Large area strain analysis using scanning transmission electron microscopy across multiple images

Nanometer scale spatial resolution is often required to elucidate the influence of morphology, chemical inhomogeneity, and/or defects on strain. For example, the high temperature mechanical properties of Ni-based superalloys rely on a microstructure consisting of ordered γ′ (space group $Pm3¯m$⁠) precipitates embedded within a continuous disordered γ (space group $Fm3¯m$⁠) matrix.¹ In part, the small difference in lattice parameters between γ′ and γ results in the formation of coherency strains near the γ′/γ interface which helps to impede dislocation motion.¹ Further, this lattice misfit strongly influences the precipitate morphology and stability under stress at high temperatures.² Therefore, understanding superalloy internal coherency strain that results from lattice misfit is critical to engineering novel superalloys for high temperature applications.¹ 

Determination of local crystallographic strain has long been possible with transmission electron microscopy techniques. For example, convergent beam electron diffraction (CBED) is a commonly applied technique with nanometer scale spatial resolution^3,4 and strain precision on the order of $±10−4$⁠. These capabilities of CBED have enabled lattice parameter and mismatch^5,6 measurements while maintaining the capability to study local changes in crystal symmetry.⁷ These diffraction techniques are, however, limited to >1 nm spatial resolution. Rather, atomic scale characterization of strain in real-space can be applied using geometric phase analysis (GPA), which determines the displacement field through Fourier analysis using two non-collinear Bragg vectors.⁸ While this approach has been widely applied to conventional high resolution transmission electron microscopy (HRTEM), changes in sample thickness or defocus and projector lens distortions can lead to significant errors.^9,10 As a result, GPA analysis is limited to measuring strain relative to a reference area within the same image.

In contrast to HRTEM, atomic resolution high-angle annular dark field scanning transmission electron microscopy (STEM) provides incoherent images with intensities that scale strongly with the atomic numbers present.¹¹ The incoherent nature of HAADF STEM limits the influence of thickness and defocus on strain analysis,⁴ opening a doorway to measure strain with a reference not contained within the image. Until recently, however, image distortion introduced by specimen drift has precluded this possibility. Overcoming this limitation, the revolving STEM (RevSTEM) method has recently been developed to accurately remove drift distortion by acquiring a series of images with scan rotation introduced between each frame.¹² This process encodes the sample drift vector across the series, which can then be measured and corrected for every frame. Through removal of drift, precise and accurate crystallographic measurements can be made regardless of a changing drift vector during a microscopy session.

In this letter, we demonstrate that strain analysis can be conducted across the electron microscopy sample using multiple atomic resolution RevSTEM images. As an example, we investigate the local internal elastic strain between the γ′ and γ phases in an undeformed NiAlCr superalloy. We show that the stability of RevSTEM images, even in the presence of fast sample drift, allows the selection of a single reference area not contained within images of other strain analysis regions. For completeness, we calculate strain using both a direct real-space approach and GPA, which are found to be in excellent quantitative agreement. Furthermore, we utilize the high spatial resolution to analyze strain at the nanometer length scale and below.

A directionally solidified ternary superalloy with nominal composition 75.7Ni-16.7Al-7.6Cr at% (NiAlCr) was used throughout the study. The as-grown crystal was annealed at 1250 °C ± 10 °C for 14 days in He atmosphere to ensure homogeneity. Samples for electron microscopy were prepared using a standard twinjet electropolisher (Struers TENUPOL-2) with an applied potential of 15 V and temperature kept between −40 and −20 °C. The electrolyte was 10% perchloric acid in 90% methanol. For HAADF STEM and energy dispersive x-ray spectroscopy (EDS), a probe-corrected FEI Titan G2 60–300 kV S/TEM equipped with a high-brightness Schottky field emission gun (X-FEG) was used and operated at 200 kV. The convergence and inner collection semi-angles for HAADF were 13 mrad and 76 mrad, respectively. Position averaged convergent beam diffraction (PACBED) patterns were recorded for thickness determination at each image location.¹³ The EDS maps were generated using the X-ray K-line for each element. For each RevSTEM series, a total of 40 1024 × 1024 frames were acquired with a dwell time of 3 μs/pixel. The scan coordinates were rotated 90° between each frame.¹² Each RevSTEM series was then post-processed and averaged with a custom MATLAB program to remove sample drift and scan distortion. Atom columns in final images were first located using a normalized cross-correlation template, fit to two-dimensional Gaussian functions for sub-pixel precision, and then indexed into a matrix representation.¹⁴ 

