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.     

Equation of state of nonadditive d-dimensional hard-sphere mixtures

An equation of state for a multicomponent mixture of nonadditive hard spheres in d dimensions is proposed. It yields a rather simple density dependence and constitutes a natural extension of the equation of state for additive hard spheres proposed by us [A. Santos, S. B. Yuste, and M. Lopez de Haro, Mol. Phys. 96, 1 (1999)]. The proposal relies on the known exact second and third virial coefficients and requires as input the compressibility factor of the one-component system. A comparison is carried out both with another recent theoretical proposal based on a similar philosophy and with the available exact results and simulation data in d=1, 2, and 3. Good general agreement with the reported values of the virial coefficients and of the compressibility factor of binary mixtures is observed, especially for high asymmetries and/or positive nonadditivities.     

Extended pressure—consistent equation for simple fluids

Abstract

A previous generalization of the Percus-Yevick (PY) and hypernetted chain (HNC) equations for simple fluids, involving a density- and temperature-dependent coefficient m, is extended by including a spatial dependence in m. The new approximation yields an exact fourth virial coefficient and, by further requirement, a consistent equation of state from both the virial and compressibility forms. Comparison of calculated results for the hard sphere potential shows an improvement over the PY, HNC, and previous pressure-consistent equations     

Rescaled density expansions and demixing in hard-sphere binary mixtures

The demixing transition of a binary fluid mixture of additive hard spheres is analyzed for different size asymmetries by starting from the exact low-density expansion of the pressure. Already within the second virial approximation the fluid separates into two phases of different composition with a lower consolute critical point. By successively incorporating the third, fourth, and fifth virial coefficients, the critical consolute point moves to higher values of the pressure and to lower values of the partial number fraction of the large spheres. When the exact low-density expansion of the pressure is rescaled to higher densities as in the Percus-Yevick theory, by adding more exact virial coefficients a different qualitative movement of the critical consolute point in the phase diagram is found. It is argued that the Percus-Yevick factor appearing in many empirical equations of state for the mixture has a deep influence on the location of the critical consolute point, so that the resulting phase diagram for a prescribed equation has to be taken with caution.     

Higher virial coefficients of four and five dimensional hard hyperspheres

The seventh and eighth virial coefficients for hard hyperspheres are calculated by Monte Carlo techniques. It is found that B(7)/B(2) (6)=0.001 43+/-0.000 13 and 0.000 44+/-0.000 12 in four and five dimensions, respectively, and that B(8)/B(2) (7)=0.000 414+/-0.000 20 in four dimensions. These values are used to investigate various proposed equations of state. Comparisons against the molecular dynamics calculations of Luban and Michels show that their proposed semiempirical form is excellent at higher densities. Moreover, we confirm Santos observation in five dimensions that a suitable linear combination of the Percus-Yevick compressibility and virial equations of state fits the molecular dynamics data nearly as well as any other proposed form.     

Molecular characterization of a hyperbranched polyester. II. Small‐angle neutron scattering

A series of fractions of a hyperbranched polyester in deutero tetrahydrofuran solution were investigated by small‐angle neutron scattering. Concentrations of polymer from 2 to 5% w/v were used, and the molecular parameters were obtained from Zimm plots of the data. Second virial coefficients were positive, and these values were confirmed by dilute‐solution light scattering on a small number of fractions with deutero tetrahydrofuran as a solvent. The small‐angle neutron scattering data exhibited the general features predicted for the particle scattering functions of nonrandomly branched polymers, but an exact fit of the theoretical equation to the data could not be obtained for all fractions of the hyperbranched polymer, particularly those of high molecular weight. Excluded volume effects were cited as a possible cause for this disagreement. A fractal dimension of ～2.5 was obtained from the scattering vector dependence of the differential scattering cross section of the polymer in deutero tetrahydrofuran solution, which agreed with the scaling exponent for the dependence of the radius of gyration on weight‐average molecular weight. Hydrogenous tetrahydrofuran solutions of the hyperbranched polymer exhibited negative second virial coefficients that were attributed to isotopic influences on the thermodynamic properties of the polymer–solvent combination. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1352–1361, 2003     

