Model Of Hard Rods Research Articles

Mesoscopic impurities in generalized hydrodynamics

This study focuses on the impurities in integrable models from the viewpoint of generalized hydrodynamics (GHD). An impurity can be thought of as a boundary condition for the GHD equation, relating the states on the left and right sides. It was found that, in interacting models, it is not possible to disentangle the incoming and outgoing states, which means that it is not possible to think of scattering as a mapping that maps the incoming state to the outgoing state. A novel class of impurities, dubbed mesoscopic impurities, was then introduced, whose spatial size is mesoscopic (i.e. their size Lmicro≪Limp≪L is much larger than the microscopic length scale Lmicro but much smaller than the macroscopic scale L). Because of their large size, it is possible to describe mesoscopic impurities via GHD. This simplification allows one to study these impurities both analytically and numerically. These impurities show interesting non-perturbative scattering behavior, such as the nonuniqueness of the solutions and a nonanalytic dependence on the impurity strength. In models with one quasi-particle species and a scattering phase shift that depends only on the difference in momenta, the scattering can be described using an effective Hamiltonian. This Hamiltonian is dressed due to the interaction between the particles and satisfies a self-consistency fixed-point equation. In the example of the hard-rod model, it was demonstrated how this fixed-point equation can be used to find almost explicit solutions to the scattering problem by reducing it to a two-dimensional system of equations that can be solved numerically.

Origins of the Failure of the Activity Virial Series

We examine the development of the virial equation of state when expressed as a series in the activity with coefficients labeled bn. Using the one-dimensional hard-rod model as a prototype, we consider steps in its development that introduce inaccuracy that leads it to form a divergent series. We discuss the role of volume dependence of the virial coefficients and present expressions and calculations for volume-dependent coefficients bn(V) for the hard-rod model up to n = 200. We examine alternative methods for computing properties from the bn. We recommend that further efforts be made to compute volume-dependent virial coefficients as a means to understand better the virial equation of state and to make it more robust in applications.

Anomalous phase behavior of quasi-one-dimensional attractive hard rods.

We study a two-state model of attractive hard rods using the transfer matrix method, where the centers of the particles are confined to a straight line, but the orientations of the rods can be parallel or perpendicular to the confining line. The rods are modeled as hard rectangles with length L and width D and decorated with attractive sites at both ends of the rectangles. We find that the particles align parallel to the line and form long chains at low densities, while they turn out of the line and form a Tonks gas at high densities. With increasing the stickiness between the rods, the structural change between parallel and perpendicular states becomes stronger and the pressure vs density curve becomes almost a horizontal line at the transition pressure. We show that such a behavior is reminiscent of the first-order phase transition. This manifests in the validity of the lever rule of the phase transitions for very sticky cases.

Euler-scale dynamical correlations in integrable systems with fluid motion

We devise an iterative scheme for numerically calculating dynamical two-point correlation functions in integrable many-body systems, in the Eulerian scaling limit. Expressions for these were originally derived in Ref. [1] by combining the fluctuation-dissipation principle with generalized hydrodynamics. Crucially, the scheme is able to address non-stationary, inhomogeneous situations, when motion occurs at the Euler-scale of hydrodynamics. In such situations, in interacting systems, the simple correlations due to fluid modes propagating with the flow receive subtle corrections, which we test. Using our scheme, we study the spreading of correlations in several integrable models from inhomogeneous initial states. For the classical hard-rod model we compare our results with Monte-Carlo simulations and observe excellent agreement at long time-scales, thus providing the first demonstration of validity for the expressions derived in Ref. [1]. We also observe the onset of the Euler-scale limit for the dynamical correlations.

Absence of logarithmic divergence of the entanglement entropies at the phase transitions of a 2D classical hard rod model

Entanglement entropy is a powerful tool to detect continuous, discontinuous and even topological phase transitions in quantum as well as classical systems. In this work, von Neumann and Renyi entanglement entropies are studied numerically for classical lattice models in a square geometry. A cut is made from the center of the square to the midpoint of one of its edges, say the right edge. The entanglement entropies measure the entanglement between the left and right halves of the system. As in the strip geometry, von Neumann and Renyi entanglement entropies diverge logarithmically at the transition point while they display a jump for first-order phase transitions. The analysis is extended to a classical model of non-overlapping finite hard rods deposited on a square lattice for which Monte Carlo simulations have shown that, when the hard rods span over 7 or more lattice sites, a nematic phase appears in the phase diagram between two disordered phases. A new Corner Transfer Matrix Renormalization Group algorithm (CTMRG) is introduced to study this model. No logarithmic divergence of entanglement entropies is observed at the phase transitions in the CTMRG calculation discussed here. We therefore infer that the transitions neither can belong to the Ising universality class, as previously assumed in the literature, nor be discontinuous.

