Simulating asymmetric colloidal mixture with adhesive hard sphere model

Monte Carlo simulation and Percus-Yevick (PY) theory are used to investigate the structural properties of a two-component system of the Baxter adhesive fluids with the size asymmetry of the particles of both components mimicking an asymmetric binary colloidal mixture. The radial distribution functions for all possible species pairs, g(11)(r), g(22)(r), and g(12)(r), exhibit discontinuities at the interparticle distances corresponding to certain combinations of n and m values (n and m being integers) in the sum nsigma(1)+msigma(2) (sigma(1) and sigma(2) being the hard-core diameters of individual components) as a consequence of the impulse character of 1-1, 2-2, and 1-2 attractive interactions. In contrast to the PY theory, which predicts the delta function peaks in the shape of g(ij)(r) only at the distances which are the multiple of the molecular sizes corresponding to different linear structures of successively connected particles, the simulation results reveal additional peaks at intermediate distances originating from the formation of rigid clusters of various geometries.     

Structure of nonuniform hard sphere fluids from shifted linear truncations of functional expansions

Percus showed that approximate theories for the structure of nonuniform hard sphere fluids can be generated by linear truncations of functional expansions of the nonuniform density rho(r) about that of an appropriately chosen uniform system. We consider the most general such truncation, which we refer to as the shifted linear response (SLR) equation, where the density response rho(r) to an external field phi(r) is expanded to linear order at each r about a different uniform system with a locally shifted chemical potential. Special cases include the Percus-Yevick (PY) approximation for nonuniform fluids, with no shift of the chemical potential, and the hydrostatic linear response (HLR) equation, where the chemical potential is shifted by the local value of phi(r). The HLR equation gives exact results for very slowly varying phi(r) and reduces to the PY approximation for hard core phi(r), where generally accurate results are found. We show that a truncated expansion about the bulk density (the PY approximation) also gives exact results for localized fields that are nonzero only in a "tiny" region whose volume V(phi) can accommodate at most one particle. The SLR equation can also exactly describe a limit where the fluid is confined by hard walls to a very narrow slit. This limit can be related to the localized field limit by a simple shift of the chemical potential, leading to an expansion about the ideal gas. We then try to develop a systematic way of choosing an optimal local shift in the SLR equation for general phi(r) by requiring that the predicted rho(r) is insensitive to small variations about the appropriate local shift, a property that the exact expansion to all orders would obey. The resulting insensitivity criterion (IC) gives a theory that reduces to the HLR equation for slowly varying phi(r) and is much more accurate than HLR both for very narrow slits, where the IC agrees with exact results, and for fields confined to "tiny" regions, where the IC gives very accurate (but not exact) results. However, the IC is significantly less accurate than the PY and HLR equations for single hard core fields. Only a small change in the predicted reference density is needed to correct this remaining limit.     

Analytic solutions for Baxter's model of sticky hard sphere fluids within closures different from the Percus-Yevick approximation

We discuss structural and thermodynamical properties of Baxter's adhesive hard sphere model within a class of closures which includes the Percus-Yevick (PY) one. The common feature of all these closures is to have a direct correlation function vanishing beyond a certain range, each closure being identified by a different approximation within the original square-well region. This allows a common analytical solution of the Ornstein-Zernike integral equation, with the cavity function playing a privileged role. A careful analytical treatment of the equation of state is reported. Numerical comparison with Monte Carlo simulations shows that the PY approximation lies between simpler closures, which may yield less accurate predictions but are easily extensible to multicomponent fluids, and more sophisticate closures which give more precise predictions but can hardly be extended to mixtures. In regimes typical for colloidal and protein solutions, however, it is found that the perturbative closures, even when limited to first order, produce satisfactory results.     

