The difficulty in designing materials includes increased/ decreased volume in natural or artificial materials. However, this phenomenon has mechanical explanations, but only for some specific materials. A general answer to the question “Why does volume sometimes decrease with the increase of temperature?” remains unclear.

A team of Penn State scientists has developed a theory to explain and then predict it: Zentropy. Zentropy is a play on entropy. It mentions the thermodynamic relationship of thermal expansion. When the volume increases due to higher temperature, it is equal to the negative derivative of entropy concerning pressure, i.e., the entropy of most material systems decreases with an increase in pressure.

Based on this, Zentropy theory predicts the change of volume as a function of temperature at a multiscale level. According to scientists, the theory could also predict anomalies of other physical properties of phases beyond volume. This could happen because entropy drives the responses of a system to external stimuli.

Zi-Kui Liu, Dorothy Pate Enright Professor of Materials Science and Engineering and primary investigator in the study, said, *“Macroscopic functionalities of materials stem from assemblies of microscopic states (microstates) at all scales at and below the scale of the macroscopic state of the investigation. These functionalities are challenging to predict because only one or a few microstates can be considered in a typical computational approach such as the predictive “from the beginning” calculations, which help determine the fundamental properties of materials.”*

*“This challenge becomes acute in materials with multiple phase transitions. This is often where the most transformative functionalities exist, such as superconductivity and giant electromechanical response.”*

The theory stacks these different scales into an entropy theory, confining other elements of an entire system. Thereby, it demonstrates a formula for the entropy of complex multiscale systems.

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

Zentropy could potentially change the way of material designing, especially those that are part of systems that are exposed to higher temperatures. It could also help understand and design materials with emergent properties, such as new superconductors and new ferroelectric materials that could lead to new electronics classes.

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

**Journal Reference:**

- Zi-Kui Liu, Yi Wang, Shun-Li Shang. Zentropy Theory for Positive and Negative Thermal Expansion. Journal of Phase Equilibria and Diffusion, 2022; DOI: 10.1007/s11669-022-00942-z