One-Dimensional model of hard rods with gravitational interactions

The study of one-dimensional hard rods with gravitational interactions was made by means of the isobaric and the canonical ensembles. The canonical partition function was obtained from the isobaric partition function by taking the inverse Laplace transform. The explicit closed forms for the canonical particle density and the higher-order molecular distribution functions were determined. The wall pressures, where the force exerted on the fluid by the walls, were obtained from both the partial derivative of the free energy with respect to the position of the walls and the contact value of the density determined from the one-particle density at the walls. The wall pressure implies that the hard rods with a gravitational interaction do not exhibit a phase transition. The partition function and the molecular distribution functions in the grand canonical ensemble are also reported.

Thermoreversible Gels Composed of Colloidal Silica Rods with Short-Range Attractions

Dynamic arrest transitions of colloidal suspensions containing nonspherical particles are of interest for the design and processing of various particle technologies. To better understand the effects of particle shape anisotropy and attraction strength on gel and glass formation, we present a colloidal model system of octadecyl-coated silica rods, termed as adhesive hard rods (AHR), which enables control of rod aspect ratio and temperature-dependent interactions. The aspect ratios of silica rods were controlled by varying the initial TEOS concentration following the work of Kuijk et al. (J. Am. Chem. Soc., 2011, 133, 2346-2349) and temperature-dependent attractions were introduced by coating the calcined silica rods with an octadecyl-brush and suspending in tetradecane. The rod length and aspect ratio were found to increase with TEOS concentration as expected, while other properties such as the rod diameter, coating coverage, density, and surface roughness were nearly independent of the aspect ratio. Ultrasmall angle X-ray scattering measurements revealed temperature-dependent attractions between octadecyl-coated silica rods in tetradecane, as characterized by a low-q upturn in the scattered intensity upon thermal quenching. Lastly, the rheology of a concentrated AHR suspension in tetradecane demonstrated thermoreversible gelation behavior, displaying a nearly 5 orders of magnitude change in the dynamic moduli as the temperature was cycled between 15 and 40 °C. The adhesive hard rod model system serves as a tunable platform to explore the combined influence of particle shape anisotropy and attraction strength on the dynamic arrest transitions in colloidal suspensions with thermoreversible, short-range attractions.

A solvable model of hard rods with gravitational interactions

We consider a simple 1D model of hard rods with gravitational interactions. First, we consider the situation where the sytem is enclosed in a box with finite size and we exactly compute the equilibrium thermodynamical quantities. Thanks to the confining nature of gravity in 1D which prevents evaporation, the box can be released and we can study an open system with its center of mass fixed. Then, we exactly compute the corresponding equilibrium density profile within the microcanonical ensemble. All those analytical results are discussed in connection with the general issue of ensemble inequivalences for systems with long-range interactions. They also provide specific tests for the reliability of the hydrostatic approach combined with a mean-field treatment of gravitational interactions. In particular, the hydrostatic approach is shown to fail for energies close to the collapse energy where a core-halo structure emerges.

Structure and growth of tetracene on Ag(111)

