Fluctuating Interfaces in Liquid Crystals

Summary: We review and compare recent work on the properties of fluctuating interfaces between isotropic and nematic liquid-crystalline phases. Molecular dynamics and Monte Carlo simulations have been carried out for systems of ellipsoids and hard rods with aspect ratio 15:1, and the fluctuation spectrum of interface positions (the capillary wave spectrum) has been analyzed. In addition, the capillary wave spectrum has been calculated analytically within the Landau-de Gennes theory. The theory predicts that the interfacial fluctuations can be described in terms of a wave vector dependent interfacial tension, which is anisotropic at small wavelengths (stiff director regime) and becomes isotropic at large wavelengths (flexible director regime). After determining the elastic constants in the nematic phase, theory and simulation can be compared quantitatively. We obtain good agreement for the stiff director regime. The crossover to the flexible director regime is expected at wavelengths of the order of several thousand particle diameters, which was not accessible to our simulations.     

Bifurcational analysis of the isotropic-nematic phase transition of rigid rod polymers subjected to biaxial stretching flow

A bifurcational analysis is performed on Doi's equation of nematodynamics that describes the non-equlibrium isotropic-nematic phase transition of rigid rod polymers in the presence of steady biaxial stretching flow. The symmetry of the flow and of the governing order parameter equations are shown to be the source of a rich bifurcation, symmetry breaking, and multistability behavior involving two physically equivalent biaxial nematic phases, one uniaxial nematic phase and one uniaxial paranematic phase. According to the relative intensity of the nematic ordering field and stretching rate, the uniaxial isotropic-biaxial nematic transition may be continuous (2nd order), discontinuous (1st order), or it may exhibit a tricritical non-equilibrium phase transition point. The solutions to the Doi equations of nematodynamics are found to be consistent with those of Khokhlov and Semenov [Macromolecules 15 , 1272 (1982)], which are based on a version of the Onsager theory of isotropic-nematic phase transitions. The present simulations provide a useful guide for orientation control in biaxial stretching flows.     

Critical point in a random side-chain nematic copolymer: Mean-field theory

A model for anisotropic phases in a random side-chain nematic copolymer is proposed and solved in the mean-field approximation. The main results of the calculation are the appearance of paranematic and nematic phases and a critical point, without the application of external electric or magnetic fields.     

Interactions between spherical colloids mediated by a liquid crystal: a molecular simulation and mesoscale study

Monte Carlo simulations and dynamic field theory (DyFT) are used to study the interactions between dilute spherical particles, dispersed in nematic and isotropic phases of a liquid crystal. A recently developed simulation method (expanded ensemble density of states) was used to determine the potential of mean force (PMF) between the two spheres as a function of their separation and size. The PMF was also calculated by a dynamic field theory that describes the evolution of the local tensor order parameter. Both methods reveal an overall attraction between the colloids in the nematic phase; in the isotropic phase, the overall attraction between the colloids is much weaker, whereas the repulsion at short range is stronger. In addition, both methods predict a new topology of the disclination lines, which arises when the particles approach each other. The theory is found to describe the results of simulations remarkably well, down to length scales comparable to the size of the molecules. At separations corresponding to the width of individual molecular layers on the particles' surface, the two methods yield different defect structures. We attribute this difference to the neglect of density inhomogeneities in the DyFT. We also investigate the effects of the size of spherical colloids on their interactions.     

Critical point in a random side-chain nematic copolymer: Mean-field theory

Abstract

A model for anisotropic phases in a random side-chain nematic copolymer is proposed and solved in the mean-field approximation. The main results of the calculation are the appearance of paranematic and nematic phases and a critical point, without the application of external electric or magnetic fields.     

