Bifurcation analysis and computer simulation of biaxial liquid crystals

Mean-field model of boomerang nematic liquid crystals with diminished coupling of molecular uniaxial and biaxial susceptibilities

The mean-field theory approach has been applied to the boomerang type particles from P. I. C. Teixeira, A. Masters, and B. Mulder [Mol. Cryst. Liq. Cryst. 323, 167 (1998)MCLCE91058-725X10.1080/10587259808048440] but with diminished strength of the interaction coefficient responsible for the coupling between molecular uniaxial and biaxial susceptibilities. For the rodlike particles, when the apex boomerang angle is larger than 107.35^{∘}, the stable uniaxial rodlike phase occurs. For smaller angles, beyond the point where the transition is of the second order (the Landau point) and for diminished parameter of molecular biaxial-uniaxial coupling, a biaxial phase is observed with the transition undergoing directly from the isotropic phase. According to the order parameters the character of this transition is of the first order. Such behavior is in accordance with the Sonnet-Durand-Virga model of the biaxial phases. The change in the type of the phase transition order is also illustrated by the changes in the equations of state and the changes in second and third derivatives of the free energy. The possibilities to tailor interaction coefficients of real molecules to obtain such a phase transition scenario are discussed.

Complex free-energy landscapes in biaxial nematic liquid crystals and the role of repulsive interactions: A Wang-Landau study

General quadratic Hamiltonian models, describing the interaction between liquid-crystal molecules (typically with D_{2h} symmetry), take into account couplings between their uniaxial and biaxial tensors. While the attractive contributions arising from interactions between similar tensors of the participating molecules provide for eventual condensation of the respective orders at suitably low temperatures, the role of cross coupling between unlike tensors is not fully appreciated. Our recent study with an advanced Monte Carlo technique (entropic sampling) showed clearly the increasing relevance of this cross term in determining the phase diagram (contravening in some regions of model parameter space), the predictions of mean-field theory, and standard Monte Carlo simulation results. In this context, we investigated the phase diagrams and the nature of the phases therein on two trajectories in the parameter space: one is a line in the interior region of biaxial stability believed to be representative of the real systems, and the second is the extensively investigated parabolic path resulting from the London dispersion approximation. In both cases, we find the destabilizing effect of increased cross-coupling interactions, which invariably result in the formation of local biaxial organizations inhomogeneously distributed. This manifests as a small, but unmistakable, contribution of biaxial order in the uniaxial phase. The free-energy profiles computed in the present study as a function of the two dominant order parameters indicate complex landscapes. On the one hand, these profiles account for the unusual thermal behavior of the biaxial order parameter under significant destabilizing influence from the cross terms. On the other, they also allude to the possibility that in real systems, these complexities might indeed be inhibiting the formation of a low-temperature biaxial order itself-perhaps reflecting the difficulties in their ready realization in the laboratory.

Molecular field theory for biaxial nematics formed from liquid crystal dimers and inhibited by the twist-bend nematic

Liquid crystal dimers with odd spacers are good candidates as materials for biaxial nematic phases (N_B). The dimers are flexible molecules sustaining biaxial conformations, and couplings between the conformational and orientational distributions could be expected to stabilise N_B. We apply a molecular field theory for flexible molecules developed elsewhere to study a simple system made up of dimers composed of two cylindrically symmetric mesogenic groups. Our model allows for two idealised conformations: one linear and one bent at a tetrahedral angle. For a restricted set of chain lengths, the model predicts a first-order reentrant phase transition from the N_B phase into a low temperature uniaxial nematic phase (N_U). However the formation of the biaxial nematic could be blocked by the appearance of a twist-bent nematic.

