The Unfortunate Tetrahedron

With all due respect to Planet Money, batteries don’t “suck”, you just don’t know why/how they work. Inherent to all batteries is a mix of (super duper) interesting and competing tradeoffs that are exceptionally sensitive to the application at hand. It is a testament to 150 years of careful research and engineering that they work at all.

You asked for this, and I am happy to tell you this. Don’t worry, we will have minimal discussion of chemistry and physics, and we’re not going to get into flame wars pitting different batteries against one another, rather we’re going to talk about what makes them similar, and how different batteries trade different attributes, like a barbarian and a rogue in your favorite RPG.

Batteries are Overloaded Reactors

A battery is a closed electrochemical reactor in which the redox potential between two dissimilar materials is exploited to provide an electrical current outside the cell while the inside of the cell moves what mass it needs to do to equilibrate (both under load and otherwise). I desperately want to draw analogies to rocks on cliffs and stretching springs but because batteries are more complicated than rocks on cliffs and stretching springs this will only make things confusing later.

Just remember this: a battery stores its energy in the volatility with which the two electrodes want to react and the mass of the electrodes which are allowed to undergo a reaction. A battery derives its power from the rate at which the above reactive masses can be converted. The power is limited by how quickly the electrodes can react (kinetics) and how quickly the reactants can get to or from the electrodes (mass transport). That’s all the science for now. Because a battery is a closed system, energy and power are strongly coupled.

When I was first in grad school and learning about batteries, the Ragone plot was king. This plot demonstrates the inherent tie between energy and power for a battery. All batteries provide less energy when discharged at a high enough rate. The Ragone plot illustrates this relationship.

Figure 1) Ragone Plot Credit: Venkat Srinivasan

While this rendition of the plot is a couple of years old, not much has changed and the order certainly hasn’t changed for commercially available batteries. It becomes clear why lithium ion technologies dominated my graduate school days: based on the Ragone plot there is no reason to use capacitors for anything but the highest power applications. Battery engineers like the Ragone plot because:

1. it provides a clear engineering goal (we want to be up and to the right)
2. it provides a clear view of engineering trade-offs when spec’ing a battery
3. it provides a clear indication, for a given battery and duration requirement, what the minimum battery size should be (more on that later)
4. it can be predicted with high accuracy for a given system across a range of electrode configurations.

But there are at least two major factors in batteries not addressed by this plot, and one is hidden in plain sight. We’ll start with that.

The Ragone plot shows energy vs. power; dimensional analysis shows that the diagonal lines must in the units of time, specifically charge or discharge duration. Point 3 tells us that for a given application duration at a given power drain we need a certain size of battery. In a perfect world we multiply power by time to figure out what the battery size should be, in an imperfect world Ragone corrects for this.

Hidden deeper is an existential concern. Upon first glance, all things being equal there is no reason not to want to hang out on the upper right. However, consider a device that can provide upwards of 400 WHr/kg in under 36 seconds. This could either be

1. a great battery or
2. a mediocre bomb

Mediocre or not, we’d rather not have to carry a potential bomb around with us if we can avoid it. I am going to catch flack for this but I posit with my hand on my heart now: outside of an external control system a battery does not know whether or not there’s a short (e.g. a wrench across the terminals of the battery or an filament within the battery) or an exceptionally low resistance load (e.g. Tesla Model S demanding to go from to 0 to 60 MPH in 4 ish seconds).

Luckily, batteries can be designed to have inherent limits on power density. This means, however, that once you design it not to give you the power density you want, you’re not going to be able to extract it later.

So hidden in the Ragone plot is the tradeoff of Safety vs. Power, or at least part of it. We want a battery with a near unlimited power density, but this must be tempered against the mediocrity of the bomb we’d like to carry around.

Missing from the Ragone plot is any sense of how often one can charge and discharge a battery (cycle life and calendar life) and how much the battery costs (per unit energy, $/kWhr and the overhead required to operate it safely). These aspects aren’t missing for reasons of ignorance.

1. Cycle life is difficult to model and is at the forefront of battery systems research.
2. Cost is a difficult metric because it is tied to commodities markets and specific manufacturing goals. Cost can also be considered as capital (how much it costs to build the battery) and operating (how much it costs to keep the battery going).

