Un théorème H pour une suspension de sphères inélastiques

The problem of a two-phase dispersed medium is studied within the framework of the kinetic theory. We propose a model for the collision operator of the Boltzmann equation in the case of small undeformable and hard spheres that have all the same radius. The collisions are supposed instantaneous, binary and inelastic. The resulting collision operator allows, after validation, to prove the existence of an H theorem in several configurations according to the assumptions made about the particles and, in particular, in the case of a diluted suspension of inelastic hard spheres with or without dimension.     

Hydrodynamics of an endothermic gas with application to bubble cavitation

The hydrodynamics for a gas of hard spheres which sometimes experience inelastic collisions resulting in the loss of a fixed, velocity-independent, amount of energy Delta is investigated with the goal of understanding the coupling between hydrodynamics and endothermic chemistry. The homogeneous cooling state of a uniform system and the modified Navier-Stokes equations are discussed and explicit expressions given for the pressure, cooling rates, and all transport coefficients for D dimensions. The Navier-Stokes equations are solved numerically for the case of a two-dimensional gas subject to a circular piston so as to illustrate the effects of the energy loss on the structure of shocks found in cavitating bubbles. It is found that the maximal temperature achieved is a sensitive function of Delta with a minimum occurring near the physically important value of Delta approximately 12,000 K approximately 1 eV.     

Transport of a heated granular gas in a washboard potential

We study numerically the motion of a one dimensional array of Brownian particles in a washboard potential, driven by an external stochastic force and interacting via short range repulsive forces. In particular, we investigate the role of instantaneous elastic and inelastic collisions on the system dynamics and transport. The system displays a locked regime, where particles may move only via activated processes and a running regime where particles drift along the direction of the applied field. By tuning the value of the friction parameter controlling the Brownian motion we explore both the overdamped dynamics and the underdamped dynamics. In the two regimes we considered the mobility and the diffusivity of the system as functions of the tilt and other relevant control parameters such as coefficient of restitution, particle size, and total number of particles. We find that while in the overdamped regime the results for the interacting systems present similarities with the known noninteracting case, in the underdamped regime the inelastic collisions determine a rich variety of behaviors among which is an unexpected enhancement of the inelastic diffusion.     

Periodic boundary condition induced breakdown of the equipartition principle and other kinetic effects of finite sample size in classical hard-sphere molecular dynamics simulation

We examine consequences of the non-Boltzmann nature of probability distributions for one-particle kinetic energy, momentum, and velocity for finite systems of classical hard spheres with constant total energy and nonidentical masses. By comparing two cases, reflecting walls (NVE or microcanonical ensemble) and periodic boundaries (NVEPG or molecular dynamics ensemble), we describe three consequences of the center-of-mass constraint in periodic boundary conditions: the equipartition theorem no longer holds for unequal masses, the ratio of the average relative velocity to the average velocity is increased by a factor of [N/(N-1)]1/2, and the ratio of average collision energy to average kinetic energy is increased by a factor of N/(N-1). Simulations in one, two, and three dimensions confirm the analytic results for arbitrary dimension.     

DynamO: a free O(N) general event-driven molecular dynamics simulator

Molecular dynamics algorithms for systems of particles interacting through discrete or "hard" potentials are fundamentally different to the methods for continuous or "soft" potential systems. Although many software packages have been developed for continuous potential systems, software for discrete potential systems based on event-driven algorithms are relatively scarce and specialized. We present DynamO, a general event-driven simulation package, which displays the optimal O(N) asymptotic scaling of the computational cost with the number of particles N, rather than the O(N) scaling found in most standard algorithms. DynamO provides reference implementations of the best available event-driven algorithms. These techniques allow the rapid simulation of both complex and large (>10(6) particles) systems for long times. The performance of the program is benchmarked for elastic hard sphere systems, homogeneous cooling and sheared inelastic hard spheres, and equilibrium Lennard-Jones fluids. This software and its documentation are distributed under the GNU General Public license and can be freely downloaded from http://marcusbannerman.co.uk/dynamo.     

