The increased demand for knowledge of small-scale behavior is making microrheology a key step in the understanding of biological systems, the design and use of advanced materials, and the novel applications of already existing materials. Most microrheological experiments and analyses to date have focused on linear viscoelastic properties, by correlating the random thermally-driven displacements of small tracers to the complex modulus through a generalized Stokes-Einstein relation, a process which is relatively well understood but which is limited in its scope to equilibrium systems. Many systems of practical interest are driven out of equilibrium and display nonlinear behaviors, but the microrheology work in this area has been scarce (conventional macroscale rheometers are used to measure both linear and nonlinear regimes), and the connection between micro and macroscale measurements is unclear. The need and interest in microscale measurements, whether due to the scarcity of the material or the size of system, makes microrheology an important technology, but unfortunately one which currently lacks fundamental understanding in many aspects. The proposed research examines the relation between micro and macrorheology through theoretical studies, with a particular and strong emphasis on the `active' (driven) and nonlinear regime. This research is best described in terms of applications to colloidal dispersions, and will focus on such systems because they offer very well-defined and well-characterized materials, allowing for comparisons to macroscale measurements. However, the impact of this research extends beyond colloidal systems as the theoretical foundation and general conclusions are extendable to many complex materials, especially biomaterials. Specific issues such as shear thickening in microrheology, the effect of the size ratio (tracer size to typical medium length scale) on the `continuum approximation' and on microscale velocity fluctuations, and the interactions between pairs of moving particles leading to structure formation are addressed. This is the first attempt to analyze the nonlinear behavior of materials within the context of microrheology and provides a fundamental validation of microrheology as a sound technique, critical for its continued application and future growth. Broader Impact: In a broader context, this research will engage PhD students who will become experts in colloidal physics and rheology, and who will go on to positions of leadership in industry and/or academia. To aid in the education of future generations of scientists and engineers, a microrheology section for the undergraduate chemical engineering laboratory at Caltech will be developed. To disseminate the research as widely as possible, in addition to publication in conventional technical journals, a website will be maintained with research results that are accessible to the general public. Since this research provides the theoretical foundation for a new experimental technique that has widespread application in science and technology, its impact is both very broad and deep.