The equation of state of isotropic fluids of hard convex bodies from a high-level virial expansion

We have calculated virial coefficients up to seventh order for the isotropic phases of a variety of fluids composed of hard aspherical particles. The models studied were hard spheroids, hard spherocylinders, and truncated hard spheres, and results are obtained for a variety of length-to-width ratios. We compare the predicted virial equations of state with those determined by simulation. We also use our data to calculate the coefficients of the y expansion [B. Barboy and W. M. Gelbart, J. Chem. Phys. 71, 3053 (1979)] and to study its convergence properties. Finally, we use our data to estimate the radius of convergence of the virial series for these aspherical particles. For fairly spherical particles, we estimate the radius of convergence to be similar to that of the density of closest packing. For more anisotropic particles, however, the radius of convergence decreases with increased anisotropy and is considerably less than the close-packed density.     

Primary electroviscous effect in a dilute suspension of charged mercury drops

The standard theory of the primary electroviscous effect in a dilute suspension of charged spherical rigid particles in an electrolyte solution (Watterson, I. G.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1981, 77, 1115) is extended to cover the case of a dilute suspension of charged mercury drops of viscosity eta(d). A general expression for the effective viscosity or the electroviscous coefficient p of the suspension is derived. This expression tends to that for the case of rigid particles in the limit of eta(d) --> infinity. We also derive an approximate analytical viscosity expressions applicable to mercury drops carrying low zeta potentials at arbitrary kappaa (where kappa is the Debye-Hückel parameter and a is the drop radius) and to mercury drops as well as rigid spheres with arbitrary zeta potentials at large kappaa. It is shown that the large-kappaa expression of p for rigid particles predicts a maximum when plotted as a function of zeta potential. This result for rigid particles agrees with the exact numerical results of Watterson and White. It is also shown that in the limit of high zeta potential the effective viscosity of a suspension of mercury drops tends to that of uncharged rigid spheres given by Einstein's formula (Einstein, A. Ann. Phys. 1906, 19, 289), whereas in the opposite limit of low zeta potential the effective viscosity approaches that of a suspension of uncharged liquid drops derived by Taylor (Taylor, G. I. Proc. R. Soc. London, Ser. A 1932, 138, 41).     

Structure factor for hard hyperspheres in higher dimensions

The structure factor for hard hyperspheres in two to eight dimensions is computed by Fourier transforming the pair correlation function obtained by computer simulation at a variety of densities. The resulting structure factors are compared to the known Percus-Yevick equations for odd dimensions and to the model proposed by Leutheusser [J. Chem. Phys. 84, 1050 (1986)] and Rosenfeld [J. Chem. Phys. 87, 4865 (1987)] in even dimensions. It is found that there is fine agreement among all these approaches at low to moderate densities but that the accuracy of the analytical models breaks down as the freezing transition is approached. The structure factor gives another insight into the decrease in the ordering of the hyperspheres as the dimension is increased.     

Second virial coefficients of asymmetric top molecules

A short self-contained derivation is given for the second virial coefficient B2(T) of a gas consisting of identical interacting asymmetric rigid rotors. The resulting expression is correct through variant Planck's over h2. First, the canonical partition function is derived by means of an variant Planck's over h expansion of exp[-H/(k(B)T)] due to Friedmann [Adv. Chem. Phys. 4, 225 (1962)]. The present work applies angular momentum operators and known facts from angular momentum theory. It is considerably more accessible than Friedmann's exposition, which is not based on angular momentum operators, but instead on explicit derivatives with respect to Euler angles. The partition function obtained from the variant Planck's over h expansion is applied to the derivation of an expression for B2(T) that is identical in appearance to the expression for symmetric rotors of T Pack [J. Chem. Phys. 78, 7217 (1983)]. The final equation in this work is valid for rigid rotors of any symmetry.     

