Polymer reference site interaction model

PRISM (polymer reference interaction-site model) method for modeling homopolymer melts... 

If each polymer is modeled as being composed of N beads (or sites) and the interaction potential between polymers can be written as the sum of site-site interactions, then generalizations of the OZ equation to polymers are possible. One approach is the polymer reference interaction site model (PRISM) theory [90] (based on the RISM theory [91]) which results in a nonlinear integral equation given by... 

PRISM polymer-reference-interaction-site model... 

The PRISM (Polymer-Reference-Interaction-Site model) theory is an extension of the Ornstein-Zernike equation to molecular systems [20-22]. It connects the total correlation function h(r)=g(r) 1, where g(r) is the pair correlation function, with the direct correlation function c(r) and intramolecular correlation functions (co r)). For a primitive model of a polyelectrolyte solution with polymer chains and counterions only, there are three different relevant correlation functions the monomer-monomer, the counterion-counterion, and the monomer-counterion correlation function [23, 24]. Neglecting chain end effects and considering all monomers as equivalent, we obtain the following three PRISM equations for a homogeneous and isotropic system in Fourier space ... 

In this section we introduce integral equation theories (IETs) and approximate closures applicable for various models of polyelectrolyte solutions. A theory for linear polyelectrolytes based on the polymer reference interaction site model has also been proposed [58, 59], but this approach will not be reviewed here. 

To describe the solubility of sodium poly(styrene sulfonate) in the presence of z-valent counterions, the electrostatic model [18,19] was introduced. This model takes into account a short-range electrostatic attraction between negatively charged monomers (1) and those carrying condensed multivalent counterions that are positively charged (z 1). Phase diagrams have been calculated using two techniques the polymer reference interaction site model (PRISM) [18] and the random phase approximation (RPA) [19]. Both techniques lead to the spinodal equation... 

Polymer Reference Interaction Site Model, PRISM... 

Preparation of blends 13-24, 214, 251, 252, 276, 324, 342-350, 640, 1024-1032,1128, 1151,1337 PRISM (Polymer reference interaction site model) 166,167 Probability density function 166... 

The infortnation provided in this chapter can be divided into four parts 1. introduction, 2. thermodynamic theories of polymer blends, 3. characteristic thermodynamic parameters for polymer blends, and 4. experimental methods. The introduction presents the basic principles of the classical equilibrium thermodynamics, describes behavior of the single-component materials, and then focuses on the two-component systems solutions and polymer blends. The main focus of the second part is on the theories (and experimental parameters related to them) for the thermodynamic behavior of polymer blends. Several theoretical approaches are presented, starting with the classical Flory-Huggins lattice theory and, those evolving from it, solubility parameter and analog calorimetry approaches. Also, equation of state (EoS) types of theories were summarized. Finally, descriptions based on the atomistic considerations, in particular the polymer reference interaction site model (PRISM), were briefly outlined. 

The older mean-field theories are often based on the lattice model of ill-defined a priori size and shape, neglecting variability of monomer structures in PO copolymers and blends. Several newer approaches have been proposed, viz., the polymer reference interaction site model (PRISM) (Schweizer and Curro 1989, 1997), the Monte Carlo (MC) simulations (Sariban and Binder 1987 Muller and Binder 1995 Weinhold et al. 1995 Escobedo and de Pablo 1999), the continuum field theory (CFT) (Fredrickson et al. 1994), or an analytical lattice models (Dudowicz and Freed 1991). The latter model leads to relatively simple mathematical expressions, which offer an insight into basic thermodynamics, but again do not predict how monomer structure affects blend miscibility. 

Over the p t several years we and our collaborators have pursued a continuous space liquid state approach to developing a computationally convenient microscopic theory of the equilibrium properties of polymeric systems. Integral equations method [5-7], now widely employed to understand structure, thermodynamics and phase transitions in atomic, colloidal, and small molecule fluids, have been generalized to treat macromolecular materials. The purpose of this paper is to provide the first comprehensive review of this work referred to collectively as Polymer Reference Interaction Site Model (PRISM) theory. A few new results on polymer alloys are also presented. Besides providing a unified description of the equilibrium properties of the polymer liquid phase, the integral equation approach can be combined with density functional and/or other methods to treat a variety of inhomogeneous fluid and solid problems. 

Recent studies have shown that structural differences among components, eg differences in tacticity, affect phase behavior (33-35). The inability of the Flory-Huggins theory to account for these experimental observations is mainly a result of neglecting the particular molecular structure of the pol3rmer chains. Extensive studies have been imdertaken in order to be able to link polymer phase behavior to molecular structure (36). The polymer reference interaction site model (PRISM) represents one of the successful approaches in imderstanding the relationship between molecular structin-e, local packing and thermodynamics. 

The static conformations on scales larger than the persistence length or the average spacing between nonbonded nearest neighbors do not depend on the specific model. For shorter distances differences occur and the lattice models are somewhat less realistic than the continuous space models. The MD simulations of Kremer and Grest are described very well by the PRISM (polymer reference interaction site model) model of Curro and Schweizer. ... 

We also note that the same Monte Carlo data have helped to sort out an inadequate approximation in the context of the polymer reference interaction site model (PRISM) theory, which yielded a relation Tc oc /N while now oc is generally accepted. It has been very difficult to provide convincing experimental evidence on this issue— true symmetrical monodisperse polymer mixtures hardly exist, and the temperature range over which Tc N) can be studied is limited by the glass transifion temperature from below and by chemical instability of the chains from... 

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