Intermolecular Correlation Functions Research Articles

Integral equations of the correlation functions for polymeric liquids

The integral equations for intramolecular and intermolecular correlation functions are derived for nonrigid polymeric (polyatomic) liquids by the device of the Kirkwood charging parameters. These integral equations are cast into mean-field-type equations by using the potential elimination method, reported previously for dense simple fluids. Based on the mean-field integral equations, we examine the superposition approximations for various levels of correlation. The present theory provides a means to make systematic corrections for superposition approximations for correlation functions of various orders. Upon using the superposition approximations for the triplet correlation functions in the Kirkwood hierarchy and an assumption or another concerning the charging parameter dependence of the cavity functions, we derive a set of generalized Percus–Yevick and hypernetted chain integral equations for the intramolecular and intermolecular pair correlation functions for beads (sites) of polymeric (polyatomic) liquids. This set of integral equations allows the intramolecular and intermolecular correlation functions to be determined self-consistently. The connection of this set of integral equations to the bead–bead (molecular) Ornstein–Zernike relation is pointed out. The integral equations for the intramolecular correlation functions will be numerically solved for some properties of a single polymer chain in the infinite dilution limit in the sequel to this paper.

Far-infrared spectra of diatomic polar molecules in simple liquids: Accounting for the anisotropic interaction

A theoretical expression giving the far-infrared spectrum of a dilute solution of diatomic polar molecules in a nonpolar liquid solvent is derived in order to analyze the influence on this spectrum from the J=1 and J=2 components of the Legendre polynomial expansion of the anisotropic solute–solvent potential. A non-Markovian spectral theory incorporating finite correlation of the interaction, mixing effects between rotational lines and quantum intermolecular potential correlation functions is used. This theory allows one to analyze the influence of each anisotropic part of the interaction in terms of a very reduced set of parameters: the strength and the width of its time correlation function. The zero and finite frequency components of the J=2 contribution are also studied. In particular, we show that the use of quantum potential correlation functions gives a zero frequency component not only in the width, but also in the shift of the different rotational lines. Numerical results for DCl, HCl, and HF in SF6 liquid at 273 K are obtained.

Quantum time correlation functions for solute-solvent intermolecular potentials : Influence on the far-infrared spectrum

A comparative study between the theoretical far-infrared spectra of HCl in Ar (105 K), Kr (125 K) and Xe (175 K) liquids obtained by using classical and quantum intermolecular potential time correlation functions (TCF) is presented The obtained results show that the consideration of quantum TCF's leads to an enhacement of the absorption in the winds of the spectra. Furthermore, whereas there are no significative diferences between widths of individual rotational lines calculated from classical and quantum TCF's, the shifts calculated from quantum TCF's are greater than those calculated with classical ones, specially for low values of j.

Local structure of model polymeric fluids: Hard-sphere chains and the three-dimensional fluctuating bond model

Monte Carlo simulations are performed for athermal polymers in the three-dimensional fluctuating bond lattice model. Polymer molecules composed of 20 and 50 freely-jointed beads are studied at volume fractions ranging from 0.2 to 0.35. Chain dimensions, intramolecular correlation functions, and intermolecular correlation functions are compared to results of off-lattice simulations for a similar freely-jointed chain model. It is found that the intramolecular correlation functions are qualitatively similar both on and off the lattice; there are quantitative differences which arise because of the larger persistence lengths of the off-lattice chains. At low densities the intermolecular correlation functions are similar in the two models, but significant differences appear at higher densities because of the limited extent of packing in the lattice fluid. We conclude that the lattice model adequately describes the effect of configurational entropy on fluid structure, but is inadequate to address packing effects which are important in the study of polymer melts. The volumetric behavior of the two model fluids is also discussed.

Neutron diffraction study of the partial pair correlation functions of liquid hydrogen sulphide

Following the results of a previous measurement, new neutron diffraction data were taken in order to extract the intermolecular partial correlation functions of liquid H2S. The experimental results are successfully compared with the results of a molecular dynamics simulation. Comparison with the pair correlation functions of liquid water at the same thermodynamic state is also discussed.

Percus-Yevick integral equation theory for athermal hard-sphere chains.

