Gaussian thread model analytic

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... 

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. 

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" ... 

The incompressible chi-parameter defined in Eq. (6.16) has also been extensively studied. Many of the numerical results for site volume and/or statistical segment length asymmetric athermal Gaussian drain blends are adequately reproduced at a qualitative level by the analytic thread model discussed in Sect. 2. For an athermal stiffness blend of very long Gaussian threads the k = 0 direct correlation functions are [23,62] ... 

An attractive virtue of PRISM theory is the ability to derive analytic solutions for many problems if the most idealized Gaussian thread chain model of polymer structure is adopted. The relation between the analytic results and numerical PRISM predictions for more chemically realistic models provides considerable insight into the question of what aspects of molecular structure are important for particular bulk properties and phenomena. Moreover, it is at the Gaussian thread level that connections between liquid-state theory and scaling and field-theoretic approaches are most naturally established. Thus, throughout the chapter analytic thread PRISM results are presented and discussed in conjunction with the corresponding numerical studies of more realistic polymer models. 

The idealized symmetric blend model is not representative of the behavior of most polymer alloys due to the artificial symmetries invoked. Predictions of spinodal phase boundaries of binary blends of conformationally and interaction potential asymmetric Gaussian thread chains have been worked out by Schwelzer within the R-MMSA and R-MPY/HTA closures and the compressibility route to the thermodynamics. Explicit analytic results can be derived for the species-dependent direct correlation functions > effective chi parameter, small-angle partial collective scattering functions, and spinodal temperature for arbitrary choices of the Yukawa tail potentials. Here we discuss only the spinodal boundary for the simplest Berthelot model of the Umm W t il potentials discussed in Section V. For simplicity, the A and B polymers are taken to have the same degree of polymerization N. 

An analytical approximation [23] can be found for the Gaussian chain melt by taking either the so-called thread or string limits. The thread-like chain model has been discussed in depth elsewhere [23]. Mathematically, it corresponds to the limit that all microscopic length scales approach zero but their ratios remain finite. In particular, the site hard core diameter d 0, but the site density pm - oo such that the reduced density, pmd , is non-zero and finite. In... 

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. 

---