The structure of the tetracene/Ag(111) interface in the coverage range \ensuremath{\theta} = 0 to 2.4 ML is studied with scanning tunneling microscopy (STM) at 8 K and with low energy electron diffraction (LEED) at T = 300 \dots{} 100 K. For \ensuremath{\theta} $\ensuremath{\lesssim}$ 0.01 ML, one-dimensional (1D) diffusion of single molecules along $\ensuremath{\langle}01\overline{1}\ensuremath{\rangle}$-directions is observed even at 8 K. For 0.1 ML \ensuremath{\theta} 0.5 ML molecules are homogeneously distributed over the surface forming a disordered phase (static at T = 8 K, dynamic at T = 25 K), indicating a repulsive intermolecular interaction (\ensuremath{\delta}-phase). For \ensuremath{\theta} $\ensuremath{\gtrsim}$ 0.5 ML, local ordering in the commensurate \ensuremath{\gamma}-phase is observed. Further increase of the coverage yields a compressed monolayer (ML) phase (\ensuremath{\theta} \ensuremath{\equiv} 1 ML) with point-on-line registry (\ensuremath{\alpha}-phase). The interaction between molecules has been calculated with the force-field approach to rationalize the molecular packing motifs in the various phases. Under most circumstances molecule-molecule interactions are repulsive, in agreement with experimental findings. A simulation of the adsorption up to \ensuremath{\theta} = 1 ML according to the random sequential adsorption (RSA) algorithm shows that the disorder-to-order transition from the \ensuremath{\delta}- to \ensuremath{\gamma}-phase occurs close to random close packing (RCP), \ensuremath{\theta} = 0.5--0.6 ML. Since tetracene molecules are a two-dimensional (2D) representation of Onsager's hard rod model, this suggests that this phase transition is driven both energetically and entropically. For \ensuremath{\theta} \ensuremath{\approx} 2.23 ML a metastable bilayer phase with point-on-line coincidence is observed (\ensuremath{\beta}-phase). The basic structural unit of this phase is a triplet of molecules that are tilted along the long molecular axis against each other; at least one of these molecules is tilted out of the surface plane. Within the \ensuremath{\beta}-phase a superstructure of alternating rotation domains is observed. This superstructure has a period of 7.4 nm. The molecular packing in the \ensuremath{\beta}-phase resembles the packing in the bulk crystal structure of tetracene, its formation can therefore be interpreted as incipient pseudomorphic growth of tetracene on Ag(111). However, pseudomorphic growth cannot be continued beyond the \ensuremath{\beta}-phase.

Nonequilibrium statistical mechanics of self-propelled hard rods

Using tools of nonequilibrium mechanics, we study a model of self-propelled hard rods ona substrate in two dimensions to quantify the interplay of self-propulsion andexcluded-volume effects. We derive a Smoluchowski equation for the configurationalprobability density of self-propelled rods that contains several modifications as compared tothe familiar Smoluchowski equation for thermal rods. As a side-product of our work, wealso present a purely dynamical derivation of the Onsager form of the mean-fieldexcluded-volume interaction among thermal hard rods.

The crossover from single file to Fickian diffusion

The crossover from single-file diffusion, where the mean-square displacement scales as <x^2> ~t^(1/2), to normal Fickian diffusion, where <x^2>~t$, is studied as a function of channel width for colloidal particles. By comparing Brownian dynamics to a hybrid molecular dynamics and mesoscopic simulation technique, we can study the effect of hydrodynamic interactions on the single file mobility and on the crossover to Fickian diffusion for wider channel widths. For disc-like particles with a steep interparticle repulsion, the single file mobilities for different particle densities are well described by the exactly solvable hard-rod model. This holds both for simulations that include hydrodynamics, as well as for those that don't. When the single file constraint is lifted, then for particles of diameter \sigma and pipe of width $L$ such that (L- 2 \sigma)/\sigma = \delta_c << 1 the particles can be described as hopping past one-another in an average time t_{hop}. For shorter times t << t_{hop} the particles still exhibit sub-diffusive behaviour, but at longer times t > t_{hop}, normal Fickian diffusion sets in with an effective diffusion constant D_{hop} ~ t_{hop}^(1/2). For the Brownian particles, t_{hop} ~ 1/\delta_c^(2) when \delta << 1, but when hydrodynamic interactions are included, we find a stronger dependence than \delta_c^{-2}. We attribute this difference to short-range lubrication forces that make it more difficult for particles to hop past each other in very narrow channels.

Enhanced Diffusion and Ordering of Self-Propelled Rods

Starting from a minimal physical model of self-propelled hard rods on a substrate in two dimensions, we derive a modified Smoluchowski equation for the system. Self-propulsion enhances longitudinal diffusion and modifies the mean-field excluded volume interaction. From the Smoluchowski equation we obtain hydrodynamic equations for rod concentration, polarization and nematic order parameter. New results at large scales are a lowering of the density of the isotropic-nematic transition and a strong enhancement of boundary effects in confined self-propelled systems.

Transfer operators for hard-rod-type models

We describe a new approach to the hard rod model and construct a Ruelle-Mayer-type transfer operator which satisfies a dynamical trace formula. This yields a meromorphic continuation of the dynamical zeta function in two variables among them the inverse temperature.

Statistical properties of granular gas under microgravity one-dimensional inelastic hard rod system

We have studied numerically statistical properties of granular gas in a one-dimensional inelastic (viscoelastic) hard rod model under microgravity, which is designed to the mimic experimental granular-vibrated beds by introducing a velocity-dependent restitution coefficient (VDRC). Our systematic simulations show that various macroscopic properties of this model are quantitatively different from a linear combination of the previous simulations based on the constant restitution coefficient (CRC). The present results are significantly important to study a vibration response and dynamics of granular gas especially in microgravity experiment.

