Molecular closure approximations blends

Structure and the molecular closure approximations. Very recent work by Gromov and de Pablo has shown for the symmetric blend model that PRISM with the R-MPY closure is in excellent agreement with continuous space simulations for the structure, mixing thermodynamic properties, and the coexistence curve. 

The simplest molecular closure based on the above ideas is one that builds in the hard core reference behavior and correctly treats the longer ranged attractive potentials in the weak coupling limit. It is called the Reference Molecular Mean Spherical Approximation (RMMSA) and is given in real space for a homopolymer blend by [68-70]... 

Extensive analytic results for the symmetric thread blend have also been derived [68,70b]. In the thread-polymer limit the hard core condition becomes irrelevant for the molecular closure relations. In particular, for the R-MMSA and R-MPY/HTA approximations the MM (k) functions are fully specified by the closure relations, and their k = 0 values are given in general by... 

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. 

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