Nonetheless, for grid scale applications one might argue that the quantities not explicitly highlighted by the Ragone plot are the most important, namely capital cost and operating costs (cycle life, and the requirements of safety).

There’s an adage I learned from the senior machinist when I was in grad school “You can have it good, fast or cheap; pick two out of three.” This started my love affair with triplets. If we add the cost of the battery to the Ragone plot, we can create a triangle where, in 2015, we can only get two out of the the three.

Figure 2) A triplet

1. A cheap, power dense battery is an electrolytic capacitor
2. A cheap, energy dense battery is a Zn/MnO2 Alkaline (i.e. standard ‘AA’)
3. An energy dense and power dense battery is the C6/NCA Lithium ion system, which is an order of magnitude (at least) more expensive per unit energy than our ‘AA’ battery and per unit power than a capacitor.

The point is we can’t have it all right now, and upon inspection figuring out why we can’t have it all right now is the fun. If we put all the possible triplet series together we get something that looks like this

Figure 3) Four triplets

In the above I’ve cheated a bit and lumped safety and cycle life into “operating costs”, but the lumping should make sense in a bit. So this is a complicated dance of many triplets to think of, but if we assemble the triangles together like so:

Figure 4) Four triplets assembled into one unfortunate tetrahedron

So now we have our tetrahedron which fully describes everything in which a battery is measured by, and the shape implies that we have to give up on one metric to achieve the rest. This is not to say that this is a law, but rather if you look at any battery on the market today its performance is a balance of the four variables of the corners of the tetrahedron. Let’s label critical edges. The energy density/power density tie line represents the Ragone plot above. I’d posit the CC/OC tie line represents power utility space, or the metrics of greatest concern to folks that make and/or use stationary energy storage devices.

The next figure is going to get me into trouble but if I assign proxies for the currently-available optimizations for each corner we can get a sense of what each battery might be:

Figure 4a) Figure 4 + Context

Where

• Energy Density => Saft Lithium-SOCl2 Primary Cell
• Power Density => A123 LFP Cell
• Capital Cost => Duracell Alkaline Cell
• Operating Cost = > Lead Acid Cell (Edit: 2017–05–25 It is likely that the A123 LFP cell is going to win here for certain applications, which is a big deal.)

Again, the tetrahedron alone doesn’t limit what batteries can and cannot do, rather it is put forth to succinctly describe the current tradeoffs inherent to batteries. The four corners emerge when we examine where different chemistries excel and fail.

Electric vehicles are so hard because, with a gasoline engine standard (and no carbon tax) in 2015, all four corners must simultaneously be optimized.

So the tetrahedron describes the space. Describing why the tetrahedron describes the space is a bit tricky.

A “Stat Mech” Treatment of Batteries

I’m getting out of my depth here but the quotes above mean this is supposed to be cute. Casual reader stay with me; this is for you!

Figure 5) A very not to scale and incomplete picture of a battery.

There are many different chemistries that can be used to make batteries and most of the states of matter (solid, liquid, gas) are represented. One can quickly become overwhelmed by the possible combinations. Let’s not worry about them right now. Instead let’s set up some unassailable relations.

1. Within a battery, there are two types of bonds, those which participate in the storing and releasing of electrical energy (i.e. the electrochemistry), and those that do not. The higher the fraction of the former bonds, the higher the energy density.
2. The rate at which the bonds can be formed and broken determines the power which can be drawn from or put into the battery.
3. The operation of a primary (non-rechargeable) battery involves accessing the maximum number of energy bonds from point (1) at the power determined by point (2)
4. The operation of a secondary (rechargeable) battery involves doing point (3) while returning to the initial condition of point (1)
5. The price of the battery is set by the price of the atoms which determine all of the bonds in point (1)

That’s about it. Everything else is about maximizing point 1 and 2 in such a manner where they do not degrade with time and/or use. Sounds simple but it is difficult to do in practice.