Gaussian core model phase diagram and pair correlations in high Euclidean dimensions

The physical properties of a classical many-particle system with interactions given by a repulsive Gaussian pair potential are extended to arbitrarily high Euclidean dimensions. The goals of this paper are to characterize the behavior of the pair correlation function g(2) in various density regimes and to understand the phase properties of the Gaussian core model (GCM) as parametrized by dimension d. To this end, we explore the fluid (dilute and dense) and crystalline solid phases. For the dilute regime of the fluid phase, a cluster expansion of g(2) in reciprocal temperature beta is presented, the coefficients of which may be evaluated analytically due to the nature of the Gaussian potential. We present preliminary results concerning the convergence properties of this expansion. The analytical cluster expansion is related to numerical approximations for g(2) in the dense fluid regime by utilizing hypernetted chain, Percus-Yevick, and mean-field closures to the Ornstein-Zernike equation. Based on the results of these comparisons, we provide evidence in support of a decorrelation principle for the GCM in high Euclidean dimensions. In the solid phase, we consider the behavior of the freezing temperature T(f)(rho) in the limit rho-->+infinity and show T(f)(rho)-->0 in this limit for any d via a collective coordinate argument. Duality relations with respect to the energies of a lattice and its dual are then discussed, and these relations aid in the Maxwell double-tangent construction of phase coexistence regions between dual lattices based on lattice summation energies. The results from this analysis are used to draw conclusions about the ground-state structures of the GCM for a given dimension.     

A perturbation method for the Ornstein-Zernike equation and the generic van der Waals equation of state for a square well potential model

We calculate the generic van der Waals parameters A and B for a square well model by means of a perturbation theory. To calculate the pair distribution function or the cavity function necessary for the calculation of A and B, we have used the Percus-Yevick integral equation, which is put into an equivalent form by means of the Wiener-Hopf method. This latter method produces a pair of integral equations, which are solved by a perturbation method treating the Mayer function or the well width or the functions in the square well region exterior to the hard core as the perturbation. In the end, the Mayer function times the well width is identified as the perturbation parameter in the present method. In this sense, the present perturbation method is distinct from the existing thermodynamic perturbation theory, which expands the Helmholtz free energy in a perturbation series with the inverse temperature treated as an expansion parameter. The generic van der Waals parameters are explicitly calculated in analytic form as functions of reduced temperature and density. The van der Waals parameters are recovered from them in the limits of vanishing density and high temperature. The equation of state thus obtained is tested against Monte Carlo simulation results and found reliable, provided that the temperature is in the supercritical regime. By scaling the packing fraction with a temperature-dependent hard core, it is suggested to construct an equation of state for fluids with a temperature-dependent hard core that mimicks a soft core repulsive force on the basis of the equation of state derived for the square well model.     

Thermodynamic and structural properties of a fluid with a rectangular well potential

Thermodynamic and structural properties of a system with a rectangular well potential are investigated in the supercritical region using the approximate Martynov-Sarkisov (MS) integral equation for the binary distribution function. It is shown that, in contrast to other approximations, in particular the Percus-Yevick equation (PY) and hypernetted chain approximation (HNC), the MS equation describes the limits of existence of the homogeneous phase.Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Pushchino. State University of New York, Mt. Sinai Medical Center, USA. Translated from Zhurnal Strukturnoi Khimii, Vol. 34, No. 2, pp. 88–95, March–April, 1993.     

Integral equation theory for hard spheres confined on a cylindrical surface: anisotropic packing entropically driven

The structure of two-dimensional (2D) hard-sphere fluids on a cylindrical surface is investigated by means of the Ornstein-Zernike integral equation with the Percus-Yevick and the hypernetted-chain approximation. The 2D cylindrical coordinate breaks the spherical symmetry. Hence, the pair-correlation function is reformulated as a two-variable function to account for the packing along and around the cylinder. Detailed pair-correlation function calculations based on the two integral equation theories are compared with Monte Carlo simulations. In general, the Percus-Yevick theory is more accurate than the hypernetted-chain theory, but exceptions are observed for smaller cylinders. Moreover, analysis of the angular-dependent contact values shows that particles are preferentially packed anisotropically. The origin of such an anisotropic packing is driven by the entropic effect because the energy of all the possible system configurations of a dense hard-sphere fluid is the same. In addition, the anisotropic packing observed in our model studies serves as a basis for linking the close packing with the morphology of an ordered structure for particles adsorbed onto a cylindrical nanotube.     

How "sticky" are short-range square-well fluids?

