Higher harmonics in colloid rheology

In this thesis we analyze a schematic Mode-Coupling Theory in two time variables for small amplitude oscillatory shear with frequency ω. We derive the second order Taylor approximation for the density correlator and formulas for its long-time behavior. We also numerically solve the resulting equations, thus confirming the analytic results. We find that the solutions for the perturbation of the equilibrium correlator behave differently depending on the distance to the glass transition . In a liquid, < 0, the solutions decay to zero for long times, with a minimum for intermediate times, and an overlying decaying oscillation with frequencies ω and 2ω. We find a power-law governing the decay towards the minimum. Approaching the glass transition, the value of the minimum goes towards minus infinity, such that, at the glass transition, = 0, we get a diverging power-law solution with apparent exponent 1/3 and an overlying finite oscillation with frequencies ω and 2ω for all times. In the glass for > 0 we again find a finite solution for the Taylor approximation, but here with a constant offset and oscillation for all times. Thereby the amplitudes of the oscillations depend on the frequency and can be calculated using the Fourier transform of the solution of the schematic model in equilibrium. We compare this Taylor approximation with the full schematic model and find that it fits well for some intermediate times, but after that the full solution decays to zero, a feat that we see in the Taylor approximation only directly at the glass transition. We analyze the decay of the full model to find the timescale that governs the decay. The Taylor approximation can be used to derive a formula for the shear stress, including the first higher harmonic. We find that the first higher harmonic is given as the Fourier transform of the equilibrium solution multiplied with an oscillatory term of our approximation, evaluated at 2ω. We compare numerical results of this formula with experimental data for the limit of vanishing shear amplitude.