Reference Interaction Site Model theory

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 ... 

Statistical thermodynamics is also changing in recent years. Newly developed RISM (Reference Interactions Site Model) theory has no restriction on the shape of solute species, in contrast to old theories in which spherical species are usually assumed. Ab initio calculations are being combined with molecular dynamic simulations. This combination becomes possible because of the improvement of high-speed computers. The polarization effect and multibody problem will be... 

Kinoshita, M. Okamoto, Y. Hirata, F. (1997) Solvation structure and stability of peptides in aqueous solutions analyzed by the reference interaction site model theory. Journal of Chemical Physics 107, 1586-1599... 

An alternative theoretical approach to determining gas solubility utilizes the reference interaction site model theory. This theory has been applied for the noble gases and is found to give reasonable agreement to experimental data... 

Kinoshita M, Hirata F (1997) Analysis of salt effects on solubility of noble gases in water using the reference interaction site model theory. J Chem Phys 106 5202-5215... 

T. Imai, A. Kovalenko and F. Hirata. Partial molar volume of proteins studied by the three-dimensional reference interaction site model theory. J. Phys. Ghem. B 109, 2005, 6658-6665. 

One important class of integral equation theories is based on the reference interaction site model (RISM) proposed by Chandler [77]. These RISM theories have been used to smdy the confonnation of small peptides in liquid water [78-80]. However, the approach is not appropriate for large molecular solutes such as proteins and nucleic acids. Because RISM is based on a reduction to site-site, solute-solvent radially symmetrical distribution functions, there is a loss of infonnation about the tliree-dimensional spatial organization of the solvent density around a macromolecular solute of irregular shape. To circumvent this limitation, extensions of RISM-like theories for tliree-dimensional space (3d-RISM) have been proposed [81,82],... 

The integral equation method is free of the disadvantages of the continuum model and simulation techniques mentioned in the foregoing, and it gives a microscopic picture of the solvent effect within a reasonable computational time. Since details of the RISM-SCF/ MCSCF method are discussed in the following section we here briefly sketch the reference interaction site model (RISM) theory. 

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... 

The original OZ equation can be applied only to liquids composed of spherically symmetrical particles, namely atoms. Chandler and Andersen proposed one of the possible extensions to treat general polyatomic cases, referred to as the reference interaction site model (RISM) or site-site OZ (SSOZ) theory [4], This has been further extended by Hirata and Rossky to be applicable to polar molecules such as water [5], In the RISM theory,... 

Sato presents an alternative method to both continuum solvation models and hybrid QM/MM or ONIOM approaches. This is represented by the reference interaction site model (RISM) formalism when combined to a QM description of the solute to give the RISM-SCF theory. 

This point of interest is brought forward by the RISM approach to the structure of molecular liquids, and a RISM model with HNC closure supports a similar result for the excess chemical potential in terms of atom-atom correlations (Singer and Chandler, 1985 Hirata, 1998). RISM - reference interaction site model - is an acronym that refers to a class of theories for the joint two-atom distributions in molecular liquids. The most basic decision of RISM models is that theories of molecular liquids should focus first on the atom-atom distributions extracted from X-ray and neutron scattering data rather than more complex possibilities this highly practical point was not so obvious in an earlier epoch when models of molecular liquids were scarcely realistic on an atomic scale. That basic decision was encapsulated by invention of a site-site (or atom-atom) Ornstein-Zernike (SSOZ) (Cummings and Stell, 1982) equation that involved intramolecular atom-atom correlations. The original suggestions (Chandler and Andersen, 1972) were sufficiently successful as to support subsequent flamboyant developments, and to be substantially impervious to more fundamental improvements (Chandler et al, 1982). For these reasons a full discussion of the RISM models wouldn t fit here. Fortunately, a devoted exposition of current RISM work is already available (Hirata, 1998). 

Integral equation ideas on the structure of monatomic liquids were first modified and applied to molecular liquids by Chandler and Andersen, Their classic work is now referred to as the reference interaction site model (RISM) of molecular liquids. Polymer RISM (PRISM) is essentially an extension of RISM theory that successfully describes the structure of flexible polymer chains in the liquid state. 

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. 

Another, a similar route that considers interactions between individual elements of a system, is the reference interaction site model, RISM. The theory involves computations of the system structure by means of the probability density function, which describes location of all N particles of the system. The binary interactions define the pair-density function ... 

As concerns the site-site approach the most important theory is the reference interaction site model , or RISM. This method applies to an intermolecular pair potential modeled by a site-site form, i.e., V(rc0iC02 ) = 5 apV p (r ) and its original intuitive derivation is based on exploring the possibility of decomposing g(r o),a)2) also in the same form, i.e., as a sum of site-site gap(r)s. 

In the weak-segregation regime, the phase behavior of a polymer melt composed of flexible-chain macromolecules can be described on the basis of the random-phase approximation (RPA) or the polymer integral equation reference interaction site model (pRISM) theory that allow finding the conditions under which the spatially homogeneous state of the system becomes unstable. 

Kezic, B. and A. Perera. 2011. Towards a more aceurate reference interaction site model integral equation theory for molecular liquids. Journal of Chemical Physics. 135, 234104. Kim, J. I. 1978. A critical study of the Ph4AsPh4B assumption for single ion thermodynamics in amphiprotic and dipolar-aprotic solvents Evaluation of physical parameters relevant to theoretical consideration. Zeitschriftfur Physikalische Chemie. 113,129. 

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. 

These integral equation ideas of monatomic liquids were generalized and applied to molecular liquids by Chandler and Andersen [6,8] to formulate the Reference Interaction Site Model or RISM theory of molecular fluids. In the RISM approach, each molecule is subdivided into spherically symmetric, interaction sites. The intermolecular pair structure of a uniform molecular liquid of M molecules is now specified through a site-site radial distribution function matrix gaY(r) ... 

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