Small amplitude oscillatory shear flow superposed on parallel or perpendicular (orthogonal) steady shear of polydisperse linear polymers: The MLD model

Abstract

The “naïve” polydisperse MLD model is used to analyze small amplitude oscillatory shear flow superposed on steady parallel shear flow [1]. Analytical results are derived for the dynamic moduli, G‖′(γ˙,ω) and G‖″(γ˙,ω), as a function of the molecular weight distribution, applied steady shear rate and the spectrum of longest orientational relaxation times of the system. The analysis reveals that there are two distinct factors contributing to deviations of the dynamic moduli from their equilibrium, no steady flow values, G′(0,ω) and G″(0,ω). The first is an orientational effect that reduces the moduli for all ω, but not uniformly, by reducing the projection of the chain segments on the applied incremental oscillatory deformation. The second effect is a relaxation spectrum cut-off phenomenon that truncates the low ω dynamic moduli selectively as the orientational relaxation time is systematically reduced by CCR. The dynamic moduli for superposed small oscillations perpendicular a steady flow, G⊥′(γ˙,ω) and G⊥″(γ˙,ω), are also calculated analytically for the naïve polydisperse MLD model again assuming there is no stretch. Under this assumption the results are identical to those for small strain oscillations parallel to a steady flow which is contrary to experiment. The relaxation spectrum cut-off phenomenon in steady shear flow is well established both experimentally and theoretically dating back to the seminal work of Bersted [2,3]. The analytic results are compared with direct numerical simulations of the diluted stretch tube polydisperse MLD model [4] in small amplitude oscillations superposed on parallel steady shear flow. Quantitative differences betwen the analytical results generated from the naïve MLD model and the diluted stretch tube MLD model are identified and interpreted using the analytic solution structure for G‖∗(γ˙,ω) and G⊥∗(γ˙,ω) which applies to both models. The combination of generally relevant analytical results and direct numerical simulation provide a direct means to interpret the dynamic moduli G‖∗(γ˙,ω) and G⊥∗(γ˙,ω) in terms of molecular parameters characterizing the system using the well-established MLD model.