When I joined my research group in early 2012, I thought I was solely going to work on experiments. The idea of trying out different formulations and materials to improve the lithium-air battery sounded fun, but whenever you jump into a new field of research without prior experience in it, you have a lot of catch-up reading to do. After all, how would you know what to put in the battery if you do not have a good understanding of the battery’s physical challenges? There was just one problem: only a select few papers gave useful insights on the fundamental chemistry; the vast majority were quick trial-and-error tests. Lithium-air batteries became the hottest topic in electrochemistry research that year, and everyone wanted a piece of it.
The hype about lithium-air batteries is somewhat justified. They can theoretically hold ten times more energy per weight compared to conventional lithium-ion batteries. If you had a working lithium-air battery in your car, you would be able to drive from San Francisco to San Diego on a single charge. There are two reasons for the exceptional energy density. First, the lithium-oxygen battery uses pure lithium metal (Li) instead of the lithium intercalation compounds (such as LiC6) found in conventional lithium-ion batteries. This saves a lot of weight because most of the atoms in an intercalation compound (for example, the six carbon atoms in LiC6) do not actually take part in the main reaction. Second, the lithium-air battery reacts with oxygen, which, besides being available from the air, is practically the lightest compound that lithium will react with.
There are problems associated with each component of the lithium-oxygen battery that need to be solved before the battery can be used in real-world applications. However, most of these problems take place in the cathode, which is where the main reaction takes place. In the cathode, lithium ions, oxygen, and electrons combine to form lithium peroxide (Li2O2) as the reaction product. Lithium peroxide is solid and electrically insulating, which are two inherently problematic properties. The fact that it’s solid means that it can block oxygen from entering the pores. The fact that it’s electrically insulating means electrons can no longer reach the reaction sites. You know you’re bound for some frustration when the reaction makes a product that immediately wants to shut itself off.
Back in 2012, there was some apparent disagreement between the theoretical and experimental research. Theoretical researchers, using calculations based on fundamental physics, found that lithium peroxide is such a poor electron conductor that it should not grow any larger than ten nanometers. Thus, they claimed that electronic insulation is the limiting factor. Most experimental researchers, however, found something different under the electron microscope. Lithium peroxide particles were growing to hundreds of nanometers, making it seem that lithium peroxide might be a better conductor than predicted by theory. Thus, most experimental papers assumed that the battery stops discharging because lithium peroxide blocks the oxygen from entering.
Essentially, all models are wrong, but some are useful.
— Box and Draper (1987)
Usually, this would be the perfect opportunity for a mathematical model to bridge the theoretical and experimental works. In their textbook on empirical model building, George Box and Norman Draper famously wrote, “Essentially, all models are wrong, but some are useful.” In other words, since making assumptions is a necessary part of model development, the assumptions should ideally not detract from the model’s ability to make good approximations of the real system. Battery models often aim to replicate the results found in experiments, but with theory-based mathematics instead of a physical device. The most common experiment performed on batteries is the galvanostatic discharge, in which the battery is discharged at a constant current (say, 0.1 milliamperes) and the voltage is measured as a function of the total charge passed. The plots of voltage versus capacity are called discharge curves. One distinguishing feature of the lithium-air battery’s discharge curve is the rapid drop in voltage off towards the end. Models would typically be fit and compared to these discharge curves as a measure of their accuracy.
However, the models published at the time did not agree on what caused the voltage drop. Most of them were formulated under the assumption that oxygen blocking was the limiting factor. Although they seemed to fit the experimental results, they contained around 20 equations and used more than five free parameters. (Enough free parameters could make any model fit with a set of data, regardless of how “correct” it is.) The theorists who calculated the conductivity of lithium peroxide also developed models based upon their findings. However, these models only seemed to fit to experiments done on flat, glassy carbon substrates instead of the rough, porous carbon used in more practical versions of the lithium-air battery. One particular model by the IBM research group simulated the effects of electronic resistance and oxygen blocking together. Whereas previous models assumed from the outset that one of the limiting factors was dominant without considering the other, their model found that electronic resistance was dominant while considering both. Their model did not fit as precisely to the experimental data as some of the other models, but the fact that it made such a definitive prediction from first principles led it to be the most influential model at the time.
The irony of research is how trying to answer a question often brings about more questions. If the IBM group was right in their prediction that electronic resistance of lithium peroxide is the main cause for the steep voltage drop, then why is lithium peroxide able to grow to such large thicknesses in experiments? Furthermore, why does a typical experimental discharge curve experience a much steeper voltage drop than what an electronic resistivity model predicts? It’s not hard to notice that all of the resistance-based models had one thing in common: they assumed that lithium peroxide grew as a flat layer on the electrode surface. However, almost everyone who has looked at lithium peroxide under an electron microscope would agree that’s not true. In most experiments, lithium peroxide was found to grow as individual particles, resulting in a very uneven layer. Considering how difficult it is to calculate resistance through an uneven layer, this assumption was most likely made to simplify this calculation. But was it a valid assumption?
