Hysteresis and Dynamic Response of a Polymer Chain under Oscillatory Force

The dynamics of polymers on the molecular scale impact the response of mechanical properties on the macroscopic level. In this study, we develop an analytical and numerical framework to describe the response of a single polymer chain subjected to an oscillatory pulling force with frequency ω. Using the Rouse model, we derive closed-form expressions for the chain extension and hysteresis area of an ideal (Gaussian) chain in the Hookean regime, as well as a scaling relation for excluded volume chains in the Pincus regime. The energy dissipated per cycle, quantified by the hysteresis area A(ω), exhibits distinct scaling regimes: A(ω) ∝ ω at low frequencies, where the chain follows the driving force adiabatically, and A(ω) ∝ ω–2ν/(2ν+1) at high frequencies, where only short-wavelength modes contribute. Here, ν = 1/2 for ideal chains and ν = 3/5 for real (self-avoiding) chains. In the latter case, excluded volume interactions lead to force-dependent relaxation times τ ∝ τRf(1–2ν)/ν deviating from the Rouse time τR. Upon appropriate normalization, data from a wide range of driving parameters, obtained by Monte Carlo and molecular dynamics simulations, collapse into a single master curve, confirming the universality of the underlying viscoelastic response. These results establish a molecular-level understanding of energy dissipation in flexible polymers under cyclic loading.