Percus-Yevick Theory Research Articles

Structure of the square-shoulder fluid

The structural properties of square-shoulder fluids are derived from the use of the rational function approximation method. The computation of both the radial distribution function and the static structure factor involves mostly analytical steps, requiring only the numerical solution of a single transcendental equation. The comparison with available simulation data and with numerical solutions of the Percus–Yevick and hypernetted-chain integral equations shows that the present approximation represents an improvement over the Percus–Yevick theory for this system and a reasonable compromise between accuracy and simplicity.

Density functional for hard hyperspheres from a tensorial-diagrammatic series

We represent the free energy functional by a diagrammatic series with tensorial coefficients indexed by powers of the length scale. For hard cores, we obtain Percus's exact functional in one dimension and the Kierlik-Rosinberg form of fundamental measures theory in three dimensions. In five dimensions, the functional describes bulk fluids better than does the Percus-Yevick theory. At planar walls, density profiles oscillate with smaller periods than in lower dimensions. Our findings open up avenues for treating both more general high-dimensional systems and three-dimensional mixtures via dimensional reduction.

Simple relationship between the virial-route hypernetted-chain and the compressibility-route Percus–Yevick values of the fourth virial coefficient

As is well known, approximate integral equations for liquids, such as the hypernetted chain (HNC) and Percus-Yevick (PY) theories, are in general thermodynamically inconsistent in the sense that the macroscopic properties obtained from the spatial correlation functions depend on the route followed. In particular, the values of the fourth virial coefficient B(4) predicted by the HNC and PY approximations via the virial route differ from those obtained via the compressibility route. Despite this, it is shown in this paper that the value of B(4) obtained from the virial route in the HNC theory is exactly three halves the value obtained from the compressibility route in the PY theory, irrespective of the interaction potential (whether isotropic or not), the number of components, and the dimensionality of the system. This simple relationship is confirmed in one-component systems by analytical results for the one-dimensional penetrable-square-well model and the three-dimensional penetrable-sphere model, as well as by numerical results for the one-dimensional Lennard-Jones model, the one-dimensional Gaussian core model, and the three-dimensional square-well model.

Correlation functions of a nonisotropic binary mixture in the Percus-Yevick theory

The correlation properties are studied of binary mixtures in the critical state of vaporization in external field. The investigations are based on the system of Percus-Yevick integral equations for pair correlation functions Gαβ(r1, r2) and the theory of scaling in critical phenomena. Expressions are obtained for pair correlation functions and correlation radius Rc, which are analyzed in various extreme cases.

Theory of the liquid state

Considerable progress towards developing satisfactory theories of the liquid state has been made in the past few years. Following a brief discussion of molecular interactions, we review, for simple liquids, four approaches in which there have been recent developments. These are machine calculation methods, the significant structure, Percus-Yevick, and perturbation theories. The Percus-Yevick theory does not appear to be useful for real fluids with attractive forces. However, we include this approach because it works very well for hard spheres. This review concludes with a brief discussion of quantum simple liquids and non-simple liquids.

THE STUDY OF GAY–BERNE FLUID: INTEGRAL EQUATIONS METHOD

We study a classical fluid of nonspherical molecules. The components of the fluid are the ellipsoidal molecules interacting through the Gay–Berne potential model. A method is described, which allows the Percus–Yevick (PY) and hypernetted-chain (HNC) integral equation theories to be solved numerically for this fluid. Explicit results are given and comparisons are made with recent Monte Carlo (MC) simulations. It is found that, at lower cutoff lmax, the HNC and the PY closures give significantly different results. The HNC and PY (approximately) theories, at higher cutoff lmax, are superior in predicting the existence of the phase transition in a qualitative agreement with computer simulation.

Fluids of core-softened particles in dimension 2: an integral equation study

An interaction model with core-softening that produces clustered phases in dimension 2 is studied by integral equation theories, and compared with corresponding simulation results. It is shown that the Hypernetted-chain (HNC) equation is surprisingly accurate and easier to solve numerically than the Percus–Yevick (PY) equation, which appears unable to operate in the cluster phase region. A comparison is made of the behaviour of the two theories in the absence of core-softening: in the high-temperature regime the Percus–Yevick theory is more accurate, whereas in the low-temperature regime it is HNC theory that becomes more accurate. It is the inclusion of an infinite class of cluster diagrams that allows the latter theory to better describe phases where local structures and small clusters play a predominant role in characterizing their macroscopic properties. The contrasting behaviour observed for continuous phases of hard and soft interactions must be due to fortuitous diagram compensations in the real system. HNC theory gives a very accurate structural description of the various cluster phases, as shown by comparing the radial distribution functions and structure factors with those from simulation.

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.

