New entropy theory can solve material design problems

Entropy is the measure of disorder in a system that occurs over a period of time with no energy invested in restoring order. Zentropy integrates entropy at multi-scale levels. Credit: Elizabeth Flores-Gomez Murray/Jennifer M. McCann, Penn State

A challenge in material design is that in both natural and man-made materials, the volume sometimes decreases or increases with increasing temperature. Although there are mechanical explanations for this phenomenon for some specific materials, a general understanding of why it sometimes occurs is still lacking.

However, a team of Penn State researchers has come up with a theory to explain and then predict it: Zentropy.

Zentropy is a play on entropy, a concept central to the second law of thermodynamics that expresses the measure of disorder of a system that occurs over a period of time when there is no energy applied to maintain order in the system. Think of a playroom in a kindergarten; if no energy is put into keeping it tidy, it quickly becomes messy with toys all over the floor, a state of high entropy. If the energy is set up via cleaning and organizing the room after the children have left, the room returns to a state of order and low entropy.

Zentropy theory notes that the thermodynamic relationship of thermal expansion, as volume increases due to higher temperature, is equal to the negative derivative of entropy with respect to pressure, i.e., the entropy of most material systems decreases with increasing pressure. This allows the Zentropy theory to predict volume change as a function of temperature at a multi-scale level, i.e. the different scales within a system. Each state of matter has its own entropy, and different parts of a system have their own entropy.

“So that’s what the Zentropy equation is all about, stacking them together. It creates a partition function which is the sum of all the entropy scales.

Zi-Kui Liu, Dorothy Pate Enright Professor of Materials Science and Engineering

“When we talk about configuration entropy (different ways that particles rearrange themselves within a system), this entropy is only part of the entropy of the system,” said Zi-Kui Liu, Dorothy Pate Enright professor of materials science and engineering and principal investigator in the study. “So you have to add the entropy of the individual components of that system into the equation and then you consider the different scales, the universe, the Earth, the people, the materials, these are different scales within different systems .”

The authors of the study published in the Journal of phase equilibria and diffusion, believe that Zentropy may be able to predict anomalies of other physical properties of phases beyond volume. Indeed, the responses of a system to external stimuli are determined by entropy.

The macroscopic functionalities of materials come from assemblies of microscopic states (microstates) at all scales at and below the scale of the macroscopic state of investigation. These features are difficult to predict because only one or a few microstates can be accounted for in a typical computational approach such as “from the start” predictive calculations, which help determine the fundamental properties of materials.

“This challenge becomes acute in materials with multiple phase transitions, which are processes that convert matter from one state to another, like the vaporization of a liquid,” Liu said. “That’s often where the most transformative features are, such as superconductivity and giant electromechanical response.”

According to Liu, zentropy theory “stacks” these different scales into a theory of entropy that encompasses the different elements of an entire system, presenting a nested formula for the entropy of complex multi-scale systems.

“You have these different scales and you can stack them with the Zentropy theory,” Liu said. “For example, atoms as a vibrational property, it’s on a small scale, and then you have an electronic interaction, that even lower scale. So now how to stack them to cover the whole system? So that’s what the Zentropy equation is all about, stacking them. It creates a partition function which is the sum of all the entropy scales.

“The Zentropy theory has the potential to be applied to larger systems because entropy drives changes in all systems, whether black holes, planets, societies, or forests.”

Zi-Kui Liu, Dorothy Pate Enright Professor of Materials Science and Engineering

This approach is something Liu’s lab has been working on for more than 10 years and five different published studies.

“The idea actually became very simple after studying and understanding it,” Liu said.

Zentropy has the potential to change the way materials are engineered, especially those that are part of systems exposed to higher temperatures. These temperatures, given the thermal expansion, could cause problems if the materials expand.

“This has the potential to enable the fundamental understanding and design of materials with emerging properties, such as new superconductors and new ferroelectric materials that could potentially lead to new classes of electronics,” Liu said. “In addition, other applications such as designing better structural materials that withstand higher temperatures are also possible.”

Although there are benefits for society in general, researchers could apply Zentropy to several areas. This is because of the way entropy is present in all systems.

“The Zentropy theory has the potential to be applied to larger systems because entropy drives changes in all systems, whether black holes, planets, societies or forests,” said Liu.

Reference: “Zentropy Theory for Positive and Negative Thermal Expansion” by Zi-Kui Liu, Yi Wang, and Shun-Li Shang, February 3, 2022, Journal of phase equilibria and diffusion.
DOI: 10.1007/s11669-022-00942-z

Besides Liu, other study authors include Yi Wang, research professor of materials science and engineering, and Shun-Li Shang, research professor of materials science and engineering. The work was supported by the National Science Foundation, the Department of Energy and the Department of Defense.

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