The effect of rotational and translational energy exchange on tracer diffusion in rough hard sphere fluids

A study is presented of tracer diffusion in a rough hard sphere fluid. Unlike smooth hard spheres, collisions between rough hard spheres can exchange rotational and translational energy and momentum. It is expected that as tracer particles become larger, their diffusion constants will tend toward the Stokes-Einstein hydrodynamic result. It has already been shown that in this limit, smooth hard spheres adopt "slip" boundary conditions. The current results show that rough hard spheres adopt boundary conditions proportional to the degree of translational-rotational energy exchange. Spheres for which this exchange is the largest adopt "stick" boundary conditions while those with more intermediate exchange adopt values between the "slip" and "stick" limits. This dependence is found to be almost linear. As well, changes in the diffusion constants as a function of this exchange are examined and it is found that the dependence is stronger than that suggested by the low-density, Boltzmann result. Compared with smooth hard spheres, real molecules undergo inelastic collisions and have attractive wells. Rough hard spheres model the effect of inelasticity and show that even without the presence of attractive forces, the boundary conditions for large particles can deviate from "slip" and approach "stick."     

The influence of bond rigidity and cluster diffusion on the self-diffusion of hard spheres with square well interaction

Hard spheres interacting through a square well potential were simulated by using two different methods: Brownian cluster dynamics (BCD) and event driven Brownian dynamics (EDBD). The structure of the equilibrium states obtained by both methods was compared and found to be almost identical. Self-diffusion coefficients (D) were determined as a function of the interaction strength. The same values were found by using BCD or EDBD. Contrary to EDBD, BCD allows one to study the effect of bond rigidity and hydrodynamic interaction within the clusters. When the bonds are flexible, the effect of attraction on D is relatively weak compared to systems with rigid bonds. D increases first with increasing attraction strength, and then decreases for stronger interaction. Introducing intracluster hydrodynamic interaction weakly increases D for a given interaction strength. Introducing bond rigidity causes a strong decrease in D which no longer shows a maximum as function of the attraction strength.     

Free energy calculations using flexible-constrained, hard-constrained and non-constrained molecular dynamics simulations.

A comparison of different treatments of bond-stretching interactions in molecular dynamics simulation is presented. Relative free energies from simulations using rigid bonds maintained with the SHAKE algorithm, using partially rigid bonds maintained with a recently introduced flexible constraints algorithm, and using fully flexible bonds are compared in a multi-configurational thermodynamic integration calculation of changing liquid water into liquid methanol. The formula for the free energy change due to a changing flexible constraint in a flexible constraint simulation is derived. To allow for a more direct comparison between these three methods, three different pairs of models for water and methanol were used: a flexible model (simulated without constraints and with flexible constraints), a rigid model (simulated with standard hard constraints), and an alternative flexible model (simulated with flexible constraints and standard hard constraints) in which the ideal or constrained bond lengths correspond to the average bond lengths obtained from a short simulation of the unconstrained flexible model. The particular treatment of the bonds induces differences of up to 2 % in the liquid densities, whereas (excess) free energy differences of up to 5.7 (4.3) kJ mol(-1) are observed. These values are smaller than the differences observed between the three different pairs of methanol/water models: up to 5 % in density and up to 8.5 kJ mol(-1) in (excess) free energy.     

Crystal nucleation of colloidal hard dumbbells

Using computer simulations, we investigate the homogeneous crystal nucleation in suspensions of colloidal hard dumbbells. The free energy barriers are determined by Monte Carlo simulations using the umbrella sampling technique. We calculate the nucleation rates for the plastic crystal and the aperiodic crystal phase using the kinetic prefactor as determined from event driven molecular dynamics simulations. We find good agreement with the nucleation rates determined from spontaneous nucleation events observed in event driven molecular dynamics simulations within error bars of one order of magnitude. We study the effect of aspect ratio of the dumbbells on the nucleation of plastic and aperiodic crystal phases, and we also determine the structure of the critical nuclei. Moreover, we find that the nucleation of the aligned close-packed crystal structure is strongly suppressed by a high free energy barrier at low supersaturations and slow dynamics at high supersaturations.     

