Hard-sphere Radial Distribution Function Research Articles

Application of the symmetric Poisson-Boltzmann theory to a model colloidal mixture.

A symmetric Poisson-Boltzmann theory is used to analyse the structure of a primitive model colloidal system which contains either 4 or 6 components. The approach symmetrizes the pair distribution function gij(r) between two asymmetric charged species with respect to an interchange of the indices. Good agreement is found with molecular dynamics simulation structural properties when the exclusion volume term is modelled by the Percus-Yevick uncharged hard sphere radial distribution function.

Application of the symmetric Poisson-Boltzmann theory to equilibrium thermodynamic properties of primitive model electrolyte mixtures

A symmetric Poisson-Boltzmann theory is applied to a primitive model electrolyte solution containing 7 different ion species. Only 7 coupled equations need to be solved for the mean electrostatic potential around the ions once the uncharged hard sphere radial distribution function is specified. Comparison is made with the individual activity and osmotic coefficients of a simulation study of a sea water mixture. Good overall agreement is found with the simulation results.

Singlet Fission in Concentrated TIPS-Pentacene Solutions: The Role of Excimers and Aggregates.

The excited-state dynamics of 6,13-bis(triisopropylsilylethynyl)pentacene is investigated to determine the role of excimer and aggregate formation in singlet fission in high-concentration solutions. Photoluminescence spectra were measured by excitation with the evanescent wave in total internal reflection, in order to avoid reabsorption effects. The spectra over nearly two magnitudes of concentration were nearly identical, with no evidence for excimer emission. Time-correlated single-photon counting measurements confirm that the fluorescence lifetime shortens with concentration. The observed rate constant grows at high concentrations, and this effect is modeled in terms of the hard-sphere radial distribution function. NMR measurements confirm that aggregation takes place with a binding constant of between 0.14 and 0.43 M-1. Transient absorption measurements are consistent with a diffusive encounter mechanism for singlet fission, with hints of more rapid singlet fission in aggregates at the highest concentration measured. These data show that excimers do not play the role of an emissive intermediate in exothermic singlet fission in solution and that, while aggregation occurs at higher concentrations, the mechanism of singlet fission remains dominated by diffusive encounters.

Chain formation in a two-dimensional system of hard spheres with a short-range anisotropic interaction.

We analyze a generalization of the hard-sphere dipole system in two dimensions in which the range of the interaction can be varied. We focus on the system in the limit in which the interaction becomes increasingly short ranged, while the temperature becomes low. By using a cluster expansion and taking advantage of low temperatures to perform saddle-point approximations, we argue that a well-defined double limit exists in which the only structures which contribute to the free energy are chains. We then argue that the dominance of chain structures is equivalent to the dominance of chain diagrams in a cluster expansion, but only if the expansion is performed around a hard-sphere system (rather than the standard ideal gas). We show that this leads to nonstandard factorization rules for diagrams and use this to construct a closed-form expression for the free energy at low densities. We then compare this construction to several models previously developed for the hard-sphere dipole system in the regime where chain structures dominate and argue that the comparison provides evidence in favor of one model over the others. We also use this construction to incorporate some finite-density effects though the hard-sphere radial distribution function and analyze how these effects impact chain length and the equation of state.

New SAFT-VR equation of state based on Morse potential

Abstract In this work, a new statistical associating fluid theory with variable range (SAFT-VR) equation of state (EOS) for the Morse fluids has been developed. Monte Carlo (MC) simulations have been performed to determine the first and second order perturbative contributions of Barker and Henderson (BH) perturbation theory over wide range of density, temperature and potential ranges. In addition to perturbative terms, pressure and internal energy of Morse fluid have been simulated and compared with theoretical results. In development of new EOS, an exact analytic solution of the Percus-Yevick (PY) integral equation for hard sphere radial distribution function (RDF) has been implemented into calculation of the first and second terms of BH theory. Our results showed that there are good agreement between MC simulation data and analytical solution of hard sphere RDF. In the case of BH first order term, excellent agreement was obtained. It is found that, the new EOS is able to predict the perturbative Helmholtz free energy, pressure and residual internal energy of Morse fluids over wide range of thermodynamic conditions. Finally, the new EOS has been utilized to correlate vapor pressure and saturated liquid density of pure components. Obtained results in this work prove that considering Morse as potential function can lead us to achieve an accurate SAFT-based EOS.

