Continuum percolation of the four-bonding-site associating fluids

Wertheim’s integral equation theory for associating fluids is reformulated for the study of the connectedness properties of associating hard spheres with four bonding sites. The association interaction is described as a square-well saturable attraction between these sites. The connectedness version of the Ornstein-Zernike (OZ) integral equation is supplemented by the PY-like closure relation and solved analytically within an ideal network approximation in which the network is represented as resulting from the crossing of ideal polymer chains. The pair connectedness functions and the mean cluster size are calculated and discussed. The condition for the percolation transition and the analytical form of the percolation threshold are derived. The connection of the percolation with the gas-liquid phase transition is discussed.     

A New Uniform Phase Bridge Functional: Test and Its Application to Non-uniform Phase Fluid

A new bridge functional as a function of indirect correlation function was proposed, which was basedon analysis on the asymptotic behavior of the Ornstein-Zernike (OZ) equation system and a series expansion whoserenormalization resulted in an adjustable parameter determined by the thermodynamics consistency condition. Theproposed bridge functional was tested by applying it to bulk hard sphere and hard core Yukawa fluid for the predictionof structure and thermodynamics properties based on the OZ equation. As an application, the present bridge functionalwas employed for non-uniform fluid of the above two kinds by means of the density functional theory methodology, theresulting density distribution profiles were in good agreement with the available computer simulation data.     

Structural and thermodynamic properties of a multicomponent freely jointed hard sphere multi-Yukawa chain fluid

The product-reactant Ornstein-Zernike approach, represented by the polymer mean-spherical approximation (PMSA), is utilized to describe the structure and thermodynamic properties of the fluid of Yukawa hard sphere chain molecules. An analytical solution of the PMSA for the most general case of the multicomponent freely jointed hard sphere multi-Yukawa chain fluid is presented. As in the case of the regular MSA for the hard sphere Yukawa fluid, the problem is reduced to the solution of a set of nonlinear algebraic equations in the general case, and to a single equation in the case of the factorizable Yukawa potential coefficients. Closed form analytical expressions are presented for the contact values of the monomer-monomer radial distribution function, structure factors, internal energy, Helmholtz free energy, chemical potentials and pressure in terms of the quantities, which follows directly from the PMSA solution. By way of illustration, several different versions of the hard sphere Yukawa chain model are considered, represented by one-Yukawa chains of length m, where m = 2, 4, 8, 16. To validate the accuracy of the present theory, Monte Carlo simulations were carried out and the results are compared systematically with the theoretical results for the structure and thermodynamic properties of the system at hand. In general it is found that the theory performs very well, thus providing an analytical route to the equilibrium properties of a well defined model for chain fluids.     

Heat capacity of square-well fluids of variable width

We have obtained by Monte Carlo NVT simulations the constant-volume excess heat capacity of square-well fluids for several temperatures, densities and potential widths. Heat capacity is a thermodynamic property much more sensitive to the accuracy of a theory than other thermodynamic quantities, such as the compressibility factor. This is illustrated by comparing the reported simulation data for the heat capacity with the theoretical predictions given by the Barker-Henderson perturbation theory as well as with those given by a non-perturbative theoretical model based on Baxter's solution of the Percus-Yevick integral equation for sticky hard spheres. Both theories give accurate predictions for the equation of state. By contrast, it is found that the Barker-Henderson theory strongly underestimates the excess heat capacity for low to moderate temperatures, whereas a much better agreement between theory and simulation is achieved with the non-perturbative theoretical model, particularly for small well widths, although the accuracy of the latter worsens for high densities and low temperatures, as the well width increases.     

13C hyperfine splitting in the benzene negative ion

A method is developed to calculate the coefficients appearing in the Høye, Stell and Waisman analytic solution of the Ornstein-Zernike equation for a hard core potential with a direct correlation function of the two Yukawa form. The method is simple and makes the choice of a physical solution easy. The compressibility, energy and virial pressures are calculated in the vicinity of the liquid-gas transition region and their dependence on changing K ₂, the amplitude of the repulsive Yukawa part, is analysed. This method offers promising possibilities for the application of the hard core, two Yukawa (HCTY) system in the theory of fluids.     

