Inelastic hard rods in a periodic potential

A simple model of inelastic hard rods subject to a one-dimensional array of identical wells is introduced. The energy loss due to inelastic collisions is balanced by the work supplied by an external stochastic heat bath. We explore the effect of the spatial nonuniformity on the steady states of the system. The spatial variations of the density, granular temperature, and pressure induced by the gradient of the external potential are investigated and compared with the analogous variations in an elastic system. Finally, we study the clustering process by considering the relaxation of the system starting from a uniform homogeneous state.     

Event-driven Brownian dynamics for hard spheres

Brownian dynamics algorithms integrate Langevin equations numerically and allow to probe long time scales in simulations. A common requirement for such algorithms is that interactions in the system should vary little during an integration time step; therefore, computational efficiency worsens as the interactions become steeper. In the extreme case of hard-body interactions, standard numerical integrators become ill defined. Several approximate schemes have been invented to handle such cases, but little emphasis has been placed on testing the correctness of the integration scheme. Starting from the two-body Smoluchowski equation, the authors discuss a general method for the overdamped Brownian dynamics of hard spheres, recently developed by one of the authors. They test the accuracy of the algorithm and demonstrate its convergence for a number of analytically tractable test cases.     

Depletion potentials in colloidal mixtures of hard spheres and rods

The depletion potential between a hard sphere and a planar hard wall, or two hard spheres, imposed by suspended rigid spherocylindrical rods is computed by the acceptance ratio method through the application of Monte Carlo simulation. The accurate results and ideal-gas approximation results of the depletion potential are determined with the acceptance ratio method in our simulations. For comparison, the depletion potentials are also studied by using both the density functional theory and Derjaguin approximations. The density profile as a function of positions and orientations of rods, used in the density functional theory, is calculated by Monte Carlo simulation. The potential obtained by the acceptance ratio method is in good agreement with that of density functional theory under the ideal-gas approximation. The comparison between our results and those of other theories suggests that the acceptance ratio method is the only efficient method used to compute the depletion potential induced by nonspherical colloids with the volume fraction beyond the ideal-gas approximation.     

Heterogeneous dynamics in columnar liquid crystals of parallel hard rods

In the wake of previous studies on the rattling-and-jumping diffusion in smectic liquid crystal phases of colloidal rods, we analyze here for the first time the heterogeneous dynamics in columnar phases. More specifically, we perform computer simulations to investigate the relaxation dynamics of a binary mixture of perfectly aligned hard spherocylinders. We detect that the columnar arrangement of the system produces free-energy barriers that the particles should overcome to jump from one column to another, thus determining a hopping-type diffusion. This phenomenon accounts for the non-Gaussian intercolumn diffusion and shows a two-step structural relaxation that is remarkably analogous to that of out-of-equilibrium glass-forming systems and gels. Surprisingly enough, slight deviations from the behavior of simple liquids due to transient cages is also observed in the direction perpendicular to this plane, where the system is usually referred to as liquidlike.     

Isotropic-nematic transition of hard rods immersed in random sphere matrices

Using replica density functional theory and Monte Carlo computer simulations we investigate a system of annealed hard spherocylinders adsorbed in a matrix of quenched hard spheres. Theoretical predictions for the partition coefficient, defined as the ratio of density of rods in the matrix and that in a reservoir, agree well with simulation results. Theory predicts the isotropic-nematic transition to remain first order upon increasing sphere packing fraction, and to shift towards lower rod densities. This scenario is consistent with our simulation results that clearly show a jump in the nematic order parameter upon increasing the rod density at constant matrix packing fraction, corresponding to the isotropic-nematic transition, even for sphere matrix packing fractions < or approximately equal to 0.3.     

Crystalline ordering in lattice models of hard rods with nearest neighbour attraction

Several lattice models for hard rods with attraction between neighbouring parallel rodes are considered. It is proved that the models for sufficiently low temperature and high fugacity exhibit a phase transition to an ordered crystalline structure. Nothing is proved about the existence of a liquid crystal state.     

