I acquired a Titanium which wouldn't turn on because one of the pads connecting to the batteries was covered with black burned looking gunk. After sanding the gunk off, it turned on.
But the screen goes black when it's turned on. Then it slowly brightens to normal. Everything else works.
How long it takes for the screen to get to normal depends on how long the calculator was off. If it was off for a minute, it'll get back to normal in a few seconds. If it was off for twenty minutes or so, it'll take half a minute or so.
Any ideas?
After 1 minute off, the screen brightens in a few seconds. After 2 minutes off, the calculator screen takes about 12 seconds to brighten up. After being off overnight, about 3 minutes to reach full brightness.

It feels like there is some capacitor that's charging too slowly.

Besides the corrosion around the + battery pad, there is no obvious damage anywhere on the PCB (my eyesight isn't as good as it used to be, but I used a magnifying glass). I tested the small number of electrolytic capacitors I could find, and they seemed OK (maybe a slightly out of tolerance, but maybe my multimeter isn't calibrated well).
The only thing that comes to mind is DC offset (that is, charge) buildup in the liquid crystal material, but I am not sure if that can even happen to the extent that would make it take 3 minutes to recover, & there is no reason for that to suddenly start happening after cleaning a battery contact. It also seems like that should make the LCD dark even when the calculator is off, but perhaps not due to some other electrode(s) not being powered at the same time.

Have you tried leaving it for different amounts of time with no batteries vs. with batteries? That would at least tell whether it is something like a capacitor needing to charge or something like charge buildup on the LCD...it should brighten slower after a period with no batteries if the former, or faster (maybe even instantly) if the latter. Another thing you could try is blowing on the screen...I occasionally find a dark spot on my TI-89 Titanium's screen that goes away when I blow on it (which seems likely to be from static buildup, but that should resemble the charge buildup possibility).

If it is charge buildup, leaving it for too long could permanently damage the LCD, but I am not sure how long that would take, nor what could be done to prevent it. (Leaving the batteries out when not using the calculator seems like a non-starter as a long-term solution.)

EDIT: Fixed wording & grammar (oops)
I measured the voltage across the C76 capacitor. When the screen is functioning at a maximum darkness, it's 16.41V, and at minimum darkness (maximum brightness) it's 17.05V. Optimal contrast is 16.96-16.99.

When I turn off the calculator for a couple of minutes and turn it back on, the voltage across C76 after turning on starts at something low like 16.86 and slowly rises to the correct value. When I turn the calculator off and on briefly, the voltage is immediately correct.

My multimeter says it's 11.6 uF (testing in-circuit). That fits with the "10-25L" marking (I am guessing it's 10 uF and 25V). So as far as I can tell, the capacitor is OK, but is for some reasoning charging up too slowly when the calculator is turned on.
It looks like there are signs of rework on that flat-flex assembly. Could there be damage to the flat-flex that's preventing the capacitor from charging (i.e. resistance)? Or maybe the silicon on the flat-flex has something blown in its charge pump?

Nice use of a macro ringlight. Have you ever tried creating a photocollage?
The web suggests that even if the capacitance value is correct, the ESR of the capacitor may be bad. I know very little about capacitors. Could that cause my issue?

I don't have an ESR meter, but the says that one can measure ESR with an oscilloscope and a function generator. I don't have a function generator, but I can get a microcontroller to generate a square wave. Is this worth trying?
Any suggestions on how to desolder the existing capacitor? I don't have much SMD experience, and all I have is a good temperature controlled iron.
A giant package like that can be easily removed using two irons, one on each side. It's how I've managed to remove multiple SOIC-8 packages without burning or lifting any pads (along with a solder bridge across all pins on a single side). The first time I did it, the device came off so easily it felt like it leapt off the board. It's almost certainly lead-free solder, so of course use a higher temperature than for lead-based solders.
Does anyone think it might be worth soldering a second (smaller?) capacitor in parallel first, as a test? Or would increasing the capacity be dangerous?
I was going to replace the capacitor when I noticed that the problem just disappeared on its own, after my son used the calculator for a few days at school. I have no idea what happened.
I spoke too soon. The problem didn't disappear, just decreased. After being off overnight, the brightening period takes about 10 seconds, which is usable, but not ideal.

I tried to measure the 10uF tantalum capacitor ESR using an oscilloscope and a pulse generator, and it seemed a bit smaller than a fresh electrolytic, so there is no point replacing it, I guess.
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