Detecting rare, abnormally large grains by x-ray diffraction
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Bimodal grain structures are common in many alloys, arising from a number of different causes including incomplete recrystallization and abnormal grain growth. These bimodal grain structures have important technological implications, such as the well-known Goss texture which is now a cornerstone for electrical steels. Yet our ability to detect bimodal grain distributions is largely confined to brute force cross-sectional metallography. The present study presents a new method for rapid detection of unusually large grains embedded in a sea of much finer grains. Traditional X-ray diffraction-based grain size measurement techniques such as Scherrer, Williamson–Hall, or Warren–Averbach rely on peak breadth and shape to extract information regarding the average crystallite size. However, these line broadening techniques are not well suited to identify a very small fraction of abnormally large grains. The present method utilizes statistically anomalous intensity spikes in the Bragg peak to identify regions where abnormally large grains are contributing to diffraction. This needle-in-a-haystack technique is demonstrated on a nanocrystalline Ni–Fe alloy which has undergone fatigue-induced abnormal grain growth. In this demonstration, the technique readily identifies a few large grains that occupy <0.00001 % of the interrogation volume. While the technique is demonstrated in the current study on nanocrystalline metal, it would likely apply to any bimodal polycrystal including ultrafine grained and fine microcrystalline materials with sufficiently distinct bimodal grain statistics.
KeywordsGrain Size Distribution Diffraction Ring Abnormal Grain Growth Nanocrystalline Metal Goss Texture
Abnormal grain growth (AGG) in polycrystalline materials is defined by a small fraction of grains that grow larger than the vast majority of their neighbors, resulting in a bimodal grain size distribution that does not evolve in a self-similar manner over time. The precise mechanism for AGG is a topic of ongoing debate , although the most common argument centers on a few grain boundaries that have exceptionally high mobility under special conditions [2, 3]. The microstructural heterogeneities created by AGG can be both detrimental and beneficial. Perhaps the most famous example of a beneficial use of AGG traces back to Goss’ 1935 work on Fe–Si where certain thermomechanical processes were found to produce extreme crystallographic texture (the ‘Goss texture’), enabled by AGG . These alloys, now known as electrical steels, have seen extensive industrial use for decades due to their high magnetic anisotropy associated with the strong texture. AGG is problematic in the sintering of undoped alumina ceramics to achieve densities >97 %, an issue that was resolved in the 1960s by the introduction of MgO .
There is interest in AGG phenomenon, especially in the case of nanocrystalline metals which appear to be particularly susceptible. While nanocrystalline metals can exhibit AGG during thermal exposure (e.g., [6, 7]), they also often show a strong propensity for AGG under mechanical loading, such as tension , indentation , wear [10, 11], and fatigue [12, 13]. Our previous studies of fatigue-induced grain growth in nanocrystalline Ni alloys  suggest that a vanishingly small fraction of grains can grow abnormally under fatigue loading from ~50 to ~500 nm resulting in crack initiation and eventual catastrophic failure. Current understanding of mechanically driven AGG in nanocrystalline metals is limited due to challenges in direct non-destructive observation. Therefore, there is an urgent need for fast non-destructive in situ survey methods to identify the onset and progression of AGG. While AGG is generally considered problematic for nanocrystalline metals, the resulting bimodal grain size may also provide beneficial properties, such as improved ductility and corrosion resistance, if properly controlled [14, 15].
X-ray diffraction methods are ideal in many ways for in situ non-destructive grain size measurement. There is substantial literature describing the relevant theory (e.g., ), as well as many documented examples of the application of these methods to in situ experiments (e.g., ). All of the common methods for grain size determination of polycrystals using diffraction, including the 100-year old Scherrer method, Williamson–Hall’s  adaptation, and the more sophisticated Warren–Averbach  approach, are based on the analysis of the widths of diffraction peaks. These techniques can be used to determine average grain sizes typically in the <500 nm regime. Determination of the width of a diffraction peak for grain sizes of 1–50 nm is straight-forward, but as the grain sizes surpass 50 nm, contributions from instrumental broadening to the peak width are considerable and need to be carefully taken into account; for this reason, grain size determination above ~200 nm can be particularly challenging. Furthermore, Williamson–Hall only provides the average value for crystallite size. For this reason, it may not be sensitive enough to identify the presence of a diffracting volume of coarse grains which only represents a small fraction (<0.1 %) of the total volume of diffracting grains. Warren–Averbach analysis, by contrast, can estimate the entire grain size distribution, but it requires not only the widths but also the shape of several higher order diffraction peaks . For this reason, the technique is applicable to only extremely high resolution, low noise diffraction data with many high-Q diffraction peaks. Warren–Averbach analysis is particularly challenging for small unit cell, high symmetry lattices, such as that of many structural metals. Furthermore, overlapping peaks additionally confound analysis. Conventional methods of grain size determination for grains larger than 100 nm, therefore, require high resolution diffractometers operating at high energies (to reach high-Q range). Even with the best diffractometer configuration, they often fail for samples with very low symmetry and large unit cell, where peak overlap is significant, or samples with high symmetry and a small unit cell, where only a few peaks are accessible. These high resolutions are almost always achieved by sacrificing intensity and they are also not very compatible with parallel detection using large area detectors. These systems, therefore, are slow, laborious and not well suited for quick in situ detection of AGG.
