Using Variant Selection to Facilitate Accurate Fitting of γ″ Peaks in Neutron Diffraction

In this work, IN718 with nominal chemical composition (wt pct): C (0.02), Cr (19.03), Nb (5.06), Ti (1.00), Al (0.54), Mo (3.06), Ni (52.16), balance Fe was used. 8-mm-diameter rod specimens with gauge length of 67 mm and threaded ends were extracted from forged IN718 bars which were solution heat treated at 1000 °C for 1 hour followed by fan cooling. Then the specimens were subjected to aging heat treatment at 790 °C for 5 hours. During the aging treatment, a 300 MPa tensile stress was applied to the specimens along the axial direction. After the heat treatment, the specimens were air cooled to room temperature. The microstructures after heat treatment were characterized using a scanning electron microscopy (SEM) (JSM-7800F FE-SEM, JEOL, Japan). The SEM specimen was cut from the cross-section of the bulk specimen, grinded and chemically etched with a mixed solution composited by 5 g CuCl₂ + 100 mL hydrochloric acid + 100 mL ethanol to remove the γ matrix. The observed microstructure displays that the γ grains are about 30 μm, and a small amount of δ phase exists at the grain boundaries. Electron backscatter diffraction (EBSD) characterization was performed to determine the grain orientations. The γ″ variant distribution in differently oriented grains was also characterized.

Neutron diffraction experiments were carried out using the ENGIN-X neutron diffractometer at ISIS Time-of-flight neutron source, UK.[33] The rod specimen was mounted on a sample stage with the specimen axial direction oriented horizontally 45 deg to the incident beam. Detectors were fixed at 90 deg to the incident beam. Radial collimators sat between the detectors and the sample, ensuring the detected neutron beams were from the same gauge volume in the sample.[34] The incident beam slit was set as 4 mm in width and 10 mm in height, the diffracted slit width was 4 mm as defined by the radial collimators. With such a slits setup, the sample gauge volume was maximized in order to gain more diffracted signals, and the pseudo strain arose from the partially filled nominal gauge volume in vertical direction was negligible.[33] The neutron scanning was performed at room temperature with acquisition time of about 20 minutes. The experiment setup is illustrated in Figure 5.

The γ″ variant distribution was characterized in three grains with their orientations close to [110], [100], and [111] directions which were selected from the EBSD map (Figure 6(a)). Dispersions of small round particles were observed in all three grains with diameter around 33 nm, these particles were γ′ particles as discerned from their shape and size. Dispersions of larger particles in all three grains were γ″ particles. The mean diameter of the disc-shaped γ″ particles was determined to be 68 nm using ImageJ. Although the thickness of γ″ particles could not be observed directly in these micrographs, it was estimated to be 16 nm using the reported aspect ratio from literature.[18,35] The γ′/γ″ volume fraction ratio was estimated to be 0.36 by counting the number of the precipitates in the SEM images, assuming the γ′ particles in spherical shape with a diameter of 33 nm and the γ″ particles in ellipsoidal shape with a major axis of 68 nm and minor axis of 16 nm. This rough estimation agrees with the γ′/γ″ volume fraction ratio of an IN718 sample reported by Cozar and Pineau,[36] which had similar Al+Ti/Nb ratio to the present study. In the grain oriented close to [110] (Figure 6(b)), the γ″ particles all aligned in one direction, these γ″ particles would belong to two variants as indicated by the inset cube. In the grain oriented close to [100] and [311] (Figure 6(c)), only the round shape γ″ particles existed, representing the only variant in this grain. In the grain oriented close to [111] (Figure 6(d)), the γ″ particles existed in three different orientations, representing the three variants of γ″. The observed variant distributions in these three grains were consistent with the analysis in Section II, proving that the variant selection manifested after the stress-aging heat treatment.

Figure 7 shows the illustration of γ″ variants distribution in the four grains viewed along the transverse direction of sample. The cross-section lattice planes of the γ″ variants in the four grains are (112), (004), (204), and (116) planes, respectively. Noted that the axial direction of the rod specimen was set parallel to the diffraction vector, with such a geometry, the {004}, {204}, and {116} γ″ peaks will have larger intensities (as illustrated in Figure 2(b)), since the amount of the variants which contribute to these peaks have increased at the expense of other variants.

