Theory Of Nematic Liquid Crystals Research Articles

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.

Molecular Model for Chain Ordering in Biomembranes

Abstract Detailed comparison between the self-consistent theory of nematic liquid crystals and membranes is made. A modified mean field Hamiltonian is so constructed as to take into account the connectivity of the chains and the constraints imposed due to the polar head surface. This gives rise to a variation of order parameter along the chain, which has been studied below the transition point. Results agree satisfactorily with experiment.

A Molecular Statistical Theory of Nematic Liquid Crystals (The Effects of a Pair Spatial Correlation Function)

Abstract The pair spatial correlation function is introduced approximately into mean field theory in terms of an orientation-averaged pair potential, and its effects on nematic-isotropic phase transition are investigated in detail. One finds that the anisotropy of the pairwise intermolecular potential and of the molecular shape account for the stability of nematic order, i.e. increase the long-range orientational order and also the transition temperature. The results of the numerical calculations are compared with experimental data for PAA and with other theoretical results.

On the determination of liquid crystal-to-wall anchoring anisotropy by the surface-plasmon polariton technique

Based on the Oseen and Frank elastic continuum theory of nematic liquid crystals, and on a generalized torque balance equation at the interface, the magnitude and the gradient of the tilt angle of liquid crystal molecules at the interface have been calculated in a homogeneous cell as a function of applied voltage and liquid crystal-to-wall anchoring anisotropy C. Consequently, the reflectivities associated with the excitation of the surface-plasmon polariton propagating along the liquid crystal-to-wall interface can be evaluated for comparison with experimental results to determine a value for C. Numerical computations of reflectivities have been shown to fit the experimental results of Sprokel, Santo, and Swalen [Mol. Cryst. Liq. Cryst. 68, 29 (1981)]. An average value of 0.11±0.05 ergs/cm2 for C has been deduced.

Elastic constants for liquid crystals of disc-like molecules

Using a previously developed theory of nematic liquid crystals we present a calculation of the order parameters and elastic constants for those liquid crystals which consist of disc-like molecules. We obtain a simple inversion as compared to rod-like molecules. In addition we remark that the Maier-Saupe theory can be considered as describing the intermediate case, in which the molecules are neither rod-like nor disc-like. They have a form such that the mutually excluded volume of a pair of molecules is always a sphere, but with a radius which depends on the relative orientation of their symmetry axes.

A theory for nematic liquids with biaxial molecules

We consider a mean field theory for nematic liquid crystals consisting of biaxial molecules. The internal degrees of freedom of the molecules are described by eight variables, three of which are the components of the angular momentum. The five remaining variables are the components of the dynamical quadrupole moment. Using the fact that these variables are the generators of SU(3), the partition function in the mean field approximation can be calculated exactly. From a numerical determination of the minimum of the free energy we then show that with decreasing temperature the fluid will have two successive transitions according to the scheme: isotropic → unaxial order → biaxial order. The order parameters and the specific heat are calculated in their dependence on the temperature and on the form of the molecules.

A theory for nematic liquid crystals

Using a Corner-type potential for the interaction between two mono-axial molecules and choosing a special form for the dependence on the relative orientations of these molecules, we derive a simple equation for the angular dependence of the single particle distribution function. Given this function we obtain algebraic expressions for the elastic constants for splay, twist and bend, in which the dependence on the four parameters, occurring in the potential, is shown explicitly. We obtain good agreement with the experimental data for the order parameters and the elastic constants.

The relationship between the second and fourth rank order parameters for liquid crystals composed of non-rigid molecules

Abstract The values of the fourth rank order parameter P4 determined by Raman scattering experiments are invariably lower than those predicted by the Maier-Saupe theory of nematic liquid crystals. However this theory is only applicable to rigid particles whereas the nematogenic molecules studies contain flexible alkyl chains. We have estimated the influence of the alkyl chain on P4 with the aid of a model which account successfully for the variation of the second rank order parameter P2 along the chain. It is found that the values of P2 and P4 calculated for the CN direction in the 4-n-alkyl-4′-cyanobiphenyls are indistinguishable from those predicted by the Maier-Saupe theory. It would appear therefore that the low values observed for P4 cannot be attributed to the non-rigidity of the alkyl chain.

