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

Random close packing of semi-flexible polymers in two dimensions: Emergence of local and global order

Through extensive Monte Carlo simulations, we systematically study the effect of chain stiffness on the packing ability of linear polymers composed of hard spheres in extremely confined monolayers, corresponding effectively to 2D films. First, we explore the limit of random close packing as a function of the equilibrium bending angle and then quantify the local and global order by the degree of crystallinity and the nematic or tetratic orientational order parameter, respectively. A multi-scale wealth of structural behavior is observed, which is inherently absent in the case of athermal individual monomers and is surprisingly richer than its 3D counterpart under bulk conditions. As a general trend, an isotropic to nematic transition is observed at sufficiently high surface coverages, which is followed by the establishment of the tetratic state, which in turn marks the onset of the random close packing. For chains with right-angle bonds, the incompatibility of the imposed bending angle with the neighbor geometry of the triangular crystal leads to a singular intra- and inter-polymer tiling pattern made of squares and triangles with optimal local filling at high surface concentrations. The present study could serve as a first step toward the design of hard colloidal polymers with a tunable structural behavior for 2D applications.

Prediction of the liquid-crystal phase behavior of hard right triangles from fourth-virial density-functional theories

We have used an extended scaled-particle theory that incorporates four-body correlations through the fourth-order virial coefficient to analyze the orientational properties of a fluid of hard right isosceles triangles. This fluid has been analyzed by computer simulation studies, with clear indications of strong octatic correlations present in the liquid-crystal phase, although the more symmetric order tetratic phase would seem to be the most plausible candidate. Standard theories based on the second virial coefficient are unable to reproduce this behavior. Our extended theory predicts that octatic correlations, associated to a symmetry under global rotations of the oriented fluid by 45^{∘}, are highly enhanced, but not enough to give rise to a thermodynamically stable phase with strict octatic symmetry. We discuss different scenarios to improve the theoretical understanding of the elusive octatic phase in this intriguing fluid.

Reduced-variance orientational distribution functions from torque sampling

We introduce a method to sample the orientational distribution function in computer simulations. The method is based on the exact torque balance equation for classical many-body systems of interacting anisotropic particles in equilibrium. Instead of the traditional counting of events, we reconstruct the orientational distribution function via an orientational integral of the torque acting on the particles. We test the torque sampling method in two- and three-dimensions, using both Langevin dynamics and overdamped Brownian dynamics, and with two interparticle interaction potentials. In all cases the torque sampling method produces profiles of the orientational distribution function with better accuracy than those obtained with the traditional counting method. The accuracy of the torque sampling method is independent of the bin size, and hence it is possible to resolve the orientational distribution function with arbitrarily small angular resolutions.

Self-Assembled Tetratic Crystals by Orthogonal Colloidal Force

Bonding simple building blocks to create crystalline materials with design has been sophisticated in the molecular world, but this is still very challenging for anisotropic nanoparticles or colloids, because the particle arrangements, including position and orientation, cannot be manipulated as expected. Here biconcave polystyrene (PS) discs to present a shape self-recognition route are used, which can control both the position and orientation of particles during self-assembly by directional colloidal forces. An unusual but very challenging two-dimensional (2D) open superstructure-tetratic crystal (TC)-is achieved. The optical properties of the 2D TCs are studied by the finite difference time domain method, showing that the PS/Ag binary TC can be used to modulate the polarization state of the incident light, for example, converting the linearly polarized light into left-handed or right-handed circularly polarized light. This work paves an important way for self-assembling many unprecedented crystalline materials.

Effect of combined roundness and polydispersity on the phase behavior of hard-rectangle fluids

We introduce a model for a fluid of polydisperse rounded hard rectangles where the length and width of the rectangular core are fixed, while the roundness is taken into account by the convex envelope of a disk displaced along the perimeter of the core. The diameter of the disk has a continuous polydispersity described by a Schulz distribution function. We implemented the scaled particle theory for this model with the aim of studying: (i) the effect of roundness on the phase behavior of the one-component hard-rectangle fluid and (ii) how polydispersity affects phase transitions between isotropic, nematic, and tetratic phases. We found that roundness greatly affects the tetratic phase, whose region of stability in the phase diagram strongly decreases as the roundness parameter is increased. Also, the interval of aspect ratios where the tetratic-nematic and isotropic-nematic phase transitions are of first order considerably reduces with roundness, both transitions becoming weaker. Polydispersity induces strong fractionation between the coexisting phases, with the nematic phase enriched in particles of lower roundness. Finally, for high enough polydispersity and certain mean aspect ratios, the isotropic-to-nematic transition can change from second (for the one-component fluid) to first order. We also found a packing-fraction inversion phenomenon for large polydispersities: the coexisting isotropic phase has a higher packing fraction than the nematic.

