Pair correlation function ideal chain

The exact calculation of the pair correlation function of an ideal chain leads to an additional factor 3/tt ... 

This description is a manifestation of the self-similarity (fractal nature) of polymers, discussed in Section 1.4. The fractal nature of ideal chains leads to the power law dependence of the pair correlation function g(r) on distance r. This treatment for the ideal chain can be easily generalized to a linear chain with any fractal dimension V. The number of monomers... 

Tiencc, Eq. (2.121) is a special case of this result, with P — 2 for an ideal chain. The fractal dimension of a rod polymer is — 1 and the pair correlation function is g(r) ... 

The probability to find a monomer within a distance r of a given monomer is called the pair correlation function g(r). For ideal linear chains, g(r) is reciprocally proportional to the distance r ... 

Figure 3.7. Pair correlation function of segments q[r) in dilute solutions (a single chain) in ideal (/) 2Uid good (S) solvent, and in semidilute solutions [3) (de Gennes, 1979) [Heprinted from Pierre-Gilles de Gennes Scaling Concepts in Polymer Physics. Copyright 1979 by Cornell University. Used by permission of the publisher, Cornell University Press]...

A single chain at the compensation point Q has a quasi-ideal behavior. The size R scales like N, and the pair correlation function g r) decreases like 1 r (for r 7 ). However, the three-body repulsive interactions remain effective even at T = 6. Their effect (in three dimensions) is to introduce some correlation between the monomers. The probability of contact between two (or three) monomers is reduced by certain logarithmic factors. These factors could show up in certain measurements which are sensitive to local properties (e.g., specific heat) and possibly in certain optical properties. 

The objective of the polarization model is to relate the material parameters, such as the dielectric properties of both the liquid and solid particles, the particle volume fraction, the electric field strength, etc., to the rheological properties of the whole suspension, in combination with other micro structure features such as fibrillatcd chains. A idealized physical model ER system—an uniform, hard dielectric sphere dispersed in a Newtonian continuous medium, is usually assumed for simplification reason, and this model is thus also called the idealized electrostatic polarization model. The hard sphere means that the particle is uncharged and there are no electrostatic and dispersion interactions between the particles and the dispersing medium before the application of an external electric field. For the idealized electrostatic polarization model, there are roughly two ways to deal with the suspensions One is to consider the Brownian motion of particle, and another is to ignore the Brownian motion and particle inertia. For both cases the anisotropic structure of such a hard sphere suspension is assumed to be represented by the pair correlation function g(r,0), derived by... 

A second case of Interest Is when chromophores are attached randomly to an Isolated linear Ideal polymer chain. Here the pair correlation function is of the Debye form... 

Specific results require two additional pieces of information. First, the single diain polymer structure as contained in the species-dependent structure factors must Ik known. In principle, these correlation fimctions should be self-consis-tently computed along with the intermolecular pair correlations. This most rigorous approach is now feasible and is briefly discussed in Sect. 10. However, all published PRISM work on blends has employed the Flory ideality ansatz discussed in Sect. 2, i.e., the required functions are presumed given and effectively independent of blend composition and proximity to the spinodal. To date, published analytical and numerical work for alloys has employed coarsegrained models sudi as the Gaussian, freely-jointed, and Kmi-fiexible chain. Second, a closure approximation is required which relates the intermolecular pair and direct site-site correlation functions outside the hard core diameter. Both traditional atomic and novel moleculaF dosures have been studied. 

---