Complementarity and clustering in a simple model mixed bilayer

A bilayer of uniform thickness containing a mixture of long and short lipids is simulated using a parallel hard-rod model to illustrate the effect of transbilayer repulsions between the tails of the long component. Monte Carlo simulations show considerable entropy-driven clustering within each layer. Demixing reaches a maximum at the highest packing fraction of the liquid state and decreases as the system orders. The formation of complementary clusters of long and short rods on opposite sides of the bilayer increases translational freedom within each cluster by reducing constraints imposed by the opposing leaflet, an effect that becomes less important as rods lock into facing hexagonally ordered arrays.

Beyond the Tonks-Girardeau Gas: Strongly Correlated Regime in Quasi-One-Dimensional Bose Gases

We consider a homogeneous 1D Bose gas with contact interactions and a large attractive coupling constant. This system can be realized in tight waveguides by exploiting a confinement induced resonance of the effective 1D scattering amplitude. By using the diffusion Monte Carlo method we show that, for small densities, the gaslike state is well described by a gas of hard rods. The critical density for cluster formation is estimated using the variational Monte Carlo method. The behavior of the correlation functions and of the frequency of the lowest breathing mode for harmonically trapped systems shows that the gas is more strongly correlated than in the Tonks-Girardeau regime.

From Random Walk to Single-File Diffusion

We report an experimental study of diffusion in a quasi-one-dimensional (q1D) colloid suspension which behaves like a Tonks gas. The mean squared displacement as a function of time is described well with an ansatz encompassing a time regime that is both shorter and longer than the mean time between collisions. The ansatz asserts that the inverse mean squared displacement is the sum of the inverse mean squared displacement for short time normal diffusion (random walk) and the inverse mean squared displacement for asymptotic single-file diffusion (SFD). The dependence of the 1D mobility in the SFD on the concentration of the colloids agrees quantitatively with that derived for a hard rod model, which confirms for the first time the validity of the hard rod SFD theory. We also show that a recent SFD theory by Kollmann leads to the hard rod SFD theory for a Tonks gas.

Isotropic-liquid crystalline phase diagram of a CdSe nanorod solution.

We report the isotropic-liquid crystalline phase diagram of 3.0 nm x 60 nm CdSe nanorods dispersed in anhydrous cyclohexane. The coexistence concentrations of both phases are found to be lower and the biphasic region wider than the results predicted by the hard rod model, indicating that the attractive interaction between the nanorods may be important in the formation of the liquid crystalline phase in this system.

Rotational dynamics of colloidal tracer spheres in suspensions of charged rigid rods

The short-time rotational dynamics of colloidal silica tracer spheres in suspensions of rigid silica rods is investigated, using time-resolved phosphorescence anisotropy, as a function of tracer radius aT, rod volume fraction φ, and the range κ−1 of the double-layer repulsions between the like-charged rods and tracer spheres. A large tracer size aT and a small screening length κ−1 appear to maximize hydrodynamic hindrance of tracer diffusion for given φ. The marked φ-dependence of the rotational dynamics is primarily determined by the large excluded volumes of the high-aspect ratio rods. Stokes–Einstein–Debye (SED) scaling of the rotational diffusion coefficients with the inverse viscosity of the rod suspensions holds fairly well, expect for small aT and large κ−1. The ionic strength dependence of deviations from SED scaling is rationalized in terms of an effective hard-rod model with the bare length L replaced by an effective length L+4κ−1.

Self-referential method for calculation of the free energy of crystals by Monte Carlo simulation.

We propose a Monte Carlo simulation method for the evaluation of free energies in crystalline systems. In principle, the method involves evaluating the free-energy difference between systems of N molecules and 2N molecules. This difference, coupled with the assumption that the free energy is extensive and thus proportional to N, provides sufficient information to obtain the absolute free energy of the crystal. The approach to doubling the system size does not involve insertion or removal of molecules in the system. Instead, the configurations of the molecules are expressed in terms of the normal-mode coordinates of a harmonic lattice. By decoupling certain of these coordinates from the molecule configurations, we obtain a transformation that in effect yields the system-size doubling. The method is examined via application to a system of hard rods in one dimension. This simple model is considered principally because of the availability of an analytic solution for its free energy, which permits accurate testing of the performance and correctness of the proposed method. In using the hard-rod model we also avoid other complications related to treatment of the temperature, and application of normal-mode coordinate decoupling in higher dimensions. The proposed method is shown to be able to provide good results for the free-energy calculation, but further development will be needed before it can be considered practical for general-purpose use.