A SAFT model for associating Lennard-Jones chain mixtures

The recently proposed equation of state of statistical associated fluid theory (SAFT) is extended to associating Lennard-Jones (LJ) chain mixtures. In this extension, a new radial distribution function (RDF) for LJ mixtures is derived around the LJ potential size (σ _ij ). The RDF expression is completely analytical and real. Comparisons with computer simulation data under various conditions indicate that the RDF is very accurate up to its first peak. The new RDF, together with a previously established equation of state for LJ mixtures, is employed to study LJ chain mixtures by combining with Wertheim's first-order perturbation theory. The resulting equation of state is tested satisfactorily against computer simulation data for both non-associating and associating LJ chain mixtures, with a performance similar to its predecessors for pure LJ chains and LJ mixtures. The SAFT model is uniquely featured by being totally mixing-rule free and by being adjustable at both chain bonding and association sites. Moreover, the SAFT model is formulated very generally, so that it is applicable to both homonuclear and heteronuclear chain mixtures.     

The Discrete N-Vector Ferromagnet: Connection to a Percolation with Frustration Features

We extend the Kasteleyn–Fortuin formalism to the discrete N-vector ferromagnet. We show that the free energy and the correlation functions of this model are related, when the number of states tends to 1, to the mean number of clusters and to the pair connectedness of a polychromatic bond percolation type problem which combines frustration and connectivity features.     

Mean spherical approximation for a model liquid metal potential

The solution of the Ornstein-Zernike equation for a direct correlation function c(x) with damped oscillations and a hard core condition imposed upon the total correlation function h(x) has been proposed by Cummings as a means of treating a simple model potential for liquid metals in the mean spherical approximation [1]. Here some numerical results are given for this model and their significance is discussed. The solution of the Ornstein-Zernike equation is also extended; the hard core condition is generalized to a soft core condition, and Yukawa terms are added to the oscillatory c(x). Ways in which these extensions can be incorporated into more accurate liquid metal models, as well as into more accurate approximations for these models, are discussed. Finally, it is shown that our solution of the Ornstein-Zernike equation, after a change in the core condition, yields the structure of a spin glass model considered by Høye and Stell in the MSA-like approximation they propose [22].     

Highly Asymmetric Electrolytes in the Associative Mean-Spherical Approximation

The associate mean-spherical approximation (AMSA) is used to derive the closed-form expressions for the thermodynamic properties of an (n+m)-component mixture of sticky charged hard spheres, with m components representing polyions and n components representing counterions. The present version of the AMSA explicitly takes into account association effects due to the high asymmetry in charge and size of the ions, assuming that counterions bind to only one polyion, while the polyions can bind to an arbitrary number of counterions. Within this formalism an extension of the Ebeling–Grigo choice for the association constant is proposed. The derived equations apply to an arbitrary number of components; however, the numerical results for thermodynamic properties presented here are obtained for a system containing one counterion and one macroion (1+1 component) species only. In our calculation the ions are pictured as charged spheres of different sizes (primitive model) embedded in a dielectric continuum. Asymmetries in charge of –10:+1, –10:+2, –20:+1, and –20:+2 and asymmetries in diameter of 2:0.4nm and 3:0.4nm are studied. Monte Carlo simulations are performed for the same model solution. By comparison with new and existing computer simulations it is demonstrated that the present version of the AMSA provides semiquantitative or better predictions for the excess internal energy and osmotic coefficient in the range of parameters where the regular hypernetted chain (HNC) and improved (associative) HNC do not yield convergent solutions. The AMSA liquid–gas phase diagram in the limit of complete association (infinitely strong sticky interaction) is calculated for models with different degrees of asymmetry.     

The kirkwood-salsburg equations for random continuum percolation

We develop two different hierarchies of Kirkwood-Salsburg equations for the connectedness functions of random continuum percolation. These equations are derived by writing the Kirkwood-Salsburg equations for the distribution functions of thes-state continuum Potts model (CPM), taking thes1 limit, and forming appropriate linear combinations. The first hierarchy is satisfied by a subset of the connectedness functions used in previous studies. It gives rigorous, low-order bounds for the mean number of clusters n _c and the two-point connectedness function. The second hierarchy is a closed set of equations satisfied by the generalized blocking functions, each of which is defined by the probability that a given set of connections between particles is absent. These auxiliary functions are shown to be a natural basis for calculating the properties of continuum percolation models. They are the objects naturally occurring in integral equations for percolation theory. Also, the standard connectedness functions can be written as linear combinations of them. Using our second Kirkwood-Salsburg hierarchy, we show the existence of an infinite sequence of rigorous, upper and lower bounds for all the quantities describing random percolation, including the mean cluster size and mean number of clusters. These equations also provide a rigorous lower bound for the radius of convergence of the virial series for the mean number of clusters. Most of the results obtained here can be readily extended to percolation models on lattices, and to models with positive (repulsive) pair potentials.     

