Molecular modeling closure approximations

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). 

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. 

Equations (7.57a) and (7.57b) provide two ways to calculate the pressure and chemical potential. The first one is to perform the appropriate derivative of the free energy, assuming that the latter can be evaluated for all states of interest. The second, more direct way is to employ the relations to the pressure and chemical potentials in the replicated system. This second strategy is particularly useful for the calculation of fit because lim o in closed form for a variety of model systems and closure relations, including the HNC approximation for molecular fluids [309, 310]. The pressure is more difficult because of the presence of the second, matrix-related term on the right side of Eq. (7.56). 

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. 

For practical applications, second-order closure models are required for the third-order diffusion correlations, the pressure-strain correlation and the dissipation rate correlation as described by Launder and Spalding [95] and Wilcox ([185], Sect. 6.3). Launder and Spalding [95] argued that the pressure diffusion terms and the molecular diffusion of turbulent momentum fluxes are smaller than the rest of the terms in the equation. These terms can thus be sufficiently approximated by a gradient... 

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