Structure factors in a two-dimensional binary colloidal hard sphere system

ABSTRACT

Hard disks are one of the simplest interacting many-body model system in two dimensions (2D). Here, we present a comprehensive set of measurements of the static structure factors for quasi-2D monodisperse fluids and two different binary colloidal hard sphere mixtures: a small size ratio (SSR) system with a negligibly small negative non-additivity and a large size ratio system with a significantly larger non-additivity. We compare the experimental results for the monodisperse and SSR systems to those calculated using density functional theory (DFT) for additive mixtures. Furthermore, we determine the zero-wavevector limits of the static structure factors for the monodisperse and binary hard sphere fluids directly from an analysis of number and concentration fluctuations. For the monodisperse case, this leads to the isothermal compressibility, which agrees very well with DFT, and is consistent with the scaled particle theory equation of state for hard disks. For the binary fluids, the partial static structure factors are used to calculate the Bhatia–Thornton structure factors, and we find qualitative agreement with DFT for the SSR mixture. Finally, the zero-wavevector limits of the Bhatia–Thornton structure factors are determined and directly related to the thermodynamic factor, the dilatation factor and the isothermal compressibility.     

A liquid-liquid phase transition in the “collapsing” hard sphere system

A liquid-liquid phase transition is discovered in a system of collapsing hard spheres using the thermodynamic perturbation theory. This is the first evidence in favor of the existence of that kind of phase transition in systems with purely repulsive and isotropic interactions.     

Electron spectral density in a disordered system of hard sphere scatterers

The spectral density of an electron propagating in a disordered system of hard sphere scatterers is studied by use of a self-consistent approximation for the self-energy. The spectral density at fixed wavenumber is found to be a single-peaked function of energy. The approximation yields a sharp wavenumber-dependent band edge. For large wavenumbers the spectral density is well approximated by a Lorentzian, but for small wavenumbers it is dominated by a characteristic square root singularity at the band edge.     

Structure of the hard sphere solid

We show that near densest packing the perturbations of the hexagonal close packed (hcp) structure yield higher entropy than perturbations of any other densest packing. The difference between the various structures shows up in the correlations between motions of nearest neighbors. In the hcp structure random motion of each sphere impinges slightly less on the motion of its nearest neighbors than in the other structures.     

Structure factor of a system with shouldered hard sphere potential

The structure factor of a system of particles interacting through a shouldered hard sphere potential is computed from a Monte Carlo simulation. The result pleads for a similar effective two-body potential in liquid metals like bismuth and confirms previous conclusions of Silbert and Young.     

Heterogeneous shear in hard sphere glasses

There is growing evidence that the flow of driven amorphous solids is not homogeneous, even if the macroscopic stress is constant across the system. Via event-driven molecular dynamics simulations of a hard sphere glass, we provide the first direct evidence for a correlation between the fluctuations of the local volume fraction and the fluctuations of the local shear rate. Higher shear rates do preferentially occur at regions of lower density and vice versa. The temporal behavior of fluctuations is governed by a characteristic time scale, which, when measured in units of strain, is independent of shear rate in the investigated range. Interestingly, the correlation volume is also roughly constant for the same range of shear rates. A possible connection between these two observations is discussed.     

Settling statistics of hard sphere particles

Direct imaging of settling, non-Brownian, hard sphere, particles allows measurement of particle occupancy statistics as a function of time and sampling volume dimension. Initially random relative particle number fluctuations, (2)>/ = 1, become suppressed, anisotropic, and dependent. Fitting to a simple Gaussian pair correlation model suggests a minute long ranged correlation leads to strong if not complete suppression of number fluctuations. Calflisch and Luke predict a divergence in velocity fluctuations with increasing sample volume size based on random (Poisson) statistics. Our results suggest this is not a valid assumption for settling particles.     

