Integral equation theories closure approximation

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. 

The structure of the integral equation approach for calculating the angular pair correlation function g(ri2C0iC02) starts with the OZ integral equation [8.76] between the total (h) and the direct (c) correlation function, which is here schematieally rewritten as h=h[c] where h[c] denotes a functional of c. Coupled to that a second relation, the so-ealled closure relation c=c[h], is introduced. While the former is exact, the latter relation is approximated the form of this approximation is the main distinction among the various integral equation theories to be described below. 

The PRISM integral equation theory has been generalized for multicomponent polymer mixtures by Curro and Schweizer [23,59-63]. Much formally exact analysis can be carried out based simply on the structure of the PRISM matrix integral equations without specifying a particular closure approximation [61]. In this section these aspects are summarized. 

Polymer miscibility—with solvents, other polymers, or in block copolymers— can be treated with integral equation methods such as PRISM (255-257) or density functional theory (252,258). Theories of this kind inevitably required approximations that are difficult to assess by independent methods, with the variety of closure relations having been developed in integral equation theories (259) illustrating the case in point. Molecular modeling can provide the detailed molecular-level information, such as pair correlation fimctions, that is needed to assess the validity of these t5q>es of theories (260,261). 

A second, entirely different class of new polymer integral equation theories have been developed by Lipson and co-workers, Eu and Gan, " and Attard based on the site-site version of the Born-Green-Yvon (BGY) equation. The earliest work in this direction was apparently by Whittington and Dunfield, but they addressed only a special aspect of the isolated polymer problem (dilute solution). The central quantity in the BGY approaches is the formally exact expressions that relate two and three (or more) intramolecular and intermolecular distribution functions. The generalized site-site Ornstein-Zernike equations and direct correlation functions do not enter. In the BGY schemes the closure approximation(s) enter as approximate relations between the two- and three-body distribution functions supplemented with exact normalization and asymptotic conditions. In the recent BGY work of Taylor and Lipson a four-point distribution function also enters. 

Integral Equation and Eield-Theoretic Approaches In addition to theories based on the direct analytical extension of the PB or DH equation, PB results are often compared with statistical-mechanical approaches based on integral equation or density functional methods. We mention only a few of the most recent theoretical developments. Among the more popular are the mean spherical approximation (MSA) and the hyper-netted chain (HNC) equation. Kjellander and Marcelja have developed an anisotropic HNC approximation that treats the double layer near a flat charged surface as a series of discrete layers.Attard, Mitchell and Ninham have used a Debye-Hiickel closure for the direct correlation function to obtain an analytical extension (in terms of elliptic integrals) to the PB equation for the planar double layer. Both of these approaches, which do not include finite volume corrections, treat the fluctuation potential in a manner similar to the MPB theory of Outhwaite. 

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 ... 

Why should one go to all this trouble and do all these integrations if there are other, less complex methods available to theorize about ionic solutions The reason is that the correlation function method is open-ended. The equations by which one goes from the gs to properties are not under suspicion. There are no model assumptions in the experimental determination of the g s. This contrasts with the Debye-Htickel theory (limited by the absence of repulsive forces), with Mayer s theory (no misty closure procedures), and even with MD (with its pair potential used as approximations to reality). The correlation function approach can be also used to test any theory in the future because all theories can be made to give g(r) and thereafter, as shown, the properties of ionic solutions. 

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