The focus of this research is on the mathematical theory of anisotropic complex fluids, such as viscoelastic materials, liquid crystals, and ionic fluids in proteins and bio-solutions. The research will include two categories of problems: (1) Mathematical analysis of the systems with basic variational structures that are common to complex fluids. (2) The application of such a general framework, in particular to the study of ionic fluids in cells and proteins. Complex fluids, including mixtures and solutions, are abundant in daily life. The complicated phenomena and properties exhibited by these materials reflect the coupling and competition between microscopic interactions and macroscopic dynamics. The project will study the underlying energetic variational structures that are common to all these materials. New mathematical approaches in the modeling, analysis and numerical simulations will be developed in order to understand these multiscale-multiphysics systems.
Major efforts will be devoted to the study of ionic solutions in proteins. Molecular biology has shown that understanding behavior of proteins and nucleic acids in biological plasma (ionic solutions) is essential for understanding and controlling many biological systems. Nearly all biological processes involve the transport of electrically charged ions. Ion channels are instrumental for a vast number of biological phenomena, with many diseases (such as cystic fibrosis) and drug treatments dependent on channel functioning. In mathematical analysis of these phenomena, variational methods show a great promise because they deal naturally with systems with many components that flow between definite boundaries. In this research we will apply energy variational methods to study biological plasma, ionic solutions of sodium, potassium, and chloride ions as they flow through specific protein channels.
Complex fluids, including mixtures and solutions, are abundant in our daily life, including wide varieties of mixtures, polymeric solutions, colloidal dispersions, biofluids, electro- rheological fluids, ionic fluids, liquid crystals and liquid crystal polymers. Unlike simple solids, ideal gases or liquids, and the diluted solutions, the model equations for complex fluids continue to evolve as new experimental evidence and applications become available. The complicated phenomena and prop- erties exhibited by these materials reflects the coupling and competition between the microscopic interactions and the macroscopic dynamics. New mathematical theories are needed to resolve the ensemble of micro-elements, their intermolecular and distortional elastic interactions, their coupling to hydrodynamics and the applied electric or magnetic fields. This focus of this proposal is to make several distinct mathematical advances and to integrate those advances into studies of these physical and biological complex fluids. The main focus is on understanding the coupling between microstructures and the macroscopic properties, such as the kinetic transport and the induced elastic stresses in viscoelasticitic materials. Major effort will be devoted in the studying of ionic solutions in proteins. The proposed research has potential impact on many disciplines. The interdisciplinary nature of the proposal also provided unique opportunities to educate and train graduate and undergraduate students as well as postdoctoral associates in physical and biological modeling, numerical simulation, and mathematical analysis of complex fluids. The major findings of this proposal include: 1) Majority of the biological environments are those of non-ideal situations. In particular, in these non-diluted solutions, particle interactions have significant contributions to the overall properties. We had developed a generalized diffusion theory based on the energetic variational approaches. The approaches distinguished the equilibrium contribution of the systems, as well as the dissipative contributions. 2) We had worked on a specic biological application of our theory for sensoring mechanism of gramicidin A peptides. The preliminary work had demonstrated the robustness of our model. 3) We had develop a effective boundary potential trap model to capture the complicated boundary effects of ionic solutions. In particular, the model is consistent with the classical theories for diluted solutions and large containers, and at the same time, allow us to extend the theory to sub-Debye length region, which are important for microfluidic and biological applications. 4) Develop a new thermodynamics consistent model and algorithms to model mixtures of Newtonian fluids with big contrast of densities. 5) We had justified the derivation of continuum description of ionic transport in nano size channels, through time averaging of the kinetic description. Moreover, this is the first step to deal with the multiple temporal scales arising naturally in biology. 6) Establish the energetic variational structures for integral master equations. This allows us to build explicit relations between stochastic processes, general diffusions and nonlinear partial differential equations.