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boundary component can reach large volume fraction values. Fig. 2.3 shows a schematic diagram of the atomic structure of a two-dimensional nanocrystalline material.

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The boundary core regions (open circles in Fig. 2.3) are characterized by a reduced atomic density and interatomic spacings deviating from the ones of a perfect lattice (black circles). Using a three-dimensional treatment involving tetrakaidecahedral grains (Fig. 2.4a), Palumbo et al. (1990) presented a detailed calculation showing the variation of the total intercrystal volume fraction, f ic, consisting of grain boundaries, f gb, and triple junctions, f tj, as a function of grain size, d (Fig. 2.4b). The volume fractions were calculated by using a grain boundary thickness, , of 1 nm,

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The intercrystal volume fraction ( f ic ) increases from a value of 0.3% at 1000 nm to more than 50% at grain sizes less than 5 nm. The triple junction volume fraction ( f tj ) displays a stronger grain size dependence when compared to the grain boundary volume fraction ( f gb ).

In the range of 100 nm to 2 nm, the triple junction volume fraction increases by three orders of magnitude, while the grain boundary volume fraction increases by a little over one order of magnitude.

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Figure 2.4.

(a) Tetrakaidecahedral grains, (b) effect of grain size on calculated volume fractions of intercrystal regions, grain boundaries and triple junctions, assuming a grain boundary thickness of 1 nm [Palumbo et al. (1990)].

2.1.3. Properties Hardness Nanocrystalline Ni produced by electrodeposition was first studied systematically by El-Sherik et al. (1992). They found that grain refinement of electroplated Ni (100 nm) resulted in unique and improved properties when compared to conventional polycrystalline

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Ni. For example, the hardness of the electrodeposited Ni ranges from about 150 VHN for deposits with ~100 μm grain size to about 650 VHN at 10 nm [El-Sherik et al. (1992)]. At grain sizes less than 30 nm, a noticeable deviation from the regular Hall-Petch relationship was observed (dashed line in Fig. 2.5). This deviation was considered to be an effect of increasing grain boundary and triple junction volume fractions similar to that described by McMahon and Erb (1989) in their study of nanocrystalline Ni-P alloys, and consequently, differences in deformation mechanisms.

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Cheung et al. (1995) developed an electroplating bath which could be used to produce nanocrystalline Ni-Fe alloys (0-28 wt.%Fe) in bulk form. Using X-ray diffraction, it was found that deposits in the range of Fe concentration studied preserved the face-centered cubic (fcc) structure. The grain size was measured using the Scherrer XRD line broadening

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technique [Cullity and Stock (2001)] and found to decrease with increasing Fe concentration (Fig. 2.6). Grimmett et al. (1993) in an earlier study and Li and Ebrahimi (2003) in a later study found similar correlations.

Figure 2.6.

Effect of Fe concentration on the grain size of electrodeposited nanocrystalline Ni-Fe alloys [Cheung et al. (1995)].

The Vickers microhardness was measured as a function of Fe concentration and compared to conventional polycrystalline materials [Cheung et al. (1995)] as shown in Fig. 2.7. The hardness of conventional polycrystalline Ni is reported to be 85 VHN and as the Fe concentration increases the hardness essentially remains constant up to 51 wt.%Fe. The hardness of the nanocrystalline counterpart, on the other hand, increases from 490 VHN to 625 VHN at 17 wt.%Fe and then decreases to about 580 VHN at 28 wt.%Fe.

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Figure 2.7.

Microhardness of electrodeposited nanocrystalline and conventional polycrystalline Ni-Fe alloys as a function of Fe concentration [Cheung et al. (1995)].

Figure 2.8.

Hall-Petch plot for electrodeposited nanocrystalline Ni-Fe alloys showing transition from regular to inverse Hall-Petch behaviour [Cheung et al. (1995)].

