Evolution of local misorientations in the γ/γ’‐microstructure of single crystal superalloys during creep studied with the rotation vector baseline (RVB) EBSD method

The present work uses the rotation vector baseline electron back scatter orientation imaging method (RVB‐EBSD) to study the evolution of small misorientations between the γ‐ and γ′‐phase in Ni‐base single crystal superalloys (SXs) during creep. For this purpose, two material states of the SX ERBO1 (CMSX4 type) were characterized after creep deformation at 850°C and 600 MPa to final strains of 1% and 2%. Obtaining reliable phase boundary misorientation (PBM), kernel average misorientation (KAM) and orientation spread (OS) data represents a challenge for electron backscatter diffraction (EBSD), not only because the method operates at its limits of lateral and angular resolution, but also because it is difficult to differentiate between the two phases merely based on Kikuchi diffraction. The two phases differ in chemical composition which gives rise to different EBSD background intensities. These can be exploited to differentiate between the two phases. In the present work, crystallographic and chemical information are combined to demonstrate that orientation imaging can be used to document the formation of dislocation networks at γ/γ′‐interfaces and the filling of γ‐channels by dislocations. These findings are in good agreement with reference results from diffraction contrast scanning transmission electron microscopy. It is also shown that misorientations evolve between small groups of equally oriented γ/γ′‐neighborhoods, on a size scale above characteristic γ/γ′‐dimensions (>0.5 μm) and below distances associated with dendritic mosaicity (<200 μm). The methodological aspects as well as the new material specific results are discussed in the light of previous work published in the literature.

resolution, but also because it is difficult to differentiate between the two phases merely based on Kikuchi diffraction.The two phases differ in chemical composition which gives rise to different EBSD background intensities.These can be exploited to differentiate between the two phases.In the present work, crystallographic and chemical information are combined to demonstrate that orientation imaging can be used to document the formation of dislocation networks at γ/γ 0 -interfaces and the filling of γ-channels by dislocations.These findings are in good agreement with reference results from diffraction contrast scanning transmission electron microscopy.It is also shown that misorientations evolve between small groups of equally oriented γ/γ 0 -neighborhoods, on a size scale above characteristic γ/γ 0 -dimensions (>0.5 μm) and below distances associated with dendritic mosaicity (<200 μm).The methodological aspects as well as the new material specific results are discussed in the light of previous work published in the literature.

Research Highlights
• Microstructure evolution during [001] tensile creep of Ni-based single-crystalline alloy.
• Separation of γ/γ 0 phases using experimental post-processing of raw EBSD data.
local dislocation densities, misorientations between γ and γ 0 phases, rotation vector baseline EBSD method, scanning transmission electron microscopy, single crystal Ni-base superalloys Plastic deformation of metallic single crystals is associated with lattice rotations (Schmid & Boas, 1935).When single slip governs the deformation of large single crystals this can lead to high lattice rotations, which Schmid and Boas (Schmid & Boas, 1935) have described in their Wurstscheibenmodell (sausage slab model).In the present work rotation phenomena are characterized which are associated with high temperature and low stress [001] tensile creep of Ni-base single crystal superalloys (SXs).Under these conditions, creep is always governed by multiple slip and macroscopic rotations are only occasionally observed after rupture, when large scale plasticity has led to necking (Ardakani et al., 2000;Basoalto et al., 2002;Cao et al., 2020;Ghosh et al., 2000;Mayr et al., 1996).Single crystal Ni-base superalloys exhibit a microstructure which is composed of small ordered γ 0 -cubes (l1 2 -phase, 75% volume fraction, average edge length of the γ 0 -cubes: 0.4 μm) separated by narrow γ-channels (FCC crystal structure, volume fraction: ≈25%, average γ-channel width: ≈50 nm).During creep dislocations enter the γ-channels and form dislocation networks close to the γ/γ 0 -interfaces.These dislocation networks have been studied in detail throughout the last decades (Carroll et al., 2008;Feller-Kniepmeier & Link, 1989;Field et al., 1992;Gabb et al., 1989;Keller et al., 1993;Kolbe et al., 1998;Lahrmann et al., 1988;Lasalmonie & Strudel, 1975;Singh et al., 1988;Xie et al., 2014;Zhang et al., 2005Zhang et al., , 2021)).Figure 1 shows a scanning transmission electron microscopy (STEM) micrograph which was taken after creep at 850 C and 600 MPa (total accumulated strain 2%).The STEM foil was cut out perpendicular to the applied stress σ, as indicated in the upper left.In Figure 1, the γ 0 -phase exhibits a brighter gray contrast than the γphase.The strain fields surrounding dislocations appear as black lines.
The arrows 1 point into γ-channels with high dislocation densities.
The arrows 2 point to locations, where dislocation networks are present at the γ/γ 0 -interface with widths of the order of 20 nm.Arrow 3 highlights a rare cutting event, where dislocations from the network cut into the ordered γ 0 -phase.Location 4 shows a dislocation network which has formed in a horizontal γ-channel.The arrangement of γ 0cubes is not regular and at location 4, the STEM foil contains a horizontal γ-channel with a network spreading out flat through the foil.
The networks are not as regular as under conditions of high temperature and low stress creep (T > 900 C, σ < 400 MPa).But at these conditions, the γ-channels are not as busy and the microstructure does not contain so many planar faults as under conditions of low temperature (T < 800 C, σ > 500 MPa) high stress creep.
Dislocation networks have been characterized by networks spacings s, which are used in Brook' 0 s formula to estimate the magnitude of the misfit between the γand the γ 0 -phase (Carroll et al., 2008;Lahrmann et al., 1988;Long et al., 2017).Frank (1951) has derived equations which allow to correlate network spacings to misorientations.A well-known text book example (Hull & Bacon, 1984) relates the misorientation angle θ between two crystal regions separated by a pure tilt boundary as where b is the magnitude of Burgers vector.
In the γ/γ 0 -microstructures of SXs, dislocation networks form in the early stages of creep.The objective of the present work is to assess whether orientation imaging scanning electron microscopy can provide new insight into the elementary processes which govern microstructural evolution in the early stages of creep.This represents a challenge, because γ-channels are narrow and the small crystallographic differences between the lattices of the γand γ 0 -phases do not yield Kikuchi line diffraction patterns (Nishikawa & Kikuchi, 1928) which allow to differentiate between the two phases.Figure 2 shows that the Kikuchi line diffraction patterns of the two phases can hardly be distinguished.Therefore, we exploited the chemical difference between the γand the γ 0 -phase (Parsa et al., 2015), as described below.
In the present work the rotation vector base line electron backscatter diffraction (RVB-EBSD) method (Thome et al., 2019), which allows to resolve orientation differences of the order of 0.03 , is upgraded in in multiple regards.Specifically optimized imaging parameters together with a virtual dark field technique and enhanced post processing algorithms (Hielscher et al., 2019;Wright et al., 2015)