Strain, ϵ, was calculated in two ways. First, strain was determined by measuring all second nearest Ni-Ni atom column separations, equal to the unit cell repeat distance, along $〈100〉$ (x-direction) and $〈010〉$ (y-direction). The strain was then taken as the average of the strain measurements originating from each cell, and determined with reference to the center of a γ′ precipitate as justified below. Second, GPA was performed using the FRWRtools plugin implemented in digital micrograph by Christoph Koch. Strain maps from GPA were generated using the 002 family of reflections. To make the reference area available to the GPA algorithm, the analyzed region and reference RevSTEM images were stitched together across their left and right boundaries, respectively. A circular aperture (mask size = 1 nm⁻¹) was utilized and smoothed by a cosine function (smoothing parameter = 0.7).

Typical of Ni-based superalloys, the NiAlCr microstructure is comprised of γ′ precipitates in a γ matrix as shown by the bright (γ) and dark (γ′) regions in the HAADF image in Figure 1(a). The difference in intensities is due to heavier Cr preferentially partitioning into the matrix, as highlighted by the accompanying EDS results in Figure 1(a) where the Al and Cr are inversely correlated. The microstructure consists approximately of 70% volume fraction of γ′ with a mean size of ∼250 nm, embedded within the matrix. The average width of the γ regions between γ′, also known as γ channels, is 30–40 nm. Atomic-resolution EDS, in Figure 1(b), reveals that Cr predominantly occupies the Al sub-lattice in the γ′ phase, in agreement with recent atom probe tomography experiments^15,16 and first-principles calculations.¹⁷ At the atomic scale, a representative γ′/γ interface is shown in Figure 1(c), where the two regions can be distinguished by the presence of the ordered Al/Cr sub-lattice atom column intensities (dark). In contrast, all atom columns in the γ region exhibit similar intensity because a random solid solution is formed on a face-centered cubic lattice. The lack of misfit dislocation formation at the boundary indicates a coherent interface.

The points for strain analysis using a representative few hundred nanometer-wide γ′ precipitate are shown in the inset of Figure 2(a). At each point, an atomic resolution ∼8 nm × 8 nm image is analyzed. The sample thickness is approximately 25 nm across the investigated area, with less than 5 nm thickness variation between γ′ and γ phases. Moreover, the measurement stability was verified by successively capturing RevSTEM images and measuring the average repeat length, i.e., the lattice parameter a along x- and y-directions. The maximum deviation observed subsequently is 0.079% and 0.04% for a_x and a_y, respectively. This stability thus enables strain measurements to within 0.1% for images across the sample, but with a common reference area.

As an initial comparison, the average lattice distortion is quantified by the measuring the average $ay/ax$ ratio from across each ∼8 nm × 8 nm image. Importantly, tetragonality of the average unit cell increases as the ratio deviates from unity. While negligible tetragonality is measured $(ayo/axo=1.002)$ within the reference area, the ratio consistently increases across the precipitate towards the γ phase to a_y/a_x = 1.009, which is indicative of increasing strain. Commonly, this lattice distortion in Ni-based superalloys results from coherency strain as reported from CBED analysis.^5,7,18 Thus with the low degree of distortion, the γ′ precipitate center (O) is selected as a common reference for strain analysis elsewhere in the sample.

