Gaussian thread

We have briefly reviewed methods which extend the self-consistent mean-field theory in order to investigate the statics and dynamics of collective composition fluctuations in polymer blends. Within the standard model of the self-consistent field theory, the blend is described as an ensemble of Gaussian threads of extension Rg. There are two types of interactions zero-ranged repulsions between threads of different species with strength /AT and an incompressibility constraint for the local density. 

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. 

Figure 1. Schematic representation of three levels of chain models considered and the coarse-graining procedure. The top level is an atomistic model of polyolefins. The second level shows two intermediate models site overlapping semiflexible chain (with bending energy e ) and freely jointed branched chain. The bottom level is the Gaussian thread chain.

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... 

The results of this approach as applied to polyethylene are shown in Figure 6. Remarkable agreement between the atomistic model g(r) and the SFC g(r) is found. Moreover, even the Gaussian thread result seems reasonable as an interpolation through the atomistic g(r). For inte-... 

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... 

Finally, field theoretic approaches have recently predicted athermal phase separation driven by nonlocal-entropic considerations for incompressible blends of Gaussian thread polymers. This prediction is at odds with PRISM theory in the thread limit. However, for the effective chi parameter PRISM theory has been shown to be equivalent to the field theory if the free energy route is employed in conjunction with the extremely simple RPA closure (not PY). The RPA closure, Cmm ( ) = -/8mmm ( ) for all r, is known to be very poor for repulsive force systems and violates the hard-core impenetrability condition. Thus, the field-theoretic prediction has been suggested to be a consequence of the combined use of a long-wavelength incompressibility approximation in conjunction with a RPA closure. ... 

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" ... 

For the symmetric Gaussian thread blend interacting via the Yukawa tail potentials of Eq. (3.16), a nearly complete analytic treatment can be carried out for all three molecular closures of Eq. (6.7) within the S potential ordering." Remarkably, all the analytically derived trends are consistent with numerical studies based on the compressibility route to the thermodynamics. Results based on the free energy route have also been obtained." ... 

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. 

In the Gaussian thread limit analytic results have been derived for copolymer fluids using the molecular closures. " The analytic results provide insights to several key questions and behaviors that emerge from the numerical PRISM studies. These Include (1) the role of nonzero monomer hard-core diameter, density fluctuations, and concentration fluctuations on dlblock liquid-phase behavior and structure (2) relationship between phenomenological field-theoretic approachesand the molecular closure-based versions of PRISM theory and (3) the influence of molecular weight, composition, solution density, and chemical and conformational asymmetries of the blocks on copolymer microphase separation temperatures. 

Figure 31. PRISM plus linearized R-MPY closure predictions" for the normalized inverse peak scattering intensity of the/ = 4 symmetric Gaussian thread diblock copolymer model. Results (top to bottom) for iV = 20, 200, 2000, and 20,000 are shown at fixed melt density and a Yukawa tail potential range parameter of a = 0.5, Here, the bare driving force for microphase separation, 1". varied by changing temperature. The inset shows the apparent exponent that describes the scaling relation between peak scattering intensity and /V as a function of inverse temperature (as extracted from the three largest chain lengths).

PRISM theory, particularly in its analytic Gaussian thread and string versions, has also been extensively employed by Schweizer and co-workers as the equilibrium input to microscopic generalized Langevin and mode-coupling theories of the dynamics of macro-molecular fluids. ... 

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