There are implications from the simplicity:

• When accounting for specific and gravimetric performance (e.g mass and volume normalized energy and power), every bond in the battery must be considered. If you look at figure 5, crudely drawn as it is, this point doesn’t mean that the anode and cathode are pure energy storage, but rather the anode and cathode, more likely than not, will have non-reacting overhead. In fact, history has shown that the lower the reacting fraction of a battery, the longer it lasts and the more power we can extract. By way of example, it doesn’t matter if a new material can store 10x the energy per unit mass if it needs 10x the support bond to keep it stable. And support bonds include binder, conductive additive, package, power management. Everything must be accounted for.
• We are already using lithium, which is the most reducing element we’ve got. There is improvement available in point one, but there’s only so high a fraction of bonds that can be used to store energy reversibly.
• Even when a bond is allocated to be an energy bond, more often than not the battery will last longer if it is never used. The more often a larger fraction of the bonds are utilized (e.g. running your battery down all the way), the fewer times they can be reformed.
• Beyond the anode and cathode, all the other stuff should not react. The packaging, the current collectors, the electrolyte. We want to minimize those components because they take up spacing and volume where we could have energy bonds, but if we don’t have enough of any of them the battery won’t work, won’t last long, and/or won’t provide the power we need.
• Where the bonds are, both structural and energetic, matters a lot. Like, a lot. Battery materials are important. But battery geometry is equally important and far easier to optimize. Consider this challenge: you can literally invent a new compound and figure out what the tradeoffs are, or you can take well understood compounds and place them carefully and with consideration. The former is what academic labs think is important, the latter is what battery companies know is important. Boom. Geometry matters.

The following points are not unassailable, but are rather being assailed on a daily basis in labs around the country. The problem is that the assaults are just reinforcing the presence of the tetrahedron. What the Planet Money folks ascribe to “batteries sucking” really represents the challenges of getting all 5 things above to happen in a reliable manner. Here’s what generally goes wrong.

• Points (1) and (2) act against each other. The rate at which energy bonds can be accessed is inversely proportional to the fraction of bonds available for electrochemistry
• Point (1) gets worse as the life expectancy of point (4) increases. As a battery cycles, active bonds (seem to) eventually become deactivated, decreasing the ratio posited in (1).
• Point (5) and point (4) often act against each other: cheap materials and construction do not generally last forever.

This is a high level comparison, but I have not found a system which directly bucks any of these trends.

Competing Corners

Getting into the dirt, and further away from the unassailable, what follows are cursory explanations of the kinds of things that compete in a battery..

Energy Density vs. Cycle Life

Simply: the more bonds within a structure are changed, the harder it is to put those bonds back to where they started.

Figure 6) Energy vs. Cycle Life. Long story short: the more bonds are broken, the more the shape can change, the more the shape can change the harder it is to get everything back in place.

Long story short: the more bonds are broken, the more the shape can change; the more the shape can change, the harder it is to get everything back in place. As currently designed, batteries with significant mechanical change do not cycle well. Many folks are trying to prove this wrong. Self assembly this and nano that may save the day.

Air Isn’t Free: Some argue that a battery with an air electrodes gets around this, but the problem is that air is not just oxygen. Combustion engines deal with impurities in air by simply passing them through at high temperature (creating NOx and SOx in the balance). As such, low temperature air breathing batteries will have to deal with N2, CO2, NOx, SOx, etc. Until a method exists which is different from a traditional scrubber, the material energy advantage will likely be outweighed by the systematic mass, volume, and monetary cost.

Energy Density vs. Power Density

The balance between energy density and power density, taken by active materials, almost mimics the previous relationship, with a couple of notable changes.

Figure 6) Power Density Considerations

The energy density and power density of a materials system is dependent on how long it takes to get reactant to a surface. In capacitors and plate metals surface is always ready to go. In other systems this isn’t generally the case. But too much is made of how “fast” a material is, what matters more is how available a material is to react. Way back in the beginning I discussed how a Ragone plot can be predicted well for a given system. This is because the real power limitation for most batteries comes from how densely packed the electrode is, and how available the electrolyte is to the electrode surface.

This is a record, I am 2,200 words in and only mentioning the electrolyte’s role now. For shame. The electrolyte is a necessary element in the battery that allows ions (massive species) to travel between electrodes while preventing electrons from traveling in the battery, forcing them through the external circuit. Newman, for his master’s thesis, formulated the mathematical framework (porous electrode model) which we still use today to determine how we can maximize energy density for a given material while achieving sufficient power. Forgive me for I am about to sin.

