«STRUCTURE AND PROPERTIES OF ELECTRODEPOSITED NANOCRYSTALLINE NI AND NI-FE ALLOY CONTINUOUS FOILS by Jason Derek Giallonardo A thesis submitted in ...»
corresponds to the most intense part of the ring which normally occurs in the middle along the horizontal direction of the image. A summary of the macrostress values for each of the stress tensor components (see Section 3.7) is given in Table 6.2. The macrostress values are calculated using the Young’s modulus values, Em, determined from the nanoindentation measurements reported in Chapter 5 (Table 5.1). Given the magnitude of the planar stress components, 11 and 22, and the relatively low shear stress component, 12, it may be concluded that these samples are in a state of biaxial stress, i.e., in-plane. Due to the limited penetration of X-rays, macrostresses measured using XRD normally belong to this stress state [He (2009)].
Macrostresses determined using diffraction methods have two main types of error associated with them: statistical error, which arises from the random arrival of X-ray photons at the detector, causing some variability in the intensity at any 2 position, and instrumental error, which can be a result of specimen alignment, effect of beam optics, etc.
[Noyan and Cohen (1987)]. More important are the practical sources of error. Prevey (1986) classifies these practical sources of error as follows: 1. instrumental and positioning errors, 2.
effect of sample geometry and microstructure (typically grain size), and 3. X-ray elastic constants.
The instrumental error is normally a result of the high precision required to determine the diffraction line position. Thus, extremely precise positioning of the diffraction apparatus or the sample to accuracies of approximately 0.025 mm is critical but often difficult. Instrumental and positioning issues can result in errors in internal stress measurement of approximately ±14 MPa for high diffraction angle techniques [Prevey (1986)]. The current study made use of a high diffraction angle technique, i.e., the (311) diffraction line, in order to minimize instrumental and positioning errors on the final results.
When put in to context, a ±14 MPa error may affect more those samples which have relatively low macrostress values, for example, samples no. 1-6 with grain sizes greater than 20 nm (Table 6.2). In this case, 14 MPa could represent a significant portion of the actual macrostress value. However, samples with high macrostresses, e.g., samples no. 7-9 would not be greatly influenced by this error since 14 MPa is rather small when compared to the experimental values (Table 6.2).
Effects of sample geometry, including roughness, pitting, or curvature of the surface within the irradiated area, can result in systematic error similar to sample positioning. All samples were generally free of roughness or pitting; however, some of the foils (e.g., samples no. 1, 8, and 9) were not without some degree of curvature. In order to minimize the effect
of the curvature, the samples were affixed to the holder as firmly and as flat as possible.
However, there was no guarantee that the curvature was completely eliminated. This factor is likely to have played a role in the noticeable differences between respective planar stress component values for samples no. 1, 8 and 9. Large grain sizes can reduce the number of crystals contributing to the diffraction line resulting in asymmetry and random error in peak location and macrostress measurement. In the current study, all samples had a relatively small grain sizes and thus, such an effect is minimized. Rather, the relatively small grain sizes provided for a good statistical sampling and fairly accurate macrostress measurement.
When analyzing the macrostress in materials that are elastically anisotropic, the use of macroscopic elastic constants is essential for obtaining true values. In this case, the bulk elastic constants should be replaced with X-ray elastic constants that depend on the lattice plane (hkl) where the measurement is performed. Included in the plane specific X-ray elastic constant calculation is a value called the radiocrystallographic anisotropy factor ( ARX ). This value typically ranges between 1 and the single crystal anisotropy factor [Lu (1996)]. For Ni-based (fcc) materials, a value of 1.52 was used [He (2009)]. In the case of the current samples, the anisotropy of the materials was analyzed earlier and shown to increase with the addition of Fe (see Chapter 5). Unfortunately, the absence of elastic constant values for each of the materials containing Fe did not allow for the use of an accurate value for ARX.
