Self-diffusion of spheres in a concentrated suspension

We calculate the concentration-dependence of the short-time self-diffusion coefficient D_s for spherical particles in suspension. Our analysis is valid up to high densities and fully takes into account the many-body hydrodynamic interactions between an arbitrary number of spheres. The importance of these many-body interactions can be inferred from our calculation of the second virial coefficient of D_s.     

Diffusion of spheres in a concentrated suspension II

We evaluate the wavevector dependent (short-time) diffusion coefficient D(k) for spherical particles in suspension, by extending a previous study of selfdiffusion (which corresponds to the case of large k). Our analysis is valid up to high concentrations and fully takes into account the many-body hydrodynamic interactions between an arbitrary number of spheres, as well as the resummed contributions from a special class of correlations. Results obtained which agree well with available experimental data.     

Optical properties of a suspension of hard spheres

The optical properties of suspensions are studied in a wide range of concentrations. An expression for the polarization operator is obtained taking into account the contributions of two-and three-particle correlations. The extinction length l and the transport length l* are calculated in terms of a model of hard spheres. A detailed comparison of the results of calculations with experimental data is performed. In calculations, the structure factor is determined in the Percus-Yevick approximation, while the form factor is taken into account in the Rayleigh-Gans approximation and in terms of the Mie theory. It is shown that taking into account the contribution of three-particle correlations improves the agreement of the theory with experiment. It is found that, in the range of high suspension concentrations, the optical parameters are more sensitive to the choice of the model for the structure factor than for the form factor.     

Application of the scaled particle theory to the glass transition of hard spheres

In a previous article [1], an improved approach to the scaled particle theory of Reiss et al. was presented. As a result, we obtained a Padé-like expression of the compressibility factor of the fluid state. That expression contains two parameters p ₁ and p ₂, which we were able to calculate. In this paper we find other sets of values for p ₁ and p ₂. Two of these sets yield equations of state which agree with the simulation results of the glassy states obtained by Woodcock and Speedy. The calculations are based on two assumptions: firstly, that second order phase transitions may occur and secondly, that a hard sphere glass contains a fluid-like and a solid-like part. As an additional result, we found an upper and a lower limit for the fluid densities at which glass transitions may occur.     

Orientational structure of dipolar hard spheres near a hard neutral wall

Spherical boundaries are used in a Monte Carlo simulation to calculate the angular structure of dipolar hard spheres near a neutral hard wall.     

The effective shear viscosity of a uniform suspension of spheres

A general theory is presented to calculate the wave vector and frequency dependent effective viscosity of a suspension of spheres. The resulting formal expression for the effective viscosity is then used to evaluate the coefficients of the linear and the quadratic terms in an expansion in the volume fraction of the spheres using stick boundary conditions. On the linear level the well-known Einstein result is obtained. On the quadratic level we find that it is not justified to neglect higher order spatial derivatives of the secondary velocity fields as done in the analysis by Peterson and Fixman. As a consequence we find a value for the Huggins coefficient which is 12% higher than their value.     

Is the percolation transition of hard spheres a thermodynamic phase transition?

In hard-sphere systems, there is a fluid-solid transition, but no gas-liquid transition. In the fluid region, however, one can find a purely geometric percolation transition, which is studied in detail. The van der Waals model of hard spheres is treated. In this model, a uniform negative background potential is added. This modification does not change the structure, but induces a gas-liquid transition. In fact, percolation and the gas-liquid transition can be related to each other.     

Critical scaling of shear viscosity at the jamming transition

We carry out numerical simulations to study transport behavior about the jamming transition of a model granular material in two dimensions at zero temperature. Shear viscosity eta is computed as a function of particle volume density rho and applied shear stress sigma, for diffusively moving particles with a soft core interaction. We find an excellent scaling collapse of our data as a function of the scaling variable sigma/|rho(c)-rho|(Delta), where rho(c) is the critical density at sigma=0 ("point J"), and Delta is the crossover scaling critical exponent. We define a correlation length xi from velocity correlations in the driven steady state and show that it diverges at point J. Our results support the assertion that jamming is a true second-order critical phenomenon.     

