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Aerogel is such nifty stuff that I'm going to stop here for the moment, add some links, and then add some more text.
The NYT article says that aerogel holds 14 records in the Guiness Book.
This stuff should work for a vacuum balloon shell. The outer surface probably needs to be given a thin coating
in order to make sure it is air-tight.
I'm not sure how small a vacuum balloon can be, if made with an aerogel shell. Perhaps a foot in diameter (about 1/3 meter). I'm quite sure a meter-diameter balloon would work; it may not even need to be fully evacuated to float at sea-level! (Also, a partial-pressure balloon wouldn't have to resist as much air pressure, so its shell could be thinner.)
A nice article at Wikipedia [Vernon, Oct 24 2008]
More on aerogel
The New York Times describes a search for a suitable material for a NASA project. [Vernon, Oct 24 2008]
A really cool picture! [Vernon, Oct 24 2008]
Pictures and links, courtesy of Google [Vernon, Oct 24 2008]
Hot Hydrogen Balloon
[MisterQED, Oct 24 2008]
(?) Develop a lighter-than-air solid
Aerogels were discussed in detail in this halfbakery idea over the past seven years. [Amos Kito, Oct 25 2008]
Aerogel art, created in a mold. [Amos Kito, Oct 25 2008]
wikipedia on microlattice, and just slightly, aerographite
[beanangel, Dec 27 2017]
||Interesting but my idea is better, see Hot Hydrogen Balloon (link). I added a glass outer shell for strength and rigidity and a process to launch the balloon from sea level and survive the pressure.
||The problem with Aerogels under tension is that they tend to shatter. It's also quite hard to shape them into anything other than lumps or squares or slabs.
||A better idea might be to fill each gel pocket with hydrogen instead of, if I remember correctly, CO2. No idea if that would work, but it might decrease the density further so much as they float by themselves.
||Oh, and aerogels are expensive. Very expensive.
||[mitxela], aerogels are expensive mostly because the stuff is not yet in such demand that lots of effort has been put into finding less expensive manufacturing methods. We also can ignore such things here at the HalfBakery, when posting Ideas. Next, I'm sure any shape that can be molded (such as a hemisphere) can be the final shape of a block of aerogel. Next, ANYTHING solid will shatter if enough force is applied; you are not saying anything new, AND you are not showing that the steady pressure against a partial-pressure aerogel-shell at low altitude, or against a vacuum-filled aerogel shell at high altitude, will be crushed. I ignore your remark, therefore.
||Aerogels are hard to form into precise shapes? That sounds like the requirement list for a new extreme sport: Aerogel Topiary.
||It's so fragile, you'd have to use a microscopic process to carve it into shapes -- maybe with a laser? It resembles a hologram and could be formed into fascinating sculptures. Hang on, somebody's done that. [Link]
||Aerogel is extremely porous glass, formed as a foam. It can be crushed by a fingertip. Perhaps the idea is to use a stronger composite material instead of silicon dioxide. Then it can be crushed by a thumb.
||Off-Topic: As to sculpture, can glass be made to "foam" in a less expensive process? A little thicker, a little stronger, but still almost transparent.
||[mitxela], it appears I left something out of my previous
annotation. If an aerogel shell is used for a vacuum balloon,
that evacuated shell will be under compression by the
outside air, not tension. So, see the "Resisting Pressure"
||And now there is microlattice! Wikipedia [link]
"Metallic microlattices are characterized by very
densities, with the 2011 record of 0.9 mg/cm3
among the lowest values of any known solid. The
previous record of 1.0 mg/cm3 was held by silica
aerogels, and aerographite is claimed to have a
density of 0.2 mg/cm3."
||At first I thought microlattice balloon, but
aerographite would be the latest.
||Nobody sneeze at the metallic microlattices factory...