| Material and mechanical testing
The material investigated in the present work was an alloy ERBO/1 (CMSX-4 type).The chemical composition and all heat treatment details of the slabs from which the specimens were taken have been published in the literature (Parsa et al., 2015).Its microstructure consists of two phases, where cuboidal γ 0 -particles (ordered cubic L1 2 crystal structure; volume fraction: close to 70%; typical γ 0 -cube edge length: 0.4 μm) are separated by thin γ-channels (face centered cubic crystal structure; typical volume fraction: close 30%; channel width: 50 nm).Miniature creep specimens machined using an iterative procedure which combines crystallographic orientation (Laue method) with spark erosion machining.Specimens with rectangular cross sections and precise crystallographic orientations (deviation from targeted crystallographic directions: <1 ) were obtained (tensile direction: [001], flat sides: [010] and [100], see fig. 3 of Wollgramm et al., 2016).
All other details of creep testing have been reported in the literature (Wollgramm et al., 2016).In the present work two crept material states were investigated, which were deformed at a temperature of 850 C under a stress of 600 MPa to strains of 1 and 2%.

| Specimen preparation
To obtain electron backscatter diffraction (EBSD)-results with high angular (<<1 ) and spatial (<50 nm) resolutions, an excellent surface quality of metallographic cross-sections is required.High quality (100) specimen surfaces were obtained using grinding, diamond paste polishing and vibro-polishing.Grinding was performed using SiC emery paper up to a grit size of P 2500.This was followed by 5 min periods of diamond paste polishing using grain sizes of 3, 1, and 0.25 μm and subsequent vibro-polishing in a Buehler VibroMet (master met 2 suspension, 2 h).Build-in functions of MTex in combination with self-written scripts were used in the application of RVB-EBSD method (Thome et al., 2019).In the present work EBSD-scans were obtained using step sizes of 10 and 20 nm (< γ-channel width).The 20 nm step sizes yielded more reliable results (larger fields of view accessible, less noise/scatter) and therefore 20 nm was chosen as a step size for the quantitative investigations performed in the present work.The RVB-EBSD method allows to measure KAM angles with high angular resolution and it has been described in the literature how it can visualize orientation spreads (OS) which occur over larger specimen regions (Thome et al., 2019).

| Virtual dark field images and accelerating voltage
It is not possible to use standard EBSD indexation procedures to reliably differentiate between the γand γ 0 -phases, as is shown in the image quality (IQ) map shown in Figure 3a.To improve contrast, one can take advantage of small differences in the illumination of the background, which are caused by differences in the chemical composition of the γand the γ 0 -phase (Parsa et al., 2015) and the associated electron scattering cross-sections of different elements.We note that the γ-phase in dendritic regions contains significantly more than the γ 0 -phase (in wt-%, respectively: 5.9, 1.7, 0.7, and 6.1) (Parsa et al., 2015).Wright et al. (2015) have shown that the EBSD detector can be used as a virtual electron diode to generate chemical contrast by integrating the electron signal in specific areas of the EBSD detector screen.The electron signal from the sample, which creates the EBSD pattern and its background, is composed of diffracted and scattered electrons.The background intensity of scattered electrons carries chemical information which can be exploited.This can be achieved by dividing the detector area into virtual front scatter diodes (FSD) (Wright et al., 2015) and assessing the resulting virtual dark field images.The EBSD Kikuchi pattern shown in Figure 3b is subdivided into five regions.From each of these regions, a virtual dark field image field can be generated (Wright et al., 2015), two of which are shown in Figure 3c,d.Both allow for a better phase separation than the standard IQ image presented in Figure 3a.However, the signal integrated in area 2 provides the best result.Area 2 conditions were used to automatically define a threshold condition for phase distinction.
Unclear results required manual adjustments, by either reconstructing missing features manually or by excluding unclear regions from further analysis.It was demonstrated earlier by Nolze et al. (2017) that this type of contrast can be used to differentiate between the γand γ 0phase in a Ni-based superalloy.In the present work, we exploit this virtual dark field technique.
As can be seen in the virtual dark field images (area 2 conditions) in Figure 4, decreasing the accelerating voltage from 20 to 15 kV results in a significant improvement of phase contrast and the associated lateral resolution.All images obtained in the present work were therefore collected using an accelerating voltage of 15 kV.

