Reaction rate laws key to unlocking lithium metal batteries for flying cars and fast charging

Venkat Viswanathan
6 min readNov 24, 2020

2020 has been the year for lithium metal. Lithium metal batteries, which promise to deliver about 50% improvement in energy delivered per weight and volume, appear to be ready for primetime. The main challenge thus far has been to stop the formation of dendrites, needle-like patterns of metal deposition that, if left unchecked, can create dangerous short-circuits, leading to fires. It now appears the battery field has dendrites under control, it is time to satisfy other important performance requirements needed for vehicles and other devices.

Lithium metal batteries are seen as a viable option for electric vehicles (EV’s) and electric vertical takeoff and landing (eVTOL) aircraft. Consumers demand that EV batteries charge fast (~15–30 minutes). Current batteries based on graphite are limited in their charging rates by lithium plating. In the case of lithium metal batteries, lithium plating is what we want. Flying cars, on the other hand, need high power for takeoff and landing. So, two questions arise: what sets the limit for fast charging in lithium metal batteries; are lithium metal cells capable of discharging fast? One limit is set by the rate at which reactions happen between the lithium anode and the electrolyte.

The rate of reactions is a topic that is taught in undergraduate curricula. Laws of chemical kinetics are well-established, but the laws of electrochemical kinetics are still evolving. When it comes to electrochemical reactions, there are two quantities that are of interest, (i) current, which tells us how many electrons are moving to participate in the reaction, and (ii) overpotential, which tells us how efficiently electrical energy is being used to make sure the reaction is happening. Early electrochemists observed that the overpotential and the current are closely related. In 1905, Julius Tafel proposed the Tafel equation which states that the current is exponentially related to the overpotential. However, experimental observations on hydrogen electrodes, the standard electrochemical apparatus of the time, showed that this exponential relationship did not always hold. In 1930, German electrochemist Max Volmer and Tibor Erdey-Grúz conducted several experiments exploring the origin of overpotential in hydrogen electrodes. Simultaneously, an English chemist John Alfred Valentine Butler was also trying to understand this phenomenon. They co-developed a new relation known as the Butler-Volmer equation, which posits a simple linear model relating the overpotential to the activation energy of reaction, a pivotal quantity controlling reaction rates. Simply put, if more energy is needed to “activate” (or kickstart) every instance of a reaction, it will proceed more slowly in the aggregate. While the Butler-Volmer equation is very simple, it turns out to work well in many cases, generally at moderate overpotentials (moderate charge/discharge rates). One problem is that it does not impose an upper limit on possible reaction rates, leading to the unphysical conclusion that if one applies more and more voltage, a reaction can proceed faster and faster without limit.

Both the Tafel and Butler-Volmer equations were proposed as empirical relations. In the mid-1950s, Canadian chemist Rudolph Marcus proposed a more detailed theoretical model for the activation energy, which worked better at faster rates, and, critically, contained an upper bound for how fast a reaction could go. Around the same time, Noel Hush was also working out a similar theory. The theory makes a curious prediction: at much higher overpotentials, the reaction rate would “invert” and slow down again! Initially, many saw this as a flaw, and assumed that the theory simply wouldn’t apply when the overpotential was that large. However, eventually, experimentalists found that under certain conditions, this so-called “inverted region” could in fact be observed! Marcus would go on to win the 1992 Nobel Prize in Chemistry.

Marcus theory looked primarily from the perspective of the (liquid) electrolyte, with the tacit assumption that most of the chemical “action” happened in the electrolyte phase. A few years after Marcus put forth his theory, German scientist Heinz Gerischer supplied another perspective that considered the electrochemistry of the (solid) electrode more explicitly. The Gerischer model reframed the problem entirely and could thus account more fully for variations in electronic states between different solids, by incorporating a quantity known as the density of states. However, despite this more detailed treatment, the Gerischer model applies only for relatively weak interactions between electrolyte and electrode.

In 2009, when I was a graduate student, I took a class on electrochemistry taught by Christopher Chidsey. The first class involved reading his 1991 paper in Science, where he took the first step towards uniting the Marcus perspective, focused on the electrolyte molecules, with Gerischer’s attention to the electronic states in the solid. His model predicted that, rather than continuing to grow forever as in the Butler-Volmer theory, or “inverting” as in Marcus theory, that the current would level out and cease to depend on voltage beyond a certain value.

In the summer of 2020, the Journal of Chemical Physics (JCP) ran a special issue to celebrate 65 years since Marcus’ original paper. This was finally the impetus my group and I needed to pay close attention to electrochemical reaction kinetics. Shashank Sripad, who had served as a teaching assistant in my electrochemical energy storage class twice, was well-versed in electrochemical rate laws and we set out to tackle the problem in the context of lithium metal electrodes. We only had about two months to make the submission for the special issue.

During this scoping out phase, we stumbled upon an incredible dataset of measurements published by David Boyle and Yi Cui’s team earlier this year. He measured the rate of lithium electrodeposition and stripping using transient voltammetry in a micro-electrode configuration. This dataset provided exactly what we needed to explore this within the lens of these theories. At this point, I reached out to Steven DeCaluwe, whose group works actively on implementing reaction kinetic models within a long-running open-source project called Cantera. Together with Steven’s graduate student, Daniel Korff, we were off to the races, trying to build a battery model using the new treatments of reaction kinetics. An innocent-looking tweet from Steven started a great discussion around rate laws and lithium metal batteries, highlighting what’s possible with open science.

Twitter thread on the kinetics of lithium metal, started by an innocent-looking tweet on Marcus kinetics

In mid-July, in our research group’s weekly meeting, Shashank gave a short talk to share the progress that he and Daniel had made towards building a model for a lithium metal battery incorporating these kinetic rate laws. The following week, in our journal club meeting, Ph.D. student Adarsh Dave led us through a detailed examination of Marcus’ original paper as well as Chidsey’s 2009 contribution. This spurred lively discussion, and postdoctoral fellow Dr. Rachel Kurchin posed the following question: Why not insert the factor of the density of states (the same quantity Gerischer had used in his model) into the Marcus-Hush-Chidsey formula? This simple question led her, with about two weeks until the special issue deadline, to quickly craft a companion piece to Shashank’s paper that explored this modification to the model along with the same Boyle dataset.

Reaction rate laws for fast discharge and fast charge relevant for eVTOLs and EVs

The results from Rachel’s paper showed that the effect of the electronic states of the electrode is less critical if a lithium-foil is used but becomes crucially important in an anode-free configuration. Thus, selecting the current collector to enable anode-free batteries needs to consider this together with the necessity that smooth deposits are formed, a problem tackled in an earlier work by my Ph.D. graduate, Dr. Vikram Pande. The results and analysis from Shashank and Daniel’s paper showed that the reaction rate limit in liquid electrolytes is well beyond (5 times) the required rate for fast charging for EVs or discharge rate needed for flying cars. This shows that there may be potential to make the charging time even shorter with lithium metal. Only time will tell if we can approach the reaction rate-limit of 2–5 minute charge time!

[This article was written with input from Dr. Rachel Kurchin and Shashank Sripad and from the BMWS community.]



Venkat Viswanathan

Associate Professor @CarnegieMellon University, Advanced Batteries, Electrochemical Devices