Application of aberration‐corrected scanning transmission electron microscopy in conjunction with valence electron energy loss spectroscopy for the nanoscale mapping of the elastic properties of Al–Li–Cu alloys

The stress and strain play an important role in strengthening of the precipitation‐hardened Aluminum (Al) alloys. Despite the determination of relationship between the mechanical properties and the precipitation existing in the microstructure of these alloys, a quantitative analysis of the local stress and the strain fields at the hardening‐precipitates level has been seldom reported. In this paper, the microstructure of a T8 temper AA2195 Al alloy is investigated using aberration corrected scanning transmission electron microscopy (AC‐STEM). The strain fields in Al matrix in the vicinity of observed precipitates, namely T1 and β′, are determined using geometric phase analysis (GPA). Young's modulus (Ym) mapping of the corresponding areas is determined from the valence electron energy loss spectroscopy (VEELS) measured bulk Plasmon energy (Ep) of the alloys. The GPA‐determined strains were then combined with VEELS‐determined Ym under the linear theory of elasticity to give rise the local stresses in the alloy. The obtained results show that the local stresses in Al matrix having no precipitates were in the range of 138 ± 2 MPa. Whereas, in the vicinity of thin and thick T1 platelet shape precipitates, the stresses were found to be about 202 ± 3 MPa and 195 ±3 MPa, respectively. The stresses measured in the vicinity of β′ spherical shape precipitates found out to be 140 ± 3 MPa which was near to the local stress value in Al matrix. Our findings suggest that the precipitate hardening in T8 temper AA2195 Al alloy predominantly stems from thin T1 precipitates.

Investigations of the relationships between the precipitation existing in the microstructure of the alloys and their mechanical properties have been conducted, and many important inroads have been made already (Bai, Liu, Hou, Zhao, & Xing, 2012;Khushaim, Boll, Seibert, Haider, & Al-Kassab, 2015;Kim et al., 2016;Liu & Robertson, 2011).
The local stresses under the elastic or Hook's Law approximation are below the yield strengths. However, before this work, even in the limit of the linear theory of elasticity, no study has been conducted to experimentally quantify the local stresses or strain fields around the nano-precipitate features in such alloy systems. It is because the strain fields as well as the elastic moduli surrounding the precipitates in the Al-matrix are required first to be measured experimentally in the direct manner. Only then, the local stresses can be obtained using the relationship between the stress (σ) and the strain (ε) in which (σ = Y m ε), where Y m is the Young's modulus (Pelleg, 2013). A significant number of efforts have been attempted on measuring the mechanical properties of nanostructured materials in the range of nanometer scale dimensions. Incidentally, some methods can be found from literature in this regard (Howe & Oleshko, 2004;Hÿtch, Snoeck, & Kilaas, 1998;Li, Zhao, Xing, Su, & Cheng, 2013;Oleshko, Murayama, & Howe, 2002;Rouvière, 2011). However, none of the studies showed a method allowing the direct determination of local stresses from experimentally measured strains and Y m . It was the case because spatially resolved determination of Y m in the range of nanometer scales was not attempted, even though strains were determined successfully in the same length scales. Therefore, for the first time, we attempted to determine the local stresses in Al-alloys by combining the experimentally measured strains and Y m in the same length scales. As step one, we measured strain fields in the precipitates containing Al-matrix regions. This was accomplished by applying a reliable and direct method of determining the atomic displacements based on a powerful technique of high-resolution transmission electron microscopy (HRTEM) (Hÿtch et al., 1998;Hÿtch & Plamann, 2001;. The method that allows carrying out this exercise of determining the strain fields at nanometer scales is called geometric phase analysis (GPA). It is a robust and straight forward technique that can be directly used to provide quantitative measurements of strain fields from HRTEM images of crystalline materials. Its algorithm is based on the implication of atomic resolution on the Fourier space technique (Hÿtch et al., 1998). A detailed explanation about this method can be found elsewhere (Rouvière et al., 2011). Measurements of strain fields with the GPA technique yield 0.003 nm sensitivity (Hÿtch, Putaux, & Pénisson, 2003) and has been applied to a wide variety of materials such as quantum dots (Sarigiannidou, Monroy, Daudin, Rouvière, & Andreev, 2005), nano-wires (Taraci et al., 2005), and low angle