Imagine you have a small region of the electrode where one half is covered by a lithium peroxide particle, but the other half is not. Since lithium peroxide has such a high resistance, electrons will want to pass through the half that is not covered by lithium peroxide. Now, let’s take it a step further and imagine what happens as you discharge the battery. More lithium peroxide particles will materialize and grow, gradually covering the surface of the electrode. You can think of the electrode surface as a circuit with lots of parallel branches, where the parts covered by lithium peroxide are branches with resistors on them. As long as some part of the surface is still uncovered, the electrons can still bypass these resistors through an alternative pathway. But once the entire surface is covered — and all branches of the circuit have resistors on them — the electrons have no choice but to pass through the lithium peroxide. Thus, the electronic resistance of lithium peroxide is bypassed during most of the discharge process until the surface is almost completely covered. Once this happens, the voltage drops steeply. Sound familiar? A steep voltage drop is exactly what we’ve been seeing at the end of a discharge curve. Uneven growth of lithium peroxide, it turns out, is something that cannot be ignored. Thus, I started working on a new model for the lithium-air battery that is based on the coverage of the electrode surface by the lithium peroxide layer, rather than the layer’s thickness, as the limiting factor.
This new coverage-based model requires that the electrons have some means of traveling to the reaction site (i.e. the surface of the lithium peroxide particles) besides passing through the bulk of the lithium peroxide itself. At the time (late 2013), the specific mechanism for how this can happen was not yet understood. There were theoretical studies that show that it’s possible for lithium peroxide surfaces to be electronically conductive, even if the bulk is insulating. There were also a couple of experiments that propose lithium superoxide (LiO2) as an intermediary species, which can dissolve into the electrolyte and combine with each other to form lithium peroxide. Either pathway would enable lithium peroxide to grow much larger than what bulk electronic conductivity would allow. It may seem scientifically irresponsible to develop a model that relies heavily on the existence of some speculative unknown mechanism… except, it doesn’t. The only thing this model assumes is that the particles are allowed to grow to large sizes, which is well-established based on the large amount of experimental evidence. The specific mechanism by which they grow is not critically important. Spoiler alert: Almost exactly a year later, Peter Bruce’s group, which is one of the leading lithium-air battery research groups, published their findings on the solubility of lithium superoxide in different electrolytes. Their findings were consistent with the model’s formulation, which made this part of the model much easier to explain.
Considering how much this model depends on particle-like growth, it should be able to predict trends in the particle’s characteristics as well. One unexplained phenomenon that has been observed ever since people started studying lithium-air batteries is the relationship between lithium peroxide particle size and the discharge current. Basically, it has been observed that a higher current leads to smaller particles. Closely related is the relationship between the number of particles and the discharge current: a higher current leads to more particles. If coverage is really the limiting factor of the battery, then these two trends might actually be different manifestations of the same phenomenon. When there are more particles on the electrode surface, each particle has less space to grow before it collides with its neighbors. Why is effect relationship important? Because larger particles means more lithium peroxide was produced, which means the battery was able to discharge a greater amount of energy for the same amount of electrode surface area. Hence, particle number, particle size, and discharge capacity are all connected.
The size vs. number relationship makes qualitative sense, but for this to be implemented in the model, a qualitative explanation is not enough. Specifically, we’re looking to describe two processes — nucleation of new particles and the overlapping of particles as they grow — in the form of equations. It turns out that the answers to both of these lie in some old papers that were written well before the first lithium-air battery has been studied. Alexander Milchev and colleagues developed the equations for electrochemical nucleation back in 1974. Developing a nucleation rate equation for the lithium-air battery was as simple as choosing the one that most closely fits the conditions of the battery and making some minor adjustments. Overlapping of growing particles was actually studied by the famous Soviet mathematician Audrey Kolmogorov back in 1937. With a clever use of probability theory, he found that the space occupied by a set of randomly distributed overlapping particles can be expressed in terms of the space they would have occupied if they did not overlap. The result was a surprisingly simple equation, which was incorprated into the model without modification. Now, with a total of just four equations and two fitting parameters, the model is complete.
One of the reactions I received when I presented the figures above was, “this almost looks too good to be true.” To be honest, that was also my reaction when I first saw the results of the fit myself, but the fact that only two fitting parameters were used for a set of five discharge curves makes it highly unlikely that these results happened by chance. To be sure, I also tested the model against data from another research group’s paper and found that it fit almost just as well.
If we follow Box and Draper’s dictum, the best assessment of a model’s value would be to test its usefulness. Up until now, there was no easy way to predict the discharge capacity of the lithium-air battery. If the lithium-air battery is to ever find its way into commercial use, being able to predict the capacity is crucial to avoid suddenly losing power to your device. Being unable to predict your capacity is almost like driving a gasoline car without a fuel gauge. But here, for the first time, we have demonstrated a reliable prediction of the capacity as a function of discharge current.
So, have we found the end-all be-all model for the lithium-air battery? Of course not. As a model, it necessarily has some degree of “wrongness,” especially when extrapolating to unfamiliar conditions. For one, it does not take into account the effect of temperature, which is also known to have a significant effect on discharge capacity and particle size. It also tends to underestimate particle sizes, despite getting the trend right. While the model does have a potentially useful application as a capacity prediction method, much of this model’s usefulness can be derived from its own development process. Many previous models only echoed existing beliefs on the battery’s limiting mechanism without scrutiny. Developing this new model, however, challenged us to seek a simpler, yet more logically consistent hypothesis based on electrode surface coverage. It introduces a new way of thinking about the problems associated with the lithium-oxygen battery, which will hopefully inspire fresh approaches to solve them. 🌍
Note: Technical details were simplified substantially to make this article suitable for a wider audience. For the complete treatment, please read the publication in Nano Letters.
Nucleation and Growth of Lithium Peroxide in the Li–O2 Battery
by Sampson Lau and Lynden A. Archer