Failure of standard density functional theory to describe the phase behavior of a fluid of hard right isosceles triangles

A fluid of hard right isosceles triangles was studied using an extension of scaled-particle density-functional theory which includes the exact third virial coefficient. We show that the only orientationally ordered stable liquid-crystal phase predicted by the theory is the uniaxial nematic phase, in agreement with the second-order virial theory. By contrast, Monte Carlo simulations predict exotic liquid-crystal phases exhibiting tetratic and octatic correlations, with orientational distribution functions having four and eight equivalent peaks, respectively. This demonstrates the failure of the standard density-functional theory based on two- and three-body correlations to describe high-symmetry orientational phases in two-dimensional hard right-triangle fluids, and it points to the necessity to reformulate the theory to take into account high-order body correlations and ultimately particle self-assembling and clustering effects. This avenue may represent a great challenge for future research, and we discuss some fundamental ideas to construct a modified version of density-functional theory to account for these clustering effects.

Orientational ordering in a fluid of hard kites: A density-functional-theory study

Using density-functional theory we theoretically study the orientational properties of uniform phases of hard kites-two isosceles triangles joined by their common base. Two approximations are used: scaled particle theory and a new approach that better approximates third virial coefficients of two-dimensional hard particles. By varying some of their geometrical parameters, kites can be transformed into squares, rhombuses, triangles, and also very elongated particles, even reaching the hard-needle limit. Thus, a fluid of hard kites, depending on the particle shape, can stabilize isotropic, nematic, tetratic, and triatic phases. Different phase diagrams are calculated, including those of rhombuses, and kites with two of their equal interior angles fixed to 90^{∘}, 60^{∘}, and 75^{∘}. Kites with one of their unequal angles fixed to 72^{∘}, which have been recently studied via Monte Carlo simulations, are also considered. We find that rhombuses and kites with two equal right angles and not too large anisometry stabilize the tetratic phase but the latter stabilize it to a much higher degree. By contrast, kites with two equal interior angles fixed to 60^{∘} stabilize the triatic phase to some extent, although it is very sensitive to changes in particle geometry. Kites with the two equal interior angles fixed to 75^{∘} have a phase diagram with both tetratic and triatic phases, but we show the nonexistence of a particle shape for which both phases are stable at different densities. Finally, the success of the new theory in the description of orientational order in kites is shown by comparing with Monte Carlo simulations for the case where one of the unequal angles is fixed to 72^{∘}. These particles also present a phase diagram with stable tetratic and triatic phases.

Evaporation-induced alignment of nanorods in a thin film

During the solvent evaporation of a thin film, Brownian rod-shaped particles self-assemble into microstructures and their orientation arrangements change while their volume fractions increase. We have studied the phenomena using a simple model which accounts for the anisotropic diffusion and the mean-field interaction of the particles. By numerically solving the Smoluchowski equation under moving boundary conditions, we obtain the spatiotemporal evolution of volume fractions and order parameters. It is shown that the evaporation dynamics alter the equilibrium orientational configuration of particles to meta-stable states. This alternation is possible by controlling either Péclet numbers or anisotropic diffusion rates. This understanding of the dynamic self-assembly of rod-shaped particles can be useful in manipulating the collective rod-arrangement in printing and coating technologies.

Emergent tetratic order in crowded systems of rotationally asymmetric hard kite particles

Considering multi-body systems of monodisperse hard Brownian particles, it remains challenging to predict the forms of order that can emerge in their dense assembled structures. Surprisingly, here, using Monte Carlo simulations, we show that tetratic-ordered phases emerge in a dense two-dimensional system of hard kites that are rotationally asymmetric and have opposite 72° and α ≈ 90° internal angles. We observe a new tetragonal rectangular crystal (TRX) phase possessing (quasi-)long-range fourfold molecular-orientational order. We propose a method based on local polymorphic configurations of neighboring particle pairs (LPC-NPPs) to understand this emergent tetratic order and show that LPC-NPPs can be useful for predicting orientational order in such systems. To examine the dependence of the tetratic order on α, we apply LPC-NPP analysis to other hard kites for 54° ≤ α ≤ 144°. Our work provides insight into the creation of novel ordered materials by rationally designing particle shape based on anticipated LPC-NPPs.

