PRISM theory

Schweizer, K. S. Prism Theory of the Structure, Thermodynamics, and Phase Transitions of Polymer Liquids and Alloys. VoL 116, pp. 319-378. 

If each polymer is modeled as being composed of N beads (or sites) and the interaction potential between polymers can be written as the sum of site-site interactions, then generalizations of the OZ equation to polymers are possible. One approach is the polymer reference interaction site model (PRISM) theory [90] (based on the RISM theory [91]) which results in a nonlinear integral equation given by... 

The wall-PRISM theory has been extended to multiple site models [95], A simple example of a multiple-site model is a vinyl polymer (e.g., polypropylene) where there are three types of united-atom sites corresponding to CH2, CH, and CH3 groups. Ignoring end effects as before the PRISM equations take the form... 

For the united atom models of realistic polymers the wall PRISM theory predicts interesting structure near the surface [95]. For example, the side chains are found preferentially in the immediate vicinity of the surface and shield the backbone from the surface. This behavior is expected from entropic considerations. Computer simulations of these systems would be of considerable interest. 

Recently the wall-PRISM theory has been used to investigate the forces between hydrophobic surfaces immersed in polyelectrolyte solutions [98], Polyelectrolyte solutions display strong peaks at low wavevectors in the static structure factor, which is a manifestation of liquid-like order on long lengths-cales. Consequently, the force between surfaces confining polyelectrolyte solutions is an oscillatory function of their separation. The wall-PRISM theory predicts oscillatory forces in salt-free solutions with a period of oscillation that scales with concentration as p 1/3 and p 1/2 in dilute and semidilute solutions, respectively. This behavior is explained in terms of liquid-like ordering in the bulk solution which results in liquid-like layering when the solution is confined between surfaces. In the presence of added salt the theory predicts the possibility of a predominantly attractive force under some conditions. These predictions are in accord with available experiments [99,100]. 

The wall-PRISM theory has also been implemented for binary polymer blends. For blends of stiff and flexible chains the theory predicts that the stiffer chains are found preferentially in the immediate vicinity of the surface [60]. This prediction is in agreement with computer simulations for the same system [59,60]. For blends of linear and star polymers [101] the theory predicts that the linear polymers are in excess in the immediate vicinity of the surface, but the star polymers are in excess at other distances. Therefore, if one looks at the integral of the difference between the density profiles of the two components, the star polymers segregate to the surface in an integrated sense, from purely entropic effects. 

Figure 12. Comparison of predictions of the (a) wall-PRISM theory [94] and (b) density functional theory of Yethiraj and Woodward [37] to Monte Carol simulations for the density profile of 20-mers at hard walls for various packing fractions (as marked). The density profiles are normalized to the average value in the cell.

The accuracy of the theory does not diminish when the external field has a slowly varying component in addition to the hard-core repulsion. Figure 13 compares predictions of the wall-PRISM theory and the YW theory to computer simulations of freely jointed hard chains at surface that include a hard-core plus a soft component that is, the external field is given by... 

Figure 13. Comparison of prediction of the wall-PRISM theory with the hypernetted chain...

Figure 16. Comparison of the wall-PRISM theory to computer simulations for the density profiles of fused-hard-sphere chains at a hard wall for N = 16.

Schweizer KS, Curro JG (1994) PRISM theory of the structure, thermodynamics, and phase transitions of polymer liquids and alloys. Adv Polym Sci 116 319-377... 

A complication arising from the extension of the theory to flexible macromolecules is that in general, the intermolecular and intramolecular radial distribution functions depend on each other.In modeling the bulk of a one-phase polymer melt, however, the situation resolves itself because the excluded volume effect is insignificant under these conditions the polymer chains assume unperturbed dimensions (see also the section on Monte Carlo simulations by Corradini, as described originally in Ref. 99). One may therefore calculate the structure of the unperturbed single chain and employ the result as input to the PRISM theory to calculate the intermolecular correlation functions in the melt. 

Application of the PRISM theory to Gaussian chains is the simplest case. Each site is bonded to its neighbors by simple entropic Hookean springs. Nonbonded interactions are neglected. In this case, one obtains... 

---