Next, the distribution of local strain within the alloy is systematically determined starting at the center of the γ′ precipitate, marked (O), through the γ′/γ interface. The average ϵ_xx and ϵ_yy from each image along paths A and B is presented in Figure 2(a). Along path A, ϵ_xx and ϵ_yy are perpendicular and parallel to the γ′/γ interface plane, respectively. With respect to the central γ′ region, the strain increases to approximately 0.1% and then suddenly drops to −0.6% within the γ phase. This relationship is reversed along path B where the large negative strain is now observed for ϵ_xx. Note, the strain is maximized in the direction parallel to the γ′/γ interfaces for both paths A and B, indicative of distortion introduced by thermal expansion mismatch between the phases. While not shown here, the shear strain, ϵ_xy, is ∼0.05%, which is well within the measurement error and thus not considered further. Also, as shown in Figure 2(b), strain variation along path C is nearly identical to that observed along path A as expected since the paths are parallel.

In contrast to A and C, the strains measured along paths B and D are inconsistent. While the strain appears negligible along path D, complete understanding of these trends requires atomic resolution real-space analysis. Inspection of the interface RevSTEM image for path D, Figure 3(a), shows that the width of the γ channel is <2 nm. Analysis of the strain within this narrow channel reveals a fairly abrupt compressive strain at the γ′/γ interface. This is further highlighted by averaging a line profile across the interface, Figure 3(b), where directly measured and GPA strain profiles show approximately −0.4% strain within the γ channel.

Inspection of atomic resolution images across γ′/γ interfaces along paths A and B reveals important features that cannot be readily captured by CBED. Comparing the γ′ reference area, (O), with the γ endpoints along paths A and B, Figure 4, shows that the unit-cell strain variation is consistently larger in the precipitate than in the matrix (note the same color scale is used for all images). To quantify this variation, we use the standard deviation, σ for ϵ_xx and ϵ_yy which are presented in the corresponding figures. On average, this represent a 32.8% and 14.7% difference in unit-cell strain along the x and y directions, respectively. We hypothesize that this variation is the result of the random chemical environments for each Al/Cr atom column. Note that some atom columns will exhibit higher concentration of Cr than others and that the atomic radius of Cr $(r≈0.127 nm)$ is approximately 13% smaller than Al $(r≈0.143 nm)$⁠.¹⁹ As a consequence, compressive strain would be introduced in the neighboring Ni sub-lattice for Cr rich atom columns.²⁰ These displacements have been indirectly observed previously using X-ray/neutron diffraction based methods, and a similar effect has recently been observed in a complex oxide solid solution using RevSTEM.^20,21 Further investigations are required to correlate chemistry with the degree of displacement but are beyond the scope of this work.

Surface relaxation of internal stresses upon thinning of TEM foils is important phenomenon to consider for strain analysis. Multiphase microstructures are further susceptible to surface relaxation if their coefficients of thermal expansion differ.^18,22 We expect the surface relaxation effect to play a role here, and is the probable cause for the slight crystal distortion at the center of the γ′ precipitate. Our tetragonality and strain variation findings are, however, consistent with those obtained from CBED analysis.^3,18 Moreover, for this sample, we did not observe a dependence of coherency strain on sample thickness, at least to within our measurement precision.

To conclude, we have reported direct lattice strain measurements across a sample using a single reference area for multiple images. This was enabled by correcting for drift and scan distortion using the RevSTEM technique. In the NiAlCr superalloy investigated, local elastic strain varied as a function of position from the γ′/γ interface. The strain was mainly concentrated in the γ matrix channels, particularly in the direction parallel to the γ′/γ interface. Finally, these results open the possibility to capture atomic scale structural information across large areas while maintaining absolutely quantitative measurements in STEM.

The authors gratefully acknowledge support for this work from the Air Force Office of Scientific Research (Grant No. FA9550-12-1-0456). We acknowledge the use of the Analytical Instrumentation Facility (AIF) at North Carolina State University, which was supported by the State of North Carolina and the National Science Foundation.