The Excluded Volume Problem in the Polymer Reference Interaction Site Model

Abstract

An intramolecular distribution function for lattice chains, accounting for the long range excluded volume, has been derived employing an Ornstein-Zernike-like integral equation approach, initially investigated for off-lattice chains by Curro, Blatz, and Pings. The intramolecular distributions, obtained for 16-, 30-, and 60-mers, are compared to theoretical results for the freely jointed chain model and to Monte Carlo simulation data. The new single chain intramolecular distribution function accurately matches the Monte Carlo data whereas the ideal chain structure factor overestimates the short-range intramolecular correlations. Upon increasing density the simulated intramolecular distributions gradually evolve toward the ideal chain results although the latter are never actually reached. The intermolecular distributions for athermal and interacting 30-mers at several densities have been calculated by combining the investigated intramolecular distribution functions with the lattice-PRISM theory. Furthermore, the equation of state for athermal 20-and 30-mers and the liquid-vapor spinodal for 16-mers have been investigated. It is found that the ideal chain distribution function results in unphysical thermodynamic behavior, reflected in the equation of state as well as in the liquid-vapor spinodal. The predicted intermolecular distribution is quite reasonable, considering the simplicity of the ideal chain model. The combination lattice-PRISM/excluded volume intramolecular distribution function offers at least qualitatively correct results both for the thermodynamic properties and intermolecular correlations.

Integral Equation Prediction of the Structure of Alternating Copolymer Nanocomposites near a Substrate

“…When the PRISM theory is extended to inhomogeneous systems, ,,,, the structure of the PNCs near a substrate can be described in a modified manner. The approximate closure equation can be improved through introducing a bridge functional to enhance the theoretical self-consistency and predictive accuracy .…”

“…Although experimental and simulation work has been performed to investigate the morphology of CPNs near a substrate, − the influence of parameters of interaction and chemistry on the microscopic dispersion and aggregation mechanism of alternating copolymer nanocomposites (ACNs) is still poorly understood; and experimental techniques become difficult in separating the different contributions from different components, and molecular dynamics (MD) simulation also has the problem of being computationally intensive in simulating asymmetric PNCs under confinement. ,,, Meanwhile, the polymer reference interaction site model (PRISM) ,,,− ,,,,− theory has been widely used to investigate the structure and properties of polymer melts, solutions, blends, and copolymers. It has also been extended to model the structure, effective interactions, and phase separation of PNCs and give some quantitative predictions with small angle scattering experiments in some realistic PNCs systems. ,− Moreover, by constructing a local bridge functional obtained from the density functional theory (DFT) in the traditional approximate closure equation, , the inhomogeneous PRISM theory was further extended to describe the structure and properties of inhomogeneous systems with a quantitative description of density distributions of nanoparticle/polymer blends, real polymer systems, and microphase separation behavior of polymers near a substrate. ,,,− Therefore, the modified inhomogeneous PRISM theory provides us with a desirable theoretical tool to investigate the structure and density profiles of PNCs near a substrate .…”

Statistical thermodynamics of polymer solutions

“…where 11 , 12 , and 22 are the energies associated with these respective pair contacts, and n 11 , n 12 , and n 22 are the number of these respective pair contacts in which n 12 ϭl. It satisfies 2n 22 ϩn 12 ϩn 02 ϭ͑zϪ2 ͒r 2 N 2 ϩ2N 2 , 2n 11 ϩn 12 ϩn 01 ϭzN 1 , 2n 00 ϩn 01 ϩn 02 ϭzN 0 ; ͑14͒…”