For colloidal particles, it remains challenging to predict the forms of order that can emerge in their dense assembled structures. Here, the authors show, by Monte Carlo simulations, that tetratic-ordered phases emerge in a dense two-dimensional system of hard kites that are rotationally asymmetric.

Anisotropic line tension of domains in lipid monolayers

We formulate a simple effective model to describe molecular interactions in a lipid monolayer and calculate the line tension between coexisting domains. The model represents lipid molecules in terms of two-dimensional anisotropic particles on the plane of the monolayer. These particles interact through forces that are believed to be relevant for the understanding of fundamental properties of the monolayer: van der Waals interactions originating from lipid chains and dipolar forces between dipole groups in the molecular heads. The model stresses the liquid-crystalline nature of the ordered phase in lipid monolayers and explains coexistence properties between ordered and disordered phases in terms of molecular parameters. Thermodynamic and interfacial properties of the model are analyzed using density-functional theory. In particular, the line tension at the interface between ordered and disordered phases turns out to be highly anisotropic with respect to the angle between the nematic director and the interface separating the coexisting phases. This important feature mainly results from the tilt angle of lipid chains and, to a lesser extent, from dipolar interactions perpendicular to the monolayer. The role of the two dipolar components, parallel and perpendicular to the monolayer, is assessed by comparing with computer simulation results for lipid monolayers.

Principal component analysis of the excluded area of two-dimensional hard particles

The excluded area between a pair of two-dimensional hard particles with given relative orientation is the region in which one particle cannot be located due to the presence of the other particle. The magnitude of the excluded area as a function of the relative particle orientation plays a major role in the determination of the bulk phase behavior of hard particles. We use principal component analysis (PCA) to identify the different types of excluded areas corresponding to randomly generated two-dimensional hard particles modeled as non-self-intersecting polygons and star lines (line segments radiating from a common origin). Only three principal components are required to have an excellent representation of the value of the excluded area as a function of the relative particle orientation for sufficiently anisotropic particles. Independent of the particle shape, the minimum value of the excluded area is always achieved when the particles are antiparallel to each other. The property that affects the value of the excluded area most strongly is the elongation of the particle shape. PCA identifies four limiting cases of excluded areas with one to four global minima at equispaced relative orientations. We study selected particle shapes using Monte Carlo simulations.

Uniform phases in fluids of hard isosceles triangles: One-component fluid and binary mixtures

We formulate the scaled particle theory for a general mixture of hard isosceles triangles and calculate different phase diagrams for the one-component fluid and for certain binary mixtures. The fluid of hard triangles exhibits a complex phase behavior: (i) the presence of a triatic phase with sixfold symmetry, (ii) the isotropic-uniaxial nematic transition is of first order for certain ranges of aspect ratios, and (iii) the one-component system exhibits nematic-nematic transitions ending in critical points. We found the triatic phase to be stable not only for equilateral triangles but also for triangles of similar aspect ratios. We focus the study of binary mixtures on the case of symmetric mixtures: equal particle areas with aspect ratios (κ_{i}) symmetric with respect to the equilateral one, κ_{1}κ_{2}=3. For these mixtures we found, aside from first-order isotropic-nematic and nematic-nematic transitions (the latter ending in a critical point): (i) a region of triatic phase stability even for mixtures made of particles that do not form this phase at the one-component limit, and (ii) the presence of a Landau point at which two triatic-nematic first-order transitions and a nematic-nematic demixing transition coalesce. This phase behavior is analogous to that of a symmetric three-dimensional mixture of rods and plates.

Ordering transitions of weakly anisotropic hard rods in narrow slitlike pores

The effect of strong confinement on the positional and orientational ordering is examined in a system of hard rectangular rods with length L and diameter D (L>D) using the Parsons-Lee modification of the second virial density-functional theory. The rods are nonmesogenic (L/D<3) and confined between two parallel hard walls, where the width of the pore (H) is chosen in such a way that both planar (particle's long axis parallel to the walls) and homeotropic (particle's long axis perpendicular to the walls) orderings are possible and a maximum of two layers is allowed to form in the pore. In the extreme confinement limit of H≤2D, where only one-layer structures appear, we observe a structural transition from a planar to a homeotropic fluid layer with increasing density, which becomes sharper as L→H. In wider pores (2D<H<3D) planar order with two layers, homeotropic order, and even combined bilayer structures (one layer is homeotropic, while the other is planar) can be stabilized at high densities. Moreover, first-order phase transitions can be seen between different structures. One of them emerges between a monolayer and a bilayer with planar orders at relatively low packing fractions.

