Polymer fluids correlation functions

Integral equations can also be used to treat nonuniform fluids, such as fluids at surfaces. One starts with a binary mixture of spheres and polymers and takes the limit as the spheres become infinitely dilute and infinitely large [92-94]. The sphere polymer pair correlation function is then simply related to the density profile of the fluid. 

If go(r), g CrX and g (r) are known exactly, then all three routes should yield the same pressure. Since liquid state integral equation theories are approximate descriptions of pair correlation functions, and not of the effective Hamiltonian or partition function, it is well known that they are thermodynamically inconsistent [5]. This is understandable since each route is sensitive to different parts of the radial distribution function. In particular, g(r) in polymer fluids is controlled at large distance by the correlation hole which scales with the radius of gyration or /N. Thus it is perhaps surprising that the hard core equation-of-state computed from PRISM theory was recently found by Yethiraj et aL [38,39] to become more thermodynamically inconsistent as N increases from the diatomic to polyethylene. The uncertainty in the pressure is manifested in Fig. 7 where the insert shows the equation-of-state of polyethylene computed [38] from PRISM theory for hard core interactions between sites. In this calculation, the hard core diameter d was fixed at 3.90 A in order to maintain agreement with the experimental structure factor in Fig. 5. 

Finally, we mention an interesting recent study by Chandler that extended the Gaussian field-theoretic model of Li and Kardar to treat atomic and polymeric fluids. Remarkably, the atomic PY and MSA theories were derived from a Gaussian field-theoretic formalism without explicit use of the Ornstein-Zernike relation or direct correlation function concept. In addition, based on an additional preaveraging approximation, analytic PRISM theory was recovered for hard-core thread chain model fluids. Nonperturbative applications of this field-theoretic approach to polymer liquids where the chains have nonzero thickness and/or attractive forces requires numerical work that, to the best of our knowledge, has not yet been pursued. 

This form for the correlation function fits a wide variety of relaxation data in complex fluids and, as in the case of many other properties, turns out to have a value close to 0.5. This form for a correlation function is often interpreted in terms of a distribution of relaxation times however in the case of model polyethylene it has been shown that the results cannot be explained in this way. Instead the result is interpreted in terms of a model for anisotropic motion in which the polymer chain is confined to a pipe formed by its neighbors. ... 

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. 

We will now discuss expansion functions of the general pair correlation function which are controlled by orientation correlations, starting with the function which represents the relative orientation of vector quantities such as the dipolar moments of molecular groups. The spatial integral over this function is directly related to the Kirkwood correlation factor, g, well-known from the theory of dielectric relaxation. The Kirkwood correlation factor was found to be of the order of 1 to 3 for strongly polar fluids such as water, methanol, etc., and to be of the order of 1 or less than 1 for less polar fluids, including nematic fluids. Similar results were obtained for amorphous polymers such as poly(methyI methacrylate), where the correlation factor is less than 1... 

FIGURE 7.23 Experimental evidences of the logarithmic decay of the density correlation functions. In the left panel, a colloid-polymer mixture with constant colloid volume fraction

fluid state between the two glasses, and the upper one is an attractive glass [39]. (From Pham et al. 2004. Phys. Rev. E 69 011503. With permission.) In the right panel, micellar systems where the attraction is induced by increasing temperature are studied [67]. (From Chen S.-H. et al. 2003. Science 300 619-622. With permission.)... 

---