Mesoscopic simulations of the diffusivity of ethane in beds of NaX zeolite crystals: comparison with pulsed field gradient NMR measurements

Mesoscopic kinetic Monte Carlo simulations and pulsed field gradient nuclear magnetic resonance (PFG NMR) measurements are compared in order to investigate the transport of ethane in a bed of NaX crystals. A novel molecular mechanics particle-based reconstruction method is employed for the digital representation of the bed, enabling for the first time a parallel study of the real system and of a computer model tailored to reproduce the void fraction, particle shape and average size of the real system. Simulation of the long-range diffusion of ethane in the bed over the Knudsen, transient, and molecular diffusion regimes is consistent with the PFG NMR measurements in yielding tortuosity factors which depend upon the regime of diffusion; more specifically, tortuosity factors defined in the conventional way are higher in the Knudsen than in the molecular diffusion regime. Detailed statistical analysis of the computed molecular trajectories reveals that this difference arises in a nonexponential distribution of the lengths and in a correlation between the directions of path segments traversed between collisions with the solid in the Knudsen regime. When the Knudsen tortuosity is corrected to account for these features, a single, regime-independent value is obtained within the error of the calculations.     

Coil to rod transition of a single polyelectrolyte chain

Monte Carlo computer simulations are used to investigate the coil to rod transition of a single polyelectrolyte in the Coulombic unscreened regime. The results are compared with the predictions of the mean field theory (MFT) of dilute solutions.     

Bifurcational analysis of the isotropic-discotic nematic phase transition in the presence of extensional flow

A bifurcational analysis is performed on a version of Doi's equation of nematodynamics that describes the non-equilibrium isotropic-discotic nematic phase transition in the presence of steady uniaxial extensional flow. The disc-like molecular geometry and the degenerate extensional flow-induced orientation are shown to be the source of a complex bifurcation and multistability behaviour involving two physically equivalent biaxial nematic phases, one uniaxial nematic phase and one uniaxial paranematic phase. Depending on the temperature and the extension rate, the isotropic-discotic nematic transition, involving the two biaxial nematic phases and the uniaxial paranematic phase, may be continuous (2nd order), discontinuous (1st order), or it may exhibit a tricritical non-equilibrium phase transition point. A validation procedure on the validity of the predictions is implemented. The predictions presented here find practical applications in the industrial spinning of mesophase carbon fibres, and also provide new results that increase the present fundamental understanding of the rheology of discotic nematic liquid crystals.     

Nanospheres in a nematic liquid crystal solvent: the influence of particle size

We use Monte Carlo simulations to investigate a simple lattice model for nematic liquid crystals containing nanospheres. The influence of particle size on the phase behaviour is studied using two different sized particles. The phase diagram is found to be topologically equivalent for both particle sizes, with a large biphasic region corresponding to coexistence between a rod-rich nematic and a rod-poor isotropic phase. For small spheres, the rod-rich nematic phase is stable for relatively large volume fractions of spheres (up to a maximum of about 16%). In contrast, the nematic phase for the system with larger spheres is constrained to a much narrower region of the phase diagram.     

Ordering transition and demixing of a binary mixture of thick and thin rodlike molecules in the presence of an external field

The effect of an external field (electric/magnetic) on the phase behavior of the binary mixture of very long thick and thin rodlike particles is studied. Both the thick and thin particles possess positive but different susceptibility anisotropics (Delta alpha). The difference in the extent of interaction between the external field and the two species is varied by means of a coupling parameter (l = Delta alpha(thick)/Delta alpha(thin)). Isotropic-nematic phase transition and demixing phase transitions taking place both in the isotropic and nematic phases are examined as a function of field strength on the level of the second virial theory of Onsager in the range of 0 < l <1. The approximate sixth order Legendre polynomial expansion method is used to represent the excluded volume interaction between the rodlike particles. It is found that the isotropic phase becomes weakly nematic (paranematic) in the presence of external field and the field orients both components in the direction of the field even if the field does not have direct interaction with the thick component (l = 0). Analytical expressions are derived for the external field induced order parameters and birefringence. The increasing field destabilizes both types of demixing transitions (isotropic-isotropic and nematic-nematic) and the paranematic-nematic phase transition. Moreover it induces closed loop immiscibility, and upper and lower critical points terminating the paranematic-nematic phase coexistence may occur for low values of the coupling parameter. It is interesting that while the phase boundaries of the paranematic-paranematic demixing and the paranematic-nematic transitions are very sensitive to the value of the coupling parameter at low pressures, the paranematic-nematic and nematic-nematic phase boundaries are practically independent of the coupling parameter at high pressures.     

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.     

Computer simulation studies of anisotropic systems. XXII. An equimolar mixture of rods and discs: A biaxial nematic?