Liquid crystals of hard rectangles on flat and cylindrical manifolds

The self-assembly of rectangular particles on flat and curved substrates was investigated using density functional theory and simulations.

Role of length polydispersity in the phase behavior of freely rotating hard-rectangle fluids

We use the density-functional formalism, in particular the scaled-particle theory, applied to a length-polydisperse hard-rectangle fluid to study its phase behavior as a function of the mean particle aspect ratio κ_{0} and polydispersity Δ_{0}. The numerical solutions of the coexistence equations are calculated by transforming the original problem with infinite degrees of freedoms to a finite set of equations for the amplitudes of the Fourier expansion of the moments of the density profiles. We divide the study into two parts. The first one is devoted to the calculation of the phase diagrams in the packing fraction η_{0}-κ_{0} plane for a fixed Δ_{0} and selecting parent distribution functions with exponential (the Schulz distribution) or Gaussian decays. In the second part we study the phase behavior in the η_{0}-Δ_{0} plane for fixed κ_{0} while Δ_{0} is changed. We characterize in detail the orientational ordering of particles and the fractionation of different species between the coexisting phases. Also we study the character (second vs first order) of the isotropic-nematic phase transition as a function of polydispersity. We particularly focus on the stability of the tetratic phase as a function of κ_{0} and Δ_{0}. The isotropic-nematic transition becomes strongly of first order when polydispersity is increased: The coexistence gap widens and the location of the tricritical point moves to higher values of κ_{0} while the tetratic phase is slightly destabilized with respect to the nematic one. The results obtained here can be tested in experiments on shaken monolayers of granular rods.

Clustering in vibrated monolayers of granular rods

We investigate the ordering properties of vertically-vibrated monolayers of granular cylinders in a circular container at high packing fraction. In line with previous works by other groups, we identify liquid-crystalline ordering behaviour similar to that of two-dimensional hard rectangular particles subject to thermal equilibrium fluctuations. However, due to dissipation, there is a much stronger tendency for particles to cluster into parallel arrangements in the granular system. These clusters behave as a polydisperse mixture of long life-time 'superparticles', and some aspects of the system behaviour can be understood by applying mean-field theories for equilibrium hard rectangles, based on two-body correlations, to these 'superparticles'. Many other features of the granular system are different: (i) for small particle length-to-breadth ratio κ, we identify tetratic ordering at moderate packing fractions and smectic fluctuations at higher packing fractions, with no sharp transition between the two states. Both types of ordering can be explained in terms of clustering. (ii) For large κ, strong clustering precludes the stabilisation of a uniaxial nematic state, and the system exhibits a mixture of randomly-oriented clusters which, as packing fraction is increased, develops into states with smectic fluctuations, again through a diffuse transition. (iii) Vorticity excitations of the velocity field compete with smectic ordering, causing dynamic fluctuations and the absence of steady states at high densities; the tetratic state, by contrast, is very stiff against vorticity, and long-standing steady states, spatially and orientationally homogeneous except for four symmetrical defects located close to the wall, can be observed.

Effect of orientational restriction on monolayers of hard ellipsoids

The effect of out-of-plane orientational freedom on the orientational ordering properties of a monolayer of hard ellipsoids is studied using the Parsons-Lee scaling approach and replica exchange Monte Carlo computer simulation. Prolate and oblate ellipsoids exhibit very different ordering properties, namely, the axes of revolution of prolate particles tend to lean out, while those of oblate ones prefer to lean into the confining plane. The driving mechanism of this is that the particles try to maximize the available free area on the confining surface, which can be achieved by minimizing the cross section areas of the particles with the plane. In the lack of out-of-plane orientational freedom the monolayer of prolate particles is identical to a two-dimensional hard ellipse system, which undergoes an isotropic-nematic ordering transition with increasing density. With gradually switching on the out-of-plane orientational freedom the prolate particles lean out from the confining plane and destabilisation of the in-plane isotropic-nematic phase transition is observed. The system of oblate particles behaves oppositely to that of prolates. It corresponds to a two-dimensional system of hard disks in the lack of out-of-plane freedom, while it behaves similar to that of hard ellipses in the freely rotating case. Solid phases can be realised by lower surface coverage due to the out-of-plane orientation freedom for both oblate and prolate shapes.

