Okay DC, and UB, heres something to shoot at, as I really dont know much about the physics involved.
Construct an envelope for the balloon from a layer of Mylar, a layer of aerogel, and a layer of nylon, in that order. The Mylar will insulate and waterproof the aerogel, the aerogel (only 3 times
heavier than air) will provide an extreme level of insulation, and the nylon will add structural integrity for both the aerogel compound, and the balloon rigging.
The envelope will be sealed, attached to the outside atmosphere through a small air pump. This is to prevent heat radiating out of the interior air. The pump is used to increase or decrease volume for rapid altitude changes. For normal altitude changes, a heat pump is used to vary volume.
Heat, to expand the volume of air inside the balloon will come from an electrically driven heat pump (or propane fired). The heat pump will draw air in near the lower extremity of the envelope and expel it almost centrally, so there is again, no exchange with the outside atmosphere. Heat will come from a thermal accumulator (aerogel insulated) that is used to either store or give-up heat for the main envelope. During the day, PV cells provide additional power to heat the accumulator, as well as charge batteries for the night portion of the trip. Using a heat pump will lighten the energy load (and thus the weight) by not using raw fuel to heat freezing high-altitude air.
The balloon envelope Im imagining will be about 1cm thick so not as convenient as a regular balloon, and somewhat more fragile.
All Ive found so far, in terms of the insulating property of an aerogel, is that the r-value can go from about 16, at 1 atmosphere, all the way up to 120 in a vacuum.
Seems, that the greater the altitude of this balloon design, the better the insulation works.
Aerogel weighs as little as 1.2 milligrams per cubic centimeter. Nylon?? Mylar, a few grams/m2.
Anyone care to take a stab at the math involved?