Primitive chain network simulations of the creep of entangled polymers

Although the behavior of entangled polymers in startup shear flows with constant shear rates has been thoroughly investigated, the response under creep has not been frequently considered. In this study, primitive chain network simulations, based on a multi-chain sliplink model, are modified so as to describe creep experiments. Creep simulations are compared to a literature dataset of an entangled polybutadiene solution, and qualitative agreement is found in the nonlinear range, i.e., under large stresses. Simulations allow one to extract details of the transient molecular motion, and results suggest that the deformation-induced disentanglement is relatively mild in the stress-controlled mode as compared to the rate-controlled one, because coherent molecular tumbling at the start of flow is disrupted.

💡 Research Summary

This paper extends the primitive chain network (PCN) simulation framework, originally developed for shear‑rate‑controlled flows, to model stress‑controlled creep experiments of entangled polymer melts. The authors base their approach on a multi‑chain slip‑link model in which each polymer is represented by a sequence of strands connected by slip‑links (binary entanglements). The dynamics of slip‑link positions and strand lengths are governed by Langevin‑type equations that balance drag, chain tension, osmotic pressure, and thermal noise. To emulate a creep test, a feedback loop is introduced: the imposed shear stress is held at a target value by continuously adjusting the background shear rate. Slip‑links are created when dangling ends encounter a nearby chain and are destroyed when a dangling end slides off a slip‑link, allowing the network topology to evolve dynamically during flow.

Parameter mapping is performed using experimental data from an entangled polybutadiene (PB) solution studied by Ge et al. (2014). The plateau modulus and entanglement molecular weight are used to estimate the average number of Kuhn segments per strand and the equilibrium strand length, which serve as the length, energy, and time units (ℓ₀, k_BT, τ₀) for the non‑dimensional simulations. Linear viscoelastic properties (storage and loss moduli) obtained from the Green‑Kubo stress autocorrelation agree well with experiment, confirming that the coarse‑grained model captures the low‑frequency dynamics despite neglecting high‑frequency Rouse and glassy modes.

The core of the study compares two simulation protocols: (i) shear‑rate‑controlled startup flows at fixed γ̇ and (ii) stress‑controlled creep where the shear stress σ_xy is held constant. For the rate‑controlled case, the simulations reproduce the experimentally observed shear‑stress overshoot and shear‑rate undershoot, reflecting the well‑known stretch‑orientation decoupling: the orientation tensor p* rises sharply, producing the stress peak, while the strand stretch m* lags behind, causing the subsequent stress decay. In contrast, the stress‑controlled simulations capture the early‑time undershoot in shear rate but reach a steady state more rapidly than the experiments, leading to an under‑prediction of the steady‑state shear rate at high stresses.

To elucidate this discrepancy, the authors analyze microscopic quantities: the number of entangled strands per molecule (h), strand stretch (m), orientation tensor (p*), and the surviving fraction of original slip‑links (s₁). In rate‑controlled flows, h decreases rapidly while m increases, and the orientation component q₊ (square of the shear‑gradient component of the end‑to‑end vector) exhibits a pronounced undershoot, indicating coherent molecular tumbling. Stress‑controlled flows, however, show a monotonic decline of q₊ without an undershoot, implying that the coherent tumbling is disrupted. Consequently, the decay of s₁ (a measure of deformation‑induced disentanglement) is slower in the stress‑controlled mode, confirming that the loss of entanglements is suppressed when the flow is driven by a constant stress rather than a constant rate.

The flow curve (steady‑state σ_xy versus γ̇) further clarifies the issue: while the simulated stress values match the experimental data at each γ̇, the simulated shear rates at a given σ_xy are systematically higher, especially in the nonlinear regime where σ_xy ∝ γ̇^0.24. This asymmetry explains why the model reproduces rate‑controlled startup data well but struggles with stress‑controlled creep.

Overall, the study demonstrates that the PCN slip‑link model can be adapted to stress‑controlled creep, providing qualitative agreement with experiments and, more importantly, offering molecular‑level insight into how the mode of deformation (rate vs. stress control) influences entanglement dynamics. The key finding is that coherent molecular tumbling, which accelerates disentanglement in rate‑controlled flows, is largely suppressed under constant‑stress conditions, leading to milder deformation‑induced disentanglement. These results have practical implications for polymer processing, where the choice between stress‑controlled and rate‑controlled operations can affect the evolution of the microstructure and thus the material properties. Future work may focus on refining parameter calibration to improve quantitative agreement in the nonlinear creep regime and extending the approach to more complex flow histories.