Triplet Correlation Function Research Articles

Calculation of the entropy for simple classical fluids

We consider the calculation of the excess entropy in the grand canonical ensemble and redevelop a systematic expansion in terms of the multiparticle correlation functions. The exact expansion for the entropy is written as two series of functions, one is directly related to the potentials of mean force as in the closed systems (generalizing ladder diagram expansion) and another is a sum of fluctuation corrections involving functional derivatives of potentials of mean force. The generalized superposition approximation is then applied to extract the leading contributions of the fluctuation terms as a sum of functions including both pair and triplet correlation functions (generalizing ring diagram expansion). The stationary entropy principle is used to derive two extensions of the hypernetted chain (HNC) equation for the triplet correlation function. An approximate expression of the entropy, which is suitable for numerical calculations in homogeneous systems, is also proposed.

First-order correction to classical nucleation theory: A density functional approach

It is shown that the classical expression for the change in grand potential of a system on formation of a cluster of radius R is modified by a factor [1−(2w+6δT)/R], to first order in 1/R, where w is a correction due to the nonzero compressibilities of liquid and vapor (near the triple point, w is approximately equal to the product of liquid compressibility and surface tension), and δT is the coefficient in the expression relating the surface tension of the droplet, γ(R), to the planar surface tension, γ∞, i.e., γ(R)=γ∞(1−2δT/R). An expression for δT is derived involving the pair and triplet correlation functions and the density profile of the planar surface. This complements the expression for δT involving the pair distribution function derived by Blokhuis and Bedeaux; the equivalence of the two expressions in the low density limit is demonstrated. Calculations of δT and w are performed using mean-field density functional theory for the Yukawa potential and an r−6 potential, as well as using the square-gradient approximation. δT is found to be negative for all conditions investigated; its magnitude depends on the potential used, and tends to increase with increasing temperature. However, the ratio δT/w is found to be relatively insensitive to potential and to temperature, being between about −1.2 and −1.5 for the conditions investigated. The effect of using a weighted density approximation in place of the local density approximation for the hard-sphere part of the potential is estimated in a square-gradient approximation and found to be small.

Symmetry based approach to triplet correlation functions

Two related approaches to the theory of inhomogeneous classical systems are introduced, both yielding analytic forms for triplet and higher-order direct correlation functions in the homogeneous limit. The present theories lead to results that exactly obey the known sum rule relating the triplet direct correlation function to the derivative of the Ornstein-Zernike function. The resulting triplet direct correlation functions are then found to be simple products in both reciprocal and real space. Agreement with simulation results for the triplet direct correlation function in the hard sphere fluid is generally found to be very good; even the simpler version of the theory agrees well with the results of the more computationally intensive weighted density approximation.

The isotropic–nematic phase transition in a fluid of dipolar hard spherocylinders

A thermodynamic perturbation theory in the high temperature expansion is presented for a system of hard spherocylinders with a permanent longitudinal dipole positioned at the center of the molecule. The free energy of the reference hard spherocylinder is described using an expression derived by Lee [J. Chem. Phys. 80, 7036 (1988)]. The dipole contribution to the free energy is evaluated using a generalization to the perturbation theory for dipolar hard spheres proposed by Larsen, Rasaiah, and Stell [Mol. Phys. 33, 987 (1977)]. The effect of increasing dipole strength on the isotropic-nematic phase transition of a fluid of spherocylinders of aspect ratio, L/D=5, is studied using a generalization to the well known Onsager theory [Ann. N. Y. Acad. Sci. 51, 627 (1949)]. The single particle orientational distribution function is approximated with a gaussian trial function while an ad hoc approximation is used for the pair and triplet correlation functions of the hard spherocylinder reference system. While these approximations seem quite severe they are not expected to affect the general features of the phase diagram. Increasing the dipole moment is found to destabilize the nematic phase with respect to the isotropic phase, shifting the phase transition to higher densities and pressures. This is in general agreement with the most recent simulation studies of the system. All other theories of this system predict a stabilization of nematic phase. It is suggested that the reason for the failure of these theories lies in the two body character of their approach to the orientational order in the liquid crystal phase. To the knowledge of the authors this is the first theory for the isotropic-nematic phase transition of dipolar hard-spherocylinders which explicitly includes three-body interactions in the orientationally ordered phase.

