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A nanomaterial has constituent particles or structural features on the nanoscale, conventionally around $1$–$100\mathrm{nm}$. A collection of nanoparticles is one example; a bulk piece containing nanoscale internal structure can also be relevant. The scientific interest lies in how dimensions and interfaces affect behavior, rather than in treating $100\mathrm{nm}$ as a universal physical threshold.^[ref]

A simple geometric calculation explains one important effect. Suppose a fixed volume $V$ of solid is divided into identical spherical nanoparticles of radius $r$. Ignoring contact between particles, their number $N$ and total surface area $A$ areReducing $r$ from $50\mathrm{nm}$ to $5\mathrm{nm}$ makes $1000$ times as many particles and ten times as much surface area, at the same total volume. More material is consequently close to an interface. Surface reactions and interactions with the surrounding medium can become much more influential; this geometric result alone does not guarantee a faster useful reaction or a more stable material.^[ref]

Size can also affect electronic behavior through quantum confinement. Which effect matters depends on the material, shape and environment. A useful comparison therefore holds composition fixed, measures particle dimensions with electron microscopy, and compares a chosen property across a size series. Such a comparison separates a size-dependent change from a change caused merely by using a different substance.^[ref]

Yi Cui develops nanomaterials and methods for studying their structure and behavior, with applications including lithium-ion batteries, catalysis and environmental technologies. His Stanford research description identifies this combination of material design and characterization.^[ref] He also coauthored the indium tin oxide study cited here.^[ref]

A useful geometric motivation for nanoscale design is that a sphere's surface-area-to-volume ratio is $3/r$. Reducing its radius tenfold increases that ratio tenfold, changing how much material lies near an interface. In an electrode, small dimensions can also shorten transport distances, while a greater exposed surface can increase unwanted reactions. Useful material design must therefore test the complete device behavior rather than assuming smaller features improve every property.