Equations for three intermolecular correlation functions for athermal hardsphere chain fluids are derived in the context of the Percus-Yevick (PY) integral equation theory. The approach employed here is based on a particle-particle description of chain molecules developed in part I of this series. Specifically, we obtained equations for the site-site total correlation function h αβ(r), the single index average total correlation function h α(r), and the average total correlation function h(r) for m-mer homonuclear hard-sphere chains. It is found that the PY theory does not yield a closed equation for the average total correlation function h(r). Solving h(r) requires additional information on the chain-end correlation functions. Two approximations (beyond the PY theory) which yield a single closed equation for h(r) are proposed and examined. Analytic expressions for these average correlation functions at contact are obtained as a function of the chain length m and hard-sphere site volume fraction η. Numerical solutions for g(r) are obtained for 4-mer and 8-mer chains and compared with computer simulation data. It is found that the PY theory is able to predict g(r) for 4-mer chains accurately; however, only qualitative agreement is obtained for 8-mer chains.

Molecular dynamics simulation of the liquid mixture CCl4/CS2

All centre-centre and atom-atom pair correlation functions have been calculated from a molecular dynamics simulation of the mixture CCl4/CS2 at three mole fractions and at room temperature. The agreement between the intermolecular correlation functions obtained and the corresponding data from a neutron scattering is excellent. The results are discussed in terms of the changes of the local environment of the CCl4 and CS2 molecules with changing mole fractions.

A kinetic theory of collision-induced depolarized light scattering

The collision-induced depolarized light scattering spectrum arising from the dipole–induced dipole (DID) interactions in atomic fluids has been calculated using kinetic theory methods. The theory contains no adjustable parameters and requires only the temperature, density, and collision diameter as input. Above 25 cm−1, the calculated line shape was found to be in excellent agreement with the spectrum obtained from a hard-sphere molecular dynamics simulation, with the same model for the interaction-induced polarizability. The domain of applicability of the approximations used in reducing the theory to a tractable form was carefully evaluated. The methods developed should be applicable to other intermolecular correlation functions, such as those observed in vibrational relaxation and interaction-induced infrared spectroscopy.

Semiclassical dynamics in Liouville space: Application to molecular electronic spectroscopy

A semiclassical theory of molecular dynamics and electronic spectroscopy, based on the propagation of a Liouville-space generating function (LGF) is developed. The method is particularly useful for treating the dynamics of systems, consisting of coupled classical and quantum degrees of freedom. Applications to intramolecular and intermolecular dynamics, correlation functions, and spectral line shapes in linear and nonlinear optical measurements are developed. The electronic absorption line shapes and Raman excitation profiles of model anharmonic molecules at finite temperatures are calculated as a demonstration.

Equilibrium theory of polymer liquids: Linear chains

An equilibrium theory of polymer melts, developed previously by us for polymer rings, is generalized to include linear polymer chains. This theory is based on the reference interaction site model (RISM) integral equation approach developed by Chandler and co-workers for molecular liquids. We are able to construct a tractable formalism for the high polymer problem by employing the fact that a polymer molecule in a melt is ideal. This leads to a set of coupled, nonlinear integral equations for the intermolecular radial distribution functions. A simple optimized perturbative scheme is developed for long linear chains based on the relative unimportance of end effects. To lowest order, the theory reduces to a single nonlinear integral equation. Chain end corrections to the site-averaged intermolecular correlation functions vanish according to N−2, where N is the degree of polymerization. Numerical techniques were used to compute the radial distribution function and structure factor for hard core linear Gaussian chains of 16 000 units over a range of densities. Calculation of higher order corrections for end effects is possible with our formalism.

Small-angle scattering by polyelectrolyte solutions: Intermolecular correlation function of poly(methacrylic acid) in salt-free semidilute solutions

Intermolecular correlations in salt-free solutions of poly(methacrylic acid) neutralized by NaOD were investigated as a function of the degree of neutralization, α. At α = 0, the relative concentration of monomer units belonging to foreign macromolecules at a distance r from a reference monomer unit, g(r) = N(r) N ̄ , is nearly equal to unity in the r range investigated. Already at α = 0.01, however, the values of g( r) are significantly smaller than unity near the origin. With increasing r, g( r) monotonically increases and reaches unity at r = 120–150 A ̊ . No oscillations indicating an ordered structure are observed. The shape of g( r) curves is in agreement with the correlation hole concept. The non-monotonic dependence of the radius of the correlation hole on α is explained by the overlap of macroions at higher degrees of neutralization.

Intermolecular correlation functions from Foerster energy-transfer experiments

Intermolecular correlation functions from Foerster energy-transfer experiments

Quantum mechanical contributions to the structure of liquid water

Path integral Monte Carlo methods are used to study the effect of quantization of the orientational degrees of freedom of water (H 2O), using the ST2 model. A comparison of the classical and quantum atom—atom intermolecular correlation functions show that significant quantitative effects are manifest in the results.

