Air is 6 times denser than aerographite but looking at pictures or videos presenting the material, I see it resting on tables rather than raising to the ceiling.
Also, since the material is made of carbon nanotubes, I assume there are empty spaces between those tubes. Why are those spaces not occupied by air?
Answer
Your two questions are connected. There is a huge amount of empty space in aerographene (and other aerogels). However this space is filled with air, and precisely because it is filled with air it doesn't float.
This is because the density reported is the density the material would have if the air was sucked out (i.e. in vacuum), and it is so low because the material is extremely porous. But in the atmosphere, the air fills the immense empty space. The effective volume of air displaced by aerographite now takes up only the volume of the constituent nanotubes of aerographite, which is extremely small. The tiny weight of this displaced air presents the buoyant force, which is not sufficient to counter the weight of the structure. Effectively because it is so porous the aerographite's density increases when not in vacuum.
On the other hand, given that graphene is known to be non-permeable for atoms, if you sucked the air out of aerographene and encased it in graphene, and if outside air didn't squish the whole thing so that it's density surpassed that of air, than the resulting balloon might float.
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