Math shows how materials meld to last

CARNEGIE MELLON (US) — A large-scale computer simulation of the evolution of grain boundaries under a variety of conditions is giving scientists insight into what is needed to precisely engineer a material.

Many materials, 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.

Using data from the simulation a team of mathematicians and material scientists observed that as the model system evolves, some of the grain boundaries get bigger, others get smaller, while some of the boundaries—and the grains—disappear completely.


The research is reported in the journal Physical Review B.

“We’ve developed a completely new theory to describe how the grain boundary network evolves into an ideal distribution over time,” says David Kinderlehrer, professor of mathematical sciences and materials science and engineering at Carnegie Mellon University.

“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.”

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 new research clarifies how the process 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 says.

“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,” says Katayun Barmak, professor of materials science and engineering.

“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 new study is an extension of a 2004 discovery 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 says.

Well-known in statistical thermodynamics, the Boltzmann distribution describes how gas molecules settle into equilibrium and how energy is distributed among molecules inside living cells.

The National Science Foundation funded the research.

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