The Polymer RISM Theory

To describe the equilibrium structure of the system, one can use the polymer integral equation RISM theory [140,141], which allows one to find collective correlation functions. For AB copolymers, the polymer RISM equation is represented in the matrix form [142,143] 

H and C are symmetric matrices whose elements are the partial total hap(r) and direct cap,(r) pair correlation functions a,ft = A,B) W is the matrix of intramolecular correlation functions wap r) that characterize the conformation of a macromolecule and its sequence distribution and p is the average number density of units in the system. Equation 17 is complemented by the closure relation corresponding to the so-called molecular Percus- 

ur/p r) describes the interaction between nonbonded units and oa is the effective unit size ( y = erg = a). In [153,154], the Yukawa potential was used 

When the block length becomes comparable with Ny distinctions between the behaviors of two copolymers practically disappear. At L 200, one observes that T Ly with y = 4/3. In this case, the characteristic scale of the microdomain structure behaves as r Ls with 5 = 1/2. This dependence is caused by the fact that flexible chains in the melt have a Gaussian conformation, and the average size of any chain section of n units is proportional to ft1/2 [75]. Hence, for sufficiently large Vs, the spatial scale of microinhomogeneities in the system is determined only by the block size. However, the behavior of the random-block copolymer at L 102 is more complicated. In particular, r has a minimum at L 10. 

Therefore, the PRISM calculations [153] show that the protein-like copolymers with A-A attractions are more inclined to self-organization than their random-block counterparts and regular multiblock copolymers with the same chemical composition and the same chain length. 

27 The values of T and r as a function of block length L for the regular and random-block copolymers at the volume fraction of macromolecules 0 = 0.8. Adapted from [153] 

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The solute solvent contribution to the free energy stabilizing the DNA poly ion can be calculated within the polymer RISM theory by a charging up process " ... 

What is most important for our discussion is the fact that the spatial scale r of the segregated structure for protein-like copolymers is appreciably larger than that for random-block copolymers with the same composition and the same average block length. Also, MIST in the protein-like copolymer system occurs at a temperature higher than that of the random-block system, which is in agreement with the prediction of the polymer RISM theory [153]. 

The polymer RISM theory relates the set of total correlation functions (r) = g(r)-l to the set of site-site direct correlation functions, (c(r), and the set of intramolecular distribution functions, w(r), via the nonlinear site-site Ornstein-Zemike-like (SSOZ) integral matrix equation [41]... 

Equations (2.3)-(2.5) lead to v v + l)/2-coupled integral equations that make up the polymer RISM theory in its simplest form appropriate for dense, repulsive force polymer fluids. The integral equations can be solved numerically using a variety of standard techniques. " ... 

F. Hirata and R. M. Levy, /. Phys. Chem., 93, 479 (1989). Salt-Induced Conformational Changes in DNA Analysis Using the Polymer 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... 

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. 

The outline of the paper is as follows. In Sect. 2 we describe the basic RISM and PRISM formalisms, and the fundamental approximations invoked that render the polymer problem tractable. The predicticms of PRISM theory for the structure of polymer melts are described in Sect. 3 for a variety of single chain models, including a comparison of atomistic calculations for polyethylene melt with diffraction experiments. The general problem of calculating thermodynamic properties, and particularly the equation-of-state, within the PRISM formalism is described in Sect. 4. A detailed application to polyethylene fluids is summarized and compared with experiment. The develojanent of a density functional theory to treat polymer crystallization is briefly discussed in Sect. 5, and numerical predictions for polyethylene and polytetrafluoroethylene are summarized. 

We have generalized the RISM formalism to describe the structure of flexible polymer chain liquids [11-15]. The application of RISM theory to polymers is... 

In essence, CPMD is a fairly typical example of the hierar-chy-of-models approach in which the electrons are treated quantum-mechanically as a massless fluid while nuclei are described as a set of classical particles. A wide scope for the application of hybrid simulations is provided by problems in polymer science. There are hybrid methods coupling a classical molecular-dynamics model with an elastic continuum modd, " RISM theory," " " and so on. The... 

Beginning in 1987, we and our co-workers have extended and widely applied the RISM concepts to the case of flexible polymer solutions and melts, polymer mixtures or blends, and block copolymers. We generically refer to this work as polymer RISM, or PRISM, theory. The connection of the elementary aspects of PRISM theory with the quantum electron work has been discussed. ... 

Numerical studies of chain molecule fluids have also been carried out by Yethiraj" using the considerably more complicated diagrammatically proper formulation of RISM theory due to Chandler et al. " Novel, even more complicated closures have been recently proposed by several workers, but numerical predictions for polymer fluids have not been established. 

Although we believe much progress has been made based on the PRISM theory approach, there remain important basic theoretical issues that require continuing attention in the future. The most obvious is the question of closure approximation. Even for the purely repulsive or hard-core polymer fluid, improved closures are desirable. Results for diatomic and polymer fluids based on the diagramatically proper Chandler-Silbey-Ladanyi (CSL) formulation of RISM theory have been obtained by Yethiraj based on the site-site PY closure. Unfortunately, for chain molecules this approach does not represent an... 

Taking into account that polyolefins can be blended in spite of only small van der Waals enthalpic interactions, in a series of papers theoretical models were developed to explain the miscibility in such nearly athermal and/or athermal polymeric mixtures. Thus Schweizer et al. [85] found by extending the RISM ( reference interaction side model ) theory, that structural asymmetry between the polymer components leads to negative interaction parameters because of significant noncombinatorial mixing entropy contributions which stabilize the polymer blend by spatially... 

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. 

The PRISM theory of Curro and Schweizer extends RISM theory to polymers by considering the intramolecular structure of flexible polymers [14, 15, 16, 69, 70, 17, 18]. The theory assumes that the Flory ideality concept is valid and polymers exhibit ideal behavior in the melt phase. This is justified by the fact that the intramolecular excluded volume is nearly balanced by intermolecular excluded volume when a chain is surrounded by identical chains, so excluded volume forces can be neglected [3, 71]. They also developed a perturbative scheme to account for chain end effects [15]. 

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