Laser flash photolysis of benzophenone in polymer films

With a nanosecond laser we studied flash photolysis of benzophenone (BP) dissolved in four different polymer films. We measured kinetics of decay of a triplet state of benzophenone (3)BP as well as kinetics of decay of benzophenone ketyl free radicals BPH(?). Polymer matrices have plenty of reactive C-H bonds, and the hydrogen abstraction by (3)BP leads to a formation of geminate pair which either recombines into molecular products or dissociates. Decay kinetics of (3)BP is well described by dispersive kinetics and in particular by the kinetic law suggested in Albery, W. J.; et al. J. Am. Chem. Soc. 1985, 107, 1854. We observed a broader distribution of rate constants in hard films. It was observed that the decay kinetics of transients radicals in the "hard" polymers is quite satisfactory described by the same law for dispersive kinetics. Kinetics of radicals decay in "soft" polymers is satisfactorily described as a diffusion-enhanced reaction. Effect of a hardness of polymer matrix on the measured kinetic parameters is discussed.     

Radiative decay of nonstationary system

When a finite quantum system, say a fluorescent molecule is attached to a bulk surface and excited by a short laser pulse, the decay dynamics of the system is modulated by the surface and the signal is enhanced due to the bulk surface. We have considered the decay dynamics of a model of displaced distorted molecule whose excited potential surface is coupled to a continuum and then this first continuum is in turn coupled to a second continuum. In the short time scale there is a coherent exchange of energy between the system molecule and the first continuum states. In the long time scale the energy of the whole system plus first continuum drains out to the final continuum states. A dendrimer nanocomposite with the gold surface shows an enhanced light emission. This can be qualitatively understood from the model we proposed here. We have numerically studied the various potential parameters of the molecule which can affect the signal. When the potential surfaces are flat, the band structure of the first continuum states along with its initial excitation has some nontrivial effect on the profile of the radiative decay.     

The relationship between physical observables and the potential energy surface in the He-HF system

This paper examines the effect of infinitesimal functional variations in a rigid rotor He-HF potential surface on several different types of observables: inelastic cross sections, rate constants, and rotational energy level populations. The dynamics and kinetic observables studied were found to be sensitive to a large number of Legendre components of the potential with the region of highest sensitivity dependent upon the energy or temperature as well as the states related by the individual observable. Sensitivity to the entire surface tends to show a large degree of structure due to competition among sensitivities to the individual potential components. Significant information loss has been observed in the transition from microscopic to macroscopic observables.     

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.     

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.     

A density functional model for the binary crystal of hard spheres with vacancies

We study the stability of a binary mixture of hard spheres in the crystalline state in which a small fraction of lattice sites in the solid structure are vacant. The optimum vacancy concentration is obtained by minimizing the free energy of the inhomogeneous solid state. We use the modified weighted density functional approximation to compute the free energy. The necessary input for the theory is the thermodynamic properties of the homogeneous state of the mixture and is obtained from the solutions of the corresponding Percus-Yevick integral equations for the binary system. We compute the free energy for the crystal having two kinds of ordered structures in which (i) both the species lie in a disordered manner on a single face-centered-cubic lattice and (ii) each of the two species lie on a separate cubic lattice. Our theoretical model obtains equilibrium vacancy fraction of O(10(-5)) near the freezing point in both cases. The vacancy concentration decreases with the increase of density in both cases.     

Decay of correlation functions in hard-sphere mixtures: structural crossover

We investigate the decay of pair correlation functions in a homogeneous (bulk) binary mixture of hard spheres. At a given state point the asymptotic decay r-->infinity of all three correlation functions is governed by a common exponential decay length and a common wavelength of oscillations. Provided the mixture is sufficiently asymmetric, size ratios q less than or approximately 0.7, we find that the common wavelength reflects either the size of the small or that of the big spheres. By analyzing the (complex) poles of the partial structure factors we find a sharp structural crossover line in the phase diagram. On one side of this line the common wavelength is approximately the diameter of the smaller sized spheres whereas on the other side it is approximately the diameter of the bigger ones; the wavelength of the longest ranged oscillations changes discontinuously at the structural crossover line. Using density functional theory and Monte Carlo simulations we show that structural crossover also manifests itself in the intermediate range behavior of the pair correlation functions and we comment on the relevance of this observation for real (colloidal) mixtures. In highly asymmetric mixtures, q< or =0.1, where there is metastable fluid-fluid transition, we find a Fisher-Widom line with two branches. This line separates a region of the phase diagram where the decay of pair correlations is oscillatory from one in which it is monotonic.     