| Applying orientation filters
In the present work orientation information is obtained using the RVB-EBSD method.KAM mapping can be used to access local misorientations, like those caused by dislocations and dislocation networks (Bachmann et al., 2010;Kamaya, 2011;Konijnenberg et al., 2015;Kuwahara et al., 1976;Wright et al., 2011).The RVB-EBSD method can reach angular resolutions of 0.03 (Thome et al., 2019).It has recently been shown that filters, like the Kuwahara filter (Kuwahara et al., 1976) and the half quadratic filter (HQF) (Bergmann et al., 2016) can significantly improve the signal to noise ratio of orientation mappings.In the present study, a half quadratic orientation filter is applied to improve the image quality of the orientation data.Figure 5a (without filter) and b (with filter) show the effect of the filtering procedure on a conventional Hough based OIM data set.
The signal to noise ratio is improved, however, the angular resolution of the input data does not allow to reveal the critical F I G U R E 3 Electron backscatter diffraction results obtained using a step size of 20 nm showing how to optimize phase contrast using the virtual dark field method of Wright et al. (2015).

| Shift correction
In order to discuss the crystallographic and chemical information obtained from the EBSD patterns, we present experimental and simulation results in Figure 6. Figure 6a shows the raw experimental KAM data presented using the color coding as indicated by the color bar.
The chemical information retrieved from the EBSD background analysis described above are shown as gray boundary lines around each γ 0particle.The region highlighted with a small white rectangle is shown at higher magnification in Figure 6b.The white arrow points to a γ-channel.From STEM micrographs like the one shown in Figure 1, it is known that dislocation activity is concentrated in the channels.This shift is systematic and can be found for all the γ-channels parallel to the tensile axis (horizontal orientation in Figure 6).In order to rationalize this shift, the interaction of electrons with the material was investigated using a Monte Carlo procedure.Monte Carlo calculations were performed using the software Casino v2.5.1.0(Casino, n.d.;Hovington et al., 1997).The calculations were performed using an accelerating voltage of 15 kV, an electron beam diameter of 10 nm, a weighted average of the densities of the two phases (γ: 11.16 and γ 0 : 9.83 g/mm 3 ) and the EBSD-geometry (electron beam tilted by 70 to the normal of the specimen surface).
50,000 material-electron interactions were considered, which are represented by blue and red trajectories in Figure 6c.As can be seen in Figure 6c, electrons can reach depths exceeding 200 nm (blue trajectories).The back scattered electrons, which reach the EBSD detector, stem from the escape volume (red trajectories).Figure 6d shows the number of electrons which can escape the specimen as a function of the depth of the secondary source.We suggest that the average source depth associated with the scattered electrons which form the background of the pattern is lower than the depth of the diffracted electrons which create the Kikuchi lines.This difference in escape depth accounts for the small shift observed between the images relying on chemical and on orientation information.We use this qualitative insight to correct the results by applying a shift such that regions of increased KAM intensities are in reasonable agreement with horizontal γ-channels.The corrected data are presented in Figure 6e,f.Figure 6b,f show that this correction effectively compensates for the discrepancy between the chemical and crystallographic information obtained for the γ-channels with the horizontal orientation in Figure 6.  between the two locations highlighted with blue plateau lines, Figure 7b.In the present study these locations were 2 steps (in the γ 0phase) and 1 step (in the γ-phase) away from the phase boundary.We will later show by direct comparison with STEM reference results, that this procedure yields reliable data.Figure 7c shows data from close to 5000 individual phase boundary misorientations (PBMs), obtained for the specimen which was deformed to 2% strain.The red histogram is obtained using the raw data (step-to-step difference, red double arrow in Figure 7b).The blue histogram is obtained when determining PBM data obtained by determining the orientation difference between the two reference points as described above (difference between the two measurement locations highlighted in blue in Figure 7b).The blue histogram shown in Figure 7c yields significantly higher misorientations.We will later show that the blue histogram represents realistic data.
2.2.7 | Classification of γ-channel direction, relative to stress In the early stages of [001] tensile creep, dislocation plasticity is mainly observed in the γ-channels perpendicular to stress (Wu et al., 2016).Therefore, PBM between the γ and γ 0 phases and KAM

| Specimen preparation and operating conditions
Thin STEM foils were prepared, cutting out slices from the crept material states perpendicular to the direction of the applied stress.
The slices cut by an Accutom 5 from Struers were subsequently ground to a thickness of 90 μm, using emery paper of 4000 mesh size.The rotation operator R and the coordinate transformation tensor T between two orthonormal coordinate systems are related as The coordinate transformation T i,j between LCCS i and LCCS j is composed of two successive transformations.First a transformation  A i,MCS from LCCS i to MCS is constructed, which is followed by a transformation B MCS,j from MCS to LCCS j such that In order to perform these two steps, one has to express the MCS unit vectors e x , e y , and e z in terms of the coordinates of the local coordinate systems LCCS i and LCCS j .For this we use the angles α and β (Figures 9b and 10b) defining the position of a crystallographic zone P in each of the two corresponding CBED patterns i and j.The three crystallographic zones which we consider are characterized by the unit vectors p 1 , p 2 , and p 3 .The system of linear equations where angles α are measured in CBED pattern i and j, allows to obtain the coordinates of the unit vector e z in the LCCS i and LCCS j .Similarly, the solution of the linear system yields the coordinates of the unit vector e x .d 1 , d 2 and d 3 are the unit vectors which have been introduced above.For zone P1, for example, the direction d 1 can be obtained by the cross product and corresponding products yield vectors d 2 and d 3 .Finally, the coordinates of the unit vector e y are given by the cross product e z Â e x .Having expressed the MCS unit vectors in terms of the two LCSSs, one can construct the transformation matrices as A i, MCS = (e x , e y , e z ) i and B MCS,j = (e x , e y , e z ) j À1 .This allows to obtain T, from where one can derive R. With R, one can determine the unit vector u, which defines the orientation of the rotation axis, and the associated rotation angle ϕ, which we use to quantify the misorientation between LCCS i and LCCS j .The unit vector u can be obtained from where λ is the real positive Eigenvalue which fulfills Equation (7).
Figure 10c illustrates how one can find the sense of the rotation (+ or À).We select an arbitrary vector a in the lattice at location i, which is not colinear with the vector u, and define a unit 3 | RESULTS