grain boundaries present in metallic materials . For the case of metals, GPA has been used to measure the displacement of the edge dislocation in Al and gold (Au) (Zhao, Xing, Bai, Hou, & Dai, 2008;Zhao, Xing, Zhou, & Bai, 2008). Particularly for Al-based alloys, GPA has demonstrated its ability to measure the strain field around the Guinier-Preston (GP) zone in Al-Zn-Cu-Mg alloy's series (Bai et al., 2012). Moreover, the local transient strain induced deformation in Al-based alloys during in-situ transmission electron microscopy (TEM) was mapped by applying the GPA method (Gammer et al., 2016). Despite numerous advantages, GPA method has few drawbacks or limitations as well such as the thickness/focus variations can affect the image contrast from one region to another and hence the overall accuracy in the generated strain maps may get compromised (Zhu, Ophus, Ciston, & Wang, 2013). That is why application of GPA method has a limited success for the case of a material whose lattice parameter is very different from that of the available reference lattice (Sanchez, 2015).
As mentioned above, despite measuring the strain fields around the precipitates in nanometer range, the engineering strain-stress curve cannot be completed without the measurements of the Y m in Al matrix around the precipitates at the same spatial resolution. Therefore, in second step, we utilized valence electron energy loss spectroscopy (VEELS) based method for the first time that allows nanoscale mapping Y m of same regions in crystalline metallic materials from strains are determined, for instance, using GPA method. This method relies on the fact that the bulk plasmon energy (E p ) in electron energy loss spectra (EELS) of metallic alloys can be related to their Y m . Moreover, the Y m maps can been generated by processing the EELS spectra if the EELS datasets had been acquired in the synchronized manner along with the usual dark field scanning transmission electron microscopy (DF-STEM) imaging. It is pertinent to note herein that determining Y m using VEELS in the TEM mode of microscope has been developed and reported already (Howe & Oleshko, 2004). However, generation of spatially resolved Y m maps was not possible to in the TEM mode. It is because the Y m mapping requires pixel-by-pixel synchronization of the scanning electron beam with the acquisition of VEELS spectra in the acquired so-called spectrum-image datasets.
Hence, this type of datasets can be realized only by performing the VEELS experiments in DF-STEM mode of the microscope. However, a strong scaling correlation between Plasmon energy (E p ) and different elastic properties of various materials had been found already by using TEM-VEELS experiment (Howe & Oleshko, 2004). In another study, the use of E p to determine the elastic modulus and microhardness of different materials recently been developed (Gilman, 1999). It is to be noted that the volume E p varies with the valance electron density (n) of a material due to single electron excitation, and hence different relationship equations relating E p to the bulk modulus (B m ), Y m and shear modulus (G m ) have been developed (Monthioux, Soutric, & Serin, 1997;. In the third and last step performed in this study, the local stresses below the yield-strength of AA2195 alloy were calculated by combining GPA and STEM-VEELS determined strain fields and Y m , respectively. In the present work, the nanoscale mapping of mechanical properties, that is, local stresses below yields strengths of T8 temper Alalloy is performed experimentally using the techniques of aberration corrected scanning transmission electron microscopy (AC-STEM) and VEELS. These Al-alloys were included in the presented study because they are excellent materials for aerospace applications due to their good characteristics of high strength, low density, good corrosion resistance, and fracture toughness. In the response to the demands of various industries for light structural materials and in order to improve the fuel efficiency for aircraft and aerospace materials, different structural designs for light materials have been developed (Jeswiet et al., 2008). An example of these light materials is a commonly used Al alloy from 2000 series, which is named as AA2195. This AA2195 alloy material has received considerable attention from both industrial and scientific communities (Pickens, Langan, & Kramer, 1989;Sanders, 1996). It is an attractive choice for direct and engineered substitution in all weight-critical applications. It has been shown that AA2195 alloy conducting T8 tempering (stretched and artificial ageing conditions) has a combination of excellent fracture toughness (60 MPa ffiffiffiffi m p ) and improve strength (~550 MPa) (Wang & Shenoy, 1998) with the Y m value of (~69 GPa) (Hertzberg, 1989).
