Theory of time-dependent and non-local microrheology

Abstract

Microrheology is the study of the rheological properties of materials by measuring the motion of tracer particles embedded within the material This thesis extends the concept of mode- coupling microrheology (MCT-MR) to incorporate time-dependent external forces. The gen- eralization was achieved under few conditions (e.g., constant direction of the force), still result- ing in a more complex mathematical structure, while maintaining strong similarities with the previous constant force case. The Zwanzig-Mori (ZM) equations of motion for the transient density correlators appearing within the integration through transients (ITT) approach, were derived successfully. The introduction of parallel relaxation channels posed further challenges, while a reasonable handling of the geometric structure is provided. The quantitative study predominantly relies on schematic models rather than a fully wave- vector-dependent theory, which is currently deemed impractical for numerical implementation. These are derived for the familiar cases of components parallel and perpendicular to the applied force. We present different numerical implementations of these models, particularly for sce- narios involving step-force protocols, such as in recoil experiments, where the tracer particle is released after a certain duration of driving. This poses a simple but immensely interest- ing example of a time-dependent microrheological protocol that has recently been studied in experiments. The linear response case is treated analytically to derive a simple relation connecting the recoil to the equilibrium mean squared displacement. This result evaluated for existing MCT data of hard spheres is found to provide a reasonable quantitative description of experiments in a visoelastic system. Furthermore, the relation is used to test the numerical implementation of the schematic models, showing only minor quantitative deviations, suggesting that the pre- sented solutions are sufficiently robust. The schematic models are evaluated numerically for a number of time-dependencies in the nonlinear regime, not only, but most notably for the recoil protocol. In this setting, we find that the models can exhibit a non-monotonic dependence of the recoil magnitude on the driving force, which we attribute to the dominance of plastic de- formations in the surrounding bath, overtaking of elastic contributions to the response. These results are assessed by comparison to recoil simulations and previous observations of a critical force in the glassy state. Additionally, the thesis introduces a framework for microrheology that considers all prod- ucts of bath and tracer density modes rather than only linear densities. The previous theory for constant forces is expanded to include the full 2×2 system of tracer-bath correlators, which will be evaluated in the future. In general the theory offers the possibility to treat microrheol- ogy in a nonlocal fashion aiming to resolve the space-dependent deformations in the bath, e.g. the local melting of a glassy material.