The diffraction pattern obtained by the bank 1 detector is shown in Figure 8 with peaks identified. As expected, {004}, {204}, and {116} γ″ peaks show relatively good peak-shaped quality. As explained in Section II, the {200}, {220}, and {312} γ″ peaks would have little intensity since the γ″ variant with their lattice planes {200}, {220}, and {312} normal to the diffraction vector did not exist as confirmed in Figure 6. The {112} γ″ peaks partly overlap with the {111} γ peak, leading to an obvious peak ‘shoulder’ at the {111} γ peak. In addition, two peaks from δ phase are identified, the existence of δ peaks implies that possible δ peaks would exist at positions that are close to the {004}, {204}, and {116} γ″ peaks. However, the integrated intensity of {211} δ peak is about half of that of {004} γ″ peak, indicating those possible δ peaks that exist close to {004}, {204}, and {116} γ″ peaks would have little intensity, and thus, have little influences to the fitting of {004}, {204}, and {116} γ″ peaks. The low γ′/γ″ volume fraction ratio estimated in Section IV–A indicates that the γ′ peaks, which have close positions to the γ peaks, can also be neglected in the fitting procedure. Therefore, the major peaks in Figure 8 can be assumed to be solely γ peaks.

The overlapped {112} γ″/{111} γ peaks lead to a large inaccuracy of fitting. Thus, the refinement and the fitting were only performed to the pattern within the d-spacing range from 0.98 to 2.04 Å for the purpose of a better accuracy. Since a small stress of 5 MPa was applied during the diffraction experiment, a Pawley refinement to the pattern was performed by fitting the peaks from γ, γ″, and δ phases using GSAS-II as shown in Figure 9. The green line gives the residual between the fitted and measured data and shows a satisfying fitting quality. The lattice parameters of both γ and γ″ phases were obtained and listed in Table III. Although the fitting uncertainties for γ″ phase are larger than that for γ phase, they still maintain an acceptable level corresponding to less than 150 microstrains. Misfit strains between the two phases along a-axis and c-axis were calculated byrespectively, and listed in Table II. As expected, the misfit along c-axis was 3.28 pct, which was significantly larger than 0.44 pct along a-axis.

In in-situ studies of mechanical performance, the plane-specific lattice strain response to an applied stress is commonly investigated by measuring d-spacing evolution of the individual lattice plane.[9,10,37] Therefore, it is worthy of fitting individual peaks to obtain d-spacing of both γ and γ″ phases. Peaks from γ′ and δ phases were neglected due to their low intensities and limited influences. Although the γ″ peaks were discerned separated from the γ peaks in the pattern, their relatively close positions required each set of γ‐γ″ peaks to be fitted simultaneously. During the fitting, the tail of the γ peak had a large influence on the fitting to the neighboring γ″ peak since the γ″ peaks were relatively small, thus, an accurate fitting to the γ peak was particularly important for the fitting of the γ″ peak. The γ peaks exhibited asymmetric peak shape due to the moderation process of the time-pulsed neutron source,[33] so that an asymmetric peak function for the correct fitting of the γ peaks was necessary. The asymmetric peak function used in this study consists of a complementary error function (erfc) convoluted with a Voigt function.[38]For the fitting of γ″ peaks, a symmetric Voigt function rather than the asymmetric function was used. This is because when both peaks are fitted using asymmetric functions, the correlation effect of asymmetric variables of the two peaks make the fitting difficult to converge. In addition, the γ″ peak was broadened by the nano-particle sizes and localized strain field, which reduced the peak asymmetry. Therefore, the use of the symmetric Voigt function gave a better γ″ peak fitting as shown in Figure 10.[33] The d-spacings of each lattice plane were determined by the peak positions and the corresponding misfit strain (ε_hkl) were calculated using the below expression, results are listed in Table III.Different lattice misfit strains existed for different hkl planes, which was not unexpected and consistent with the misfit strains in Table II.

