Is there a free app of digital scales weighing grams using my iPhone 5 screen?

Is there a free app of digital scales weighing grams using my iPhone 5 screen?

No I'm afraid not, good idea for an app though, but its impossible without a feature designed by apple.

There's no scale built in to an iPhone, so no app will be able to do this!

No. The iphone screen is a capacitance touchscreen. It can't measure strain or weight, just if you are touching the screen or not. Digital scales use strain gauge loadcells to measure weight by measuring the stress on the loadcell body as it is bent slightly by the weight on the platform.

There are a few iphone apps out there that claim to weigh like a scale using the iphones accelerometer (tilt sensor). You set the iphone on a pen or pencil and balance it so that it teeters like a see-saw creating a balance. The accuracy of this setup is very low because balances depend on sharp fulcrums or knives to balance on and reduce friction. The pencil or pen as a fulcrum will not allow you to weigh grams accurately, plus the weight distribution inside the iphone is not perfect and finding the middle / balance point of the phone is not a very precise process. Therefore these apps are mostly a novelty and don't provide any practical use.