Micrometer Slider Cubes, Yet Again
The MSC are aluminosilicate molecular nanotechnology, ASMNT, and mass about three grams per cc. AlSiO, is 94% of the mass of the Earth, so we're not going to run out; the problem, if any, is the other atoms, carbon, hydrogen, nitrogen, and rarer elements, which would make up some of the 'working parts', where AlSiO is mostly about structure. It is the bones of the planet, after all.
(Mind, this is all just a thought experiment, if I can even dignify it with that much credibility. I'm vague on a lot of this, because there are good reasons why we might not be able to treat atoms as 'solid' structural members; that's also why I'm describing these assembler/disassemblers as 'machines' approximately a thousand atoms on a side, and half a billion atoms, half empty, half the specific density of the Earth at 2.8 g/cc)
Triangular in cross-section this time around, one millionth of a meter tall and four of them together make up a micrometer cube. A manipulator places each atom in its' place, in theory, at the rate of one million times per second, 1 megahertz. If the triangle is one thousand atoms and one thousand nanometers across, that would be one millionth of a meter, one micrometer. Four of them to make up a one micrometer square, a quarter of a million atoms to place in a quarter of a second. One thousand layers would take 250 seconds, a little over four minutes. That would be four in a one micrometer cube ten thousand times smaller than a centimeter, 10^4 cubed or 10^12 per cc. One cc of them is 1 m^2, one layer thick, or 1/4 m^2 four layers thick.
I'm going with 13 watts/joules per second per square meter of photovoltaic surface, for seven good hours of sunlight on an average day. and about 10 MJ/kg to convert aluminosilicates, rocks, into ASMNT. As I mentioned, 47 out of every 50 atoms which make up the Earth are aluminum, oxygen or silicon, rock, sand, dirt without the organics which make good farmland. One cc of this masses about 3 g/cc, at 10^4 J/g, 30 KJ into 13 * 60^2 * 7 is 91 * 3600 is ~32 cc of the micrometer things per day, only we're going with four layers, so it's only ~8. Divide that by 100 (14.4/hr * 7 hrs), and compound a 1.0819 fold increase, 100 times, is a 2600 fold increase per day... 47.3 trillion fold in four days, which, starting at 1/4 m^2 and ending up with over 11 m. Km^2 is a couple of percent of the Earths' surface (shades of grey goo, yes). It needs to spread at ~58 m/s to reach the ends of the Earth in that time, a hundred and thirty miles per hour!
How does it spread out? I want it to be four layers thick for redundancy; the 'cubes' are sliding past each other, dropping onto the ground but moving over and next to other 'cubes'. They first need to spread out to get 1/4 m^2 and ~3 watts/joules per second, which can happen in under one second @ 1m/s, a little slower than walking speed. But that's just area, not a shape. I'm partial to spreading out, in curls, even. The layers just need to flex a little, stretching a little longer to the outside of the curve. Fractal, 'ferny' shapes branching off into infinity, and beyond!
If the stuff spreads out into two circles four layers wide and thick, the two circles are each about 10^4 meters across, 20 by 10 km. 20 circles, one km across four by five six-pack fashion would be five by four kilometers, or like a honeycomb is five by seven, roughly.
Another way is to float...
A one meter sphere has a surface area of pi and a volume of about half a cubic meter. Four layers thick, times pi, times 2.8 g/cc, is about 35 grams. The air in the sphere needs to be more than 2.5% warmer than ambient, of about 288 K, 7.2 degrees C, 72 degrees F. A four meter version has sixty-four times the lift @ 2.24 kg, but masses sixteen times as much, .56 kg, a net lift of 1.68 kg. At 18 degrees C above ambient, that's 5.6 minus .56 kg, ~5 kg net. The air inside is blood warm, but since hot air rises, the bottom of the 4 meter, 13 foot, sphere is somewhere between 98 and 59 degrees F, eighty-some degrees. A 20 meter sphere has 700 kg less 14 kg. Fifty-seven layers masses nearly 200 kg, with a net lift of half a tonne. This is actually a bit of an upside down tear-drop shape, where the mass of passengers and cargo pull it into a bit of point.
Another (and sillier) way to look at this is make a 10 meter cube with an internal temperature ten degrees above the ambient of 288 K, 15 C. About 1200 kg of air at 15 C, displaced by roughly 1160 kg of 25 C air, for 40 kg of lift. The cube masses 600 m^2 * 11.2 g, for four layers, or about 7 kg, so there's about 33 kg of lift, not enough to lift an adult, but a 70 pound child, yes.
I don't have a good idea how the design and production of my MSC would work, but my wild ass guess is that the molecular architects would need to place an atom every second for over a hundred thousand work-hours; 1/2 billion divided by 60^2 is ~140,000. That would be 70 work-years, a lifetime. Even crowd-sourced to ten thousand people, that'd be couple of weeks if the average person only had an hour a day to help. By a team working forty-hour weeks for $100,000 per year, that's $7 million for labor, plus the capital investment is probably an order of magnitude higher. Since we don't actually know how to do any of this, $70 million easily becomes $700 million...
A billionaire could do this, maybe. Not quietly, not without a cover-story, but the story I want to tell always goes with a secret roll-out and some benevolent soul wanting to provide for the basic needs of the entire species. And that fairy tale is one I still want, badly.
A little over one hundred cubic centimeters of my MSC weighs less than five ounces and are the product of nearly half a decade of work by a small team, a few dozen people. One bright spring morning, someone takes them up onto the roof, where, with a good view of the local city and the surrounding river valley (which could be Philadelphia, London, St. Louis, Portland, Sydney, Buenos Aires, or Pittsburgh...) the package is set down and a program automatically causes it to spread out and then blow itself into three bubbles, three one meter spheres which warm and lift off into the breeze.
The three are programmed to spread out, seeking different altitudes and blown in different directions, dropping one cc off every hour, miles apart. The 'submunitions' each form lacy bits with a quarter square meter of surface area. When they touch rock or dirt, they soak up sunlight by day and replicate, spreading out until they have enough substance to blow another one meter sphere and take off again, which only takes a few hours. If the MSC lands in water, it rolls itself into a bobbing float and waits until it touches rock, sand, or dirt again, then spreads out and grows into a hot air balloon...
About 70 percent of the time, the balloons will fall to water, but eventually touch land, and ten percent of the time, they will fall on ice. About 20 percent of the time they will find land again. Even if they only do this once a day, that means an eight-fold increase every day, and after thirty days, there would be roughly three billion of them, probably many more, since the waterlogged ones add up over time... another WAG, but 9^30 is reasonable, I think, should be somewhere in the neighborhood of one hundred billion (9/8ths is 1.125, 9/8^2 ~ 1.25; 1.25^10 is 10, so 9/8^30 is ~ 1.25^15 is ~31 8-P ).
With an average wind-speed of 5 m/s, 18 kph or 11 mph, these balloons would spread around the planet, but at ~180 miles per day, that'd be be only a few thousand kilometers, three thousand miles, in a month. Billions over North America, or Europe, but regional, not global. Delivery of one of these to every person on the planet would take months by wind power...