Polymer Reference Site Interaction Model PRISM

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... 

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... 

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. 

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... 

More modem approaches borrow ideas from the liquid state theory of small molecule fluids to develop a theory for polymers. The most popular of these is the polymer reference interaction site model (PRISM) theory " which is based on the RISM theory of Chandler and Andersen. More recent studies include the Kirkwood hierarchy, the Bom-Green-Yvon hierarchy, and the perturbation density functional theory of Kierlik and Rosinbeig. The latter is based on the thermodynamic perturbation theory of Wertheim " where the polymeric system is composed of very sticky spheres that assemble to form chains. For polymer melts all these liquid state approaches are in quantitative agreement with simulations for the pair correlation functions in short chain fluids. With the exception of the PRISM theory, these liquid state theories are in their infancy, and have not been applied to realistic models of polymers. 

The central problem in the liquid state theory of polymers is the determination of a)ay(r) and gay(r). This cannot be done exactly (except by a many-molecule computer simulation) and many approximate schemes have been proposed which extend theories for simple liquids to polymers. Perhaps the most widely used approach is that of Curro and Schweizer, " known as the polymer reference interaction site model (PRISM) theory which is based on the RISM theory of Chandler and Andersen, I will describe this approach and compare its predictions to computer simulations and other theories. 

An alternative approach that combines the Gaussian thread model of polymers with liquid-state theory is known as the polymer reference interaction site model (PRISM) approach [34-38[. This approach has the merit that phenomena such as the de Gennes [3] correlation hole phenomena and its consequences are incorporated in the theoretical description, and also one can go beyond the Gaussian model for the description of intramolecular correlations of a polymer chain, adding chemical detail (at the price of a rather cumbersome numerical solution of the resulting integral equations) [37,38[. An extension to describe the structure of colloid-polymer mixtures has also become feasible [39, 40]. On the other hand, we note that this approach shares vhth other approaches based on liquid state theories the difficulty that the hierarchy of exact equations for correlation functions needs to be decoupled via the so-called closure approximation [34—38]. The appropriate choice of this closure approximation has been a formidable problem [34—36]. A further inevitable consequence of such descriptions is the problem that the critical behavior near the critical points of polymer solutions and polymer blends is always of mean-field character ... 

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