In principle, binary mixtures of rod-like and disc-like particles should exhibit a biaxial nematic phase, but in practice phase separation into two uniaxial nematic phases prevents this. Here, we report the results of a computer simulation study of an equimolar mixture of rods and discs in which phase separation is not allowed. The particles are confined to the sites of a simple cubic lattice in which each rod is surrounded by six discs and vice versa. Neighbouring particles interact such that they prefer to align with their respective symmetry axes orthogonal to each other. In contrast, the interaction between next nearest neighbours, which are either rods or discs, is such that their symmetry axes tend to be parallel. Monte Carlo simulations of this model mixture show that an orientationally ordered phase exists at low temperatures. This nematic phase has overall uniaxial symmetry and the particles have a negative second rank orientational order parameter, indicating that they tend to align at right angles to the director. The two interpenetrating sub-lattices containing either rods or discs, however, exhibit a biaxial nematic phase. The results of the simulation are found to be in reasonable agreement with the predictions of a molecular field theory for this model mixture. We have also investigated the behaviour of this mixture when the rods and discs are allowed to exchange between their lattice sites. The mixture is found to separate into two uniaxial nematic phases composed essentially of either rods or discs, as expected.     

Isotropic-nematic interfacial tension of hard and soft rods: application of advanced grand canonical biased-sampling techniques

Coexistence between the isotropic and the nematic phase in suspensions of rods is studied using grand canonical Monte Carlo simulations with a bias on the nematic order parameter. The biasing scheme makes it possible to estimate the interfacial tension gamma(IN) in systems of hard and soft rods. For hard rods with LD=15, we obtain gammaIN approximately 1.4kBT/L2, with L the rod length, D the rod diameter, T the temperature, and kB the Boltzmann constant. This estimate is in good agreement with theoretical predictions, and the order of magnitude is consistent with experiments.     

Computer simulation studies of anisotropic systems. The influence of external fields on the nematic—isotropic transition

The results of Monte Carlo simulations are reported for a model nematogen in the presence of an intense magnetic field. The predictions of a molecular field theory for the system are found to be in reasonable accord with the simulations. It would appear therefore that the critical field for a real nematogen may be unobservably high.     

Orientational ordering in hard rectangles: The role of three-body correlations

We investigate the effect of three-body correlations on the phase behavior of hard rectangle two-dimensional fluids. The third virial coefficient B3 is incorporated via an equation of state that recovers scaled particle theory for parallel hard rectangles. This coefficient, a functional of the orientational distribution function, is calculated by Monte Carlo integration, using an accurate parametrized distribution function, for various particle aspect ratios in the range of 1-25. A bifurcation analysis of the free energy calculated from the obtained equation of state is applied to find the isotropic (I)-uniaxial nematic (N(u)) and isotropic-tetratic nematic (N(t)) spinodals and to study the order of these phase transitions. We find that the relative stability of the N(t) phase with respect to the isotropic phase is enhanced by the introduction of B3. Finally, we have calculated the complete phase diagram using a variational procedure and compared the results with those obtained from scaled particle theory and with Monte Carlo simulations carried out for hard rectangles with various aspect ratios. The predictions of our proposed equation of state as regards the transition densities between the isotropic and orientationally ordered phases for small aspect ratios are in fair agreement with simulations. Also, the critical aspect ratio below which the N(t) phase becomes stable is predicted to increase due to three-body correlations, although the corresponding value is underestimated with respect to simulation.     

Monte Carlo simulation of the self-assembly and phase behavior of semiflexible equilibrium polymers

Grand canonical Monte Carlo simulations of a simple model semiflexible equilibrium polymer system, consisting of hard sphere monomers reversibly self-assembling into chains of arbitrary length, have been performed using a novel sampling method to add or remove multiple monomers during a single MC move. Systems with two different persistence lengths and a range of bond association constants have been studied. We find first-order lyotropic phase transitions between isotropic and nematic phases near the concentrations predicted by a statistical thermodynamic theory, but with significantly narrower coexistence regions. A possible contribution to the discrepancy between theory and simulation is that the length distribution of chains in the nematic phase is bi-exponential, differing from the simple exponential distribution found in the isotropic phase and predicted from a mean-field treatment of the nematic. The additional short length-scale characterizing the distribution appears to arise from the lower orientational order of short chains. The dependence of this length-scale on chemical potential, bond association constant, and total monomer concentration has been examined.     

