Introduction to Nanotechnology

140 BULK NANOSTRUCTURED MATERIALS
where B, is the fhctional stress opposing dislocation movement, K is a constant, and
d is the grain size in micrometers. Hardness can also be described by a similar
equation. Figure 6.9 plots the measured yield strength of Fe-Co alloys as a function
of d-(’’’), showing the linear behavior predicted by Eq. (6.1). Assuming that the
equation is valid for nanosized grains, a bulk material having a 50-nm grain size
would have a yield strength of 4.14 GPa. The reason for the increase in yield strength
with smaller grain size is that materials having smaller grains have more grain
boundaries, blocking dislocation movement. Deviations from the Hall-Petch beha-
vior have been observed for materials made of particles less than 20 nm in size. The
deviations involve no dependence on particle size (zero slope) to decreases in
yield strength with particle size (negative slope). It is believed that conventional
dislocation-based deformation is not possible in bulk nanostructured materials with
sizes less than 30 nm because mobile dislocations are unlikely to occur. Examination
of small-grained bulk nanomaterials by transmission electron microscopy during
deformation does not show any evidence for mobile dislocations.
Most bulk nanostmctured materials are quite brittle and display reduced ductility
under tension, typically having elongations of a few percent for grain sizes less than
30 nm. For example, conventional coarse-grained annealed polycrystalline copper is
very ductile, having elongations of up to 60%. Measurements in samples with grain
sizes less than 30nm yield elongations no more than 5%. Most of these measure-
ments have been performed on consolidated particulate samples, which have large
residual stress, and flaws due to imperfect particle bonding, which restricts disloca-
1300~1 I I I I I I I I I I I I I 1 I I I I I I I I 14
1//d (microns)-’’2
Figure 6.9. Yield strength of Fe-Co alloys versus l/d’/‘, where d is the size of the grain.
[Adapted from C.-H. Shang et al., J. Mater. Res. 15, 835 (2000).]