PRISM self-consistent

Potential of mean force 60 Pressure tensor 18 PRISM, self-consistent 211, 222 equation 218 theory 58, 64, 210, 217... 

Figure B3.6.5. Phase diagram of a ternary polymer blend consisting of two homopolymers, A and B, and a synnnetric AB diblock copolymer as calculated by self-consistent field theory. All species have the same chain length A and the figure displays a cut tlirough the phase prism at%N= 11 (which corresponds to weak segregation). The phase diagram contains two homopolymer-rich phases A and B, a synnnetric lamellar phase L and asynnnetric lamellar phases, which are rich in the A component or rich in the B component ig, respectively. From Janert and Schick [68].

Formally, the self-consistent PRISM theory is easily generalized to treat polymer blends and copolymers [92] where significant non-ideal conformational effects may occur which intensify as phase separation is approached. 

The further development of the self-consistent version of PRISM theory will be particularly important in two areas (i) liquids of flexible conjugated polymers where the electron delocalization length and interchain dispersion forces are strongly coupled to chain conformation [100], and (ii) polymer alloys where... 

Finally, we mention that very recently three other integral equation approaches to treating polymer systems have been proposed. Chiew [104] has used the particle-particle perspective to develop theories of the intermolecular structure and thermodynamics of short chain fluids and mixtures. Lipson [105] has employed the Born-Green-Yvon (BGY) integral equation approach with the Kirkwood superposition approximation to treat compressible fluids and blends. Initial work with the BGY-based theory has considered lattice models and only thermodynamics, but in principle this approach can be applied to compute structural properties and treat continuum fluid models. Most recently, Gan and Eu employed a Kirkwood hierarchy approximation to construct a self-consistent integral equation theory of intramolecular and intermolecular correlations [106]. There are many differences between these integral equation approaches and PRISM theory which will be discussed in a future review [107]. 

VIII. Solvation Potentials and Self-Consistent PRISM 1. Solvation Potential Theories... 

In our application of PRISM theory to flexible polymer systems, one expects that the in/ramolecular structure, represented by Eq. (2.4), depends on the in/ermolecular structure specified in Eq. (2.3) and vice versa. " " Thus, in a rigorous calculation the intramolecular and inter-molecular structure must be determined in a self-consistent manner... 

VIII. SOLVATION POTENTIALS AND SELF-CONSISTENT PRISM... 

Implementation of self-consistent PRISM theory requires addressing the difficult technical question of how to iteratively solve the effective A/-body... 

The first self-consistent PRISM studies by Schweizer et al. considered only the HNC-style solvation potential and were based on an optimized perturbative, not variational, determination of the ideal reference system effective bending energy. The starting point is a simple functional expansion of the true single-chain free energy about an ideal reference system" ... 

The best test of self-consistent PRISM theory and the different solvation potential approximations is via comparison of its predictions against exact computer simulation studies of the same model. The drawback is that present computer power limits such comparisons to short and intermediate length chains (/V less than roughly 200). Many detailed comparisons have been carried out at all levels of approximation discussed in Section VIII.B. Here we give a few examples along with summarizing remarks. The reader is referred to the original studies for details and a complete discussion. 

We begin with the most rigorous version of self-consistent PRISM based on a Monte Carlo evaluation of the effective single-chain problem. Theoretical predictions of Grayce and co-workers" are compared with many-chain simulation results for the mean-square end-to-end distance of the hard-core chain model as a function of polymer packing fraction in... 

Figure 33. The change with polymer density of the mean-square end-to-end distance of hard-core chains of length (a) N =20 and (b) /V = 100. The data points are exact many-chain simulation results and the solid (dash-dot) lines are the self-consistent PRISM/Monte Carlo (free energy generating fuiKtional) predictions using the two solvation potentials." The dashed horizontal line is the value of / for an ideal freely joined chain with a minimum next nearest neighbor bending angle of 60°, which mimics the local hard-core repulsion.

Figure 34. Self-consistent PRISM structural predictions for (a) average intramolecular structure factor plotted in Kratky form, and (b) site-site intermolecular radial distribution function for N = 100 hard-core chains at a concentrated solution packing fraction of 0.3. The points are the many chain simulation results and the lines are the PRISM results based on the PY-style solvation potential and the simplified version of the variational generating functional method of Grayce et al. discussed in the appendix of Ref. 47.

Application and generalization of the self-consistent PRISM theory to flexible trimer fluids, and detailed comparison with many molecule simulations, has also been performed by both Grayce and dePablo and Yethiraj. ... 

There are many other physical problems and macromolecular systems for which the self-consistent PRISM approach should be useful. The following represents an incomplete list of problems for which preliminary work has been done or which appear to be attackable based on the present state of the art. 

The above physical features imply that a fully self-consistent treatment of intramolecular and intermolecular pair correlations is more important for star polymers than linear chains, and the concept of ideality is expected to be of much less utility even at high melt densities. The treatment of star polymers within a self-consistent PRISM formalism has been very recently pursued by Grayce and Schweizer. Here we give a brief description of some of the essential theoretical modifications... 

The predicted nonideal conformational effects can be probed by SANS experiments, and theoretical/experimental comparisons are given elsewhere. A detailed physical picture of the origin of the nonideal conformational behavior in terms of the thermodynamic forces a star experiences has been constructed. Comparison of the self-consistent PRISM theory results with phenomenological scaling, and other coarsegrained polymer physics approaches have also been presented, and distinctive qualitative and quantitative differences have been identified. ... 

Intramolecular correlations are handled in different approximate manners in the various BGY approaches. Taylor and Lipson treat pair correlations on the same chain as input to the theory in a manner similar to PRISM theory. In contrast, the formulations of Eu and Gan, " and also Attard, yield closed integral equations for both the intra- and intermolecular pair distribution functions. Thus, in a sense the intra- and intermolecular pair correlations are treated on an equal footing, and a self-consistent integral equation theory is naturally obtained. Eu and Gan have recently presented a comparison between their BGY approach and self-consistent and non-self-consistent PRISM theory, in both general conceptual terms and within the context of numerical predictions for specific model hard-core systems. For the jointed hardcore chain model studies, the theory of Eu and Gan appears quantitatively superior to PRISM predictions, particularly for the equation of state. ... 

Analytic and numerical self-consistent PRISM theories of polymer blends and diblock copolymers... 

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