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Nanotechnology Engineering DISCUSSION
Every introduction to nanomaterials says that properties change at the nanoscale because of the high surface-to-volume ratio, but usually without numbers. I would like to be able to calculate it.
For spherical silica particles of 20 nm diameter, what is the surface area per gram, and how does it compare with 10 µm particles of the same material? And how can I estimate what fraction of the atoms in a small metal particle, say gold at 3 to 50 nm, actually sits at the surface?
For a sphere of diameter d, the surface area is π × d² and the volume is π × d³ / 6, so the ratio is 6 / d. Dividing by density gives the specific surface area: SSA = 6 / (ρ × d).
Silica with ρ = 2200 kg/m³ and d = 20 nm: SSA = 6 / (2200 × 20 × 10^-9) = 136,000 m²/kg, or about 136 m²/g. For d = 10 µm the same formula gives 0.27 m²/g. The area per gram scales with 1 / d, so going from 10 µm to 20 nm multiplies it by 500. About two grams of the nanopowder has the surface area of a tennis court.
The relation is used in reverse all the time. Gas adsorption (the BET method) measures SSA directly, and an equivalent spherical diameter follows from d = 6 / (ρ × SSA). A titania powder (ρ about 4000 kg/m³) with a measured 50 m²/g gives d = 6 / (4000 × 50,000) = 30 nm, remembering to convert m²/g to m²/kg.
Compare that with the size from electron microscopy. If the BET diameter is much larger than the primary particle size seen in the microscope, the particles are fused into hard aggregates with necks between them and part of the surface is lost. If it is much smaller, the particles are porous or rough.
For the fraction of surface atoms, a simple shell estimate works well: treat the outermost atomic layer as a shell one atom thick. The fraction is then 1 - (1 - 2a / d)³, where a is the atomic diameter. For gold, a is about 0.29 nm.
d = 50 nm: 1 - (1 - 0.0116)³ = 3.4 percent. d = 10 nm: 16 percent. d = 5 nm: 31 percent. d = 3 nm: 47 percent. Below about 5 nm a large share of the atoms are surface atoms with fewer neighbours than in the bulk, and that is the size range where properties such as melting point and catalytic activity depart clearly from the bulk values.
Why the surface matters physically: surface atoms have unsatisfied bonds, so they carry extra energy. A powder with 136 m²/g stores a significant amount of surface energy and lowers it any way it can. That shows up as strong agglomeration, sintering at temperatures far below those of coarse powders, faster dissolution, higher reactivity and strong adsorption of water and organics from the air.
It is also why nanopowders need stabilisers or surface coatings in suspension, and why a measured property can change simply because the powder was stored open to the atmosphere.
The same numbers are behind the handling precautions. Many metal nanopowders (aluminium, iron, titanium) oxidise so fast on their large surface that they can ignite spontaneously in air or form explosive dust clouds, and dry nanopowders in general become airborne easily and stay suspended for a long time.
Handle dry powders in a fume cupboard, glove box or ventilated enclosure with suitable filtration, keep them as suspensions or pastes where the process allows, avoid ignition sources with reactive metals, and follow the supplier's safety data sheet. Weighing a few milligrams on an open bench is poor practice.
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