Ordering in granular-rod monolayers driven far from thermodynamic equilibrium

The orientational order in vertically agitated granular-rod monolayers is investigated experimentally and compared quantitatively with equilibrium Monte Carlo simulations and density functional theory. At sufficiently high number density, short rods form a tetratic state and long rods form a uniaxial nematic state. The length-to-width ratio at which the order changes from tetratic to uniaxial is around 7.3 in both experiments and simulations. This agreement illustrates the universal aspects of the ordering of rod-shaped particles across equilibrium and nonequilibrium systems. Moreover, the assembly of granular rods into ordered states is found to be independent of the agitation frequency and strength, suggesting that the detailed nature of energy injection into such a nonequilibrium system does not play a crucial role.

Hard-body models of bulk liquid crystals

Hard models for particle interactions have played a crucial role in the understanding of the structure of condensed matter. In particular, they help to explain the formation of oriented phases in liquids made of anisotropic molecules or colloidal particles and continue to be of great interest in the formulation of theories for liquids in bulk, near interfaces and in biophysical environments. Hard models of anisotropic particles give rise to complex phase diagrams, including uniaxial and biaxial nematic phases, discotic phases and spatially ordered phases such as smectic, columnar or crystal. Also, their mixtures exhibit additional interesting behaviours where demixing competes with orientational order. Here we review the different models of hard particles used in the theory of bulk anisotropic liquids, leaving aside interfacial properties and discuss the associated theoretical approaches and computer simulations, focusing on applications in equilibrium situations. The latter include one-component bulk fluids, mixtures and polydisperse fluids, both in two and three dimensions, and emphasis is put on liquid-crystal phase transitions and complex phase behaviour in general.

Entropic chiral symmetry breaking in self-organized two-dimensional colloidal crystals

Long-range chiral symmetry breaking (CSB) has been recently observed in 2D self-organized rhombic crystals of hard, achiral, 72 degree rhombic microparticles. However, purely entropic selection of a CSB crystal in an idealized system of hard achiral shapes, in which attractions are entirely absent and the shape does not dictate a chiral tiling, has not yet been quantitatively predicted. Overcoming limitations of a purely rotational cage model, we investigate a translational-rotational cage model (TRCM) of dense systems of hard achiral rhombs and quantitatively demonstrate that entropy can spontaneously drive the preferential self-organization of a chiral crystal composed of achiral shapes that also tile into an achiral crystal. At different particle area fractions, ϕA, we calculate the number of accessible translational-rotational microstates, Ω, of a mobile central rhomb in a static cage of neighboring rhombs, which can have different orientation angles, γ, relative to the bisector of the crystalline axes. As we raise ϕA, two maxima emerge in Ω(γ) at non-zero cage orientation angles, ±γmax. These maxima correspond to additional translational microstates that become accessible in the CSB crystalline polymorph through reduced translational tip-tip interference. Thus, entropy, often associated with structural disorder, can drive CSB in condensed phase systems of non-attractive achiral objects that do not tile into chiral structures. The success of the TRCM in explaining the entropic origin of CSB in systems of hard rhombs indicates that the TRCM will have significant utility in predicting the self-organized behavior of dense systems of other hard shapes in 2D.

Domain walls in two-dimensional nematics confined in a small circular cavity

Using Monte Carlo simulation, we study a fluid of two-dimensional hard rods inside a small circular cavity bounded by a hard wall, from the dilute regime to the high-density, layering regime. Both planar and homeotropic anchoring of the nematic director can be induced at the walls through a free-energy penalty. The circular geometry creates frustration in the nematic phase and a polar-symmetry configuration with a distorted director field plus two +1/2 disclinations is created. At higher densities, a quasi-uniform structure is observed with a (minimal) director distortion which is relaxed via the formation of orientational domain walls. This novel structure is not predicted by elasticity theory and is similar to the step-like structures observed in three-dimensional hybrid slit pores. We speculate that the formation of domain walls is a general mechanism to relax elastic stresses under the conditions of strong surface anchoring and severe spatial confinement.