Hierarchy of orientational phases and axial anisotropies in the gauge theoretical description of generalized nematic liquid crystals

The paradigm of spontaneous symmetry breaking encompasses the breaking of the rotational symmetries O(3) of isotropic space to a discrete subgroup, i.e., a three-dimensional point group. The subgroups form a rich hierarchy and allow for many different phases of matter with orientational order. Such spontaneous symmetry breaking occurs in nematic liquid crystals, and a highlight of such anisotropic liquids is the uniaxial and biaxial nematics. Generalizing the familiar uniaxial and biaxial nematics to phases characterized by an arbitrary point-group symmetry, referred to as generalized nematics, leads to a large hierarchy of phases and possible orientational phase transitions. We discuss how a particular class of nematic phase transitions related to axial point groups can be efficiently captured within a recently proposed gauge theoretical formulation of generalized nematics [K. Liu, J. Nissinen, R.-J. Slager, K. Wu, and J. Zaanen, Phys. Rev. X 6, 041025 (2016)2160-330810.1103/PhysRevX.6.041025]. These transitions can be introduced in the model by considering anisotropic couplings that do not break any additional symmetries. By and large this generalizes the well-known uniaxial-biaxial nematic phase transition to any arbitrary axial point group in three dimensions. We find in particular that the generalized axial transitions are distinguished by two types of phase diagrams with intermediate vestigial orientational phases and that the window of the vestigial phase is intimately related to the amount of symmetry of the defining point group due to inherently growing fluctuations of the order parameter. This might explain the stability of the observed uniaxial-biaxial phases as compared to the yet to be observed other possible forms of generalized nematic order with higher point-group symmetries.

Reexamination of the mean-field phase diagram of biaxial nematic liquid crystals: Insights from Monte Carlo studies

Investigations of the phase diagram of biaxial liquid-crystal systems through analyses of general Hamiltonian models within the simplifications of mean-field theory (MFT), as well as by computer simulations based on microscopic models, are directed toward an appreciation of the role of the underlying molecular-level interactions to facilitate its spontaneous condensation into a nematic phase with biaxial symmetry. Continuing experimental challenges in realizing such a system unambiguously, despite encouraging predictions from MFT, for example, are requiring more versatile simulational methodologies capable of providing insights into possible hindering barriers within the system, typically gleaned through its free-energy dependences on relevant observables as the system is driven through the transitions. The recent paper from this group [Kamala Latha et al., Phys. Rev. E 89, 050501(R) (2014)], summarizing the outcome of detailed Monte Carlo simulations carried out employing an entropic sampling technique, suggested a qualitative modification of the MFT phase diagram as the Hamiltonian is asymptotically driven toward the so-called partly repulsive regions. It was argued that the degree of (cross) coupling between the uniaxial and biaxial tensor components of neighboring molecules plays a crucial role in facilitating a ready condensation of the biaxial phase, suggesting that this could be a plausible factor in explaining the experimental difficulties. In this paper, we elaborate this point further, providing additional evidence from curious variations of free-energy profiles with respect to the relevant orientational order parameters, at different temperatures bracketing the phase transitions.

Detection of an intermediate biaxial phase in the phase diagram of biaxial liquid crystals: entropic sampling study

We investigate the phase sequence of biaxial liquid crystals, based on a general quadratic model Hamiltonian over the relevant parameter space, with a Monte Carlo simulation which constructs equilibrium ensembles of microstates, overcoming possible (free) energy barriers (combining entropic and frontier sampling techniques). The resulting phase diagram qualitatively differs from the universal phase diagram predicted earlier from mean-field theory (MFT), as well as the Monte Carlo simulations with the Metropolis algorithm. The direct isotropic-to-biaxial transition predicted by the MFT is replaced in certain regions of the space by the onset of an additional intermediate biaxial phase of very low order, leading to the sequence N(B)-N(B1)-I. This is due to inherent barriers to fluctuations of the components comprising the total energy, and may explain the difficulties in the experimental realization of these phases.