The aim of this work is to investigate to what extent the structural properties of a short-range square-well (SW) fluid of range lambda at a given packing fraction eta and reduced temperature T* = kBT/epsilon can be represented by those of a sticky-hard-sphere (SHS) fluid at the same packing fraction and an effective stickiness parameter tau(T*,lambda). Such an equivalence cannot hold for the radial distribution function g(r) since this function has a delta singularity at contact (r = sigma) in the SHS case, while it has a jump discontinuity at r = lambda sigma in the SW case. Therefore, the equivalence is explored with the cavity function y(r), i.e., we assume that [formula: see text]. Optimization of the agreement between y(SW) and y(SHS) to first order in density suggests the choice tau(T*,lambda) = [12(e(1/T* - 1)(lambda - 1)](-1). We have performed Monte Carlo (MC) simulations of the SW fluid for lambda = 1.05, 1.02, and 1.01 at several densities and temperatures T* such that tau(T*,lambda) = 0.13, 0.2, and 0.5. The resulting cavity functions have been compared with MC data of SHS fluids obtained by Miller and Frenkel[J. Phys.: Condens. Matter 16, S4901 (2004)]. Although, at given values of eta and tau, some local discrepancies between y(SW) and y(SHS) exist (especially for lambda = 1.05), the SW data converge smoothly toward the SHS values as lambda-1 decreases. In fact, precursors of the singularities of y(SHS) at certain distances due to geometrical arrangements are clearly observed in y(SW). The approximate mapping y(SW)-->y(SHS) is exploited to estimate the internal energy and structure factor of the SW fluid from those of the SHS fluid. Taking for y(SHS) the solution of the Percus-Yevick equation as well as the rational-function approximation, the radial distribution function g(r) of the SW fluid is theoretically estimated and a good agreement with our MC simulations is found. Finally, a similar study is carried out for short-range SW fluid mixtures.     

The equation of state of hard hyperspheres in nine dimensions for low to moderate densities

The equation of state of hard hyperspheres in nine dimensions is calculated both from the values of the first ten virial coefficients and from a Monte Carlo simulation of the pair correlation function at contact. The results are in excellent agreement. In addition, we find that the virial series appears to be dominated by an unphysical singularity or singularities on or near the negative density axis, in qualitative agreement with the recently solved Percus-Yevick equation of state in nine dimensions.     

Structure of penetrable-rod fluids: exact properties and comparison between Monte Carlo simulations and two analytic theories

Bounded potentials are good models to represent the effective two-body interaction in some colloidal systems, such as the dilute solutions of polymer chains in good solvents. The simplest bounded potential is that of penetrable spheres, which takes a positive finite value if the two spheres are overlapped, being 0 otherwise. Even in the one-dimensional case, the penetrable-rod model is far from trivial, since interactions are not restricted to nearest neighbors and so its exact solution is not known. In this paper the structural properties of one-dimensional penetrable rods are studied. We first derive the exact correlation functions of the penetrable-rod fluids to second order in density at any temperature, as well as in the high-temperature and zero-temperature limits at any density. It is seen that, in contrast to what is generally believed, the Percus-Yevick equation does not yield the exact cavity function in the hard-rod limit. Next, two simple analytic theories are constructed: a high-temperature approximation based on the exact asymptotic behavior in the limit T--> infinity and a low-temperature approximation inspired by the exact result in the opposite limit T--> 0. Finally, we perform Monte Carlo simulations for a wide range of temperatures and densities to assess the validity of both theories. It is found that they complement each other quite well, exhibiting a good agreement with the simulation data within their respective domains of applicability and becoming practically equivalent on the borderline of those domains. A comparison with numerical solutions of the Percus-Yevick and the hypernetted-chain approximations is also carried out. Finally, a perspective on the extension of our two heuristic theories to the more realistic three-dimensional case is provided.     

Vapor pressure measurements on nonaqueous electrolyte solutions. Part 6. Some remarks on integral equations

Various combinations of the hypernetted chain (HNC) equation with the mean spherical approximation (MSA) and the Percus-Yevick (PY) equation are compared both for a well-known aqueous 2-2 electrolyte model solution and for real acetonitrile solutions. Belloni's self-consistency test shows that classical HNC calculations yield the best compressibility data for the two systems despite an apparently unrealistic g₊₊ maximum in the case of the aqueous solution. Effective concentration-dependent potentials making use of the dependence of the solution permittivity on electrolyte concentration are used for HNC calculations of osmotic coefficients for methanol solutions.     

