Research Articles

Neutron scattering from mixtures of isotopically labelled molecules. A new method for determining the orientational distribution function in liquid crystals

Abstract Neutron scattering from a mixture of normal hydrogeneous and perdeuteriated 4-n-pentyloxybenzylidene-4′-n-heptyl quiline (50.7) has been shown to contain a component which depends only on the structure and orientational distribution of the molecules. It has been used to determine the orientational order parameters P n for n = 2, 4 or 6 in the smectic and nematic phases of 50.7. These values were much less than predicted by the Maier-Saupe molecular field theory, but in qualitative agreement with a model based on biaxial intermolecular interactions. Comparison with the P 2 values obtained by analysing the smearing into arcs of intermolecular interfence features, suggested that the correlation of orientation of neighbouring molecules is not very high so that the method tends to overestimate the order parameters. Other possible applications of the technique of neutron scattering from isotopically labelled mixtures are discussed.

Remarks on the temperature dependence of the optical Friedericksz transition

We report measurements of the nonlinear phase shift induced on a light beam by nematic liquid crystal films above the threshold of the optical Friedericksz transition for different temperatures, showing an excellent agreement between the experimental data and the predictions of the Maier-Saupe theory both for the threshold behavior and for the saturation behavior with temperature.

Electric field poling of acceptor–donor molecules

The Maier–Saupe theory for liquid crystals is extended to describe poling by an electric field of side-chain polymers with π-conjugated acceptor–donor molecules as pendant groups. Applications of the theory are in the field of nonlinear-optical materials. By using a quasi-one-dimensional description of the acceptor–donor molecules, our model is capable of going beyond the linear approximation and is intended for the strong-field regime. The theory takes both the dipole moment and the anisotropic polarizability of the side chains into account. A field-induced phase transition to the nematic phase is predicted for isotropic materials. This leads to enhanced polar order and thus to a larger nonlinear susceptibility than is predicted by the simpler isotropic model. The difference between initially liquid-crystalline and initially isotropic materials becomes smaller if the field-induced axial order is taken into account.

Theory of the transition of two-dimensional systems to the orientationally ordered state

From the application of the theory of the self-consistent mean field (Maier-Saupe theory) it is shown that in two-dimensional systems with a continuous distribution of the orientations of the elements passage from the isotropic to the ordered state occurs as a phase transition of the second kind (as distinct from three-dimensional systems where the corresponding transition is a phase transition of the first kind). Because of fluctuational instability of far order in a continual two-dimensional system the results of the meann field theory may be applied only to portions of finite size. In two-dimensional systems with discrete orientations of the elements (lattice models) transitions to the ordered state may be both of the first and second kinds depending on the symmetry of the lattice.

Optical and Thermodynamic Properties of Mixtures of Pleochroic Dye with Some Alkyl-Cyanobiphenyl Liquid Crystals

Abstract The temperature dependence of the refractive indices, guest and host order parameters as well as the melting and clearing points for the binary mixtures of the pleochroic dye in some members of the homologous series of the alkyl-cyanobiphenyl have been measured. Odd-even effect in some thermo- dynamic properties of the mixtures investigated was observed. A good compatibility between the shift in the nematic-isotropic transition temperature after dye addition and activity coefficient ratio on the one hand and the mutual connection between guest and host order parameter in the vicinity of the clearing point on the other have been found. The experimental results were compared in some details with calculations done on the basis of the Maier-Saupe theory for binary mixtures. It was demonstrated a good agreement between experiment and theory taking into account the flexibility of the end-chain of the host.

Critical Points and Re-Entrance in Liquid Crystalline Side-Chain Polymers

Abstract Competing nematic and smectic A phases of comb-like polymers are described by combining Maier-Saupe and McMillan type theories with the worm concept of semi-flexible polymers. There are as few material parameters as possible: main chain flexibility and phenomenological coupling constants. New qualitative predictions, the existence of critical points and re-entrant nematic phases, as well as quantitative predictions for comparison with experiments are provided. A new mechanism for thermotropic nematic biaxiality which is specific of side-chain polymeric liquid crystals is found.

Anomalous density and refractive index variation of a nematic liquid crystal

Abstract This paper reports the results of the measurement of refractive indices and densities of p-ethoxyphenyl trans-4-butyl cyclohexane carboxylate at different temperatures. The molecular polarizabilities have been calculated from refractive indices using Vuks' formula and orientational order parameters are determined from the polarizability values. The order parameter values have been compared with those obtained from Maier-Saupe theory. The compound shows anomalous behaviour regarding variation of density, ordinary refractive index and order parameter values with temperature. This is consistent with our previous X-ray study of this compound. The possible reason for this peculiar behaviour has been discussed.

On the relationship of the nematic-isotropic transition temperature of an oligomer to that of its constituent units

Abstract We consider the nematic-isotropic transition temperature for a rigid oligomer formed by linking identical mesogenic units together. In particular, we seek to relate the transition temperature for the oligomer to that of the constituent units, within the framework of the Maier-Saupe molecular field theory and its various parameterizations.