Depletion potential in the infinite dilution limit

The depletion force and depletion potential between two in principle unequal "big" hard spheres embedded in a multicomponent mixture of "small" hard spheres are computed using the rational function approximation method for the structural properties of hard-sphere mixtures [S. B. Yuste, A. Santos, and M. Lopez de Haro, J. Chem. Phys. 108, 3683 (1998)]. The cases of equal solute particles and of one big particle and a hard planar wall in a background monodisperse hard-sphere fluid are explicitly analyzed. An improvement over the performance of the Percus-Yevick theory and good agreement with available simulation results are found.

Fortran codes for the correlation functions of hard sphere fluids

Our Fortran codes for hard sphere fluids and their mixtures for the correlation functions that arise from the Percus–Yevick theory and the Verlet–Weis semi-empirical correction have proven useful during a period of nearly four decades and continue to be useful. In order to make these codes even more widely available, a brief summary is presented here and listings of these codes are given in the electronically accessible Supplementary Material to this paper.

Direct correlation function of the hard-sphere fluid

Two analytic approximations for the direct correlation function of a hard-sphere fluid are considered. The first follows from a generalization of the Percus–Yevick result in d dimensions, whereas the second arises in the Rational Function Approximation (RFA) method. Both approximations require the equation of state of the hard-sphere fluid as input. The results, derived after use of the Carnahan–Starling and the Padé 4,3 equations of state in both approaches, are compared with simulation data. The comparison shows that the first approximation is rather accurate in the region inside the core, but inherits the limitation of the Percus–Yevick theory for distances beyond the hard-sphere diameter. On the other hand, the results of the RFA method are also accurate inside the core and capture well the initial part of the tail beyond the hard-sphere diameter, but fail to account for the subsequent oscillations observed in the simulations. Other merits and limitations of the two approaches are reported.

Pair correlation functions and a free energy functional for the nematic phase

In this paper we have presented the calculation of pair correlation functions in a nematic phase for a model of spherical particles with the long-range anisotropic interaction from the mean spherical approximation (MSA) and the Percus-Yevick (PY) integral equation theories. The results found from the MSA theory have been compared with those found analytically by Holovko and Sokolovska [J. Mol. Liq. 82, 161 (1999)]. A free energy functional which involves both the symmetry conserving and symmetry broken parts of the direct pair correlation function has been used to study the properties of the nematic phase. We have also examined the possibility of constructing a free energy functional with the direct pair correlation function which includes only the principal order parameter of the ordered phase and found that the resulting functional gives results that are in good agreement with the original functional. The isotropic-nematic transition has been located using the grand thermodynamic potential. The PY theory has been found to give a nematic phase with pair correlation function harmonic coefficients having all the desired features. In a nematic phase the harmonic coefficient of the total pair correlation function h(x1,x2) connected with the correlations of the director transverse fluctuations should develop a long-range tail. This feature has been found in both the MSA and PY theories.

Radial distribution function of penetrable sphere fluids to the second order in density

The simplest bounded potential is that of penetrable spheres, which takes a positive finite value epsilon if the two spheres are overlapped, being zero otherwise. In this paper we derive the cavity function to second order in density and the fourth virial coefficient as functions of T* identical with k(B)T/epsilon (where k(B is the Boltzmann constant and T is the temperature) for penetrable sphere fluids. The expressions are exact, except for the function represented by an elementary diagram inside the core, which is approximated by a polynomial form in excellent agreement with accurate results obtained by Monte Carlo integration. Comparison with the hypernetted-chain (HNC) and Percus-Yevick (PY) theories shows that the latter is better than the former for T* < or similar to 1 only. However, even at zero temperature (hard sphere limit), the PY solution is not accurate inside the overlapping region, where no practical cancellation of the neglected diagrams takes place. The exact fourth virial coefficient is positive for T* < or similar to 0.73, reaches a minimum negative value at T* approximately 1.1, and then goes to zero from below as 1/T(*4) for high temperatures. These features are captured qualitatively, but not quantitatively, by the HNC and PY predictions. In addition, in both theories the compressibility route is the best one for T* < or similar to 0.7, while the virial route is preferable if T* > or similar to 0.7.

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.

Liquid‐like structure of polymer solutions near the overlap concentration

AbstractThe light scattering structure factor S(q, c) has been measured for a series of concentrations near the overlap value c* for solutions of high molecular weight poly(α‐methyl styrene) in the good solvent toluene. Scattering functions near and above overlap are characterized by a maximum as a function of scattering vector q. Scattering functions have also been calculated for these conditions using the measured second virial coefficient and radius of gyration, as reported previously for dilute solutions. The scattering function is factored into an intramolecular part that is described by a Debye function with no adjustable parameters and an intermolecular part that depends on the coil–coil pair correlation function, as suggested by Flory and Bueche. The pair correlation function is calculated using the Percus–Yevick theory of liquids and the Flory–Krigbaum potential for coil–coil interactions, as suggested by Frank Stillinger. Good agreement is obtained for the most concentrated dilute solutions, but as the overlap concentration is approached significant discrepancies are observed. The thermodynamic value of the scattering function, S(0, c), is overestimated by the theory. This discrepancy is discussed in terms of the importance of three‐body interactions, the failure of the Flory–Krigbaum potential in semidilute solutions and the limited precision of the standard protocol for calculating the measured scattering function in nondilute solutions. The observed maximum in the scattering function near overlap is not quantitatively reproduced by the theory. This discrepancy is discussed in terms of the failure of the shape of the Flory–Krigbaum potential to accurately reflect the energy of overlap for chains separated by distances near twice the radius of gyration. The mean‐field nature of the potential ignores the increased probability of interactions of linear neighboring segments. Well into the overlap region, the calculated scattering function poorly describes the observed results. The failure of the Flory–Bueche approximation in semidilute solutions is discussed as well as the effect of a changing radius of gyration as a function of concentration on the intramolecular scattering function. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 703–710, 2006

