Lebwohl-Lasher Model Research Articles

Phase transition and director fluctuations in the three-dimensional Lebwohl-Lasher model of liquid crystals

Monte Carlo computer simulation techniques are used to study the orientational phase transition in the three-dimensional Lebwohl-Lasher model which couples molecular rotors placed on a cubic lattice by the potential P 2(cos ϑ ij ). The orientational transition is a model of the nematic-isotropic phase transition in liquid crystals. The simulations involve the determination of energy and order parameter distribution functions which permit free energy functions to be derived. The data have been analysed by finite size scaling methods to reveal the nature of the phase transition which is found to be weakly first order with stability limits of the nematic and isotropic phases being extremely close to the equilibrium transition temperature, in good agreement with experimental studies of room temperature nematogens. Results are reported for the specific heat, the axial and biaxial susceptibilities, as well as the enthalpy and nematic order parameter discontinuity at the transition. It is shown that the inclusio...

Computer simulation of liquid crystal films

In this paper we present a computer simulation study of thin film behaviour performed using the Lebwohl-Lasher liquid crystal model. In this, we study systems with a slab geometry of L parallel planes. We have measured the shifts in the bulk transition temperature that result from variation both of L and of an aligning surface field ϵs. From these, we have used a Kelvin-like equation to calculate the nematic-isotropic surface tension. We have also investigated the nature of the shifted bulk transition in these thin film systems. From detailed comparison of energy distribution functions, we have found that the ϵs = 0 system has a critical thickness, 8 < L c < 16, below which there is no bulk transition. We note that mean field theories fail to predict this critical point and suggest that its cause is related to the pretransitional divergence in orientational correlations seen at the bulk 3d transition.

Ordering dynamics of microscopic models with nonconserved order parameter of continuous symmetry.

Numerical Monte Carlo temperature-quenching experiments have been performed on two three-dimensional classical lattice models with continuous ordering symmetry: the Lebwohl-Lasher model [Phys. Rev. A 6, 426 (1972)] and the ferromagnetic isotropic Heisenberg model. Both models describe a transition from a disordered phase to an orientationally ordered phase of continuous symmetry. The Lebwohl-Lasher model accounts for the orientational ordering properties of the nematic-isotropic transition in liquid crystals and the Heisenberg model for the ferromagnetic-paramagnetic transition in magnetic crystals. For both models, which have a nonconserved order parameter, it is found that the linear scale, R(t), of the evolving order, following quenches to below the transition temperature, grows at late times in an effectively algebraic fashion, R(t)\ensuremath{\sim}${\mathit{t}}^{\mathit{n}}$, with exponent values which are strongly temperature dependent and furthermore vary for different measures of the time-dependent length scale. The results are discussed in relation to modern theories of ordering dynamics in systems with continuous order-parameter symmetry.

Are Monte Carlo simulation results affected by sample shape? An investigation based on the Lebwohl-Lasher model

We investigate the sample shape dependence of periodic boundary conditions and cluster Monte Carlo simulation results for a Lebwohl-Lasher model with 1000 particles. We find marked differences between the two simulations, with pronounced shape effects when using periodic boundary conditions and we discuss their significance.

Fractional Brownian motion of director fluctuations in nematic ordering.

Temporal director fluctuations in nematic ordering were studied by computer simulation on the Lebwohl-Lasher model as well as by neutron-scattering experiments on the nematogen d-PAA. The time-series data have been analyzed by the rescaled-range method and in terms of the power spectrum in order to determine the Hurst exponent H. Theory and experiment are in good agreement. A value of H\ensuremath{\simeq}1 was found for the nematic phase, characterizing fractional Brownian motion, whereas H\ensuremath{\simeq}0.5, reflecting ordinary Brownian motion, applies in the isotropic phase. Field-induced crossover from fractional to ordinary Brownian motion was observed in the nematic phase.

ON THE SWITCHING FROM THREE TO ONE DIMENSIONAL BEHAVIOR IN A MODEL LIQUID CRYSTAL

A Monte Carlo study of a Lebwohl-Lasher lattice model with anisotropic interaction is presented. Six different values of out of plane to the in-plane coupling ratio have been taken into account. Comparison with mean field prediction and with analytic results, when possible, are made.