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The observed variation in microhardness cannot be explained by solid-solution strengthening and therefore the grain size was considered using a Hall-Petch plot [Cheung et al. (1995)], as shown in Fig. 2.8. Initially, decreasing the grain size increased the microhardness to a peak value of 625 VHN (average grain size, 14 nm) and then decreased with further decrease in grain size leading to an inverse Hall-Petch behavior similar to that observed in other nanocrystalline materials produced by electrodeposition [McMahon and Erb (1989), El-Sherik et al. (1992)]. Similar results were found in a later study by Li and Ebrahimi (2003). Young’s Modulus The earliest studies on the influence of grain size on the Young’s modulus produced inconsistent and often conflicting results. Significant reductions (50%) of Young’s modulus were observed in nanocrystalline materials produced by compaction of inert gas condensed precursor powders, e.g., Nieman et al. (1991). However, this synthesis technique is conducive to producing materials with a substantial amount of porosity [Krstic et al.

(1993), Zugic et al. (1997)]. The effect of porosity is illustrated in Fig. 2.9 which shows a plot of the normalized Young’s modulus, Em / E0, where Em is the measured Young’s modulus value and E0 is the published value for the polycrystalline counterpart, as a function of grain size [Zhou et al. (2003b)]. Two regions are defined in Fig. 2.9: Region I (materials with negligible porosity) and Region II (materials which contain considerable porosity).

Shen et al. (1995) measured the Young’s modulus using nanoindentation over a large grain size range of fully dense nanocrystalline Fe produced by mechanical milling/alloying

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Figure 2.9.

Compilation of normalized Young’s modulus E / E0, where E is the measured Young’s modulus value and E0 is the published value for the polycrystalline counterpart, as a function of grain size [Zhou et al. (2003b)].

and showed that the Young’s modulus values between 80 and 20 nm are practically the same as for polycrystalline material. Only smaller grain sizes ( 20 nm) revealed some reductions up to about 6%. Shen et al. (1995) also suggested that the effect of the intercrystal defects can be estimated by using a simple rule of mixtures taking into consideration three main components: grain boundary ( E gb ), triple junction ( Etj ), and grain interior ( E g ). The volume fraction is dependent on the grain size, d, grain boundary thickness, , and grain shape based on Eq. 2-1, 2-2, and 2-3 [Palumbo et al. (1990)]. Zhou et al. (2003a,b) studied fully dense nanocrystalline Ni-P alloys produced by electrodeposition with decreasing grain size from about 30 to 4 nm and in the same manner as Shen et al. (1995), they considered the effect of increasing intercrystal volume fraction with decreasing grain size due to grain

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Figure 2.10.

Comparison between the Young’s modulus and the interfacial component volume fractions [Zhou et al. (2003a)].

boundaries and triple junctions. Below 20 nm, a continuous but very gradual decrease in the Young’s modulus was observed down to about 86% of the value for polycrystalline Ni at a grain size of 4 nm (see Fig. 2.10). This reduction was explained in terms of the sharp increase in grain boundary and triple junction volume fractions. Using the composite model suggested by Shen et al. (1995) and Young’s modulus values for materials with different grain sizes, calculations by Zhou et al. (2003a) showed that E g = 204 GPa, E gb = 184 GPa and Etj = 143 GPa. The reduced Young’s modulus values for the grain boundaries ( E gb ) and triple junctions ( Etj ) may be attributed to the increase in free volume in the interfacial region (see Fig. 2.3). Assuming the same interatomic force and bonding curves in the interfacial region as in a perfect crystal lattice, the increase in interatomic spacing would result in a lower Young’s modulus. This hypothesis was further studied by Zhou et al. (2009) by

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positron annihilation spectroscopy. The measurements showed that there are indeed excess free volumes in the grain boundaries and triple junctions which are smaller than a vacancy in a perfect lattice.

Another important factor that is known to influence Young’s modulus is the preferred crystallographic orientation, or texture. The Young’s modulus of conventional electroplated Ni has been measured by several authors [e.g., Mazza et al. (1996), Sharpe et al. (1997), Cho et al. (2003)], however a rather large range of values were obtained (160 to 247 GPa). Fritz et al. (2003) also measured the Young’s modulus using different measuring techniques including the laser-acoustic method, instrumented microindentation, and resonance frequency of laterally and vertically swinging cantilevers. They also found a large range of Young’s modulus values and on closer examination the variability was attributed to the texture of the electrodeposited Ni. The lower values corresponded to (100) textured electrodeposits while a higher value corresponded to a nearly isotropic Ni or weak (110) textures. Torrents et al.