The difficulty in characterizing a bimodal grain size distribution by X-ray diffraction has been discussed in several previous publications (e.g., ). It is noted in that prior work that a minority of “coarse grains do not produce measurable profile broadening” . Therefore, it is necessary to consider alternative analysis methods to determine bimodal grain size distributions, such as those caused by AGG.
The total (integrated) scattered intensity from a given volume of material is independent of the grain size (under a kinematic approximation). As the grains grow, the scattered intensity gets focused into sharper peaks. Thus, the diffraction peak width decreases with grain size, as suggested by Scherrer, Williamson, Warren, etc., but it also grows in height proportional to the grain size (e.g., see Fig. 4 in ). For a small fraction of abnormally large grains among a larger volume of much finer grains, anomalous intensity spikes are much easier to detect than sharpening of the peak width. Based on this observation, we propose a new approach to identify very small fractions of abnormally large grains in a matrix of much finer grains. To demonstrate this approach, we use a previously prepared sample of nanocrystalline Ni–Fe where cyclic fatigue loading of the material resulted in the presence of a few abnormally large grains. A series of X-ray transmission diffraction experiments are described, which investigate the conditions for the use of such a non-destructive method in both post-mortem and in situ analysis. In our proposed approach to an in situ AGG investigation, the sample would be subjected to gradually increasing external stimulus (e.g., temperature, strain, or impact), or increasing repetition of cyclic loading. The sample would be continuously surveyed to find new pockets of AGG by a technique similar to the one suggested below. As soon as the survey detects initiation of AGG, the in situ experiment would be halted and the microstructure, grain orientation, grain size distribution, and strain in the pocket of AGG and its surroundings would be investigated in detail, including Warren–Averbach and Williamson–Hall methods, if needed.
X-ray diffraction measurements
The beam spot size was controlled by an aperture composed of horizontal and vertical slits which could be independently positioned at micron intervals. The experiments described here utilized equal horizontal and vertical slit spacings of 50, 100, or 200 µm. With the 10 μm sample thickness, these slits correspond to nominal sampling volumes of 2.5 × 104, 1 × 105, and 4 × 105 μm3. Since the sample itself was much larger than the X-ray footprint, the sample was rastered in-plane at increments equal to the nominal X-ray beam size defined by the size of the slits. The sample was known to have at least one cluster of abnormally large grains, so at least one position during rastering was expected to produce an anomalous X-ray peak.
The selected beamline was configured for transmission powder diffraction with a monochromatic beam and a 2-circle goniometer holding the broken fatigue sample. This configuration is not ideally suited for identifying the single-crystal signature associated with the anomalous large grain region. A single large grain will create a lattice array of intensity spikes in reciprocal space (actually a few depending on the multiplicity of that reflection ). However, in a monochromatic experiment, the single-crystal spikes may not intersect the permissible Bragg diffraction rings. Stated from the perspective of a Ewald-sphere construction, there is no guarantee that the thin shell of the Ewald-sphere will intersect the single-crystal lattice spots in reciprocal space. There are three possible strategies for identifying the presence of abnormal single crystals: (1) interrogate many diffraction rings through either high-energy X-rays and/or close detector working distance, (2) span X-ray energy either through a polychromatic beam or rastering a monochromator, and (3) rotate or tilt the sample during the measurement. While any of these three options are viable, the limitations of the beamline in the current study only afforded the ability to tilt the specimen with respect to the incoming beam. Unlike most conventional single-crystal diffraction studies, it is not necessary to capture many diffraction spots from the single crystal: while more diffraction spots provide further confirmation, only one definitive diffraction spike is needed to identify the presence of an abnormally large grain.