| Misorientations measured with STEM
In the present study the STEM results serve as a basis for assessing the results obtained with the modified RVB-EBSD method and are presented first.The axis u /angle φ pairs together with a sense of rotation (+ or À) obtained for all ten γ/γ 0 -misorientations highlighted in Figure 9a,b are listed in Table 1.The Table 1 data for Figure 9a are shown in the enlarged micrograph of Figure 12.
In summary, the Table 1 STEM results obtained after 2% creep deformation yield φ-angles between 0.14 and 0.61, which are all smaller than the angular resolution limit of the conventional Hough transformation based EBSD technique.In the following we consider how well these results can be reproduced using the modified RVB-EBSD method and where the RVB-EBSD method can provide additional information.

| Phase boundary misorientations
In Figure 13 EBSD results obtained for the γ/γ 0 phase boundary misorientations are presented.Figure 13a    the initial preference for the ⊥ γ-channels plasticity.These RVB-EBSD results are thus in excellent agreement with experimental data on the evolution of dislocation densities in the early stages of SX creep published so far in the literature, see for example (Wu et al., 2016).creep strain: dashed lines, 2% creep strain: full lines).Figure 14c shows that the γ-phase always shows higher KAM-values than the γ 0phase.A difference Δ in % was determined for a cumulative frequency of 0.5.As indicated in Figure 14c, this difference increases from 27% after 1% creep strain to 39% after 2% creep strain.On Figure 14 d and e, the KAM values are divided based on the k and ⊥ γ-channel character (same as for the PBM).As expected, the ⊥ to stress γ-channels give a slightly higher KAM value than k γ-channels.

| Kernel average misorientations
Unlike the PBM, the difference is pronounced also in the case of 2% deformed sample.suggest that the spread of crystallographic orientations is not only associated with differences between γ-channels and γ 0 -particles.It also represents a larger scale phenomenon, microstructural regions consisting of groups of γ 0 -particles which are significantly larger than the average γ 0 -particle size (>0.5 μm) but also significantly lower than the dendrite spacing (< 200 μm) are slightly misoriented to each other.This is evident in Figure 15a,b.
Originally, EBSD was used for the characterization of polycrystalline materials with high angle grain boundaries, for example (Engler, 2009;McAuliffe et al., 2020;Thome et al., 2022;Thome et al., 2023).In recent years advanced imaging technology and indexing algorithms have helped to increase the angular resolution of the EBSD method by more than 1 order of magnitude (from about 1 to below 0.1 ) (Chen et al., 2015;Nolze et al., 2016;Thome et al., 2019;Wilkinson et al., 2006).This allows to explore microstructural phenomena which are associated with small processing or deformation induced lattice rotations in single crystalline materials, for example, single crystal mosaicity and crystal plasticity, for example (Pantleon, 2008;Scholz et al., 2021).In the present work we apply the rotation vector baseline EBSD (RVB-EBSD) technique to study small misorientations in Nibase superalloy single crystals, with microstructures which feature small ordered γ 0 -cubes which have coherently precipitated in a FCC γ-matrix.As outlined in the introduction, in the early stages of medium and high temperature creep dislocations enter the narrow γchannels (typical average width: 50 nm) separating the small cuboidal γ 0 -particles (typical average edge length: 400 nm) (Carroll et al., 2008;Feller-Kniepmeier & Link, 1989;Field et al., 1992;Gabb et al., 1989;Keller et al., 1993;Kolbe et al., 1998;Lahrmann et al., 1988;Lasalmonie & Strudel, 1975;Singh et al., 1988;Xie et al., 2014;Zhang et al., 2005;Zhang et al., 2021).They form dislocation networks which cause a small misorientation between γ-channels and the adjacent γ 0particles.Convergent beam electron diffraction (CBED) in the scanning transmission electron microscope (STEM) can easily resolve these details.
However, for orientation imaging scanning electron microscopy this represents a challenge.One must work at the limits of the lateral and angular resolutions of the method.Moreover, it is difficult to use Kikuchi maps to differentiate between the two phases, Figure 2. To overcome this problem, crystallographic results were successfully combined with chemical results from energy dispersive x-ray spectroscopy (EDS) reported in literature, for example (McAuliffe et al., 2020).Due to the limited lateral resolution of EDS mappings this does not work for SX-microstructures with small γ 0 -particles (<1 μm) and narrow γ-channels (<<1 μm).As described above this can be overcome by using the detector screen as an imaging tool (Nolze et al., 2017;Wright et al., 2015).Nolze et al. (2017)  (lower average Z-value due to lower concentrations of Re and Ta).
The small magnitude of misorientations which evolve in the early stages of creep requires the application of an EBSD method with a high angular resolution.If the angular resolution is not sufficient, even advance filtering procedures can yield erroneous results, Figure 5a,b.
There is a need to optimize the accelerating voltage used in the EBSD measurements.A best compromise has to be found between optimizing the lateral resolution by minimizing the interaction volume, which requires low accelerating voltages.On the other hand, the accelerating voltage has to be high enough to provide a sufficient signal to noise ratio in the Kikuchi line diffraction patterns (Steinmetz & Zaefferer, 2010).In the present study we work with 15 keV, which yields the image quality shown in Figure 4. Novel detector hardware, such as direct electron detectors, might be able to further increase the lateral resolution of EBSD be decreasing the accelerating voltage even more in the future (Steinmetz & Zaefferer, 2010;Wang et al., 2021).Using a different geometric setup of the EBSD system is another option to further increase the lateral resolution, for example, by characterizing thin transmission Kikuchi diffraction specimen (Keller & Geiss, 2011).However, even with this best compromise one cannot avoid superposition of crystallographic information from both phases and one has to carefully think about how to evaluate misorientations, as schematically illustrated in Figure 7.In the present we work we therefore do not consider differences between adjacent EBSD results obtained with 20 nm step size.An effort has been made to evaluate misorientations between locations which are far enough from the interface to minimize the effect of superposition of crystallographic information.As shown in Figure 7b, one cannot exclude that the measured PBM-values slightly underpredict the real PBM*-values.This is why the STEM validation of the EBSD-procedure performed in the present work is important.As can be seen in Figure 13, the misorientations obtained with the EBSD-procedure are in very good agreement with the precise STEM CBED measurements.

