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Rheology is the study of the flow of matter. Microrheology extends rheology to a microscopic scale by observing the motion of a microscopic tracer particle. This reduces the required amount of the sample and additionally provides information about the local viscoelastic properties. Microrheology is called active, when the tracer is moved by means of an external force. This allows to probe the nonlinear regime of the rheological quantities, i.e. when the viscosity becomes a function of the applied force. However, the relation of micro- and macrorheological quantities is not yet fully understood.A mode coupling theory (MCT) for active microrheology has been developed by Gazuz et al. We extend this model for systems of hard spheres to higher forces. The novel approach is to decompose the memory kernel into parallel relaxation channels. In this thesis we will elaborate this new model. First, we give a derivation of the MCT equations of motion for the tracer density mode correlator. Throughout this thesis we will focus on the the long time limit of this quantity, also called tracer nonergodicity parameter. This quantity is only nonzero, when we are in the glassy state, i.e. the tracer remains localized for all times. We develop and implement a discretization scheme for the numerical solution of the long time limit. We propose several methods to find the critical force, where a delocalization transition of the tracer particle occurs and compare their results. Furthermore, we analyze the characteristics of the nonergodicity parameter. We find that the mean as well as the mean squared displacement increase strongly near the critical force. The reason for this is the evolution of an exponential tail in direction of the applied force in the probability distribution of the tracer particle. We propose a simple model to explain this tail. Finally, we can identify this feature also in the Fourier space solution.

A relation between equilibrium, steady-state, and waiting-time dependent dynamical two-time correlation functions in dense glass-forming liquids subject to homogeneous steady shear flow is discussed. The systems under study show pronounced shear thinning, i.e., a significant speedup in their steady-state slow relaxation as compared to equilibrium. An approximate relation that recovers the exact limit for small waiting times is derived following the integration through transients (ITT) approach for the nonequilibrium Smoluchowski dynamics, and is exemplified within a schematic model in the framework of the mode-coupling theory of the glass transition (MCT). Computer simulation results for the tagged-particle density correlation functions corresponding to wave vectors in the shear-gradient directions from both event-driven stochastic dynamics of a two-dimensional hard-disk system and from previously published Newtonian-dynamics simulations of a three-dimensional soft-sphere mixture are analyzed and compared with the predictions of the ITT-based approximation. Good qualitative and semi-quantitative agreement is found. Furthermore, for short waiting times, the theoretical description of the waiting time dependence shows excellent quantitative agreement to the simulations. This confirms the accuracy of the central approximation used earlier to derive fluctuation dissipation ratios (Phys. Rev. Lett. 102, 135701). For intermediate waiting times, the correlation functions decay faster at long times than the stationary ones. This behavior is predicted by our theory and observed in simulations., Comment: 16 pages, 12 figures, submitted to Phys Rev E

We present a detailed analysis of the fluctuation dissipation theorem (FDT) close to the glass transition in colloidal suspensions under steady shear using mode coupling approximations. Starting point is the many-particle Smoluchowski equation. Under shear, detailed balance is broken and the response functions in the stationary state are smaller at long times than estimated from the equilibrium FDT. An asymptotically constant relation connects response and fluctuations during the shear driven decay, restoring the form of the FDT with, however, a ratio different from the equilibrium one. At short times, the equilibrium FDT holds. We follow two independent approaches whose results are in qualitative agreement. To discuss the derived fluctuation dissipation ratios, we show an exact reformulation of the susceptibility which contains not the full Smoluchowski operator as in equilibrium, but only its well defined Hermitian part. This Hermitian part can be interpreted as governing the dynamics in the frame comoving with the probability current. We present a simple toy model which illustrates the FDT violation in the sheared colloidal system., 21 pages, 13 figures, submitted to Phys. Rev. E

