Continuous relaxation spectra for constitutive models in medium-amplitude oscillatory shear

Abstract

We derive and demonstrate how analytical solutions for strain-controlled medium-amplitude oscillatory shear (MAOS) can be coupled with a continuous distribution of relaxation times. This applies generally to the vast library of existing MAOS models, including those that are not time-strain separable. The theoretical results are important for improved certainty in model parameters, since their number can be reduced dramatically compared to a discrete distribution of relaxation times. We exemplify this continuous spectrum MAOS approach by modeling experimental data for a transient network formed from an aqueous semidilute unentangled solution of poly(vinyl alcohol) and sodium tetraborate (Borax). The full frequency-dependent MAOS signatures are fit well by only five parameters: three linear parameters for a log-normal spectral distribution and two nonlinear parameters for the strength of the nonlinearity and its cutoff time scale. Remarkably, longer modes (τ>τw) are not activated in the asymptotically nonlinear regime. Although this may be compatible with the possible mechanisms for the shear nonlinearities, the reason for the hard cutoff of time scales is currently unknown. Our results also suggest that the sign change location for the third-harmonic viscous nonlinearity may be sensitive to large-scale structural features (such as molar mass distribution or long-chain branching) that control the polydispersity of terminal relaxation times.