## Thursday, March 31, 2011

### The Three Laws of Batteries (and a Bonus Zeroth Law)

The Three Laws of Batteries (and a Bonus Zeroth Law): "

Any field has to have laws that define it. Some of these will be real laws (think laws of thermodynamics or Newton’s laws) while others will be correlations that are not fundamentally true, but are true enough (think Tom Friedman’s First law of Petropolitics or Moore’s Law, or the ever popular Murphy’s law). Still others are laws that don’t (yet) have any practical use (like Issac Asimov’s Three laws of Robotics).

Suffice to say, you will need a few laws for batteries. Batteries are governed by the laws of thermodynamics and are made more complex by kinetics and transport. So any fundamental laws that define batteries are only derivatives of the laws in these other areas.

So instead of focusing on anything fundamental, I’ve decided to list my own laws of batteries that are observations on how batteries operate.

Here are my three laws for batteries. I will explain each as we go forward, and I have the bonus Zeroth law in the end, so stick around.

First law. In any battery, energy and power will play against each other; increasing one will lead to the loss of the other.

Second law. Any battery that is widely commercialized will operate at a voltage higher than its thermodynamic stability window.

Third law. Of the four metrics batteries are graded on for a given application (i.e., performance, cost, life, and safety), typically, only two can be simultaneously achieved. If the battery is designed to also perform satisfactorily on a third metric, it will fail spectacularly on the fourth.

Before I explain these laws, I should point out that some of the concepts in this blog post are based on previous posts have I made in my blog, This Week in Batteries. Click on the links to read the background.

## The First Law

Let’s start with the first law, which states: In any battery, energy and power will play against each other; increasing one will lead to the loss of the other.

When a battery veteran gives an overview talk, there’s a customary slide that has to be shown; this plots specific energy on the x-axis and specific power on the y-axis. (Some flip this, but it is probably better to use the independent variable in the x-axis.) This is called a Ragone plot.

The picture below is an example of a Ragone plot.

Before we get to the law, it’s probably best that we understand the difference between energy and power. I find people use the two interchangeably. For example, they’ll say “my mobile phone lasts only 1 hour. I need a more powerful battery”. What they really mean is they need a battery with more energy.

Power, on the other hand, is how quickly you use the energy.

Here’s one way to remember the difference: If you own an electric car (you probably don’t, but use your imagination), then having more energy means you can drive more miles before you have to recharge. Having more power means you can accelerate faster from, say, 0 to 60 mph.

So a Ragone plot captures how much energy the battery can give you at the power at which we use the energy.

Just so you know, the ratio of energy to power is the time of discharge of the battery. These are the diagonal lines you see in the figure.

What the first law states is that if you try to get the energy out quickly (at high power), then the energy will be lower than if you ask for the energy to come out slowly (at low power).

The reason for this is rooted in the losses inherent in a battery. When the power increases, the losses increase, and this in turn decreases the “effective” voltage of the battery. The lower the voltage is, the lower the energy.

Hence the law that states that energy and power will play against each other.

But why is this the first law?

It’s because any battery design starts with understanding what the time of discharge requirements are. If you have a mobile phone, then you may say it’s three-hour continuous operation. If you need to use it in your Prius, you may come up with 10 seconds.

Once this is defined, a battery engineer can design the system around this requirement. In a mobile phone, the power requirement is minimal, so you find a way to get as much energy as you can in a small package. In a Prius, your need a lot of power to get the car moving, so you’ll need to maximize the power.

Battery design is so rooted in this concept that the energy-power interplay is, in my mind, the first law of batteries.

## The Second Law

On to the second law, which states: Any battery that is widely commercialized will operate at a voltage higher than its thermodynamic stability window.

Anyone who has worked on batteries will have, at some point in their career, experienced what alcoholics refer to as a moment of clarity (to quote Samuel L. Jackson in Pulp Fiction). The epiphany is that every battery we know of exists because of a freak of nature. For me, the realization came when I was thinking about the Ni-MH battery, but a more glaring example is the lead-acid battery.

First, some background. Water electrolysis is a process by which water is converted to hydrogen and oxygen in an electrochemical cell. Remember the hydrogen economy? The one that was supposed to power our world in 2005 (or was it 2010? I can’t remember, but it was sometime in the past).

The environmentalist’s dream of the hydrogen economy involved using solar panels to make electricity; the electricity was then to be used to split water into hydrogen and oxygen via electrolysis, then the hydrogen from this was to be used in a fuel cell to get electricity again, which was then to be used to power our cars.

This water electrolysis process occurs when the voltage of an electrochemical cell goes beyond 1.23 V in a water-based electrolyte.

In other words, anytime a water-based electrochemical cell operates above 1.23 V, there is a very distinct possibility of water electrolysis.

The voltage of a lead-acid battery is ~ 2 V. The electrolyte in this battery is water-based.

How do you have a battery operating at 2 V when at 1.23 V water starts to split?

It also turns out that nature gives us a break, because the rate of the reaction that splits water to make oxygen is very poor, so this reaction isn’t that dominant. (Incidentally, the inability to get the oxygen reaction to go in reverse has been one of the many issues that have prevented fuel cells from taking off).

What this means is it’s more favorable to oxidize lead sulfate than to oxidize the water. Voila: We have a lead-acid battery instead of a water electrolysis cell.

