Analysis of medium amplitude oscillatory shear data of entangled linear and model comb polymers

Studying the mechanical response of nearly monodisperse linear and comb polystyrene (PS) melts to medium amplitude oscillatory shear (MAOS), Hyun and Wilhelm (Macromolecules 42, 411 (2009)] identified two important scaling relations: (1) The relative intensity I-3/1 of the third harmonic compared to the first harmonic scales with the strain amplitude according to gamma(2)(0). Consequently, a new nonlinear coefficient Q I-3/1 / gamma(2)(0) as well as the so-called intrinsic nonlinearity Q(0) lim gamma(0)-> 0 Q was introduced. (2) In the terminal relaxation regime, the intrinsic nonlinearity Q(0) (omega) scales with omega(2) and was found to be a very sensitive measure regarding molecular topology by identifying and separating relaxation processes in model branched polymers. A constitutive analysis based on a general single integral constitutive equation, which includes the Doi-Edwards model without (DE) and with (DEIA) independent alignment assumption as well as the molecular stress function (MSF) model, confirms both scaling relations. We show that the nonlinear viscoelastic moduli can be expressed as sums of their linear-viscolelastic counterparts at frequencies of omega, 2 omega, and 3 omega. The absolute value of Q(0)(omega) depends on the difference (alpha-beta) between the third order orientational effect (parameter alpha) according to the DE or DE IA model and the second order isotropic stretching effect (parameter beta) according to the MSF model. When comparing MAOS data to constitutive models, the apparent values of Q(0)(omega) measured in parallel-plate geometry have to be rescaled in order to take the non-uniform shear deformation into account. Both the DE and DE IA models fail to describe the experimental data. The data of four linear PS melts are quantitatively described by the MSF model with nonlinear parameters alpha = 5/21 (corresponding to the DE IA model) and beta=0.12 in the terminal relaxation regime. For the comb polymers, with the same orientational parameter of alpha=5/21, stretch parameters of beta=0.14