Calamitic and antinematic orientational order produced by the generalized Straley lattice model

We consider here a classical model, consisting of D_{2h}-symmetric particles in a three-dimensional simple-cubic lattice; the pair potential is isotropic in orientation space, and restricted to nearest neighbors. The simplest potential model is written in terms of the squares of the scalar products between unit vectors describing the three interacting arms of the molecules, as proposed in previous literature. Two predominant antinematic couplings of equal strength (+1) are perturbed by a comparatively weaker calamitic one, parameterized by a coupling constant -z ranging in [-1,0]. This choice rules out thermodynamically stable phases endowed with macroscopic biaxiality. The antinematic terms favor states with the corresponding molecular axes mutually orthogonal. Although the low-temperature phase of the special case with null calamitic term (PP0) is uniaxial and antinematically ordered, in the general case presented here both Monte Carlo and molecular-field approaches show that, for z close to zero, the models exhibit a low-temperature uniaxial nematic phase, followed by an antinematic one, and finally by the orientationally disordered one. On the other hand, for sufficiently large values of z, we only find evidence of uniaxial calamitic behavior, as expected by following the limiting cases.

Antinematic orientational order produced by an extreme case of the generalized Straley lattice model

We address here a special, extreme case of the quadratic pair interaction potential between classical, D(2h)-symmetric particles (the generalized Straley model) on a three-dimensional simple cubic lattice. The model involves predominant antinematic couplings and it has been studied by Monte Carlo simulation and a molecular field treatment. The obtained results show a second-order transition between the isotropic phase and the low-temperature one, exhibiting uniaxial antinematic order.

Depletion-induced biaxial nematic states of boardlike particles

With the aim of investigating the stability conditions of biaxial nematic liquid crystals, we study the effect of adding a non-adsorbing ideal depletant on the phase behavior of colloidal hard boardlike particles. We take into account the presence of the depletant by introducing an effective depletion attraction between a pair of boardlike particles. At fixed depletant fugacity, the stable liquid-crystal phase is determined through a mean-field theory with restricted orientations. Interestingly, we predict that for slightly elongated boardlike particles a critical depletant density exists, where the system undergoes a direct transition from an isotropic liquid to a biaxial nematic phase. As a consequence, by tuning the depletant density, an easy experimental control parameter, one can stabilize states of high biaxial nematic order even when these states are unstable for pure systems of boardlike particles.

Orientationally ordered phase produced by fully antinematic interactions: a simulation study

We consider here a classical model, consisting of D(2h) symmetric particles, whose centers of mass are associated with a three-dimensional simple-cubic lattice; the pair potential is isotropic in orientation space, and restricted to nearest neighbors. Two orthonormal triads define orientations of a pair of interacting particles; the simplest potential models proposed in the literature can be written as a linear combination involving the squares of the scalar products between corresponding unit vectors only, thus depending on three parameters, and making the interaction model rather versatile. A coupling constant with negative sign tends to keep the two interacting unit vectors parallel to each other, whereas a positive sign tends to keep them mutually orthogonal (antinematic coupling). We address here a special, extreme case of the above family, involving only antinematic couplings: more precisely, three antinematic terms whose coefficients are set to a common positive value (hence the name PPP model). The model under investigation produces a doubly degenerate pair ground state; the nearest-neighbor range of the interaction and the bipartite character of the lattice can propagate the pair ground state and increase the overall degeneracy, but without producing frustration. The model was investigated by a simplified molecular field treatment as well as by Monte Carlo simulation, whose results suggested a second-order transition to a low-temperature biaxially ordered phase; ground-state configurations producing orientational order have been selected by thermal fluctuations. The molecular field treatment also predicted a continuous transition, and was found to overestimate the transition temperature by a factor 2.

Mesogenic lattice models with partly antinematic interactions producing uniaxial nematic phases

