A new way to think about the movement of atoms in a material

University of Illinois materials scientist, Dallas Trinkle, has developed a comparative framework for evaluating models of diffusion, creating opportunities for the development of new or combined models. The paper, “Variation Principle for Mass Transport,” is published in Physical Review Letters, and has been highlighted by the American Physical Society.

The ongoing quest to build better batteries, create more dependable alloy materials, and reduce corrosion in metals relies on an understanding of how fast atoms diffuse in solid materials.

“When we make aluminum alloys for automotive products, we put them through heat treatment,” explains Dallas Trinkle, professor of materials science and engineering at the University of Illinois Urbana-Champaign. “And the length of time required in heat treatment is controlled by the diffusion of magnesium or copper atoms.”

“Atoms can move when they are next to an open space, a vacancy. The concentration of vacancies influences the total diffusion, which is really difficult to determine experimentally or when processing a material in a factory.” Because the true value of diffusion for a given material is unknown, optimization is based on estimates.

“There are experimental ways we can measure diffusion and they are often quite complicated with many different variables,” said Trinkle. “Temperature is one variable, but it also depends on other defects, such as what was done to a material prior to the experiment, or missing atoms.”

Diffusion can also be influenced by the presence of other elements. “For example, copper may diffuse differently in the presence of magnesium atoms.”

So many different variables need to be taken into consideration in experiments, that may be hard to control, which is where mathematical modeling supplements problems with approximations.

“Modeling diffusion has been a complicated problem for a long time,” said Trinkle. “But in a sense, it is also a simple problem, mathematically, in that we are adding up small displacements, or movements, and over time you get large amounts of motion from that.”

Now, there are several models used to approximate total diffusion, or diffusivity, in various materials.

“The challenge is that, so far, it has been impossible to know which of the many models to choose from will give you the best approximation for a specific problem.”

“All of these models make different approximations.  You don’t know either before or after a simulation, which one is the best, most accurate, choice.”

Trinkle’s paper shows, mathematically, that a whole series of commonly used, yet very different, computational approximations are actually versions of the same thing. “You can say that they are all trying to solve a particular problem in a particular way.”

“The paper shows that diffusivity has a minimum value, the true value minimizes something and so each approximation by definition will be above that minimum value, so you know that the approximation with the smallest amount of diffusivity is the most accurate model among them. You don’t know how much error there is, but you do know which one is the most accurate in that set for that problem,” said Trinkle. “It’s a variational approach to diffusion problems.”

As proof of concept, the paper compared different diffusion problems, finding the approximation that minimized the diffusion value without ever knowing what the true value of the diffusion.

The method can be used to evaluate which approximation model will be most accurate per case and provides a framework for new models and combinations of existing approximation models.

“It’s ultimately a different way to model and think about mass transport, or the movement of atoms in a material.”