Liquid-crystal patterns of rectangular particles in a square nanocavity

Using density-functional theory in the restricted-orientation approximation, we analyze the liquid-crystal patterns and phase behavior of a fluid of hard rectangular particles confined in a two-dimensional square nanocavity of side length H composed of hard inner walls. Patterning in the cavity is governed by surface-induced order as well as capillary and frustration effects and depends on the relative values of the particle aspect ratio κ≡L/σ, with L the length and σ the width of the rectangles (L≥σ), and cavity size H. Ordering may be very different from bulk (H→∞) behavior when H is a few times the particle length L (nanocavity). Bulk and confinement properties are obtained for the cases κ=1, 3, and 6. In bulk the isotropic phase is always stable at low packing fractions η=Lσρ_{0} (with ρ_{0} the average density) and nematic, smectic, columnar, and crystal phases can be stabilized at higher η depending on κ: For increasing η the sequence of isotropic to columnar is obtained for κ=1 and 3, whereas for κ=6 we obtain isotropic to nematic to smectic (the crystal being unstable in all three cases for the density range explored). In the confined fluid surface-induced frustration leads to fourfold symmetry breaking in all phases (which become twofold symmetric). Since no director distortion can arise in our model by construction, frustration in the director orientation is relaxed by the creation of domain walls (where the director changes by 90^{∘}); this configuration is necessary to stabilize periodic phases. For κ=1 the crystal becomes stable with commensurate transitions taking place as H is varied. These transitions involve structures with different number of peaks in the local density. In the case κ=3 the commensurate transitions involve columnar phases with different number of columns. In the case κ=6 the high-density region of the phase diagram is dominated by commensurate transitions between smectic structures; at lower densities there is a symmetry-breaking isotropic to nematic transition exhibiting nonmonotonic behavior with cavity size. Apart from the present application in a confinement setup, our model could be used to explore the bulk region near close packing in order to elucidate the possible existence of disordered phases at close packing.

Entropic crystal-crystal transitions of Brownian squares

When a monolayer of hard microscale square platelets, produced lithographically, is osmotically concentrated in a flat plane to raise the particle area fraction Φ(A), an order-order transition occurs between a hexagonal rotator crystal and a rhombic crystal. Strikingly, phases having fourfold symmetry are not observed at any Φ(A). The rhombic lattice angle α increases continuously with Φ(A), as the system maximizes its total rotational and translational entropy. A cage model, based on packing rotationally swept squares, or "squaroids," reasonably predicts the measured α(Φ(A)), indicating that rotational entropy and the square particle shape combine to produce the rhombic unit cell.

Angelo SK, González P, Keating CD, Fichthorn KA. Assembly of gold nanowires by sedimentation from suspension: Experiments and simulation

We investigated the ordering of gold nanowires that settled from aqueous suspension onto a glass substrate due to gravity. The nanowires, ca. 300 nm in cross-sectional diameter and ca. 2, 4, or 7 microns in length, were coated with 2-mercaptoethanesulfonic acid to provide electrostatic repulsion and prevent aggregation. The layer of nanowires in direct contact with the substrate was examined from below using optical microscopy and found to exhibit smectic-like ordering. The extent of smectic ordering depended on nanowire length with the shortest (2 μm) nanowires exhibiting the best ordering. To understand the assembly in this system, we used canonical Monte Carlo simulations to model the two-dimensional ordering of the nanowires on a substrate. We accounted for van der Waals and electrostatic interactions between the nanowires. The simulations reproduced the experimental trends and showed that roughness at the ends of the nanowires, which locally increased electrostatic repulsion, is critical to correctly predicting the experimentally observed smectic ordering.

Frustrated rotator crystals and glasses of Brownian pentagons

Two-dimensional Brownian dispersions of microscale pentagonal platelets exhibit rich structural and dynamical behavior as the particle area fraction, phi(A), is increased. As phi(A) is raised above 0.66, a rotator crystal forms, and while in an equilateral hexagonal lattice, pentagons still explore all angles as they rotationally diffuse. At larger phi(A), the interference of the tips of neighboring pentagons causes rotational dynamical heterogeneity; particle rotations become nonergodic, the hallmark of a frustrated rotator crystal. Upon further compression, the quenched-in rotational disorder and inability of pentagons to fully tile a flat plane creates spatial defects, precluding access to a dense striped crystalline packing.

Theory and simulation of two-dimensional nematic and tetratic phases

Recent experiments and simulations have shown that two-dimensional systems can form tetratic phases with fourfold rotational symmetry, even if they are composed of particles with only twofold symmetry. To understand this effect, we propose a model for the statistical mechanics of particles with almost fourfold symmetry, which is weakly broken down to twofold. We introduce a coefficient kappa to characterize the symmetry breaking, and find that the tetratic phase can still exist even up to a substantial value of kappa. Through a Landau expansion of the free energy, we calculate the mean-field phase diagram, which is similar to the result of a previous hard-particle excluded-volume model. To verify our mean-field calculation, we develop a Monte Carlo simulation of spins on a triangular lattice. The results of the simulation agree very well with the Landau theory.