| Assessment of KAM resolution
It has been proposed by Kamaya (Kamaya, 2011) that one can estimate the angular resolution of an EBSD maps by considering pixel neighborhoods at different distances from the central pixel.Figure 16 shows a Kamaya plot of the data obtained after 1% and 2% strain.It shows that straight lines with positive slopes are obtained when plotting the average misorientations as a function of increasing neighbor distances, numbered from 1 (20 mn) to 5 (100 nm).Kamaya (2011) suggested that the intersection of these lines with the y-axis at x = 0 yields the angular resolution.In our case the two series obtained for the crept material states which were deformed to 1% and 2% yield angular resolution values of 0.01 and 0.03 , respectively.This is of the same order as the value of 0.

| STEM CBED results-sense of rotation
Considering the misorientation results obtained between channel positions 4 and 5 and their adjacent γ 0 -neighbors in Figure 9b (4: 3 and 9; 5: 2 and 8), we find that the rotation angles φ vary between 0.14 and 0.61 and the rotation axis have different u-vectors.However, the sense of the rotations is often the same on both sides of one γ-channel.During creep, dislocation loops enter γ 0 -channels and leading screw segments deposit dislocations of opposite sign in the γphase close to the γ/γ 0 -interfaces (Probst-Hein et al., 1999).The interface dislocations can have a 60 character (Probst-Hein et al., 1999), which have a 2/3 edge character.This is considered in the schematic illustration of Figure 17, where the interface dislocations are represented as simple edge dislocations.The figure demonstrates that the deposited segments of opposite signs create lattice rotations of the same sense on both sides of the channel.
Once the angle φ and the Burgers vector of the associated microscopic crystallographic slip system is known, one can estimate the dislocation spacing d using Equation (1).The STEM/CBED measurements listed in Table 1 (few high precision data) yield a mean φ-value of 0.265 .This is in excellent agreement with the average value of 0.244 obtained from the RVB-EBSD data (value based on more than 5000 measurements, better statistics) from the present work.From this misorientation angle one can determine an average spacing between dislocations of 54 nm.This value is in excellent agreement with dislocation network spacings which have been reported in the literature (Carroll et al., 2008;Gabb et al., 1989;Probst-Hein et al., 1999;Singh et al., 1988).
F I G U R E 1 6 Kamaya plots (Kamaya, 2011) of average KAMvalues yielding angular resolutions of 0.01 (material state which was deformed to 1%) and 0.03 (material state deformed to 2%).For details see text.

| Evolution of local orientations
In the early days of Ni-base single crystal research, evolutions of local lattice misorientations during processing and creep did not receive much attention.SXs crystals which formed during directional solidification were merely characterized by deviation angles from targeted <100> solidification directions (Nazé et al., 2021;Reed, 2006).Only in the final stages of creep, where necking occurred during final rupture, large lattice rotations (> 20 ) were reported (Ardakani et al., 2000;Basoalto et al., 2002;Cao et al., 2020).The RVB-EBSD method (Thome et al., 2019) was developed to study crystal mosaicity associated with the growth of dendrites during solidification (Hallensleben et al., 2017;Hallensleben et al., 2019;Scholz et al., 2021).It was found that the nature of dendritic solidification resulted in orientation spreads of up the order of 5 (Scholz et al., 2021;Thome et al., 2019).
In the present work, three new additional EBSD results were obtained.First, the misorientations at the γ/γ 0 -interface associated with the formation of interface dislocation networks (arrow 2 in   respectively) are significantly smaller than those reported for dendritic crystal mosaicity (≈ 5 (Hallensleben et al., 2017;Hallensleben et al., 2019;Scholz et al., 2021;Thome et al., 2019)).At the same time, they are significantly larger than the misorientations measured across γ/γ 0 -phase boundaries (0.25 , present work).In a recent phase field study (Ali et al., 2023), which was performed for the same alloy investigated in the present work but crept at a higher temperature and lower stress (950 C, 350 MPa), maximum deviation angles of the order of 1.5 were observed after 1% strain.However, as can be seen in Figure 12 of the phase study (Ali et al., 2023), this misorientations were only locally detected in isolated γ-channels.γ 0 -particles were assumed to behave purely elastic.The phenomenon shown in Figure 15 of the present work was not predicted by the phase field study, which was performed in a different stress temperature range.
However, the phase field predictions of angular misorientations are of the same order of magnitude as the maximum orientation spreads presented in Figure 15.