Glassy dynamics and kinetic vitrification of isotropic suspensions of hard rods

A nonlinear Langevin equation (NLE) theory for the translational center-of-mass dynamics of hard nonspherical objects has been applied to isotropic fluids of rigid rods. The ideal kinetic glass transition volume fraction is predicted to be a monotonically decreasing function beyond an aspect ratio of two. The functional form of the decrease is weaker than the inverse aspect ratio. Vitrification occurs at lower volume fractions for corrugated tangent bead rods compared to their smooth spherocylinder analogs. The ideal glass transition signals a crossover to activated dynamics, which is estimated to be observable before the nematic phase boundary is encountered if the aspect ratio is less than roughly 25. Calculations of the glassy elastic shear modulus and absolute yield stress reveal a roughly exponential growth with volume fraction. The dependence of entropic barriers and mean barrier hopping times on concentration for rods of variable aspect ratios can be collapsed quite well based on a difference volume fraction variable that quantifies the distance from the ideal glass boundary. Full numerical solution of the NLE theory via stochastic trajectory simulation was performed for tangent bead rods, and the results were compared to their hard sphere analogs. With increasing shape anisotropy the characteristic length scales of the nonequilibrium free energy increase and the magnitude of the localization well and entropic barrier curvatures decreases. These changes result in a significant aspect ratio dependence of dynamical properties and time correlation functions including weaker intermediate time subdiffusive transport, stronger two-step decay of the incoherent dynamic structure factor, longer mean alpha relaxation time, and stronger wavevector-dependent decoupling of relaxation times and the self-diffusion constant. The theoretical results are potentially testable via computer simulation, confocal microscopy, and dynamic light scattering.     

Orientational ordering in hard rectangles: The role of three-body correlations

We investigate the effect of three-body correlations on the phase behavior of hard rectangle two-dimensional fluids. The third virial coefficient B3 is incorporated via an equation of state that recovers scaled particle theory for parallel hard rectangles. This coefficient, a functional of the orientational distribution function, is calculated by Monte Carlo integration, using an accurate parametrized distribution function, for various particle aspect ratios in the range of 1-25. A bifurcation analysis of the free energy calculated from the obtained equation of state is applied to find the isotropic (I)-uniaxial nematic (N(u)) and isotropic-tetratic nematic (N(t)) spinodals and to study the order of these phase transitions. We find that the relative stability of the N(t) phase with respect to the isotropic phase is enhanced by the introduction of B3. Finally, we have calculated the complete phase diagram using a variational procedure and compared the results with those obtained from scaled particle theory and with Monte Carlo simulations carried out for hard rectangles with various aspect ratios. The predictions of our proposed equation of state as regards the transition densities between the isotropic and orientationally ordered phases for small aspect ratios are in fair agreement with simulations. Also, the critical aspect ratio below which the N(t) phase becomes stable is predicted to increase due to three-body correlations, although the corresponding value is underestimated with respect to simulation.     

Long-time tails of translational and rotational Brownian motion in a suspension of hard spheres

The long-time translational and rotational Brownian motion of a sphere in a suspension of hard spheres is studied on the basis of the linearized Navier-Stokes equations and the fluctuation-dissipation theorem. It is shown that for the rotational long-time coefficient an effective medium conjecture is incorrect. There are short-range velocity correlations that decay at the same rate as the macroscopic flow pattern used in the effective medium conjecture. An estimate of the short-range correction is made on the basis of the pair term in the cluster expansion of the rotational admittance.     

