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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.

We derive, from the dimensional-crossover criterion, a fundamental-measure density functional for parallel hard curved rectangles moving on a cylindrical surface. We derive it from the density functional of circular arcs of length σ with centers of mass located on an external circumference of radius R(0). The latter functional in turn is obtained from the corresponding two-dimensional functional for a fluid of hard disks of radius R on a flat surface with centers of mass confined onto a circumference of radius R(0). Thus the curved length of closest approach between the two centers of mass of hard disks on this circumference is σ=2R(0)sin(-1)(R/R(0)), the length of the circular arcs. From the density functional of circular arcs, and by applying a dimensional expansion procedure to the spatial dimension orthogonal to the plane of the circumference, we finally obtain the density functional of curved rectangles of edge lengths σ and L. Along with the derivation, we show that, when the centers of mass of the disks are confined to the exterior circumference of a circle of radius R(0),(i) for R(0)>R, the exact Percus one-dimensional (1D) density functional of circular arcs of length 2R(0)sin(-1)(R/R(0)) is obtained, and (ii) for R(0)R, the obtained functional is equivalent to that of parallel hard rectangles on a flat surface of the same lengths, except that now the density profile of curved rectangles is a periodic function of the azimuthal angle, ρ(φ,z)=ρ(φ+2π,z). The phase behavior of a fluid of aligned curved rectangles is obtained by calculating the free-energy branches of smectic, columnar, and crystalline phases for different values of the ratio R(0)/R in the range 1

The surface phase behavior of binary mixtures of colloidal hard rods in contact with a solid substrate (hard wall) is studied, with special emphasis on the region of the phase diagram that includes the smectic A phase. The colloidal rods are modeled as hard cylinders of the same diameter and different lengths, in the approximation of perfect alignment. A fundamental-measure density functional is used to obtain equilibrium density profiles and thermodynamic properties such as surface tensions and adsorption coefficients. The bulk phase diagram exhibits nematic-smectic and smectic-smectic demixing, with smectic phases having different compositions; in some cases they are microfractionated. The calculated surface phase diagram of the wall-nematic interface shows a very rich phase behavior, including layering transitions and complete wetting at high pressures, whereby an infinitely thick smectic film grows at the wall via an infinite sequence of stepwise first-order layering transitions. For lower pressures complete wetting also obtains, but here the smectic film grows in a continuous fashion. Finally, at very low pressures, the wall-nematic interface exhibits critical adsorption by the smectic phase, due to the second-order character of the bulk nematic-smectic transition.

We report the nematic and smectic ordering in an aqueous suspension of monolayer alpha -Zirconium phosphate platelets possessing a high polydispersity in diameter but uniform thickness. We observe an isotropic-nematic transition as the platelet volume fraction increases, followed by the formation of a smectic, an elusive phase that has been rarely seen in discotic liquid crystals. The smectic phase is characterized by x-ray diffraction high-resolution transmission electron microscopy, and optical microscopy. The phase equilibria in this highly polydisperse suspension are rationalized in terms of a theoretical approach based on density-functional theory.

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.

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.

We obtain a fundamental-measure density functional for mixtures of parallel hard cylinders. For this purpose we first generalize to multicomponent mixtures the fundamental-measure functional proposed by Tarazona and Rosenfeld for a one-component hard disk fluid, through a method alternative to the cavity formalism of those authors. We show the equivalence of both methods when applied to two-dimensional fluids. The density functional so obtained reduces to the exact density functional for one-dimensional mixtures of hard rods when applied to one-dimensional profiles. In a second step, we apply an idea put forward some time ago by two of us, based again on a dimensional reduction of the system, and derive a density functional for mixtures of parallel hard cylinders. We explore some features of this functional by determining the fluid-fluid demixing spinodals for a binary mixture of cylinders with the same volume, and by calculating the direct correlation functions.

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.

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.

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.

The phase diagram of prolate and oblate particles in the restricted orientations approximation (Zwanzig model) is calculated. Transitions to different inhomogeneous phases (smectic, columnar, oriented, or plastic solid) are studied through minimization of the fundamental measure functional (FMF) of hard parallelepipeds. The study of parallel hard cubes (PHC's) as a particular case is also included motivated by recent simulations of this system. As a result a rich phase behavior is obtained which include, apart from the usual liquid crystal phases, a very peculiar phase (called here discotic smectic) which was already found in the only existing simulation of the model, and which turns out to be stable because of the restrictions imposed on the orientations. The phase diagram is compared at a qualitative level with simulation results of other anisotropic particle systems.