Crystallization mechanism of hard sphere glasses

In supercooled liquids, vitrification generally suppresses crystallization. Yet some glasses can still crystallize despite the arrest of diffusive motion. This ill-understood process may limit the stability of glasses, but its microscopic mechanism is not yet known. Here we present extensive computer simulations addressing the crystallization of monodisperse hard-sphere glasses at constant volume (as in a colloid experiment). Multiple crystalline patches appear without particles having to diffuse more than one diameter. As these patches grow, the mobility in neighboring areas is enhanced, creating dynamic heterogeneity with positive feedback. The future crystallization pattern cannot be predicted from the coordinates alone: Crystallization proceeds by a sequence of stochastic micronucleation events, correlated in space by emergent dynamic heterogeneity.     

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.     

Jastrow correlations and the hard sphere fermi gas

A diagrammatic expansion for the variational energy of a Fermi fluid using state-dependent correlations is described. It is used to obtain the leading order terms (up to (k_Fa)²) in the low density expansion of a hard sphere Fermi gas using both state-dependent and state-independent correlations.     

SAFT-D theory for hard sphere and hard disc chain fluids

SAFT-dimer (SAFT-D) theory is reformulated to yield an improved equation of state for the hard sphere chain fluid. Two sets of the equation of state are proposed by employing Chiew's expressions for the contact values of the m hard sphere site-site correlation function g(σ). Comparison with molecular simulation data shows that the improved SAFT-D equation of state predicts the compressibility factor more accurately than Ghonasgi and Chapman's equation of state. It has been shown that SAFT-dimer theory can be applied readily to fused hard sphere chain fluids by considering the correct value of the effective chain length (m*). SAFT-dimer theory is also extended to the 2-dimensional tangent and fused hard disc chain fluids. For the fused hard disc dimer fluid, the SAFT equation of state is found to be more accurate than the Boublik hard disc dimer equation of state. For tangent hard disc chain fluids, the results obtained from SAFT-dimer theory are compared with Monte Carlo results for 5-mers and with GFD theory for 4-mers, 8-mers and 16-mers.     

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.     

Calculation of entropy of mixing of liquid metal alloys in a hard sphere system

A semi-empirical model, based on the hard sphere system, is used to determine the entropy of mixing of simple as well as compound-forming alloys. For the compound-forming liquid solutions, the method leads to fairly accurate results, showing thereby that the usual theory of hard spheres mixtures can be applied to compound forming alloys also. It has been shown that the compound formation is very sensitive to the temperature of the mixture. Numerical applications are attempted for NaHg and NaGa.     

Phase transitions of hard sphere lattice gases

A unified proof is given for the existence of a phase transition for any two or three dimensional lattice gas with hard cores excluding nearest neighbor occupancy, provided only that the lattice is an open one (possessing two sublattices). It is not required that one sublattice be a translate of the other. Consequently the proof applies to the plane hexagonal and to the diamond lattices, as well as the cubic lattices previously proved to have phase transitions. The models are converted to equivalent Ising spin 1/2 systems on one sublattice by a partial trace over the other. The spin system has many-spin interactions including some of odd order, which destroys up-down symmetry, but recent work of Pirogov and Sinai on such systems is shown to be applicable and to prove the existence of the phase transition.Supported in part by National Science Foundation Grant No. GP 33535X.     

Note on the fractal dimension of hard sphere trajectories

Using Monte Carlo molecular dynamics, a new, careful study is made of the approach of the trajectory of a typical particle in a hard sphere fluid to that of a Brownian particle, discussed before by Powles and Quirke and Rapaport. The apparent fractal dimension of the trajectory, as a function of reduced length scale,(), characterizes the transition from mechanical to Brownian motion and differs markedly from 2 in all present computer simulations.     

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.     

Dynamics of a hard sphere granular impurity

An impurity particle coupling to its host fluid via inelastic hard sphere collisions is considered. It is shown that the exact equation for its distribution function can be mapped onto that for an impurity with elastic collisions and an effective mass. The application of this result to the Enskog-Lorentz kinetic equation leads to several conclusions: (1) every solution in the elastic case is equivalent to a class of solutions in the granular case; (2) for an equilibrium host fluid the granular impurity approaches equilibrium at a different temperature, with a dominant diffusive mode at long times; (3) for a granular host fluid in its scaling state, the granular impurity approaches the corresponding scaling solution.