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Is it possible to create a container for helium that it can't permeate through, like a permanent balloon? Or, if not, what kind of material would have the lowest permeability for that application?

I don't just mean stuff that we could build today either, for instance, could it permeate through diamond? Like if we had molecular manufacturing or something, in the future, and built a box out of diamond, of the appropriate dimensions (like total mass vs volume inside) so that it would float, and constructed it so that there were no openings...well, say we build it almost all the way and then stop, and there's a hole in the top, and we drop some solid, frozen helium in, the amount we would need it to be filled with, and then built the rest of the box, sealing it in (all in a vacuum chamber of course, since that's how those hypothetical fabricators are typically depicted). Would that be permeable? Atomically precise diamond lattice or whatever, would that be permeable?

Hypothetically speaking...

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Yes, this is feasible in practice for a fair definition of "permanent". Monolayers (sheets a single atom thick) of graphene are impermeable to any gas, including helium. To be more precise, the bulk permeability rate for helium through graphene is very low, of the order of $10^{-15}\ mm\ g\ cm^{-2}\ s^{-1}\ bar^{-1}$. For comparison, it means a stack of graphene layers $100\ \mu m$ thick (about the thickness of paper or a human hair) will leak less helium than a $10\ cm$ thick slab of glass

In fact, the original article which investigated gas retention of graphene monolayers specifically mentioned making microscopic balloons, including helium balloons. I don't see why a macroscopic balloon wouldn't work, other than the difficulty involved in preparing a large flawless sheet of the material. There is currently a ton of research behind large scale production of graphene, so something of the sort may actually happen eventually.

Even more interestingly, a closely related yet quite distinct material called graphene oxide has been shown to also create films which are essentially impermeable to helium, just like graphene, but water passes through the graphene oxide film unimpeded, as if there were no barrier at all!

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