Solution of the Percus-Yevick equation for hard disks

The authors solve the Percus-Yevick equation in two dimensions by reducing it to a set of simple integral equations. They numerically obtain both the pair correlation function and the equation of state for a hard disk fluid and find good agreement with available Monte Carlo results. The present method of resolution may be generalized to any even dimension.     

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.     

The equation of state of hard hyperspheres in four and five dimensions

The equation of state of hard hyperspheres in four and five dimensions is calculated from the value of the pair correlation function at contact, as determined by Monte Carlo simulations. These results are compared to equations of state obtained by molecular dynamics and theoretical approaches. In all cases the agreement is excellent.     

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.     

A coarse-grained model for polyethylene glycol polymer

A coarse-grained (CG) model of polyethylene glycol (PEG) was developed and implemented in CG molecular dynamics (MD) simulations of PEG chains with degree of polymerization (DP) 20 and 40. In the model, two repeat units of PEG are grouped as one CG bead. Atomistic MD simulation of PEG chains with DP = 20 was first conducted to obtain the bonded structural probability distribution functions (PDFs) and nonbonded pair correlation function (PCF) of the CG beads. The bonded CG potentials are obtained by simple inversion of the corresponding PDFs. The CG nonbonded potential is parameterized to the PCF using both an inversion procedure based on the Ornstein-Zernike equation with the Percus-Yevick approximation (OZPY(-1)) and a combination of OZPY(-1) with the iterative Boltzmann inversion (IBI) method (OZPY(-1)+IBI). As a simple one step method, the OZPY(-1) method possesses an advantage in computational efficiency. Using the potential from OZPY(-1) as an initial guess, the IBI method shows fast convergence. The coarse-grained molecular dynamics (CGMD) simulations of PEG chains with DP = 20 using potentials from both methods satisfactorily reproduce the structural properties from atomistic MD simulation of the same systems. The OZPY(-1)+IBI method yields better agreement than the OZPY(-1) method alone. The new CG model and CG potentials from OZPY(-1)+IBI method was further tested through CGMD simulation of PEG with DP = 40 system. No significant changes are observed in the comparison of PCFs from CGMD simulations of PEG with DP = 20 and 40 systems indicating that the potential is independent of chain length.     

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.     

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     

Molecular correlation functions for uniaxial ellipsoids in the isotropic state

We perform event-driven molecular dynamics simulations of a system composed by uniaxial hard ellipsoids for different values of the aspect ratio and packing fraction. We compare the molecular orientational-dependent structure factors previously calculated within the Percus-Yevick approximation with the numerical results. The agreement between theoretical and numerical results is rather satisfactory. We also show that, for specific orientational quantities, the molecular structure factors are sensitive to the particle shape and can be used to distinguish prolate from oblate ellipsoids. A first order theoretical expansion around the spherical shape and a geometrical analysis of the configurations confirms and explains such an observation.     

Constructing a new closure theory based on the third-order Ornstein-Zernike equation and a study of the adsorption of simple fluids

The third-order Ornstein-Zernike equation (OZ3) is used in the construction of a bridge functional that improves over conventional liquid-theory closures (for example, the hypernetted chain or the Percus-Yevick equations). The OZ3 connects the triplet direct correlation C((3)) to the triplet total correlation h((3)). By invoking the convolution approximation of Jackson and Feenberg, we are able to express the third-order bridge function B(3) as a functional of the indirect correlation γ. The resulting expression is generalized to higher-order bridge terms. This new closure is tested on the adsorption of Lennard-Jones fluid on planar hard surfaces by calculating the density profiles and comparing with Monte Carlo simulations. Particular attention is paid to the cases where molecular depletion on the substrate is evident. The results prove to be highly accurate and improve over conventional closures.     

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.     

Mapping a homopolymer onto a model fluid

We describe a linear homopolymer using a grand canonical ensemble formalism, a statistical representation that is very convenient for formal manipulations. We investigate the properties of a system where only next neighbor interactions and an external, confining, field are present and then show how a general pair interaction can be introduced perturbatively, making use of a Mayer expansion. Through a diagrammatic analysis, we shall show how constitutive equations derived for the polymeric system are equivalent to the Ornstein-Zernike and Percus-Yevick equations for a simple fluid and find the implications of such a mapping for the simple situation of Van der Waals mean field model for the fluid.     

Rotational dynamics and conformational kinetics in liquid crystals

Two particular aspects of solute dynamics in ordered media are analysed on the basis of the solution of multivariate diffusion equations: the effects of the solvation dynamics on the rotational motions of dipolar probes in liquid crystal solvents, and the alteration of reaction pathways in isomerization kinetics caused by the solvent order. The introduction of a suitable solvent coordinate allows the interpretation of high frequency contributions in the rotational correlation functions observed by spectroscopic techniques, namely dielectric dispersion, IR and Raman spectroscopy, ESR lineshapes and optical Kerr effect. For molecular systems undergoing conformational changes, a method is offered to evaluate the modification of the torsional barriers resulting from the anisotropic torques modulated by the molecular shape changes along the reaction coordinate.     