Analytical representation of the density derivative of the Percus–Yevick hard-sphere radial distribution function

ABSTRACTExplicit analytical expressions are presented for the density derivative, ∂gHS(R; ρ)/∂ρ, of the Percus–Yevick approximation to the hard-sphere radial distribution function for R ≤ 6σ, where σ is the hard-sphere diameter and ρ = (N/V)σ3 is the reduced density, where N is the number of particles and V is the volume. A FORTRAN program is provided for the implementation of these for R ≤ 6σ, which includes code for the calculation of gHS(R; ρ) itself over this range. We also present and incorporate within the program code convenient analytical expressions for the numerical extrapolation of both quantities past R = 6σ. Our expressions are numerically tested against exact results.

Unified SAFT-VR theory for simple and chain fluids formed of square-well, triangular-well, Sutherland and Mie segments

The thermodynamic properties of simple and chain fluids, whose molecules interact through the three following basic hard-core intermolecular potentials: the square-well (SW), the triangular-well (TW) and the Sutherland (SU) [−ε(d/r)γ] potentials are presented in a unified mathematical framework. In order to achieve this goal we use the Barker–Henderson perturbation theory with the macroscopic compressibility approximation (BH-MCA) of monomer fluids combined to a simple, analytic expression of the hard-sphere radial distribution function (RDF) derived by Sun [Can. J. Phys. 83 (2005) 55], which allows the first-order perturbation term of the Helmholtz free energy, f1, to be obtained in terms of the same generic analytic function for the three potentials. This function depends on three parameters which assume different values according to the potential considered. Other thermodynamic quantities such as the compressibility factors can be derived from f1, which thus plays a pivotal role in the equation of state (EOS) obtained from perturbation theory. Numerical values of f1 obtained from this generic function are in good agreement with recent Monte Carlo simulations for the three aforementioned basic potentials, as well as with the various parameterizations of f1 in terms of an effective packing fraction ηeff for the square-well and Sutherland potentials commonly used in the statistical associating fluid theory for potentials of variable range (SAFT-VR). For the corresponding chain fluids, Wertheim first-order thermodynamic perturbation theory (TPT1) is used. It requires the SAFT-VR expressions of the potentials RDF at the hard-core contact d, g(d), and its derivative with respect to the packing fraction, which can be expressed in terms of the generic function giving f1. Numerical values of g(d) are also in agreement with the corresponding Monte Carlo results. The equations of state (EOS) of the SW, TW and SU chain fluids are then obtained in simple, analytical forms. The above three basic potentials can then be combined to give more complicated molecular interactions. One important example of combination is studied. It involves the linear combination of two Sutherland potentials with γ=α, β to obtain the Mie(α,β) class of interactions, of which the Lennard–Jones LJ(12,6) or Mie(12,6) is a particular example. Numerical values of: a) the Mie(12,6) first-order perturbation term of the residual Helmholtz free energy, f1Mie, b) the radial distribution function at contact, gMie(σ), [uMie(σ)=0], and c) the reduced pressure or compressibility factor of the monomer, dimer and tetramer Mie(12,6) fluids, obtained from the generic function, f1, corresponding to the combined Sutherland potentials, are compared to Monte Carlo and molecular dynamics simulations and to the recent enhanced SAFT-VR Mie 20013 EOS of Lafitte et al. [J. Chem. Phys. 139 (2013) 154504], whose improvements with respect to the previous SAFT-VR Mie 2006 EOS are highlighted in detail. The results obtained from the generic function f1 are slightly more accurate for the Mie(12,6) monomer fluid, while the results from Lafitte et al., which uses a new f1 parameterization of ηeff for the Sutherland potentials, valid for 5<γ≤100, are more accurate for the Mie(12,6) dimer and tetramer fluids.

