Dynamical properties of heterogeneous nucleation of parallel hard squares.

We use the Dynamic Density-Functional Formalism and the Fundamental Measure Theory as applied to a fluid of parallel hard squares to study the dynamics of heterogeneous growth of non-uniform phases with columnar and crystalline symmetries. The hard squares are (i) confined between soft repulsive walls with a square symmetry, or (ii) exposed to external potentials that mimic the presence of obstacles with circular, square, rectangular or triangular symmetries. For the first case the final equilibrium profile of a well commensurated cavity consists of a crystal phase with highly localized particles in concentric square layers at the nodes of a slightly deformed square lattice. We characterize the growth dynamics of the crystal phase by quantifying the interlayer and intralayer fluxes and the non-monotonicity of the former, the saturation time, and other dynamical quantities. The interlayer fluxes are much more monotonic in time, and dominant for poorly commensurated cavities, while the opposite is true for well commensurated cells: although smaller, the time evolution of interlayer fluxes is much more complex, presenting strongly damped oscillations which dramatically increase the saturation time. We also study how the geometry of the obstacle affects the symmetry of the final equilibrium non-uniform phase (columnar vs. crystal). For obstacles with fourfold symmetry, (circular and square) the crystal is more stable, while the columnar phase is stabilized for obstacles without this symmetry (rectangular or triangular). We find that, in general, density waves of columnar symmetry grow from the obstacle. However, additional particle localization along the wavefronts gives rise to a crystalline structure which is conserved for circular and square obstacles, but destroyed for the other two obstacles where columnar symmetry is restored.

Ordering of hard rectangles in strong confinement.

Using transfer operator and fundamental measure theories, we examine the structural and thermodynamic properties of hard rectangles confined between two parallel hard walls. The side lengths of the rectangle (L and D, L>D) and the pore width (H) are chosen such that a maximum of two layers is allowed to form when the long sides of the rectangles are parallel to the wall, while only one layer is possible in case the rectangles are perpendicular to the wall. We observe three different structures: (i) at low density, the rectangles align mainly parallel to the wall, (ii) at intermediate or high density, two fluid layers form in which the rectangles are parallel to the wall, and (iii) a dense single fluid layer with rectangles aligned mainly perpendicular to the wall. The transition between these structures is smooth without any non-analytic behaviour in the thermodynamic quantities; however, the fraction of particles perpendicular (or parallel) to the wall can exhibit a relatively sudden change if L is close to H. In this case, interestingly, even three different structures can be observed with increasing density.

Biaxial nematic phase stability and demixing behaviour in monolayers of rod-plate mixtures.

We theoretically study the phase behaviour of monolayers of hard rod-plate mixtures using a fundamental-measure density functional in the restricted-orientation (Zwanzig) approximation. Particles can rotate in 3D but their centres of mass are constrained to be on a flat surface. In addition, we consider both species to be subject to an attractive potential proportional to the particle contact area on the surface and with adsorption strengths that depend on the species type. Particles have board-like shape, with sizes chosen using a symmetry criterion: same volume and same aspect ratio κ. Phase diagrams were calculated for κ = 10, 20 and 40 and different values of adsorption strengths. For small adsorption strengths the mixtures exhibit a second-order uniaxial nematic-biaxial nematic transition for molar fraction of rods 0 ≤x≲ 0.9. In the uniaxial nematic phase the particle axes of rods and plates are aligned perpendicular and parallel to the monolayer, respectively. At the transition, the orientational symmetry of the plate axes is broken, and they orient parallel to a director lying on the surface. For large and equal adsorption strengths the mixture demixes at low pressures into a uniaxial nematic phase, rich in plates, and a biaxial nematic phase, rich in rods. The demixing transition is located between two tricritical points. Also, at higher pressures and in the plate-rich part of the phase diagram, the system exhibits a strong first-order uniaxial nematic-biaxial nematic phase transition with a large density coexistence gap. When rod adsorption is considerably large while that of plates is small, the transition to the biaxial nematic phase is always of second order, and its region of stability in the phase diagram considerably widens. At very high pressures the mixture can effectively be identified as a two-dimensional mixture of squares and rectangles which again demixes above a certain critical point. We also studied the relative stability of uniform phases with respect to density modulations of smectic, columnar and crystalline symmetry.

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.

Capillary and winding transitions in a confined cholesteric liquid crystal.

We consider a Lebwohl-Lasher model of chiral particles confined in a planar cell (slit pore) under different boundary conditions, and solve it using mean-field theory. The phase behaviour of the system with respect to temperature and pore width is studied. Two phenomena are observed: (i) an isotropic-cholesteric transition, which exhibits an oscillatory structure with respect to pore width, and (ii) an infinite set of winding transitions caused by commensuration effects between cholesteric pitch and pore width. The latter transitions have been predicted and analysed by other authors for cholesterics confined in a fixed pore and subjected to an external field promoting the uniaxial nematic phase; here we induce winding transitions solely from geometry by changing the pore width at zero external field (a setup recently explored in atomic-force microscopy experiments). In contrast with previous studies, we obtain the phase diagram in the temperature vs. pore width plane, including the isotropic-cholesteric transition, the winding transitions and their complex relationship. In particular, the structure of winding transitions terminates at the capillary isotropic-cholesteric transition via triple points where the confined isotropic phase coexists with two cholesterics with different helix indices. For symmetric and asymmetric monostable plate anchorings the phase diagrams are qualitatively similar.

