Relationships among coarse-grained field theories of fluctuations in polymer liquids

Two closely related field-theoretic approaches have been used in previous work to construct coarse-grained theories of corrections to the random phase approximation for correlations in block copolymer melts and miscible polymer blends. The "auxiliary field" (AF) approach is based on a rigorous expression for the partition function Z of a coarse-grained model as a functional integral of an auxiliary chemical potential field. The "effective Hamiltonian" (EH) approach is instead based on an expression for Z as a functional integral of an observable order parameter field. The exact effective Hamiltonian H(eff) in the EH approach is defined as the free energy of a system with a constrained order parameter field. In practice, however, H(eff) has often been approximated by a mean-field free energy functional, yielding what we call a mean-field effective Hamiltonian (MFEH) approximation. This approximation was the starting point of both the Fredrickson-Helfand analysis of fluctuation effects in diblock copolymers and earlier work on the Ginzburg criterion in polymer blends. A more rigorous EH approach by Holyst and Vilgis used an auxiliary field representation of the exact H(eff) and allowed for Gaussian fluctuations of this field. All applications of both AF and EH approaches have thus far relied upon some form of Gaussian, or "one-loop" approximation for fluctuations of a chemical potential and/or order parameter field about a mean-field saddle-point. The one-loop EH approximation of Holyst and Vilgis and the one-loop AF theory are equivalent to one another, but not to the one-loop MFEH theory. The one-loop AF and MFEH theories are shown to yield predictions for the inverse structure factor S(-1)(q) that (in the absence of further approximations to either theory) differ by a function that is independent of the Flory-Huggins interaction parameter χ. As a result, these theories yield predictions for the peak scattering intensity that exhibit a similar χ-dependence near a spinodal. The Fredrickson-Helfand theory for the structure factor in disordered diblock copolymer melts is an asymptotic approximation to the MFEH one-loop theory that captures the dominant asymptotic behavior of very long, symmetric copolymers very near the order-disorder transition.