Detergent Foam Images as Analogue/Digital Model Fluids. 1. Phase Behavior and Effective Pair Potentials

Detergent foam images are investigated as analogue/digital (A/D) models for simple fluids. Results of foam preparation/image processing experiments with quasi-two-dimensional systems are illustrated. In the analysis, the pair and triplet correlation functions, g(r) and ξ(ϑ,s;r), are measured for the vertexes of plateau border networks. The phase morphology is characterized by the amplitude and extent of oscillations in g(r). Effective pair potentials u(r) are calculated via the lowest order equation in the Born−Yvon−Green hierarchy. The u(r) are further investigated via Monte Carlo calculations and the Morse function. The experimental data are used to calculate the pair correlation entropy and the compressibility factor as functions of particle number density. These and other observables enable comparisons between the model systems and real atomic fluids. Overall, the results show foam images to mimic fluid structure at the atomic level in several respects. For the experimental conditions discussed, succe...

Hydration of an α-Helical peptide: comparison of theory and molecular dynamics simulation

We present a statistical mechanical description of biomolecular hydration that accurately describes the hydrophobic and hydrophilic hydration of a model alpha-helical peptide. The local density of water molecules around a biomolecule is obtained by means of a potential-of-mean-force (PMF) expansion in terms of pair- and triplet-correlation functions of bulk water and dilute solutions of nonpolar atoms. The accuracy of the method is verified by comparing PMF results with the local density and site-site correlation functions obtained by molecular dynamics simulations of a model alpha-helix in solution. The PMF approach quantitatively reproduces all features of the peptide hydration determined from the molecular dynamics simulation. Regions of hydrophobic hydration near the C alpha and C beta atoms along the helix are well reproduced. The hydration of exposed polar groups at the N- and C-termini of the helix are also well described by the theory. A detailed comparison of the local hydration by means of site-site radial distribution functions evaluated with the PMF theory shows agreement with the molecular dynamics simulations. The formulation of this theory is general and can be applied to any biomolecular system. The accuracy, speed of computation, and local character of this theory make it especially suitable for studying large biomolecular systems.

Hydration of an α‐Helical peptide: comparison of theory and molecular dynamics simulation

We present a statistical mechanical description of biomolecular hydration that accurately describes the hydrophobic and hydrophilic hydration of a model α-helical peptide. The local density of water molecules around a biomolecule is obtained by means of a potential-of-mean-force (PMF) expansion in terms of pair- and triplet-correlation functions of bulk water and dilute solutions of nonpolar atoms. The accuracy of the method is verified by comparing PMF results with the local density and site-site correlation functions obtained by molecular dynamics simulations of a model α-helix in solution. The PMF approach quantitatively reproduces all features of the peptide hydration determined from the molecular dynamics simulation. Regions of hydrophobic hydration near the Cα and Cβ atoms along the helix are well reproduced. The hydration of exposed polar groups at the N- and C-termini of the helix are also well described by the theory. A detailed comparison of the local hydration by means of site-site radial distribution functions evaluated with the PMF theory shows agreement with the molecular dynamics simulations. The formulation of this theory is general and can be applied to any biomolecular system. The accuracy, speed of computation, and local character of this theory make it especially suitable for studying large biomolecular systems. © 1997 Wiley-Liss Inc.

Theoretical description of biomolecular hydration. Application to A-DNA.

The local density of water molecules around a biomolecule is constructed from calculated two- and three-points correlation functions of polar solvents in water using a Potential-of-Mean-Force (PMF) expansion. As a simple approximation, the hydration of all polar (including charged) groups in a biomolecule is represented by the hydration of water oxygen in bulk water, and the effect of non-polar groups on hydration are neglected, except for excluded volume effects. Pair and triplet correlation functions are calculated by molecular dynamics simulations. We present calculations of the structural hydration for ideal A-DNA molecules with sequences [d(CG)5]2 and [d(C5G5)]2. We find that this method can accurately reproduce the hydration patterns of A-DNA observed in neutron diffraction experiments on oriented DNA fibers (P. Langan et al. J. Biomol. Struct. Dyn., 10, 489 (1992)).

Concentration dependence of the local order in amorphous NiNb alloys

Abstract The atomic structure of amorphous NiNb alloys has been studied via molecular-dynamic simulations on the basis of effective interatomic potentials calculated using the hybridized nearly-free-electron-tight-binding-bond technique. The partial static structure factors and pair correlation functions at four different compositions are compared with results from neutron diffraction with isotopic substitution. Theory and experiment agree in finding a certain chemical order (similar to that in the crystalline NiNb compounds) and strong NiNb bonds that are contracted compared with the average NiNi and NbNb bond lengths. The chemical short-range order depends only weakly on composition. In addition to the laboratory experiment, the computer simulation also yields information on the partial triplet correlation functions. The bond-angle distributions indicate that the local atomic arrangement is essentially of a polytetrahedral type, with broad peaks in the total bond-angle distribution close to the icosahedral bond-angles. In the partial bond-angle distributions, the peaks are slightly shifted as a consequence of size differences and chemical order. This shift is again similar to the bond angles in the coordination polyhedra of the crystalline Frank-Kasper phase. Compared with NiZr and NiY and Ni-metalloid glasses, NiNb glasses are found to be more random and show no sign of the trigonal prismatic order characteristic of these systems. The physical origin of these structural trends is discussed.