The effect of steepness of soft-core square-well potential model on some fluid properties

The effect of repulsive steepness of the soft-core square well (SCSW) potential model on the second virial coefficient, critical behaviour (two- phase region and the position of critical point), and coordination number are investigated. The soft-core thermodynamic perturbation theory (TPT) presented by Weeks-Chandler-Anderson (WCA) recently developed by Ben-Amotz and Stell (BAS) has been used for the reference system, and the Barker-Henderson TPT for the perturbed system. The Barker-Henderson macroscopic compressibility approximation has been used for all order perturbation terms in which the second-order one is improved by assuming that the molecules in every two neighbouring shells are correlated upon the original assumption. By using the hard-sphere isothermal compressibility consistency for the radial distribution function (RDF), an analytical closed expression has been derived for the Helmholtz free energy function contained effective hard-sphere diameter. The accuracy of the model has been examined for the hard-core system, and an appropriate range found for the attractive width of the potential well (R), then the effect of steepness parameter on the critical quantities, coordination number, and the inversion temperature of the second virial coefficient, has been investigated qualitatively. The predicted results are in good agreement with the computer simulation data for the critical constants, and coordination number at the limit of the hard-core square-well potential model at least qualitatively, and for the attractive range 1.55 ≤ R ≤ 1.7, quantitatively. It was found that the steepness of the potential model has a marginal effect on the critical behaviour, and also every thermodynamic quantity at low and medium temperatures for which the molecular penetration is negligible, but since the penetration at high temperatures is significant, the role of the steepness of potential on the inversion temperature of the second virial coefficient and coordination number is highlighted.     

A core-softened fluid model in disordered porous media. Grand canonical Monte Carlo simulation and integral equations

We have studied the microscopic structure and thermodynamic properties of a core-softened fluid model in disordered matrices of Lennard-Jones particles by using grand canonical Monte Carlo simulation. The dependence of density on the applied chemical potential (adsorption isotherms), pair distribution functions, as well as the heat capacity in different matrices are discussed. The microscopic structure of the model in matrices changes with density similar to the bulk model. Thus one should expect that the structural anomaly persists at least in dilute matrices. The region of densities for the heat capacity anomaly shrinks with increasing matrix density. This behavior is also observed for the diffusion coefficient on density from independent molecular dynamics simulation. Theoretical results for the model have been obtained by using replica Ornstein-Zernike integral equations with hypernetted chain closure. Predictions of the theory generally are in good agreement with simulation data, except for the heat capacity on fluid density. However, possible anomalies of thermodynamic properties for the model in disordered matrices are not captured adequately by the present theory. It seems necessary to develop and apply more elaborated, thermodynamically self-consistent closures to capture these features.     

An accurate theory for the square-well potential: from long to short well width

Density function theory (DFT) is combined with the first-order mean spherical approximation (FMSA) to study the radial distribution function (RDF) of the square-well (SW) potential. The combination (DFT + FMSA) is based on the direct correlation function (DCF) of the FMSA. Upon comparison with computer simulation data, DFT + FMSA is shown to give better performance than FMSA for mid- and long-range attractions. For short-range and very short-range attractions, the theory successfully corrects the deficiencies of the original FMSA. Comparisons include the evaluation of contact values, gap height at a discontinuity and profiles of the RDF. This work provides an accurate and consistent way to handle the SW potential.     

Self-diffusion of particles interacting through a square-well or square-shoulder potential

The diffusion coefficient and velocity autocorrelation function for a fluid of particles interacting through a square-well or square-shoulder potential are calculated from a kinetic theory similar to the Davis-Rice-Sengers theory and the results are compared to those of computer simulations. At low densities the theory yields too low estimates due to the neglect of correlations between subsequent partial collisions of identical pairs; in particular, the neglect of boundstate effects appears important. At intermediate densities the theory makes reasonable predictions and at high densities it produces too high values, due to the neglect of ring terms and other correlated collision events. The results for the square-shoulder potential generally exhibit better agreement between theory and simulations than do those for the square-well potential.     

A Global Investigation About Hard Core Attractive Yukawa Approximation and Adhesive Hard Sphere Approximation for Structure of Colloidal Dispersion Systems

The accuracy of hard core attractive Yukawa (HCAY) potential and adhesive hard sphere (AH) potential in representing the structure factor of short range square well potential and Asakura and Oosawa (AO) depletion potential is examined by comparing theoretical predictions with the existing simulation data and the present numerical results from the non-linear optimized random phase approximation closure for Ornstein-Zernike equation. For the case of square-well (SW) potential, it is shown that the structure factor of HCAY potential based on a recently proposed semi-analytical expression for the radial distribution function can describe the structure factor of SW potential with reduced well width λ≤2 only if the reduced contact potential βε_SW≤0.25, while the analytical expression for the structure factor of AH potential under Percus-Yevick (PY) approximation completely fails for the case of λ>1.2. For the case of AO depletion potential, the domain of validity of both HCAY potential and AH potential is complementary. With the above analysis and considering the solid-liquid transition of the AH potential with an adhesive parameter τ below 1.31 cannot be predicted by modified weighted density approximation, the role played by the HCAY potential about the mapping manipulation should not be ignored.     

