Equation of state and correlation function contact values of a hard sphere mixture

Recently, there has been considerable activity in the study of the equation of state and correlation functions of a hard sphere mixture when the concentration of the large spheres is exceedingly small. This system is of considerable interest both as a simple prototype of an asymmetric mixture and as a simple but useful model of a colloidal suspension. We review our ad hoc formulae for the correlation function contact values and discuss the evidence, both pro and con, from exact and nearly exact theorems, computer simulations and experiment. Our conclusion is that our formulae are, on the whole, correct but the large sphere–large sphere correlation function contact value seems to need revision when the concentration of the large spheres is small but not exceedingly small. We propose a modest revision that preserves the seemingly correct aspects of our formula and this is in good agreement with the most recent simulations.     

Contact values for disparate-size hard-sphere mixtures

A universality ansatz for the contact values of a multicomponent mixture of additive hard spheres is used to propose new formulae for the case of disparate-size binary mixtures. A comparison with simulation data and with a recent proposal by Alawneh and Henderson for binary mixtures shows reasonably good agreement with the predictions for the contact values of the large–large radial distribution functions. A discussion on the usefulness and limitations of the new proposals is also presented.     

Second-order Percus–Yevick theory for the radial distribution functions of a mixture of hard spheres in the limit of zero concentration of the large spheres

The radial distribution functions of a mixture of hard spheres are quite interesting when the ratio of diameters is large and the concentration of the large spheres is very small. In this regime, the radial distrbution functions change rapidly with concentration. The usual PercusYevick theory, which is adequate over most of the concentration range, fails at low concentrations of the large spheres. Values are reported of the radial distribution functions for zero concentration of the large spheres using the most accurate theory presently available, secondorder Percus-Yevick theory. Agreement with recent formulae for the contact values of these functions is very good except for the contact value for a pair of large spheres, where the agreement is fairly good. It is possible that the radial distribution function for a pair of large spheres may be a little larger than the already large values given by this recent formula.     

Contact values of highly asymmetric hard sphere mixtures from Fundamental Measure Density Functional Theory

The White Bear version of Fundamental Measure Theory (FMT-WB) has been tested in binary mixtures of hard spheres in the vicinity of the colloidal limit, where the size ratio of the two species is exceedingly large and the large sphere mole fraction is infinitely low. Contact values of large–large sphere radial distribution functions have been calculated and compared with molecular dynamics simulations and previously proposed theoretical formulas. In contrast to the failure of BMCSL (Boublik, Mansoori, Carnahan, Starling, Leland equation of state) predictions, FMT-WB gives good agreement with simulation for a range of species size ratios and mole fractions. The performance of BMCSL is qualitatively related to one of its model parameters, which could indicate the reliability of the BMCSL result. Our results confirm the accuracy of FMT-WB in the colloidal limit for the first time and suggest that BMCSL contact values must be applied carefully to account for chain connectivity when studying certain cases with classical Density Functional Theories.     

Phase behavior of binary hard-sphere mixtures from perturbation theory

Using a first-order perturbation theory, we have studied the phase diagram of a binary mixture of hard spheres for different values of the size ratio. Recent models for the two-body depletion potential between large spheres are used to take into account the role of the small spheres. The theory predicts a complex phase diagram including a fluid-solid transition at high packing fraction of small spheres, metastability of fluid-fluid demixing, an isostructural solid-solid transition at high packing fraction of the large spheres for sufficiently small values of the size ratio q of the spheres, and the tendency to sticky-sphere behavior in the limit q-->0. The agreement with recent simulation results is quite good. We also show that this phenomenology was already implicit in the pioneering work of Asakura and Oosawa.     

Simulation of binary hard-sphere systems with 1 : 5 and 1 : 10 size ratios

Hard-sphere molecular dynamics simulations were performed on asymmetric binary systems to obtain pair-correlation functions. Particle-diameter ratios of 1?:?5 and 1?:?10 were applied to mimic the colloid/solvent situation, where the large particles were the colloids and the small ones represented the solvent molecules. The packing fractions were chosen systematically between 0.1–0.5 for the large spheres and 0.0–0.5 for the small ones. We focused on two questions: a) To what extent can one approximate the inter-particle structure of the large spheres in binary systems with one-component hard-sphere data. b) How reliable are the theoretical pair-correlation functions of the Percus-Yevick and Rational Function Approximation models. Overlap integrals of pair-correlation function pairs were calculated to answer quantitatively. The results supported quantitatively the limits of the one-component hard-sphere approximation. On contrary, the selected theoretical methods reproduced the simulation results satisfactorily, if the packing fraction of the solvent was not more than 0.2. The theoretically important contact values of the pair-correlation functions were calculated, too.     