Self-consistent integral-equation theory of chain-molecular liquids: Structure and thermodynamics

Self-consistent integral equations for the pair intramolecular and intermolecular correlation functions are derived from a general hierarchy of integral equations for chain-molecular liquids. These coupled equations are obtained by using superposition approximations for the triplet correlation functions, an approximate translational symmetry for the site–site intramolecular correlation functions and the equivalence of sites for intermolecular correlation functions. In addition to this self-consistent set of integral equations, the polymer reference interaction site model (PRISM) integral equation is also made self-consistent by coupling this intermolecular equation to the equations for the intramolecular correlation functions derived in the present theory. The intra- and intermolecular correlation functions of the self-consistent schemes considered in this work obey integral equations, and they are different from the other self-consistent schemes proposed in the literature. Self-consistent solutions for the structural properties, such as intra- and intermolecular correlation functions and structure factor, and macroscopic properties, such as chain expansion factor and thermodynamic functions of athermal polymer melts, are compared with available Monte Carlo results and other theories. For the properties examined, self-consistent solutions yield better results than the non-self-consistent calculations with ad hoc, ideal Gaussian inputs for the intramolecular correlation functions.

Evidence for domains in deposited lipid bilayers

We have modeled dimyristoylphosphatidylcholine bilayer membranes deposited on argon-sputtered glass or polyacrylamide substrates in order to understand the dependence of the lipid self-diffusion coefficient upon temperature. We used a lattice model that has been successful in understanding some of the thermodynamics of lipid bilayers. We considered two modifications of this model : model I in which the bilayer was described by a distribution of noninteracting bilayer domains within and between which lipids could move and model II in which lipid-lipid interactions were weakened in a random way thereby permitting large diffusion coefficients at low temperatures. We modeled the lateral exchange oflipids as being permitted only when either an adjacent pair of lipids or a mutually adjacent trio of lipids were in their excited states. We carried out a direct computer simulation of lipid movement or related the self-diffusion coefficient to pair or triplet correlation functions. We found that the results predicted by the unperturbed model are in general agreement with measurements of the self-diffusion coefficient on oxidized silica wafers as a prototype of a polished surface. This permitted us to apply the perturbed models to study the other cases for which we found that only model I could account for measurements on argon-sputtered glass and on polyacrylamide films. Argon-sputtered glass yielded domain distributions with a most probable size of ∼250 to 600 lipids per half-bilayer, while polyacrylamide films yielded a most probable domain size of ∼600 lipids per half-bilayer. On the basis of these results, we have made predictions about the average number of gauche bonds per molecule as a function of temperature.

A new geometrically based integral equation hierarchy for fluids of hard-sphere systems

We use a new hierarchy of integral equations for the background correlation functions of hard spheres previously derived to calculate the thermodynamic and structural properties of the hard-disc fluid. The virial coefficients, compressibility factors, and chemical potentials calculated using the second-order closure agree excellently with computer simulation data. Values of the background pair correlation function, and the triplet correlation function for all three discs in contact are also given. For the latter functions, at high densities the second-order closure agrees well with the BGY2 integral equation theory, and predicts large deviations from the superposition approximation. The zeroth-order form of the theory is shown to be equivalent to the BGYM theory of Chae, D. G., Ree, F. H., and Ree, T., 1969, J. chem. Phys., 70, 1581.

Possible quantitative measures of order/disorder in models of liquid and amorphous structures

The variance of the pair correlation function, the triplet correlation function, g(3)(r,s,t), and the three-body contribution to the configurational entropy, S3, were studied as possible tools for distinguishing between three different models of amorphous silicon (a-Si) with (nearly) identical pair correlation functions. All of these quantities were shown to be able to make the distinction unequivocally. For practical reasons, short-range S3 is suggested as the best quantitative measure of disorder, in terms of higher-order correlations. For the first time, various projections of the three-body correlation function of a complicated amorphous material are also shown. On the basis of the current study it is proposed that the most realistic structural model of a-Si is the reverse Monte Carlo fit, starting from the well known Wooten model.