DETECTING PHASE EQUILIBRIA IN MODELS OF THERMOTROPIC AND LYOTROPIC LIQUID CRYSTALS

New ways are discussed of numerically detecting the nature of phase transitions and the position of phase equilibria in models of thermotropic and lyotropic liquid crystals where the phase transitions are dominated by strong fluctuations. The cases of continuous transitions (critical points), first-order transitions, as well as the absence of transitions, are considered. It is shown how computer-simulation techniques, which operate on a free-energy level by using reweighting (histogram) techniques in combination with finite-size scaling theory, provide an effective tool for unambiguously determining the nature of the transition and the position of associated phase equilibria. Three specific models are considered: the Lebwohl–Lasher model of the nematic–isotropic transition in thermotropic liquid crystals, the mismatch model of the main chain-melting phase transition in lipid-bilayer lyotropic liquid crystals, and a model for critical mixing in a lipid-bilayer lyotropic liquid crystal incorporated with trans-bilayer polypeptides.

Weak first-order orientational transition in the Lebwohl-Lasher model for liquid crystals.

The nature of the orientational phase transition in the three-dimensional Lebwohl-Lasher model of liquid crystals has been studied by computer simulation using reweighting techniques and finite-size scaling analysis. Unambiguous numerical evidence is found in favor of a weak first-order transition and the presence of pseudospinodal points, ${\mathit{T}}_{\ifmmode\pm\else\textpm\fi{}}^{\mathrm{*}}$, which are extremely close to the equilibrium transition temperature, \ensuremath{\Vert}${\mathit{T}}_{\mathit{c}}$-${\mathit{T}}_{\ifmmode\pm\else\textpm\fi{}}^{\mathrm{*}}$\ensuremath{\Vert}/${\mathit{T}}_{\mathit{c}}$\ensuremath{\lesssim}0.5\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$, in good agreement with experimental data for the nematic-isotropic transition.

Computer simulations of the elastic properties of liquid crystals

We have carried out extensive computer simulations of the Lebwohl-Lasher model of a nematic liquid crystal to determine the Frank elastic constant K for this system as a function of temperature. We use equilibrium, k-dependent, fluctuation expressions, valid at small wave vector k, and also measure the response to applied, spatially varying fields that couple to the k-dependent molecular orientation tensor. We find good agreement between these two methods. The ratio K/P\ifmmode\bar\else\textasciimacron\fi{} $_{2}^{2}$, where P${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{2}$ is the nematic order parameter, adopts its molecular-field value at low temperatures, but rises by over 20% on approaching the nematic-isotropic transition temperature ${\mathit{T}}_{\mathit{N}\mathrm{\ensuremath{-}}\mathit{I}}$. We note that this temperature dependence cannot be accounted for by molecular-field arguments. However, spin-wave theory correctly relates the k-dependent elastic constant with the order parameter at low temperatures, and is quite accurate up to T/${\mathit{T}}_{\mathit{N}\mathrm{\ensuremath{-}}\mathit{I}}$\ensuremath{\approxeq}0.8. Finally, direct simulations of the Fr\'eedericksz transition give results in reasonable agreement with the value of K as calculated by the other methods; however, we find that this is not a very efficient route to K.

The effect of confinement on the isotropic-nematic transition

The Lebwohl-Lasher model of a liquid crystal is examined in a slab geometry, with free top and bottom surfaces, using standard molecular-field and a new Bethe-type approximation that includes some effects of correlations. The results are compared with computer simulations for a range of slab widths L. We find that the approximate treatments, while predicting a nematic-isotropic transition that is too strongly first-order, are in semiquantitative agreement with the simulation results. The results of the approximate treatments show that the Kelvin equation for the shift in transition temperature due to confinement is accurate for L ≥ 64 layers. For large L a wetting film of the disordered, isotropic phase intrudes between each surface and the nematic phase. The film thickness, evaluated at the transition in the slab, increases as ln L in accord with the theory of complete wetting for systems with short-ranged forces.