Improved analytical thermodynamic properties of the triangular-well fluid from perturbation theory

In this short communication, it is shown that the triangular-well fluid analytic equation of state (EOS) recently proposed by Betancourt-Cárdenas et al. [Mol. Phys. 105 (2007) 2987] within the framework of the Barker–Henderson perturbation theory of fluids with the macroscopic compressibility approximation (BH-MCA) can be improved in two ways. First it is cast into a more compact form, which avoids the calculations of seventy-five numerical coefficients. Secondly, by using a more accurate expression of the hard-sphere radial distribution function (HS-RDF), gHS(r), the accuracy of the analytical expression of the free energy second-order term is improved, being in agreement with its BH-MCA numerical evaluation. For short-range triangular-well potentials, the present EOS also leads to second-order residual internal energies and reduced pressures in better agreement with the molecular simulations than the Betancourt-Cárdenas et al. results.

A new simple analytic equation of state for square-well chain fluids with variable width, 1.1 < λ < 2, based on perturbation theory and an analytic representation of the hard-sphere radial distribution function g HS(r)

A new, simple and analytic perturbation theory equation of state for square-well chain fluids of variable well width (1.1 < λ < 2) is proposed. It is based on the recently developed, infinite-order Barker and Henderson perturbation theory for the thermodynamic properties of the reference monomer square-well fluid [J. Chem. Phys. 124 (2006) 154505], the statistical associating fluid theory to account for chain formation, and the new analytical expression of the radial distribution function of hard spheres developed by Sun in terms of a polynomial expansion of nonlinear base functions [Can. J. Phys. 83 (2005) 55]. The various integrals appearing in the theory are then simply evaluated in closed form. The compressibility factor and the internal energy thus obtained are in overall good agreement with existing computer simulation data for dimer, 4-mer, 8-mer and 16-mer square-well chain fluids at different well-width. Moreover the compressibility factors also compare favorably with those obtained from six other square-well chain fluids equations of state.

Hard-sphere radial distribution function from the residual chemical potential

A method of determining the radial distribution function on the basis of expressions for the residual chemical potential of hard spheres and of the infinitely diluted mixture of a hard dumbbell (originated from overlapping of two spheres) in hard spheres is revised. The enlarged hard dumbbell (instead of the standard one used in our previous study [T. Boublík, Molec. Phys. 59, 775 (1986)]) is considered for determining its geometric functionals—a volume, surface area and mean radius; these quantities characterize hard-body geometry in the (self-consistent) expressions used for the residual chemical potentials. New formulas for the characteristic geometric quantities and improved expressions for the residual chemical potential result in an improvement of predictions of the radial distribution function for the larger reduced distances and enable an application of the method to inhomogeneous systems; the case of hard-sphere fluids near a hard wall is studied. The present version of the theory was successfully applied to pure hard-sphere fluids and binary mixtures with densities as high as ρ*= 0.925.

First derivative of the hard-sphere radial distribution function at contact

Molecular dynamics simulations have been carried out of the radial distribution function ofthe hard sphere fluid for a range of densities in the equilibrium fluid and just into themetastable region. The first derivative of the hard-sphere radial distribution function atcontact was computed and its density dependence fitted to a simple analytic form.Comparisons were made with semi-empirical formulae from the literature, and of thesethe formula proposed by Tao et al (1992 Phys. Rev. A 46 8007) was found to be inbest agreement with the simulation data, although it slightly underestimatesthe derivative at the higher packing fractions in excess of about 0.45. Close tocontact, within a few per cent of the particle diameter, the radial distributionfunction can be represented well by a second order polynomial. An exponentialfunction, which has some useful analytic features, can also be applied in this region.