Nitroxide spin exchange due to re-encounter collisions in a series of n-alkanes

Bimolecular collisions between perdeuterated 2,2,6,6-tetramethyl-4-oxopiperidine-l-oxyl molecules in three alkanes have been studied by measuring the electron paramagnetic resonance (EPR) spectral changes induced by spin exchange. We define an "encounter" to be a first-time collision followed by a series of re-encounters prior to the diffusing pair's escaping each other's presence. The present work stems from a recent proposal [B. L. Bales et al., J. Phys. Chem. A 107, 9086 (2003)] that an unexpected linear dependence of the spin-exchange-induced EPR line shifts on spin-exchange frequency can be explained by re-encounters of the same probe pair during one encounter. By employing nonlinear least-squares fitting, full use of the information available from the spectral changes allows us to study encounters and re-encounters separately. The encounter rate constants appear to be dominated by hydrodynamic forces, forming a common curve for hexane, decane, and hexadecane when plotted against T/eta, where eta is the shear viscosity. Unexpectedly, encounters are not dependent on the ratio mu = a/a(s), where a and a(s) are the van der Waals radii of the nitroxide probe and the solvent, respectively. It is argued that the near coincidence of the resulting encounter rate constant with the hydrodynamic prediction is likely due to a near cancellation of terms in the general diffusion coefficient. Thus, the semblance of hydrodynamic behavior is coincidental rather than intrinsic. In contrast, the mean times between re-encounters do depend on the relative sizes of probe and solvent. For hexane at lower temperatures, the Stokes-Einstein equation apparently describes re-encounters well; however, at higher temperatures and for decane and hexadecane, departures from the hydrodynamic prediction become larger as mu becomes smaller. This is in qualitative agreement with the theory of microscopic diffusion of Hynes et al. [J. Chem. Phys. 70, 1456 (1979)]. These departures are well correlated with the free volume available in the solvent; thus, the mean times between re-encounters form a common curve when plotted versus the free volume. Because free volume is manifested macroscopically by the isothermal compressibility, it is expected and observed that the re-encounter rate also forms a common curve across all three solvents when plotted with respect to compressibility. The existence of a common curve for alkanes raises the prospect of using EPR to determine the compressibility of substances such as fossil fuels and biological membranes.     

Structural thermodynamics of lamellar cationic lipid-DNA complex: DNA compressibility modulus

We have studied theoretically the compressibility modulus B of DNA and complexation adsorption isotherms of DNA and lipids, as a function of DNA spacing d(DNA) and NaCl electrolyte concentration, respectively, in isoelectric states of lamellar DNA/cationic lipid (CL) self-assemblies. The electrostatic free energy derived from the Poisson-Boltzmann theory predicts partial agreement with measured B values for interhelical separations d(DNA)>33 A when use is made of a fit of hydration repulsion from bulk DNA hexagonal phases in solution. For lower interchain separations the prediction worsens due to the hydration interaction that overcomes the electrostatic contribution. An exact match of the system's counterion electrochemical potentials and the coions of salt in aqueous phase leads to the electrostatic part of the free energy that renders isotherms of d(DNA) versus ionic strength in qualitative consistency with general trends of available experimental data of CL-DNA complexes.     

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.     

Percus-Yevick theory for the structural properties of the seven-dimensional hard-sphere fluid

The direct correlation function and the (static) structure factor for a seven-dimensional hard-sphere fluid are considered. Analytical results for these quantities are derived within the Percus-Yevick [Phys. Rev.110, 1 (1958)] theory.     

Dissolution of rubbery and glassy polymers

A mathematical model is formulated for solvent dissolution of rubbery and glassy polymers. An exact solution to the problem is derived for the constant diffusivity case, and a weighted residual solution is developed for the case of a concentration-dependent diffusion coefficient. The solution is used to calculate concentration profiles, dissolution curves, dissolution half-times, and pseudointerface positions at various times. © 1998 John Wiley & Sons, Inc. J. Polym. Sci. B Polym. Phys. 36: 2607–2614, 1998     