Phase ordering of hard needles on a quasicrystalline substrate

Quasicrystals possess long-range positional and orientational order. However, they cannot be periodic in space due to their non-crystallographic symmetries such as a 10-fold rotational axis. We perform Monte Carlo simulations of two-dimensional hard-needle systems subject to a quasiperiodic substrate potential. We determine phase diagrams as a function of density and potential strength for two needle lengths. With increasing potential strength short needles tend to form isolated clusters that display directional order along the decagonal directions. Long needles create interacting clusters that stabilize the nematic phase. At large potential strengths the clusters position themselves on two interwoven Fibonacci sequences perpendicular to the cluster orientation. Alternatively, one obtains extended domains of needle clusters which are aligned along all decagonal symmetry directions.     

3D Monte Carlo simulations of a magnetic disk-like particle dispersion

We have investigated the aggregate structures of a colloidal dispersion composed of ferromagnetic disk-like particles with a magnetic moment normal to the particle axis at the particle center, by means of 3D Monte Carlo simulations. Such disk-like particles have been modeled as a circular disk-like particle with the side section shape of spherocylinder. We have attempted to clarify the influences of the magnetic field strength, magnetic interactions between particles and volumetric fraction of particles. In order to discuss quantitatively the aggregate structures of clusters, we have focused on the radial distribution and orientational pair correlation functions, etc. For no applied magnetic field cases, long column-like clusters come to be formed with increasing magnetic particle–particle interactions. The internal structures of these clusters clearly show that the particles incline in a certain direction and their magnetic moments alternate in direction between the neighboring particles in the clusters. For applied magnetic field cases, the magnetic moment of each particle inclines in the magnetic field direction and therefore the column-like clusters are not formed straightforwardly. If the magnetic field is much stronger than magnetic particle–particle interactions, the particles do not have a tendency to form the clusters. As the influence of magnetic particle–particle interactions is significantly strong, thick chain-like clusters or column-like clusters or brick-wall-like clusters come to be formed along the magnetic field direction.     

Thermodynamic properties of small rare gas clusters by path integral Monte Carlo simulations: a preliminary study

A strategy for reducing the risk of non-ergodic simulations in Monte Carlo calculations of the thermodynamic properties of clusters is discussed with the support of some examples. The results obtained attest the significance of the approach for the low-temperature regime, as non-ergodic sampling of potential energy surfaces is a particularly insidious occurrence. Fourier path integral Monte Carlo techniques for taking into account quantum effects are adopted, in conjunction with suitable tricks for improving the procedure reliability. Applications are restricted to Lennard-Jones clusters of rare-gas systems.     

An analytical approximation for the orientation-dependent excluded volume of tangent hard sphere chains of arbitrary chain length and flexibility

Onsager-like theories are commonly used to describe the phase behavior of nematic (only orientationally ordered) liquid crystals. A key ingredient in such theories is the orientation-dependent excluded volume of two molecules. Although for hard convex molecular models this is generally known in analytical form, for more realistic molecular models that incorporate intramolecular flexibility, one has to rely on approximations or on computationally expensive Monte Carlo techniques. In this work, we provide a general correlation for the excluded volume of tangent hard-sphere chains of arbitrary chain length and flexibility. The flexibility is introduced by means of the rod-coil model. The resulting correlation is of simple analytical form and accurately covers a wide range of pure component excluded volume data obtained from Monte Carlo simulations of two-chain molecules. The extension to mixtures follows naturally by applying simple combining rules for the parameters involved. The results for mixtures are also in good agreement with data from Monte Carlo simulations. We have expressed the excluded volume as a second order power series in sin?(γ), where γ is the angle between the molecular axes. Such a representation is appealing since the solution of the Onsager Helmholtz energy functional usually involves an expansion of the excluded volume in Legendre coefficients. Both for pure components and mixtures, the correlation reduces to an exact expression in the limit of completely linear chains. The expression for mixtures, as derived in this work, is thereby an exact extension of the pure component result of Williamson and Jackson [Mol. Phys. 86, 819-836 (1995)].