Pathologies in the sticky limit of hard sphere Yukawa models for colloidal fluids: a possible correction

A known ‘sticky hard sphere’ model, starting from a hard sphere Yukawa potential and taking the limit of infinite amplitude and vanishing range with their product remaining constant, is shown to be ill-defined. This is because its Hamiltonian (which we call SHS2) leads to an exact second virial coefficient which diverges, unlike that of Baxter's original model (SHS1). This deficiency has never been observed so far, since the linearization implicit in the ‘mean spherical approximation’ (MSA), within which the model is analytically solvable, partly masks such a pathology. To overcome this drawback and retain some useful features of SHS2, we propose both a new model (SHS3) and a new closure (‘modified MSA’), whose combination yields an analytical solution formally identical with the SHS2-MSA solution. This mapping allows the recovery of many results derived from SHS2, after a re-interpretation within a correct framework. Possible developments are indicated.     

Transport coefficients of hard sphere fluids

New calculations have been made of the self-diffusion coefficient D, the shear viscosity η_s, the bulk viscosity η_b and thermal conductivity λ of the hard sphere fluid, using molecular dynamics (MD) computer simulation. A newly developed hard sphere MD scheme was used to model the hard sphere fluid over a wide range up to the glass transition (～0.57 packing fraction). System sizes of up to 32 000 hard spheres were considered. This set of transport coefficient data was combined with others taken from the literature to test a number of previously proposed analytical formulae for these quantities together with some new ones given here. Only the self-diffusion coefficient showed any substantial N dependence for N < 500 at equilibrium fluid densities (ε 0.494). D increased with N, especially at intermediate densities in the range ε ～ 0.3–0.35. The expression for the packing fraction dependence of D proposed by Speedy, R. J., 1987, Molec. Phys., 62, 509 was shown to fit these data well for N ～ 500 particle systems. We found that the packing fraction ε dependence of the two viscosities and thermal conductivity, generically denoted by X, were represented well by the simple formula X/X ₀ = 1/[1 ? (ε/ε₁)]^m within the equilibrium fluid range 0 > ε > 0.493. This formula has two disposable parameters, ε and m, and X ₀ is the value of the property X in the limit of zero density. This expression has the same form as the Krieger-Dougherty formula (Kreiger, I. M., 1972, Adv. Colloid. Interface Sci., 3, 111) which is used widely in the colloid literature to represent the packing fraction dependence of the Newtonian shear viscosity of monodisperse colloidal near-hard spheres. Of course, in the present case, X _o was the dilute gas transport coefficient of the pure liquid rather than the solvent viscosity. It was not possible to fit the transport coefficient normalized by their Enskog values with such a simple expression because these ratios are typically of order unity until quite high packing fractions and then diverge rapidly at higher values over a relatively narrow density range. At the maximum equilibrium fluid packing fraction ε = 0.494 for both the hard sphere fluid and the corresponding colloidal case a very similar value was found for η_s/η_o ?30–40, suggesting that the ‘crowding’ effects and their consequences for the dynamics in this region of the phase diagram in the two types of liquid have much in common. For the hard sphere by MD, D_o/D ～ 11 at the same packing fraction, possibly indicating the contribution from ‘hydrodynamic enhancement’ of this transport coefficient, which is largely absent for the shear viscosity. Interestingly the comparable ratio for hard sphere colloids is the same.     

一个简单硬球碰撞问题中的混沌

对于一个在铅垂线上的两个硬球及刚性地面组成的硬球碰撞体系,计算了它的Lyapunov指数．因为该Lyapunov指数的正定性,所以该体系是一个混沌体系．     

Local size segregation in polydisperse hard sphere fluids

The structure of polydisperse hard sphere fluids, in the presence of a wall, is studied by the Rosenfeld density functional theory. Within this approach, the local excess free energy depends on only four combinations of the full set of density fields. The case of continuous polydispersity thereby becomes tractable. We predict, generically, an oscillatory size segregation close to the wall, and connect this, by a perturbation theory for narrow distributions, with the reversible work for changing the size of one particle in a monodisperse reference fluid.     

Fortran codes for the correlation functions of hard sphere fluids

Our Fortran codes for hard sphere fluids and their mixtures for the correlation functions that arise from the Percus–Yevick theory and the Verlet–Weis semi-empirical correction have proven useful during a period of nearly four decades and continue to be useful. In order to make these codes even more widely available, a brief summary is presented here and listings of these codes are given in the electronically accessible Supplementary Material to this paper.     