Analytical coarse-grained description for polymer melts

Starting from the Ornstein-Zernike equation the authors derive an analytical theory, at the level of pair correlation functions, which coarse grains polymer melts into liquids of interacting soft colloidal particles. Since it is analytical, the presented coarse-graining approach will be useful in developing multiscale modeling procedures to simulate complex fluids of macromolecules. The accuracy of the theory is tested by its capacity to reproduce the liquid structure, as given by the center-of-mass intermolecular total pair correlation function. The theory is found to agree well with the structure predicted by molecular dynamics simulations of the liquid described at the united atom level as well as by molecular dynamics simulations of the liquid of interacting colloidal particles. The authors perform simulations of the liquid of interacting colloidal particles having as input the potential obtained from their analytical total pair correlation function by enforcing the hypernetted-chain closure approximation. Tests systems are polyethylene melts of chains with increasing degrees of polymerization and polymer melts of chains with different chemical architectures. They also discuss the effect of adopting different conventional approximations for intra- and intermolecular monomer structure factors on the accuracy of the coarse-graining procedure, as well as the relevance of higher-order corrections to their expression.     

Rotational dynamics and conformational kinetics in liquid crystals

Abstract

Two particular aspects of solute dynamics in ordered media are analysed on the basis of the solution of multivariate diffusion equations: the effects of the solvation dynamics on the rotational motions of dipolar probes in liquid crystal solvents, and the alteration of reaction pathways in isomerization kinetics caused by the solvent order. The introduction of a suitable solvent coordinate allows the interpretation of high frequency contributions in the rotational correlation functions observed by spectroscopic techniques, namely dielectric dispersion, IR and Raman spectroscopy, ESR lineshapes and optical Kerr effect. For molecular systems undergoing conformational changes, a method is offered to evaluate the modification of the torsional barriers resulting from the anisotropic torques modulated by the molecular shape changes along the reaction coordinate.     

Tests of an approximate scaling principle for dynamics of classical fluids

We used molecular dynamics computer simulations to test an approximate scaling principle that conjectures that two equilibrium atomic liquids have very similar dynamical properties if they have the same density and similar static pair correlation functions when the length scales of the two liquids are adjusted appropriately, even if they have different interatomic potentials and different temperatures. The simulations were performed on two types of model atomic liquids at various temperatures at the same density. In the first type, the interatomic potential is the Lennard-Jones potential (LJ). In the second type, the interatomic potential is the repulsive part of the Lennard-Jones potential (RLJ). We identified pairs of systems that have very similar pair correlation functions despite the fact that they had different potentials. Each pair consisted of an LJ liquid at a specific temperature and a corresponding RLJ liquid at a lower temperature. We compared various time correlation functions and transport coefficients of the two systems in each pair. Many dynamical properties are very similar in each pair, in accordance with the approximate scaling principle, whereas others are significantly different. The results indicate that certain dynamical properties are very insensitive to large changes in the interatomic potential that leave the pair correlation function largely unchanged, whereas other dynamical properties are much more sensitive to such changes in the potential. The transport coefficients for diffusion and viscosity are among the dynamical properties that are insensitive to such changes in the potential, and this may be part of the reason transport properties of many fluids have been calculated or rationalized in terms of a simple hard sphere model of liquids.     

Ab initio molecular dynamics simulation of liquid Al88Si12 alloys

First-principles molecular dynamics simulations are carried out to study the structures, dynamics, and electronic properties of liquid Al88Si12 in the temperature ranging from 898 to 1298 K. The temperature dependence of static structure factors, pair correlation functions, and electronic density-of-states are investigated. The structural properties obtained from the simulations are in good agreement with the x-ray diffraction experimental results.     

Collective motions in computer models of water. Large-scale and long-time correlations

Two-particle correlation functions describing the simultaneous motion of a pair of molecules initially separated by a given distance R0 are calculated to study collective effects in the diffusive motion of water molecules in molecular dynamics models. Various types of such functions and their dependences on the interaction potential, temperature, and the number of particles in the model are considered. At short times (of the order of ten picoseconds), these functions exhibit irregular behavior depending on R0. The most nontrivial and unexpected result was the detection of correlations in the displacements of pairs of particles that extend for tens of angstroms and last for hundreds of picoseconds. Such correlations are not observed in the random walk models of noninteracting particles. It is suggested that the observed large-scale correlations reveal the vortex-like motions of the molecules.     

New conditions for validity of the centroid molecular dynamics and ring polymer molecular dynamics

Validity of the centroid molecular dynamics (CMD) and ring polymer molecular dynamics (RPMD) in quantum liquids is studied on an assumption that momenta of liquid particles relax fast. The projection operator method allows one to derive the generalized Langevin equation including a memory effect for the full-quantum canonical (Kubo-transformed) correlation function. Similar equations for the CMD and RPMD correlation functions can be derived too. The comparison of these equations leads to conditions under which the RPMD and CMD correlation functions agree approximately with the full-quantum canonical correlation function. The condition for the RPMD is that the memory effects of the full-quantum and RPMD equations vanish quickly with the same time constants. The CMD correlation function requires additional conditions concerning static correlation.