Entropic forces and directed alignment of hard squares in suspensions of rods and disks

We use Monte Carlo simulations in two dimensions to study the depletion forces between two hard squares in a suspension of hard rods or disks. We determine the effects of size and concentration of rods and disks on the potential of mean force between the squares. Both rods and disks produce a short-range depletion attraction between the two squares. The depletion interaction can be strong enough to outweigh the (rotational) entropic repulsion between the squares at certain sizes and concentrations of the rods and disks. We also probe the relative orientation that two squares adopt as they approach each other and we observe rich behavior, in which the relative orientation depends on the size, concentration, and shape of the depletion agent. Simple models based on the ideas of Asakura and Oosawa [J. Chem. Phys. 22, 1255 (1954)] can explain trends in the potentials of mean force obtained from the simulations.     

Brownian dynamics simulations with stiff finitely extensible nonlinear elastic-Fraenkel springs as approximations to rods in bead-rod models

A very stiff finitely extensible nonlinear elastic (FENE)-Fraenkel spring is proposed to replace the rigid rod in the bead-rod model. This allows the adoption of a fast predictor-corrector method so that large time steps can be taken in Brownian dynamics (BD) simulations without over- or understretching the stiff springs. In contrast to the simple bead-rod model, BD simulations with beads and FENE-Fraenkel (FF) springs yield a random-walk configuration at equilibrium. We compare the simulation results of the free-draining bead-FF-spring model with those for the bead-rod model in relaxation, start-up of uniaxial extensional, and simple shear flows, and find that both methods generate nearly identical results. The computational cost per time step for a free-draining BD simulation with the proposed bead-FF-spring model is about twice as high as the traditional bead-rod model with the midpoint algorithm of Liu [J. Chem. Phys. 90, 5826 (1989)]. Nevertheless, computations with the bead-FF-spring model are as efficient as those with the bead-rod model in extensional flow because the former allows larger time steps. Moreover, the Brownian contribution to the stress for the bead-FF-spring model is isotropic and therefore simplifies the calculation of the polymer stresses. In addition, hydrodynamic interaction can more easily be incorporated into the bead-FF-spring model than into the bead-rod model since the metric force arising from the non-Cartesian coordinates used in bead-rod simulations is absent from bead-spring simulations. Finally, with our newly developed bead-FF-spring model, existing computer codes for the bead-spring models can trivially be converted to ones for effective bead-rod simulations merely by replacing the usual FENE or Cohen spring law with a FENE-Fraenkel law, and this convertibility provides a very convenient way to perform multiscale BD simulations.     

Brownian dynamics simulations of the self- and collective rotational diffusion coefficients of rigid long thin rods

Recently a microscopic theory for the dynamics of suspensions of long thin rigid rods was presented, confirming and expanding the well-known theory by Doi and Edwards [The Theory of Polymer Dynamics (Clarendon, Oxford, 1986)] and Kuzuu [J. Phys. Soc. Jpn. 52, 3486 (1983)]. Here this theory is put to the test by comparing it against computer simulations. A Brownian dynamics simulation program was developed to follow the dynamics of the rods, with a length over a diameter ratio of 60, on the Smoluchowski time scale. The model accounts for excluded volume interactions between rods, but neglects hydrodynamic interactions. The self-rotational diffusion coefficients D(r)(phi) of the rods were calculated by standard methods and by a new, more efficient method based on calculating average restoring torques. Collective decay of orientational order was calculated by means of equilibrium and nonequilibrium simulations. Our results show that, for the currently accessible volume fractions, the decay times in both cases are virtually identical. Moreover, the observed decay of diffusion coefficients with volume fraction is much quicker than predicted by the theory, which is attributed to an oversimplification of dynamic correlations in the theory.     

Thermodynamics of attractive hard rods: a test of mean field density functional theory

Mean field density functional theory (MFDFT) has been employed to calculate the free energy of a pair of attractive hard rods on a ring. The results for homogeneous and optimal inhomogeneous density profiles have been compared with the exact free energy as a test of the approach. We discuss the problems in applying MFDFT to small systems and suggest modifications which allow a reasonably accurate treatment of this particular, rather extreme, case.     