Quantum corrections to thermodynamics of polar hard sphere fluids

The quantum corrections to the thermodynamic properties of polar hard sphere fluids and fluid mixtures are estimated taking into account the influence of dipole and quadrupole moments. Expressions are given for the second virial coefficient, free energy and pressure and results are given for different values ofμ* andϑ*. The first order quantum correction arises due to the translational contribution only. The quantum effect increases with density,μ* andϑ*. Numerical results are also estimated for binary mixtures of (i) hard spheres and dipole hard spheres and (ii) hard spheres and quadrupole hard spheres. The ‘excess’ free energy for dipole hard sphere binary mixture is also reported. It is found that the ‘excess’ quantum effect depends on the concentration and the particle diameter ratio and increases with increase ofμ* andϑ*.     

The fifth virial coefficients of fused hard sphere fluids

The virial coefficients up toB ₅ of 34 fused hard sphere models have been computed using about (4–10)×10⁵ trial configurations of a modified Monte Carlo integration method. The principle of the conformal behaviour of hard particle systems is tested against these data. It is found that while for the fourth coefficient the principle is satisfied in all the cases, the fifth coefficients exhibit more complicated behaviour and do not conform, in general, to this principle.     

Two-dimensional chiral segregation of the bent hard needles model with Lennard-Jones sites

Assuming the aggregates of a single chiral component as a pure phase, chiral segregation can be considered as a coexistence of two phases, therefore the formalisms of phase transitions in Statistical Mechanics can be applied. That is, chiral segregation can be considered as phase equilibrium. The standard mechanism to understand phase equilibrium considers the complete van der Waals forces, including repulsions and attractions. However, during the last few years, a particular kind of fluid-fluid equilibrium, explained in terms of only excluded volume effects, has gained important interest. This kind of phenomenon is known as depletion forces. In this paper, the contribution of these two mechanisms is investigated. In order to analyze the effect that repulsions and attractions have on chiral segregation we study a two-dimensional model, bent hard needles model. In one case only infinitely repulsive interactions are considered while in the second case, one and two Lennard-Jones (LJ) sites are added to model the complete van der Waals forces. In the first case, because of the nature of the model, the cause for chiral segregation is due to excluded volume effects that appear when the density of the system is sufficiently high. In the second case, the model proposed is more complex and besides the density the temperature also affects the phase separation. Monte Carlo simulations in the Gibbs ensemble (GMC) are used to perform this comparison. As a result, the segregation phase diagram is obtained for the infinitely repulsive model and that with two LJ sites. The important conclusions are that for some particular molecular geometries, repulsions are not able to produce chiral segregation at all, independently of density. However, when attractions are included the effect of repulsion is complemented and phase separation can be achieved even for those molecular geometries where infinitely repulsions were not able.     

On the compressibility equation of state for multi-component adhesive hard sphere fluids

The compressibility equation of state (EOS) for a multi-component sticky hard sphere model alternative to Baxter's one is investigated within the mean spherical approximation (MSA). For this model and this closure, as well as for a more general class of models and closures leading to Baxter functions q_ij(r) with density-independent stickiness coefficients, no compressibility EOS can exist for mixtures, unlike the one-component case (in view of this, an EOS recently reported in the literature turns out to be incorrect). The reason is the failure of the Euler reciprocity relation for the mixed second-order partial derivatives of the pressure with respect to the partial densities. This is in turn related to the inadequacy of the approximate closure (in particular, the MSA). A way out to overcome this drawback is presented in a particular example, leading to a consistent compressibility pressure, and a possible generalization of this result is discussed.     

Classical and quantum hard sphere fluids: Theory and experiment

A comprehensive picture is presented of what can be reconstructed of the equations of state, at all densities, of both classical and quantum hard sphere fluids (the latter in their ground states) just from the first few coefficients which are known of the low-density expansions. Extrapolation techniques, mostly Padé and generalizations thereof, are employed and comparisons with computer simulation studies are made wherever these are available.     

On the critical non-additivity driving segregation of asymmetric binary hard sphere fluids

A previously proposed version of thermodynamic perturbation theory, appropriate for singular pair interactions between particles, is applied to binary mixtures of hard spheres with non-additive diameters. The critical non-additivity Δ_C required to drive fluid–fluid phase separation is determined as a function of the ratio ξ ≤ 1 of the diameters of the two species. Δ_C(ξ) is found to decrease with ξ and to go through a minimum for ξ ? 0.015 before increasing sharply as ξ → 0, irrespective of the total packing-fraction η of the mixture. These results are the basis of an estimate of the range of size ratios for which a binary mixture of additive hard spheres exhibit a fluid–fluid miscibility gap. This range is conjectured to be 0.01 ? ξ ? 0.1.     

Modified Parsons-Lee theory for fluids of linear fused hard sphere chains

The Parsons-Lee theory has been modified to study the liquid-crystalline phase behaviour of the linear fused hard sphere chain fluids. The modification of the Parsons-Lee theory is based on the application of the so-called effective molecular volume instead of the real molecular volume. This alteration does not mean any change for the Parsons-Lee treatment of the hard convex bodies but does for the non-convex ones. The results of the modified Parsons-Lee theory show very good agreement with simulations not only for the location of the isotropic-nematic phase transition but for the equation of state.     

