Everything we make has some minimum mechanical strength to keep it from breaking at undesirable moments, whether a spoon, a skyscraper, a scarf, a sedan, or a circuit; and this strength has several different components, including resistance to tension, compression, shear, flexion, and impact, which trade off against one another to some extent. Also, we make many things with more material than is needed for its mere mechanical strength, in order to make them more rigid. As described in Sandwich theory, scaling laws are such that many of our large structures fail by buckling, which is limited by rigidity rather then tensile strength, compressive strength, shear strength, or impact toughness; and we take advantage of this by producing structures like I-beams, structural tubing, and sandwich panels which have greater rigidity for the same mass.
But I got to thinking about the implications for very small structures, and I came to some astonishing conclusions about the possibilities of massively parallel automated fabrication, even without molecular nanotechnology.
Think about the aspect ratio of the legs of a daddy-long-legs.
But consider scaling down a highway sign. It’s cantilevered out over the highway on a steel truss to support its weight and occasional wind loadings. The truss is made of angle irons, L-shaped profiles extruded or bent from sheet steel. To make up some dimensions, let’s suppose the angle irons are 6 mm thick, and each leg of the L is 30 mm long, and the truss is in the form of a square beam 500 mm across and 5 m long, any given cross-section of which has 8 angle irons running through it for a total of almost three thousand square millimeters of steel out of the 250,000 square millimeters of the cross-section. (This puts the weight of the truss itself in a bit over 100 kg.) Perhaps the truss can safely support a tonne of road sign and whatnot at this 5-meter lever arm, despite being almost 99% empty space.
Now, suppose we scale it down by a linear factor of 100. Now it is 50 mm long and 5 mm across, weighing a bit over 100 milligrams, and the road sign has shrunk from a tonne to a gram. The moment at the base of the truss has gone from 49 kN m to .00049 N m, a decrease of not six but eight orders of magnitude. But the steel cross-section in tension at the top of the truss has diminished by only four orders of magnitude. So, while previously it was near stressed to near its yield stress, now it is four orders of magnitude away from its yield stress. Suppose its Young’s modulus is 200 GPa; its elongation might previously have been 0.07%, or about 3 mm, but now it is 0.000007%, or about 3 nanometers.
This means that we can thin out our angle irons quite a bit. We scaled them down from being 6 mm thick to being 60 microns thick, which is about six times as thick as common aluminum foil (see Single-point incremental forming of aluminum foil). If we managed to scale them down by four orders of magnitude, they would be 6 nanometers thick, and would still be able to hold up the one-gram model highway sign, with the same relative deformation and safety factor under its scaled-down load as the original full-scale highway sign.
This change would change our model truss from being 99% empty space, or rather air space, to being 99.9999% empty space. Instead of 100 milligrams, it would weigh 10 micrograms, which it turns out is about 0.8% of the weight of the air in its air space. It’s 99.2% air by mass now, lighter than the lightest silica aerogel.
However, removing 99.99% of its remaining solid material has also removed 99.99% of its tensile and compressive strength. While previously its tensile and compressive strength was proportionally 100 times greater than those of the full-scale highway sign, now they are 100 times less. So perhaps it will fail in some other way when trying to support the model highway sign; we might need as much as a full milligram of steel to provide the necessary tensile and compressive strength. But at this scale we are no longer obliged to provide massive amounts of unnecessary tensile and compressive strength merely to get adequate rigidity and flexural strength.
(The very light model highway sign would probably be unable to withstand much of a breeze.)
Now, we can’t actually do this with steel; it isn’t malleable enough to roll that thin, and in contact with air, I doubt 6-nanometer-thick steel foil would last long.
However, we can do it with gold leaf, which is typically 0.2 microns thick (according to Compressed sensing microscope) and stable in Earth’s atmosphere. Normally we think of gold as being an extremely expensive material to build things out of; it currently costs US$1480 per troy ounce. But if we only need a milligram of gold to build our model, that’s only 5 cents at that price. We also think of it as being impractically soft and weak, but with the relative strength boost we get from scaling down in this way, we no longer need the brute strength of steel for most things.
Gold leaf is delicate enough that you need to use special hand tools and blowing of air to manipulate it without breaking it. More practical for many uses may be gold foil, which comes in thicknesses of 1 micron to 10 microns.
Carbon nanotubes are thinner than gold leaf as well as stronger, and they are also stable in air. They may provide a better material than gold leaf or gold foil.
These materials are also stable in air and are stronger than gold as well as lighter; short enough spun fibers (or cut crystals) will not buckle under compression. You can build trusses out of them if you can make joints, but often at these scales the problem is not so much getting things to stick together as avoiding catastrophic accidental stiction and cold-welding.
More generally we can think of the existing nearly-isotropic bulk materials we routinely build things from — steel, cement, brass, glass, and so on — as meeting their rigidity requirements at a low space cost by virtue of spending a lot of mass on the problem. Historically we’ve been able to sometimes get lower mass (and lower cost!) by using wood instead, when we can afford a larger volume. Wood is a nanostructured metamaterial, but it, too, is somewhat optimized for low volume, perhaps so that trees can resist wind but I think largely so that they can resist predation. Balsa wood, pith-core trees and the remarkable moringa demonstrate the existence of other possibilities.
Suppose that by using trusses made of gold, glass fiber, or carbon nanotubes, we can produce metamaterials with much better stiffness-to-density ratios. Will this enable us to use only enough material to provide the tensile and compressive strength and impact resistance we need, in macroscopic structures? Maybe not, because the rigidity of a structure and the modulus of elasticity of a material (or metamaterial) are different things.
In Plastic cutters I say ASTM A36 steel has a Young’s modulus of 200 GPa, and like iron and steels in general, a density of 7.9 g/cc; that makes its stiffness-to-density ratio 25 kJ/g. Balsa wood has density ranging up to 0.38 g/cc and axial Young’s modulus up to 9 GPa, giving a surprisingly lower, and similar, 24 kJ/g.
Why are these so similar? Maybe because balsa wood’s axial elasticity, like steel’s, comes from straining crystal lattices and atomic bonds, and the particular crystal lattices and atomic bonds involved aren’t enormously different, perhaps by a factor of 2. Balsa wood spreads them out over a larger cross-sectional area, the rest of which is air, which contributes insignificant mass and stiffness; it thus decreases its density and its Young’s modulus proportionally. The same is true of, for example, steel tubing, which is as light as balsa wood once its width is on the order of 100 times the thickness of its walls.
There are really three contexts where we have to brute-force thicken things up to get the rigidity we need instead of just spreading the mass over a larger area. One is where there’s some mechanical constraint that makes the extra space unavailable: your truck has to fit under bridges, your boring bar has to fit into the hole being bored, your axle needs to fit through the bearing. A second is where we need something like hardness: the force that must be resisted is being applied at a point or over a small area, and so we need a concentration of material in that area to resist it. A third is where making the truss or honeycomb structure or whatever is difficult or expensive because of the limitations of our fabrication technology, and in general that’s a question of things being very small.
(Balsa wood is still better than any artificial material so far, though.)
Very-low-density nanostructured materials made out of thin members have a long history in flight; dandelion seeds, feathers, and parachuting spiders are three examples. Aside from the possibility of lighter-than-air flight (a gold-leaf balloon made of two sheets sealed together at the edges should be able to fly if you can fill its middle with at least 4 mm of hydrogen or helium) the possibility of ultralightweight metamaterials enabling insect-scale structures would seem to offer many fascinating options. Also, structures so delicate that they could easily be blown away might be best off if far from the ground.