Effect of shape biaxiality on the phase behavior of colloidal liquid-crystal monolayers.

We extend our previous work on monolayers of uniaxial particles [J. Chem. Phys., 2014, 140, 204906] to study the effect of particle biaxiality on the phase behavior of liquid-crystal monolayers. Particles are modelled as board-like hard bodies with three different edge lengths σ1 ≥ σ2 ≥ σ3, and the restricted-orientation approximation (Zwanzig model) is used. A density-functional formalism based on the fundamental-measure theory is used to calculate phase diagrams for a wide range of values with the largest aspect ratio κ1 = σ1/σ3 ∈ [1,100]. We find that particle biaxiality in general destabilizes the biaxial nematic phase already present in monolayers of uniaxial particles. While plate-like particles exhibit strong biaxial ordering, rod-like ones with κ1 > 21.34 exhibit reentrant uniaxial and biaxial phases. As particle geometry is changed from uniaxial- to increasingly biaxial-rod-like, the region of biaxiality is reduced, eventually ending in a critical-end point. For κ1 > 60, a density gap opens up in which the biaxial nematic phase is stable for any particle biaxiality. Regions of the phase diagram, where packing-fraction inversion occurs (i.e. packing fraction is a decreasing function of density), are found. Our results are compared with the recent experimental studies on nematic phases of magnetic nanorods.

Interplay between columnar and smectic stability in suspensions of polydisperse colloidal platelets.

The phase behavior of a model suspension of colloidal polydisperse platelets is studied using density-functional theory. Platelets are modelled as parallel rectangular prisms of square section l(2) and height h, with length and height distributions given by different polydispersities δ(l) and δ(h). The model is intended to qualitatively represent experimental colloidal platelet suspensions at high densities with a high degree of orientational ordering. We obtain the phase behavior of the model, including nematic, smectic and columnar phases and its dependence on the two polydispersities δ(l) and δ(h). When δ(l) > δ(h) we observe that the smectic phase stabilises first with respect to the columnar. If δ(h) > δ(l) we observe the opposite behavior. Other more complicated cases occur, e.g. the smectic stabilises from the nematic first but then exists a first-order transition to the columnar phase. Our model assumes plate-rod symmetry, but the regions of stability of smectic and columnar phases are non-symmetric in the δ(l) - δ(h) plane due to the different dimensionality of ordering in the two phases. Microsegregation effects, i.e. different spatial distribution for different sizes within the periodic cell, take place in both phases and, in each case, is more apparent in the variable associated with ordering.

Phase behaviour of liquid-crystal monolayers of rod-like and plate-like particles.

Orientational and positional ordering properties of liquid crystal monolayers are examined by means of Fundamental-Measure Density Functional Theory. Particles forming the monolayer are modeled as hard parallelepipeds of square section of size σ and length L. Their shapes are controlled by the aspect ratio κ = L/σ (>1 for prolate and <1 for oblate shapes). The particle centers of mass are restricted to a flat surface and three possible and mutually perpendicular orientations (in-plane and along the layer normal) of their uniaxial axes are allowed. We find that the structure of the monolayer depends strongly on particle shape and density. In the case of rod-like shapes, particles align along the layer normal in order to achieve the lowest possible occupied area per particle. This phase is a uniaxial nematic even at very low densities. In contrast, for plate-like particles, the lowest occupied area can be achieved by random in-plane ordering in the monolayer, i.e., planar nematic ordering takes place even at vanishing densities. It is found that the random in-plane ordering is not favorable at higher densities and the system undergoes an in-plane ordering transition forming a biaxial nematic phase or crystallizes. For certain values of the aspect ratio, the uniaxial-biaxial nematic phase transition is observed for both rod-like and plate-like shapes. The stability region of the biaxial nematic phase enhances with decreasing aspect ratios for plate-like particles, while the rod-like particles exhibit a reentrant phenomenon, i.e., a sequence of uniaxial-biaxial-uniaxial nematic ordering with increasing density if the aspect ratio is larger than 21.34. In addition to this, packing fraction inversion is observed with increasing surface pressure due to the alignment along the layers normal. At very high densities the nematic phase destabilizes to a nonuniform phases (columnar, smectic, or crystalline phases) for both shapes.

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.

Effect of polydispersity, bimodality, and aspect ratio on the phase behavior of colloidal platelet suspensions.