Thermodynamically Consistent Adaptation of Scaled Particle Theory to an Arbitrary Hard-Sphere Equation of State

Scaled particle theory (SPT) is adapted to an arbitrary hard-sphere equation of state (EOS) while utilizing an SPT interpolation function that satisfies all the exact conditions of SPT and, as such, is thermodynamically consistent. The new modification of SPT is tailored to predict with high accuracy the reversible work of forming a cavity in a hard-sphere fluid, but it also estimates the surface tension and excess surface adsorption. The application of additional SPT conditions also provides estimates of the first two derivatives of the hard-sphere radial distribution function (RDF) at contact. When various accurate equations of state are used, in particular, the Carnahan−Starling EOS, the resulting work of cavity formation is nearly identical to accurate Monte Carlo simulation data and the predictions of the recent SPT6 [Heying and Corti, J. Phys. Chem. B 2004, 108, 19756]. The estimates of the surface tension and surface adsorption at moderate to high packing fractions deviate somewhat from both previous theoretical techniques and the simulation. Meanwhile, EOS-based predictions of the initial slope of the RDF are very close to simulation estimates. Predictions of the initial curvature, like those of SPT6, deviate from simulation, but not drastically so. The overall accuracy of our version of SPT demonstrates that a thermodynamically consistent modification of SPT provides correct predictions of the work of cavity formation and is, therefore, suitable for use as the ideally repulsive reference state in various solvation theories of liquids.

Scaled Particle Theory Revisited: New Conditions and Improved Predictions of the Properties of the Hard Sphere Fluid

We revisit the successful scaled particle theory (SPT) of hard particle fluids, originally developed by Reiss, Frisch, and Lebowitz (J. Chem. Phys. 1959, 31, 369). In the initial formulation of SPT, five exact conditions were derived that constrained the form of the central function G. Only three of these conditions, however, were employed to generate an equation of state. Later, the number of relations used to determine G was increased to five (Mandell, M. J.; Reiss, H., J. Stat. Phys. 1975, 13, 113). The resulting equation of state was an improvement over the original formulation, although its accuracy was still limited at high densities. In an effort to increase the accuracy of SPT predictions, we propose two new formally exact conditions on the form of G. These sixth and seventh conditions relate exactly known derivatives of G to the slope and curvature of the hard sphere radial distribution function at contact, g‘(σ+) and g‘ ‘(σ+), respectively. To apply the new conditions, we derive, again within the framework of SPT, physically and geometrically based approximations to g‘(σ+) and g‘ ‘(σ+). These additional restrictions on the function G yield markedly improved predictions of the pressure, excess chemical potential, and work of cavity formation for the hard sphere fluid, now making SPT competitive with other existing equations of state.

Binary hard-sphere solute-solvent radial distribution function in the colloidal limit: exact calculation from an equation of state

An exact formula is derived relating the contact value of the solute-solvent radial distribution function for an additive binary hard-sphere (HS) mixture at infinite dilution, g ∞ 12(d 12), to the mixture equation of state (EOS) (1 denotes the solvent and 2 denotes the solute). This result can also be considered to be a consistency condition involving approximations for g ∞ 12(d 12) and for the mixture EOS. Employing three approximate HS mixture equations of state from the literature, we use our formula to derive corresponding analytical approximations for g ∞ 12(d 12) In addition, new computer simulations were performed to obtain accurate results for g ∞ 12(d 12) and for g ∞ 12(r 12) at the solute-solvent diameter ratios {1, 3, 5, 7, 10, 20} and the reduced solvent density ρ* = 0.8. We compare our results for g ∞ 12(d 12) with the simulation results and with the results of approximate analytical expressions for g ∞ 12(d 12) proposed by several workers. The results obtained from our formula in conjunction with two of the EOS expressions considered are more accurate than all previously proposed approximations, with the exception of the approximation of Matyushov and Ladanyi [1997, J. chem. Phys., 107, 5815], which is of comparable accuracy.

A simplified analytical expression for the first shell of the hard-sphere fluid radial distribution function

An analytical expression for the first coordination cell of the radial distribution function (RDF) of the hard-sphere fluid is derived. It is based on a series expansion of the analytical expression of the Percus–Yevick solution of the RDF of the hard-sphere fluid derived by Chang and Sandler [J. Chang, S.I. Sandler, Mol. Phys. 81 (1994) 745.]. The expansion is carried out both in terms of the packing fraction and in terms of the radial distance. Expressions are also obtained for the coordination number and its first and second derivatives as functions of radial distance and packing fraction. These expressions, which are useful in perturbation theory, are simpler to use than those obtained from the starting equation, while giving nearly the same results. They also provide close agreement with simulation data.

