A generalisation of the Onsager trial-function approach: describing nematic liquid crystals with an algebraic equation of state

Self-Assembly of All-DNA Rods with Controlled Patchiness

Double-stranded DNA (dsDNA) fragments exhibit noncovalent attractive interactions between their tips. It is still unclear how DNA liquid crystal self-assembly is affected by such blunt-end attractions. It is demonstrated that stiff dsDNA fragments with moderate aspect ratio can specifically self-assemble in concentrated aqueous solutions into different types of smectic mesophases on the basis of selectively screening of blunt-end DNA stacking interactions. To this end, this type of attractions are engineered at the molecular level by constructing DNA duplexes where the attractions between one or both ends are screened by short hairpin caps. All-DNA bilayer and monolayer smectic-A type of phases, as well as a columnar phase, can be stabilized by controlling attractions strength. The results imply that the so far elusive smectic-A in DNA rod-like liquid crystals is a thermodynamically stable phase. The existence of the bilayer smectic phase is confirmed by Monte-Carlo simulations of hard cylinders decorated with one attractive terminal site. This work demonstrates that DNA blunt-ends behave as well-defined monovalent attractive patches whose strength and position can be potentially precisely tuned, highlighting unique opportunities concerning the stabilization of nonconventional DNA-based lyotropic liquid crystal phases assembled by all-DNA patchy particles with arbitrary geometry and composition.

Algebraic equations of state for the liquid crystalline phase behavior of hard rods

Based on simplifications of previous numerical calculations [H. Graf and H. Löwen, Phys. Rev. E 59, 1932 (1999)1063-651X10.1103/PhysRevE.59.1932], we propose algebraic free energy expressions for the smectic-A liquid crystal phase and the crystal phases of hard spherocylinders. Quantitative agreement with simulations is found for the resulting equations of state. The free energy expressions can be used to straightforwardly compute the full phase behavior for all aspect ratios and to provide a suitable benchmark for exploring how attractive interrod interactions mediate the phase stability through perturbation approaches such as free-volume or van der Waals theory.

Effect of Size Polydispersity on the Pitch of Nanorod Cholesterics

Many nanoparticle-based chiral liquid crystals are composed of polydisperse rod-shaped particles with considerable spread in size or shape, affecting the mesoscale chiral properties in, as yet, unknown ways. Using an algebraic interpretation of Onsager-Straley theory for twisted nematics, we investigate the role of length polydispersity on the pitch of nanorod-based cholesterics with a continuous length polydispersity, and find that polydispersity enhances the twist elastic modulus, K 2 , of the cholesteric material without affecting the effective helical amplitude, K t . In addition, for the infinitely large average aspect ratios considered here, the dependence of the pitch on the overall rod concentration is completely unaffected by polydispersity. For a given concentration, the increase in twist elastic modulus (and reduction of the helical twist) may be up to 50% for strong size polydispersity, irrespective of the shape of the unimodal length distribution. We also demonstrate that the twist reduction is reinforced in bimodal distributions, obtained by doping a polydisperse cholesteric with very long rods. Finally, we identify a subtle, non-monotonic change of the pitch across the isotropic-cholesteric biphasic region.

Isotropic-Nematic Transition and Demixing Behavior in Binary Mixtures of Hard Spheres and Hard Spherocylinders Confined in a Disordered Porous Medium: Scaled Particle Theory

We develop the scaled particle theory to describe the thermodynamic properties and orientation ordering of a binary mixture of hard spheres (HS) and hard spherocylinders (HSC) confined in a disordered porous medium. Using this theory, the analytical expressions of the free energy, the pressure, and the chemical potentials of HS and HSC have been derived. The improvement of obtained results is considered by introducing the Carnahan-Starling-like and Parsons-Lee-like corrections. Phase diagrams for the isotropic-nematic transition are calculated from the bifurcation analysis of the integral equation for the orientation singlet distribution function and from the conditions of thermodynamic equilibrium. Both the approaches correctly predict the isotropic-nematic transition at low concentrations of hard spheres. However, the thermodynamic approach provides more accurate results and is able to describe the demixing phenomena in the isotropic and nematic phases. The effects of porous medium on the isotropic-nematic phase transition and demixing behavior in a binary HS/HSC mixture are discussed.

