Time is length in self-similar logarithmic aging of physically cross-linked semiflexible polymer networks

TL;DR

We study aging in reversible crosslinked semiflexible polymer networks using molecular dynamics simulations and identify ultra-slow, self-similar aging with a single coarsening length $L(t)\sim \ln t$ that governs structure, energies, and dynamics. The internal clock is the mean filament length $L_f$, which grows as $L_f(t_w)\approx a \ln(t_w/t_0)$ and enables data collapse of two-time correlators and the MSD when expressed via the ratio $L_f(t+t_w)/L_f(t_w)$, signaling superuniversal aging. The analysis argues for thermally activated filament breakage within a droplet-theory framework, predicting $ au_s\sim t_w \ln(t_w/t_0)$ and robust logarithmic coarsening across bending stiffness $\kappa$; large-scale relaxation times become age dependent while small-scale dynamics exhibit similar scaling. The findings provide a unifying picture of aging in soft disordered materials and may inform memory effects in biomimetic networks and pathological fiber assemblies.

Abstract

Physical aging in polymers is a fundamental yet poorly understood phenomenon, as diverse macromolecular systems exhibit remarkably similar slow dynamics. Through molecular dynamics simulations of physically crosslinked networks composed of semiflexible polymers, we identify a previously unexplored class of self-similar aging. The network undergoes ultra-slow coarsening characterized by a logarithmically growing mesh size, $L(t)\sim \ln t$, which governs the spatial organization, cohesive and bending energies, and the aging dynamics of the system. This single time-dependent length scale defines an internal clock, giving rise to spatio- temporal self-similarity of both structure and dynamics - offering a perspective on aging in soft and disordered materials.

Figures (7)

• Figure 1: Mean filament length $L_f$ vs. waiting time $t_\textrm{w}\ge t_p$ for $\kappa=50$. The black-yellow line shows Eq. \ref{['Lf_logtw']} with $a=0.74$ and $t_0=0.0012$. Inset: Mean number of clusters vs. $t_\textrm{w}$ (particles that are bonded via permanent or temporary bonds belong to the same cluster, as in kroger_ultra-slow_2025). Snapshots show two-dimensional projections at the respective waiting times $t_\textrm{w}$. Each chain has its own color.
• Figure 2: (a) Self-part of the incoherent scattering function, $C_q(t,t_\textrm{w})$, vs. time $t$ on a logarithmic scale for $q=1$ and $\kappa=50$. Different values of the waiting time $t_\textrm{w}\ge t_p$ are color coded (see colorbar). (b) Same quantity as shown in (a) with the same color code but plotted vs $L_f(t+t_\textrm{w})/L_f(t_\textrm{w})$. The wave vector $q$ increases from top to bottom as $q=0.05, 0.1, 0.2, 0.3, 0.5, 1.0$. (c) The MSD $\Delta(t,t_\textrm{w})$ vs. time $t$ on a double-logarithmic scale (linear plot in Fig. S5citeSI) for different waiting times (see colorbar). (d) Same quantity as shown in (c) with the same color code but plotted vs. the ratio of the corresponding filament lengths $L_f(t+t_\textrm{w})/L_f(t_\textrm{w})$.
• Figure 3: (a) The relaxation times $\tau_q$ and $\tau_q'$, respectively, determined from Eq. \ref{['powerML']} versus $t_\textrm{w}$ for $\kappa=50$ and different $q\ge 0.2$ (see colorbar). (b) The same data are shown versus $q$ and different $t_\textrm{w}\le 2\times 10^5$ (see colorbar). As before, data for $t_\textrm{w}<t_p$ are shown in dark gray.
• Figure 4: The large-scale relaxation time $\tau^\prime_0$ (full symbols) obtained from $C_q(t,t_\textrm{w})$ and the separating time $\tau_\textrm{s}$ for coarsening (open symbols) obtained from $L_f(t_\textrm{w})$ as function of $t_\textrm{w}$ on a double-logarithmic scale.
• Figure 5: (a) Illustration of the potentials governing the bead--spring chains. Cohesion energy $E_{\rm coh}$ is directly related to the cutoff distance $r_c$, which determines temporary reversible bonds in addition to the permanent FENE bonds. Inset: FENE (dashed black), LJ (blue), and combined potentials (black). (b) Geometric network descriptors based on the polymer skeleton (thinning algorithm), including strand length, thickness, junctions, chord lengths, and pore sizes. Reprinted with permission from kroger_ultra-slow_2025.