Cheap roofs are mostly made out of corrugated galvanized sheet steel, but this has several disadvantages. A typical price is AR$2070/1.1m² (US$12/m²) for 500-μm-thick (25-gauge, in the Argentine system) corrugated galvanized steel, so the cost is not insignificant. The metal has no real insulating properties, reradiating all the heat of the sunlight that it absorbs, and although its finish is initially quite reflective, soon after installation it corrodes enough to absorb a lot of sunlight. It’s kind of heavy (4.4 kg/m²). It tends to make a lot of noise when things fall on it. You have to drill holes in it to fasten it to things, which creates water leaks when it rains. It’s a pain to bend or cut.
Maybe you could make a better material out of a sandwich panel. Take a layer of aluminum foil (50¢/m², see Aluminum foil) and lay down a layer of aluminum window screen on top of it; a 1.2 m × 30 m roll costs AR$13244, US$81, US$2.25/m². Or 200-μm-thick galvanized steel window screen, at AR$9053 for 1 m × 30 m, US$1.90/m². Or fiberglass window screen; a 1 m × 30 m roll costs AR$5377, US$33, US$1.10/m² — though that vendor says it’s actually not glass but plastic! On top of the window screen, add a 50mm layer of expanded vermiculite (0.1 kg/ℓ, US$0.23/ℓ, according to Potential local sources and prices of refractory materials), previously moistened with waterglass (US$2/kg) and a second layer of window screen. Now press down the whole pile to ensure good contact among the vermiculite particles, and solidify the waterglass, either by letting it dry or by gassing it with CO₂.
(Dehydrated alabaster is a possible alternative binder, and it’s cheaper at US$0.30/kg, but it won’t coat the vermiculite grains as nicely, so you might end up using a lot more of it.)
(A layer of chicken wire (US$40 per roll of 1 m × 25 m, thus US$1.60/m²) or hardware cloth (US$82 per roll, thus US$3.30/m²) might provide additional strength and stiffness. 220g/m² woven fiberglass cloth for composites is US$3.50/m².)
The window screens provide tensile stiffness, and the somewhat springy vermiculite provides shear strength and impact absorption. The aluminum foil reflects the sunlight and adds a little extra stiffness, and the vermiculite provides insulation. The waterglass sticks it all together, and in particular keeps the aluminum foil from flapping in the wind.
Let’s guess the weight of the waterglass is about the same as that of the vermiculite, which turns out to be 5 kg/m². So 1 m² is US$0.50 (foil) + US$4 (screens) + US$11.50 (5 kg vermiculite) + US$10 (5 kg waterglass) = US$26. So our panels weigh 10 kg/m² and cost US$26/m², each twice as much as the corrugated steel we were hoping to improve on. But now we have insulation and corrosion resistance, the panels absorb sound and can be cut with a box cutter, and we can drive screws into them without impairing their water resistance. And they’re still fireproof.
Loose vermiculite might conduct heat at 0.06 W/m/K, but with the waterglass it’s probably more like 0.1 W/m/K. So if our vermiculite roof is at 45° and the indoors is at 20°, it will conduct about 50 W/m², which seems like a lot, even if it’s only 5% of what enters through an open window. This is a U-value of 2 W/m²/K.
What’s the flexural strength of the panels? It seems like it ought to be something reasonable, but I’m not sure how to calculate it.
In this form, we haven’t yet achieved a Pareto improvement but only a tradeoff; however, we’ve come within a stone’s throw of the price and weight of the standard approach. Can we improve these panels further?
Because we don’t need super high temperature resistance, it’s probably better to use fiberglass insulation (US$0.026/ℓ, about 0.025 W/m/K, and also lower density, 0.02 kg/ℓ) or, if properly fireproofed, styrofoam at US$1/m² for 20mm (US$0.05/ℓ, 0.033 W/m/K); both of these have several times lower thermal conductivity than vermiculite. (Fiberglass is much less rigid, though, so perhaps it should be installed below the roofing panels rather than integrated into them.) Then maybe we could use a thinner vermiculite layer, just to provide a little stiffness, or a layer of gypsum. Or two layers of gypsum separated by a lower-density layer.
You could use perlite, LECA, or pumice, instead of vermiculite; all of these are stiffer than vermiculite but also denser and more expensive. Fired-clay ceramic made porous by the burnout of organic materials like sawdust or yerba mate is another possibility; though not commercially available, it's easy to make, and even at 75% or 80% burned-out filler, it’s still pretty solid.
In a similar way, it ought to be possible to dilute the vermiculite with another granulated material of about the same granulometry, but much lower density and strength, without interrupting the continuous network of vermiculite grains in the finished composite. Let’s call this bulking additive “filler”. Crude reasoning suggests that the dilution could be up to a factor of 3: in a close packing of spheres, each sphere is in contact with 12 others, and to form a 3-dimensional network rather than a 2-dimensional one, at least 4 of those spheres need to be non-filler. See Glass foam for one filler possibility; styrofoam and foamed starch are two others.
If the vermiculite grains form a continuous network, these filler grains could be removed once the material has solidified. Organics could be simply burned out; foamed starch could be washed out with water much more quickly than the water would affect dried waterglass; anhydrous calcium chloride is a candidate filler that could be washed out with water and would also instantly, irreversibly harden any waterglass it came in contact with during the mixing process, keeping the bulk of each calcium chloride grain from being dissolved. Calcium chloride is fairly cheap (US$1.60/kg) and could be reused.
There are other candidate aqueous binder systems that could be similarly activated by surface contact with grains of a water-soluble “filler”, including aqueous solutions of soluble carbonates and phosphates (I wrote about some of these in my Dercuano note on powder-bed 3-D printing processes) and Sorel cement (where the “filler” grains would be magnesium chloride). Reversing the roles, aqueous calcium chloride itself is capable of forming a thin coating on grains of aggregate such as vermiculite, and then being hardened by surface contact with “filler” grains of anhydrous soluble carbonate or phosphate, or possibly of solid potassium silicate.
By carrying out such a “dilution” at multiple granulometry scales, it might be possible to get much lower vermiculite densities. Suppose 2-mm expanded vermiculite grains mixed 1:1 with 2-mm filler grains can successfully form a continuous vermiculite network, with a little binder, which seems likely. Then if the resulting 50%-vermiculite mixture is mixed 1:1 with 5-mm filler grains, the same process should repeat at larger scales: the vermiculite-network regions should be able to form a continuous-phase network around the 5-mm filler grains, for a 25%-vermiculite solid. A third stage of dilution with 12-mm filler grains would repeat the process, leaving a 12½%-vermiculite solid network with “pores” at different scales of 12 mm, 5 mm, and 2 mm, filled with useless filler particles, effectively a foamed foam.
Such high-porosity solids might be useful for a variety of purposes other than construction insulation.
A thin layer of quartz sand under the aluminum foil, once bonded with waterglass or something similar, would harden the aluminum-foil surface greatly against abrasion and impact, as well as providing a great deal more stiffness per dollar than metal reinforcement could. Construction sand (relatively pure quartz) costs US$0.03/kg and weighs about 2.4 g/cc. A 500-μm-thick layer would thus cost about US$0.04/m² per side and weigh 1.2 kg/m² (2.4 kg/m² on both sides). This would probably require the panel as a whole to be very stiff, because this layer of effectively mortar would crack off very easily if the surface flexed much. Gypsum is more expensive, but lighter and more flexible, so it might be a better option. Gypsum used as a binder for other lightweight, stiff aggregate such as expanded perlite might be better still.
Sandwich panel optimization has notes on how to get the cheapest sandwich panels for a given stiffness.