Nematic and crystalline ordering in the lattice model of hard rods with a mean field interaction

A lattice model of long hard rods with a nearest neighbour attraction proportional to the length of overlap is considered. It is shown, that in the van der Waals approximation the model exhibits non only the isotropic and nematic phases obtained with the approximative treatment of the hard rod interaction but also a crystalline phase.     

One-dimensional model for water and aqueous solutions. III. Solvation of hard rods in aqueous mixtures

A simple one-dimensional model for aqueous solution is applied to study the solvation thermodynamics of a simple solute (here, a hard-rod particle) in mixtures of waterlike particles and a cosolvent. Two kinds of cosolvents are considered, one that stabilizes and one that destabilizes the "structure of water." The results obtained for the Gibbs energy, entropy, enthalpy, and heat capacity of solvation are in qualitative agreement with experimental data on the solvation of argon and methane in mixtures of water and ethanol and of water and p-dioxane.     

Isotropic-nematic interfacial tension of hard and soft rods: application of advanced grand canonical biased-sampling techniques

Coexistence between the isotropic and the nematic phase in suspensions of rods is studied using grand canonical Monte Carlo simulations with a bias on the nematic order parameter. The biasing scheme makes it possible to estimate the interfacial tension gamma(IN) in systems of hard and soft rods. For hard rods with LD=15, we obtain gammaIN approximately 1.4kBT/L2, with L the rod length, D the rod diameter, T the temperature, and kB the Boltzmann constant. This estimate is in good agreement with theoretical predictions, and the order of magnitude is consistent with experiments.     

Surface phase transitions in athermal mixtures of hard rods and excluded volume polymers investigated using a density functional approach

Using fundamental measures' density functional framework based on Wertheim's first order perturbation theory [J. Chem. Phys. 87, 7323 (1987)] we study the surface phase transitions in athermal polymer-needle mixtures, which demix in bulk into the isotropic polymer-rich (rod-poor) and polymer-poor (rod-rich) phases. We find that the polymer-rich (rod-poor) phase wets the hard wall at coexistence and the wetting transition is of first order. In the partial wetting regime we find a sequence of layerings but these transitions are gradually suppressed as the chain length increases. For long enough chains we detect the prewetting line. Rods exhibit pronounced ordering at the wall in the polymer-rich phases. Our results imply that experiments on the (isotropic) wetting transition for colloidal rod-polymer mixtures should be easier to carry out than those for the colloidal rod-sphere mixtures because the wetting transition occurs at lower rod densities. On the other hand, layerings in sphere-needle mixtures may turn out to be difficult to observe experimentally because some of them will be metastable with respect to the freezing transition, whereas the remaining ones are located very close to the binodal.     

Brownian dynamics simulation of the diffusion of rods and wormlike chains in a gel modeled as a cubic lattice: application to DNA

The translational diffusion constant of a particle, D, in a congested medium or a gel can be written as the product of two terms that account for long-range hydrodynamic interaction between the gel or congested medium and the particle, DEM, and a short-range "steric" term, S. For particles of arbitrary shape, DEM has been examined previously within the framework of the effective medium, EM, model (S. Allison et al., J. Phys. Chem. B 2008, 112, 5858-5866). In the present work, we examine S for rod- and wormlike chain models of duplex DNA in the size range of 100 to over 2000 base pairs. The gel is modeled explicitly as a cubic lattice, and Brownian dynamics simulation is used to examine S for a wide range of rod/wormlike chain and gel parameters. For wormlike chains with P = 50 nm, an empirical formula is derived for S that should be valid over a wide range of wormlike chain/gel parameters. For duplex DNA in the size of several hundred to several thousand base pairs in an agarose gel of 2% or less, fair agreement between modeling and experiment is obtained. However, modeling overestimates the length dependence of D observed experimentally. Finally, the reduction of D of DNA (100 to over 1000 base pairs in length) in cytoplasm relative to water can be accounted for quite well using the effective medium plus steric correction approach.