Renormalization of the one-loop theory of fluctuations in polymer blends and diblock copolymer melts

摘要

Attempts to use coarse-grained molecular theories to calculate corrections tothe random-phase approximation (RPA) for correlations in polymer mixtures havebeen plagued by an unwanted sensitivity to the value of an arbitrary cutofflength, {\it i.e.}, by an ultraviolet (UV) divergence. We analyze the UVdivergence of the inverse structure factor $S^{-1}(k)$ predicted by a`one-loop' approximation similar to that used in several previous studies. Weconsider both miscible homopolymer blends and disordered diblock copolymermelts. We show, in both cases, that all UV divergent contributions can beabsorbed into a renormalization of the values of the phenomenologicalparameters of a generalized self-consistent field theory (SCFT). Thisobservation allows the construction of a UV convergent theory of corrections toSCFT phenomenology. The UV-divergent one-loop contribution to $S^{-1}(k)$ areshown to be the sum of: (i) a $k$-independent contribution that arises from arenormalization of the effective $\chi$ parameter, (ii) a $k$-dependentcontribution that arises from a renormalization of monomer statistical segmentlengths, (iii) a contribution proportional to $k^{2}$ that arises from asquare-gradient contribution to the one-loop fluctuation free energy, and (iv)a $k$-dependent contribution that is inversely proportional to the degree ofpolymerization, which arises from local perturbations in fluid structure nearchain ends and near junctions between blocks in block copolymers.