The problem had stuck with him for more than 15 years before Penn State Professor of Mathematics Xiantao Li thought of a whole new way of approaching it. Rather than trying to fix equations that don’t hold up at the nanoscale, he could start over and work from the ground up—or rather from the atom up.

Li was turned on to the problem by his mentor when he was working as a research associate at Princeton University in 2002. The problem? Textbook equations and models that describe how heat moves through materials don’t work very well when applied to nanomaterials. But understanding heat conductivity, or flow, in nanomaterials is critical for materials scientists working with nanoscale electronics devices. For example, they might want to understand how power dissipation, which is related to temperature, affects the performance of these devices.

“Usually we see that heat conductivity is related to a temperature gradient in a material, with heat moving predictably from higher temperature areas to lower temperature areas,” said Li. “But when people started to look at heat conductivity in nanomaterials, they found that the standard heat equation—Fourier’s law—and our typical heat conduction models break down. We knew about this problem for a long time but had no idea where to start. It was only a few years ago that I started thinking about it in a different way.”

Rather than “fixing” Fourier’s law and the usual models, he would start from scratch at the atomic level. Fundamentally, heat is generated and transferred through the vibrations of atoms, so Li used molecular dynamics simulations to study the movements of atoms and molecules, forming the basis of new models that would hold up at the nanoscale.

“By starting with the underlying physics, we can trust that no incorrect assumptions have been made,” said Li. “Then the challenge is to move from the microscopic level to a larger scale, to scale up the system.”

Starting at the atomic level necessitates keeping track of many different atoms and mathematical factors, so Li used a method called coarse graining to keep things under control, or in mathematical terms, to reduce the number of degrees of freedom. This method simplifies the complexity, for example by modeling the behavior of groups of atoms instead of individual atoms.

“The approach helps us pick out the things that are most relevant to the problem under study,” said Li. “But we can’t just throw away everything else. We found a way to project how everything else would behave as a whole and combined this into one term describing the net influence on the system.”

One of Li’s graduate students, Weiqi Chu, used a similar approach in her previous research.

“For my undergraduate research, I used a projection-related method to reduce the number of degrees of freedom in numerical algorithms,” said Chu, who first met Li during a joint program between Penn State and Peking University in China that brought her to University Park for three weeks during the summer between her junior and senior years. “It was really interesting to make connections between my original work and the current problem of nanoscale heat conduction and to use this method in a more applied setting.”

Ultimately, Li and Chu produced a suite of mathematical equations that describes many elements of heat conduction, including how heat fluctuates, how heat flow might be delayed, and characteristics of wave propagation–like behavior, such as the notation of a second sound speed, which describes how temperature can propagate like sound waves. They demonstrated the use of these equations using a nanotube system, and they hope to do so again with other nanomaterials, such as graphene, to investigate heat conduction in two-dimensional materials.

“Materials scientists don’t always have a way to directly measure parameters like relaxation time or second sound speed in nanomaterials,” said Li. “Now we have a formula, and scientists can use the resulting parameters when simulating the behavior of these important materials.”

Chu, who graduates in 2019 but will continue to collaborate on the project, is currently working with Li to incorporate stochasticity, or random events, into their models. Next, they plan to incorporate the movement of electrons, as heat conductivity and electron conductivity are often related but could be individually optimized to improve material performance.

“This is not a typical problem in applied mathematics, but the applied mathematics community is increasingly becoming interested in this kind of methodology,” said Li. “The National Science Foundation and my colleagues in the Department of Mathematics are very open minded and have been incredibly supportive of our work.”