Scaling laws in the rheology of colloidal dispersions

In this thesis, starting from a schematic mode-coupling model, the dynamics and the ow behavior of colloidal dispersions under stationary shearing close to the glass transition is analyzed. Both analytical and numerical methods are applied. The most important control parameters of the model are the shear rate and the separation parameter measuring the distance from the glass transition point. A negative separation parameter de nes a liquid-like state and a positive value for the separation parameter leads to a glassy state. After discussing the basic properties of the model and the corresponding numerical algorithm, the main part of the thesis subdivides into three chapters: The β-relaxation process After motivating the equation of motion and discussing the range of validity, a useful aspect, the two-parameter scaling law, is discussed. This allows a precise de nition of the liquid-, transitionand the yielding glass region and the introduction of natural time scales. Analytical expressions are derived to describe the dynamics in the three di erent regions for all relevant time scales and the results are tested numerically. The dynamics can be described by di erent generalized power series and an exponential function. Four of these power series are necessary to describe the dynamics in the liquid region. The dynamics in the transition region can be described by two power series. In the yielding glass region, three power series and an exponential function are necessary to describe the dynamics on all time scales. The short time asymptotes and the shear-dominated long time asymptotes are not dependent on the separation parameter. Flow curves In this chapter the steady state shear stress as a function of the shear rate is analyzed. After discussing the basic properties of the ow curves, an asymptotic expansion using the analytical expression for the long time dynamics and the two-parameter scaling law for the β-relaxation process leads to an analytical expression for the ow curves, the Λ model. This model can be evaluated analytically in three special cases. The results are power series with non-integer exponents. In the general case approximations using numerically motivated formulae are required. For small positive separation parameters the Λ model describes the ow curves correctly for sufciently small shear rates. For small negative separation parameters the Λ model describes the ow curves correctly in some nite shear rate windows. Finally, the in ection point occurring for negative separation parameters in a double logarithmic plot of the ow curves is analyzed. An approximation of the ow curve by the corresponding in ection tangent leads to an e ective power law.