Mathematicians Resolve a Long-Standing Problem in Materials Research-Mellon College of Science - Carnegie Mellon University

Monday, April 25, 2011

Mathematicians Resolve a Long-Standing Problem in Materials Research

The polycrystalline structure of a material evolves through a series of changes between the grain boundaries.
The polycrystalline structure of a material evolves through a series of changes between the grain boundaries.
PITTSBURGH—From the nanoscale wiring of a computer to the macroscale beams of steel in buildings and bridges, most materials are aggregates of crystals. The underlying theories that govern how these polycrystalline structures evolve to form the final end product have been of immense interest to materials scientists for years, but fully understanding the process — called coarsening — is exceedingly complex. Now, new research from a team of Carnegie Mellon University mathematicians and materials scientists has shed some light on how these polycrystalline structures evolve.  

This finding, reported in the April 1 issue of "Physical Review B" as an Editor's Selection, represents an enormous step forward in reducing the complexity of what scientists and engineers know about materials systems, and provides new insights into developing the predictive theories needed to precisely engineer a material.

Many materials used in engineering, whether natural or man-made, arise from a myriad of crystals, or grains, that grow as the material is heated. The types of boundaries between the grains and the manner in which they are connected affect a wide range of properties and, ultimately, a material's performance and lifetime.

"We've developed a completely new theory to describe how the grain boundary network evolves into an ideal distribution over time. Quantifying this and understanding how it happens leads to predictability of the process and is a step toward developing strategies for influencing these characteristics in predictable ways," said David Kinderlehrer, Alumni Professor of Mathematical Sciences, professor of Materials Science and Engineering and a member of CMU's Materials Research Science and Engineering Center (MRSEC).   

Together with Materials Science and Engineering Professor Katayun Barmak and Mathematical Sciences Professor Shlomo Ta'asan, Kinderlehrer developed a large-scale computer simulation of the evolution of grain boundaries under many diverse conditions. By harvesting vast amounts of data from the simulation, the MRSEC team was able to observe that, as the model system evolves, some of the grain boundaries get bigger, others get smaller, while some of the boundaries — and thus the grains — disappear completely. A version of the simulation can be seen at http://www.math.cmu.edu/video/red1-1.mov.

The grain boundaries, which are regions of "mismatch," are at higher energies compared to the more orderly regions within the grains. Eventually, during coarsening, low-energy boundaries outnumber the high-energy boundaries as the system as a whole strives to achieve the lowest energy state possible. The MRSEC team's latest findings explain how this happens. 

"The way in which the system drives the energy down and settles into equilibrium is consistent with its behaving like a solution to a differential equation called the Fokker-Planck equation," Kinderlehrer said.

"Experiments over the past decade have shown that there are more low-energy boundaries than high-energy boundaries, but quantifying this and understanding how it happens leads to predictability of the process," Barmak said. "We have not, in materials science, had all of the predictive theories that we need to truly engineer a material. Our work is a huge step forward in understanding the mechanistic origins of the evolution of the grain boundary character distribution."

The current research is an extension of a previous discovery by the MRSEC research group in 2004 that uncovered a special property related to the evolution of grain boundaries.

"As grains evolve, the constant fluctuation changes the nature of the type and frequency of grain boundaries you see. A few years ago we discovered that this grain boundary character distribution settles into something that is close to the Boltzmann distribution," Kinderlehrer said.

The Boltzmann distribution is very well known in statistical thermodynamics, describing how gas molecules settle into equilibrium. At equilibrium, the probability for a molecule to be in a given energy state goes down exponentially with increasing energy. The Boltzmann distribution also explains how energy is distributed among molecules inside living cells, so it is equally applicable to biological systems.

The question the MRSEC team addressed in the current research is how and why the grain boundary character distribution approaches a Boltzmann distribution in the first place — a long-standing problem in the materials science and mathematics fields.

In addition to Barmak, Kinderlehrer and Ta'asan, the authors include former CMU postdoctoral associates Eva Eggeling, Maria Emelianenko, Yekaterina Epshteyn and Richard Sharp.

This work was supported by grants from the National Science Foundation.

By: Jocelyn Duffy, jhduffy@andrew.cmu.edu, 412-268-9982