## Hall Petch Strengthening Mechanism

The experimental Hall-Petch relationship applied to hardness shows that the hardness of microcrystalline materials increases with a decrease in the average diameter of the grains [328-332]:

where d is the average diameter of the grains, H is the hardness, H0 is the hardness of the monocrystal when d ^ x>, and kH is the Hall-Petch intensity constant.

Extrapolation of the Hall-Petch relationship to nanostruc-tured materials should show a much higher hardness than that of coarse-grained materials having the same composition. However, limitations should be taken into account [199] since (1) hardness cannot increase indefinitely, (2) relaxation processes in grain boundaries can decrease hardness, and (3) in nanometer-sized grains the Hall-Petch strengthening is not valid for nanograins less than 2 nm [333]. Regarding nanolayered composites, the Hall-Petch strengthening is not as applicable [334] when the bilayer repeat period approaches a few nanometers in size.

Originally, the empirical Hall-Petch relationship [328, 329], which was obtained for coarse-grained microcrystalline mild steel, ingot iron, and spectrographic iron with grain sizes larger than 1 ¡im, was based on the conception of dislocation pileup in individual grains. Therefore, the Hall-Petch strengthening for coarse-grained structures is explained in terms of dislocation propagation [335, 336]. For microsized grains, it is easier to activate the dislocation multiplication and propagation at lower stress levels. But dislocation sources [337-340], for example, the Frank-Read dislocation source, can operate if grains are larger than 15-100 nm [341]. Grains are free of dislocations when grain sizes are smaller than [341, 342]:

where a1 is a geometrical constant with value of about 0.51, a is an appropriate stress, b is the numerical value of Burger's vector, and G is the shear modulus. Similar evaluation of the critical grain size has been given [343]:

where H is the hardness and v is the Poisson ratio. For grains smaller than dc, material softening should occur because of the introduction of a new deformation mechanism due to refinement of the nanostructure [344].

From the amorphous state through the nanocrystalline state to the microcrystalline state, a schematic diagram of microhardness versus modulation period, which is represented by a bilayer repeat period (A) for nanolayered composites and an average nanograin size (d) for nanocrys-talline composites, is presented in Figure 5. Interval I of the modulation periods reflects the microhardness anomaly of nanocomposites when the nanostructure and composition of the composites becomes optimal. The critical modulation period corresponds to maximum microhardness. Interval II of the modulation periods corresponds to transformation of the nanostructure into an amorphous state. Interval III corresponds to the classical Hall-Petch relationship.

Various attempts have been made to explain [198, 311, 346-357] the hardness behavior of nanostructured coatings and to modify [333, 341, 343, 358-366] the classical Hall-Petch model, adapting it to nanostructured materials. The critical average nanograin size (dc), when the hardness reaches a maximum, was calculated [360], and the hardness for grain size d > dc and d < dc was evaluated. The hardness values are described by the following Hall-Petch equation adapted for nanocrystalline coatings [360, 367, 368]:

where a = a/(Gb2) is a dimensionless constant, a is the applied shear stress, G is the shear modulus, b is the magnitude of Burger's vector of dislocation, and reff is the effective cutoff radius.

This model predicts the slope formation and magnitude of the microhardness dependence on grain size. The inverse (or abnormal, or negative) Hall-Petch H2 gives a good phenomenological agreement with the experimental softening of nanocrystalline materials for d < dc [333, 369373]. The inverse Hall-Petch region is sometimes referred to as "Coble creep" [333]. Good agreement with a large set of experimental data has been demonstrated [310, 374]. Thus, the standard Hall-Petch strengthening relationship was adapted to nanostructured coatings. It was proposed [375] that a deformation mechanism such as grain boundary sliding appears at very fine grain sizes approaching the amorphous state (d < dc). The proposed mechanism considers sliding along suitable grain boundaries. The model offers a physical explanation for the softening of nanomaterials at very small grain sizes.

It has been clearly shown for various materials [161, 342, 376-384] that optimization of the nanostructure of the composite coatings, for example, to be free of dislocations, is more important than interatomic bonds in the crystal lattice in order to obtain superhard nanocomposites. The major principles on which the superhardness of the nanostructured coatings is based are as follows:

• Crystal domains of nanoscale size are free of any dislocations or have a small number of dislocations in critical pileup.

• Propagation of dislocations through phase interfaces is strongly hampered.

• Dislocation multiplication sources cannot operate in domains of nanoscale size.

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