Theory of nematic backbone polymer phases and conformations

The nematic-isotropic phase transition of long flexible polymers using mean-field theory with the Maier-Saupe expression of the van der Waals interaction are described. The order parameter of nematic polymers is calculated as a function of reduced temperature, kB/T(a epsilon )1/2, where a is the quadrupolar mean-field strength per unit length of the molecules and epsilon is the bend elasticity of the polymers. The reduction factor (a epsilon )1/2 is in contrast to the factor in the Maier-Saupe theory of conventional nematics, a, indicating via epsilon the polymer aspect to the problem. The transition temperature, critical-order parameter and latent heat are given. For very long chains the transition between the nematic and isotropic phases occurs when T=0.38775 (a epsilon )1/2/kB where the order parameter is equal to 0.35642. The authors predict the temperature dependence of the order parameter and transition behaviour for different lengths of the polymer chain. A detailed procedure is suggested for comparison with experiment involving order parameters, latent entropy and configurational properties as revealed by small-angle neutron and X-ray scattering.

Bogolyubov-Tyablikov-like approximation for nematic liquid crystals: Bridge between continuum and mean-field theories.

A procedure analogous to the Bogolyubov-Tyablikov approximation is proposed in order to bridge between the continuum theory and the Maier-Saupe model of nematic liquid crystals. The equation obtained for the order parameter, which results in a first-order transition, reduces to the Maier-Saupe equation in the limit of infinite-range interactions. A prescription for the evaluation of the free energy in the framework of the Bogolyubov-Tyablikov treatment is presented, and the resulting Maxwell construction is used to determine the first-order transition temperature. The transition temperature, the range of metastability, and the value of the order parameter at the transition are all considerably lower than in Maier-Saupe theory.

Fluctuations in the nematic-isotropic phase transition.

We study the fluctuations in a lattice model of the nematic-isotropic phase transition, whose mean-field approximation follows the Maier-Saupe theory. We show that fluctuations significantly change the order parameter, the transition entropy, and the precocity of the phase transition (${T}_{c}$-${T}^{\mathrm{*}}$)/${T}_{c}$. A comparison with Monte Carlo simulation data shows that the fluctuation corrections remove most of the deficiencies of Maier-Saupe approximation.

Rod to coil transitions in nematic polymers

The order parameter and anisotropy (elongation) of the configurations of a nematic polymer in the nematic phase are calculated. At low temperatures exponentially rapid growth of chain dimensions as a function of inverse temperature is found. In the nematic direction the chain eventually adopts a rod-like state. The thermal activation of hairpins (abrupt reversals in chain directions) causes this behaviour. However, at even lower temperatures the deviation from rod-like alignment is governed by gentle meandering away from the mean field direction. A Maier-Saupe mean field theory of the nematic-isotropic transition is constructed calculating for long chains the transition temperature as a function of chain properties and predicting a universal jump in the order parameter at the transition, Delta Sni=1/3.

The order parameters < P2 > and < P 4 > in nematic p-alkyl-p' -cyano-biphenyls : polarized Raman measurements and the influence of molecular association

Values for the order parameters and from polarized Raman measurements are reported for the homologous series of nemaric alkylcyanobiphenyls. With increasing chain length a change of behaviour is observed from «low» -values to higher values corresponding to the Maier-Saupe mean-fied theory. In addition, correlation factors describing the anti-parallel molecular association in these compounds have been derived from permittivity measurements. A tentative qualitative model is given in which the low -values are attributed to this association. This leads to a certain proportion of «dimers», for which the long molecular axis is not parallel to the direction of the Raman-vibration anymore. In addition, steric interactions can be expected to lead to non-parallelism of the long axis of a dimer and of a neighbouring monomer. For specific parameters of the dimer, the combination of these two effects can lead to a marked broadening of f(β), and thus to low or negative values for En fonction de la longueur des chaines, on observe une augmentation de correspondant a la theorie de champ moyen de Maier et Saupe. Evaluation des facteurs de correlation decrivant l'association moleculaire antiparallele en vue de mesures de permittivite

Computer simulation studies of anisotropic systems. XI. Second- and fourth-rank molecular interactions

Abstract The Maier-Saupe theory for nematic liquid crystals provides a reasonable account of their orientational order and its temperature dependence. The theory is based on second-rank anisotropic interactions and its predictions can be improved by the introduction of higher-rank terms as in the Humphries-James-Luckhurst theory. However comparison with the properties of real nematogens does not allow an unambiguous test of the theory because the form of the pair potential is unknown. This is not the case for computer simulations where the intermolecular potential is defined. We have therefore undertaken a Monte Carlo study of the influence fourth-rank interactions on nematic behaviour and report the results of our simulations here. The model nematogen used as a reference is that developed by Lebwohl and Lasher in which the particles are confined to the sites of a simple cubic lattice and interact via a second-rank anisotropic potential. The simulation gives the internal energy, the heat capacity at constant volume and the second-rank order parameter as a function of temperature, as well as the nematic-isotropic transition temperature. These results are used to provide the first unambiguous test of the Humphries-James-Luckhurst theory. We also discuss the changes in the transition temperature which are caused by the introduction of the fourth-rank term in the pair potential using thermodynamic perturbation theory for the Helmoltz free energy.

