Flory mean-field theory, incompressible

Incompressible Flory mean-field theory is recovered from Eqs. (5.4) and (5.5) if one assumes the following (i) no excess volume of mixing (ii) a blend composition-independent total packing fraction (iii) the... 

While the Flory-Huggins mean-field theory [13, 14, 15] of Sect. 2.1 describes the generic, qualitative behavior of incompressible polymer h-solvent mixtures, it invokes three important simplifications that restrict its application ... 

The well-known mean-field incompressible Flory-Huggins theory of polymer mixtures assumes random mixing of polymer repeat units. However, it has been demonstrated that the radial distribution functions gay(r) of polymer melts are sensitive to the details of the polymer architecture on short length scales. Hence, one expects that in polymer mixtures the radial distribution functions will likewise depend on the intramolecular structure of the components, and that the packing will not be random. Since by definition the heat of mixing is zero for an athermal blend, Flory-Huggins theory predicts athermal mixtures are ideal solutions that exhibit complete miscibility. 

There have been sophisticated calculations of x these systems. In the Flory-Huggins model x is independent of molecular weight and composition. In reality, this parameter has been shown, by scattering experiments and by theoretical calculations to be a function of Af, T and 0. 3,40,49 Theoretical attempts, which are beyond the scope of the mean field predictions of the Flory-Huggins approach which applies stri ct to incompressible systems, have been made to address these questions. 3,4o 49 The theories of Bates and Muthukumar 3 and of Schweizer and Curro °> both have predictions which may be written in the following form... 

Wall interfaces have also been found to alter the phase separation of nearby symmetric block copolymer chains. Fredrickson used SCF theory to demonstrate that a block copolymer melt in the vicinity of a solid wall or free surface (one with selective attraction) possessed a modified Flory-Huggins interaction parameter. Due to the connectivity of the blocks and the incompressibility of the material (an assumption of the calculation), the calculated interaction parameters have an oscillatory component with period 2njdo, normal to the wall plane, which decays exponentially from the interface. Milner and Morse also predirted this oscillatory profile normal to the surface for bulk-cylindrical morphology as well (corresponding to thickness commensurability), though they also observed that the decay length is longer closer to the mean-field critical point. 

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