Integral equation theory of hard sphere liquids on two-dimensional cylindrical surfaces

An integral equation theory has been developed to elucidate the structure of hard sphere liquids on the two dimensional (2D) surface of a cylinder. The 2D cylindrical coordinate breaks the spherical symmetry. Hence, the pair correlation function is reformulated as a function of two variables to account for particles packing along and around the cylinder. Both Percus–Yevick (PY) and Hypernetted chain (HNC) closures are employed to solve the integral equation theory numerically. For dense sphere concentrations, the calculated pair correlation function displays an oscillatory structure, along and around the cylinder, due to liquid like ordering. The HNC theory predicts a more pronounced oscillatory structure than PY, and the solvation shells obtained from the HNC theory shift to smaller distances due to tighter packing, along and around the cylinder, predicted by the theory. Also, the deviation between the PY and HNC theory becomes more significant at higher densities.

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.

Microscopic theory of glassy dynamics and glass transition for molecular crystals

We derive a microscopic equation of motion for the dynamical orientational correlators of molecular crystals. Our approach is based upon mode coupling theory. Compared to liquids we find four main differences: (i) the memory kernel contains Umklapp processes if the total momentum of two orientational modes is outside the first Brillouin zone, (ii) besides the static two-molecule orientational correlators one also needs the static one-molecule orientational density as an input, where the latter is nontrivial due to the crystal's anisotropy, (iii) the static orientational current density correlator does contribute an anisotropic, inertia-independent part to the memory kernel, and (iv) if the molecules are assumed to be fixed on a rigid lattice, the tensorial orientational correlators and the memory kernel have vanishing l,l(') = 0 components, due to the absence of translational motion. The resulting mode coupling equations are solved for hard ellipsoids of revolution on a rigid sc lattice. Using the static orientational correlators from Percus-Yevick theory we find an ideal glass transition generated due to precursors of orientational order which depend on X(0) and psi, the aspect ratio and packing fraction of the ellipsoids. The glass formation of oblate ellipsoids is enhanced compared to that for prolate ones. For oblate ellipsoids with X(0) < or = 0.7 and prolate ellipsoids with X(0) < or = 4, the critical diagonal nonergodicity parameters in reciprocal space exhibit more or less sharp maxima at the zone center with very small values elsewhere, while for prolate ellipsoids with 2 < or = X(0) < or = 2.5 we have maxima at the zone edge. The off-diagonal nonergodicity parameters are not restricted to positive values and show similar behavior. For 0.7 < or = X(0) < or = 2, no glass transition is found because of too small static orientational correlators. In the glass phase, the nonergodicity parameters show a much more pronounced q dependence.

Effect of shape anisotropy on the phase diagram of the Gay-Berne fluid

We have used the density functional theory to study the effect of molecular elongation on the isotropic-nematic, isotropic-smectic A and nematic-smectic A phase transitions of a fluid of molecules interacting via the Gay-Berne intermolecular potential. We have considered a range of length-to-width parameter 3.0 < or = x(0) < or = 4.0 in steps of 0.2 at different densities and temperatures. Pair correlation functions needed as input information in density functional theory are calculated using the Percus-Yevick integral equation theory. Within the small range of elongation, the phase diagram shows significant changes. The fluid at low temperature is found to freeze directly from isotropic to smectic A phase for all the values of x(0) considered by us on increasing the density while the nematic phase stabilizes in between isotropic and smectic A phases only at high temperatures and densities. Both isotropic-nematic and nematic-smectic A transition density and pressure are found to decrease as we increase x(0). The phase diagram obtained is compared with computer simulation result of the same model potential and is found to be in good qualitative agreement.

Simulating colloids with Baxter’s adhesive hard sphere model

The structure of the Baxter adhesive hard sphere fluid is examined using computersimulation. The radial distribution function (which exhibits unusual discontinuities due tothe particle adhesion) and static structure factor are calculated with high accuracy over arange of conditions and compared with the predictions of Percus–Yevick theory. Wecomment on rigidity in percolating clusters and discuss the role of the model in thecontext of experiments on colloidal systems with short range attractive forces.