An explanation to the abnormal Hall-Petch relation in nanocrystalline materials

Introduct ion For conventional polycrystalline materials, the hardness (or the strength), Hv, is related to the average grain size, d, in the well-known Hall-Petch relationship [1]: H,, = Ho + kd- ' /2 (1) where H0 and k are constants. Normally the value of k is positive, or the hardness of a polycrystal increases with a refinement of grains. This relationship has been confirmed in both theory and practice in many metallic materials usually with grains coarser than micrometers [2]. In recent years, a new catalogue of polycrystalline materials with ultrMine grains (usually in namometer scale), or called nanocrystalline (NC) materials, have been synthesized [3], which extends understanding of the structure-property elationship in polycrystalline solids. Among the investigations on properties of the NC materials, several cases of the abnormal H-P relationship where hardness decreases with a reduction of grain size, or a negative k value appears, have been detected in some NC pure metals [4,12], metallic alloys [5,6-,,10], and oxides NC materials [11], which were synthesized using various methods, including the inert gas condensa-tion and consolidation of the ultrafine powders (UFP) [3], ball-milling, crystallization from amorphous solids [13], electrodeposition [6], and so on. The evidence which deviates from the normal H-P relationship indicate