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It's harder than tooth enamel and can cause erosion.
Wait, what? They're not putting diamonds in there. Charcoal is softer than graphite, which is a soft mineral to start with.
Charcoal ranges from 50-100 on the Rockwell scale while tooth enamel is around 51.
Charcoal can also contain silica since trees absorb it, which is definitely harder than enamel.
Silica is a common ingredient in most toothpastes.
Hmm. Maybe we've found another dimension where brand matters, then.
Something in between enamel and plaque in hardness seems like the most advisable approach.
Is this the paper you're looking at?
If you found something else, can I have a citation? It's actually really hard to find hardness information on charcoal. Graphite is 0.5 on the Mohs scale, while tooth enamel is 5, for reference.
I can't rule out that it's actually harder than graphite and just seems softer due to being full of voids and crumbly. Then again, the activated charcoal in toothpaste isn't exactly the same thing as the wood charcoal I'd be familiar with (or cow bone charcoal, for some reason).
Silica could wreck you, if there's a significant amount. The silica in normal food is probably a big contributor to tooth wear. If like the other poster says it's a common ingredient I wonder why.