Empty smectics of hard nanorings: insights from a second-virial theory

Inspired by recent simulations on highly open liquid crystalline structures formed by rigid planar nanorings we present a simple theoretical framework explaining the prevalence of smectic over nematic ordering in systems of ring-shaped objects. The key part of our study is a calculation of the excluded volume of such non-convex particles in the limit of vanishing thickness to diameter ratio. Using a simple stability analysis we then show that dilute systems of ring-shaped particles have a strong propensity to order into smectic structures with an unusual antinematic order while solid disks of the same dimensions exhibit nematic order. Since our model rings have zero internalvolume these smectic structures are essential empty, resembling the strongly porous structures found in simulation. We argue that the antinematic intralamellar order of the rings plays an essential role in stabilizing these novel smectic structures

Liquid crystal phase behaviour of colloidal mixtures

The central aim of this thesis is to theoretically investigate the effects of mixing anisometric colloidal particles with different shapes on their (lyotropic) liquid crystal phase behaviour. Many of the studies to be described in this thesis have been triggered off by recent experimental observations in mixtures of colloids with well-controlled shapes and interactions. In particular, we mention the experimental work of Van der Kooij [F. M. van der Kooij, thesis, Utrecht University, 2000] who investigated a vast number of mixtures which display many interesting phenomena left open for theoretical interpretation. One of our primary goals in this work is to account for these experimental observations by constructing simple, yet realistic, models for the colloidal systems under consideration and by scrutinizing relevant aspects of their phase behaviour. The first part of this thesis will be devoted to binary mixtures of anisometric particles. In Chapter 2 a simple model is proposed that allows to qualitatively explain the recently observed isotropic-nematic density inversion in polydisperse systems of colloidal platelets. In the next two chapters we shall be concerned with mixtures of rods and platelets and provide a theoretical underpinning for the low-concentration part of the experimental phase diagram. We also establish the possible stability of the disputed biaxial nematic phase in experimentally realizable mixtures. In Chapter 5 we conclude the first part with an overview on demixing transitions within the isotropic and nematic phases of binary mixtures of particles whose size differs only in one particle dimension. Previously published results for rodlike particles will be combined with new results for platelets to compare phase diagram topologies and demixing mechanisms pertaining to the various mixtures. In the second part of this thesis we address the more challenging issue of calculating phase equilibria in polydisperse mixtures of anisometric particles. In Chapter 6 we present a study of isotropic-nematic phase coexistence in systems of length-polydisperse hard rods, focussing in particular on fractionation effects and the possibility of a demixing of the nematic phase. Chapter 7 deals with polydisperse systems of thickness-polydisperse platelets. The binary model, introduced in Chapter 2, is extended to a polydisperse one which allows us to provide a more realistic, albeit still qualitative, description of the experimental observations. In Chapter 8 we provide a preliminary calculation on the competition between smectic and columnar ordering in systems of polydisperse hard rods. As a first-order approximation we consider an artificial model system of perfectly aligned cylinders. The possibilities of extending the approach towards a more realistic one will be discussed. The contents of Chapter 9 of this thesis differ somewhat from the rest because of the introduction of an external field. Inspired by recent experimental observations of a significant sedimentation in dispersions of platelets we illustrate the drastic effect of gravity on the phase behaviour of colloidal mixtures. As an example we consider a system of sedimenting platelets mixed with non-sedimenting ideal polymers, as studied in experiment. Also here, the results of the calculations reveal an improved description of the experimentally observed behaviour. Finally, in Chapter 10 we present a free-volume theory for a columnar state of hard platelets by combining the traditional cell model with an appropriate fluid description, accounting for the rotational freedom of the particles in the (one-dimensional) direction of the phase. Excellent quantitative agreement is found with recent computer simulation results for various thermodynamic and structural properties of a dense columnar state

Active Brownian Particles. From Individual to Collective Stochastic Dynamics

We review theoretical models of individual motility as well as collective dynamics and pattern formation of active particles. We focus on simple models of active dynamics with a particular emphasis on nonlinear and stochastic dynamics of such self-propelled entities in the framework of statistical mechanics. Examples of such active units in complex physico-chemical and biological systems are chemically powered nano-rods, localized patterns in reaction-diffusion system, motile cells or macroscopic animals. Based on the description of individual motion of point-like active particles by stochastic differential equations, we discuss different velocity-dependent friction functions, the impact of various types of fluctuations and calculate characteristic observables such as stationary velocity distributions or diffusion coefficients. Finally, we consider not only the free and confined individual active dynamics but also different types of interaction between active particles. The resulting collective dynamical behavior of large assemblies and aggregates of active units is discussed and an overview over some recent results on spatiotemporal pattern formation in such systems is given.Comment: 161 pages, Review, Eur Phys J Special-Topics, accepte

Sedimentation and multi-phase equilibria in mixtures of platelets and ideal polymer

The role of gravity in the phase behaviour of mixtures of hard colloidal plates without and with non-adsorbing ideal polymer is explored theoretically. By analyzing the (macroscopic) osmotic equilibrium conditions, we show that sedimentation of the colloidal platelets is significant on a height range of even a centimeter. Gravity enables the system to explore a large density range within the height of a test tube which may give rise to the simultaneous presence of multiple phases. As to plate-polymer mixtures, it is shown that sedimentation may lead to a four-phase equilibrium involving an isotropic gas and liquid phase, nematic and columnar phase. The phenomenon has been observed experimentally in systems of colloidal gibbsite platelets mixed with PDMS polymer

Isotropic-nematic density inversion in a binary mixture of thin and thick hard platelets

We study the phase behavior of a binary mixture of thin and thick hard platelets, using Onsager’s second virial theory for binary mixtures in the Gaussian approximation. Higher virial terms are included by rescaling the excluded volume part of the Onsager free energy using a modified form of the Carnahan-Starling free energy for hard spheres (Parsons’ approach). Our calculations provide a simple explanation for the isotropicnematic (I-N) density inversion, as experimentally observed in systems of polydisperse gibbsite platelets by Van der Kooij et al. (J. Phys. Chem. B 2001, 105, 1696). In these systems, a nematic upper phase was found to coexist with an isotropic bottom phase. We confirm the original conjecture of the authors, which states that the phenomenon originates from a pronounced fractionation in thickness between the phases, such that the thick platelets are largely expelled from the nematic phase and preferentially occupy the isotropic phase. Our calculations show that the inverted state is found in a major part of the I-N coexistence region. In addition, a nematic-nematic demixing transition is located at sufficiently high osmotic pressures for any thickness ratio L2/L1 > 1. The N-N coexistence region is bounded by a lower critical point which shifts toward lower values as the thickness ratio is increased. At high thickness ratios (L2/L1 > 3.3), a triphasic coexistence is found at which two nematic phases coexist with an isotropic phase. We show that the demixing transition is driven by a small Φ(L/D) contribution to the excluded volume entropy