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The Hall-Petch relationship or Hall-Petch Law is a relation in materials science that deals with the connection between the grain size, or crystallite size, and the yield point of a material. This relation says that the larger the grain size of a crystalline material, the weaker it is, that is the smaller its yield strength. The relation is described mathematically by the Hall-Petch equation which is

where k_y is the fitting parameter (a material constant), σ_o is a materials constant for the starting stress for dislocation movement, d is the grain diameter, and σ_y is the yield stress.

Contents 1 History 1.1 Hall 1.2 Petch 2 Reverse Hall-Petch Relationship 3 References

In the early 1950s two groundbreaking series of papers were written independently on the relationship between grain boundaries and strength.

In 1951, while at the University of Sheffield, E.O. Hall wrote three papers which appeared in volume 64 of the Proceedings of the Physical Society. In his third paper, Hall showed that the length of slip bands or crack lengths correspond to grain sizes and thus a relationship could be established between the two. Hall concentrated on the yielding properties of mild steels.

Based on his experimental work carried out in 1946-1949, N.J. Petch of the University of Leeds, England published a paper in 1953 independent from Hall's. Petch's paper concentrated more on brittle fracture. By measuring the variation in cleavage strength with respect to ferritic grain size at very low temperatures, Petch found a relationship exact to that of Hall's. Thus this important relationship is named after both Hall and Petch.

The Hall-Petch relationship has been well established experimentally for grain sizes in the millimeter through submicron regimes. Consequently it was thought that nanosized grains would produce materials with even greater mechanical integrity. Modern computer simulations, however, have shed light on what may actually occur in materials with grains 10-20 nm in size ^[1]. Consequently, it has been proposed that there is a "reverse Hall-Petch relationship" at ever-decreasing grain sizes, in which material hardness and yield stress likewise decrease.

The classic Hall-Petch relationship is based on the idea that grain boundaries act as obstacles to dislocation glide. Dislocations require greater amounts of energy to overcome these barriers to motion. Because dislocations are carriers of plastic deformation, this phenomenon manifests itself macroscopically as an increase in material strength. For very small grains (~12 nm) the deformation mechanism is different. It has been proposed that plastic deformation is no longer dominated by dislocation motion but by atomic sliding of grain boundaries^[1]. In this small grainsize regime, this sliding effect would tend to dominate because of the larger ratio of grain boundary to crystal lattice. This mode of deformation leads to observed softening of a material.

Other explanations that have been proposed to rationalize the apparent softening of metals with nanosized grains include poor sample quality and the suppression of dislocation pileups^[2].

Many of the early measurements of a reverse Hall-Petch effect were likely the result of unrecognized pores in samples. The presence of voids in nanocrystalline metals would undoubtedly lead to their having weaker mechanical properties.

The pileup of dislocations at grain boundaries is a hallmark mechanism of the Hall-Petch relationship. Once grain sizes drop below the equilibrium distance between dislocations, though, this relationship should no longer be valid. Nevertheless, it is not entirely clear what exactly the dependency of yield stress should be on grain sizes below this point.

1. ^ ^{a b} J. Schiotz, K.W. Jacobsen. A maximum in the strength of nanocrystalline copper. Science. 301 (2003), pp. 1357-1359.
2. ^ J. Schiotz, F.D. Di Tolla, K.W. Jacobsen. Softening of nanocrystalline metals at very small grains. Nature. 391 (1998), p.561.

• Meier, Mike. The Hall-Petch Relationship. Sept. 13, 2004. Depeartment of Materials Science at UC-Davis. Nov. 6, 2005 http://www.matsci.ucdavis.edu/MatSciLT/EMS-174L/Files/HallPetch.pdf.