Brownian simulations for fracture of star polymer phantom networks

Based on a recent simulation study [Masubuchi et al., Macromolecules, 56, 9359 (2023)], the cycle rank plays a significant role in determining the fracture characteristics of network polymers. However, the study only considered energy-minimized networks without the effects of thermal agitation. We conducted Brownian dynamics simulations at various stretch rates to address this gap. The results showed that even with Brownian motion, the strain and stress at the break obtained for different node functionalities and conversion ratios exhibited master curves if plotted against cycle rank. These master curves were dependent on the strain rate, with the curves tending to approach those observed in energy-minimized simulations as the strain rate decreased, even though the fracture process was affected by the competition against Brownian motion, elongation, and bond degradation.

💡 Research Summary

In this work the authors extend previous investigations of fracture in star‑polymer phantom networks by explicitly incorporating thermal fluctuations through Brownian dynamics. Networks are built from Rouse‑type star polymers with a variable number of arms (functionality f = 3–8) and a fixed arm length of five beads. After equilibrating the polymer solution, end‑linking reactions are introduced with a prescribed reaction distance and probability, generating networks with conversion ratios E ranging from 0.6 to 0.95. The resulting networks are subjected to uniaxial stretching at three different strain rates (𝛾̇ = 10⁻³, 10⁻⁴, 10⁻⁵ in simulation time units). Particle motion follows the Langevin equation, with a non‑linear FENE‑type spring potential that prevents premature thermal bond breakage; a bond is removed when its length exceeds a critical value b_break.

The authors record true stress–true strain curves, identify the peak stress (Y₄) and the corresponding strain (F₄) as fracture indicators, and monitor the fraction of broken strands (E₄₅) throughout deformation. Several key observations emerge: (i) Even with Brownian motion, the peak stress and strain for networks of different functionality f and conversion E collapse onto master curves when plotted against the cycle rank χ (the number of independent loops in the network). (ii) These master curves shift systematically with strain rate; slower rates bring the curves closer to those obtained from energy‑minimized (deterministic) simulations, indicating that the competition between flow, structural relaxation, and thermal bond scission governs the observed behavior. (iii) At high strain rates the stress after the peak does not decay to zero because residual clusters generate a drag force, leading to a non‑vanishing tail in the stress–strain response. (iv) The broken‑strand fraction grows continuously at high rates, whereas at low rates it saturates, reflecting the balance between bond breaking and network relaxation. (v) Power‑law scaling of Y₄/ρ_branch and F₄ with χ is observed, with exponents –0.16 and 0.42 respectively, independent of strain rate but distinct from the –½ and 1/3 reported in earlier energy‑minimized studies. The discrepancy is attributed to a more conservative choice of the thermal degradation parameter b_break in the present simulations.

System size effects are also examined: larger systems require slower strain rates to achieve convergence with the energy‑minimized master curves, consistent with longer Rouse relaxation times for larger clusters. The authors conclude that the cycle rank remains a robust structural descriptor of fracture, even when thermal fluctuations and kinetic effects are present. However, they acknowledge that additional physical ingredients—excluded volume, entanglements, chain rigidity, osmotic forces—are absent from the current phantom‑chain model, and that mapping the simulation results to experimental gels will require incorporating these factors. Ongoing work aims to relate χ to other topological measures such as higher‑order loop density, minimum path length, and loop opening length, thereby deepening the theoretical understanding of how network topology controls macroscopic fracture.