Hard Rods Research Articles

Lattice Fundamental Measure Theory Beyond 0D Cavities: Dimers on Square Lattices

Using classical density functional theory, we study the behavior of dimers, i.e. hard rods of length L=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L=2$$\\end{document}, on a two-dimensional cubic lattice. For deriving a free energy functional, we employ Levy’s prescription which is based on the minimization of a microscopic free energy with respect to the many-body probability under the constraint of a fixed density profile. Using that, we recover the "0D cavity" functional originally found by Lafuente and Cuesta and derive an extension by applying a more general "cluster density functional theory" method introduced by Lafuente and Cuesta as well. Moreover, we introduce a new free energy functional, which is based on approximated configurational probabilities. Both derived free energy functionals are exact on cavities that can hold at most two particles simultaneously. The first functional allows to improve the prediction of the free energy in bulk and both of them improve the prediction in highly confined systems, especially for high packing fractions, in comparison to the "0D cavity" functional.

Soft and Deformable Thermoresponsive Hollow Rod-Shaped Microgels.

Depending on their aspect ratio, rod-shaped particles exhibit a much richer 2D and 3D phase behavior than their spherical counterparts, with additional nematic and smectic phases accompanied by defined orientational ordering. While the phase diagram of colloidal hard rods is extensively explored, little is known about the influence of softness in such systems, partly due to the absence of appropriate model systems. Additionally, investigating higher volume fractions for long rods is usually complicated because non-equilibrium dynamical arrest is likely to precede the formation of more defined states. This has motivated us to develop micrometric rod-like microgels with limited sedimentation that can respond to temperature and reversibly reorganize into defined phases via annealing and seeding procedures. A detailed procedure is presented for synthesizing rod-shaped hollow poly(N-isopropylacrylamide) microgels using micrometric silica rods as sacrificial templates. Their morphological characterization is conducted through a combination of microscopy and light scattering techniques, evidencing the unconstrained swelling of rod-shaped hollow microgels compared to core-shell microgel rods. Different aspects of their assembly in dispersion and at interfaces are further tested to illustrate the opportunities and challenges offered by such systems that combine softness, anisotropy, andthermoresponsivity.

Hyperdensity Functional Theory of Soft Matter.

We present a scheme for investigating arbitrary thermal observables in spatially inhomogeneous equilibrium many-body systems. Extending the grand canonical ensemble yields any given observable as an explicit hyperdensity functional. Associated local fluctuation profiles follow from an exact hyper-Ornstein-Zernike equation. While the local compressibility and simple observables permit analytic treatment, complex order parameters are accessible via simulation-based supervised machine learning of neural hyperdirect correlation functionals. We exemplify efficient and accurate neural predictions for the cluster statistics of hard rods, square-well rods, and hard spheres. The theory allows one to treat complex observables, as is impossible in standard density functional theory.

Exploring anisotropic pressure and spatial correlations in strongly confined hard-disk fluids: Exact results.

This study examines the transverse and longitudinal properties of hard disks confined in narrow channels. Employing an exact mapping of the system onto a one-dimensional polydisperse, nonadditive mixture of hard rods with equal chemical potentials, we compute various thermodynamic properties, including the transverse and longitudinal equationsof state, along with their behaviors at both low and high densities. Structural properties are analyzed using the two-body correlation function and the radial distribution function, tailored for the highly anisotropic geometry of this system. The results are corroborated by computer simulations.

One dimensional lattice fluid mixture with nearest neighbour interactions

We present an exact derivation of the free energy functional of a fluid mixture of hard rods with arbitrary sizes on a one-dimensional lattice. Our approach is based on the Wertheim cluster theory which consists of mapping a system with finite range interactions to the system with pure hard-core interaction but with modified activities. We show that the free energy functional has the same form as the fundamental measure functional. The interactions part of the free energy has two contributions, one from the one-particle cavity restricted to the hard rod or hard-sphere diameter and a second from the two-particle cavity which includes the finite range of the interaction. In the limit of a one-component system, our results reduce to the one derived using the Markov chain approach. For vanishing interactions, the density functionals coincide exactly with the previously derived for the mixture of hard rods with pure hard-core interaction.