A Theoretical and Experimental Study of the Total Light Scattering Technique for particle size analysis

The total light scattering method for particle size measurement has a series of outstanding features, e.g. very simple instrumental arrangement, no serious requirements for its optical, photodetecting and electronic systems and its ease of use. Nevertheless, some problems still remain unsolved which should be dealt with to improve its further applications. An improvement has been made in this work. Comprehensive theoretical studies showed that this method is applicable to particle size analysis in the range 0.1–10 μm, i.e. for those particles whose diameter lies between the lower limit of the diffraction method and the upper limit of photon correlation spectroscopic (PCS) technique. Experimental studies with different monodisperse polystyrene latex spheres as standard reference material give good agreement with their nominal values.     

Clustering of continuous distributions of charge

An investigation is carried out of the association and clustering of equimolar mixtures of oppositely charged Gaussian charge distributions (CDs) of the form ～ exp ( ? r²?/2α²), where r is the separation between the centres of charge and α governs the extent of charge spreading (α→0 is the point charge limit). The results of molecular dynamics (MD) and Ornstein–Zernike integral equation with the mean spherical approximation (MSA) and hypernetted-chain (HNC) closures are compared for these systems. The MD and HNC radial distribution functions, g(r), agree very well for not too small α. The MD and MSA, g(r), also agree well for α ≈ 1 and greater. The potential energy per particle for the three methods also agrees well over a wider range of α values, better than might be expected from inspection of the radial distribution functions, because the dominant contributions to U come predominantly from intermediate and long distance ranges where there is good agreement between the g(r) from the MSD and HNC closures. The nature of the association and clustering of the charges as a function of α is explored through the mean nearest neighbour distance for unlike and like species and the mean and root-mean-square force. The velocity and force autocorrelation functions are also calculated; they show increasingly oscillatory behaviour in the small α limit, originating in vibrations of a pair of CDs of opposite sign.     

Statistical fluid theory for associating fluids containing alternating heteronuclear chain molecules

Statistical associating fluid theory of homonuclear dimerized chain fluids and homonuclear monomer-dimer mixture chain fluids are extended to fluids containing alternating heteronuclear chain molecules separately. The proposed models account for the appropriate site-site correlation functions at contact. The modified equations of state show a good agreement with generalized Flory dimer theory and MD simulation data for small and medium size ratio of hard sphere diameters.     

Contact condition for the density profiles in spherical and cylindrical double layers

Exact sum rules involving the contact values of the density profiles and bulk osmotic pressure in spherical and cylindrical electric double layers are formulated. When the radius of curvature in these systems tends to infinity, the contact conditions reduce to the well-known contact condition in planar double layer due to Henderson, Blum, and Lebowitz (1979). However, unlike the latter relation, the contact conditions in the non-planar geometries are non-local, and require for their implementation full knowledge of the electrode–ion singlet distribution functions.     

Fluid in contact with semipermeable vesicle: second-order integral equation approach

Secondorder Ornstein–Zernike integral equations in conjunction with the Lovett–Mou–Buff–Wertheim equation for the density profile are used to investigate a mixture of hard spheres in contact with a semipermeable membrane of spherical symmetry. Theoretical predictions are compared with grand canonical Monte Carlo simulations for several parameters, and reasonable agreement has been found. The pair functions for the systems considered are also determined and discussed.     

Novel silicon-carbon fullerene-like cages

We present model calculations of kinetic energy releases and fission barriers in asymmetric fission of C₆₀^{r +} ions, using a simple electrostatic model where the fragments are treated as conducting spheres. The kinetic energy releases are calculated using two different approaches for deducing the radii of the spheres. Both approaches give results in qualitative agreement with experimental results. The fission barriers, on the other hand, depend strongly on the model radii and the activation energies for neutral fragment emission. A comparison between the model calculations shows that the choice of the finite size of the smaller fragments become important as r increases and have large influences on the prediction for the C₆₀^{r +} stability limit. The competition between neutral (evaporation) and charged-fragment emission (fission) are discussed within a static over-the-barrier model for electron transfer between conducting spheres.Received: 10 December 2003, Published online: 27 January 2004PACS: 34.70. + e Charge transfer - 36.40.Qv Stability and fragmentation of clusters - 36.40.Wa Charged clusters     

Metal-ceramic contact angles determined by thin film techniques

A method of estimating contact angles is described in which 1000 A thick layers of metals are condensed from the vapour. If condensation takes place at temperatures greater than about 0.8 T_m the film is observed to break up into drops which are approximately segments of spheres, and the contact angle may be calculated from measurements on the drop size when the average film thickness is known. The experimental method is straightforward and relatively very fast, yet gives results in good agreement with earlier measurements. Results for the following metals on sapphire are presented: Al, Ag, Au, Cu, In, Sn. In addition results for Al on quartz are discussed, and it is suggested that the method may give meaningful results even when, as in this case, chemical reaction takes place.     

