Some updates on this project
The solar cell phone charger started to take shape. At the moment I'm using my award winning DIY power supply to simulate the solar cells. I'm following the drawing posted earlier: Li-Ion batteries are being charged through the TP4056 regulator and a step-up converter provides 5 Volts to charge the cell phone. I made a battery bank out of four cell phone batteries, wired them in parallel, each around 1000 mAh capacity and charged it all the way up.
The 1350 mAh battery of my Nokia X1-01 was down to the level of 1 bar and with the first try I charged it up to 5 bars out of 6. Then I waited to go down to 1 bar again and with second try I gave it a full charge. After this the battery bank still had 3.6 Volts in it so I took a Samsung cell phone which barely turned on because of the low battery and I gave that one almost a full charge. That's when the battery bank depleted. Now I'm charging it back again to do some more testing.
I also encountered some problems or I should say "birrbert wasn't paying attention.
First, I killed one step-up converter. I didn't pay attention and I connected the battery bank to the converter the wrong way and it didn't work anymore. It's allowing electricity to pass, but it's not boosting. I went online for information and discovered that indeed they don't have reverse polarity protection. That's that. Luckily I had two: the green one is dead, the blue one is till OK so I'm using that now.
Second, I think I damaged the charge regulator. I wanted to check its temperature by putting my finger on the board, but unfortunately I was charged with static electricity and there were some serious sparks upon contact. Since then the LED of the regulator is green, which normally indicates that charging is done, but in reality charging is not done, the battery bank is still being charged, because I can see current draw on the LCD of my power supply. So, the LED should be red and turn green only when charging is finished. The LED remains green even when battery bank is not connected; in this case it should be blinking. Again, lucky that I have two more of this regulator, though they are different versions and this was the one I liked most.
Lastly, I did a quick research on the Internet about step-up converters. Please find attached the .txt file which lists the boost converters I think can be useful. There are cheap ones that say they work even with an iPhone and then there are two more expensive ones from engineers I already mentioned in this topic. If you think I missed some, please post here.
In the next round I will prepare everything for a test out in the sun. I'll be back with photos too (my camera is not working at the moment). Cheers!