On the thermodynamics of hard spheres near a soft repulsive wall

We extend the variational method based on the Gibbs-Bogolioubov inequality to the case of fluids against a wall. We investigate the influence of the softness of the wall on the free energy of the system. For small packing fraction we consider a density expansion. The variational results are compared with the exact ones which are given by a direct expansion of the free energy. A comparison between variational and perturbation methods has been done for small packing fraction and also for a case corresponding to the liquid state. The accuracy of the present extension of the variational method to a surface phenomena is found as good as in the bulk fluid. A very simple expression is given for the change on surface tension when we go from the perfect hard wall to soft repulsive wall.     

Freezing and glass transition of hard spheres in cavities

Dynamical and static properties of N=13-4000 hard spheres in spherical cavities with smooth and rough walls have been calculated by molecular-dynamics computer simulations. We use a dynamical criterion to distinguish between fluidlike and solidlike states. The associated crossover densities show a strong dependence both on the system size and on the surface roughness. For large N, these crossover densities tend to the bulk glass transition density for rough walls and to the bulk crystallization density for smooth walls. The crossover densities for finite N are found to be significantly smaller than the corresponding bulk densities. A detailed examination of the layer-resolved radial- and tangential mean-square displacements reveals qualitatively different dynamics for smooth and rough cavities.     

Phase transition in gas of hard core spheres

We present a new method of analyzing the gas of hard core spheres. We investigate analytic properties of the thermodynamic function over the circle of convergence of the cluster expansion and describe the way in which phase transition occurs.     

Temperature dependence of viscosity near the glass transition

Real glass transitions, where activated processes are included, are analysed near Whitney fold and cusp singularities. In particular the temperature dependence of the viscosity is studied. For sufficiently strongly coupled systems there is close to a fold singularity located atT=T _c , a crossover from an algebraic divergence (T–T _c )^– forT>T _c , to an Arrhenius dependence exp(E/k _B T) forT<T _c . If a parameter vector specifying the system also passes close to a cusp singularity, atT ₀<T _c , there may be a temperature region where the viscosity also shows Fogel-Fulcher like dependence. The results for the viscosity are intimately connected with the dynamics for the so-called -relaxation process, where relaxation occurs via a fractal time process in the delevant time region.     

Glass transition line for the system of charged hard spheres

We locate the glass transition line of the charged-hard-sphere system in the density-temperature plane, using a mean-field hypernetted chain approximation within the replica-symmetry-breaking scenario [S. Franz and G. Parisi, Phys. Rev. Lett. 79 (1997) 2486. [11]]. Our results demonstrate a dominant role of the steric factor and explain the ineffectiveness of purely Coulombic interactions in driving phase transitions.     

Molecular dynamics calculations of shear viscosity time-correlation functions for hard spheres

The time-correlation function for shear viscosity is evaluated for hard spheres at volumes of 1.6 and 3 times the close-packed volume by a Monte Carlomolecular dynamics technique. At both densities, the kinetic part of the timecorrelation function is consistent, within its rather large statistical uncertainty, with the long-timet ^–3/2 tail predicted by the mode-coupling theory. However, at the higher density, the time-correlation function is dominated by the cross and potential terms out to 25 mean free times, whereas the mode-coupling theory predicts that these are asymptotically negligible compared to the kinetic part. The total time-correlation function decays roughly ast ^–3/2, with much larger than the mode-coupling value, similar to the recent observations by Evans in his nonequilibrium simulations of argon and methane. The exact value of the exponent is, however, not very precisely determined. By analogy with the case of the velocity autocorrelation function, for which results are also presented at these densities, it is argued that it is quite possible that at high density the asymptotic behavior is not established until times substantially longer than those attainable in the present work. At the lower density, the cross and potential terms are of the same magnitude as the kinetic part, and all are consistent with the mode-coupling predictions within the relatively large statistical uncertainties.Work performed under the auspices of U.S. Department of Energy.     