The present paper considers nematogenic lattice models, involving particles of D_{2h} symmetry, whose centers of mass are associated with a three-dimensional simple cubic lattice; the pair potential is isotropic in orientation space and restricted to nearest neighbors. Let two orthonormal triads define orientations of a pair of interacting particles; the simplest potential models proposed in the literature can be reduced to a linear combination involving the squares of the scalar products between corresponding unit vectors only and depending on three parameters. By now, various sets of potential parameters have been proposed and studied in the literature, some of which capable of producing biaxial orientational order at sufficiently low temperature. On the other hand, in experimental terms, mesogenic biaxial molecules mostly produce uniaxial mesophases; thus we address here two very simple cases, involving a nematic (calamitic) term as well as one (model P0M) or two (model PPM) antinematic ones, whose coefficients are set equal in magnitude; when only one antinematic coefficient is used, the third one is set to zero. The calamitic term favors the alignment of two corresponding molecular axes, whereas antinematic terms or geometric constraints tend to keep two other pairs of axes mutually orthogonal. The models were investigated by molecular-field treatments and Monte Carlo simulation and found to predict a first- or second-order transitions between uniaxial nematic and isotropic phases; the molecular-field treatments yielded results in reasonable agreement with simulation.

Computer simulations of biaxial nematics

Biaxial nematic (N(b)) liquid crystals are a fascinating condensed matter phase that has baffled, for more than thirty years, scientists engaged in the challenge of demonstrating its actual existence, and which has only recently been experimentally found. During this period computer simulations of model N(b) have played an important role, both in providing the basic physical properties to be expected from these systems, and in giving clues about the molecular features essential for the thermodynamic stability of N(b) phases. However, simulation studies are expected to be even more crucial in the future for unravelling the structural features of biaxial mesogens at the molecular level, and for helping in the design and optimization of devices towards the technological deployment of N(b) materials. This review article gives an overview of the simulation work performed so far, and relying on the recent experimental findings, focuses on the still unanswered questions which will determine the future challenges in the field.

Biaxial and uniaxial phases produced by partly repulsive mesogenic models involving D2h molecular symmetries

The present paper considers biaxial nematogenic lattice models, involving particles of D2h symmetry, whose centers of mass are associated with a three-dimensional simple-cubic lattice. The pair potential is isotropic in orientation space and restricted to nearest neighbors. Let two orthonormal triads define orientations of a pair of interacting particles. The investigated potential models are quadratic with respect to the nine scalar products between the two sets of unit vectors. Actually, based on available geometric identities, these expressions can be reduced to diagonal form containing only the scalar products between corresponding unit vectors and depending on three parameters. Over the years, this comparatively simple functional form has also proven to be rather versatile. By now, various sets of potential parameters capable of producing mesogenic behavior of some kind have been proposed and studied in the literature. A new and simplified form was recently proposed and investigated by Sonnet, Virga, Durand, and De Matteis [A. M. Sonnet, E. G. Virga, and G. E. Durand, Phys. Rev. E 67, 061701 (2003); G. De Matteis and E. G. Virga, Phys. Rev. E 71, 061703 (2005)] and is known to support a biaxial phase at sufficiently low temperature. Following the idea of the above authors, we have studied a more extended range of parameters, including cases where biaxiality cannot be sustained in the pair ground state. In cases where a biaxial phase survives, an appropriate mean-field analysis may predict the existence of a direct second-order transition to the isotropic phase as well as a second-order sequence isotropic-to-uniaxial-to-biaxial. A second-order phase transition is also predicted, which involves isotropic and uniaxial phases only. Monte Carlo simulations have been carried out as well, for a few points in the parameter space, and found to produce results which partly confirm mean-field predictions.

Dominant biaxial quadrupolar contribution to the nematic potential of mean torque

Within the general quadrupolar model for biaxial nematic liquid crystals, whose potential of mean torque extends that in the Maier-Saupe theory with two extra interaction terms, we propose a quantitative criterion to identify the dominant biaxial interaction. We show that the ratio of the biaxial-to-uniaxial and uniaxial-to-isotropic transition temperatures is almost independent of one interaction parameter, thus indicating the other as dominant. We also show that there is a significant mismatch between the principal orientational order parameters predicted by the theory and those measured for the biaxial phase of a tetrapode.