Order-disorder transition in the solid phase of a charged hard sphere model

We investigate the solid phases of the restricted primitive model (RPM). Monte Carlo simulations show the existence of an order-disorder transition from a substitutionally disordered face centered cubic lattice (fcc) to a new ordered fcc structure which is proposed as the ground state of the RPM at the close packing density. Our results suggest that the new phase might turn out in a new triple point in the RPM phase diagram involving three solid phases: CsCl, fcc ordered and fcc disordered structures. The order-disorder transition is also studied using the cell theory. The theory shows good agreement with the simulation results and suggests that the transition is weakly first order.     

Spatial arrangements of particles with different mobility tendencies in a model glass-former

The use of the isoconfigurational method has enabled one to determine the existence of particles with high and low dynamic propensity (tendency to be mobile) and particles with preferred directionality for motion (directional particles) in supercooled liquids. On the other hand, dynamical studies have shown that the relaxation of such systems evolves by means of rapid crossings between metabasins of the potential energy surface (a metabasin being a group of mutually similar, closely related structures which differ markedly from the ones belonging to other metabasins), as collectively relaxing units (d-clusters) take place. Here we determine the spatial arrangements of such particles in a model three dimensional glass-forming system. We show that both the highest and the lowest propensity particles form compact clusters, which are separated from each other by the high directionality particles. The particles of this interfacial region seem to behave as to help make room for the enhanced (expansion) movement of the high propensity cluster and to keep the local density constant. Finally, we also find that only the high propensity particles (but not the directional ones) exhibit a great tendency to take part in d-cluster events.     

Static structure factor for a fluid with interaction of hard spheres plus two Yukawa tails

We determine the static structure factor S(k) for a fluid of hard spheres with two-Yukawa interactions through the application of the mean spherical approximation (MSA) to a multi-component system composed of hard-spheres plus double Yukawa interactions (HSDY). This S(k) depends on scaling parameters Γ_n that satisfy a system of nonlinear equations. We report explicit results for a mono-dispersed HSDY fluid and show that the hard-sphere contributions control the main peak of the S(k),while for wave vectors approaching zero, we predict a cluster peak which could be identified with that of recent experimental results of Liu et al. [Y. Liu, W.-R. Chen, S.H. Chen, J. Chem. Phys. 122 (2005) 044507-1].     

Liquid mixture excess properties by the hard sphere model

The excess properties of mixing were calculated for seven systems Ar-CH₄; Ar-N₂; Ar-O₂; Ar-CO; CO-CH₄; O₂-N₂; N₂-CO, using the Caranhan and Starling equation of state of rigid spheres and the Longuet-Higgins and Widom model. Two sets of calculations were done one using the experimentalG ^E to calculateS ^E ,H ^E andV ^E , and the other making use of Miller’s cross parameter values. The calculated values are compared with those of Snider and Herrington and Miller’s values and also with the experimental values. The agreement was found to be good.     

The Smoluchowski limit for a simple mechanical model

We consider a vertical stick constantly accelerated along thex-axis by a forceF and which elastically collides with point particles of the same mass (atoms). The atoms are initially Poisson distributed and are allowed to have four velocities only. It is shown that under suitable scaling of the system the displacementQ(t) of the stick satisfies a nontrivial CLT:Q(t)=vFt+D ^1/2 W(t) (Smoluchowski equation), where the values ofv andD depend on the fact that one atom may collide several times.     

二分量带电胶体悬浮系统的等效硬球模型

研究由无限稀薄的靶粒子散布于有限浓度(体积分数为)的主粒子悬浮液中而组成的二分量带电胶体系统,计算了靶粒子的短时间平动和转动自扩散系数.当系统中的粒子浓度和电解质浓度都不太高时,只考虑流体力学相互作用对扩散张量的首项两体贡献.为了计算体系的对分布函数,在数值计算的基础上发展了一个等效硬球模型,近似地把主粒子和靶粒子看作等效半径为δEHS的相同硬球粒子.结果表明,靶粒子的自扩散系数随两种粒子尺寸比和主粒子体积分数变化的关系可以很好地用等效硬球模型来解释 关键词： 胶体悬浮系统 自扩散 等效硬球 流体力学作用 关联函数     

An effective method for hard sphere diameters of real fluids along the coexistence line vapour-liquid

The variation of the effective hard sphere diameter with the temperature along the equilibrium line vapour-liquid has been analyzed for a great variety of substances including: (a) atomic simple liquids; (b) inorganic and organic molecular liquids and (c) liquid metals, following several procedures, based in the analogy with a hard sphere fluid. Though the comparison among available methods is generally satisfactory, the method based on internal pressure seems to be more effective.