PRISM theory with molecular closures

Detailed analytical and numerical studies of the above questions are in progress, and a very rich and nonadditive dependence of the phase behavior on the precise nature of the attractive potentials, single chain architecture, and thermodynamic state is found [67, 72]. A full understanding of these issues would provide a scientific basis for the rational molecular design of polymeric alloys. The influence of asymmetries on the spinodal phase boundary of simple model polymer alloys using analytic PRISM theory with molecular closures has been derived by Schweizer [67]. In this section a few of these results are briefly discussed. 

An example of a comparison by Honnell et al. of PRISM theory with molecular dynamics simulations are shown in Figure 3. Details of the model are given elsewhere. Briefly, a meltlike density was studied for /V = 50-150 unit chains. The linear polymers were modelled as freely jointed beads with a purely repulsive, shifted Lennard-Jones interaction between all segment pairs. The corresponding chain aspect ratio is F-1.4. PRISM theory with the PY closure (plus a standard correction... 

Although essentially all studies to date using PRISM and the molecular closures have involved macromolecules, it is conceivable such closures may be of value even for small or intermediate-sized flexible and/or rigid molecules. A careful documentation of the accuracy of the new molecular closures as a broad function of thermodynamic state and molecular fluid type remains an important future direction. In addition, recent interesting alternative approaches to liquid theory for polymer mixtures with attractions have been developed within the general PRISM framework by Melenkevitz and Curro based on the optimized RPA(ORPA) approach, and Donley et. ah " based on density functional theory and also from a field-theoretic perspective by Chandler. Application of these approaches to treat the effect of attractive interactions on fluid structure and phase transitions remains to be worked out. 

Extensive studies of the predications of the new molecular dosure to the blend PRISM theory for the symmetric binary blend have been carried out by Yethiraj and Schweizer [68-70]. Here, a few of their major results are summarized, beginning with the numerical studies. The PRISM equations with the molecular closures can be solved using standard Picard iteration methods and the fast Fourier transform [5,70]. 

Finally, for PRISM/R-MPY theory the thermodynamic consistency between the free energy and compressibility route calculations of the chi parameter and spinodal phase boundaries has been shown to be remarkably good. Moreover, in the long-chain limit the predicted chi parameter and phase boundary appear to be exactly equivalent, which is a unique circumstance for liquid-state theories. However, this is not a general feature of PRISM with the molecular closures but rather derives from the fact that in the long-chain limit the critical temperature becomes arbitrarily high, the HTA is rigorous, and thus the symmetric blend reference system reduces to a composition-independent homopolymer melt. 

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. 

With the development of molecular closures, PRISM theory has shown the ability to predict a x parameter with composition and degree of polymerization dependence that is consistent with simulation results [114]. Smdies of symmetric block copolymer liquids show qualitative agreement with Monte Carlo simulation, but both the R-MMSA and R-MPY closures fail to predict a point of spinodal decomposition for finite degrees of polymerization [73, 74], These results are for the somewhat unrealistic system of a symmetric blend model where each species has the same chain length and site diameter and the interactions between monomers of the same type are purely repulsive while the cross term has an attractive tail. 

Polymer miscibility—with solvents, other polymers, or in block copolymers— can be treated with integral equation methods such as PRISM (255-257) or density functional theory (252,258). Theories of this kind inevitably required approximations that are difficult to assess by independent methods, with the variety of closure relations having been developed in integral equation theories (259) illustrating the case in point. Molecular modeling can provide the detailed molecular-level information, such as pair correlation fimctions, that is needed to assess the validity of these t5q>es of theories (260,261). 

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