The model is a coupled solver, but it boils down to this: imagine a parking garage with a fixed amount of space. You can store cars in that space; this is the main job of the space. But the cars have to be able to come and go. It doesn’t matter where the cars go, or in what order they go, but they all have to be able to get out.

As the garage designer, you have two extremes,

1. Pack them in like sardines. Filling and emptying the garage will take a long time but you will maximize the space.
2. Allow only 1 car per floor. You are minimizing the number of cars in the garage, but each car can enter/exit quickly.

So how do you do it? Well that’s a hard question, but you know your priorities. You want to pack as many cars as possible in while ensuring the cars can leave and enter in a specified minimum amount of time. If the cars can leave faster than they need to, you’re losing money. If the cars can’t leave fast enough, the garage is useless.

In designing a porous electrode we do the same thing. We design the battery electrode with the least porosity possible such that we can get the maximum amount of energy in a sufficient duration. I can go on and on about this, but the point here is that what the reacting material (i.e. the parking spot) is is generally less important than where the material is (or isn’t, i.e. the everywhere but the parking spots) when it comes to power. Consider this when someone sells you a new battery material that provides “fast charging”.

Safety (Operating Costs)

Safety is a tricky thing. Batteries don’t catch fire until they do. Cars catch fire all the time. More with gasoline than without. But batteries have an extra challenge. Recall the trick with the Ragone plot; the whole bomb thing.

We want a battery that stores the most energy in the least amount of mass. Thermodynamics tells us that if we lose all the energy at once (or, say in about 36s), that must result in a temperature rise as dictated by:

So a battery stores all of its energy where it converts it to power. A gasoline engine does not (a gas tank stores the energy, the engine converts it).

It boils down to: imagine asking Chevrolet to design an engine with the constraint that all of the fuel must be stored in the cylinder walls. If we want batteries that have the energy density of gasoline, this is what we want. Now you should send a tweet to @teslamotors and congratulate on them on their remarkable safety record.

Capital Cost

This is tricky. The bonds that form and break cost money. We want those to be as cheap as possible. If you use one electron from Ni, then the cost contribution of Ni is $40/kAhr (NiMH), and if we use two electrons (as is the case with the GE Durathon) $20/kAhr (Na/NiCl).

For example, Zinc is $2/kAhr, but it is harder to cycle than nickel. Al is ~$0.50/kAhr, but we don’t really know how to recharge it (really, we don’t). What is the cost of balance of plant to keep the system “safe” and “cyclable”? Is it high temperature? Is it volatile? Non-aqueous? These add up. Here’s where operating cost (cycle and safety maintenance) can and do butt against capital outlay.

There’s an easier way to think about cost. Again, not by what’s the cheapest, but what is cheap enough. Remember Newman, the guy that laid out the math that explains why batteries do what they do in 1962? In 2012 he said:

We are willing to store electric energy in a watch battery for 2 or more years because we value the energy highly for its convenience and small size. Energy for the utility grid is inherently valued at its sale price, a much smaller number.

The more portable something has to be, the more cost we can deal with. See my other ramblings on this here.

Batteries Don’t Suck

Batteries are complex tools that get better in spite of themselves. When we make one thing better, many things tend to get worse. But if we have a sense of what we need from the battery, in terms of power, cost and cycle life, we can maximize the energy stored given those constraints. Regardless of the materials used, the rules are the same.

To recap, because a battery stores energy where it converts it, the relation of transport to material properties determines:

• The maximum bond utilization of the battery (i.e. energy)
• The rate at which those bonds can be utilized (i.e. power)
• The reversibility of the aforementioned reaction (i.e. cycle life)
• The extent of local equilibrium (i.e. stability)

Here’s my challenge to you, internet, show me a battery that doesn’t fall into the Unfortunate Tetrahedron. Please. Because I haven’t found it. I’ve been looking for a while. It is hard to make one thing better without making the other worse. VC’s, sell me on your latest “game changer”. Because if you can get out of the tetrahedron, then I will declare you have changed the game. And my soul weeps just typing the words “game changer”.

I’ve made a battery out to be a Faustian bargain; it giveth and it taketh away. But making a better battery is possible, we just have to know what the battery is for. Making one that is safer, or cheaper, requires understanding what can give and what cannot for a given application. In applications where it seems like nothing can give (like electric vehicles), well….

Something can always give.