Nonuniform liquid-crystalline phases of parallel hard rod-shaped particles: From ellipsoids to cylinders

In this article we consider systems of parallel hard superellipsoids, which can be viewed as a possible interpolation between ellipsoids of revolution and cylinders. Superellipsoids are characterized by an aspect ratio and an exponent alpha (shape parameter) which takes care of the geometry, with alpha=1 corresponding to ellipsoids of revolution, while alpha=infinity is the limit of cylinders. It is well known that, while hard parallel cylinders exhibit nematic, smectic, and solid phases, hard parallel ellipsoids do not stabilize the smectic phase, the nematic phase transforming directly into a solid as density is increased. We use computer simulation to find evidence that for alpha>or=alpha(c), where alpha(c) is a critical value which the simulations estimate to be approximately 1.2-1.3, the smectic phase is stabilized. This is surprisingly close to the ellipsoidal case. In addition, we use a density-functional approach, based on the Parsons-Lee approximation, to describe smectic and columnar orderings. In combination with a free-volume theory for the crystalline phase, a theoretical phase diagram is predicted. While some qualitative features, such as the enhancement of smectic stability for increasing alpha and the probable absence of a stable columnar phase, are correct, the precise location of coexistence densities is quantitatively incorrect.     

Landau theory for uniaxial nematic,biaxial nematic,uniaxial smectic-A,and biaxial smectic-A phases

We use the Landau theory of phase transitions to obtain the global phase diagram concerning the uniaxial nematic, biaxial nematic, uniaxial smectic-A and biaxial smectic-A phases. The transition between the biaxial nematic and biaxial smectic is continuous as well as the transition between the nematic phases and the transition between the smectic phases. The transition from uniaxial nematic and uniaxial smectic is continuous with a tricritical point. The tricritical point may be absent and the entire transition becomes continuous. The four phases meet at a tetracritical point.     

Mesophase formation in a system of top-shaped hard molecules: density functional theory and Monte Carlo simulation

We present the phase diagram of a system of mesogenic top-shaped molecules based on the Parsons-Lee density functional theory and Monte Carlo simulation. The molecules are modeled as a hard spherocylinder with a hard sphere embedded in its center. The stability of five different phases is studied, namely, isotropic, nematic, smectic A, smectic C, and columnar phases. The positionally ordered phases are investigated only for the case of parallel alignment. It is found that the central spherical unit destabilizes the nematic with respect to the isotropic phase, while increasing the length of the cylinder has the opposite effect. Also, the central hard sphere has a strong destabilizing effect on the smectic A phase, due the inefficient packing of the molecules into layers. For large hard sphere units the smectic A phase is completely replaced by a smectic C structure. The columnar phase is first stabilized with increasing diameter of the central unit, but for very large hard sphere units it becomes less stable again. The density functional results are in good agreement with the simulations.     

Stability of smectic phases in hard-rod mixtures

Using density-functional theory, we have analyzed the phase behavior of binary mixtures of hard rods of different lengths and diameters. Previous studies have shown a strong tendency of smectic phases of these mixtures to segregate and, in some circumstances, to form microsegregated phases. Our focus in the present work is on the formation of columnar phases which some studies, under some approximations, have shown to become thermodynamically stable prior to crystallization. Specifically we focus on the relative stability between smectic and columnar phases, a question not fully addressed in previous work. Our analysis is based on two complementary perspectives: on the one hand, an extended Onsager theory, which includes the full orientational degrees of freedom but with spatial and orientational correlations being treated in an approximate manner; on the other hand, we formulate a Zwanzig approximation of fundamental-measure theory on hard parallelepipeds, whereby orientations are restricted to be only along three mutually orthogonal axes, but correlations are faithfully represented. In the latter case novel, complete phase diagrams containing regions of stability of liquid-crystalline phases are calculated. Our findings indicate that the restricted-orientation approximation enhances the stability of columnar phases so as to preempt smectic order completely while, in the framework of the extended Onsager model, with full orientational degrees of freedom taken into account, columnar phases may preempt a large region of smectic stability in some mixtures, but some smectic order still persists.     