Heterogeneous crystallization of hard and soft spheres near flat and curved walls

Crystallization represents a long-standing problem in statistical physics and is of great relevance for many practical and industrial applications. It often occurs in the presence of container walls or impurities, which are usually unavoidable or might even be desirable to facilitate crystallization by exploiting heterogeneous nucleation. Heterogeneous nucleation relies on a seed. Here we discuss the role of the seed and concentrate on a very generic situation, namely crystallization of hard and soft colloidal spheres in the presence of flat or curved hard walls. Curvature serves as a simple means to introduce a tunable mismatch between the seed-induced crystal lattice and the thermodynamically-favoured lattice. The mismatch induces distortions and elastic stress, which accumulate while the crystallite grows. This has an important consequence: once the crystallite reaches a critical size, it detaches from the seed allowing it to relax. The relaxed crystal continues to grow in the bulk, but crystallization ceases before reaching the seed, which now represents an impurity. Therefore, while seeds favour nucleation, any mismatch, like the seed curvature or an incommensurate structure, induces unfavourable distortions and can lead to the detachment of the crystallite. An additional mechanism to relax distortions is available to soft spheres, which can exploit their interaction potential and possibly deform. The different multi-step processes have been investigated by confocal microscopy, which provides particle-level information, and compared to computer simulations and theoretical results.     

A quantum field theory of viscosity near the liquid-glass transition

The two band model is considered for randomly distributed atoms in the harmonic potential approximation. Taking into account the scattering processes due to random eigenfrequencies and random hopping matrices contributing to the self-energy parts and vertex parts in the correlation functions of the density fluctuations in the interband, we obtain the generalized frequency-dependent viscosity in the viscoelastic theory. The viscosity is proportional to the relaxation time of the atoms, the inverse of which consists of the mean-square deviation of random eigenfrequencies and that of random hopping matrices. The former is almost constant and the latter obeys the Vogel-Fulcher law. In the vicinity of the transition they cross over.     

Universal properties of bulk viscosity near the QCD phase transition

We extract the bulk viscosity of hot quark–gluon matter in the presence of light quarks from the recent lattice data on the QCD equation of state. For that purpose we extend the sum rule analysis by including the contribution of light quarks. We also discuss the universal properties of bulk viscosity in the vicinity of a second-order phase transition, as it might occur in the chiral limit of QCD at fixed strange quark mass and most likely does occur in two-flavor QCD. We point out that a chiral transition in the O(4)$O(4)$ universality class at zero baryon density as well as the transition at the chiral critical point which belongs to the Z(2)$Z(2)$ universality class both lead to the critical behavior of bulk viscosity. In particular, the latter universality class implies the divergence of the bulk viscosity, which may be used as a signature of the critical point. We discuss the physical picture behind the dramatic increase of bulk viscosity seen in our analysis, and devise possible experimental tests of related phenomena.     

A first-order phase transition defines the random close packing of hard spheres

Randomly packing spheres of equal size into a container consistently results in a static configuration with a density of ∼64%. The ubiquity of random close packing (RCP) rather than the optimal crystalline array at 74% begs the question of the physical law behind this empirically deduced state. Indeed, there is no signature of any macroscopic quantity with a discontinuity associated with the observed packing limit. Here we show that RCP can be interpreted as a manifestation of a thermodynamic singularity, which defines it as the “freezing point” in a first-order phase transition between ordered and disordered packing phases. Despite the athermal nature of granular matter, we show the thermodynamic character of the transition in that it is accompanied by sharp discontinuities in volume and entropy. This occurs at a critical compactivity, which is the intensive variable that plays the role of temperature in granular matter. Our results predict the experimental conditions necessary for the formation of a jammed crystal by calculating an analogue of the “entropy of fusion”. This approach is useful since it maps out-of-equilibrium problems in complex systems onto simpler established frameworks in statistical mechanics.