Anomalous intensity determination
Q data for the (111), (200), and (220) peaks
Q range analyzed (Å−1)
The caked diffraction ring data were then analyzed using a MATLAB script to automatically find potential diffraction peaks due to large crystallites in the diffracting volume. For this step, the caked data, imported as row (χ), column (Q), and intensity values, were searched to find the intensity maximum for each χ value. The peak intensity values were then plotted as a function of χ and fitted with a smoothing spline to establish a local mean of the dataset, as shown in Fig. 5b.
For example, an intensity spike that stands three standard deviations above the average intensity has a z-ratio of z = 3 and a confidence interval of 99.73 %. The interpretation of such a result would be that the intensity spike is statistically distinguished as an outlier that is not a result of Gaussian noise, with a confidence of 99.73 %.
The use of basic statistical tools described in the previous section allows the objective identification of statistical outliers in the z-ratio data. The confidence interval defines how likely it is that a datum is a true outlier in the statistical sense. Defined inversely, the CI represents the odds of a datum not belonging to the normal distribution. As an example, consider a datum which lies just outside the threshold of a CI of 99.9999 %, corresponding to a z-ratio of 4.9. The value is a statistical outlier with 99.9999 % certainty—conversely, the odds of it not being anomalous are 1/(1 − CI) or approximately 1000000–1. These types of measures should not be considered just in themselves, but in relation to the distribution size. A CI of 99.99 % (z = 3.9) implies that approximately 1 out of 10000 data points would be expected to fall outside the normal distribution. Therefore, if several values fall outside this range, and they occur adjacent to each other, there is a high statistical likelihood that a large crystallite is causing the diffraction anomaly. Due to the difficulty in establishing an accurate representation of the true varying mean of local maximum intensities with χ, a more conservative CI threshold may be set to prevent mistakenly identifying outliers in lower quality datasets. For this purpose, a CI of 99.9999 % (z = 4.9) was used for the anomalous peak identification in the current study.
Grid measurements and efficiency tradeoffs
The smallest interrogation spot size of 50 μm not only suffered from reduced intensity associated with the partitioned beam, but also exhibited poor spatial discrimination. Specifically, while the anomalous spike was present in the expected location, the same anomalous peak with identical diffraction coordinates (Q, χ, ϕ) was present at neighboring grid locations above, below, or to the side of the expected location. This is most likely due to beam spillover. While the slit aperture set a beam size of 50 μm, beam divergence and/or Fresnel diffraction from the slit sides caused the projected beam on the sample to be larger. This is a limit of the spatial selectivity of the beamline in use, as other beamlines are capable of defining X-ray spot sizes of 1μm or smaller.
Aperture size and count time directly affect the measured intensity of diffraction data in an experiment, which in turn affect the statistical discriminating power to identify anomalous peaks associated with unusually large grains. For fast initial scans, a short duration per diffraction pattern is desirable so as to cover a large area on the sample quickly to determine whether to continue heating/straining or stop to perform a more detailed scan. The following series of diffraction experiments examine the impact of aperture size and counting statistics using a single sample location containing known abnormally large grains. 50 µm aperture openings were paired with durations of 300, 800, and 1200 s, while 100 μm aperture openings were paired with durations of 60 and 300 s. Since there was a fixed 120 s delay associated with the detector readout and transmitting detector data after a diffraction measurement, durations shorter than 60 s do not result in a significant time savings for the current experimental setup.
Optimal configurations for this method
While the present paper describes a new concept in observing AGG phenomenon, the experiments were not conducted under ideal circumstances. The beamline employed had been selected for the ill-advised purpose of conducting traditional grain size determination. This monochromatic 2-circle configuration was clearly not well suited for a detailed single-crystal study. A more ideal configuration would involve a 4-circle goniometer, polychromatic or scannable monochromatic X-rays, and a close working distance, high-pixel-count, fast area detector. Such a setup is more commonly found in a microdiffraction beamline, although the small size of a typical micron-sized microdiffraction beam would render large area surveys difficult or impossible. Actually, the new method performed admirably well with even a 200 μm spot size, the largest size available on the beamline—even with a 1 min count time, there was ample signal to identify the presence of abnormal grains. For the purposes of surveying a larger swath of microstructure (a larger ‘haystack’), it appears to be feasible to use even a 1 mm or larger spot size.