Phase diagram of the uniaxial and biaxial soft-core Gay-Berne model

Classical molecular dynamics simulations have been used to explore the phase diagrams for a family of attractive-repulsive soft-core Gay-Berne models [R. Berardi, C. Zannoni, J. S. Lintuvuori, and M. R. Wilson, J. Chem. Phys. 131, 174107 (2009)] and determine the effect of particle softness, i.e., of a moderately repulsive short-range interaction, on the order parameters and phase behaviour of model systems of uniaxial and biaxial ellipsoidal particles. We have found that isotropic, uniaxial, and biaxial nematic and smectic phases are obtained for the model. Extensive calculations of the nematic region of the phase diagram show that endowing mesogenic particles with such soft repulsive interactions affect the stability range of the nematic phases, and in the case of phase biaxiality it also shifts it to lower temperatures. For colloidal particles, stabilised by surface functionalisation, (e.g., with polymer chains), we suggest that it should be possible to tune liquid crystal behaviour to increase the range of stability of uniaxial and biaxial phases (by varying solvent quality). We calculate second virial coefficients and show that they are a useful means of characterising the change in effective softness for such systems. For thermotropic liquid crystals, the introduction of softness in the interactions between mesogens with overall biaxial shape (e.g., through appropriate conformational flexibility) could provide a pathway for the actual chemical synthesis of stable room-temperature biaxial nematics.     

Solidification-induced and shear-induced band texture in thermotropic liquid crystalline polymer films

Two phase diagrams of a six-ring double-swallow-tailed compound are presented where the mixing components involve electron-acceptor compounds. In the mixed phase region, nematic, smectic A and C, cubic and columnar phases are induced. In this way transitions between lamellar, cubic and columnar phases can be realized by variation of the con- centration. The mesophases occurring in these systems have been characterized by X-ray investigations.     

Thermotropic uniaxial and biaxial nematic and smectic phases in bent-core mesogens

Two azo substituted achiral bent-core mesogens have been synthesized. Optical polarizing microscopy and synchrotron X-ray scattering studies of both compounds reveal the existence of the thermotropic uniaxial and biaxial nematic and three smectic phases at different temperatures in these single component small molecule systems. The transition from the uniaxial to biaxial nematic phase is confirmed to be second order. The transitions from the biaxial nematic to the underlying smectic phase and between the smectic phases have barely discernible heat capacity signatures and thus are also second order.     

High pressure study on induced SA phases in binary liquid crystal mixtures

Binary mixtures of terminal polar and non-polar liquid crystals exhibiting induced smectic phases are studied under high pressure. For terminal polar compounds with smectic phases, there are two types of T, x phase diagrams known up to now. Diagrams with a nematic gap between the induced phase and the smectic phase of the terminal polar compound and diagrams with an uninterrupted miscibility of the smectic phases. We find a continuous transformation between these phase diagrams with pressure. At a certain pressure, the phase transition lines form a cross separating two nematic and two smectic phases.     

Induced smectic phases in phase diagrams of binary nematic liquid crystal mixtures

To elucidate induced smectic A and smectic B phases in binary nematic liquid crystal mixtures, a generalized thermodynamic model has been developed in the framework of a combined Flory-Huggins free energy for isotropic mixing, Maier-Saupe free energy for orientational ordering, McMillan free energy for smectic ordering, Chandrasekhar-Clark free energy for hexagonal ordering, and phase field free energy for crystal solidification. Although nematic constituents have no smectic phase, the complexation between these constituent liquid crystal molecules in their mixture resulted in a more stable ordered phase such as smectic A or B phases. Various phase transitions of crystal-smectic, smectic-nematic, and nematic-isotropic phases have been determined by minimizing the above combined free energies with respect to each order parameter of these mesophases. By changing the strengths of anisotropic interaction and hexagonal interaction parameters, the present model captures the induced smectic A or smectic B phases of the binary nematic mixtures. Of particular importance is the fact that the calculated phase diagrams show remarkable agreement with the experimental phase diagrams of binary nematic liquid crystal mixtures involving induced smectic A or induced smectic B phase.     