Empirical equations for meniscus depression by particle attachment

In this paper the problem of calculating the depression of the gas-liquid meniscus by the particle attachment was solved. The analytical approximate equations obtained for small and large radii, r(tpc), of the three-phase contact were analyzed and compared to the available numerical results. The Derjaguin equation for small r(tpc) and the analytical results for large r(tpc) are accurate for r(tpc)/L< or =0.2 and r(tpc)/L> or =2, respectively, where L is the capillary length. For the meniscus depression with r(tpc)/L from 0.2 to 2, the empirical equations were obtained based on the asymptotic analysis of the analytical approximate solutions. The empirical numerical constants were obtained by fitting to the exact numerical results. The empirical equations together with the analytical approximate equations provide the accurate predictions for the meniscus depression for the whole range of the radius of the three-phase contact and are expected to be useful for modeling the detachment interaction in the flotation separation processes.     

Isothermal compressibility and correlation radius near the critical point: Numerical analysis of the Ornstein-Zernike equation

Solution of the Ornstein-Zernike equation is analyzed numerically in the Percus-Yevick and hyperchain approximations for a system of Lennard-Jones particles in a critical region. The temperature dependences of correlation functions, isothermal compressibility η, and correlation radius of density fluctuations ζ are investigated at a critical density; the corresponding critical indices are determined. It is shown that the Percus-Yevick approximation yields satisfactory results when the correlation functions are calculated within a range corresponding to approximately 50 atomic (molecular) diameters. In this case, with ≈5% deviations from the critical temperature, the calculated and experimental values of η and critical indices are in good agreement. Tver State University. Translated fromZhurnal Strukturnoi Khimii, Vol. 36, No. 5, pp. 799–807, September–October, 1995. Translated by I. Izvekova     

Configurational statistics of macromolecules in solution from correlations of liquid molecules

We address the relevant quest for a simple formalism describing the microstructure of liquid solutions of polymer chains. On the basis of a recent relativistic-type picture of self-diffusion in (simple) liquids named Brownian relativity (BWR), a covariant van Hove's distribution function in a Vineyard-like convolution approximation is proposed to relate the statistical features of liquid and chain molecules forming a dilute polymer solution. It provides an extension of the Gaussian statistics of ideal chains to correlated systems, allowing an analysis of macromolecular configurations in solution by the only statistical properties of the liquid units (and vice versa). However, the mathematical solution to this issue is not straightforward because, when the liquid and polymer van Hove's functions are equated, an inverse problem takes place. It presents some conceptual analogies with a scattering experiment in which the correlation of the liquid molecules acts as the radiation source and the macromolecule as the scatterer. After inverting the equation by a theorem coming from the Tikhonov's approach, it turns out that the probability distribution function of a real polymer can be expressed from a static Ornstein-Uhlenbeck process, modified by correlations. This result is used to show that the probability distribution of a true self-avoiding walk polymer (TSWP) can be modeled as a universal Percus-Yevick hard-sphere solution for the total correlation function of the liquid units. This method suits in particular the configurational analysis of single macromolecules. The analytical study of arbitrary many-polymer systems may require further mathematical investigation.     

Pair structure of the hard-sphere Yukawa fluid: an improved analytic method versus simulations, Rogers-Young scheme, and experiment

We present a comprehensive study of the equilibrium pair structure in fluids of nonoverlapping spheres interacting by a repulsive Yukawa-like pair potential, with special focus on suspensions of charged colloidal particles. The accuracy of several integral equation schemes for the static structure factor, S(q), and radial distribution function, g(r), is investigated in comparison to computer simulation results and static light scattering data on charge-stabilized silica spheres. In particular, we show that an improved version of the so-called penetrating-background corrected rescaled mean spherical approximation (PB-RMSA) by Snook and Hayter [Langmuir 8, 2880 (1992)], referred to as the modified PB-RMSA (MPB-RMSA), gives pair structure functions which are in general in very good agreement with Monte Carlo simulations and results from the accurate but nonanalytical and therefore computationally more expensive Rogers-Young integral equation scheme. The MPB-RMSA preserves the analytic simplicity of the standard rescaled mean spherical (RMSA) solution. The combination of high accuracy and fast evaluation makes the MPB-RMSA ideally suited for extensive parameter scans and experimental data evaluation, and for providing the static input to dynamic theories. We discuss the results of extensive parameter scans probing the concentration scaling of the pair structure of strongly correlated Yukawa particles, and we determine the liquid-solid coexistence line using the Hansen-Verlet freezing rule.     

