Transforming Energy Sources
Mathematical research on new materials is key to generate clean and renewable
energy and to help manage problems from existing energy sources. Partial differential
equations, the calculus of variations, continuum mechanics and numerical analysis,
among other mathematical areas, are well positioned to address these challenges.
Mathematics and Managing of Existing Supplies
Carbon Sequestration: The computational mathematics community has had a
huge impact on enhanced oil recovery techniques by developing efficient and accurate
models of multiphase flow through porous media. Darcy and Buckley-Leverett equations
for the flow in porous media have been used in Glimm, et al (2004). The new problem of
carbon dioxide subsurface reservoirs involves many of the same issues, but with
additional complexities inherent from the underlying chemistry.
Cellular and Granular Networks: These are ubiquitous in nature. They exhibit
behavior on many different length and time scales and are often found to be metastable.
The energetics and connectivity of the ensemble of the grain and the boundary network
during evolution plays a crucial role in determining the properties of a material across a
wide range of scales. Questions that arise include: What is the nature of patterns? One
view is that patterns are stable statistics of metastable systems. Can we predict the
pattern dynamics and the evolution of the microstructure? The challenge, from this
perspective, is to understand how to identify and validate such statistics, and to use
them for predictive theories (see Barmak, et al., 2008).
Mathematics and Design of New Materials for Sustainable Energy and Energy
Conversion
Energy Conversion: Electro-chemical systems such as batteries and fuel cells
require the transport of electrons, ions and multiple fluids in a controlled manner, through
a multiphase arrangement (electrodes and electrolytes each of which may itself be a
multiphase system). Further, important reactions occur at triple junctions or in the
presence of catalysts, ionic conductors may not be good electronic conductors, etc.
There have been dramatic improvements in individual components in recent years, but
this has not been manifested at the macroscopic/device level. It is here, in the design of
new materials for energy storage and conversion, that microstructure optimization will
make a fundamental contribution.
Recently, the prediction of hysteresis has acquired fresh significance in
connection with materials for energy conversion, since the efficiency of a conversion
process often depends on the size of an associated hysteresis loop. For a solid-to-solid
phase transformation, thermal hysteresis refers to a transformation temperature on
cooling that differs from that on heating. Hysteresis also occurs during stress-induced
transformation, with the stress needed to induce the forward transformation being
different from that causing the reverse transformation. Moving forward in this area will
require theoretical progress and experimental verification.
Nonlinear analysis and sharp interface models will play a pivotal role in this area
(see Delville, et al, 2011, Zhang, et al. 2009).
Photovoltaics: This a method of generating electrical power by converting solar
radiation into electricity using semiconductors that exhibit the photovoltaic effect. Here
one wants to maximize photon paths (to maximize the probability of capture) but