Molecular cluster decay viewed as escape from a potential of mean force

We show that evaporation from a quasistable molecular cluster may be treated as a kinetic problem involving the stochastically driven escape of a molecule from a potential of mean force. We derive expressions for the decay rate, and a relationship between the depth of the potential and the change in system free energy upon loss of a molecule from the cluster. This establishes a connection between kinetic and thermodynamic treatments of evaporation, but also reveals differences in the prefactor in the rate expression. We perform constant energy molecular dynamics simulations of cluster dynamics to calculate potentials of mean force, friction coefficients and effective temperatures for use in the kinetic analysis, and to compare the results with the directly observed escape rates. We also use the simulations to estimate the escape rates by a probabilistic analysis. It is much more efficient to calculate the decay rate by the methods we have developed than it is to monitor escape directly, making these approaches potentially useful for the assessment of molecular cluster stability.     

Dynamics of swelling/contracting hard spheres surmised by an irreversible Langevin equation

The diffusion of molecules through uniform homogeneous materials can readily be described by Brownian motion or generalizations thereof. The further generalization of these models to describe molecular diffusion through heterogeneous and nonstationary solvents is much less understood. Phenomenological nonstationary generalizations of the generalized Langevin equation (GLE) have earlier been developed satisfying the fluctuation-dissipation relationship in quasi-equilibrium limits while exhibiting somewhat complex behavior away from equilibrium. This reduced-dimensional representation should be capable of describing the diffusion of a particle through a colloidal suspension whose average particle size is tuned by an external driving force such as pH. A simple particle model of such a process involves the motion of a hard-sphere particle in an explicit environment of swelling hard spheres. The velocity autocorrelation functions observed in a large number of simulations of the particle model under various swelling rates agree precisely with those of a single form of the nonstationary phenomenological model. Though this procedure is not an explicit projection of the mechanical system onto the nonstationary GLE, it does show that the latter correctly describes the dynamics of the projected coordinate--namely, diffusion of the solute--under nonequilibrium conditions. Both nonequilibrium solvent models lead to behavior reminiscent of beta-relaxation processes at packing fractions substantially below that of the glass transition.     

Percolation, phase separation, and gelation in fluids and mixtures of spheres and rods

The relationship between kinetic arrest, connectivity percolation, structure and phase separation in protein, nanoparticle, and colloidal suspensions is a rich and complex problem. Using a combination of integral equation theory, connectivity percolation methods, nai?ve mode coupling theory, and the activated dynamics nonlinear Langevin equation approach, we study this problem for isotropic one-component fluids of spheres and variable aspect ratio rigid rods, and also percolation in rod-sphere mixtures. The key control parameters are interparticle attraction strength and its (short) spatial range, total packing fraction, and mixture composition. For spherical particles, formation of a homogeneous one-phase kinetically stable and percolated physical gel is predicted to be possible, but depends on non-universal factors. On the other hand, the dynamic crossover to activated dynamics and physical bond formation, which signals discrete cluster formation below the percolation threshold, almost always occurs in the one phase region. Rods more easily gel in the homogeneous isotropic regime, but whether a percolation or kinetic arrest boundary is reached first upon increasing interparticle attraction depends sensitively on packing fraction, rod aspect ratio and attraction range. Overall, the connectivity percolation threshold is much more sensitive to attraction range than either the kinetic arrest or phase separation boundaries. Our results appear to be qualitatively consistent with recent experiments on polymer-colloid depletion systems and brush mediated attractive nanoparticle suspensions.     

A practical integral equation for the structure and thermodynamics of hard sphere Coulomb fluids

A closure for the Ornstein-Zernike equation is presented, applicable for fluids of charged, hard spheres. From an exact, but intractable closure, we derive the radial distribution function of nonlinearized Debye-Hückel theory by subsequent approximations, and use the information to formulate a new closure by an extension of the mean spherical approximation. The radial distribution functions of the new closure, coined Debye-Hückel-extended mean spherical approximation, are in excellent agreement with those resulting from the hyper-netted chain approximation and molecular dynamics simulations, in the regime where the latter are applicable, except for moderately dilute systems at low temperatures where the structure agrees at most qualitatively. The method is numerically more efficient, and more important, convergent in the entire temperature-density plane. We demonstrate that the method is accurate under many conditions for the determination of the structural and thermodynamic properties of homogeneous, symmetric hard-sphere Coulomb systems, and estimate it to be a valuable basis for the formulation of density functional theories for inhomogeneous or highly asymmetric systems.