Phase behaviour of parallel hard rods in confinement: an Onsager theory study

Structural and dynamical equilibrium properties of hard board-like particles in parallel confinement

We performed Monte Carlo and dynamic Monte Carlo simulations to model the diffusion of monodispersed suspensions composed of impenetrable cuboidal particles, specifically hard board-like particles (HBPs), in the presence of parallel hard walls. The impact of the walls was investigated by adjusting the size of the simulation box while maintaining constant packing fractions, fixed at η = 0.150, for systems consisting of HBPs with prolate, dual-shaped, and oblate geometries. We observed that increasing the distance between the walls led to the recovery of an isotropic bulk phase, while local particle organization near the walls remained stable. Due to their shape, oblate HBPs exhibit more efficient anchoring at wall surfaces compared to prolate shapes. The formation of nematic-like particle assemblies near the walls, confirmed by theoretical calculations based on density functional theory, significantly influenced local particle dynamics. This effect was particularly pronounced to the extent that a modest portion of cuboids near the walls tended to diffuse exclusively in planes parallel to the confinement, even more efficiently than observed in the bulk regions.

Confinement effects on the random sequential adsorption packings of elongated particles in a slit

The behavior of a system of two-dimensional elongated particles (discorectangles) packed in a slit between the two parallel walls was analyzed using a simulation approach. The packings were produced using the random sequential adsorption model with continuous positional and orientational degrees of freedom. The aspect ratio (length-to-width ratio, ɛ=l/d) of the particles was varied within the range ɛ∈[1;32] while the distance between the walls was varied within the range h/d∈[1;80]. The properties of deposits in jammed state [the coverage, the order parameter, and the long-range (percolation) connectivity between particles] were studied numerically. The values of ɛ and h significantly affected the structure of the packings and the percolation connectivity. Particularly, the observed nontrivial dependencies of the jamming coverage φ(ɛ) or φ(h) were explained by the interplay of the different geometrical factors related to confinement, particle orientation degrees of freedom and excluded volume effects.

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.

Orientational ordering of confined hard rods: the effect of shape anisotropy on surface ordering and capillary nematization

We examine the ordering properties of rectangular hard rods with length L and diameter D at a single planar wall and between two parallel hard walls using the second virial density-functional theory. The theory is implemented in the three-state Zwanzig approximation, where only three mutually perpendicular directions are allowed for the orientations of hard rods. The effect of varying shape anisotropy is examined at L/D=10,15,and20. In contact with a single hard wall, the density profiles show planar ordering, damped oscillatory behavior, and a wall-induced surface ordering transition below the coexisting isotropic density of a bulk isotropic-nematic (I-N) phase transition. Upon approaching the coexisting isotropic density, the thickness of the nematic film diverges logarithmically, i.e., the nematic wetting is complete for any shape anisotropy. In the case of confinement between two parallel hard walls, it is found that the continuous surface ordering transition depends strongly on the distance between confining walls H for H<L, while it depends weakly on H for H>L. The minimal density at which a surface ordering transition can be realized is located at around H∼2D for all studied shape anisotropies due to the strong interference effect between the two hard walls. The first-order I-N phase transition of the bulk system becomes a surface ordered isotropic I_{B} to capillary nematic N_{B} phase transition in the slit pore. This first-order I_{B}-N_{B} transition weakens with decreasing pore width and terminates in a critical point for all studied shape anisotropies.

Hard rods in a cylindrical pore: the nematic-to-smectic phase transition

The effect of cylindrical confinement on the phase behaviour of a system of parallel hard rods is studied using Onsager's second-virial theory. The hard rods are represented as hard cylinders of diameter D and length L, while the cylindrical pore is infinite with diameter W. The interaction between the wall and the rods is hard repulsive, and it is assumed that molecules are parallel to the surface of the pore (planar anchoring). In very narrow pores (D < W < 2D), the structure is homogeneous and the system behaves as a one-dimensional Tonks gas. For wider pores, inhomogeneous fluid structures emerge because of the lowering of the average excluded volume due to the wall-particle interaction. The bulk nematic-smectic A phase transition is replaced by a transition between inhomogeneous nematic and smectic A phases. The smectic is destabilized with respect to the nematic for decreasing pore width; this effect becomes substantial for W < 10D. For W > 100D, results for bulk and confined fluids agree well due to the short range effect of the wall (∼ 3-4D).

Suppression of phase transitions in a confined rodlike liquid crystal

The nematic-to-isotropic, crystal-to-nematic, and supercooled liquid-to-glass temperatures are studied in the liquid crystal 4-pentyl-4'-cyanobiphenyl (5CB) confined in self-ordered nanoporous alumina. The nematic-to-isotropic and the crystal-to-nematic transition temperatures are reduced linearly with the inverse pore diameter. The finding that the crystalline phase is completely suppressed in pores having diameters of 35 nm and below yields an estimate of the critical nucleus size. The liquid-to-glass temperature is reduced in confinement as anticipated by the model of rotational diffusion within a cavity. These results provide the pertinent phase diagram for a confined liquid crystal and are of technological relevance for the design of liquid crystal-based devices with tunable optical, thermal, and dielectric properties.