The challenge is that Earth's air isn't dense enough. That said, the math is "easy" density is volume/mass, vacuum adds no mass, so it is just material mass / volume.
A 20cm diameter diamond sphere that was .5mm thick would have a mass of about 41g, a 20cm diameter sphere of air at sea level and "room" temperature is about 50g. So you would get 9g of "lifting force" from such a balloon (assuming I did all the calculations correctly). And experience about 2,900lbs of compression force.
If you're interested in some fiction with this concept: In Neal Stephenson's 'Diamond Age' humans have mastered assembly of individual atoms in basically arbitrary ways. They use it (among other things) to create flying airships with an envelope made of diamond with a vacuum inside. Found it to be a fun read in general.
I've really enjoyed several of Stephenson's books, but 'Diamond Age' was one I couldn't get into. I need to give it another shot.
On the subject of Sci-Fi, there's also a vacuum airship featured in one of Edgar Rice Burroughs' Tarzan novels - 'Tarzan at the Earth's Core'. It is of course much less scientific than Stephenson's version though.
He really likes to drop readers in with minimal explanation into a world, he's gotten much better at it over time but Diamond Age is both one of his earlier works and one of his most esoteric worlds with the odd neo-Victorians etc.
That's always a fun blast from the tech hype past where nanotechnology was going to be pure magic with tiny machines capable of doing anything. That was an exciting possible future while it lasted, nanotechnology is doing neat stuff but it's nowhere close to the wild promises.
Yeah we're starting to do that but the individual pieces are pretty large and I think we're much more level headed about how small they can be shrunk now.
This is a plot element in Neal Stephenson's The Diamond Age, as well as several other fictional appearances (Edgar Rice Burroughs, Azhar Abidi, Peter Watts, and Iain M. Banks all use the trope).
There are no materials known with sufficient strength to withstand compressive and buckling forces. Not even diamond.
Lightweight stiff structures (honeycomb, something resembling aerogel, perhaps) are other options, but seem unlikely as well.
Gas-filled airships or balloons benefit from the fact that low-density gasses (by either chemical composition or temperature in the case of hot-air balloons) exert a countervailing pressure to balance atmospheric pressure, but with a lower mass, hence providing buoyancy. The internal pressure actually provides some (or much) of the structural rigidity of most airship variants. Any vacuum airship would have to make up for this factor, again, increasing strength (and material mass) requirements.
I am following https://www.o-boot.com/en/ with interest. They claim to have solved the buckling issues by using a roman arch-like structure.
The idea sounds relatively sound to me. One could construct a sphere out of aerogel cones, and the external pressure would reinforce it. Not sure how practical it would be to build such a thing, though.
s/since volume grows slower than volume/since surface area grows slower than volume/
Square-cube law. Mass scales with surface area, lift scales with volume. Large balloons (conventional, not vacuum) are simpler and more efficient than small ones.
As to materials, a tremendous problem with areogels are that they're exceedingly friable. Any friction, flexing, or stress will crumble the gell to a powder. I suspect this is why the gel hasn't taken off as it had been projected to. I did some very-early 1990s work in the space and aerogels were a noted emerging technology thought to have applications in, e.g., mobile home, pre-fab housing, and RV designs. For the most part, fibre-based insulation or expanded-foam (polystyrene) insulation remains the standard, largely because road and other vibration don't reduce your insulation to a few inches of fine dust at the bottom of wall cavities.
Material properties are complex, and represent interesting trade-offs between afforded capabilities and imposed constraints.
That’d be pretty cool!
How huge would it have to be? Would it have to be made of titanium? Aluminum? Stainless steel?
How good are those materials at withstanding a vacuum (or partial vacuum) if the diameter was say... 50m?
I feel like at some size it must work, as the volume of air displaced goes up with r^3 and the surface area of metal only goes up with r^2.
If people here can do the math, maybe I could build one in time for next burning man...!