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

TL;DR

This work tackles the fundamental question of what constitutes a statistical segment and an entropic spring in polymer melts, focusing on a polyethylene system. Using atomistic MD, end-to-end distance distributions of C–C bond blocks are fitted to Gaussian forms via a grid search over block sizes and step divisions, and an inner-confinement domain is introduced to identify three Kuhn-scale substructures: ACS, RCS, and CE. The key findings are that a single Kuhn segment is statistically uncorrelated but non-Gaussian; the minimal statistical segment is a single Kuhn segment, the minimal entropic spring requires two Kuhn segments, and the minimal Gaussian segment requires five Kuhn segments; blocks with ten or more Kuhn segments are fully Gaussian. The three substructures exhibit distinct relaxation: ACS shows stretched-exponential decay with $β\approx0.5$ (slow, quasi-one-dimensional localized modes), while RCS and CE have $β\approx0.7$ (faster). The results provide a molecular interpretation of stretched-exponential relaxation via localized modes and CTRW, refine the definitions of statistical and Gaussian segments, and have implications for how entanglements and tube-model parameters are interpreted in polymer melts.

Abstract

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~Å) to extended (approximately 14~Å), 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 $β\approx 0.5$, consistent with quasi-one-dimensional, defect-mediated localized modes, whereas RCS and CE relax with $β\approx 0.7$. 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.

Figures (9)

• Figure 1: Average segment lengths obtained for the C250 system (560 chains) at $T=600$ K taken over $6.0\times 10^4$ ps. The characteristic ratio is $C_\infty = 7.32$, giving $l_\text{p} = 6.13$ Å, $b = 5.89$ Å [Eq. (\ref{['eq:b-definition']})], and $l_k = 13.80$ Å [Eq. (\ref{['eq:l-kuhn']})] . Kuhn segment molecular weight: $M_\text{k} = 153.901 \pm 5.038$ g mol^-1. The consistency between $l_k$ values obtained by different procedures demonstrates the reliability of the molecular dynamics results.
• Figure 2: End-to-end internal distance distribution for a segment with $n_\text{cc} = 6$
• Figure 3: End-to-End distance distribution and Gaussian fit for different $n_\text{cc}$ blocks: (a) 11 C--C bonds, 1 step; (b) 22 bonds, 2 steps; (c) 55 bonds, 5 steps and (d) 110 bonds and 10 steps. The number of bonds of the step size equals those of one Kuhn segment.
• Figure 4: Statistical variables and high-order moments used to characterize the segment-size crossover to Gaussian statistics, plotted as a function of the block size $s \equiv n_{\text{cc}}$. The first four data points are those shown in Table S3 in the Supplementary Material (6, 11, 17 and 22). (a) The non-Gaussian parameter, $\alpha_2(s)$: The regions indicated, Gaussian, transition to Gaussian (TG), mild non-Gaussian (MnG), and non-Gaussian were defined based on the criteria described in the Methods Section. (b) The apparent segment mean-squared displacement per bond, $\langle R^2(s)/s\rangle$:This quantity probes the characteristic segment stiffness. The interception of two tangent lines, to the plateau and to the initial data, is used to define the onset of mild non-Gaussian behavior $n_\text{cc}\approx20$. (c) Skewness and excess kurtosis: Tangent lines to the data at large block sizes, and large number of steps, are used to define the onset of Gaussinity.
• Figure 5: Distribution of perpendicular distances $P(d)$ of backbone atoms to the local Kuhn segment axis in a polyethylene chain. The Kuhn confinement radius $r_{\text{k}}$ and the critical distance $d_{\text{crit}} = (1-e^{-1})\,r_{\text{k}}$ (with $\pm10\%$ variation band) used to define the inner confinement domain are indicated. Insets: (Bottom right) Schematic of the distance measurement for a helical conformation formed by one Kuhn segment. The green confinement surface confines all atoms of the chain. (Center) Chain snapshot illustrating Aligned Chain Segments (ACS, inside the inner-domain), Random Conformational Sequences (RCS) and Chain Ends (CE) (outside), and a single non-Gaussian Kuhn segment. The normalization factor is the number of atoms in the segments.