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Author: Münch, Andreas × End date: 2022 × Start date: 2020 × Type: Text × Type: report × Subject: asymptotic analysis ×

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    Self-consistent field theory for a polymer brush. Part II: The effective chemical potential
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Münch, Andreas; Wagner, Barbara
    The most successful mean-field model to describe the collective behaviour of the large class of macromolecular polymers is the self-consistent field theory (SCFT). Still, even for the simple system of a grafted dry polymer brush, the mean-field equations have to be solved numerically. As one of very few alternatives that offer some analytical tractability the strong-stretching theory (SST) has led to explicit expressions for the effective chemical potential and consequently the free energy to promote an understanding of the underlying physics. Yet, a direct derivation of these analytical results from the SCFT model is still outstanding. In this study we present a systematic asymptotic theory based on matched asymtptotic expansions to obtain the effective chemical potential from the SCFT model for a dry polymer brush for large but finite stretching.
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    Self-consistent field theory for a polymer brush. Part I: Asymptotic analysis in the strong-stretching limit
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Münch, Andreas; Wagner, Barbara
    In this study we consider the self-consistent field theory for a dry, in- compressible polymer brush, densely grafted on a substrate, describing the average segment density of a polymer in terms of an effective chemical potential for the interaction between the segments of the polymer chain. We present a systematic singular perturbation analysis of the self-consistent field theory in the strong-stretching limit, when the length scale of the ratio of the radius of gyration of the polymer chain to the extension of the brush from the substrate vanishes. Our analysis yields, for the first time, an approximation for the average segment density that is correct to leading order in the outer scaling and resolves the boundary layer singularity at the end of the polymer brush in the strong-stretching limit. We also show that in this limit our analytical results agree increasingly well with our numerical solutions to the full model equations comprising the self-consistent field theory.