Broadband acoustic absorption at low frequencies by slabs and clusters made of hard cylindrical rods

Within the low-frequency limit, this work analyzes the viscous absorbing properties of circular clusters and semi-infinite slabs made of rigid scatterers embedded in a fluid, such as air or water. These structures are made of rigid scatterers distributed in a hexagonal lattice, and they are proposed as useful absorbing devices in the core of acoustic black holes or acoustic metasurfaces. It is demonstrated that in both types of structures, an optimum value of the filling fraction produces the maximum absorption in a given frequency band. To avoid heavy numerical simulations, the broadband absorbing power has been obtained using a homogenization theory providing not only the effective acoustic parameters (effective mass and effective sound speed) but also the decay coefficient due to viscosity. An enhancement of the broadband sound absorption can be obtained by using a refractive index gradient allowing an increase of the acoustic energy into the semi-infinite slabs. The theoretical predictions are well supported by numerical simulations based on the finite-element and boundary-element methods, respectively, for the semi-infinite slabs and clusters. These predictions have potential applications in the design of structures and metasurfaces with enhanced absorbing power at low frequencies. The analytical model is further supported by experiments made with a 3D-printed sample.

Thermalization and Hydrodynamics in an Interacting Integrable System: The Case of Hard Rods

We consider the relaxation of an initial non-equilibrium state in a one-dimensional fluid of hard rods. Since it is an interacting integrable system, we expect it to reach the Generalized Gibbs Ensemble (GGE) at long times for generic initial conditions. Here we show that there exist initial conditions for which the system does not reach GGE even at very long times and in the thermodynamic limit. In particular, we consider an initial condition of uniformly distributed hard-rods in a box with the left half having particles with a singular velocity distribution (all moving with unit velocity) and the right half particles in thermal equilibrium. We find that the density profile for the singular component does not spread to the full extent of the box and keeps moving with a fixed effective speed at long times. We show that such density profiles can be well described by the solution of the Euler equations almost everywhere except at the location of the shocks, where we observe slight discrepancies due to dissipation arising from the initial fluctuations of the thermal background. To demonstrate this effect of dissipation analytically, we consider a second initial condition with a single particle at the origin with unit velocity in a thermal background. We find that the probability distribution of the position of the unit velocity quasi-particle has diffusive spreading which can be understood from the solution of the Navier–Stokes (NS) equation of the hard rods. Finally, we consider an initial condition with a spread in velocity distribution for which we show convergence to GGE. Our conclusions are based on molecular dynamics simulations supported by analytical arguments.

Revised Enskog equation for hard rods

We point out that Percus’s collision integral for one-dimensional hard rods (Percus 1969 Phys. Fluids 12 1560–3) does not preserve the thermal equilibrium state in an external trapping potential. We derive a revised Enskog equation for hard rods and show that it preserves this thermal state exactly. In contrast to recent proposed kinetic equations for dynamics in integrability-breaking traps, both our kinetic equation and its thermal states are explicitly nonlocal in space. Our equation differs from earlier proposals at third order in spatial derivatives and we attribute this discrepancy to the choice of collision integral underlying our approach.

Three-stage thermalization of a quasi-integrable system

We consider a system of classical hard rods or billiard balls in one dimension, initially prepared in a Bragg-pulse state at a given temperature and subjected to external periodic fields. We show that at late times the system thermalizes in the thermodynamic limit via a three-stages process characterized by: an early phase where the dynamics is well described by Euler hydrodynamics, a subsequent phase where a (weak) turbulent phase is observed and where hydrodynamic gradient expansion can be broken, and a final one where the gas thermalizes according to viscous hydrodynamics. As the hard rod gas shares the same large-scale hydrodynamics as other quantum and classical integrable systems, we expect these features to universally characterize all many-body integrable systems in generic external potentials. Published by the American Physical Society 2024

A confined rod: mean field theory for hard rod-like particles

ABSTRACT In this paper, we model the configurations of a system of hard rods by viewing each rod in a cell formed by its neighbours. By minimising the free energy in the model and performing molecular dynamics, where, in both cases, the shape of the cell is a free parameter, we obtain the equilibrium orientational order parameter, free energy and pressure of the system. Our model enables the calculation of anisotropic stresses exerted on the walls of the cell due to shape change of the rod in photoisomerisation. These results are a key step towards understanding molecular shape change effects in photomechanical systems under illumination.

Layer topology of smectic grain boundaries

ABSTRACT Grain boundaries in extremely confined colloidal smectics possess a topological fine structure with coexisting nematic and tetratic symmetry of the director field. An alternative way to approach the problem of smectic topology is via the layer structure, which is typically more accessible in experiments on molecular liquid crystals. Here, we consider exemplary density functional theory results for two-dimensional hard rods to illustrate how the concept of endpoint defects, which appear as tetratic disclinations of quarter charge in director topology, translates to smectic layer topology built around the networks formed by density maxima and minima. By doing so, we elaborate on further advantages of a topological concept evolving around the layer structure rather than the director field, such as providing insight in the structure of edge dislocations or virtual defects at the confining walls.

