The strongest size in the inverse Hall-Petch relationship
  Incorporating the bond-order-length-strength correlation mechanism [Sun CQ, Prog Solid State Chem 35, 1 -159 (2007)] and Borns criterion for melting [J. Chem. Phys. 7, 591(1939)] into the conventional Hall-Petch relationship has turned out an analytical expression for the size and temperature dependence of the mechanical strength of nanograins, known as the inverse Hall-Petch relationship (IHPR), that has long been a topic under debate regarding the possible mechanisms. Reproduction of the measured IHPR of Ni, NiP and TiO2 nanocrystals revealed that: (i) the size induced energy densification and cohesive energy loss of nanograins originates the IHPR that could be activated in the contact mode of plastic deformation detection; (ii) the competition between the inhibition of atomic dislocations, via the surface energy density gain and the strain work hardening, and the activation for dislocations through cohesive energy loss determine the entire IHPR profile of a specimen; (iii) the presence of a soft quasisolid phase is responsible for the size-induced softening and the superplasticity as well of nanostructures; (iv) the bond nature involved and the T/Tm ratio between the temperature of operating and the temperature of melting dictate the measured strongest sizes of a given specimen. 
Chang Q. Sun