Reference interaction site approximation

Theories based on the solution to integral equations for the pair correlation fiinctions are now well developed and widely employed in numerical and analytic studies of simple fluids [6]. Furtlier improvements for simple fluids would require better approximations for the bridge fiinctions B(r). It has been suggested that these fiinctions can be scaled to the same fiinctional fomi for different potentials. The extension of integral equation theories to molecular fluids was first accomplished by Chandler and Andersen [30] through the introduction of the site-site direct correlation fiinction c r) between atoms in each molecule and a site-site Omstein-Zemike relation called the reference interaction site... 

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... 

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. 

Among the applications of integral equation theory to solutions, Koga et al. (1996) used a reference interaction-site model (RISM), an approximation due to Chandler and Andersen (1972), to calculate the local solvation behavior of naphthalene in supercritical carbon dioxide. They used a ten-site model for naphthalene and a three-site model for carbon dioxide, in which partial charges were placed on the sites in addition to Lennard-Jones potentials acting between sites on different molecules. Their potential function was a sum over the ten naphthalene sites and the three carbon dioxide sites of terms of the form of Equation 5.11 ... 

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... 

FH = Flory-Huggins GF = generalized Flory GFD = generalized Flory dimer HNC = hypemetted chain HTA = high temperature approximation IFJC = ideal freely joined chain ISM = interaction site model LCT = lattice cluster theory MS = Martynov-Sarkisov PMMA = polymethyl methacrylate PRISM = polymer reference interaction site model PVME = polyvinylmethylether PS = polystyrene PY = Percus-Yevick RMMSA = reference molecule mean spherical approximation RMPY = reference molecular Percus-Yevick SANS = small angle neutron scattering SFC = semiflexible chain TPT = thermodynamic perturbation theory. 

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 ... 

The MFA [1] introduces the perturbation due to the solvent effect in an averaged way. Specifically, the quantity that is introduced into the solute molecular Hamiltonian is the averaged value of the potential generated by the solvent in the volume occupied by the solute. In the past, this approximation has mainly been used with very simplified descriptions of the solvent, such as those provided by the dielectric continuum [2] or Langevin dipole models [3], A more detailed description of the solvent has been used by Ten-no et al. [4], who describe the solvent through atom-atom radial distribution functions obtained via an extended version of the interaction site method. Less attention has been paid, however, to the use of the MFA in conjunction with simulation calculations of liquids, although its theoretical bases are well known [5]. In this respect, we would refer to the papers of Sese and co-workers [6], where the solvent radial distribution functions obtained from MD [7] calculations and its perturbation are introduced a posteriori into the molecular Hamiltonian. 

The strategy for explicitly formulating the molecular closures was guided by three considerations." (1) Use of the commonly employed reference approach. The successful site-site PY closure is retained to describe the repulsive force reference fluid but a molecular closure scheme is adopted to describe the attractive, slowly varying forces. (2) The approximation scheme is required to provide an exact description of the structural consequences of the tail potentials at the two-molecule level in the weak coupling limit [/3umm-W 1]- (3) Use of an appropriate site-site approximation for the direct attractive interaction contribution motivated by experience in simple fluids. ... 

Taken as a whole, the ideas discussed led Yethiraj and Schweizer to propose the following reference molecular closure approximations for site interaction potentials consisting of a hard core plus tail" ... 

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