GLOBAL KNOWLEDGE ARK

Materials Science · Size, surfaces and nanoscale behaviour

Surface-to-volume ratio and interfaces

Useful background: Atoms, bonding, structure and defects; Measuring and checking material claims. Each worked case explains the quantities it uses.

Back to the chapter

After reading, you can

  • Calculate how surface-to-volume ratio changes with size.
01

Small particles expose more surface per volume

For a sphere of radius r, surface area is 4πr² and volume is 4πr³/3. Their ratio is 3/r. Reducing radius therefore increases surface per volume without changing the chemical identity of the atoms.

02

Surface processes can gain importance

Adsorption, oxidation, surface energy and interfacial transport can become more influential. Actual particles may agglomerate or carry coatings, so accessible surface differs from a calculation for isolated smooth spheres. Smaller size alone does not guarantee a useful property improvement.

A sphere has surface-to-volume ratio 3/rThe isolated smooth-sphere model gives 0.03 nm⁻¹ at radius 100 nm and 0.3 nm⁻¹ at radius 10 nm: a tenfold increase. Circle radii have a 10:1 drawing ratio.r = 100 nm; S/V = 0.03 nm⁻¹r = 10 nmS/V = 0.3 nm⁻¹
The isolated smooth-sphere model gives 0.03 nm⁻¹ at radius 100 nm and 0.3 nm⁻¹ at radius 10 nm: a tenfold increase. Circle radii have a 10:1 drawing ratio.

Worked example

Work through the case

An ideal spherical particle’s radius falls from 100 nm to 10 nm. Compare surface-to-volume ratio.

  1. At r = 100 nm, S/V = 3/(100 nm) = 0.03 nm⁻¹.
  2. At r = 10 nm, S/V = 3/(10 nm) = 0.3 nm⁻¹.
  3. Divide 0.3 by 0.03 to obtain 10.

Ten times greater for isolated smooth spheres.

Try the reasoning

  1. Does that imply ten times the accessible area in a clumped powder?

    Show the answer and reasoning

    Not necessarily.

    Contacts and aggregation can block surface.

  2. What must accompany the geometric argument?

    Show the answer and reasoning

    A mechanism linking surface to the target response.

    Geometry alone does not identify performance.

What to carry forward

  • Use geometry to quantify surface fraction.
  • Distinguish ideal from accessible surface.

References

Teaching resources and research supporting this lesson’s explanations and examples.

  1. What Is So Special about Nano?National Nanotechnology Coordination OfficeReferencenano.govRead source