Numerical solution of RISM for homonuclear vibrating hard-dumbells

Numerical solutions of the reference interaction site model (RISM) for site-site radial distribution functions gαβ(r) of a fluid of vibrating harddumbells are carried out, using a method of direct substitution in Fourier space. Comparisons with molecular dynamics (MD) results for the same system are given. The intramolecular distribution functions obtained by MD are used in RISM to obtain the corresponding intermolecular site-site gαβ(r), which in turn are compared against the MD results. These numerical results and further algebraic analysis indicate that the intermolecular site-site correlation functions calculated with RISM, do not depend noticeably on the form of the intramolecular correlation function used in the calculation. Additional evidence on the equivalence of vibrating and rigid hard dumbells is given.

Fourth order intermolecular correlation functions from spectroscopic data and molecular dynamics calculations

The shape of collision induced absorption spectra of noble gas mixtures and the broadening of rotational lines of dipolar molecules dissolved in noble gas fluids are described in terms of time dependent perturbation theory up to fourth order. In combination with second and fourth order intermolecular correlation functions obtained from molecular dynamics calculations, conclusions are made about the importance of these functions for spectroscopic observations. The fourth order correlation functions are highly structured. The form taken by the connected part of the fourth order function depends markedly on the range of the probe interaction.

The role of capillary wave fluctuations in determining the liquid-vapour interface

We investigate the role played by capillary-wave like fluctuations in the determination of the equilibrium density profile and the intermolecular correlation functions of various models of a planar liquid-vapour interface in an external gravitational field. We show that the interfacial profile calculated from the van der Waals model, which is the simplest (squaregradient) approximation in a density functional theory of inhomogeneous fluids, is stable against capillary-wave fluctuations and therefore cannot be used to describe the ‘bare’ interface in a capillary wave model. The density-density correlation function of the van der Waals model exhibits long-ranged correlations in the interface. These are of precisely the same form as those predicted by Wertheim and can be associated with capillary-wave like fluctuations of the Gibbs dividing surface. These fluctuations do not lead to a ‘diffuse’ interfacial profile in the limits of infinite interfacial area and zero gravity; the interfacial thickness remains ...

Dipolar spin relaxation of a pair of like spins attached to anisotropically reorienting molecules. A group theoretical approach

Intra- and intermolecular correlation function and spin-lattice relaxation time ( T 1 and T 1 Q ) expressions are developed for dipolarly coupled spin pairs using group theoretical methods. The spin-carrying molecules being fixed with centers of gravity on equivalent sites of a crystalline lattice are assumed to perform reorientational jumps between the minima of a n-fold equal-wells potential. In particular, single-axis reorientations between wells which are related by the operations of the cyclic groups C n are considered and explicit relaxation time formulas obtained.

Statistical mechanics of chemical equilibria and intramolecular structures of nonrigid molecules in condensed phases

An exact classical statistical mechanical theory is developed which describes how intermolecular forces alter the average intramolecular structures of nonrigid molecules and how these forces affect the equilibrium constant of chemically reacting species. The theory lends itself to computationally convenient approximations. Three illustrative applications of the theory are given. First, the equilibrium constant for the reaction N2O2?2NO2 is studied. Second, the shift in the chemical bond lengths of N2 and O2 from their gas phase values to those in the liquid are calculated from the theory. Third, the average conformational structure of n-butane in various dense fluid solvents is predicted. The basic methods used in the derivation of the theory are the techniques of physical cluster series (as opposed to the usual Mayer ’’mathematical’’ cluster expansions), and topological reductions. The formalism gives rise to a renormalization of chemical bonding Boltzmann factors so that the theoretical expressions are not swamped by essentially infinite and unnormalized cluster functions. Along with the results on chemical equilibria and intramolecular structures, the interaction site cluster series for intermolecular correlation functions is rederived in a more general manner than the derivation originally presented by Ladanyi and Chandler.

Structure of liquids. Part 7.—Determination of intermolecular potential functions and correlation functions in fluid argon by X-ray diffraction techniques

Methods of obtaining the radial distribution function and the direct correlation function for simple liquids from X-ray diffraction data are reviewed. A new theoretical analysis demonstrates that certain of the cluster integrals appearing in the density expansion of the radial distribution function can be evaluated directly from Fourier transforms of appropriate powers of the X-ray scattering function. These results are used to develop expressions which permit determination of the intermolecular potential function from moderately dense fluids using essentially only X-ray diffraction data. Non-additivity effects may also be treated. Experimental data available are inadequate for conclusive determination of the potential function; however, experimental results from one state of argon give plausible values for the lowest-order cluster integral.