The electrical conductivity of binary disordered systems,percolation clusters,fractals and related models

We review theoretical and experimental studies of the AC dielectric response of inhomogeneous materials, modelled as bond percolation networks, with a binary (conductor-dielectric) distribution of bond conductances. We first summarize the key results of percolation theory, concerning mostly geometrical and static (DC) transport properties, with emphasis on the scaling properties of the critical region around the percolation threshold. The frequency-dependent (AC) response of a general binary model is then studied by means of various approaches, including the effective-medium approximation, a scaling theory of the critical region, numerical computations using the transfer-matrix algorithm, and several exactly solvable deterministic fractal models. Transient regimes, related to singularities in the complex-frequency plane, are also investigated. Theoretical predictions are made more explicit in two specific cases, namely R-C and RL-C networks, and compared with a broad variety of experimental results, concerning, for example, granular composites, thin films, powders, microemulsions, cermets, porous ceramics and the viscoelastic properties of gels.     

On the Nature of Concentration Limits of Combustion Wave Propagation in Powdered and Pelletized Ti + C + <Emphasis Type="Italic">x</Emphasis>Al<Subscript>2</Subscript>O<Subscript>3</Subscript> Mixtures

The applicability of the percolation theory to describe the combustion of powdered and pelletized Ti + C mixtures in the vicinity of the concentration limits of the combustion wave propagation using different methods of dilution with an inert material—fine and coarse Al₂O₃ particles—has been studied. It has been shown that the pelletized mixtures diluted with coarse inert particles by more than 50% undergo incomplete combustion; at the combustion limit, the incompleteness achieves 50%; this finding is in qualitative agreement with the percolation theory. It has been found that the obtained concentration limit of combustion (75 wt %) and the ratio of the combustion velocities of the undiluted mixture and the mixture at the propagation limit (2.6) correspond to the predictions of the percolation theory. The possibility of flame propagation at the calculated combustion temperature of the mixture below the melting point of titanium is attributed to the presence of a percolation cluster. Necessary conditions for the applicability of the percolation theory to describe the combustion processes in condensed gasless systems have been formulated.     

Perturbation theory based direct correlation function for fluids with hard core potentials

Using second-order Barker–Henderson perturbation theory we are able to derive an explicit expression for the direct correlation function of fluids with hard core potentials. Using the obtained direct correlation function, one can explicitly calculate all thermodynamic properties of simple fluids with hard core potentials. Comparisons with computer simulation data show good agreement for both thermodynamic properties and the static structure factor of the hard core double Yukawa potential.     

Production of secondaries in high-energy <Emphasis Type="Italic">d</Emphasis>+Au collisions

In the framework of the quark–gluon string model we calculate the inclusive spectra of secondaries produced in d+Au collisions at intermediate (CERN SPS) and at much higher (RHIC) energies. The results of numerical calculations at intermediate energies are in reasonable agreement with the data. At RHIC energies numerically large inelastic screening corrections (percolation effects) should be accounted for in the calculations. We extract these effects from the existing experimental data of RHIC on minimum-bias and central d+Au collisions. The predictions for p+Au interactions at LHC energy are also given.     

Phase diagrams of Ising fluids with Yukawa-Lennard-Jones interactions from an integral equation approach

An integral equation approach is developed to investigate phase coexistence properties of Ising spin fluids with Yukawa ferromagnetic and Lennard-Jones nonmagnetic interactions in the presence of an external field. The calculations are carried out on the basis of the Duh and Henderson closure with a specific Duh-like partitioning of the total potential. The coupled set of the Ornstein-Zernike equation, the closure relation and the external field constraint are solved using an efficient numerical algorithm. The phase diagrams are evaluated in a wide range of varying the external field and the ratio of strengths of Yukawa to Lennard-Jones interactions. Different types of the phase diagram topology as well as various external field dependencies of critical temperatures and densities are identified. The complexity with respect to simple Lennard-Jones fluids is explained by coupling between spatial and spin degrees of freedom in the system. A comparison of the obtained theoretical results with simulation data is made and a good agreement is observed.