Mathematics and Mechanics in the Search for New Materials (13w5004)

Arriving in Banff, Alberta Sunday, July 14 and departing Friday July 19, 2013


(University of Oxford)

Kaushik Bhattacharya (California Institute of Technology)

(International School for Advanced Studies, Trieste, Italy)


The scientific objective of this workshop is to bring together mathematicians and physical scientists to discuss mathematical problems that arise from materials science.

Materials science is undergoing rapid development. A number of new techniques for synthesis have recently been introduced, and it is possible today to synthesize compounds today that were impossible a decade ago. New techniques of characterization, including scanning probe microscopy, are now available and they have provided a new view on material microstructure. Finally, the availability of computational power has made it possible to study a number of previously inaccessible problems.

Yet, most materials are still discovered through inspired accident and improved through expert empiricism. The essential reason for this is the complexity and the range of interactions between the electronic, atomistic, microstructural and macroscopic scales that determine the properties of materials. This complexity and range make it very hard to develop a systematic and predictive theory of behavior. Indeed, quantum mechanics (Schödinger's equation) is used at the electronic scale, molecular dynamics at the atomistic scales, phase-field or similar theories at the microstructural scales and phenomenological constitutive laws at the macroscopic scales. These theories involve fundamentally different variables and different concepts, and how one is subsumed by the other is not completely clear. A further difficulty is that interesting behavior often occurs at isolated compositions or microstructures.

Mathematics can, and has, contributed to understanding this complexity. Homogenization theory has provided insights into composite materials, meta-materials and in understanding discrete (atomistic) to continuum limits. Regularity of the solutions of partial differential equations has enabled an understanding of thin films (especially what is now called rigidity) and failure through cavitation and fracture. The calculus of variations has provided a clear understanding of why microstructure arises in phase transitions, and tools like Young measures have led to methods which give an implicit understanding of microstructure. Bifurcation theory is leading to new understanding of pattern formation during microstructure evolution. In each of these examples, essential questions of materials science modeled in a rigorous manner have led to fundamental and deep questions of mathematics, and progress in mathematics has led to non-trivial insights into material behavior. Further, mathematical insights have led to new computational approaches. Indeed, we are now beginning to see examples where this interactive advance is leading to the development of entirely new materials and structures.

Our objective in this workshop is to address this challenge with a special focus on two classes of materials -- active materials and energy-related materials. Active materials like shape-memory alloys, ferroelectrics, and ferromagnetic materials combine mechanical, electrical, magnetic and other properties. Their interesting properties make them the to be of interest in a number of applications, while at the same time, they have motivated a number of mathematical works. Further, ideas developed in this context have found applications in other areas of materials science. Much of the mathematical work on these materials has focussed on the interesting microstructural scale, but it has recently begun to address atomistic and electronic aspects as well. Thus this class of materials provide a very interesting test-bed for the development of broader ideas. Materials of interest for energy conversion and storage including fuel cells, batteries and hydrogen storage have similar issues. Therefore given their current scientific and technological interest, they provide a natural direction for expanding this discussion.

We propose to invite a cross-disciplinary group of researchers including mathematicians, mechanicians, materials scientists and physicists.

A social objective of this workshop is to celebrate the 60th birthday of Richard James, University of Minnesota. James has been one of the leaders of this field, and we believe that the connection to his birthday will attract the pre-eminent researchers in the field. Thus, we believe that this social objective will in fact further the scientific objective.