[Image above] After inspiring almost eight decades of research, researchers in the U.S. and Brazil argue it may be time to retire the Kauzmann paradox. Credit: PxHere
Knowing when to call it quits is not just a struggle for people in romantic relationships. Any relationship or activity you’ve invested time, money, and energy into is hard to step away from, especially if you previously benefitted from it.
In the scientific community, researchers often face the hard choice of continuing or ending investigations into different hypotheses or theories. The decision can be complicated when the hypothesis or theory is widely known and referenced, or it helped spark studies that led to great discoveries.
In glass research, one hypothesis that has sparked decades of debate is the Kauzmann paradox. Initially proposed in the 1940s by a U.S. chemist who was simply sharing his ponderings, the controversial conjecture has inspired hundreds of follow-up papers in a variety of scientific disciplines.
Despite this “extraordinary impact,” it may be time to put the paradox to rest, a team of researchers in the U.S. and Brazil argue in a new paper.
The researchers are led by ACerS Fellows Edgar D. Zanotto and John C. Mauro. Zanotto is senior professor at the Federal University of São Carlos (UFSCar) and director of the Center for Research, Technology, and Education in Vitreous Materials (CeRTEV). Mauro is Dorothy Pate Enright Professor and associate head for graduate education at The Pennsylvania State University.
Individually and together, Zanotto and Mauro have published numerous papers that have helped guide the field of glass science. For example, a 2017 paper provided a new definition of glass, a 2022 paper clarified the definition and role of nucleating agents within glass-ceramic systems, and two papers this year described factors that affect the atomic structure of glass-ceramics and glass, respectively.
Now, their new paper on the Kauzmann paradox provides a detailed look at this conjecture and the 75 years of research it inspired. With this thorough review, they hope to show why “future [glass] work should be focused elsewhere.”
The Kauzmann paradox: A mid-20th century take on the nature of glass
Prior to and during the 1940s, glass was viewed as an “undefined and haphazard product” that was “inaccessible to research because of its state of non-equilibrium,” as described by Bergler in 1932.
At the time, most knowledge about glasses came from calorimetry and viscosity measurements due to the scarcity of spectroscopy techniques. Based on these limited data gathering capabilities, researchers believed the glass transition could be characterized as a second-order thermodynamic transition, i.e., a process that involves an abrupt change in a material’s thermal expansion coefficient and heat capacity.
With the advancement of various spectroscopy techniques, researchers now know that while the glass transition does lead to a change in thermodynamic properties, this change occurs gradually rather than abruptly. In other words, the glass transition is inherently a kinetic transition rather than a thermodynamic transition.
On the other hand, if a supercooled liquid crystallizes rather than transitions into a glass, this process is a first-order thermodynamic phase transition, i.e., a process that involves an abrupt change in a material’s first-order thermodynamic properties, such as density.
In 1948, Princeton University professor Walter Kauzmann published a paper containing his ponderings about the nature of glass. Because of the misconception described above regarding the glass transition, Kauzmann found himself debating a potentially troublesome question—if given enough time, could a liquid remain in the liquid state at nearly all temperatures? In other words, could it avoid transitioning to a glassy or crystal state if cooled at a slow enough rate?
The troublesome nature of this question was illustrated in the entropy vs. temperature diagrams that Kauzmann was using. These graphs, which were based on theoretical extrapolations, appeared to show that yes, a liquid could be cooled slowly enough to 0 K to avoid a phase transition. But for some compounds, such as glucose and lactic acid, these liquids would reach zero entropy well above 0 K.
“If continued extrapolation of the supercooled liquid line were to occur, then the entropy would become negative at a non-zero temperature,” the Zanotto and Mauro-led team explain in their paper.
In other words, it suggested that a material could have a lower entropy in the liquid phase than crystal phase, which is not possible.
To avoid this paradox, Kauzmann proposed that the supercooled liquid had no choice but to crystallize—a “forced” crystallization, if you will. This crystallization would happen so quickly that there was no time to study the properties of the supercooled liquid before the phase transformation. So, to measure the supercooled liquid’s properties, it would need to be formed into a glass.
On this note, Kauzmann theorized that there existed two different metastable liquid states: one respective of the supercooled liquid to the glass, and one respective of the supercooled liquid to the crystal. He deemed the temperature at which the free energy barriers for each case became equal—i.e., the point at which a supercooled liquid had an equal probability of turning into either a crystal or glass—as a “pseudocritical point,” which would later be known as the Kauzmann temperature (TK).
Kauzmann posited that TK could be above or below the glass transition temperature. However, at any point below TK, the supercooled liquid would immediately crystallize as the barrier to crystallization was assumed to be lower than the barrier for relaxation (turning into a glass).
The Kauzmann paradox debate
Kauzmann could not have known that his ponderings in 1948 would serve as a launchpad for glass research over the next 75 years. As of 2022, Kauzmann’s seminal article has received nearly 3,500 citations, and his thought-provoking interpretations of supercooled liquids have resulted in more than 400 subsequent research articles.
Throughout all these papers, there are three main approaches to investigating Kauzmann’s so-called “entropy catastrophe.”
- Given the theoretical nature of the Kauzmann paradox and TK, researchers have used energy landscapes and the corresponding concept of an “ideal glass transition” to answer the questions brought forth by Kauzmann.
- In response to Kauzmann’s proposal of a “forced” crystallization, experimental and atomistic simulation approaches primarily focused on crystal nucleation to explore the paradox.
- Researchers explored the implications of relating the Vogel–Fulcher–Tammann (VFT) model for viscosity, which was developed in the 1920s and known by Kauzmann, to the Adam–Gibbs formalism for configurational entropy, which was developed in the 1960s and became a popular method for understanding supercooled liquid dynamics in the 1980s.
Overall, these investigations with more advanced and informed models found no conclusive evidence to support Kauzmann’s paradox and the accompanying Kauzmann temperature.
“Most work from crystallization and energy landscape analysis shows little support that TK refers to a meaningful temperature despite the initial conceptions provided via VFT,” the Zanotto and Mauro-led team write. “With development of the AM [Avramov–Milchev] and MYEGA [Mauro–Yue–Ellison–Gupta–Allan] viscosity models, it is not necessary to incorporate TK in any manner.”
Additionally, “Simulation work has also supported a smooth and gradual decrease in entropy to 0 K with no inclination of forced crystallization or vitrification,” they add.
Because of these findings, Zanotto, Mauro, and their colleagues conclude that, despite the “extraordinary impact” that Kauzmann’s interpretations had on the glass literature, with no conclusive evidence to support his concepts, “future work should be focused elsewhere.”
The paper, published in Acta Materialia, is “Cracking the Kauzmann paradox” (DOI: 10.1016/j.actamat.2023.118994).