| Parallel and perpendicular γ-channels
The PBM and KAM distributions evaluated separately for ⊥ and k γchannels and presented in Figures 13d and 14d,e confirm an expected preference for localization of dislocation plasticity into ⊥ γ-channels in the early stages of [001] tensile creep.This is documented by a slight shift of the PBM and KAM distributions obtained for the ⊥ γchannels towards higher misorientation angles.In relation to Equation (1), the shift can be interpreted assuming smaller dislocation spacings and thus higher dislocation densities in the ⊥ γ-channels.
Indeed, these results agree with the data reported by Wu et al. (Wu et al., 2016), who documented the higher dislocation density in the ⊥ γ-channels after 0.2% creep strain accumulated during [001] tensile creep at 750 C and 800 MPa.The same authors also investigated the evolution of dislocation density in the γ-channels with increasing creep strain and showed that, after a steep growth, the density saturates at about a strain level of 2% (Wu et al., 2016).The overlapping PBM distributions evaluated after 2% strain for both ⊥ and k γ/γ 0 interfaces suggest that the initial imbalance between these interfaces F I G U R E 1 7 Lattice rotations between the γand the γ 0 -phase by +ϕ (same positive sense of rotation) on both sides of one γ-channel.This is associated with the deposition of dislocations of opposite sign on the two sides of the γ-channel.
gradually ceases and the spacing between dislocations deposited at the interfaces may saturate.Nevertheless, the KAM distributions, which characterize a deformation state inside the γ-channels, still indicate a slight bias in favor of ⊥ γ-channels after 2% of creep strain, see Figure 14e.This may suggest slower processes of dislocation recovery inside the γ-channels as compared to the γ/γ 0 interfaces.
However, such interpretations must be examined with care in view of the small misorientation differences between the PBM and KAM distributions and still insufficient statistics covering only small microstructural regions.

| SUMMARY AND CONCLUSIONS
In the present work the RVB-EBSD method, which was originally developed to study crystal mosaicity associated with dendritic solidification, is used to characterize the evolution of small misorientations in the γ/γ 0 -microstructures of a single crystal Ni-base superalloy during creep.The objective of the present work was to identify misorientations at different length scales, including misorientations across γ/γ 0 -phase boundaries (PBM angles), misorientations associated with γ-channel dislocation activity (KAM angles) and orientation spreads in regions containing several γ 0 -particles (OS angles).This represents a challenge, not only because the microstructural features are so small (γ 0 -particle size: 400 nm, γ-channel width: 50 nm) that the EBSD method operates at the limits of its lateral and angular resolution but also because it is not straightforward to use EBSD to differentiate between the γand γ 0 -phase.The initial microstructure of the material investigated in the present work and the details of creep testing were previously documented in the literature.Two material states were compared which were creep deformed at 850 C and 600 MPa to total strains of 1% and 2%.From the results obtained in the present the following conclusions can be drawn: 1.In order to obtain viable results on a relevant γ/γ 0 -length scale, one must optimize the specimen preparation route and find optimum EBSD imaging parameters.Moreover, appropriate data processing needs to be performed to obtain viable crystallographic information.Last but not least, diffraction contrast transmission electron microscopy is required for validation.Three parameters which quantify orientation information were considered in the present work: phase boundary misorientation (PBM), kernel average misorientation (KAM) and orientation spreads (OS). 5. From a methodological point of view, it is important to highlight that using Kamaya's method and comparing the results with STEM reference data, it can be confirmed that the RVB-EBSD method provides an angular resolution between 0.01 and 0.03 for specimens creep deformed to 1% and 2% strain under challenging experimental conditions.
are implemented to resolve the orientation features of the γ/γ 0 -microstructure.As a complementary method diffraction contrast scanning transmission electron microscopy STEM was performed at selected interfaces, to obtain images like the one shown in Figure1, which provides evidence for a high dislocation activity in the γ-channels and a low dislocation activity in the γ 0 -phase.Complementary STEM results also validate the orientation measurements obtained by the RVB-EBSD method.We show that advanced analytical scanning electron microscopy orientation imaging (SEM OIM) can be used to interpret the strain related evolution of different microstructural parameters in the early stages of creep: The increase of small misorientations between the γand the γ 0 -phase, the increase of orientation spread and the confinement of plastic deformation to the γ-channels.F I G U R E 1 Scanning transmission electron microscopy micrograph of a γ/γ 0 -microstructure after [001] tensile creep deformation at 850 C, 600 MPa.Strain level at interruption of creep test: 2%.View direction: [001], parallel to the direction of the applied stress.g = (200).
SEM investigations were performed using a Zeiss Leo Gemini 1530VP equipped with a field emission gun (FEG), an in-lens BSE detector and an EBSD system from EDAX equipped with a Digiview Camera running with TSL software.The post processing of the EBSD maps is carried out in MatLab R2022a (MATLAB, 2023) expanded with the MTex 5.7.0 toolbox (MATLAB, 2023) for processing of the EBSD data sets.