Anisotropic Dispersion Interactions In Liquid Crystals

Abstract The polarizability and shape anisotropics of a new class of aliphatic liquid crystals, 4,4′-di-n-alkyl-bibicyclo[2.2.2]octanes (n, n-BBCO) have been calculated as a function of the alkyl chain length and its degree of flexibility to inquire about the relative importance of attractive and repulsive interactions in stabilizing the liquid crystalline phase. The calculated polarizability anisotropics are found to be much smaller than that of conventional mesogens. In the context of the Maier-Saupe theory, this indicates that anisotropic dispersion interactions make only a minor contribution to the stability of this class of liquid crystals. The cylindrical symmetry exhibited by the n, n-BBCO homologues and their rigid molecular structure suggest that they might be ideally suited for critical testing of existing theories of the nematic-isotropic phase transition.

Field-Induced Order in Nematic Liquid Crystals

Abstract The effect of an external magnetic field on the orientational order of a nematic liquid crystal has been examined using both Landau-de Gennes and Maier-Saupe theories. In the Maier-Saupe approach a rotationally invariant form of the pseudo-potential is introduced, which in the absence of an external field leads to three degenerate isomorphic solutions for the order parameter, corresponding to alignment along three principal axes; a similar result is obtainable from the Landau-de Gennes theory. Application of a magnetic field lifts the degeneracy of these solutions, and for materials having a positive diamagnetic susceptibility anisotropy, the uniaxial solution with alignment along the field direction is always energetically favorable. For materials with a negative susceptibility anisotropy, a biaxial solution minimizes the free energy at low temperatures, but on increasing the temperature there is a transition from a biaxial phase to an uniaxial phase. The field dependence of the transition tempe...

Molecular Theories of Nematic Liquid Crystals

Abstract In this paper, the current state of the art of molecular theories of nematic mesophases is reviewed. For this purpose, the many theories of nematogens which have been proposed are grouped into two classes: (1) the Maier-Saupe theory and its extensions and variants, which assume that nematic long-range order is primarily due to anisotropic intermolecular attractions and (2) the hard-rod and van der Waals theories, which assume that nematic order results primarily from excluded volume effects. These two types of theories are discussed in turn, with particular emphasis on the basic assumptions and approximations, the relative strengths and weaknesses, and the most fruitful applications of each of them. Finally, prospects for improving and extending current molecular theories are examined briefly.

Zvetkow Twist Viscosity Measurements of Some Nematic Liquid Crystals

Abstract Zvetkow twist viscosity measurements are reported for a series of cyanobiphenyl liquid crystals and their eutectic mixtures in the nematic phase. The values obtained exhibit an odd-even variation. If the data are plotted against a reduced density, they divide into two groups, one corresponding to the odd and the other to the even series. This is attributed to effects of the configuration of the alkyl chain on the molecular packing density in the nematic phase. Using the Maier-Saupe theory, interaction energies are derived.

Computer simulation studies of anisotropic systems. IX. The Maier—Saupe theory for nematic liquid crystals and the molecular field approximation

The first real test of the molecular field approximation employed in the Maier—Saupe theory of nematic liquid crystals was provided by the Lebwohl—Lasher Monte-Carlo simulation of an idealized lattice model. However the system employed in the simulation is not uniquely consistent with the Maier—Saupe theory and so does not provide an unambiguous test of the molecular field approximation. We describe here three new lattice models which are also compatible with the theory and report the temperature dependence of the internal energy, heat capacity and the second-rank orientational order parameter determined using the Monte-Carlo technique. In addition the second- and fourth-rank orientational pair correlation functions were also evaluated at selected temperatures for one of the models. All of the results are compared with the predictions of the Maier—Saupe theory. We find that the validity of the molecular field approximation improves as the coordination number and the range of the anisotropic pair potential employed in the model increase. This improvement is especially evident for the nematic—isotropic transition temperature although the agreement with the order parameter at the transition is not so impressive and that for the entropy of transition is poor.

Computer simulation studies of anisotropic systems

The Lebwohl-Lasher model of a nematogen provides a simple system with which to study the order-disorder transition and to examine the properties of the nematic phase. We have investigated the model containing 203 particles over a small temperature range characteristic of that for real nematogens using the Monte Carlo method to evaluate the internal energy, the heat capacity and the second rank orientational order parameter. Our results are in reasonable agreement with those of other simulations and are compared with the predictions of theories based on a cluster expansion of the free energy. The temperature dependence of the order parameter is used to investigate the validity of the singlet orientational distribution function predicted by the Maier-Saupe theory. The potential of mean torque is found to be linear in the order parameter as required by the theory although the orientational free energy is in error. The simulated properties are in reasonable but not complete accord with those of a real nematog...