Phase behavior of hard rods between two walls: Phase change of particles without occurring phase transition

Phase behavior of hard rods between two walls: Phase change of particles without occurring phase transition

Unusual ergodic and chaotic properties of trapped hard rods.

We investigate ergodicity, chaos, and thermalization for a one-dimensional classical gas of hard rods confined to an external quadratic or quartic trap, which breaks microscopic integrability. To quantify the strength of chaos in this system, we compute its maximal Lyapunov exponent numerically. The approach to thermal equilibrium is studied by considering the time evolution of particle position and velocity distributions and comparing the late-time profiles with the Gibbs state. Remarkably, we find that quadratically trapped hard rods are highly nonergodic and do not resemble a Gibbs state even at extremely long times, despite compelling evidence of chaos for four or more rods. On the other hand, our numerical results reveal that hard rods in a quartic trap exhibit both chaos and thermalization, and equilibrate to a Gibbs state as expected for a nonintegrable many-body system.

Janus helices: From fully attractive to hard helices.

The phase diagram of hard helices differs from its hard rods counterpart by the presence of chiral "screw" phases stemming from the characteristic helical shape, in addition to the conventional liquid crystal phases also found for rod-like particles. Using extensive Monte Carlo and Molecular Dynamics simulations, we study the effect of the addition of a short-range attractive tail representing solvent-induced interactions to a fraction of the sites forming the hard helices, ranging from a single-site attraction to fully attractive helices for a specific helical shape. Different temperature regimes exist for different fractions of the attractive sites, as assessed in terms of the relative Boyle temperatures, that are found to be rather insensitive to the specific shape of the helical particle. The temperature range probed by the present study is well above the corresponding Boyle temperatures, with the phase behaviour still mainly entropically dominated and with the existence and location of the various liquid crystal phases only marginally affected. The pressure in the equation of state is found to decrease upon increasing the fraction of attractive beads and/or on lowering the temperature at fixed volume fraction, as expected on physical grounds. All screw phases are found to be stable within the considered range of temperatures with the smectic phase becoming more stable on lowering the temperature. By contrast, the location of the transition lines do not display a simple dependence on the fraction of attractive beads in the considered range of temperatures.

Charge-driven liquid-crystalline behavior of ligand-functionalized nanorods in apolar solvent.

Concentrated colloidal suspensions of nanorods often exhibit liquid-crystalline (LC) behavior. The transition to a nematic LC phase, with long-range orientational order of the particles, is usually well-captured by Onsager's theory for hard rods, at least qualitatively. The theory shows how the volume fraction at the transition decreases with increasing aspect ratio of the rods. It also explains that the long-range electrostatic repulsive interaction occurring between rods stabilized by their surface charge can significantly increase their effective diameter, resulting in a decrease in the volume fraction at the transition, as compared to sterically stabilized rods. Here, we report on a system of ligand-stabilized LaPO4 nanorods, of aspect ratio ≈11, dispersed in apolar medium exhibiting the counter-intuitive observation that the onset of nematic self-assembly occurs at an extremely low volume fraction of ≈0.25%, which is lower than observed (≈3%) with the same particles when charged-stabilized in polar solvent. Furthermore, the nanorod volume fraction at the transition increases with increasing concentration of ligands, in a similar way as in polar media where increasing the ionic strength leads to surface charge screening. This peculiar system was investigated by dynamic light scattering, Fourier-transform infrared spectroscopy, zetametry, electron microscopy, polarized light microscopy, photoluminescence measurements, and X-ray scattering. Based on these experimental data, we formulate several tentative scenarios that might explain this unexpected phase behavior. However, at this stage, its full understanding remains a pending theoretical challenge. Nevertheless, this study shows that dispersing anisotropic nanoparticles in an apolar solvent may sometimes lead to spontaneous ordering events that defy our intuitive ideas about colloidal systems.

Pre-Smectic Ordering and the Unwinding Helix in Monte Carlo Simulations of Cholesteric Liquid-Crystals.