However, an exact and experimental determination of the strain field, the Y m , and hence local stress around observed nanoprecipitates in this alloy system has yet to be achieved.
In the context of above-mentioned applications, the objective of this study is (a) to evaluate the strain fields via the atomic displacements around each type of hardening precipitate in the microstructure of AA2195. It has been accomplished by the acquisition of AC-STEM images followed by the GPA application to the acquired images.
(b) Mapping of the Y m around each type of hardening precipitate within the microstructure of AA2195 alloy. This task has been completed by extracting the E p from acquired STEM-VEELS spectrum-image datasets.
(c) In the end, a direct evaluation of local stresses of multi-phases AA2195 alloy with having spatial resolution in the nanometers and stresses resolutions on the order of a few MPa was obtained. Quantification of stress and strain fields at nanoscale truly facilitates the design and hence the improvement of such structural materials.

| MATERIALS AND METHODS
AC-STEM is a method that has been used in this study to characterize, measure, map, and estimate the strain field and Y m in the vicinity of precipitates and Al-matrix of AA2915 alloy. As mentioned earlier, a commercially available alloy AA2195 in T8 temper has been used as a case study. The nominal chemical composition of this alloy is as fol- slow scan camera of model US1000 from Gatan, Inc.
In order to realize strain maps, GPA was applied to several high res- ) and the component of the displacement field (u(r)). This relationship can be expressed by (P g (r) = − 2πgu (r)), where g is a reciprocal lattice vector (Hÿtch et al., 1998). (f) calculating the displacement field using the relationship for two independent phase images P g1 (r) and P g2 (r). In this case, the two-dimensional displacement field can be calculated by Hÿtch et al. (1998): where g x , g y are two parameters of g in the reciprocal space, and u x , u y are atomic displacement fields in normal coordinate. (g) obtaining the strain field by differentiating the displacement field as the following (Hÿtch et al., 1998): These steps for applying GPA to obtain the maps of strain fields were achieved using a GPA Software Package (or plugin) from HREM Research, Inc. particularly developed for Gatan's digital micrograph environment of GMS3.2 version.
The value of Y m in AA2195 alloy was determined from its E p by using the following relationship between E p and Y m (Howe & Oleshko, 2004): It is to be noted that the generation of Y m maps requires the application of Equation (3) in the following way.
The stresses (σ) fields or values now can be calculated theoretically multiplying the Equation (2) with Equation (3) under the linear theory of elasticity and is written in the following equation.
The actual form of Equation (5) is tensor form in which both σ and ε vectors and Y m is a matrix. Using the GPA technique, it is possible to generate ε fields in vector form, that is, in perpendicular and parallel to the specimen plane. However, the STEM-VEELS technique yields in the isotropic (or scalar) form is insensitive to orientation because of the plasmons are isotropic. Consequently, the application of Equation (5) and STEM-VEELS datasets will give rise the stresses (and then strains) in vector form only and this is what is performed in this study.

| RESULTS AND DISCUSSION
The microstructure of the specimen after conducting T8 temper was found to be enriched with platelets precipitates and spherical precipitates, as shown in Figure 1. It contains AC-STEM images depicting the respective microstructure of those precipitates.