Freezing of simple liquid metals

Freezing of simple liquid metals and the relative stabilities of competing crystalline solids are investigated using thermodynamic perturbation theory, the interactions between ions being modeled by effective pair potentials derived from pseudopotential theory. The ionic free energy of the solid phase is calculated, to first order in the perturbation potential, using classical density-functional (DF) theory and an accurate approximation to the hard-sphere (HS) radial distribution function (RDF). Free energy calculations for Na, Mg, and Al yield well-defined freezing transitions and structural free energy differences for bcc, fcc, and hcp crystals in qualitative agreement with experiment.

An analytic expression for the first shell of the radial distribution function

A simple analytical expression for the first shell of the radial distribution function (RDF) is proposed. This expression, which has only three adjustable parameters, satisfies all the limiting cases of the hard-sphere RDF at high temperatures, the ideal gas RDF at zero density, the dilute-gas RDF at low densities, and the location of the peak in the first shell. The only requirement is the introduction of a potential function into the model. This theory has been applied to the Lennard Jones, Kihara, and square-well pair intermolecular potential energy functions. The first-shell RDF results are in good agreement with the available computer simulation data for the RDF of the Lennard Jones fluid and the experimental data for argon. By introducing the radius of truncation for the RDF, it is shown that information on the first shell of the RDF is sufficient to predict macroscopic properties of fluids. Calculations of radii of truncation of RDF for various properties indicate that they are always in the range of the first shell of RDF.

Statistical associating fluid theory for chain molecules with attractive potentials of variable range

A version of the statistical associating fluid theory (SAFT) is developed for chain molecules of hard-core segments with attractive potentials of variable range (SAFT-VR). The different contributions to the Helmholtz free energy are evaluated according to the Wertheim perturbation theory. The monomer properties are obtained from a high-temperature expansion up to second order, using a compact expression for the first-order perturbation term (mean-attractive energy) a1. Making use of the mean-value theorem, a1 is given as the van der Waals attractive constant and the Carnahan and Starling contact value for the hard-sphere radial distribution function in terms of an effective packing fraction. The second-order perturbation term a2 is evaluated with the local compressibility approximation. The monomer cavity function, required for the calculation of the free energy due to the formation of the chains and the contribution due to association, is given as a function of a1. We analyse the equation of state for chain molecules with three different types of monomer hard-core potentials with variable attractive range: square-well (SW), Yukawa (Y), and Sutherland (S). The theory for the hard-core potentials can easily be generalised to soft-core systems: we develop a simple equation of state for Mie m−n potentials, of which the Lennard-Jones (LJ) 6-12 potential is a particular case. The equations of state, expressed in terms of reduced variables, are explicit functions of the reduced temperature, the packing fraction, the number of monomers segments forming the chain, and the parameter λ which characterises the range of the the attractive potential. The relevance of the last parameter in the application of the theory to n-alkanes and n-perfluoroalkanes is explicitly shown with the SW expressions. An accurate description of the vapour pressure and the saturated liquid densities is obtained, with a simple dependence of the parameters of the monomer potential on the number of carbons. The extension of our SAFT-VR expressions to mixtures is also presented in terms of a simple expression for the mean-attractive energy for mixtures, based on a straightforward generalisation of the theory for pure components.

Derivative of the hard-sphere radial distribution function at contact.

A simple and accurate formula for the first derivative of the hard-sphere radial distribution function at contact as a function of density is proposed, suggested by the solution of the Percus-Yevick equation for a fluid of hard spheres.

Calculation of the hard sphere radial distribution function

We describe a recursive formula for calculating the hard sphere radial distribution function under the Percus-Yevick approximation. It provides a simple and exact way to evaluate the expressions arising in Wertheim's solution of the hard sphere radial distribution function. The method does not increase in complexity at large distances. It has been used to evaluate the function at distances up to 20 times the hard sphere diameter. The same recursive formula can be applied to calculate derivatives of the distribution function which are useful in the thermodynamic analysis of equations of state obtained from perturbation theories.