The volume fraction of γ″ phase can be estimated from the intensity ratio of {004} γ″ to {200} γ peaks. The intensity of diffraction peaks at a time-of-flight diffractometer was expressed as Reference 39:where I₀ is the incident intensity, λ the neutron beam wave length, j the multiplicity, and F the structure factor. Considering a two-phase system, the unit cell volume V₀ and volume fraction f of each phase need to be considered, and thus Eq. [5] can be revised aswhere V_v is the gauge volume, \( \eta = \frac{F}{{V_{0} }} \) the scattering length density. Following Eq. [6], the relation between intensity and volume fraction was derived[6]:The {004}γ″/{200}γ peak intensity ratio \( I_{004}^{{\gamma^{\prime\prime}}} /I_{200}^{\gamma } \)is 0.092/0.908 as calculated from the integrated intensities of these two peaks. The scattering length densities of phases, η_γ = 7.206 × 10¹⁰ cm⁻² and \( \eta_{{\gamma^{\prime\prime}}} = 7.328 \times 10^{10} \,{\text{cm}}^{ - 2} \) were reported in reference,[25] \( \eta_{\text{p}} = \sum\nolimits_{i} {\frac{{b_{i} x_{i} }}{{V_{\text{p}} }}} \), where b_i is the scattering length of each individual atom, x_i the atomic concentration which was measured by atom probe tomography in their study, and V_p the unit cell volume calculated from lattice parameters. The alloy composition in our study is similar to that used in their study, in addition, they reported chemical compositions of γ″ phase were similar after different aging heat treatments.[25] Thereby, we assumed the compositions of γ and γ″ phases in our study were similar to those in reference and adopted these scattering length densities to the estimation of γ″ volume fraction in the current study. The diffracted neutron beam wavelength is proportional to the d-spacing when the diffraction angle is fixed according to the Bragg’s law:Therefore, the volume fraction ratio f_γ″/f_γ was calculated to be 0.086 and the γ″ volume fraction f_γ″ was 0.079, the small volume fractions of γ phase and δ phase were not considered. The volume fraction uncertainty was estimated to be about 10 pct from the uncertainty of the γ″ peak intensity which was about 10 pct.

Stress-induced variant selection effect has been well studied by many researchers in reported literature,[15,29‐32] which showed that it is a result of interaction between the applied stress and elastic strains of different variants. During nucleation and coarsening, the evolution of morphology of a particle follows a way that the free energy decreases the fastest. The free energy consists of surface energy and elastic strain energy. A study by Li et al.[32] showed that for the θ′ precipitates in Al-Cu alloys, the surface energy is dominant at the early stage when particle size is less than around 1 nm. Since the surface energy is isotropic, the particle possesses a spherical shape at the early stage. When the particle size is larger than the critical value, the elastic strain energy, which is anisotropic, becomes dominant and leads to a disc-like shape of the θ′ particle. The γ″ precipitates would have a similar critical size since the misfit strains, elastic moduli, and surface energy are not in much difference compared to the θ′ precipitates in Al-Cu alloys. The γ″ precipitate forms in a spherical shape at the early stage and then grows into a disc-like shape when the precipitate is larger than the critical size. The critical size is much smaller than the normally observed size of γ″ precipitate (a few ten nanometres), thus it is the elastic strain energy dominant in most cases. When an external stress is applied, the different variants are selected for preferential growth due to the anisotropic elastic strain energy. Qin et al.[29] found that the variant selection effect was not apparent after stress aging for 2 hours, suggesting the variant selection effect is more apparent during the coarsening stage when the elastic strain is dominant compared to the nucleation stage. However, from the simulation by Li et al., variant selection happens at the both the nucleation and coarsening stages. Such an issue may be able to clarify by in-situ measurements of evolution of lattice spacing of both the γ and γ″ phases during stress aging of IN718.