According to the annular bright-field STEM (ABF-STEM) image in Figure 1a and HAADF-STEM image in Figure 1b, the distribution of precipitates was present throughout the Al matrix. Although, rod-like precipitates were particularly found to be accumulating more in the grain boundary regions. Furthermore, the presented images have triple grain-boundary junction formed by grains labeled as "1," "2," and proposed for the T 1 platelet precipitates is found to have an extremely low lattice mismatch with the {111} Al atomic planes (Howe, Lee, & Vasudevan, 1988). The crystal structure of the β 0 phase is the L1 2 structure. The β 0 phase usually appears to be fully coherent with the Al matrix because its lattice parameter a = 0.400 nm is close to that of Al matrix (a = 0.405 nm) (Starke Jr & Staley, 1996).
Several HR-STEM images of both platelet and spherical shape precipitates in Al-matrix were acquired to visualize their structure at atomic scales. The imaging conditions were selected in such a way that the acquired images exhibit optimized image and diffraction contrasts from precipitates as shown by the images presented in Figure 2.
The acquired HR-STEM image in Figure 2a contains both T 1 platelet and spherical β 0 precipitates and the atomic structure of the precipitates is clearly resolved in that image. In the same vein, the HR-STEM images of Figure 2b,c convey the same information but only on the atomic structure of a thin and a thick T 1 precipitates, respectively. The As mentioned earlier, GPA is an efficient method to measure the strain fields in materials from their HRTEM and/or HR-STEM images.
The recent studies show that the application of GPA to HR-STEM F I G U R E 1 Observation of microstructure evolution of AA2195 alloy using AC-STEM (a) An ABF-STEM image acquired showing bright-field contrast of precipitates along with the Al-matrix. The grain boundaries among the grain labeled as "1," "2," and "3" can be seen in the image. HR-TEM images. Therefore, the next step is to apply the GPA technique on the obtained HR-STEM images such as presented in Figure 2. GPA was applied to images where Al-matrix contained T 1 and β 0 precipitates. However, before carrying out the analysis, the procedure to apply this technique is demonstrated on a T 1 platelet and the results are shown in Figure 3. The HR-STEM image of the microstructure that contains a T 1 platelet precipitate and Al-matrix in which the FFT was calculated is shown in Figure 3a. Furthermore, the calculated FFT of the image is shown in Figure 3b. The geometric phase images were then calculated with respect to a reference selected in Al-matrix as shown by a box in Figure 3c. The resulting phase images from spot 1 and 2 are shown in Figure 3c,d, respectively. After obtaining the geometric phase images, the displacement fields u x , u y can be calculated according to Equation (1).
Finally, the strain fields can accordingly be numerically calculated in terms of the derivative of the displacement by using Equation (2) (Balluffi et al., 2005).

F I G U R E 3
Step-by-step application of GPA analysis on HR-STEM images for yielding the strain maps. precipitates due to a complete or coherent matching between the atomic layers of both matrix and precipitate. The β 0 phase with L1 2 structure usually appears to be fully coherent with the Al-matrix in perpendicular direction because its lattice parameter (a = 0.400 nm) is close to that of Al-matrix (a = 0.405 nm) and therefore leads to formation of coherent interfaces (Strake, 1983). Even though, the β 0 phase usually acts to control the grain size and the resistance to recrystallization (Strake, 1983), it also has a clear role in the strengthening mechanism of our respective alloy. That's mainly because of the role of the β 0 phase on providing heterogeneous nucleation sites for the T 1 phase (Lee, Lee, & Hiraga, 1998). As presented earlier that the strain  Figure 5b and Figure 5c, respectively, in which the edge dislocations in the core of T 1 platelet precipitate are marked by circles therein.
