Activating the Electrochemical Magic: A twist in the tale
Electrochemical reactions are at the core of the electrification transformation in electric cars, aircraft, and electrolysis for hydrogen production. The rate of electrochemical reactions governs a number of key metrics: fast charging in electric cars, the ability to take-off and land in electric vertical take-off and landing aircraft and unit economics for electrolysis. While the theories for rates of chemical reactions involving fossil fuels are well-established, the theories for rates of electrochemical reactions are still evolving. Understanding them is critical to the path towards net-zero emission energy systems. In this article, we cover the story behind our recent work in collaboration with the Bediako Group that was published in Nature Chemistry.
The key parameter determining the rate of a chemical reaction is the activation energy, i.e. how much energy needs to be put in to “kickstart” the process of atoms reconfiguring their bonds in new ways. The key distinguishing factor of electrochemical reactions is that electrons are transferred between atomic species, meaning that (because electrons carry a charge) these activation energies depend upon the electric potential (voltage) at which the reaction is taking place. The history of models for activation energies (and hence rates) of electrochemical reactions begins in the early 20th century with the Butler-Volmer equation. This empirical model proposes an activation energy that varies linearly with the applied voltage, also called an overpotential. As we know from the theory of Taylor series in introductory calculus, such a linear model can be expected to work well for small values of the input, but may start to deviate from reality at larger values. Starting mid-century, Rudolph Marcus built a more sophisticated model that can be thought of as including the next-order term in the Taylor series, where the fully reduced and oxidized states are modeled as harmonic energy wells and the activation energy to move between them thus depends quadratically on the overpotential.
In 1991, Christopher Chidsey made a further adaptation to Marcus theory, observing that electronic occupation of states follows a known (Fermi-Dirac) distribution, and that by turning the Marcus model into an integral over these energy states, we could account for which energies electrons would or wouldn’t be available. Known as Marcus-Hush-Chidsey (or MHC) theory (Hush being Noel Hush, who developed a similar theory to Marcus around the same time), this model removed the mysterious “inverted region” predicted by Marcus theory, where beyond a certain voltage, further increases in overpotential would result in less current, a paradoxical prediction that, while observable in some particular systems, is not in agreement with the more common behavior of reaction rates “plateau-ing” out after a given overpotential, consistent with MHC kinetics.
Last year, Dr. Rachel Kurchin introduced a new modification on top of MHC kinetics that accounts not only for the probability of occupying a state at a given energy via the Fermi-Dirac distribution, but also for how many states are available at each energy via a quantity known as the density of states, or DOS, (see this earlier post for more details of this story). In some systems (e.g. electrodeposition of metallic lithium), this DOS effect is fairly small, and predictions don’t vary much from standard MHC. However, in systems where the DOS has dramatic variations across energy, predicted reaction rates can vary quite drastically!
One such system is the twisted bilayer graphene superlattice, which consists of two sheets of graphene rotated with respect to each other by a small angle. In recent years, researchers discovered that there exist specific “magic” twist angles for which the number of electronic states of the bilayer will massively increase near the charge neutrality point (CNP), a particular energy level of note in solid-state systems. At the magic angle of 1.1 degrees, for example, Cao et al. found that the number of states near the CNP was greater than that of freestanding graphene by over three orders of magnitude. Figure 1a similarly shows that this same twist results in an increase in the number of states compared to untwisted bilayer graphene. The DOS enhancement is not constant, but rather a function of the twist angle. With the framework introduced by Dr. Kurchin, PhD students Holden Parks and Mohammad Babar could predict the kinetics of the system. In collaboration with the experimental team of Dr. D. Kwabena Bediako, we were able to show that the electroreduction rate of ruthenium hexamine on the twisted graphene system could be explained by this DOS enhancement as a function of the twist angle (Figure 1b). This extraordinary result implies the electrochemistry of the system can be controlled simply by adjusting the twist angle of the bilayer, which is much easier to control compared to other methods of achieving similar effects on the DOS, such as by introducing dopants or structural defects. With an ever-growing library of 2D materials and the generality of the twist approach to changing the system DOS, this method could be used in the future for tailoring electrocatalytic reactions.
To perform the analyses described above, we used a package Dr. Kurchin has developed, ElectrochemicalKinetics.jl, that implements her MHC+DOS model as well as other electrochemical kinetic models such as Butler-Volmer. This software allows for easy fitting of models to experimental data, comparison of the predictions of different models, and data visualization. The package remains under active development, with more functionality continually added. The eventual vision is to integrate it with cell and pack modeling software such as PyBAMM and liionpack so that battery modelers at any scale can take advantage. Expect another post here with a deep-dive soon!
[This article was co-written with Dr. Rachel Kurchin and Holden Parks]