The present study examines AGG in nanocrystalline alloys. Pure nanocrystalline metals have notoriously unstable grain boundaries—as such, there are concerted efforts to develop stabilized nanocrystalline alloys (e.g., ). However, the technique described in this proof-of-concept demonstration should also extend to other scenarios that involve AGG, even in the conventional microcrystalline regime. The technique relies on a sufficiently distinct populations of crystallographic texture. In the present scenario, the parent grains had a relatively modest texture, whereas the grain growth region was populated with essentially a perfect single-crystal texture. A similar distinctly bimodal structure at the macroscale is that of the Goss texture in secondary recrystallized Fe–Si alloys. Indeed, numerous prior studies have shown the emergence of special orientations as a result of heat treatments in X-ray pole figure texture maps (e.g., ). The distinction here is that with an intense X-ray source to provide sufficient counting statistics, it is possible to identify even one or a few abnormal grains, rather than relying on distributed bimodal grain growth. A related concern with scaling the technique to microcrystalline grains is that the sampling volume defined by the beam size should scale accordingly to maintain similar counting statistics. In the present scenario, the beam size was ~4000 times larger than the parent grain size. For a hypothetical microcrystalline metal with 25 μm grain size, the scaled beam size should be 100 mm, roughly an order of magnitude larger than conventional X-ray diffraction instruments. With reduced number of grain sampled, the powder ring will be more heterogeneous, rendering discrimination of abnormal events more difficult.
A mode of bimodal grain determination distinct from existing methods
There are several well-established techniques for grain size determination including Warren–Averbach, Williamson–Hall, and even the simplistic Scherrer formulation. These traditional formulations were applied to the integrated diffraction patterns described in the current study and were not able to detect the presence of these isolated abnormally large grains. This result is not surprising, since the peak breadth changes only infinitesimally due to the presence of a single large grain, as do the average grain size and the breadth of grain sizes integrated over a large volume of material. The current technique is also distinguished from conventional peak breadth techniques in that this new technique is conceptually scalable to be used for bimodal grain growth in the microcrystalline regime, a regime where instrumental broadening overwhelms grain size determination by breath measurements. The current proof-of-concept offers an alternative analysis methodology for identifying the onset of AGG. The initial motivation and the confirmation experiments in this study focused on AGG associated with high-cycle fatigue of nanocrystalline metals. However, the technique clearly will apply to thermally induced grain growth in nanocrystalline metals , and could likely provide better understanding of AGG phenomena in both metals (e.g., ) and ceramics (e.g., ). Unlike transmission electron microscopy studies of AGG (e.g., ) which are constrained to a relatively small sampling volume, the current technique provides the ability to identify exceptionally rare events in a much larger volume.
Based on the results presented, the use of X-ray synchrotron diffraction to identify abnormal coarse grains in a nanocrystalline matrix shows great potential as a non-destructive and in situ technique. The method presented here of identifying statistically anomalous peak intensities is very different from the line broadening analysis associated with Warren–Averbach or other traditional methods. While those methods can be used to obtain average grain size, and in some cases information regarding the breadth of the grain size distribution, the present method is better suited to identify an astonishingly small fraction of abnormally large grains. The method has been shown to be capable of identifying 350 nm coarse grains in a matrix with an average grain size of 49 nm. In this demonstration, the large (abnormal) grains are identified despite occupying merely ~0.00001 % of the interrogation volume, and are identified with a statistical confidence ≫99.9999 % (5σ). In the method demonstrated in this work, it is possible to miss a large crystallite when it is not properly oriented to diffract, but that limitation can be overcome through several very tractable pathways. The use of statistical methods provides confidence in the measurements, while the incorporation of programming scripts greatly aids in automating the search for coarse grains in a microstructure and helps avoid user bias in identifying significant peaks. The choice of large apertures and short collection times per scan makes the method suitable for in situ experiments when the onset of coarsening is unclear and the location of the coarse grains lends to a “needle in a haystack” problem.
This work was funded by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering. X-ray diffraction experiments were performed at the Stanford Synchrotron Radiation Lightsource, an Office of Science User Facility operated for the U.S. Department of Energy (DOE). The authors thank Dr. Mark Rodriguez for internal peer review. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.
Compliance with Ethical Standards
Conflict of interest
The authors declare that they have no conflict of interest.
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