Structure and Interfacial Tension of a Hard-Rod Fluid in Planar Confinement

The structural properties and interfacial tension of a fluid of rodlike hard-spherocylinder particles in contact with hard structureless flat walls are studied by means of Monte Carlo simulation. The calculated surface tension between the rod fluid and the substrate is characterized by a nonmonotonic trend as a function of the bulk concentration (density) over the range of isotropic bulk concentrations. As suggested by earlier theoretical studies, a surface-ordering scenario is confirmed by our simulations: the local orientational order close to the wall changes from uniaxial to biaxial nematic when the bulk concentration reaches about 85% of the value at the onset of the isotropic-nematic phase transition. The surface ordering coincides with a wetting transition whereby the hard wall is wetted by a nematic film. Accurate values of the fluid-solid surface tension, the adsorption, and the average particle-wall contact distance are reported (over a broad range of densities into the dense nematic region for the first time), which can serve as a useful benchmark for future theoretical and experimental studies on confined rod fluids. The simulation data are supplemented with predictions from second-virial density functional theory, which are in good qualitative agreement with the simulation results.

Perturbation Theory versus Thermodynamic Integration. Beyond a Mean-Field Treatment of Pair Correlations in a Nematic Model Liquid Crystal

The phase behavior and structure of a simple square-well bulk fluid with anisotropic interactions is described in detail. The orientation dependence of the intermolecular interactions allows for the formation of a nematic liquid-crystalline phase in addition to the more conventional isotropic gas and liquid phases. A version of classical density functional theory (DFT) is employed to determine the properties of the model, and comparisons are made with the corresponding data from Monte Carlo (MC) computer simulations in both the grand canonical and canonical ensembles, providing a benchmark to assess the adequacy of the DFT results. A novel element of the DFT approach is the assumption that the structure of the fluid is dominated by intermolecular interactions in the isotropic fluid. A so-called augmented modified mean-field (AMMF) approximation is employed accounting for the influence of anisotropic interactions. The AMMF approximation becomes exact in the limit of vanishing density. We discuss advantages and disadvantages of the AMMF approximation with respect to an accurate description of isotropic and nematic branches of the phase diagram, the degree of orientational order, and orientation-dependent pair correlations. The performance of the AMMF approximations is found to be good in comparison with the MC data; the AMMF approximation has clear advantages with respect to an accurate and more detailed description of the fluid structure. Possible strategies to improve the DFT are discussed.

Isotropic-nematic phase transition of uniaxial variable softness prolate and oblate ellipsoids

Onsager's theory of the isotropic-nematic phase separation of rod shaped particles is generalized to include particle softness and attractions in the anisotropic interparticle force field. The procedure separates a scaled radial component from the angular integral part, the latter being treated in essentially the same way as in the original Onsager formulation. Building on previous treatments of more idealised hard-core particle models, this is a step toward representing more realistic rod-like systems and also allowing temperature (and in principle specific chemical factors) to be included at a coarse grained level in the theory. The focus of the study is on the coexisting concentrations and associated coexistence properties. Prolate and oblate ellipsoids are considered in both the small and very large aspect ratio limits. Approximations to the terms in the angular integrals derived assuming the very large (prolate) and very small (oblate) aspect ratios limits are compared with the formally exact treatment. The approximation for the second virial coefficient matches the exact solution for aspect ratios above about 20 for the prolate ellipsoids and less than ca. 0.05 for the oblate ellipsoids from the numerical evaluation of the angular integrals. The temperature dependence of the coexistence density could be used to help determine the interaction potential of two molecules. The method works at temperatures above a certain threshold temperature where the second virial coefficient is positive.

Percolation in suspensions of polydisperse hard rods: Quasi universality and finite-size effects

We present a study of connectivity percolation in suspensions of hard spherocylinders by means of Monte Carlo simulation and connectedness percolation theory. We focus attention on polydispersity in the length, the diameter, and the connectedness criterion, and we invoke bimodal, Gaussian, and Weibull distributions for these. The main finding from our simulations is that the percolation threshold shows quasi universal behaviour, i.e., to a good approximation, it depends only on certain cumulants of the full size and connectivity distribution. Our connectedness percolation theory hinges on a Lee-Parsons type of closure recently put forward that improves upon the often-used second virial approximation [T. Schilling, M. Miller, and P. van der Schoot, e-print arXiv:1505.07660 (2015)]. The theory predicts exact universality. Theory and simulation agree quantitatively for aspect ratios in excess of 20, if we include the connectivity range in our definition of the aspect ratio of the particles. We further discuss the mechanism of cluster growth that, remarkably, differs between systems that are polydisperse in length and in width, and exhibits non-universal aspects.

Connectedness percolation in isotropic systems of monodisperse spherocylinders

The dependence upon aspect ratio and connectedness range of the percolation threshold for monodisperse spherocylinders as calculated from an analogy between lattice and continuum percolation is compared with findings from: (i) recent Monte Carlo (MC) simulations, and: (ii) a recent elaboration of an integral equation-based approach that uses the connectedness Ornstein-Zernike equation. The results from each of these approaches are found to be in reasonable semi-quantitative agreement, especially for the case of particles that have a finite hard core diameter.

