Beyond the Static Kuhn Length: Conformational Substructures and Relaxation Dynamics in Flexible Chains

The statistical "monomer-based" segment length $b$ and the Kuhn length $l_k$ are central to polymer physics, yet the minimal size required for a truly statistical segment - Gaussian, uncorrelated, and valid as an entropic spring - is not rigorously established. Using atomistic simulations of entangled polyethylene, we re-evaluate these foundational quantities. By fitting end-to-end distance distributions of C--C bond blocks and validating with higher-moment analyses, we identify for the first time the minimal sizes corresponding to a statistical segment and an entropic spring. A single Kuhn segment (approximately 11 bonds) is the smallest statistically uncorrelated unit, but its distance distribution is strongly non-Gaussian, while the monomer-based segment $b$, used in rheology and classical tube-theory formulations, is not statistical at all. True Gaussianity emerges only for blocks containing multiple Kuhn segments. At the Kuhn scale, we uncover a previously unresolved conformational heterogeneity. Each segment samples a broad range of conformations, from coiled (approximately 4~\AA) to extended (approximately 14~\AA), giving rise to three distinct substructures: aligned chain segments (ACS), random conformational sequences (RCS), and chain ends (CE). These exhibit distinct dynamical signatures. ACS relax with a stretched-exponential exponent $\beta \approx 0.5$, consistent with quasi-one-dimensional, defect-mediated localized modes, whereas RCS and CE relax with $\beta \approx 0.7$. By connecting these results to localized-mode theory and continuous-time random-walk models, we provide a molecular interpretation of stretched-exponential relaxation in polymer melts.