Enhanced stability of the tetratic phase due to clustering

We show that the relative stability of the nematic tetratic phase with respect to the usual uniaxial nematic phase can be greatly enhanced by clustering effects. Two-dimensional rectangles of aspect ratio kappa interacting via hard interactions are considered, and the stability of the two nematic phases (uniaxial and tetratic) is examined using an extended scaled-particle theory applied to a polydispersed fluid mixture of n species. Here the ith species is associated with clusters of i rectangles, with clusters defined as stacks of rectangles containing approximately parallel rectangles, with frozen internal degrees of freedom. The theory assumes an exponential cluster size distribution (an assumption fully supported by Monte Carlo simulations and by a simple chemical-reaction model), with fixed value of the second moment. The corresponding area distribution presents a shoulder, and sometimes even a well-defined peak, at cluster sizes approximately corresponding to square shape (i.e., i approximately kappa), meaning that square clusters have a dominant contribution to the free energy of the hard-rectangle fluid. The theory predicts an enhanced region of stability of the tetratic phase with respect to the standard scaled-particle theory, much closer to simulation and to experimental results, demonstrating the importance of clustering in this fluid.

Biaxial nematic and smectic phases of parallel particles with different cross sections

We have calculated the phase diagrams of one-component fluids made of five types of biaxial particles differing in their cross sections. The orientation of the principal particle axis is fixed in space, while the second axis is allowed to freely rotate. We have constructed a free-energy density functional based on fundamental-measure theory to study the relative stability of nematic and smectic phases with uniaxial, biaxial, and tetratic symmetries. Minimization of the density functional allows us to study the phase behavior of the biaxial particles as a function of the cross-section geometry. For low values of the aspect ratio of the particle cross section, we obtain smectic phases with tetratic symmetry, although metastable with respect to the crystal, as our Monte Carlo simulation study indicates. For large particle aspect ratios and in analogy with previous work [A. G. Vanakaras, M. A. Bates, and D. J. Photinos, Phys. Chem. Chem. Phys. 5, 3700 (2003)], we have found a four-phase point where four spinodals, corresponding to phase transitions between phases with different symmetries, meet together. The location of this point is quite sensitive to particle cross section, which suggests that optimizing the particle geometry could be a useful criterion in the design of colloidal particles that can exhibit an increased stability of the biaxial nematic phase with respect to other competing phases with spatial order.

Monte Carlo simulation of two-dimensional hard rectangles: confinement effects

We use orientational-bias Monte Carlo simulations to examine the phase behavior of two-dimensional hard rectangles in the bulk and under confinement by hard walls. For all of the rod aspect ratios and area fractions studied, we find that confinement increases the degree of nematic ordering over the bulk, as confined rods tend to align their long axes parallel to the confining walls. The extent of nematic ordering increases as the separation between the confining walls decreases. If the aspect ratio of the rectangles is sufficiently large, they exhibit nematic ordering in both the bulk and under confinement, where the nematic director is set by the walls. Rods with a small aspect ratio are isotropic in the bulk and exhibit weak tetratic tendencies for sufficiently high densities. From studies of density profiles, angular distributions, and orientational correlation functions for confined, low-aspect-ratio rods, it is apparent that they align their long axes parallel to the wall in the near-wall region, where layering occurs for sufficiently high rod densities. However, confined rods with low aspect ratios still exhibit weak tetratic (isotropic) tendencies near the center of the confined region for all but the smallest wall separations. We note that although our studies probe the ordering of hard rectangles, the entropic tendencies that we observe here will be present for rods with energetic interactions. Thus, these studies serve as a general starting point for understanding and controlling the assembly of rods in two-dimensional confining geometries.

Nematic and almost-tetratic phases of colloidal rectangles

Nonspherical colloids can exhibit liquid-crystalline phases with different degrees of broken orientational and translational symmetry. Here we investigate hard rectangles consisting of photolithographically prepared disks standing on edge. We observe a conventional Kosterlitz-Thouless transition from isotropic to nematic with almost smectic behavior at high density. But just on the isotropic side of the isotropic to nematic transition we observe an unusual regime where short-range tetratic correlations dominate over nematic correlations. This occurs due to the proliferation of Ising-like pi/2 grain boundaries that disrupt nematic order, but preserve tetratic correlations, at lengths shorter than the spacing between free disclinations.

