Modeling of Multiscale Inhomogeneous Materials with Periodic and Random Microstructure

Proposal #0204637

PI: Leonid Berlyand

Institution: Penn State University

Title: Modeling of Multiscale Inhomogeneous Materials with Periodic and Random Microstructure

ABSRACT

The scientific core of this proposal is centered around four areas. Homogenization is the common theme of all four areas, and it is expected that the proposed work will result in the development of new homogenization techniques and tools. The first area concerns the discrete network approximation for highly packed high-contrast composites. The main objective is to characterize the dependence of the effective transport properties of composites on the shapes and random locations of the filling particles in a rigorous mathematical framework with a controlled error estimate. Our objective in the second area is to obtain analytical formulas for the effective properties of composites. Such formulas reveal the explicit dependence of the effective properties on geometrical and physical parameters and provide a valuable physical insight, which can be used to test numerical algorithms developed for generic situations. The important practical issue of polydispersity will be addressed from different perspectives in the first as well as the second area, and the results will be compared. The third area concerns the rheology of complex fluids such as polymeric composites, suspensions and micellar fluids. The main features here are: (i) the interaction between micellar tubes or balls, which leads to a drastic change in the effective constitutive equations as compared with the constitutive law of the phases, (ii) the laminarization of the flow and drag reduction in viscoelastic flows (reduction in pressure). The fourth area is the exploration of novel features of homogenization for some nonlinear problems with nonstandard boundary conditions arising in modeling of superconductors and liquid crystals. The main objective is to characterize the dependence of the homogenization limit on the domain size and to explore the ramification of this size effect in physical problems.

Composite materials are of critical technological importance. The modeling and design of these materials raises fundamental questions of physics, materials science, and mathematics. Many of these questions are not yet answered, and mathematics has much to contribute. This project will advance our understanding of composite materials through a theoretical effort of the principal investigator, his collaborators and advisees coordinated with experimental studies by materials scientists. The long-term goal is to enhance the contribution from mathematics to very contemporary technological problems. This will be done with an emphasis on fostering interdisciplinary connections across neighboring disciplines, as well as between academia, laboratories and industries. The results of this research will be used in developing new materials with superior properties for various industrial needs. The main applications include the design of thermal protection packages for electronic industries, which will address the need for further miniaturization of modern electronic devices (e.g., cell phones); the use of fluids with polymer and micellar additives for cooling of various devices (e.g., reactors) and more efficient transport of oil; and the optimization of transport properties of polydispersed suspensions.