Hin Hark Gan

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.

Application of the integral equation theory of polymers : distribution function, chemical potential, and mean expansion coefficient

A recursive integral equation for the intramolecular correlation function of an isolated linear polymer of N bonds is derived from the integral equations presented in the preceding paper. The derivation basically involves limiting the density of the polymer to zero so that polymers do not interact with each other, and thus taking into account the intramolecular part only. The integral equation still has the form of a generalized Percus–Yevick integral equation. The intramolecular correlation function of a polymer of N bonds is recursively generated by means of it from those of polymers of 2, 3,..., (N−1) bonds. The end‐to‐end distance distribution functions are computed by using the integral equation for various chain lengths, temperatures, and bond lengths in the case of a repulsive soft‐sphere potential. Numerical solutions of the recursive integral equation yield universal exponents for the mean square end‐to‐end distance in two and three dimensions with values which are close to the Flory results: 0.7...

Integral equation theory of polymers: Translational invariance approximation and properties of an isolated linear polymer in solution

In this paper, we continue investigations on the solution methods for the generalized Percus–Yevick equations for the pair correlation functions of polymers, which were formulated in the previous papers of this series [J. Chem. Phys. 99, 4084, 4103 (1993)]. Previously, they were reduced to recursive integral equations and solved numerically. In this paper, a translational invariance approximation is used to reduce the number of integral equations to solve. In this approximation, only N integral equations out of N2 integral equations are required for a polymer consisting of N beads (monomers). The behavior of an isolated polymer is studied with three different potential models, a soft sphere, a hard sphere, and a Lennard‐Jones potential. The main motivation for considering these three potential models is in testing the idea of universality commonly believed to hold for some properties of polymers. We find that the universality holds for the power law exponent for the expansion factor of polymers at high te...

Influence of the solvent on the conformation of a chain molecule

Effects of the solvent on the conformation of a polymeric chain molecule are examined by using a set of polymer–solvent integral equations for correlation functions for the polymer and the solvent. Solutions of the integral equations are used for computing the polymer–solvent distribution, chain conformations, and scaling properties associated with polymer swell and collapse in good and poor solvents. The variation of chain properties with the solvent density and the solvent quality is examined for chains having up to 100 bonds.

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 t...

On integral equations for the pair correlation function

The Kirkwood integral equations for triplet or higher-order correlation functions are examined with a view to obtain some useful approximate sets of integral equations for correlation functions. Based on a set where the potential energies are eliminated from the integral equations for triplet or higher-order correlation functions, the Percus-Yevick integral equation is obtained. The derivation is rather simple but requires a pair of assumptions. This derivation implies that the Percus-Yevick equation is essentially founded on the Kirkwood superposition approximation which is one of the assumptions. A couple of plausible extensions of the Percus-Yevick equation are then suggested for pair correlation function and their consequences are examined in the light of some exact relations for pair correlation function. One of such generalized equations is numerically solved to assess its accuracy.

Self‐consistent integral‐equation theory of chain‐molecular liquids. II. Improved intermolecular equations

Improved self‐consistent intermolecular integral equations for a chain‐molecular liquid are derived from the polymer Kirkwood hierarchy. The present work is a continuation of our recent work reported in a previous paper [J. Chem. Phys. 103, 2140 (1995)]. It is shown that the reference interaction site model (RISM) equation and extensions thereof can be obtained from the new intermolecular equations. The solutions of the new self‐consistent set of intra‐ and intermolecular equations are compared with computer simulation data for chains with repulsive interaction potentials whose lengths N vary from 4 to 100 sites. The intermolecular correlation functions obtained from simulations are accurately reproduced. Comparisons with simulation data for the pressure equation of state and excess chemical potential show that the predictions of the self‐consistent theory are accurate for packing fractions up to 0.4. These thermodynamic functions are found to scale as N for N≳16, implying that results obtained for short ...

Conformation and thermodynamic properties of repeated‐block copolymers

Conformational properties of isolated linear copolymers are studied by means of the integral equation theory of polymers. We examine two‐letter copolymers that have repeated‐block symmetry; the potential between like monomers is repulsive and that between unlike ones is a Lennard‐Jones potential. This class of copolymer sequences satisfies an approximate translational invariance symmetry for the correlation functions. Conformational behavior of any given copolymer is analyzed by computing its configurational and thermodynamic properties from the information of its sequence and potentials of interaction. All properties calculated show that they are independent of sequence heterogeneity at high temperatures. The influence of sequence heterogeneity becomes significant below the theta point. Sequences that are compact and have low excess entropy generally exhibit two key features: (a) their monomer composition is symmetric and (b) the unlike monomer species tend to alternate in the sequence. These conditions ...

Integral equation theory of single-chain polymers: Comparison with simulation data for hard-sphere and square-well chains

A systematic comparison of computer simulation data for linear hard-sphere and square-well chains with the results of single-chain integral equation is reported. The single-chain integral equation is derived from the polymer Kirkwood hierarchy for site–site or pair distribution functions. Quantities compared include radius of gyration, end-to-end distance, and internal energy. We examine chain lengths up to 1000 sites for hard-sphere chains. The radius of gyration and end-to-end distance from the theory are found to agree quantitatively with Monte Carlo simulation data. Results for square-well chains with the range λ=1.5 are compared with Monte Carlo and constant temperature molecular dynamics simulation data for chains having up to 64 sites. The radius of gyration and internal energy generally deviate from simulation data by about 10% for reduced temperatures greater than 1. The values of the radius of gyration at reduced temperatures below 1 are larger than those from simulations.

Model protein conformations via pair correlation functions, distance matrix, and embedding algorithm

A method of constructing three-dimensional structures of model protein conformations is presented. The method consists of self-consistent field integral equations for pair correlation functions of constituent units in a heteropolymer chain and the use of the distance matrix and the embedding algorithm for constructing conformations. The pair correlation functions obey integral equations that are derived from the Kirkwood hierarchy by applying closure approximations; they appear as a generalized form of the liquid-state Percus–Yevick integral equation. Model protein sequences that exhibit the formation of secondary-like patterns and tertiary-like structures are examined. These structural features are formed at low temperatures and they are stabilized by strong hydrogen bonding forces. To obtain the structure in three dimensions, the method of distance geometry is used to refine the distance matrix of a folded structure which is then embedded in the three-dimensional space by an embedding algorithm.