Effect of orientational restriction on monolayers of hard ellipsoids

Packing and emergence of the ordering of rods in a spherical monolayer

Spatially ordered systems confined to surfaces such as spheres exhibit interesting topological structures because of curvature induced frustration in orientational and translational order. The study of these structures is important for investigating the interplay between the geometry, topology, and elasticity, and for their potential applications in materials science, such as engineering directionally binding particles. In this work, we numerically simulate a spherical monolayer of soft repulsive spherocylinders (SRSs) and study the packing of rods and their ordering transition as a function of the packing fraction. In the model that we study, the centers of mass of the spherocylinders (situated at their geometric centers) are constrained to move on a spherical surface. The spherocylinders are free to rotate about any axis that passes through their respective centers of mass. We show that, up to moderate packing fractions, a two dimensional liquid crystalline phase is formed whose orientational ordering increases continuously with increasing density. This monolayer of orientationally ordered SRS particles at medium densities resembles a hedgehog-long axes of the SRS particles are aligned along the local normal to the sphere. At higher packing fractions, the system undergoes a transition to the solid phase, which is riddled with topological point defects (disclinations) and grain boundaries that divide the whole surface into several domains.

From n-layer planar ordering to the monolayer homeotropic structure of confined hard rods: The effect of shape anisotropy and wall-to-wall separation

Using the Parsons-Lee theory, we examined the effect of shape anisotropy and the wall-to-wall separation (H) on the phase behavior of the hard parallelepiped rods with dimensions L, D, and D (L>D) in such narrow slitlike pores which only one homeotropic layer can form. The phase structures, including biaxiality, planar nematic layering transition as well as planar to homeotropic, were studied for some separations in the range 2.5D≤H≤10.0D for H-D≤L<H.

Phase diagram of hard squares in slit confinement

This work shows a complete phase diagram of hard squares of side length σ in slit confinement for H < 4.5, H being the wall to wall distance measured in σ units, including the maximal packing fraction limit. The phase diagram exhibits a transition between a single-row parallel 1-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□ and a zigzag 2-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\diamond }$$\end{document}◇ˆ structures for H_c(2) = (2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{{\bf{2}}}$$\end{document}2 − 1) < H < 2, and also another one involving the 1-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□ and 2-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□ structures (two parallel rows) for 2 < H < H_c(3) (H_c(n) = n − 1 + \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{{\bf{2}}{\boldsymbol{n}}-{\bf{1}}}$$\end{document}2n−1/n is the critical wall-to-wall distance for a (n − 1)-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□ to n-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\diamond }$$\end{document}◇ transition and where n-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\diamond }$$\end{document}◇ represents a structure formed by tilted rectangles, each one clustering n stacked squares), and a triple point for H_t \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\simeq }}$$\end{document}≃ 2.005. In this triple point there coexists the 1-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□, 2-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□, and 2-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\diamond }$$\end{document}◇ˆ structures. For regions H_c(3) < H < H_c(4) and H_c(4) < H < H_c(5), very similar pictures arise. There is a (n − 1)-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□ to a n-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\diamond }$$\end{document}◇ strong transition for H_c(n) < H < n, followed by a softer (n − 1)-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□ to n-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□ transition for n < H < H_c(n + 1). Again, at H \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\gtrsim }}$$\end{document}≳ n there appears a triple point, involving the (n − 1)-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□, n-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□, and n-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\diamond }$$\end{document}◇ structures. The similarities found for n = 2, 3 and 4 lead us to propose a tentative phase diagram for H_c(n) < H < H_c(n + 1) (n ∈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{{\mathbb{N}}}}$$\end{document}ℕ, n > 2), where structures (n − 1)-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□, n-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\square }}$$\end{document}□, and n-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\diamond }$$\end{document}◇ fill the phase diagram. Simulation and Onsager theory results are qualitatively consistent.

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.