6.3.2. Effect of Grain Size on Macrostress In order to investigate a possible dependency of macrostresses on grain size, the three stress components from Table 6.2 are plotted in Fig. 6.5 (shear stress component, 12 ) and
Fig. 6.6 (planar stress components, 11 and 22 ) as a function of grain size. The shear stress component, 12, tends to have a relatively low value over the entire grain size range. The planar components, 11 and 22, remain relatively low down to about 20 nm. However, below 20 nm, there is a sharp increase in the compressive macrostress. The results of ElSherik et al. (2005), who used the sin 2 method are also plotted in Fig. 6.6. Their values, which were also determined in the as-deposited state, were compressive and fit well with the data points from the current study. The observed trend for the planar components, 11 and 22, with grain size suggests the presence of an inherent dependency on grain size, especially when it is sufficiently lower than 20 nm.
This dependency on grain size is not consistent with studies of measured macrostresses using XRD techniques on materials produced by other synthesis methods. For example, Sanders et al. (1995) measured the macrostress in nanocrystalline Pd prepared by inert gas condensation and warm compaction using the sin2ψ method and found the macrostress in the as-deposited state to be compressive and in the range of -20 to -40 MPa.
The corresponding grain size range was approximately 3 to 26 nm. It should be noted that the nanocrystalline Pd was produced by a method which is known to produce materials with considerable amounts of porosity [e.g., Nieman et al. (1991)]. The presence of porosity was also the root cause of large reductions in the Young’s modulus values [Krstic et al. (1993), Zugic et al. (1997), Sanders et al. (1997)]. The relatively low macrostress values reported by Sanders et al. (1995) may also be related to the presence of porosity. Although the current
materials were not analyzed for porosity, it is well known that the electrodeposition technique produces materials with relatively low porosity when compared to materials produced by the inert gas condensation technique [Haasz et al. (1995), Van Petegem et al.
(2003), Zhou et al. (2009)]. Currently, it is not known what the effects of porosity is on macrostress, however, it is conceivable that long range stress fields in a material could diminish in the presence of sufficient porosity leading to a lower compressive macrostress value due to the presence of free internal surfaces in the grain boundary matrix. That is, stress relaxation may occur on the free internal surfaces associated with porosity in such materials. On the other hand, in fully dense materials, no internal surfaces for stress relaxation are available and therefore, stresses remain cumulative.
As shown in Fig. 6.6, when the grain size decreases below 20 nm, there is a sharp and distinct increase in the compressive macrostress values. Consequently, there is likely a relationship between the measured macrostresses and corresponding increasing intercrystal volume fraction or rather the number of interfaces in the solid. Such solid-solid interfaces are associated with stresses [Brooks (1952), Cahn and Larche (1982)]. According to Cahn and Larche (1982), an interface stress may be defined as: the reversible work per unit area to either elastically deform one phase relative to the other or both phases equally. In this case, phases may be interpreted as neighbouring grains which join to form the solid-solid interface, i.e., grain boundaries. This interpretation of grain boundaries is similar to that of Valiev et al.
(1986) who describe the “non-equilibrium” grain boundary. Such grain boundaries have long-range elastic fields since local elastic deformation is necessary for joining of the crystals.
This particular interpretation has been used to describe HR-TEM observations of localized
strains at grain boundaries [Wunderlich et al. (1990), Ping et al. (1995), Li et al. (2000), Valiev et al. (2000)] which are similar to the observations made in Chapter 4 (Section 4.4.2).
Considering the concepts of Cahn and Larche (1982), Cammarata and Eby (1991) developed a thermodynamic model which allows for the estimation of internal surface stresses via the determination of strain values. This model was initially used to determine strain values by detecting variations in lattice parameters for lamellar structures. Cammarata and Eby (1991) proposed an extension of the thermodynamic model’s application to very fine-grained metals and ceramics. That is, if the grains are considered to be spheres, the strain will depend on the grain size [Cammarata (1994), Cammarata (1997)],
where, is the strain, K is the bulk modulus, f is the interface stress associated with grain boundaries, and d is the grain size.
