text: The exceptional yield strength of nanocrystalline metals is due to grain boundary strengthening, as grain boundaries are extremely effective at blocking the motion of dislocations. Yielding occurs when the stress due to dislocation pileup at a grain boundary becomes sufficient to activate slip of dislocations in the adjacent grain. This critical stress increases as the grain size decreases, and these physics are empirically captured by the Hall-Petch relationship, 
  where  is the yield stress,  is a material-specific constant that accounts for the effects of all other strengthening mechanisms,  is a material-specific constant that describes the magnitude of the metal's response to grain size strengthening, and  is the average grain size. Additionally, because nanocrystalline grains are too small to contain a significant number of dislocations, nanocrystalline metals undergo negligible amounts of strain-hardening, and nanocrystalline materials can thus be assumed to behave with perfect plasticity.
  As the grain size continues to decrease, a critical grain size is reached at which intergranular deformation, i.e. grain boundary sliding, becomes more energetically favorable than intragranular dislocation motion. Below this critical grain size, often referred to as the “reverse” or “inverse” Hall-Petch regime, any further decrease in the grain size weakens the material because an increase in grain boundary area results in increased grain boundary sliding. Chandross & Argibay modeled grain boundary sliding as viscous flow and related the yield strength of the material in this regime to material properties as 
  where  is the enthalpy of fusion,  is the atomic volume in the amorphous phase,  is the melting temperature, and  is the volume fraction of material in the grains vs the grain boundaries, given by , where  is the grain boundary thickness and typically on the order of 1 nm. The maximum strength of a metal is given by the intersection of this line with the Hall-Petch relationship, which typically occurs around a grain size of  = 10 nm for BCC and FCC metals.
  Due to the large amount of interfacial energy associated with a large volume fraction of grain boundaries, nanocrystalline metals are thermally unstable. In nanocrystalline samples of low-melting point metals (i.e. aluminum, tin, and lead), the grain size of the samples was observed to double from 10 to 20 nm after 24 hours of exposure to ambient temperatures. Although materials with higher melting points are more stable at room temperatures, consolidating nanocrystalline feedstock into a macroscopic component often requires exposing the material to elevated temperatures for extended periods of time, which will result in coarsening of the nanocrystalline microstructure. Thus, thermally stable nanocrystalline alloys are of considerable engineering interest. Experiments have shown that traditional microstructural stabilization techniques such as grain boundary pinning via solute segregation or increasing solute concentrations have proven successful in some alloy systems, such as Pd-Zr and Ni-W.