The changes in free energies and entropies from analytical radial distribution functions

The changes in Helmholtz free energies and entropies in dense fluids have been evaluated using three known analytical expressions for radial distribution functions (RDFs) of Lennard–Jones (L-J) fluid. This method provides a simpler and a more expeditious way for the calculation of free energy and entropy in L-J dense fluids through statistical mechanics. Previously, integral equations or perturbation theories were used for this purpose. Such approach not only tests the power of analytical distribution functions in predicting the changes in Helmholtz free energies and entropies, but also specifies better expressions in determining these properties. The results are compared with experimental data and an accurate analytic equation of state for the L-J fluid. It is shown if an expression properly presents RDFs as a function of interparticle distance, density and temperature, it is possible to calculate the changes in Helmholtz free energies and entropies from analytical distribution functions.     

Improvement on the Carnahan-Starling Equation of State for Hard-sphere Fluids

Making use ofWeierstrass's theorem and Chebyshev's theorem and referring to the equations of state of the scaled-particle theory and the Percus-Yevick integration equation, we demon-strate that there exists a sequence of polynomials such that the equation of state is given by the limit of the sequence of polynomials. The polynomials of the best approximation from the third order up to the eighth order are obtained so that the Carnahan-Starling equation can be improved successively. The resulting equations of state are in good agreement with the simulation results on the stable fluid branch and on the metastable fluid branch.     

Bridge function for the dipolar fluid from simulation

The exact bridge function of the Lennard-Jones dipolar (Stockmayer) fluid is extracted from Monte Carlo simulation data. The projections g(mnl)(r) onto rotational invariants of the non-spherically symmetric pair distribution function g(r, Ω) are accumulated during simulation. Making intensive use of anisotropic integral equation techniques, the molecular Ornstein-Zernike equation is then inverted in order to derive the direct correlation function c(mnl)(r), the cavity function y(mnl)(r), the negative excess potential of mean force lny|(mnl)(r), and the bridge function b(mnl)(r) projections. b(r, Ω) presents strong, non-universal anisotropies at high dipolar coupling. This simulation data analysis may serve as reference and guide for approximated bridge function theories of dipolar fluids and is a valuable step towards the case of more refined, nonlinear water-like geometries.     

Equation of state of a seven-dimensional hard-sphere fluid. Percus-Yevick theory and molecular-dynamics simulations

Following the work of Leutheusser [Physica A 127, 667 (1984)], the solution to the Percus-Yevick equation for a seven-dimensional hard-sphere fluid is explicitly found. This allows the derivation of the equation of state for the fluid taking both the virial and the compressibility routes. An analysis of the virial coefficients and the determination of the radius of convergence of the virial series are carried out. Molecular-dynamics simulations of the same system are also performed and a comparison between the simulation results for the compressibility factor and theoretical expressions for the same quantity is presented.     

Particle connectedness and cluster formation in sequential depositions of particles: integral-equation theory

We applied the integral-equation theory to the connectedness problem. The method originally applied to the study of continuum percolation in various equilibrium systems was modified for our sequential quenching model, a particular limit of an irreversible adsorption. The development of the theory based on the (quenched-annealed) binary-mixture approximation includes the Ornstein-Zernike equation, the Percus-Yevick closure, and an additional term involving the three-body connectedness function. This function is simplified by introducing a Kirkwood-like superposition approximation. We studied the three-dimensional (3D) system of randomly placed spheres and 2D systems of square-well particles, both with a narrow and with a wide well. The results from our integral-equation theory are in good accordance with simulation results within a certain range of densities.