Liquid crystalline materials with complex mesophase morphologies

Liquid crystals, which combine order and mobility on a molecular and supramolecular level are increasingly accepted as a fourth state of matter. Besides the well-established nematic, smectic and columnar mesophases, more complex mesophase morphologies attracted increasing interest during the recent years. These are bicontinuous and discontinuous cubic mesophases and other two- and three-dimensionally ordered intermediate phases, superstructures induced by molecular chirality or by polar order of bent core molecules, novel biaxial smectic phases, and novel mesophase morphologies of polyphilic block molecules and dendrimers.     

Density functional theory of freezing of a system of conjugated oligomers parameterised via Gay–Berne potential

Density functional theory (DFT) of freezing is used to study the isotropic–nematic, isotropic–smectic A and nematic–smectic A phase transitions in a system of large, semi-flexible conjugated oligomers parameterised within Gay–Berne (GB) potential. The pair correlation functions of the isotropic fluid, used as structural inputs in the DFT, are calculated by solving the Percus–Yevick integral equation theory. Large number of spherical harmonic coefficients of each orientation-dependent functions has been considered to ensure the numerical accuracy at different densities and temperatures for the system of these model GB ellipsoids having large aspect ratio (length-to-breadth ratio). We found that the system of GB ellipsoids parameterised for conjugated oligomers shows stable isotropic, nematic and smectic A phases. At low temperatures, on increasing the density, isotropic fluid makes a direct transition to smectic A phase. Nematic phase get stabilised in between the isotropic and smectic A phases on increasing the temperature. Using the transition parameter obtained through the DFT, we have plotted the temperature–density and pressure–temperature phase diagrams which are found to be qualitatively similar to the one obtained in simulations for the systems with low aspect ratio GB particles.     

Biaxial phases in mineral liquid crystals

A review is given of liquid crystals formed in colloidal dispersions, in particular those consisting of mineral particles. Starting with the historical development and early theory, the characteristic properties related to the colloidal nature of this type of liquid crystals are discussed. The possibility to find biaxial nematic and smectic phases is described for mixtures of rods and plates and recent examples are given of biaxial liquid crystal phases of mineral particles with inherent biaxial shape.     

Liquid‐Crystalline Properties of trans‐A2B2‐Porphyrins with Extended π‐Electron Systems

Incorporation of four trialkoxyphenyl substituents combined with extending the π‐conjugated system has allowed porphyrins to display liquid‐crystalline columnar phases at room temperature. 2D and 3D columnar structures were observed as well as a biaxial smectic phase.     

Smectic, nematic, and isotropic phases in binary mixtures of thin and thick hard spherocylinders

A second-virial Onsager theory, based on Parsons-Lee rescaling and suitably extended to deal with multicomponent systems and smectic phases, has been used to calculate the phase diagram of a collection of binary mixtures of thin and thick hard spherocylinders. In particular, two types of phase diagrams are investigated. First, a number of binary mixtures where the two components have the same total length have been considered; in addition, the phase diagram of a binary mixture where the two components have the same volume has been calculated. For the particles of one of the two components, the length of the cylindrical part and the diameter have always been set equal to 5 and 1, respectively. Spherocylinders of the same total length and different diameter tend to demix considerably as soon as the diameter ratio deviates from unity. This happens especially at high pressures, when at least the phase richer in the thicker component is smectic. In the case where the two components have equal volumes, demixing is further increased due to the disparity not only in particle diameter but also in particle lengths. The incorporation of inhomogeneous layered phases is seen to alter significantly the phase diagrams calculated if only homogeneous phases are allowed, since transitions to a smectic phase often preempt those to a nematic or an isotropic phase. The apparent versatility of the recent experimental techniques suggests that the phase diagram features predicted by the theory might be also observed in real systems.     

Effect of particle geometry on phase transitions in two-dimensional liquid crystals

Using a version of density-functional theory which combines Onsager approximation and fundamental-measure theory for spatially nonuniform phases, we have studied the phase diagram of freely rotating hard rectangles and hard discorectangles. We find profound differences in the phase behavior of these models, which can be attributed to their different packing properties. Interestingly, bimodal orientational distribution functions are found in the nematic phase of hard rectangles, which cause a certain degree of biaxial order, albeit metastable with respect to spatially ordered phases. This feature is absent in discorectangles, which always show unimodal behavior. This result may be relevant in the light of recent experimental results which have confirmed the existence of biaxial phases. We expect that some perturbation of the particle shapes (either a certain degree of polydispersity or even bimodal dispersity in the aspect ratios) may actually destabilize spatially ordered phases thereby stabilizing the biaxial phase.     