Triplet correlation functions for hard-spheres: Computer simulation results

We present results for the triplet distribution function g(3)(r,s,t) of hard-spheres obtained in extensive molecular-dynamics simulations; the packing fractions we have investigated range from 0.15 to 0.45. The simulation data have been compared to results for g(3)(r,s,t) which we calculated via some recently proposed analytical and numerical methods; two of these methods are based on density-functional theory and the Wertheim–Thiele solution of the Percus–Yevick equation; another method, proposed by Barrat, Hansen, and Pastore uses a factorization ansatz for the pair direct correlation function and the last approximation is based on a formal density expansion of g(3)(r,s,t), truncated after second order. Furthermore we compared, simulation results to data obtained by the ‘‘source-particle method’’ (or PY3 method) proposed a few years ago by Attard. Attard’s method shows an extremely good agreement not only for general configurations, but in particular for particles at direct contact; this approximation has to be considered as the most reliable method available for the numerical determination of the triplet-structure of a simple liquid. Concerning the results of the other methods discrepancies with the simulation data are observed in particular near the contact and for very close triplet-configurations. Apart from Attard’s approximation the second order density expansion gives the best agreement. For less close configurations, i.e., if particles are separated by 1.5 to 2 hard-sphere diameters, the results of all the methods investigated practically coincide.

Equations of state for hard chain molecules

Equations of state for hard sphere chain molecules are derived within the framework of second order perturbation theory of polymerization (TPT2). The present formulation differs from the original Wertheim’s treatment by dealing with molecules with a fixed number of beads. However, the two versions yield very close numerical results for chains composed of freely jointed spheres, showing the insensitivity of the pressure of these fluids to polydispersity. The case of star-like molecules and chain mixtures is also considered. Theoretical predictions are compared to available simulation data, using recent results for the triplet correlation function for hard spheres in rolling contact. Some predictions for the pressure of freely rotating chains and that of trimers as a function of the bond angle are given.

Triplet correlation function for hard sphere systems

Monte Carlo simulation data are presented for the triplet correlation function g (3) (r,s,t), for a hard sphere system. Results are reported for reduced densities ρσ3 between 0·3 and 0·88, and reduced distances r/σ from 1·02 to 2·42 for all possible configurations. Comparisons with the PY3 theory show excellent agreement for all cases studied. A table of the complete simulation results is available separately.

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.

Isothermal density derivative of S(k) and the triplet correlation function: an experimental study in liquid Kr

Abstract Recent accurate measurements performed on liquid Kr have been used to test some of the models proposed in the literature for the triplet correlation function g 3 ( r, s ). It turns out that neither the superposition approximation suggested by Kirkwood (J.P. Hansen and I.R. McDonald, Theory of simple liquids, Academic Press, London, 1986), nor a modification of it which partially takes into account linear density effects (H. Fredrikze, Ph.D. Thesis, Delft, 1985) are able to satisfactorily reproduce the experimental isothermal derivatives of the structure factor. Also the uniform fluid model (P.A. Egelstaff, D.I. Page and C.R.T. Heard, J. Phys. C, 4 (1971) 1453), gives a poor representation especially at lower densities. It is concluded that, in order to correctly describe triplet correlation effects, a full account of the irreducible contributions to g 3 ( r, s ) is necessary.

Triplet correlation functions in liquid metals

The triplet correlation function, g (3)( r, r′) , was calculated for two liquid metals, rubidium and germanium, both near their respective melting points (373 K and 1250 K). The results have been obtained (i) from a molecular dynamics study and (ii) using a numerical technique, which is based on a formal density expansion of the triplet correlation function. Results of both methods are found to be in good agreement. In a closer analysis of g (3)( r, r′) in terms of angle distributions, it was found that in the case of germanium a tetragonal coordination is preferred for triangle configurations with sides near to nearest neighbour distance, whereas for rubidium rather close-packed structures are encountered.

Structure of binary metallic glasses

The short range atomic structure of binary amorphous alloys is described by the interatomic distances and the partial coordination numbers. These quantities are deduced from partial structure factors via pair correlation functions. Some methods to determine these partial functions are described. A review of experimental results obtained with metal-metalloid as well as metal-metal alloys is given. The amorphous alloys from the NiNb system are treated more in detail. Some special topics such as the dependence of the amorphous structure on the production process, refractory metallic glasses, hydrogen in metallic glasses, modelling, and triplet correlation functions are discussed.