Computer Simulation of Liquid Crystal Films

Abstract We present the results of extensive Monte Carlo simulations of liquid crystal films of various thicknesses. A simple nearest-neighbour lattice model, the Lebwohl-Lasher model, is employed, with periodic boundaries in two directions and free, planar, surfaces in the third. Particular attention is devoted to locating the temperature of the order-disorder (nematic-isotropic) phase transition. Weak first-order behaviour apparently persists in systems as thin as 8 layers across, but below this the transition cannot be detected. The shift of the transition temperature from its bulk value approaches the expected asymptotic linear dependence on inverse thickness, but significant deviations from this are seen for films of 10 layers thickness and less. These results enable an accurate estimate to be made of the bulk phase transition temperature in the thermodynamic limit, and the result is consistent with that extrapolated from systems with full periodic boundaries.

A theoretical study of the Lebwohl-Lasher model in higher dimensions

We re-examine the Lebwohl-Lasher model of the isotropic-nematic liquid crystal phase transition as a function of dimensionality d, using the Bethe-Peierls approximation. Our results suggest that the transition temperature approaches the mean-field limit asymptotically as d -1, not, as suggested by recent computer simulations, as d -1·5.

Nematic liquid crystals

The Lebwohl-Lasher model of the isotropic-nematic phase transition in liquid crystals is studied in higher dimensions, by Monte Carlo computer simulations on simple cubic lattices. The transition is found to become increasingly first-order, and rapidly approaches the mean-field results, which are exact in infinitely many dimensions. The discontinuities in internal energy and order parameter are measured by careful analysis of the computer simulation data, and the free energy of each system is calculated. An empirical scaling law is also found to describe the shift of the transition temperature from the mean-field theory result. To distinguish the effects of coordination number from those of dimension and connectivity, a simulation of an f.c.c. lattice in d = 3 dimensions is performed. In addition, a partial analysis is made of finite-size effects in the six-dimensional simulation.

TheP4model and its orientational phase transition

A simple generalization of the Lebwohl-Lasher model, where fourth rank, rather than second rank, interactions are involved is investigated. This model was first put forward and studied some years ago using molecular field theory (Zannoni, C., 1979, Molec. Crystals liq. Crystals Lett., 49, 247). There it was found that there should be a temperature interval where the fourth rank order parameter is higher than the second rank one. This unusual behaviour has been found by various groups to be consistent with fluorescence depolarization data for diphenylhexatriene in DPPC and DMPC membrane vesicles. In this paper we investigate more thoroughly the P 4 model using Monte Carlo simulations with periodic boundary conditions on a 10 × 10 × 10 lattice and with the recently proposed Cluster Monte Carlo method on a 6 × 6 × 6 and a 10 × 10 × 10 lattice. Our results are consistent with a first order transition. We find that the results for the transition temperature and for the second and fourth rank order parameters are well approximated by two site cluster theory.

Analysis of the migdal transformation for models with Heisenberg spins on a d-dimensional lattice

We show for classical Heisenberg spins, with a general nearest neighbour interaction, that in the Migdal approximation the only low-temperature phase transitions are Ising ones (ferror antiferromagnetic). For d=2 neither the pure Heisenberg model nor the Lebwohl-Lasher model show a phase transition at a finite temperature. For d>2 transitions do exist at intermediate temperature and the complete flow diagram together with a two-parameter phase diagram is obtained numerically for d=3. Apart from critical temperatures and thermal exponents, also the magnetic exponents (for both Heisenberg and XY spins) are calculated. The latter are in very good agreement with exact results.

Can Monte Carlo detect the absence of ordering in a model liquid crystal?

Abstract We investigate the possibility of inferring the absence of an ordered phase using Monte Carlo simulations. The example we have chosen is that of a one dimensional Lebwohl-Lasher model, where an analytic solution is available. We argue that Monte Carlo can be of help even in this delicate sector notwithstanding the complications created by periodic boundary conditions.

