Intramolecular correlation function

The equations that describe a blend or mixture of homopolymers is similar to that shown for multisite systems above. The only difference is that the matrix of intramolecular correlation functions is different and is given by... 

Combining the above results, we write the intramolecular correlation function as... 

Here, 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-... 

The matrix elements of the total correlation function, h, are related to all pairs of atoms. The intramolecular correlation function, to, introduced here represents the shape of the molecule. 8(r) in the diagonal element is the Dirac delta function and represents the position of an atom. The function appearing in the off-diagonal element is given by,... 

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 ... 

The new function w appearing in the derivation of (10.11) is called the intramolecular correlation function, which is defined for a pair of atoms a and 7 in a molecule by... 

Here ujxx (k) is the Fourier transform of the solute intramolecular correlation function, and... 

As we discussed in Section II.B, site-site correlation functions provide a very useful formalism for describing the structure of fluids modeled with interaction site potentials. In this formalism, information equivalent to g l,2) is obtained from the set of site-site correlation functions and intramolecular correlation functions. For this reason, a great deal of effort has been put into the development of integral equation theories for these correlation functions. The seminal contribution in this area was made by Chandler and Andersen, who sought to write an integral equation of the Ornstein-Zernike form in which the set of site-site total correlation functions were related to a set of site-site direct correlation functions. Their equation has the form... 

For large N, this equation is expected to behave similarly to that for an infinite linear chain, although the convergence depends upon the form chosen for the intramolecular correlation functions, IF,y(r). Equation (8.1.1) can be generalized to the case of a single ring in a solvent of spheriqal molecules... 

In polymer melts, the intramolecular correlation functions can be determined in a rather simple fashion by invoking the Flory ideality hypothesis, This hypothesis states that the intramolecular forces that tend to expand a chain are nominally cancelled by the intermolecular forces that tend to contract the chain. As a result the chain statistics follow those of a random walk and can be determined a priori. 

Solution of the PRISM equation for a particular system requires determining the intramolecular correlation functions. Day (k), and specifying a closure approximation. For rigid molecules, Qay (k) can be determined analytically. For example, the... 

These models provide a simple means of determining the intramolecular correlation function even for multicomponent systems, but the assumption of a freely jointed chain severely limits their predictive capabilities. Although they perform adequately over long length scales, MD simulations have shown that these models are incapable of capturing structural details on a monomeric length scale [76]. A more accurate... 

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