In this paper, molecular chirality is studied for liquid-crystal fluids represented by hard rods with the addition of an attractive chiral dispersion term. Chiral forces between molecular pairs are assumed to be long-ranged and are described in terms of the pseudotensor of Goossens [W. J. A. Goossens, Mol. Cryst. Liq. Cryst. 1971, 12, 237-244]. Following Varga and Jackson [S. Varga and G. Jackson, Chem. Phys. Lett. 2003, 377, 6-12], this is combined with a hard-spherocylinder core. We investigate the relationship between molecular chirality and the helical pitch of the system, which occurs in the absence of full three-dimensional periodic boundary conditions. The dependence of the wavenumber of this pitch on the thermodynamic variables, temperature, and density is measured. We also explore the use of a novel surface boundary interaction model. As a result of this approach, we are able to lower the temperature of the system without the occurrence of nematic droplets, which would interfere with the formation of a uniaxial pitch. Regarding the theoretical predictions of Wensink and Jackson [H. H. Wensink and G. Jackson, J. Chem. Phys. 2009, 130, 234911], on the one hand, we have qualitative agreement with the observed non-monotonic density dependence of the wavenumber. Initially increasing with density, the wavenumber reaches a maximum, before falling as the density moves toward the point of phase transition from cholesteric to smectic. However, further analysis for shorter rods, in the presence of novel boundary conditions, reveals some disagreement with the theory, at least in this case; the unwinding of the cholesteric helix in the cholesteric phase occurs simultaneously with subtle increases in smectic ordering. These pre-smectic fluctuations have not been accounted for so far in theories on cholesterics but turn out to play a key role in controlling the pitch of cholesteric phases of rod-shaped mesogens with a small to moderate aspect ratio.

Structural properties of hard-disk fluids under single-file confinement.

The structural properties of confined single-file hard-disk fluids are studied analytically by means of a mapping of the original system onto a one-dimensional mixture of non-additive hard rods, the mapping being exact in the polydisperse limit. Standard statistical-mechanical results are used as a starting point to derive thermodynamic and structural properties of the one-dimensional mixture, where the condition that all particles have the same chemical potential must be taken into account. Analytical results are then provided for the nth neighbor probability distribution function, the radial distribution function, and the structure factor, a very good agreement being observed upon comparison with simulation data from the literature. Moreover, we have analyzed the scaling form for the disappearance of defects in the zigzag configuration for high pressure and have obtained the translational correlation length and the structural crossover in the oscillation frequency for asymptotically large distances.

On the elusive saddle-splay and splay-bend elastic constants of nematic liquid crystals.

The elastic behavior of nematics is commonly described in terms of the three so-called bulk deformation modes, i.e., splay, twist, and bend. However, the elastic free energy contains also other terms, often denoted as saddle-splay and splay-bend, which contribute, for instance, in confined systems. The role of such terms is controversial, partly because of the difficulty of their experimental determination. The saddle-splay (K24) and splay-bend (K13) elastic constants remain elusive also for theories; indeed, even the possibility of obtaining unambiguous microscopic expressions for these quantities has been questioned. Here, within the framework of Onsager theory with Parsons-Lee correction, we obtain microscopic estimates of the deformation free energy density of hard rod nematics in the presence of different director deformations. In the limit of a slowly changing director, these are directly compared with the macroscopic elastic free energy density. Within the same framework, we derive also closed microscopic expressions for all elastic coefficients of rodlike nematics. We find that the saddle-splay constant K24 is larger than both K11 and K22 over a wide range of particle lengths and densities. Moreover, the K13 contribution comes out to be crucial for the consistency of the results obtained from the analysis of the microscopic deformation free energy density calculated for variants of the splay deformation.

Matrix Product Symmetries and Breakdown of Thermalization from Hard Rod Deformations.

We construct families of exotic spin-1/2 chains using a procedure called "hard rod deformation." We treat both integrable and nonintegrable examples. The models possess a large noncommutative symmetry algebra, which is generated by matrix product operators with a fixed small bond dimension. The symmetries lead to Hilbert space fragmentation and to the breakdown of thermalization. As an effect, the models support persistent oscillations in nonequilibrium situations. Similar symmetries have been reported earlier in integrable models, but here we show that they also occur in nonintegrable cases.

Finite-temperature equilibrium density profiles of integrable systems in confining potentials.

We study the equilibrium density profile of particles in two one-dimensional classical integrable models, namely hard rods and the hyperbolic Calogero model, placed in confining potentials. For both of these models the interparticle repulsion is strong enough to prevent particle trajectories from intersecting. We use field theoretic techniques to compute the density profile and their scaling with system size and temperature, and we compare them with results from Monte Carlo simulations. In both cases we find good agreement between the field theory and simulations. We also consider the case of the Toda model in which interparticle repulsion is weak and particle trajectories can cross. In this case, we find that a field theoretic description is ill-suited and instead, in certain parameter regimes, we present an approximate Hessian theory to understand the density profile. Our work provides an analytical approach toward understanding the equilibrium properties for interacting integrable systems in confining traps.