We use a fundamental-measure density functional for hard board-like polydisperse particles, in the restricted-orientation approximation, to explain the phase behaviour of platelet colloidal suspensions studied in recent experiments. In particular, we focus our attention on the behavior of the total packing fraction of the mixture, η, in the region of two-phase isotropic-nematic coexistence as a function of mean aspect ratio, polydispersity, and fraction of total volume γ occupied by the nematic phase. In our model, platelets are polydisperse in the square section, of side length σ, but have constant thickness L (and aspect ratio κ ≡ L∕<σ> < 1, with <σ> the mean side length). Good agreement between our theory and recent experiments is obtained by mapping the real system onto an effective one, with excluded volume interactions but with thicker particles (due to the presence of long-ranged repulsive interactions between platelets). The effect of polydispersity in both shape and particle size has been taken into account by using a size distribution function with an effective mean-square deviation that depends on both polydispersities. We also show that the bimodality of the size distribution function is required to correctly describe the huge two-phase coexistence gap and the nonlinearity of the function γ(η), two important features that these colloidal suspensions exhibit.

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.

Phase transitions in nanoconfined binary mixtures of highly oriented colloidal rods.

We analyse a binary mixture of colloidal parallel hard cylindrical particles with identical diameters but dissimilar lengths L(1) and L(2), with s = L(2)/L(1) = 3, confined by two parallel hard walls in a planar slit-pore geometry, using a fundamental-measure density functional theory. This model presents nematic (N) and two types of smectic (S) phases, with first- and second-order N-S bulk transitions and S-S demixing, and surface behaviour at a single hard wall which includes complete wetting by the S phase mediated (or not) by an infinite number of surface-induced layering (SIL) transitions. In the present paper the effects of confinement on this model colloidal fluid mixture are studied. Confinement brings about profound changes in the phase diagram, resulting from competition between the three relevant length scales: pore width h, smectic period d and length ratio s. Four main effects are identified: (i) second-order bulk N-S transitions are suppressed; (ii) demixing transitions are weakly affected, with small shifts in the mu(1)-mu(2) (chemical potentials) plane; (iii) confinement-induced layering (CIL) transitions occurring in the two confined one-component fluids in some cases merge with the demixing transition; (iv) surface-induced layering (SIL) transitions occurring at a single surface as coexistence conditions are approached are also shifted in the confined fluid. The trends with pore size are analysed by means of complete mu(1)-mu(2) and p-x[combining macron] (pressure-mean pore composition) phase diagrams for particular values of pore size. This work, which is the first one to address the behaviour of liquid-crystalline mixtures under confinement, could be relevant as a first step to understand the self-assembling properties of mixtures of metallic nanoparticles under external fields in restricted geometry.

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.

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.

Highly confined mixtures of parallel hard squares: A density-functional-theory study.

Using the fundamental-measure density-functional theory, we study theoretically the phase behavior of extremely confined mixtures of parallel hard squares in slit geometry. The pore width is chosen such that configurations consisting of two consecutive big squares, or three small squares, in the transverse direction, perpendicular to the walls, are forbidden. We analyze two different mixtures with edge lengths of species selected so as to allow or forbid one big plus one small square to fit into the channel. For the first mixture we obtain first-order transitions between symmetric and asymmetric packings of particles: Small and big squares are preferentially adsorbed at different walls. Asymmetric configurations are shown to lead to more efficient packing at finite pressures. We argue that the stability region of the asymmetric phase in the pressure-composition plane is bounded so that the symmetric phase is stable at low and very high pressure. For the second mixture, we observe strong demixing between phases which are rich in different species. Demixing occurs in the lateral direction, i.e., the dividing interface is perpendicular to the walls, and phases exhibit symmetric density profiles. The possible experimental realization of this behavior (which in practical terms is precluded by jamming) in strictly two-dimensional systems is discussed. Finally, the phase behavior of a mixture with periodic boundary conditions is analyzed and the differences and similarities between the latter and the confined system are discussed. We claim that, although exact calculations exclude the existence of true phase transitions in (1+ε)-dimensional systems, density-functional theory is still successful in describing packing properties of large clusters of particles.

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.

Fundamental-measure density functional for mixtures of parallel hard cylinders.

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.

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.

Positional ordering of hard adsorbate particles in tubular nanopores.

The phase behavior and structural properties of a monolayer of hard particles is examined in such a confinement where the adsorbed particles are constrained to the surface of a narrow hard cylindrical pore. The diameter of the pore is chosen such that only first- and second-neighbor interactions occur between the hard particles. The transfer operator method of [Percus and Zhang, Mol. Phys. 69, 347 (1990)MOPHAM0026-897610.1080/00268979000100241] is reformulated to obtain information about the structure of the monolayer. We have found that a true phase transition is not possible in the examined range of pore diameters. The monolayer of hard spheres undergoes a structural change from fluidlike order to a zigzaglike solid one with increasing surface density. The case of hard cylinders is different in the sense that a layering takes place continuously between a low-density one-row and a high-density two-row monolayer. Our results reveal a clear discrepancy with classical density functional theories, which do not distinguish smecticlike ordering in bulk from that in narrow periodic pores.