Hard chain models PRISM theory

The wall-PRISM equation has been implemented for a number of hard-chain models including freely jointed [94] and semiflexible [96] tangent hard-sphere chains, freely rotating fused-hard-sphere chains [97], and united atom models of alkanes, isotactic polypropylene, polyisobutylene, and polydimethyl siloxane [95]. In all implementations to date, to my knowledge, the theory has been used exclusively for the stmcture of hard-sphere chains at smooth structureless hard walls. 

We also note that the same Monte Carlo data have helped to sort out an inadequate approximation in the context of the polymer reference interaction site model (PRISM) theory, which yielded a relation Tc oc /N while now oc is generally accepted. It has been very difficult to provide convincing experimental evidence on this issue— true symmetrical monodisperse polymer mixtures hardly exist, and the temperature range over which Tc N) can be studied is limited by the glass transifion temperature from below and by chemical instability of the chains from... 

In summary, the predictions of analytic PRISM theory [67] for the phase behavior of asymmetric thread polymer Uends display a ly rich dependence on the single chain structural asymn try variables, the interchain attractive potential asymmetries, the ratio of attractive and repulsi interaction potential length scales, a/d, and the thermodynamic state variaUes t) and < ). Moreover, these dependences are intimately coupled, which mathematically arises within the compressible PRISM theory from cross terms between the repulsive (athermal) and attractive potential contributions to the k = 0 direct correlations in the spinodal condition of Eq. (6.6). The nonuniversality and nonadditivity of the consequences of molecular structural and interaction potential asymmetries on phase stability can be viewed as a virtue in the sense that a great variety of phase behaviors are possible by rational chemical structure modification. Finally, the relationship between the analytic thread model predictions and numerical PRISM calculations for more realistic nonzero hard core diameter models remains to be fully established, but preliminary results suggest the thread model predictions are qualitatively reliable for thermal demixing [72,85]. 

An important question is whether PRISM theory can predict the packing in athermal blends with the same good accuracy found for one-component melts. To address this question Stevenson and co-workers performed molecular dynamics simulations on binary, repulsive force blends of 50 unit chains at a liquidlike packing fraction of -17 = 0.465. The monomeric interactions were very similar to earlier one-component melt simulations which served as benchmark tests of melt PRISM theory. Nonbonded pairs of sites (both on the same and different chains) were taken to interact via shifted, purely repulsive Lennard-Jones potentials. These repulsive potentials were adjusted so that the effective hard site diameters, obtained from Eq. (3.12), were 1-015 and = 1.215 for the chains of type A or B, respectively. Chain connectivity was maintained using an intramolecular FENE potential between bonded sites on the same chain. The resulting chain model has nearly constant bond lengths that are nearly equal to the effective hard-core site diameter. 

The reduction of thread PRISM with the R-MMSA closure for the idealized fully symmetric block copolymer problem to the well-known incompressible RPA approach " is reassuring. However, in contrast with the blend case, for copolymers that tend to microphase separate on a finite length scale, the existence of critical or spinodal instabilities is expected to be an artifact of the crude statistical mechanical approximations. That is, finite N fluctuation effects are expected to destroy all such spinodal divergences and result in only first-order phase transitions in block copolymers [i.e., Eq. (7.3) is never satisfied]. Indeed, when PRISM theory is numerically implemented for finite thickness chain models using the R-MMSA or R-MPY/HTA closures spinodal divergences do not occur. Thus, one learns that even within the simpler molecular closures, the finite hard-core excluded volume constraint results in a fluctuation effect that destroys the mean-field divergences. 

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. ... 

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. 

Equations of state for square-well chains and for linear alkanes have been obtained using perturbation theory. Figure 6 depicts the equation of state of square-well chains from Monte Carlo simulations and perturbation theory. (In the square-well chain model the site-site interaction potential is given by u r) = oo for r < d, u r) = —s for d < r < kd, and u(r) =0 otherwise.) Three reduced temperatures, T = kT/e), are depicted in Figure 6, all for k = 1.5. Theoretical predictions were obtained using the GFD equation for the hard chain pressure and PRISM with the PY closure for the... 

---