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I G U R E 2 Examples for electron backscatter diffraction (EBSD) patterns obtained from the material state deformed to 1%.(a) EBSD pattern from γ-region.(b) EBSD pattern from γ 0 -region.Co, Cr, Re and W (in wt-%, respectively: 17.8, 17.1,12.7,and 10.1) (a) Conventional IQ-image with poor phase contrast.(b) Subdivision of the detector area collecting the elastically and inelastically scattered electrons into 5 regions.(c) and (d) Virtual dark field images generated by signal integration over areas 2 and 4, respectively.microstructural features.More importantly, artifacts are created which may result in misinterpretations.As can be seen in Figure 5c,d, starting with RVB-EBSD raw data improves the situation.Microstructural details can already be seen in the unfiltered image, Figure 6c.Applying the filter further reduces the background noise, Figure 5d.All further EBSD studies in the present work rely on the combination of RVB EBSD with HQF filtering.

Figure
Figure6bdoes not fully reflect this finding, the positions of the γ 0boundaries parallel with the tensile axis, as detected by the diffracted and background electrons, do not coincide.There is a small vertical shift between chemical phase and crystallographic orientation information, which is not observed for the γ 0 -particle boundaries perpendicular to the tensile axis.This can be clearly seen in the higher magnification image of Figure6b, where regions of high KAM values

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I G U R E 6 Compensation of small shifts between chemical and orientation information.(a) and (b) Uncorrected raw data at lower and higher magnifications, respectively.(c) Electron trajectories calculated with a Monte Carlo method.(d) Number frequency of electrons as a function of their secondary source depth.(e) and (f) Corrected data.For details see text.

2. 2
.6 | Sharp γ/γ 0 -interfaces, size of interaction volume and EBSD step size Assuming local thermodynamic equilibrium at an atomistical sharp γ/γ 0 -phase boundary and neglecting the presence of a small 20 nm wide dislocation network next to the γ/γ 0 -interface (see locations highlighted with arrows 2 in STEM micrograph of Figure1), one expects a sudden change of crystallography when crossing the γ/γ 0phase boundary.This is schematically illustrated by the step function in Figure7a.The real phase boundary misorientation PBM* is indicated by a double arrow.As has been shown in Figure6, the interaction volume from where electrons are backscattered towards the EBSD detector extends over a region of ≈100 nm, significantly larger than the EBSD step size of 20 nm.This information is schematically integrated in Figure7a.Therefore, the experimental raw data do not reproduce the step function.Instead, one obtains the continuous green curve, which results from a superposition of orientation information from both phases.In Figure7bwe show this green curve together with small horizontal plateau lines which represent the results obtained in individual EBSD data points.Figure7billustrates that the misorientation between two adjacent locations right next to the phase boundary (red plateau lines) yields a step-to-step difference (red double arrow) which is significantly smaller than PBM*.In order to obtain results which are as close as possible to PBM*, one can determine the orientation difference PBM (blue double arrow) Figure 8, a numerical algorithm identifies interface image pixels situated either inside the ⊥ window, which are classified as ⊥ -type interface, or inside the k window which form the k-type interface.The windows are then manually adjusted and moved over to other similar locations in the γ/γ 0 microstructure.We note that the manual control over the window size and position allows to address only straight portions of the interfaces and avoid their crooked segments as well as particle corners and thus exclude these regions from the analyses.
Figure 9a,b show two STEM micrographs which were taken from the material state deformed in [001] tensile creep to 2% strain.The micrographs were taken using contrast conditions of g = (À200) and g = (020), respectively.In both images, locations are marked, where Kikuchi line diffraction patterns were acquired.Note that Figure 9a,b all together contain 10 pairs of adjacent γ/γ 0 -regions.

Figure 10
Figure 10 shows a Kikuchi pattern which was obtained for the γchannel position 4 in Figure 9a.There is the [001] crystallographic zone in the lower right and pairs of sharp bright and dark lines map the crystallography of associated lattice planes.Sharp Kikuchi lines are obtained from regions where the foil thickness exceeds at least two extinction distances (De Graf, 2003; Probst-Hein et al., 1999; Reimer & Kohl, 2008).In Figure 10 the pattern is shown two times.Close to the center of Figure 10a, small open circles highlight systematic reflections associated with g = (À200).One circle is fitted to a reflection which exhibits good contrast.The precise positions of the centers of the other circles are obtained by shifting this circle by reciprocal space distances corresponding to the Bragg angle perpendicular to the associated Kikuchi lines.Thus, one obtains a row of systematic reflections which allows to precisely locate the center of the transmitted beam, marked with a full black circle and an O letter, for origin.As indicated in Figure 10a, the precise positions of three crystallographic zones P 1 = [001], P 2 = [1 2 11] and P 3 = [À1 2 11] are

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I G U R E 1 0 Kikuchi line convergent beam electron diffraction pattern taken at γ-channel position 4 of Figure 8a.g = (À200).(a) Center of the transmitted beam and positions of three crystallographic zones P 1 = [001], P 2 = [1 2 11] and P 3 = [À1 2 11].(b) Local coordinate system x, y, z. d 1 , d 2 , and d 3 represent unit vectors which point from the origin O to the three zones.The angles α 1 and β 1 define the position of P 1 .For details see text.

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I G U R E 1 1 Schematic illustrations explaining the evaluation of small misorientations from Kikuchi line diffraction patterns.(a) Fixed microscope coordinate system (MCS) with the axis x, y and z and the corresponding unit vectors e x , e y , and e z .Center of the transmitted electron beam: O. (b)The two angles α and β which are associated with one particular crystallographic zone P in the MCS.(c) Schematic drawing illustrating the rotation axis RA, its unit vector u, the sense of rotation (+: right handed, À: left handed) and the rotation angle φ.
vector b = [a -(aÁu) u]/jja -(aÁu) ujj normal to the rotation axis, see Figure 10c.A relation between the vector b and its rotated version b r = RÁb yields the sense of rotation (+: right handed, À: left handed) and the rotation angle ϕ.The sense of the rotation is then given by the sign of the product u Á (b Â b r ).The rotation angle can be calculated as ϕ = arccos[bÁb r ].
,b show, respectively, images form material states which were creep deformed to 1 and 2% strain, T A B L E 1 STEM CBED results obtained for the present work.Misorientations between the ten γ/γ 0 -neighbors highlighted in Figure 8a,b.