The next step involved the mapping of Y m for thin and thick T 1 platelet precipitates as well as for β 0 spherical precipitates in order to complete the task of evaluating the local stresses around these observed nano-precipitates below the yield strength of Al-matrix. The determination of specimen thickness around nano-features is required for determining a more accurate value of local stresses. However, the absolute thickness is impossible to determine using conventional optical methods. Therefore, we utilized low-loss STEM-EELS spectrum imaging to determine the absolute foil thickness (Iakoubovskii et al., 2008). After the determination of absolute foil thickness, E p can be extracted from the acquired STEM-EELS datasets as described in the experimental section. This can be the utilized to deduce the values of Y m of Al matrix around any respective precipitates, for example, T 1 , containing T 1 and β 0 precipitates as compared to the matrix region with no precipitates. The acquired VEELS spectra were also utilized to determine absolute thickness map of Al matrix containing T 1 platelets and β 0 precipitates and the obtained results are shown in Figure 6c. It can be noticed from the results presented in Figure 6c that even though the foil-thickness was in the range of 70 to 100 nm across the entire field-of-view, the thickness around T 1 and β 0 precipitates was also found out to be close to Al-matrix. By performing analysis of the EELS spectra for specific positions of Al-matrix, it was found that the average value of E p of the Al matrix is (14.95 ± 0.1) eV,in agreement with the typical literature values . In the same way, in the vicinity of T 1 precipitates, the average E p value of Al matrix is slightly lower for both cases of thin and thick T 1 platelets. In fact, these values found out to be E p = 14.8 ± 0.1 eV and E p = 14.6 ± 0.1 eV for thin and thick T 1 platelet, respectively. While around the regions of β 0 precipitates, the average value of E p for the Al matrix was 15.06 ± 0.1 eV. As mentioned earlier, an E p map was generated from acquired low-loss STEM-EELS datasets by applying NLLS routine. Afterwards, a procedure of generating the Y m map as per Equation (3)  tively. This finding is contrary to expected values and therefore can be interpreted as a higher Y m value occurs in Al matrix when T 1 platelets interfere with β 0 precipitates. These β 0 precipitates usually act as preferred nucleation sites for the T 1 precipitates (Itoh, Cui, & Kanno, 1996). Thus, it can be concluded from the result presented in Figure 6 that every precipitate does not necessarily results in the enhancement of Y m of Al matrix. By following the methodology described in Figure 6, we generated Ym map from a larger area of the alloy that contained both thin T1, thick T1, and β 0 precipitates.
The obtained results are shown in Figure 7 that gives an estimae on the statistical significance of determined Ym maps. The Ym results presented in Table 1 are compiled by taking the average of about 5 precipitates of each type.
In the end, to examine the strain distributions in Al-matrix quantitatively within vicinity of the observed precipitates and to assess the reliability of GPA, we performed the analysis of strain maps shown in represent small distortions correspond to a shear, and are on average zero (Hÿtch et al., 1998). Therefore, local stresses in Al matrix caused by each nano-precipitate feature can be estimated using the linear theory elasticity (σ = Y m ε) in a similar way done by Gammer et al. (2016).
This was possible to do now since the Y m is known experimentally from Figure 6. In this way, the determined values of local stress in Al matrix around thin, thick T 1 platelets, Al matrix alone, and β 0 precipitates were calculated, and the obtained results are shown in Table 1. It is important to note herein that the Y m determined in Figure 6 is a scalar quantity since the Plasmon energy is independent of crystal orientation. Therefore, the determined local strain and Y m values of Al-matrix can be directly compared with their bulk counterparts. Our finding of the local stress at the thin T 1 precipitates (  (Kim et al., 2016). In our study, the contribution of every observed precipitate to the local stress of Al matrix was estimated as shown in Table 1. Thus, the presented study demonstrates the ability of AC-STEM to measure the local stresses in Al matrix near each respective precipitate. This capability of AC-STEM to allow measuring local stress experimentally at the nanometer scales in Al alloys can be extended to other metallic alloys and hence will also allow determining their mechanical properties (Figure 7).

| CONCLUSIONS
Using AC-STEM, the strain fields and Y m of the hardening precipitates in Al alloy AA2195 at T8 temper were experimentally mapped at nanometer scale spatial resolutions. GPA is a powerful way of mapping strain fields in Al matrix in the vicinity of the T 1 and β 0 precipitates. It was found that the average strain around these precipitates along the

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.