Statistical theories of nematic liquid crystals

Nematic liquid crystals are composed of molecules with moderate shape anisotropy, and are characterized by purely orientational but no translational long range order. It is necessary to take into account both the anisotropic attractions between the molecules and the excluded volume effects to give a proper statistical theory of this phase. Indeed a reasonably satisfactory semiquantitative description has emerged only in the past two years. The paper reviews recent developments in the field.

A statistical mechanical treatment of fatty acyl chain order in phospholipid bilayers and correlation with experimental data. A. Theory

A theoretical model has been developed in order to describe the organization of acyl chains in phospholipid bilayers. Since the model is intended to reproduce highly quantitative experimental results such as the deuterium magnetic resonance (NMR) data and to supplement the experimental information, all the rotameric degrees of freedom, the excluded volume interactions and the van der Waals interactions have been considered. The model is a direct extension of a generalized van der Waals theory of nematic liquid crystals to flexible molecules. In this picture, the anisotropy of the short-range repulsive forces which are treated by a hard core potential is introduced as the dominant factor governing intrinsic order among the chains. The anisotropy of the attractive forces, which are approximated by a molecular field, plays a somewhat secondary role. The dependence of the energy of interaction on the relative chain conformations is approximated by two order parameters reflecting respectively the ‘average shape’ of the molecules and the ‘average shape’ in a ‘mean orientation’. The influence of the interactions in the polar region on the lateral chain area is accounted for by an effective lateral pressure. In certain aspects the model has features in common with the Marčelja theory.

Viscometry of Nematic Liquid Crystals

Abstract Viscometric experiments have played a significant part in establishing continuum theory for nematic liquid crystals. Such theory supports the hypothesis that observed non-Newtonian behaviour stems from competition between the aligning influences of flow and solid surfaces, and as a consequence predicts rather unusual scaling for the apparent viscosity. This paper first describes such scaling and its subsequent experimental confirmation emphasising the full implications of the latter. The relevance of the theory thus established, there follows a simple analysis of alignment in shear flow which leads to conditions on material coefficients necessary to ensure consistency with observations. Our discussion turns next to an account of solutions exhibiting non-Newtonian behaviour, and also mention of a recent stability analysis which attempts to discriminate between the different solutions that are possible. The final section considers oscillatory shear flow and presents an analysis of an instability wh...

Quadrupole Analog of the Ising Model Spin-S : Bragg-Williams Approximation

Abstract An analog of the spin-S Ising Model with quadrupolar interaction has been solved in the Bragg-Williams Approximation. We got a first order phase transition for special values of the parameters. We compare with results obtained by other authors and we recover the same results given by Maier-Saupe Theory for Nematic Liquid Crystals when the spin-S becomes infinite.

Orientational order of triradical spin probes in nematic liquid crystals

The trigonal symmetry of organic triradicals makes them important candidates for spin probes of liquid crystals. Here we develop a lineshape model for such spin probes, within the limit of fast motion, and show how the E.S.R. spectra depend on the orientation of the director with respect to the magnetic field. The theory is employed to study the lineshapes of trisverdazyl radicals in 4,4′-dimethoxyazoxybenzene, aligned by a high frequency electric field. An analysis of the angular-dependent linewidths provides reliable values for the order parameter P 4. They are in good agreement with statistical theories of nematic liquid crystals.