(2010) studied the effect of annealing on the Young’s modulus of nanocrystalline Ni. Asreceived electrodeposited nanocrystalline Ni samples were annealed at a series of different temperatures ranging from 323 to 693 K. After annealing, the samples were rapidly cooled in air and then characterized. In this particular case, an increasing Young’s modulus with increasing anneal temperature was attributed to the preferred orientation along the [200] direction that was observed in the as-received samples which continuously diminished with increasing temperature. Auerswald and Fecht (2010) produced a series of nanocrystalline Ni-W samples using different inhibitors to evaluate their effectiveness at minimizing internal stress. The Young’s modulus for each of the samples was also measured using

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nanoindentation. The highest values were consistent with the sample that had a predominant (111) texture.

2.1.4. Release of Stored Energy Due to the high concentration of interfacial defects in nanocrystalline materials, there is strong driving force for grain growth which makes the structure unstable at elevated temperatures. Therefore, the thermal stability of nanocrystalline materials is a matter of importance since it may impose a limit on the number of possible applications. Reviews on the thermal stability of nanocrystalline electrodeposits are given by Hibbard et al. (2002), Aust et al. (2003) and Aust et al. (2004). Thermal stability is often studied using anisothermal annealing methods, such as differential scanning calorimetry (DSC) or modulated differential scanning calorimetry (MDSC). These methods allow for a thorough study of the release of stored energy in the material. The release of stored energy corresponds to the area under the anisothermal anneal curve which is developed during the analysis.

In conventional polycrystalline metals energy is normally stored after they are cold worked. Most of the cold work is expended in the form of heat; however, during the cold working process a fraction of the work is stored as strain energy which corresponds to lattice defects created during the deformation process. When cold worked metals are annealed they normally go through a transformation process in three stages: 1. recovery, 2. recrystallization, and 3. grain growth. During the recovery process some of the stored energy is relieved by dislocation reconfiguration. The result is some reduction in dislocation density and/or

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dislocation rearrangements and an overall lower stored or strain energy. Even when this is complete, the grains are still in a relatively high strain energy state. The next stage – recrystallization – results in the formation of a new set of strain-free and, usually, equiaxed grains. The driving force to achieve this new structure is simply the difference in internal energy between the strained and unstrained material. It is also at this point that the mechanical properties of the metal are restored to the pre-cold worked state. Finally, if left at elevated temperatures, the strain-free grains will grow during the process of grain growth.

The driving force for grain growth is the reduction of the total grain boundary energy of the system.

When compared to an unstrained metal, the internal (free) energy of a strained metal, is approximately equal to the stored energy. Cold work increases the entropy of a metal, however it is small when compared to the increase in internal energy and thus, the -TΔS term is negligible. Therefore,

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where, G is the free energy resulting from the cold work, H is the enthalpy, or stored energy, S is the entropy increase due to the cold work, and T is the absolute temperature [e.g., Reed-Hill and Abbaschian (1994)]. An example of one of the earlier studies on the release of stored energy in a cold worked polycrystalline metal was that of Clarebrough et al.

(1955), who subjected cold worked Ni to an anisothermal heat treatment at 6oC/min (see Fig.

2.11). They defined three distinct heat release regions:

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A(a’) – disappearance of vacancies into the dislocations forming sub-grain boundaries; no change in hardness, slight decrease in electrical resistivity.

A(a”) + B – movement of dislocations from within grains into the boundary regions, mutual annihilation of dislocations of opposite sign (dislocation annihilation), and rearrangement of the dislocations into configurations of lower energy (dislocation polygonization); no change in hardness, slight decrease in electrical resistivity.

C – recrystallization; rapid decrease in hardness and electrical resistivity.

Figure 2.11.

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