Molecular closures

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 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]. 

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... 

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. 

PRISM theory based on the new molecular closures has recently been generalized by David and Schweizer [86] to treat periodic block copolymers. For... 

The same molecular closures proposed for homopolymer blends [68-70] apply to copolymers but the intramolecular structure factor matrix is now non-diagonal. Equation (8.8) becomes [86]... 

VI. Beyond Thermodynamic Perturbation Theory Molecular Closure Approximations... 

VI. BEYOND THERMODYNAMIC PERTURBATION THEORY MOLECULAR CLOSURE APPROXIMATIONS... 

The development of new molecular closure schemes was guided by analysis of the nature of the failure of the MSA closure. In particular, the analytic predictions derived by Schweizer and Curro for the renormalized chi parameter and critical temperature of a binary symmetric blend of linear polymeric fractals of mass fractal dimension embedded in a spatial dimension D are especially revealing. The key aspect of the mass fractal model is the scaling relation or growth law between polymer size and degree of polymerization Ny cr. The non-mean-field scaling, or chi-parameter renormalization, was shown to be directly correlated with the average number of close contacts between a pair of polymer fractals in D space dimensions N /R if the polymer and/or... 

PRISM with the molecular closures has been applied both numeri-... 

The strategy for explicitly formulating the molecular closures was guided by three considerations." (1) Use of the commonly employed reference approach. The successful site-site PY closure is retained to describe the repulsive force reference fluid but a molecular closure scheme is adopted to describe the attractive, slowly varying forces. (2) The approximation scheme is required to provide an exact description of the structural consequences of the tail potentials at the two-molecule level in the weak coupling limit [/3umm-W 1]- (3) Use of an appropriate site-site approximation for the direct attractive interaction contribution motivated by experience in simple fluids. ... 

Taken as a whole, the ideas discussed led Yethiraj and Schweizer to propose the following reference molecular closure approximations for site interaction potentials consisting of a hard core plus tail" ... 

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" ... 

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. 

In this section some of the recent results obtained for the symmetric blend based on PRISM with the molecular closures are summarized." ... 

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. 

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. 

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." ... 

Numerical PRISM studies based on finite thickness SFC models of the stiffness and interaction binary blend, and molecular closures and compressibility route, have not yet been widely pursued. Preliminary... 

The effect of attractions on the structure of dense one-component polymer melts. According to the van der Waals ideas, attractions should have very little effect. Surprisingly, we are unaware of simulations that have probed this question, although they are now in progress. Recent PRISM studies by Butler and Schweizer using atomic and molecular closures have been carried out. Repulsive force screening of the effects of attractions on structure is recovered for many, but not all, closure approximations. 

The effect of attractions on solvent quality (good, theta, poor), and low temperature polymer-solvent phase separation. Some tentative analytic work has been done on the latter problem by Schweizer and Yethiraj based on the molecular closures. Non-mean-field dependences of the critical polymer volume fraction on N have been found. 

As was found for the analogous blend, the application of site-site atomiclike closure (e.g., MSA) to the diblock copolymer fluid problem predicts a qualitatively incorrect N dependence of the long-wavelength thermal concentrations fluctuations. Thus, use of the molecular closure... 

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. 

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. 

The linearized R-MPY version of the thread molecular closure condition of Eq. (6.7) can be shown to result in a nonlinear, self-consistent integral equation for the effective chi parameter [or equivalently the concentration fluctuation part of the collective structure factor 5(A )]... 

Development and application of molecular closures to treat attractive forces in polymer solutions and melts (including the liquid-vapor transition). ... 

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