Turns out that the lead-acid battery actually does split water into oxygen pretty much continuously, but in small amounts. Those of you who are older will remember when you had to add deionized (or distilled) water into your car battery to “top it off”. This was essential, because any oxygen you made increases the pressure, then a vent opened and you lost, in effect, water from the cell. “Topping it off” got this water back into the system.

Li-ion cells are so much better than water-based systems because they have no water. This means the voltage window can be expanded dramatically from 1.23 V without the risk of electrolysis. Remember: The higher the voltage, the higher the energy of the battery.

But every electrolyte has a voltage limit after which you’ll destroy it. The window in a Li-ion cell is anywhere from 2.6 V to 3.3 V (depending on who you talk to).

A typical Li-ion battery operates at 3.7 V with the maximum voltage hovering around 4.2 V. So something is happening to the electrolyte. That “something” is the side reaction I have alluded to in the past. These reactions lead to fading of the capacity and increase in the resistance in these cells. Turns out the electrolyte pretty much continuously decomposes during the life of the battery.

So can’t we find materials that operate within the voltage window?

We can. For example, if one were to take a lithium titanate anode and a lithium iron phosphate cathode, then you’d have a cell that could, for the most part, stay within the voltage window of the electrolyte. This system should (at least on paper) give you good cycle/calendar life.

But the voltage of this system is around 1.9 V. Compare that to a typical Li-ion cell that has a voltage of 3.7 V, and you begin to see that this system, while being within the voltage window, will have less energy: almost half the energy of a typical Li-ion cell.

And in batteries, energy is king.

Hence, the second law is essentially a commentary on our expectations from our energy-storage devices. If we’re satisfied with a laptop with two hours of run time, then we can stay within the voltage window. But, we overwhelmingly prefer one that operates for four hours, so we operate outside the voltage window.

Even in applications where space isn’t a constraint (e.g., a stationary battery), we prefer a higher voltage. Without getting into details, this is because it’s hard to get high energy efficiency from a low voltage system.

You can obviously try to make a commercial success out of a system that doesn’t follow this law, but history suggests this may not be a good idea. Hence, the caveat that this law applies to widely commercialized batteries.

This law, while not being a fundamental law, is so true that it may as well be one.

## The Third Law

The third law is, in some sense, also a commentary on our expectations from our batteries and it states: Of the four metrics batteries are graded on for a given application (i.e., performance, cost, life, and safety), typically, only two can be simultaneously achieved. If the battery is designed to also perform satisfactorily on a third metric, it will fail spectacularly on the fourth.

This law can be summarized in a single sentence: Batteries are all about compromise.

If you decide to start a battery company, you go to the various VC firms and tell them you have a system that has the highest power, highest energy, longest life, best safety, and lowest cost of all systems out there.

Once you get money and start doing development work, you’ll find out very soon that you’re doing well in a couple of metrics (say energy and safety), but your other metrics aren’t looking so good, like your cycle life is really bad and your costs are through the roof.

You then hit upon an amazing idea. Why not decrease the voltage to which you cycle the batteries to increase the life? This decreases the energy, but by this time, reality has set in and you (and your VCs) are willing to compromise on energy. So you lose some energy, but you gain a lot in life.

But in doing so, your cost (\$/kWh) just went up from astronomical to even more astronomical. But infinity+ anything is infinity, so I guess it’s all fine.

This compromise scenario is the reason why there’s no one battery chemistry that has dominated the market for all applications. No system has yet replaced the lead-acid battery for starting your car. It’s not that the lead acid is an amazing system (try leaving your car parked with the lights on and you’ll know how flawed this chemistry is), but it’s the best system for that application, despite its shortcomings.

It’s the same law that describes our lack of clarity about which Li-ion battery chemistry will dominate the electric vehicle and plug-in hybrid vehicle market .

And it’s the same law that prevents Moore’s law-like improvements in the battery space. Try to increase the energy density too much, and your cycle life will fail spectacularly.

The first and second laws, in some sense, can be considered a subset of this law in that we compromise between energy and power when we design our battery. And we compromise between energy density and cycle/calendar life and self-discharge when we decide to maximize the voltage to go above the voltage window of the system.

The third law is something we all should remember when we try to understand how to develop a business plan around batteries, or when we are trying to identify the best battery for a given application.

However, one should also be cognizant that there is no fundamental reason why a compromise is needed. Researchers in national labs, companies, and in universities are working hard everyday to find the magic combination where no compromise is needed.

All this leads me to a law that supersedes all these laws. It captures the nature of batteries and how we design them, our decision to push the envelope in terms of voltage despite the risks, the constant compromises when it comes to batteries, and the expectations we have from our inventions.

I call this the Zeroth law, and it states:

The performance of any battery will fall (just) short of our expectations irrespective of the complexity of the device it is powering.

I shall not even bother elaborating this one.

Note: If I had written this blog post three months ago, I would have said the Zeroth law doesn’t apply to me. I consider myself a pretty evolved human with limited expectations from my energy storage devices. I had a car battery that was seven years old, a computer battery at four years, and a Bluetooth headset battery at two years all working very well and to my satisfaction. And then my world crumbled (twice), and I’m now a firm believer in the Zeroth law.

Venkat Srinivasan is a Staff Scientist at Lawrence Berkeley National Lab and writes about batteries on his site This Week In Batteries

Images courtesy of scalespeeder, moria, Tesla, akeg, Argonne National Labs, warrenski,

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