An important issue for the applications of IN718 at elevated temperatures is the thermal stability which correlates to the formation of the δ phase. The precipitation of δ phase, as reported by Sundararaman et al.[40] in IN718 and IN625 at temperatures ranged from 750 °C to 950 °C, happened on grain and twin boundaries via heterogeneous nucleation, and from transformation of γ″ to δ. The precipitation kinetic for the δ phase is much slower than that for the γ″ phase, and the transformation follows a well-accepted sequence: super-saturated solid solution − γ″ to δ, as reported by Oradei-Basile and Radavich.[41] For IN718, a small amount of the δ precipitates at the grain boundaries would be beneficial to the creep resistance. It is the transformation from γ″ to δ that significantly affects the thermal stability, and thereby such a transformation is of great concern. The transformation from γ″ to δ would be related to the formation of stacking faults in the γ″ precipitates according to the studies by Sundararaman et al.[40] The formation of stacking faults can be correlated to the loss of coherency during coarsening when the γ″ precipitates reach a critical size, saying 50 – 100 nm.[42] Therefore, the formation of δ phase depends on the coarsening kinetic of the γ″ precipitates. The effect on coarsening kinetic of the γ″ precipitates has been reported by Qin et al.,[29] the γ″ precipitate sizes after stress aging are similar to those after stress-free-aging, but the number density of precipitates is higher for the former than the latter, suggesting the applied stress during aging has little effects on coarsening kinetics, but would promote nucleation of the γ″ precipitates. Therefore, the applied stress during aging would have little effects on promoting the precipitation of the δ phase.

Stress partitioning would happen between grains that have different orientations when under loading due to the elastic anisotropy. Thus, the magnitude as well as the direction of the localized stress that experienced by an individual grain would differ from the applied stress, affecting the variant selection effect by the applied stress. In our study, the applied stress was of a magnitude of 300 MPa, which was believed large enough to induce variant selection compared to the 69 MPa tensile stress applied in the experiment by Oblak et al.[15] In fact, it is the direction of the localized stress that would affect the variant selection, since the angle between the c-axis of the γ″ and the stress axis plays the important role in determining which variant is promoted, and which is suppressed by the stress. However, a determination of the direction of the localized stress for each individual grain would be impossible. Consider a microstructure with a weak texture, an individual grain is surrounded by many grains with random orientations, the effect on the localized stress direction would be small for most cases, suggesting the intergranular stress effect would be not significant enough to affect the variant selection in {100} and {311} oriented grains where the angle θ for the selected variant is much smaller than that for the suppressed variants. For those grains in which the difference in angle θ for different variants is small, for example {111} oriented grains, the change in direction of the localized stress may have an apparent influence on the variant selection.

In the Pawley algorithm, the quality of fitting can be assessed by the weighted profile R-factor (R_wp) and by the goodness of fit (GoF).[43] The R_wp and GoF of the Pawley refinement in this study, given by the GSAS-II, are 9.15 pct and 1.00, which are acceptable for a good quality refinement.[43] From the graphic fitting in Figure 9, the computed intensities agree well with the observed intensities, the only exception being the points around the tip of γ peaks. The visual assessment together with the numerical statistics gives us confidence that a good refinement is achieved. The high R-square coefficients “R²” and the good agreement between the computed and observed intensities (Figure 10) indicate that good fittings are achieved in terms of single peak fitting.

The measured lattice spacing in Table III can be used to calculate the lattice parameter of both phases since \( \frac{1}{{d_{hkl}^{2} }} = \frac{{h^{2} + k^{2} + l^{2} }}{{a^{2} }} \) for the γ phase, or \( \frac{1}{{d_{hkl}^{2} }} = \frac{{h^{2} + k^{2} }}{{a^{2} }} + \frac{{l^{2} }}{{c^{2} }} \) for the γ″ phase. The calculated lattice parameters retain a difference of 0.2 pct compared to the lattice parameters in Table II. The existence of this systematic discrepancy is not surprising since different d-spacing ranges and fitting functions were used, as well as the constraints from other peaks were not the considered in the single peak fitting routine compared to the Pawley refinement. Such discrepancy would not degrade the use of single peak fitting in studies such as elastic moduli and thermal expansion, where the lattice strain is of higher interest than the absolute value of lattice parameters, since the systematic error will be canceled out when calculating lattice strain using the fitting results from the same fitting method.