A Monte Carlo investigation of the planar Lebwohl-Lasher lattice model

A Monte Carlo computer simulation of a planar version of the Lebwohl-Lasher model is presented. The model consists of a set of interaction centres forming a simple square lattice. The pair potential is nearest-neighbours, attractive, and varies as a second Legendre polynomial of the relative orientation between the two particles. Five lattice sizes, 5 × 5, 10 × 10, 20 × 20, 60 × 60 and 80 × 80, of this two-dimensional system have been simulated with Monte Carlo and periodic boundary conditions. A study of the orientational pair correlation function indicates a power law decay in the ordered phase and an exponential decay above the pseudo-transition temperature. Our results are consistent with the absence of a true phase transition but also indicate a low-temperature phase with long short-range order. Comparisons are made with one existing simulation and with the mean field theory results.

A Monte Carlo simulation of the inhomogeneous Lebwohl-Lasher lattice model

Abstract A Monte Carlo computer simulation of a modified Lebwohl-Lasher model is presented. The model consists of a set of interaction centres placed at the sites of a cubic lattice. The angular part of the pair potential is a second Legendre polynomial of the relative orientation between the two particles, like that of the Lebwohl-Lasher model. Each particle interacts with its six nearest neighbours with an attractive anisotropic potential differing in strength for the four horizontal and the two vertical neighbours. Various values of the in-plane to out-of-plane coupling ratio δ have been considered, i.e. δ = 0.75, 0.5, 0.1, 0.0. The latter case corresponds to the limiting situation of a two-dimensional lattice. Systems with a 1000 particles have been simulated for δ = 0.75, 0.5 and 0.1 while a sample of 3600 particles has been investigated for the two-dimensional lattice. Comparisons are made with available simulations and with mean field theory. We find that the molecular field theory predictions worsen as the effective coordination number is decreased. Energy, specific heat, second and fourth rank order parameters have been evaluated for the various models. We also present, for the first time, a way of approximately reconstructing the pair distribution, G(r12, ω12), using maximum entropy and second and fourth rank two particle order parameters.

Computer simulation studies of anisotropic systems

We have studied the surface behaviour of the Lebwohl-Lasher model nematogen using the Monte Carlo technique of computer simulation. Two states of the nematogen have been investigated, one with normal periodic boundary conditions and the other with boundary conditions employed for just two dimensions thus forming two free surfaces orthogonal to the third dimension. We have determined the variation of the internal energy and the second rank orientational order parameter, associated with individual layers, with their distance from the surface, as a function of the temperature. The order in the surface is found to be less than in the bulk nematogen although this difference is far less for the isotropic phase than for the nematic. This suppression of the surface orientational order is in accord with the surface tension simulated for the nematogen. Thus the surface tension decreases rapidly with increasing temperature in the nematic mesophase but then is essentially independent of temperature in the isotropic p...

A Monte Carlo investigation of the Lebwohl-Lasher lattice model in the vicinity of its orientational phase transition

The Lebwohl-Lasher model of nematogens consists of a system of particles placed at the sites of a cubic lattice and interacting with a pair potential U i,j = -ϵ ij P 2(cos β ij ), where ϵ ij is a positive constant, ϵ, for nearest neighbours particles i and j and β ij is the angle between the axis of these two molecules. We have investigated this model using Monte Carlo simulations with periodic boundary conditions on a 30 × 30 × 30 lattice while the largest system previously studied was 20 × 20 × 20. The number of simulation runs near the transition is also significantly higher than in previous simulations to allow a more precise determination of the orientational transition temperature T * NI. The transition temperature has been placed at kT/ϵ = 1·1232 ± 0·0006, refining previous estimates. Orientational order parameters <P 2>, <P 4> have been calculated and a new algorithm is proposed for the computation of <P 4>. Particular attention has been devoted to pretransitional properties. Pair correlations G 2(r), G 4(r) as well as the second rank g 2-factor are reported. g 2 is found to diverge at a temperature T * NI* = kT/ϵ = 1·1201 ± 0·0006 and to fit the Landau-de Gennes behaviour except near the transition. Our results indicate therefore that the Lebwohl-Lasher model can yield the small difference between the nematic-isotropic transition temperature and the isotropic phase limiting instability temperature without the need of introducing additional terms in the potential.