Random geometric graph description of connectedness percolation in rod systems

The problem of continuum percolation in dispersions of rods is reformulated in terms of weighted random geometric graphs. Nodes (or sites or vertices) in the graph represent spatial locations occupied by the centers of the rods. The probability that an edge (or link) connects any randomly selected pair of nodes depends upon the rod volume fraction as well as the distribution over their sizes and shapes, and also upon quantities that characterize their state of dispersion (such as the orientational distribution function). We employ the observation that contributions from closed loops of connected rods are negligible in the limit of large aspect ratios to obtain percolation thresholds that are fully equivalent to those calculated within the second-virial approximation of the connectedness Ornstein-Zernike equation. Our formulation can account for effects due to interactions between the rods, and many-body features can be partially addressed by suitable choices for the edge probabilities.

Generalized Onsager theory for strongly anisometric patchy colloids

The implications of soft "patchy" interactions on the orientational disorder-order transition of strongly elongated colloidal rods and flat disks is studied within a simple Onsager-van der Waals density functional theory. The theory provides a generic framework for studying the liquid crystal phase behaviour of highly anisometric cylindrical colloids which carry a distinct geometrical pattern of repulsive or attractive soft interactions localized on the particle surface. In this paper, we apply our theory to the case of charged rods and disks for which the local electrostatic interactions can be described by a screened-Coulomb potential. We consider infinitely thin rod like cylinders with a uniform line charge and infinitely thin discotic cylinders with several distinctly different surface charge patterns. Irrespective of the backbone shape the isotropic-nematic phase diagrams of charged colloids feature a generic destabilization of nematic order at low ionic strength, a dramatic narrowing of the biphasic density region, and a reentrant phenomenon upon reducing the electrostatic screening. The low screening regime is characterized by a complete suppression of nematic order in favor of positionally ordered liquid crystal phases.

Liquid crystal phase behaviour of attractive disc-like particles

We employ a generalized van der Waals-Onsager perturbation theory to construct a free energy functional capable of describing the thermodynamic properties and orientational order of the isotropic and nematic phases of attractive disc particles. The model mesogen is a hard (purely repulsive) cylindrical disc particle decorated with an anisotropic square-well attractive potential placed at the centre of mass. Even for isotropic attractive interactions, the resulting overall inter-particle potential is anisotropic, due to the orientation-dependent excluded volume of the underlying hard core. An algebraic equation of state for attractive disc particles is developed by adopting the Onsager trial function to characterize the orientational order in the nematic phase. The theory is then used to represent the fluid-phase behaviour (vapour-liquid, isotropic-nematic, and nematic-nematic) of the oblate attractive particles for varying values of the molecular aspect ratio and parameters of the attractive potential. When compared to the phase diagram of their athermal analogues, it is seen that the addition of an attractive interaction facilitates the formation of orientationally-ordered phases. Most interestingly, for certain aspect ratios, a coexistence between two anisotropic nematic phases is exhibited by the attractive disc-like fluids.

The isotropic-nematic phase transition of tangent hard-sphere chain fluids--pure components

An extension of Onsager's second virial theory is developed to describe the isotropic-nematic phase transition of tangent hard-sphere chain fluids. Flexibility is introduced by the rod-coil model. The effect of chain-flexibility on the second virial coefficient is described using an accurate, analytical approximation for the orientation-dependent pair-excluded volume. The use of this approximation allows for an analytical treatment of intramolecular flexibility by using a single pure-component parameter. Two approaches to approximate the effect of the higher virial coefficients are considered, i.e., the Vega-Lago rescaling and Scaled Particle Theory (SPT). The Onsager trial function is employed to describe the orientational distribution function. Theoretical predictions for the equation of state and orientational order parameter are tested against the results from Monte Carlo (MC) simulations. For linear chains of length 9 and longer, theoretical results are in excellent agreement with MC data. For smaller chain lengths, small errors introduced by the approximation of the higher virial coefficients become apparent, leading to a small under- and overestimation of the pressure and density difference at the phase transition, respectively. For rod-coil fluids of reasonable rigidity, a quantitative comparison between theory and MC simulations is obtained. For more flexible chains, however, both the Vega-Lago rescaling and SPT lead to a small underestimation of the location of the phase transition.