It is worthy to emphasize that the uncertainty has been kept lower than 150 microstrians, which is acceptable for quantitative strain analysis. An uncertainty of this magnitude is approximately equal to a thermal strain caused by a 10 °C variation in temperature, assuming the thermal expansion coefficient is around 15 microstrains per degree,[9] or equivalent to an elastic strain response to an applied stress of 24 MPa, assuming the elastic modulus is 160 GPa.[12] Such small uncertainty combined with the relatively short data acquisition time (20 minutes) allows mechanical properties related to the γ″ phase to be studied in-situ.

The small amount of the δ phase along the grain boundaries contributes to the δ peaks in the diffraction pattern. Although the δ peaks have low intensities, the single peak fitting to the γ″ peaks might be affected by those δ peaks which are located near the γ″ peaks. Such influence can be reduced by reducing the fraction of δ phase via solution heat treatment at a higher temperature. However, according to the consistency between the results from the single peak fitting and the Pawley refinement, the influence from δ phase is marginal. Similarly, the γ′ peaks are ignored in the fittings since the volume fraction of γ′ is small, as analyzed in Section IV–A. In addition, the γ′ has a similar lattice parameter to the γ phase, causing the fundamental γ′ peaks to completely overlap with the γ peaks. Therefore, the assumption that the major peaks are solely γ peaks is reasonable and the neglection of γ′ has little influence on the whole pattern fitting.

Many of the reported γ″ lattice parameters were measured on extracted residues.[17,36,44,45] The extraction leads to a relief in the constrain on the γ″ precipitates from the surrounding matrix. The relaxed γ″ lattice parameters would be larger than those of γ″ embedded in the matrix. The current measured lattice parameters of both γ and γ″ phases are comparable to those measured by Slama et al.[17] on extracted γ″ residues from IN718 samples, which had similar chemical composition and was heat treated under similar temperatures to the present study. On the other hand, the lattice parameters of the embedded γ″ in the bulk IN718 sample were measured by Wagner et al.[46] using high-resolution neutron powder diffractometer, leading to \( a_{\gamma } = 3.59592\,{\text{\AA}} \), and \( a_{{\gamma^{\prime\prime}}} = 3.59881\,{\text{\AA}} \). The lattice parameters of γ phase agree well between the two studies; however, the lattice parameter of γ″ in the present study is much larger. The disagreeing γ″ lattice parameters could be due to various reasons, such as a difference in the coherency strain of the γ″ precipitates and/or a difference in the chemical composition of the precipitates.

Volume fractions of γ″ phase in IN718 obtained using diverse measuring approaches or simulations are presented in Table IV. The current measured volume fraction lies within the range of the reported values. The uncertainty of current measurement majorly arises from the fitting of the {004} γ″ peak which retains a low height-to-background ratio, and the scattering lengths density which largely depends on the composition and atomic occupancies of γ″ phase. However, compared to other methods, the measurement using neutron diffraction is more straightforward. Furthermore, once the above two measurements become more accurate, the volume fraction measurement will become more reliable.

The deconvolution of the partially overlapped {111} γ/{112} γ″ peaks was not successfully performed using single peak fitting routine. This is due to the large number of variables used for the fitting and the low intensity of γ″ peak, and consequently, the d-spacing for {112} γ″ peak was not obtained.

The size and the lattice strain of the precipitates have large influences on the mechanical performance of materials and are of great concerns to researchers. The nano-particle size and lattice strain will generate broadening effect to the diffraction peaks. In turn, the mean particle size and the lattice strain can generally be estimated from the peak broadening using the Williamson–Hall relation,[47] which gives a linear plot of peak width against d-spacing if the precipitate lattice strain distribution is isotropic. However, in this experiment, the disc-shaped γ″ precipitates retain high lattice strain anisotropy, the plot of the peak width against the d-spacing is not linear, resulting in the failure of obtaining the size and lattice strain of the precipitates.