Orientation-averaged pair correlations in a molecular theory of nematic liquid crystals. II

In a previous paper (I) we carried out a model calculation for nematic liquid crystals which treats spatial correlations between molecules as for a classical isotropic liquid and orientational correlations in the meanfield approximation. Generalized Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) equations were derived for the density function and an orientationally averaged pair-correlation function. They were solved for a simple pairwise potential chosen to fit certain experimental data on $p$-azoxyanisole (PAA). A free-energy expression was obtained by identifying its Euler-Lagrange equation to the first BBGKY equation. In this paper we reformulated the molecular theory of nematic liquid crystals rigorously on the basis of a general probability distribution function ${P}_{N}(1,\dots{},N)$. It was shown that both the mean-field theory (MFT) and the above-stated orientation-averaged pair-correlation theory (OAPC) arise from applying low-order approximations to ${P}_{N}$. In the case of MFT, ${P}_{N}$ was approximated as a product of single-particle functions. In the case of OAPC, spatial correlations were introduced, but decoupling of spatial and orientational dependences were assumed. It is now clear how higher-order-approximation schemes can be systematically constructed. Furthermore, for each scheme chosen, the present formulation offers a uniquely consistent formula for the free energy. This places our previous work on a firm theoretical foundation. The theory in OAPC approximation was applied to model systems described by a more general pairwise potential than that employed in I. Potential parameters were chosen to mimic PAA and methoxybenzylidene butylaniline. For each substance we calculated the isotropic-nematic coexistence curve, temperature dependence of the order parameter, latent heat, and volume change at transition. The results were compared to experimental data.

Orientation-averaged pair correlations in a molecular theory of nematic liquid crystals

Orientation-averaged pair correlations in a molecular theory of nematic liquid crystals

Generalized van der Waals theory of nematic liquid crystals: Requirements for self-consistency

In the generalized van der Waals (GVDW) theory of nematic liquid crystals, nematogenic molecules are assumed to interact via hard core repulsions and longer ranged attractive forces, and the latter are treated in the mean field approximation. The resulting expression for the configurational Helmholtz free energy Ac as a functional of the orientational distribution function f (Ω) is N−1Ac[ρ,T;f (Ω)]=N−1Ac[ρ,T;f (Ω)] + N−1V(a)N(ρ) =N−1Ac[ρ,T;f (Ω)] + 1/2ℱf (Ω) ? (ρ,Ω) dΩ, where ρ is the number density, Ac[f (Ω)] is the free energy functional for a system of molecules with only hard core repulsions, ?(a)N is the GVDW average of the total potential energy of attraction V(a)N, and ? (ρ,Ω) is the pseudopotential describing the interaction between a molecule with orientation Ω and the mean field produced by intermolecular attractions. In this paper, it is shown that the theory exhibits statistical mechanical self-consistency if and only if both N−1?(A)Nand?(Ω)are directly proportional to the first power of the density ρ. This result is independent of the shape of the molecular hard cores and of the assumed orientation dependence of the pseudopotential ?.

Why is the Maier–Saupe theory of nematic liquid crystals so successful?

THE Maier–Saupe theory of nematic liquid crystals is founded on a molecular field treatment of long-range contributions to the intermolecular potential and ignores the important shortrange forces. Nonetheless this theory is particularly successful in predicting the orientational properties of real nematics. Here we propose a model which accounts both for the success of the Maier–Saupe theory and the failure of theories based exclusively on short-range forces.

Generalized van der Waals theory of nematic liquid crystals: An alternative formulation

An alternative formulation of the generalized van der Waals theory of nematic liquid crystals is presented and discussed briefly. The derivation is considerably more straightforward than that of Gelbart and Baron and is closely analogous to the usual derivation of the van der Waals equation of state.

Viscometric flows of anisotropic fluids

A brief introduction to the continuum theory for nematic liquid crystals is given. The problem of simple shearing flow is examined in some detail using this theory and some of the difficulties that arise are discussed. It is shown that under reasonable simplifying assumptions, useful asymptotic formulae can be found for the apparent viscosity in many viscometric flows.

Consistency of Mean Field Theories of Nematic Liquid Crystals

The Maier-Saupe theory of nematic liquid crystals, which uses a one-body mean field potential of the form φ2 nP2(cos θ), where θ is the angle between the molecular long axis and the nematic directo...