Figure
Figure γ-Channel location γ 0 -particle location Rotation axis u Rotation angle φ in Sense of rotation Figure13d,e using, respectively, red and blue colors.In comparison to the PBMÀ k distribution shown in Figure13d, the PBM À ⊥ data measured after 1% creep strain exhibit a systematic shift towards higher PBM values.A similar shift of the PBM À ⊥ distribution is not present in Figure13eshowing data collected after 2% strain.This may suggest that, after 1% creep strain, the γ/γ 0 microstructure still reflects a preferential filling of the ⊥ γ-channels by dislocations, which may cause systematically higher PBM values measured cross the ⊥ γ-channels interfaces.Such imbalance is no longer present after 2% strain (Figure13e) where an evolution of dislocation structure and related internal stress fields(Probst-Hein et al., 1999) cancels

Figure 14
Figure 14 shows the KAM results which were obtained in the present work.Figure 14a,b look similar, but it must be kept in mind that the two figures use different color coding.In fact, in Figure 14a, determined after 1% creep deformation, KAM values are significantly lower than after 2% creep deformation in Figure 14b.Kernel average misorientations increase with increasing creep strain.It is clear from Figure 14a,b that γ-channels show higher KAM values than γ 0particles.This was quantitatively evaluated in Figure 14c, where the cumulative frequency of KAM values is shown for the two phases (γchannels: red, γ 0 -particles: blue).And the two creep experiments (1% Figure15.The RVB-EBSD technique allows to resolve these deviations using a relative pole figure color coding(Thome et al., 2019).As can be seen in Figure15, the small angular spreads increase in intensity with increasing creep strain.The results presented in Figure15a,b have shown how one can obtain good chemical contrast between the γand the γ 0phase in the single crystal superalloy CMSX-10.The results obtained in the present work show, that the approach proposed by Nolze et al. (2017) also works for CMSX-4, even though it exhibits a less pronounced difference in chemical contrast between the two phases F I G U R E 1 5 Evaluation of orientation spreads (OS).(a) Color coded orientations after 1% creep.(b) Color coded orientations after 2% creep.(c) Cumulative frequencies of orientations in both phases after the two creep experiments.
03 , which Thome et al. (2019) had estimated by simulating pattern shifts on the EBSD detector.The fact that the two resolutions obtained applying the method of Kamaya (2011) are different for the two creep strains is related to different Kikuchi pattern image qualities, associated with different dislocation densities.Higher dislocation densities after 2% deformation allow lower angular resolution (0.03 ) than lower dislocation densities after 1% deformation (0.01 ).Both resolutions are, however, significantly better than angular resolution which can be reached using conventional Hough based EBSD indexation (0.5 -1 ).

Figure 1 )
Figure1) are rationalized by phase boundary misorientation (PBM) angles, Figure13.After 2% creep strain SEM and STEM data (average values) yield misorientations of ≈ 0.25 .Second, the filling of γ-channels with dislocations (arrow 1 in Figure1) is captured by the distribution of Kernel average misorientation angles, Figure14.The average KAM angles associated with an increasing γ-channel dislocation density increase from 0.02 after 1% to 0.06 after 2% strain (at 20 nm step size).In view of previous diffraction contrast STEM work, these two results are not unexpected.While the SEM method does not allow to resolve individual dislocations, it allows to document misorientations with much better statistics.While TEM studies allow to provide precise local data and to study individual dislocation parameters, they typically do not provide orientation information from more than 10 γ/γ 0 -phase boundaries(Figures 1, 9, and 12 and Figure 15.The images shown in Figure 15a,b allow to distinguish groups of particles with similar orientations.It is important to highlight that this finding contributes to the magnitude of the orientation spreads, presented in the pole figures in Figure 15c.Most importantly, the characteristic length scale associated with these equioriented particle/channel-groups (≈ 5 μm) is significantly larger/smaller than characteristic γ/γ 0 -dimensions (≈ 0.5 μm) / distances associated with dendritic mosaicity (≈ 200 μm).The maximum deviation angles associated with this phenomenon (≤1 and ≤2 for 1% and 2% creep strain,

2.
The average PBM angles of 0.244 obtained by the RVB-EBSD method is in excellent agreement with the average value of 0.265 from CBED STEM.The location of phase boundary locations was identified using a virtual dark field imaging technique, relying on the difference in average Z values of the two phases which affects the Kikuchi pattern background intensities.3.The color-coded presentation of KAM angles (20 nm step size: 1% creep strain: 0.02 , 2% creep strain: 0.06 ) allows to appreciate the local plastic deformation and the associated increase of dislocation densities in the γ-channels during creep.This finding is in good qualitative agreement with results from diffraction contrast STEM work.4. Color coded OS mappings revealed a new phenomenon, which has not been reported so far.It was found that groups of several γ-particles, clustering over distances which are significantly smaller than spacings associated with dendritic crystal mosaicity (<200 μm) and significantly larger than individual γ/γ 0 -dimensions (>0.5 μm) vary in orientation with respect to each other.

Table 1 )
, often only one or two situations are analyzed.In contrast, as can be seen in Figures13 and 14, EBSD studies not only allow to study significantly more γ/γ 0 -neighborhoods, but provides many more