Polymers integral equation theories

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. 

A third class of new polymer integral equation theories have been proposed by Kierlik and Rosinberg. Their work is an extension of a density functional theory of inhomogeneous polyatomic fluids to treat the homogeneous phase. The Wertheim thermodynamic pertubation theory of polymerization is employed in an essential manner. Applications to calculate the intermolecular structure of rather short homopolymer solutions and melts have been made. Good results are found for short chains at high densities, but the authors comment that their earlier theory appears to be unsuited for long chains at low to moderate (semidilute) densities. ... 

Sohweizer K S and Curro J G 1997 Integral equation theories of the struoture, thermodynamios and phase transitions of polymer fluids Adv. Chem. Phys. 98 1... 

Heine, D. R., Grest, G. S. and Curro,J. G. Structure of Polymer Melts and Blends Comparison of Integral Equation theory and Computer Sumulation. Vol. 173, pp. 209-249. 

Integral equation theories are widely used in the theoretical study of liquids. There are two broad classes of integral equation theories those based on the Bom-Green-Yvon (BGY) hierarchy and those based on the Omstein-Zemike (OZ) equation [88]. Although the formalism is exact in both classes, it is generally easier to fashion approximations in the case of the OZ-equation-based approach, and this type of theory has therefore been more popular. Surprisingly, the BGY approach has never been implemented for nonuniform polymers, and this section is therefore restricted to a discussion of the OZ-equation-based approach. 

The integral equation theory is a simple means of studying the density profiles of dense polymer melts at surfaces where the structure is dominated by... 

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]... 

Moreover the components of vector A change to become Aa = aA/vA, etc. An Ornstein-Zemike (OZ) approach (referred to as the integral equation theory) describing multicomponent compressible polymer blend mixtures has been extensively investigated [35]. The multicomponent OZ equation relates the direct correlations matrix C and the total (i.e., direct and indirect) correlations matrix H as ... 

K. S. Schweizer and J. G. Curro, Phys. Rev. Lett., 58, 246 (1987). Integral-Equation Theory of the Structure of Polymer Melts. 

Comparison between Integral Equation Theory and Molecular Dynamics Simulations of Dense, Flexible Polymer Liquids. 

H. P. Deutsch and K. Binder, Europhys. Lett., 17, 697 (1992). Evidence Against the Integral Equation Theory of Polymer Blends. 

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. 

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. 

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