Demixing and orientational ordering in mixtures of rectangular particles

Using scaled-particle theory for binary mixtures of two-dimensional hard particles with orientational degrees of freedom, we analyze the stability of phases with orientational order and the demixing phase behavior of a variety of mixtures. Our study is focused on cases where at least one of the components consists of hard rectangles, or a particular case of these, hard squares. A pure fluid of hard rectangles has recently been shown to exhibit, aside from the usual uniaxial nematic phase, an additional oriented phase, called tetratic phase, possessing two directors, which is the analog of the biaxial or cubatic phases in three-dimensional fluids. There is evidence, based on computer simulation studies, that the tetratic phase might be stable with respect to phases with lower translational symmetry for rectangles with low aspect ratios. As hard rectangles are mixed, in increasing concentration, with other particles not possessing stable tetratic order by themselves, the tetratic phase is destabilized, via a first- or second-order phase transition, to uniaxial nematic or isotropic phases; for hard rectangles of low aspect ratio (hard squares, in particular), tetratic order persists in a relatively large range of volume fractions. The order of these transitions depends on the particle geometry and dimensions, and also on the thermodynamic conditions of the mixture. The second component of the mixture has been chosen to be hard disks or discorectangles, the geometry of which is different from that of rectangles, leading to packing frustration and demixing behavior, or simply rectangles of different aspect ratio but with the same particle area, or different particle area but with the same aspect ratio. These mixtures may be good candidates for observing thermodynamically stable tetratic phases in monolayers of hard particles. Finally, demixing between fluid (isotropic-tetratic or tetratic-tetratic) phases is seen to occur in mixtures of hard squares of different sizes when the size ratio is sufficiently large.

Density-functional theory for complex fluids

Density-functional theory (DFT) and its variations provide a fruitful approach to the computational modeling of the microscopic structures and phase behavior of soft-condensed matter. The methodology takes deep root in quantum mechanics but shares a mathematical similarity with a number of classical approaches in statistical mechanics. This review discusses different strategies commonly used to formulate the free-energy functional of complex fluids for either phenomena-oriented applications or as a generic description of the thermodynamic nonideality owing to various components of intermolecular forces. We emphasize the connections among different schemes of DFT approximations, their underlying assumptions, and inherent limitations. We also address extensions of equilibrium DFT to phenomenological theories for the dynamic properties of complex fluids and for the kinetics of phase transitions. In addition, we highlight the recent literature concerning applications of DFT to diverse static and time-dependent phenomena in complex fluids.

Capillary ordering and layering transitions in two-dimensional hard-rod fluids

We calculate the surface phase diagram of a two-dimensional hard-rod fluid confined between two hard lines. In the first stage we study a semi-infinite system consisting of an isotropic fluid in contact with a single hard line. We have found complete wetting by the columnar phase at the wall-isotropic fluid interface. When the fluid is confined between two hard walls, capillary columnar ordering occurs via a first-order phase transition. For higher chemical potentials the system exhibits layering transitions even for very narrow slits (near the one-dimensional limit). The theoretical model used was a density-functional theory based on the fundamental-measure functional applied to a fluid of hard rectangles in the restricted-orientation approximation (Zwanzig model). The results presented here can be checked experimentally in two-dimensional granular media made of rods, where vertical motions induced by an external source and excluded volume interactions between the grains allow the system to explore those stationary states which entropically maximize packing configurations. We claim that some of the surface phenomena found here can be present in two-dimensional granular-media fluids.

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.

Demixing behavior in two-dimensional mixtures of anisotropic hard bodies

Scaled particle theory for a binary mixture of hard discorectangles and for a binary mixture of hard rectangles is used to predict possible liquid-crystal demixing scenarios in two dimensions. Through a bifurcation analysis from the isotropic phase, it is shown that isotropic-nematic demixing is possible in two-dimensional liquid-crystal mixtures composed of hard convex bodies. This bifurcation analysis is tested against exact calculations of the phase diagrams in the framework of the restricted-orientation two-dimensional model (Zwanzig model). Phase diagrams of a binary mixture of hard discorectangles are calculated through the parametrization of the orientational distribution functions. The results show not only isotropic-nematic, but also nematic-nematic demixing ending in a critical point, as well as an isotropic-nematic-nematic triple point for a mixture of hard disks and hard discorectangles.