Structure formation in doped discotic polymers and low molar mass model systems

Doping of low molar mass materials or polymers, possessing disc-like units, with electron acceptors leads to the stabilization of columnar discotic phases or even to the induction of such phases in compounds which either display a nematic discotic phase or only an amorphous phase in the absence of the electron acceptor. The induced columnar phase corresponds frequently to a hexagonally ordered one. We have observed, however, in addition the induction of new columnar phases such as the rectangularly ordered (D^ro) and the columnar nematic phase (N^c). The enhancement of the tendency towards the formation of columnar phases is a consequence of electron acceptor—electron donor complex formation. Using a model proposed by de Jeu to describe the induction of smectic phases by complex formation we are able to account qualitatively for the experimental findings.     

Phase diagrams of binary mixtures of biaxial nematogens

A mean field theory is used to describe nematic phases of binary mixtures of biaxial molecules. Using a general pseudopotential consistent with the D_2h symmetry of the constituent particles, the theory is used to calculate the elements of the order tensors necessary to describe the orientational order in binary mixtures in both uniaxial and biaxial nematic phases. For a single component, the model only requires one parameter, r₂, a ratio of anisotropic interaction strengths, to predict the temperature dependence of the four order parameters. The temperature dependence of the orientational distribution functions is illustrated for both rod-like and plate-like molecules. For binary mixtures, three anisotropic interaction strengths, r₁, r₂, and r₃, are needed to calculate the order parameters of both components as a function of concentration and temperature. The free energy is evaluated to predict the phase stability of the mixture. By systematically varying the anisotropic interaction strengths, temperature-concentration phase diagrams for a variety of molecular shapes are presented. The theoretical predictions suggest that binary mixtures of molecules with highly asymmetric shapes will display stable biaxial nematic phases.     

Smectic ordering in athermal systems of rodlike triblock copolymers

The phase behavior of the system of parallel rigid triblock copolymers is examined using the second virial density functional theory. The triblock particle consists of two identical infinitely thin hard rods of finite lengths on the opposite ends of one central hard cylinder with nonzero length and diameter. Stability analyses and free energy calculations show that the system of parallel particles can form not only uniform nematic and smectic A phases but also a smectic C phase. The stability and structure of the tilted structure are controlled by only the diameter and the length of the central cylinder segment. Interestingly, the diameter affects only the layer tilting and the periodicity, but not the packing fraction of the nematic to smectic-C transition. For all values of cylinder length the usual smectic A and smectic C transitions compete with each other and no nematic-columnar transition is observed. At low and high cylinder lengths the smectic A phase is stabilized first, while the smectic C is the most stable for intermediate length values.     

Enantiotropic nematics from cross-like 1,2,4,5-tetrakis(4'-alkyl-4-ethynylbiphenyl)benzenes and their biaxiality studies

The theoretically predicted optimum length/breadth/width ratio for maximizing shape biaxiality was investigated experimentally by the facile and successful synthesis of cross-shaped compound 3, which showed enantiomeric nematic phase behavior. This cross-like core structure could alternatively be viewed as two fused V-shaped mesogens, which have recently immerged as a new direction in biaxial nematic research, at the bending tips that can act as a new structure for biaxial investigations. Whilst the thermal analysis data of compound 3 did not meet the expected theoretical values for biaxial nematics, surface-induced biaxiality was evidenced by optical studies. Cluster-size analysis within the nematic phase of compound 3 revealed the formation of meta-cybotactic nematics, which approached the cluster sizes of cybotactic nematics. The split small-angle 2D X-ray diffraction patterns of magnetic-field-aligned samples indicated that the nematic phase was composed of small smectic?C-like clusters with the tilting of molecules within the clusters. The wide-temperature-range enantiomeric nematic phase of cross-like compound 3 enabled the molecular skeleton to serve as an alternative skeleton to bent-rod mesogens, which exhibited nematic phases with the potential competition of transitions to higher-order liquid-crystalline phases and crystallization, for future biaxial investigations.