In this thesis, different properties of colloidal suspensions near the glass transition under shear are analyzed theoretically. This system is in a non-equilibrium stationary state. Focus is put on the fluctuation dissipation theorem (FDT) which connects the response of the slightly perturbed system to the thermal fluctuations of the unperturbed system. It is violated under shear. In computer simulations, a very peculiar violation of the FDT was found for glasses under shear. The equilibrium form of the FDT seems to hold if the temperature is replaced by a different value, the so called effective temperature.We start with the analysis of different non-equilibrium correlation functions, namely the transient, the two-time and the stationary correlator. The difference between these correlators have so far only been studied in simulations and experiments. In chapter 3, the dynamics of a tagged particle in the sheared system is analyzed, e.g. the mean squared displacement.Concerning the FDT under shear, we achieve an approximative connection between the response function and measurable time derivatives of correlation functions and compare our results successfully with the simulations. In the simplest approximation, we find a universal FDT-violation, which is in agreement with the notion of an effective temperature. Corrections depend on the considered variable and hence contradict the notion of an effective temperature. We finish with an illustrative discussion of the FDT-violation and employ a simple toy model to illustrate the studied effects.

The fluctuation dissipation theorem (FDT) is studied close to the glass transition in colloidal suspensions under steady shear. Shear breaks detailed balance in the many-particle Smoluchowski equation, and gives response functions in the stationary state which are smaller at long times than estimated from the equilibrium FDT. During the final shear-driven decay, an asymptotically constant relation connects response and fluctuations, restoring the form of the FDT with, however, a ratio different from the equilibrium one., Comment: accepted to Phys. Rev. Lett.

We discuss the nonlinear rheology of dense colloidal dispersions at the glass transition. A first principles approach starting with interacting Brownian particles in given arbitrary homogeneous (incompressible) flow neglecting hydrodynamic interactions is sketched. It e.g. explains steady state flow curves for finite shear rates measured in dense suspensions of thermosensitive core-shell particles consisting of a polystyrene core and a crosslinked poly(N-isopropylacrylamide)(PNIPAM) shell. The exponents of simple and generalized Herschel Bulkley laws are computed for hard spheres., 3 pages, 1 figure; contribution to The XVth International Congress on Rheology, August 3-8, 2008, Monterey, California; submitted to J. Rheol

We consider the nonlinear rheology of dense colloidal suspensions under a time-dependent simple shear flow. Starting from the Smoluchowski equation for interacting Brownian particles advected by shearing (ignoring fluctuations in fluid velocity) we develop a formalism which enables the calculation of time-dependent, far-from-equilibrium averages. Taking shear-stress as an example we derive exactly a generalized Green-Kubo relation, and an equation of motion for the transient density correlator, involving a three-time memory function. Mode coupling approximations give a closed constitutive equation yielding the time-dependent stress for arbitrary shear rate history. We solve this equation numerically for the special case of a hard sphere glass subject to step-strain., 4 pages

Large scale molecular dynamics simulations are performed to study the steady state yielding dynamics of a well established simple glass. In contrast to the supercooled state, where the shear stress, $\sigma$, tends to zero at vanishing shear rate, $\gammadot$, a stress plateau forms in the glass which extends over about two decades in shear rate. This strongly suggests the existence of a finite dynamic yield stress in the glass, $\sigma^+ (T) \equiv \sigma(T; \gammadot \to 0) >0$. Furthermore, the temperature dependence of $\sigma^+$ suggests a yield stress discontinuity at the glass transition in agreement with recent theoretical predictions. We scrutinize and support this observation by testing explicitly for the assumptions (affine flow, absence of flow induced ordering) inherent in the theory. Also, a qualitative change of the flow curves enables us to bracket the glass transition temperature $T_c$ of the theory from above and (for the first time in simulations) {\it from below}. Furthermore, the structural relaxation time in the steady state behaves quite similar to the system viscosity at all studied shear rates and temperatures., Comment: 4 pages, 5 figures

A first principles approach to the nonlinear flow of dense suspensions is presented which captures shear thinning of colloidal fluids and dynamical yielding of colloidal glasses. The advection of density fluctuations plays a central role, suppressing the caging of particles and speeding up structural relaxation. A mode coupling approach is developed to explore these effects., 4 pages, 2 figures; slightly corrected version; Phys. Rev. Lett., to be published (2002)