High-resolution calorimetric study of a liquid crystalline organo-siloxane tetrapode with a biaxial nematic phase

High-resolution adiabatic scanning calorimetry and differential scanning calorimetry have been employed to study the thermal behavior of an organo-siloxane tetrapode reported to exhibit a biaxial nematic phase. No signature of the uniaxial to biaxial nematic phase transition could be retraced in sequential heating and cooling runs under different scanning rates, within the experimental resolution. The results obtained reveal that an extremely small heat should be involved in the uniaxial to biaxial nematic phase transition. The isotropic to uniaxial nematic transition at 318+/-0.01 K is very stable, and it is weakly first order with a rather small latent heat of 0.20+/-0.02 J/g .

Computer simulation study of a simple tetrahedratic mesogenic lattice model

Over the last 12 years, the possible existence of a tetrahedratic mesophase, involving a third-rank orientational order parameter and no positional order, has been addressed theoretically and predicted in some cases; no experimental realizations of a purely tetrahedratic phase are known at the time being, but various pieces of evidence suggest that interactions of tetrahedral symmetry do play a significant role in the macroscopic properties of mesophases resulting from banana-shaped (bent-core) mesogens. We address a very simple tetrahedratic mesogenic lattice model, involving continuous interactions; we consider particles possessing T(d) symmetry, whose centers of mass are associated with a three-dimensional simple-cubic lattice; the pair potential is taken to be isotropic in orientation space and restricted to nearest-neighboring sites; we let the two orthonormal triads {u alpha, alpha=1,2,3} and {v gamma, gamma=1,2,3} define the orientations of a pair of interacting particles; we let the unit vectors u alpha be combined to yield four unit vectors {e(j), j=1,2,3,4}, arranged in a tetrahedral fashion; we let the unit vectors v gamma be similarly combined to yield the four unit vectors {f(k), k=1,2,3,4}; and finally we let h(jk)=(e(j)f(k)). The interaction model studied here is defined by the simplest nontrivial (cubic) polynomial in the scalar products h(jk), consistent with the assumed symmetry and favoring orientational order; it is, so to speak, the tetrahedratic counterpart of the Lebwohl-Lasher model for uniaxial nematics. The model was investigated by molecular field (MF) theory and Monte Carlo simulations; MF theory predicts a low-temperature, tetrahedrically ordered phase, undergoing a second-order transition to the isotropic phase at higher temperature; on the other hand, available theoretical treatments point to the transition being driven first order by thermal fluctuations. Simulations showed evidence of a first-order transition.

Uniaxial rebound at the nematic biaxial transition

Over the last few years, renewed interest has been raised by the simplified general interaction models proposed by Straley for mesogenic molecules possessing the D{2h} symmetry and capable of producing biaxial nematic order. It has already been shown that, in the presence of certain special symmetries, just two out of the four order parameters that are in general necessary, suffice for the description of a biaxial phase. For some other range of parameters, these reducing symmetries do not hold, and, moreover, a mean-field treatment has to be suitably changed into a minimax strategy, still producing a transition to a low-temperature biaxial phase. Upon studying the general parameter range, we identify as a common feature the behavior of a uniaxial order parameter, attaining a local minimum at the biaxial-to-uniaxial transition temperature, and recognizably increasing away from it. This finding is confirmed by a Monte Carlo simulation.

Quadrupolar projection of excluded-volume interactions in biaxial nematic liquid crystals

We compute the quadrupolar approximation to the excluded-volume interaction between hard spherocuboids, which applies to both platelets and spheroplatelets as special cases. We show that this approximation can be written as the superposition of two London interactions: one attractive and the other repulsive. This conclusion also proves why the phase diagram for the excluded-volume interaction of spherocuboids is expected to feature a direct isotropic-to-biaxial transition at a single Landau point.

Universal mean-field phase diagram for biaxial nematics obtained from a minimax principle

We study a class of quadratic Hamiltonians which describe both fully attractive and partly repulsive molecular interactions, characteristic of biaxial liquid crystal molecules. To treat the partly repulsive interactions we establish a minimax principle for the associated mean-field free energy. We show that the phase diagram described by Sonnet [Phys. Rev. E 67, 061701 (2003)] is universal. Our predictions are in good agreement with the recent observations on both V-shaped and tetrapodal molecules.