The current nanocrystalline Ni and Ni-Fe alloys data may be analyzed using the relationship given by Eq. 6-1. The left side of the equation can be rearranged to include the bulk modulus, K. By substituting in the relationship between the bulk modulus, K, and the elastic modulus, E,
where, is the Poisson ratio. If it is assumed that Hooke’s law applies, then the macrostress values 11 and 22 can be substituted and related to the grain size using the thermodynamic model of Cammarata and Eby (1991). By rearranging Eq. 6-3, the following expression is
Using the plot of the macrostress, 11 / 22, with the grain size, d, a curve can be fitted based on Eq. 6-4 and the coefficient, 4 f (1 2 ), can be determined and used to estimate a value for the interface stress associated with grain boundaries. In the calculation, the Poisson ratio,
nanocrystalline Ni and Ni-Fe alloys with the best fit curve based on Eq. 6-4. In general, the
Plot of the planar components 11 and 22 for the nanocrystalline Ni samples as a function of grain size with the best fit (dashed) curve based on Eq. 6-4.
data fits well with the relationship. For illustrative purposes, the total intercrystal volume fraction based on Eq. 2-1 [Palumbo et al. (1990)] is also shown in Fig. 6.7 (solid line) to emphasize the significant increase in these defects (i.e., grain boundaries and triple junctions) with decreasing grain size.
The constant value from the best fit yields an interface stress value, f, of approximately 3.56 N/m. There are very few reports on experimental interface stress values.
For example, experimental values for surface stress of Ni have been reported to be 4.2 N/m and 2.1 N/m for the  and  directions, respectively [Lehwald et al. (1987)]. On the other hand, there are a number of calculations for surface stress values which for Ni have been reported to be 1.27 N/m and 0.43 N/m for the (100) and (111) surfaces, respectively.
The interface stress value determined for the nanocrystalline Ni and Ni-Fe alloys in the current study is in relatively good agreement with the interface stress value determined by Jiang et al. (2001) who calculated a value of 3.05 N/m for Ni. Jiang et al. (2001) obtained this value by developing thermodynamic equations that relate the interface excess free energy to the interface stress. As described in Chapter 2 (Section 2.1.4) the total enthalpy (stored energy), H, from a strained metal is approximately equal to the excess free energy, G (Eq. 2-5). The thermal analysis in Chapter 4 (Section 4.6) showed that the total enthalpy (stored energy) measured in the electrodeposited nanocrystalline Ni and Ni-Fe alloys are primarily due to intercrystal defects (i.e., grain boundaries and triple junctions). As the volume fraction of these intercrystal defects increases or the grain size decreases, the total enthalpy (stored energy) in these materials increases accordingly. If interface excess free
energy is related to the interface stress, then in a similar manner, an increase in intercrystal defects would cause an increase in the internal stress.
Weissmuller and Cahn (1997) also emphasized the interaction of grain boundaries and triple junctions and their relationship to interface stress which results from the deformed state of the interface; these interface stresses must be equilibrated by homogeneous bulk stresses or analogously macrostresses. The bulk stresses brought on by interface stresses in polycrystalline materials are rather low; however, when the volume fraction of intercrystal defects is high, as in the case of nanocrystalline materials and especially at grain sizes less than 20 nm, these interface stresses become significant. Birringer et al. (2002) developed a method based on the theory of Weissmuller and Cahn (1997) for experimentally measuring the interface stress, f. The calculation relies on the detection of a lattice parameter change which is related to the strain in the material that is induced on the order of f / d. Birringer et al. (2002) analyzed nanocrystalline Pd samples produced by the inert gas condensation method and determined an interface stress of 1.2 N/m. The lattice parameters were found to decrease with decreasing grain size which indicates the presence of a compressive bulk stress.
This is consistent with the compressive macrostresses identified in the current study for the electrodeposited nanocrystalline Ni and Ni-Fe samples.