Enhanced Landau-de Gennes potential for nematic liquid crystals from a systematic coarse-graining procedure

The macroscopic theory of nematics is conveniently described in terms of the phenomenological Landau-de Gennes free energy. Here we show how such an effective free energy can be obtained explicitly from a microscopic model via the help of a systematic coarse-graining procedure. We test our approach for the two- and three-dimensional Lebwohl-Lasher model of nematics. The effective free energy that we obtain is consistent with the phenomenological Landau-de Gennes form for weak orientational ordering and the Maier-Saupe theory of the isotropic-nematic transition. For strong orientational ordering, however, the effective free energy increases rapidly and diverges logarithmically near the fully oriented state. The explicit form for the regularized Landau-de Gennes potential proposed here restricts the order parameter to physical admissible values and reproduces our numerical data accurately.

Isotropic and nematic phases of the hard spheroid fluid: a virial approach

We use a high level virial expansion to investigate the properties of the isotropic and nematic phases of the hard spheroid fluid. We use the Monte Carlo techniques described previously to calculate the virial coefficients up to seventh order and we represent the dependence of these coefficients on particle orientations via a spherical harmonic expansion. The expansion coefficients are determined using Lebedev quadrature which carries out the angular integration required exactly. For fairly spherical spheroids (1/3<a/b<3, where a and b are the semiaxes of the spheroid) the virial expansion appears to be convergent in both the isotropic and nematic phases up to high density. For more aspherical particles, the predictions varied in a more erratic fashion upon increasing the number of virials. The use of a finite angular basis set to represent the orientationally dependent virial coefficients could be a contributory cause to this.

Cholesteric order in systems of helical Yukawa rods

We consider the interaction potential between two chiral rod-like colloids which consist of a thin cylindrical backbone decorated with a helical charge distribution on the cylinder surface. For sufficiently slender helical rods a simple scaling expression is derived which relates the chiral 'twisting' potential to the microscopic properties of the particles, such as the internal helical pitch, charge density and electrostatic screening parameter. To predict the behaviour of the macroscopic cholesteric pitch of the fluid bulk phase we invoke a simple second-virial theory generalized to treat anisotropic states with weakly twisted director fields. It is shown that, while particles with weakly coiled helices always form a cholesteric phase whose helical sense is commensurate with that of the internal helix, more strongly coiled rods lead to the formation of a cholesteric state of opposite sense. The correlation between the helical symmetry at the microscopic and macroscopic scale is found to be very sensitive to the pitch of the Yukawa helix. Mixing helical particles of sufficiently disparate length and internal pitch may give rise to a demixing of the uniform cholesteric phase into two fractions with a different macroscopic pitch. Our findings could be relevant to the interpretation of experimental observations in systems of cellulose and chitin microfibres, DNA and fd virus rods.

Liquid crystalline and antinematic behavior of shape-persistent macrocycles from molecular-dynamics simulations

In this work we present a molecular-dynamics study of a coarse-grained (CG) model for a system of planar shape-persistent macrocycles (SPMs). SPMs are synthetic organic rigid macromolecules typically comprised of meta- and para-aromatics groups connected by acetylene and/or diacetylene units. In the CG model, each SPM is represented as a rigid hexagonal arrangement of 24 soft-repulsive spheres, resembling a large ring or hoop. The supramolecular arrangement of these macrocycles at high pressures is studied using N-P-T molecular-dynamics simulation both by expansion of an initial hexagonal lattice structure and also by compression of an isotropic phase. In both cases, systems under consideration exhibit an isotropic-smectic-A phase transition, which is detected by monitoring relevant order parameters and analyzing snapshots of equilibrium configurations. The smectic-A phase is unique; although the molecules form layers, the system presents antinematic order where the orientation of the molecular axes is perpendicular to the direction of the layers themselves. Due to their planar geometry, the SPM molecules would be expected to form columnar or nematic phases. On the contrary, these phases seem suppressed by a novel smectic-A phase, formed by the mutual interpenetration of the cycles. These results are a unique example of how molecular nonconvexity can, by itself, induce mesomorphism in anisotropic systems.

Supersaturated dispersions of rodlike viruses with added attraction

The kinetics of isotropic-nematic (I-N) and nematic-isotropic (N-I) phase transitions in dispersions of rodlike fd viruses are studied. Concentration quenches were applied using pressure jumps in combination with polarization microscopy, birefringence, and turbidity measurements. The full biphasic region could be accessed, resulting in the construction of an experimental analog of the bifurcation diagram. The N-I spinodal points for dispersions of rods with varying concentrations of depletion agent (dextran) were obtained from orientation quenches using cessation of shear flow in combination with small-angle light scattering. We found that the location of the N-I spinodal point is independent of the attraction, which was confirmed by theory. Surprisingly, the experiments showed that also the absolute induction time, the critical nucleus, and the growth rate are insensitive of the attraction if the concentration is scaled to the distance to the phase boundaries.