TY - JOUR

T1 - Renormalization of the one-loop theory of fluctuations in polymer blends and diblock copolymer melts

AU - Grzywacz, Piotr

AU - Qin, Jian

AU - Morse, David C.

PY - 2007/12/14

Y1 - 2007/12/14

N2 - Attempts to use coarse-grained molecular theories to calculate corrections to the random-phase approximation (RPA) for correlations in polymer mixtures have been plagued by an unwanted sensitivity to the value of an arbitrary cutoff length, i.e., by an ultraviolet (UV) divergence. We analyze the UV divergence of the inverse structure factor S-1 (k) predicted by a "one-loop" approximation similar to that used in several previous studies. We consider both miscible homopolymer blends and disordered diblock copolymer melts. We show, in both cases, that all UV divergent contributions can be absorbed into a renormalization of the values of the phenomenological parameters of a generalized self-consistent field theory (SCFT). This observation allows the construction of an UV convergent theory of corrections to SCFT phenomenology. The UV-divergent one-loop contribution to S-1 (k) is shown to be the sum of (i) a k -independent contribution that arises from a renormalization of the effective χ parameter, (ii) a k -dependent contribution that arises from a renormalization of monomer statistical segment lengths, (iii) a contribution proportional to k2 that arises from a square-gradient contribution to the one-loop fluctuation free energy, and (iv) a k -dependent contribution that is inversely proportional to the degree of polymerization, which arises from local perturbations in fluid structure near chain ends and near junctions between blocks in block copolymers.

AB - Attempts to use coarse-grained molecular theories to calculate corrections to the random-phase approximation (RPA) for correlations in polymer mixtures have been plagued by an unwanted sensitivity to the value of an arbitrary cutoff length, i.e., by an ultraviolet (UV) divergence. We analyze the UV divergence of the inverse structure factor S-1 (k) predicted by a "one-loop" approximation similar to that used in several previous studies. We consider both miscible homopolymer blends and disordered diblock copolymer melts. We show, in both cases, that all UV divergent contributions can be absorbed into a renormalization of the values of the phenomenological parameters of a generalized self-consistent field theory (SCFT). This observation allows the construction of an UV convergent theory of corrections to SCFT phenomenology. The UV-divergent one-loop contribution to S-1 (k) is shown to be the sum of (i) a k -independent contribution that arises from a renormalization of the effective χ parameter, (ii) a k -dependent contribution that arises from a renormalization of monomer statistical segment lengths, (iii) a contribution proportional to k2 that arises from a square-gradient contribution to the one-loop fluctuation free energy, and (iv) a k -dependent contribution that is inversely proportional to the degree of polymerization, which arises from local perturbations in fluid structure near chain ends and near junctions between blocks in block copolymers.

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U2 - 10.1103/PhysRevE.76.061802

DO - 10.1103/PhysRevE.76.061802

M3 - Article

AN - SCOPUS:37249035667

SN - 1539-3755

VL - 76

JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

IS - 6

M1 - 061802

ER -