Polarizability of two conducting spheres

Exact series expressions are obtained for the longitudinal and transverse polarizabilities of a system of two conducting spheres, for arbitrary sphere radii and separation. In the important case of close approach, where the exact series converge slowly, approximate analytical expressions are obtained. These analytical expressions become more accurate as the distance between the spheres gets small compared to the sphere radii, and reduce to the known analytic values at sphere contact.     

Computationally efficient recurrence relations for one-dimensional Franck–Condon overlap integrals

Calculations based on analytical expressions for the harmonic oscillator Franck–Condon factors often yield numerically unstable and erroneous results for large values of the oscillator quantum numbers. This instability arises from inherent machine precision limits and large number round-off associated with the products and ratios of factorial and gamma functions in these expressions; the analytical expressions themselves are exact. This paper presents, first, efficient, exact recurrence relations to evaluate Franck–Condon factors for the harmonic oscillator model. The recurrence relations, which are similar to those originally found by Manneback, Wagner and Ansbacher avoid the direct use of the factorial and gamma functions. Second, a variational strategy for the evaluation of Franck–Condon factors for the Morse oscillator is proposed. The Schrödinger equation for the Morse model is solved variationally with a large enough basis set of one-dimensional harmonic oscillator functions to get good agreement with the analytic eigenvalues of the Morse potential itself. The eigenvectors of this analysis are then used together with the associated harmonic oscillator Franck–Condon overlap matrix elements to evaluate the overlap for the Morse potential. This approach allows one, in principle, to estimate Franck–Condon overlap up to states near to the dissociation limit of the Morse oscillator.     

Infrared optical constants of n-type silicon

The infrared optical constants of n-type Si are determined from reflectance and transmittance spectra. The Drude formula with empirically adjusted, concentration dependent, parameters is used. Its low-wavelength limit is in agreement with recent mobility results. A pronounced difference between As-, P-, and Sb-doped Si is found for the free electron concentration above 10¹⁹ cm^–3. Simple empirical formulae are given for the optical constants as functions of both wavenumber and concentration.     

Diffraction of light by adjacent fundamental and second or fourth harmonic ultrasound beams: Comparison of exact and simplified formulae with experiment

Theoretical expressions and their approximate formulae for the intensities of light diffracted by two adjacent ultrasonic beams are discussed and compared with experimental data for the case when the second beam is an even harmonic of the first (fundamental). In the special case of the fundamental and its second or fourth harmonic, relatively simple formulae are found for the first and second diffraction orders describing the variation of light intensity with phase shift and intensity ratio.The approximate formulae describe the experiments satisfactorily only for small values of ζ₁ and α_n. For higher values the exact formulae give better agreement. Only the Raman-Nath region is considered.     

Invariant Measures for Passive Scalars in the Small Noise Inviscid Limit

We study the singular values of the product of two coupled rectangular random matrices as a determinantal point process. Each of the two factors is given by a parameter dependent linear combination of two independent, complex Gaussian random matrices, which is equivalent to a coupling of the two factors via an Itzykson-Zuber term. We prove that the squared singular values of such a product form a biorthogonal ensemble and establish its exact solvability. The parameter dependence allows us to interpolate between the singular value statistics of the Laguerre ensemble and that of the product of two independent complex Ginibre ensembles which are both known. We give exact formulae for the correlation kernel in terms of a complex double contour integral, suitable for the subsequent asymptotic analysis. In particular, we derive a Christoffel–Darboux type formula for the correlation kernel, based on a five term recurrence relation for our biorthogonal functions. It enables us to find its scaling limit at the origin representing a hard edge. The resulting limiting kernel coincides with the universal Meijer G-kernel found by several authors in different ensembles. We show that the central limit theorem holds for the linear statistics of the singular values and give the limiting variance explicitly.     

Effective interaction between two giant spheres suspended in a size polydisperse hard-sphere fluid

Analytic expressions for the Laplace transform of the interaction energy and force between two exceedingly large hard spheres at infinite dilution in a polydisperse hard-sphere suspending fluid are presented. The equations are based on the Percus–Yevick approximation for the many-component suspending fluid, supplemented by the hypernetted chain approximation for the correlation function of the suspended spheres. By applying the Derjaguin approximation, the energy and force results for two spheres are related to the energy per unit area and the disjoining pressure between two flat walls suspended in a polydisperse fluid. Numerical results for the representative Schultz distributions of the diameters of the species comprising the suspending fluid are presented and discussed.     

The range of validity of the Rayleigh limit for computing mie scattering and extinction efficiencies

For the small particle limit for scattering by either absorbing or dielectric spheres, values of the size parameter are determined below which various approximate expressions for the scattering and extinction efficiencies are accurate to within 1%. Simple, analytical criteria for ensuring 1% accuracy are given in terms of the size parameter and the complex refractive index. A much broader range of refractive indices is considered than has been studied previously.