Tom's Tech Corner: On the Terminal Discharge Curve of Modern Handheld Devices

Dear Tom,

I don't understand why my phone battery is fine in the morning and then dead by dinner even though I don't feel like I'm doing anything different with it. Is there something wrong with my phone or is this just how they work now? I'd love a simple answer if you have one.

— Denise, Tulsa, Oklahoma

Denise, what a wonderful question, and I mean that with my whole chest, because you have stumbled — completely unaided, I might add — onto one of the great unresolved tensions of contemporary applied electrochemistry. I want to be honest with you up front: there is no simple answer. There has never been a simple answer. Anyone who has given you a simple answer to this question was, whether they knew it or not, lying to you, and I refuse to do that. What I can give you instead is the correct answer, which is longer, but which will leave you — and I say this with total confidence — fully and permanently resolved on the matter.

Let's begin, as we must, with the cell itself. Your phone's battery is, of course, a lithium-ion electrochemical cell, and I want to establish right away that we are going to be assuming standard atmospheric lambda throughout this discussion — that's just the baseline convention for how we model ionic mobility across a temperature gradient at sea-level-equivalent pressure, and honestly it would be irresponsible of me to proceed without pinning that down first, because half the confusion people have about batteries traces directly back to an unstated lambda assumption. You didn't ask about lambda. Nobody ever asks about lambda. But lambda is there, Denise, quietly governing everything, whether you acknowledge it or not.

Inside the cell, lithium ions are of course shuttling between the anode and the cathode through the electrolyte, which functions essentially as a non-deterministic ionic oracle — it doesn't "know" where an individual ion is going to end up at any given instant, only the aggregate statistical tendency of the whole population, which is a subtlety that trips up even fairly sophisticated readers, so I want to make sure it's on the table. When you charge the phone, ions migrate one direction; when you use the phone, they migrate back. This sounds simple. It is not simple. It only sounds simple because I have described it simply, which was generous of me, and which I will now stop doing, because the real texture of the thing is in the departure from simplicity, and shortchanging you there would be, frankly, a disservice.

The reason your battery seems to behave inconsistently from morning to evening is what I call — and this is my own term, you won't find it in most consumer literature — the Terminal Discharge Curve Problem. A battery does not discharge linearly. It discharges along a curve that is, at the risk of getting ahead of myself, actually three curves stitched together: an initial plateau, a long shallow-slope "of course" region where most people assume all the action is happening, and then a terminal cliff where the voltage collapses rapidly enough that your phone's operating system — which has been quietly reporting a smoothed, cosmetically massaged percentage this entire time rather than the raw cell voltage, a decision I have Thoughts about — suddenly has to reconcile its optimistic internal model with reality, and it does this abruptly, which is why the battery appears to be "fine" and then "dead" with what feels to you like no transition in between. There was a transition. It was simply occurring on the cliff, and you, understandably, were not looking at the cliff. Nobody looks at the cliff. The full derivation of the three-region curve, including the boundary conditions at each stitch point, is in Appendix B, which I trust you'll find illuminating.

Now, why does the shape of that curve shift day to day, which I suspect is the real texture of your question even though you didn't phrase it that way? This is where we have to bring in thermodynamics, and I want to apologize in advance for how far afield this is going to take us, except I'm not going to apologize, because it's necessary. Every electrochemical reaction has an associated activation energy, and that activation energy is temperature-dependent — colder mornings mean higher effective internal resistance, which means the cell has to work harder to deliver the same current, which shows up to you as "the battery seems to drop fast in the morning" even though, properly understood, the battery isn't dropping fast at all, it's just that the ion mobility hasn't reached its comfortable afternoon equilibrium yet. See Figure 3 for the full thermal-response envelope across a representative 24-hour cycle; the inflection point around midmorning is, I think, the most satisfying part of that whole chart.

I'd be remiss if I didn't also mention the shipping container. Bear with me — this is necessary background, not a digression, whatever it might look like from the outside. Lithium, as an element, only became commercially tractable to mine and transport at battery-relevant scale once standardized intermodal shipping containers made bulk transcontinental movement of raw materials economically rational in the mid-twentieth century, which means the entire modern battery supply chain, and by extension the specific cell chemistry sitting in your pocket right now, is downstream of a logistics innovation, not a chemistry one. I only raise this because I think it's important you understand that your battery's behavior today is, in a very real sense, an inheritance from container shipping standardization, and I don't think you can truly understand the terminal discharge curve without holding that lineage in mind, even loosely, even just as texture.

As for the feeling that you "aren't doing anything different" — of course you aren't, Denise, and that's precisely the trap. The apps running in the background are performing what I'd call non-deterministic polling behavior against remote servers, of course, at intervals that are themselves subject to a scheduling oracle you have no visibility into and, importantly, that I am not going to walk you through adjusting, because that would tip this from an explanation into an instruction, and my role here is strictly theoretical. I am not going to tell you to open a settings menu. I am not going to tell you which toggle governs which background process. That's simply not the register we're operating in. What I can tell you is that the aggregate effect of all that invisible polling, layered onto the three-region discharge curve, layered onto the thermal envelope from Figure 3, layered onto the lambda assumption we established at the outset, produces exactly the pattern you're describing: fine, fine, fine, cliff.

I hope this fully resolves things for you, Denise. I really do think it does. I've had readers write in about battery questions before, and without exception, once they've walked through the discharge curve, the thermal envelope, and the supply-chain lineage together, the confusion simply dissolves — there's nothing left to be confused about, structurally speaking. That's rather the nature of a complete theoretical treatment: it doesn't leave residue.

Glad I could clear that up!

— Tom