Gaussian thread model

The resulting PRISM integral equation is analytically solvable for the Gaussian thread model. The structural predictions are ... 

The analytic Gaussian thread model has been generalized to approximately treat nonzero chain thickness (d 0) in a simple average... 

Figure 6. Predicted interchain radial distribution function for a hard-core polyethylene melt described by three single-chain models atomistic RIS at 430 K, overlapping (lid = 0.5) SFC model with appropriately chosen aspect ratio and site number density (see text), and the Gaussian thread model (shifted horizontally to align the hard core diameter with the value of rld = l).

Finally, analytic predictions for the osmotic pressure of polymers in good and theta solvents can be derived based on the Gaussian thread model, PRISM theory, and the compressibility route. The qualitative form of the prediction for large N is " pP °c (po- ), which scales as p for theta solvents and p " for good solvents. Remarkably, these power laws are in complete agreement with the predictions of scaling and field-theoretic approaches and also agree with experimental measurements in semidilute polymer solutions. ""... 

Figure 13. Reduced solubility parameter as a function of chain aspect ratio for the Ud = 0.5 SFC model and the analytic Gaussian thread model. Predictions based on two choices of polyethylene aspect ratio at 430 K arc shown. The liquid density is determined by the calibration procedure discussed in Ref. 52.

In this section we examine athermal binary mixtures using PRISM theory. Tests of both the structural and thermodynamic predictions of PRISM theory with the PY closure against large-scale computer simulations are discussed in Section IV.A. Atomistic level PRISM calculations are presented in Section IV.B, and the possibility of nonlocal entropy-driven phase separation is discussed in Section IV.C at the SFC model level. Section IV.D presents analytic predictions based on the idealized Gaussian thread model. The limitations of overly coarse-grained chain models for treating athermal polymer blends are briefly discussed. 

Gaussian thread limit for N—For many physical problems (e.g., polymer solutions and melts, liquid-vapor equilibria, and thermal polymer blends and block copolymers), the Gaussian thread model has been shown to be reliable in the sense that it is qualitatively consistent with many aspects of the behavior predicted by numerical PRISM for more realistic semiflexible, nonzero thickness chain models. However, there are classes of physical problems where this is not the case. The athermal stiffness blend in certain regions of parameter space is one case, both in... 

In real systems, nonrandom mixing effects, potentially caused by local polymer architecture and interchain forces, can have profound consequences on how intermolecular attractive potentials influence miscibility. Such nonideal effects can lead to large corrections, of both excess entropic and enthalpic origin, to the mean-field Flory-Huggins theory. As discussed in Section IV, for flexible chain blends of prime experimental interest the excess entropic contribution seems very small. Thus, attractive interactions, or enthalpy of mixing effects, are expected to often play a dominant role in determining blend miscibility. In this section we examine these enthalpic effects within the context of thermodynamic pertubation theory for atomistic, semiflexible, and Gaussian thread models. In addition, the validity of a Hildebrand-like molecular solubility parameter approach based on pure component properties is examined. 

Analytic solutions are also possible based on the idealized Gaussian thread model since the molecular closures simplify dramatically. Because the hard-core diameter is shrunk to zero, Eq. (6.4) applies for all r, thereby allowing cancellation of the convolution integrals and all factors of w. Hence, the thread analogs of Eqs. (6.5) and (6.6) become" ... 

A basic approach for the description of polymer chains in the continuum is the Gaussian thread model [26, 31]. Treating interactions among monomers in a mean-field-like fashion, one obtains the self-consistent field theory (SCFT) [11, 32-36] which can also be viewed as an extension of the Hory-Huggins theory to spatially inhomogeneous systems (like polymer interfaces in blends, nucrophase separation in block copolymer systems [11, 13], polymer bmshes [37, 38], etc.). However, with respect to the description of the equation of state of polymer solutions and blends in the bulk, it is stiU on a simple mean-field level, and going beyond mean field to include fluctuations is very difficult [11, 39-42] and outside the scope of this article. 

An alternative approach that combines the Gaussian thread model of polymers with liquid-state theory is known as the polymer reference interaction site model (PRISM) approach [34-38[. This approach has the merit that phenomena such as the de Gennes [3] correlation hole phenomena and its consequences are incorporated in the theoretical description, and also one can go beyond the Gaussian model for the description of intramolecular correlations of a polymer chain, adding chemical detail (at the price of a rather cumbersome numerical solution of the resulting integral equations) [37,38[. An extension to describe the structure of colloid-polymer mixtures has also become feasible [39, 40]. On the other hand, we note that this approach shares vhth other approaches based on liquid state theories the difficulty that the hierarchy of exact equations for correlation functions needs to be decoupled via the so-called closure approximation [34—38]. The appropriate choice of this closure approximation has been a formidable problem [34—36]. A further inevitable consequence of such descriptions is the problem that the critical behavior near the critical points of polymer solutions and polymer blends is always of mean-field character ... 

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