On the structure of sulfur-stabilized allyllithium compounds in solution

A combination of density functional calculations (B3LYP/6-31+G(d) level of theory) and experimental investigations (NMR and cryoscopic measurements) lead to structural assignments in solution for a series of three sulfur stabilized allyllithium compounds 1-3. All three lithium species are monomers in THF under the experimental conditions studied here and exist exclusively in an endo conformation. Increasing the oxidation state of sulfur (thiol --> sulfoxide --> sulfone) causes a change in the solution state structure of the allyllithium compounds. In the case of 1-thiophenylallyllithium 1, a fast equilibrium between two eta(1)-species is present with the equilibrium favoring the eta(1) C(alpha)-Li contact ion pair. The preference for this conformation can be attributed to the charge stabilizing properties of the sulfur substituent. For the lithiated sulfoxide 2, this equilibrium is frozen on the NMR time scale and two different lithium species (a eta(1) C(alpha)-Li and a eta(1) C(gamma)-Li contact ion pair), each possessing an intramolecular Li-O contact, coexist in d(8)-THF. In the case of the lithiated sulfone 3, several solvated conformations are in rapid equilibrium with each other on the NMR time scale in solution. The presence of two chelating oxygen atoms allows the lithium to form a OLiO scissor-like contact ion pair that competes with the eta(1) C(alpha)-Li and the eta(1) C(gamma)-Li contact ion pairs also calculated for compound 3.     

A fundamental measure theory for the sticky hard sphere fluid

We construct a density functional theory (DFT) for the sticky hard sphere (SHS) fluid which, like Rosenfeld's fundamental measure theory (FMT) for the hard sphere fluid [Y. Rosenfeld, Phys. Rev. Lett. 63, 980 (1989)], is based on a set of weighted densities and an exact result from scaled particle theory (SPT). It is demonstrated that the excess free energy density of the inhomogeneous SHS fluid Φ(SHS) is uniquely defined when (a) it is solely a function of the weighted densities from Kierlik and Rosinberg's version of FMT [E. Kierlik and M. L. Rosinberg, Phys. Rev. A 42, 3382 (1990)], (b) it satisfies the SPT differential equation, and (c) it yields any given direct correlation function (DCF) from the class of generalized Percus-Yevick closures introduced by Gazzillo and Giacometti [J. Chem. Phys. 120, 4742 (2004)]. The resulting DFT is shown to be in very good agreement with simulation data. In particular, this FMT yields the correct contact value of the density profiles with no adjustable parameters. Rather than requiring higher order DCFs, such as perturbative DFTs, our SHS FMT produces them. Interestingly, although equivalent to Kierlik and Rosinberg's FMT in the case of hard spheres, the set of weighted densities used for Rosenfeld's original FMT is insufficient for constructing a DFT which yields the SHS DCF.     

Surface tension of a Lennard-Jones liquid under supersaturation

A formally exact Kirkwood-Buff virial formula for the surface tension of a supersaturated interface is derived. A modified Gibbs ensemble method is given that allows the creation of interacting supersaturated phases of equal chemical potential, and which enables the Kirkwood-Buff formula to be applied. The methods are tested by Monte Carlo simulation of a supersaturated Lennard-Jones fluid with a planar liquid-vapor interface. The Kirkwood-Buff results for the supersaturated surface tension are found to be in reasonable agreement with new results obtained here using the recently developed, formally exact, ghost interface method, [M. P. Moody and P. Attard, J. Chem. Phys., 2002, 117, 6705]. The surface tension is obtained as a function of supersaturation at four temperatures, and it is found to decrease with increasing supersaturation, and to vanish at the vapor spinodal. The relevance of the present results to the nucleation of droplets in a supersaturated vapor is discussed.     

Contact values of the particle-particle and wall-particle correlation functions in a hard-sphere polydisperse fluid

The contact values g(sigma,sigma') of the radial distribution functions of a fluid of (additive) hard spheres with a given size distribution f(sigma) are considered. A "universality" assumption is introduced, according to which, at a given packing fraction eta,g(sigma,sigma')=G(z(sigma,sigma')), where G is a common function independent of the number of components (either finite or infinite) and z(sigma,sigma')=[2sigmasigma'/(sigma+sigma')]mu2/mu3 is a dimensionless parameter, mu n being the nth moment of the diameter distribution. A cubic form proposal for the z dependence of G is made and known exact consistency conditions for the point particle and equal size limits, as well as between two different routes to compute the pressure of the system in the presence of a hard wall, are used to express Gz in terms of the radial distribution at contact of the one-component system. For polydisperse systems we compare the contact values of the wall-particle correlation function and the compressibility factor with those obtained from recent Monte Carlo simulations.