Ph.D. Research
My Ph.D. thesis is available here. You can download the entire thesis here (35 MB).
We typically learn in grade school that all materials can be classified as either being a solid, liquid, or a gas. However this is a vast simplification of real life. Materials we use daily in critical applications are often neither solid nor liquid: they can be soft, squishy, stretchy, slimy, tacky, stringy, etc. How does one classify chewing gum, or silly putty, or even toothpaste? How about a sand dune that remains stable in a mound for centuries under quiescent conditions - much like a solid - but can flow downhill like a liquid when a strong breeze destabilizes the dune? See this fascinating video of how sand can flow like a liquid.
Our simple grade school classification of materials into the three buckets of solids, liquids and gasses is woefully inadequate to capture the richness of materials we use daily. More generally, these are called complex fluids, and comprise a diverse class of materials such as emulsions, foams, colloidal dispersions, suspensions, polymer melts and solutions, chemically and physically cross-linked gels, etc. Ultimately, these complex behaviors arise from the constituent building blocks of the material: the chemical identity of the molecules and the material microstructure, i.e., manner in which the building blocks are arranged.
Because complex fluids are used in advanced applications, it is important to possess mathematical models that can predict their mechanical behavior upon stretching, squeezing or applying a force to it. Developing such relationships between applied deformation and resulting stresses (or vice-versa) for the complex fluid is called constitutive modeling. My thesis developed constitutive models for complex fluids that have fractal or multiscale microstructures.
Many systems in nature contain patterns or networks with multiple length scales. For example, the rocky faces of the great mountain ranges of the Earth exhibit cracks, fractures and fissures that are of many different sizes. There are long and thick cracks that run down the face of the mountain, from which shorter and thinner cracks branch off. From these smaller cracks, in turn, still smaller cracks originate, and this process continues over many different size scales (or length scales) along the mountain side. We term such systems that possess features across multiple length scales as multiscale systems. A few examples of multiscale systems found in nature are river deltas, with streams breaking up into smaller and smaller tributaries, the perimeter of rain clouds, continental coastlines, and galaxy clustering in the cosmos. Such multiscale systems often share the properties of self-similarity, or being invariant under a change of scale. In other words, they look the same when inspected from nearby or from afar. The mathematical ideal of such scale-invariant systems are fractal geometries. Mathematicians have considered the concepts of scale-invariance and scale-free patterns through various striking and beautiful geometrical constructions such as the Koch snowflake, the Sierpinski Gasket, and Peano curves.
In my thesis, I was primarily interested in understanding multiscale physical systems whose microstructures approximate this mathematical ideal of perfect fractals. I developed compact constitutive models to characterize, describe and predict the mechanical behavior of multiscale complex fluids under a single powerful mathematical framework utilizing fractional derivatives or non-integer order derivatives. The results of my work are useful for the advanced applications of a diverse range of complex fluids including crosslinked polymer networks, microgel dispersions, foams, colloidal suspensions, and soft glassy materials.