Particle size in batteries

I thought you might be interested to hear about what I've been working on for Martin over the last few days. We start of with a battery where the particles of the active material (such as lithium iron phosphate) are of different sizes. The process of using the battery involves the insertion of lithium into these particles. We assume that we are at a voltage plateau, i.e. a region where the voltage of the battery doesn't depend on how full of lithium the particles are. We also assume that the conductivity of the surface of a lithium particle is constant, so the amount of current that can be inserted is directly proportional to the surface area.

If we imagine unfolding the spherical particles to form cylinders, with the sphere's surface area as the base, the cylinders will now fill up at a constant rate - like putting different sized, but the same shape, buckets in the rain (when you live in Cambridge MA, it's very easy to think of rain as a simile, because we get plenty!). Nothing much happens until you completely fill the smallest particle. Suddenly, you can no longer put any more lithium in it. You've now effectively removed its surface area. Since current is proportional to voltage times surface area, you need to increase the voltage across the surface by an amount corresponding to the loss in area, in order to keep current constant. This voltage over the surface is being 'wasted', so the voltage that can be output by the battery is decreased by this amount.

So the difference between what the voltage actually is and what it could be were it not for this surface resistance is inversely proportional to the surface area remaining once we exclude the particles that have been completely filled. I call this Phi_A. The final question is, what size particles have been completely filled at a particular time? The answer is relatively simple: if we plot Phi_A as a function of the radius of particles that have been completely filled, then the area under the curve up to this radius is proportional to the time for which we have been discharging.

Now I have the ability to predict the voltage profile for a particular distribution of particle sizes. The graph on the left shows that for a distribution that falls off like r^-5, which is about as slowly as it can fall off. The graph on the right shows experimental results. The fit is pretty good, particularly for the fast currents. For the slow currents, the increase is too sudden right near complete discharge. I expect this is due to the interfacial resistance going up as we approach complete filling (essentially, it's harder to put an extra lithium in because most of the spaces are filled up). Perhaps I can work out how to model this, although it might be difficult.