TEM nano-Moiré evaluation for an invisible lattice structure near the grain interface.

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TEM nano-Moiré evaluation for invisible lattice structure near grain interface Received 00th January 20xx, Accepted 00th January 20xx DOI: 10.1039/x0xx00000x www.rsc.org/

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Hongye Zhang, Huihui Wen, Zhanwei Liu,* Qi Zhang, and Huimin Xie,*

Moiré technique is a powerful, important and effective tool for scientific research, from nano-scale to macro-scale, of which is essentially the interference between two or more periodic structures with similar frequency. In this study, an inverse transmission electron microscope (TEM) nano-Moiré method was proposed, for the first time, to reconstruct the invisible lattice structure near grain interface, where only one kind of lattice structure and Moiré fringe was visible in a high resolution TEM (HRTEM) image simultaneously. The inversion process was introduced in detail. Three rules were put forward to ensure the uniqueness of the inversion result. The HRTEM image of a top-coat/thermally grown oxide interface in a thermal barrier coatings (TBCs) structure was observed with coexisting visible lattice and Moiré fringes. Using the inverse TEM nano-Moiré method, the invisible lower layer lattice was inversed and the 3-dimensional structure near the interface was also reconstructed to some degree. The real strain field of oriented invisible and visible lattices and the relative strain field of Moiré fringe in the grain and near the grain boundary were obtained simultaneously through the subset geometric phase analysis method. The possible failure mechanism and position of the TBCs spallation from nanoscale to micro-scale were discussed.

1. Introduction Moiré patterns are common in our daily life and often occur when two periodic lattices with similar periodicities overlap with each other. The Moiré phenomenon is omnipresent also at the nano-scale1. The Moiré fringes have been shown to be a powerful tool for the generation of micro- and nano-scale patterns and supperlattices. Studies of various nano-materials and nano-devices are conducted using electron microscopy and Moiré method, such as atomic force microscopy (AFM)2-6, laser scanning confocal microscopy (LSCM)7, scanning electron microscopy (SEM)8-15, and scanning tunneling microscopy (STM)16-21. In most cases, these electron microscopy works in scanning mode. The observed Moiré can all be called scanning Moiré, where the reference lattice is the scanning line. Moiré patterns form between the interference of the surface structure of the tested sample and the scanning line. So the surface information is contained in the Moiré fringes. By using the scanning Moiré fringes, researchers can measure the structure of the butterfly wing7, evaluate mesoporous structures properties in a large field of view using the statistics-based scanning Moiré method9, and even characterize the nanoporous alumina structures in multiple domains using secondary Moiré patterns10. Apart from the techniques mentioned above, there is

another electron microscopy, transmission electron microscopy (TEM), which is popular and widely used for the characterization of materials in nano-scale. In many research articles, Moiré patterns were captured and discussed in TEM 22-36 images . The latest TEM can also operate in scanning mode, of which the machine is called scanning transmission electron microscopy (STEM). STEM Moiré patterns were also observed 37-40 and used for characterization of material property . Here, the STEM Moiré will not be discussed in detail, as well as the TEM Moiré formed by the interference between the scanning 41 lines in monitor screen and the specimen lattice . In the conventional TEM images, Moiré patterns will appear simultaneously with the material lattice in a proper condition. It is the lattice in different layers of the tested sample in 23 thickness direction overlap with each other. Jin S studied the vertical heterostructures of layered metal chalcogenides, where beautiful TEM Moiré patterns were observed containing both lattice fringes and Moiré fringes. The angle offset between two stacked layers in the heterostructures was calculated using fast Fourier transform (FFT) filtering and 1 Rayleigh relation . TEM Moiré patterns were always found in 27 42 graphene study , bilayer grapheme, trilayer graphene , and 43 42 even multilayer graphene . Park J and his group members qualitatively analyzed the local strain using the formed elaborate TEM Moiré patterns. The Moiré patterns indicate continuous lateral displacement between the top and bottom layers. Taking from different diffraction peaks and excluding the peak diffracted by the other rotated layers, three unidirectional Moiré patterns were obtained. It is concluded that the curvature of the Moiré lines indicating both pure-

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strain and shear-strain elements are present in the graphene 42 43 sample . Singh MK and his colleagues observed rotational misorientation within the flake in the high resolution TEM (HRTEM) images of several layers of graphene sheets benefit from Moiré patterns. The grapheme lattice of each sheet was reconstructed and the relative rotation between consecutive graphene layers up to six separate sheets was determined by filtering in the frequency domain using FFT. Coincidentally, 24 Sakurai S, et al studied the cylindrical microdomains of block copolymers using FFT and inverse FFT methods. In numerous literatures, TEM Moiré patterns were used for qualitative analysis of materials properties instead of quantitative analysis. However, quantitative analysis in nanoscale measurement is of great importance in understanding the materials’ properties more detail. The aforementioned researches about TEM Moiré were all related to Moiré and almost all those articles did not have any quantitative measurement, like displacement/strain or stress measurement. In a word, the TEM Moiré patterns are interference patterns resulting from the rotation between two different layers of any regular lattice or two crystals. The different lattices or crystals are in diverse layers, meaning that depth information is contained in the TEM Moiré images (TEM images). In fact, the 24, 42, different layers of lattice have been acquired in literature 43 to some degree. The samples are always fabricated using 42 43 deposition method or colloidal suspensions method . It is relatively easy to control the rotation and thickness of different lattice layer. So in most cases, the structure or relative rotation between lattice layers are known beforehand. Owning to the fabrication method, the tested samples are not always come from a real and macroscopical structure or material. However, in macroscopical structure, nonuniform strain or stress concentration may exist even in a nano-scale area. Quantitative analysis of this strain/stress is crucial for the failure initiation study and structure reliability research, even other material properties. When a TEM sample was fabricated and tested, intriguing and perfect Moiré patterns as in the aforementioned literatures will not be observed easily. The distribution of the lattice and/or crystal in a real structure with various elements is much more complicated. Usually, TEM Moiré patterns occur with only one specific lattice visible and the other lattices in disparate directions will not arise, like the Moiré patterns in the TEM study of thermal barrier coatings 34, 35 (TBCs) . We think there are two reasons that only one direction lattice and Moiré existing in a relatively complex TEM sample. One is that the zone axis of one of the two overlap crystal deviated from the appropriate position a bit more and the lattice cannot appear. The other is that there do have lattices in different directions existing. The information in a specific direction is strong and the other directions are too weak to appear meaning that the thickness of the crystals in these directions is dramatically thin. The weak lattices maybe appear in some local area in the TEM image. In this condition, 24 the FFT method in literature will not work, since there are not enough sets up of diffraction points in the power spectrum. 24 The FFT method can only perform in a perfect Moiré pattern, where the Moiré patterns and lattices in all directions coexist

together. So advanced method need to be developed to solve this issue, together with the invisible lattice reconstruction in the only one direction lattice and Moiré existing condition. Besides, the peculiar TBCs, yttria partially stabilized zirconia (YSZ) coatings are widely used on the turbine blades to protect the aero-engine at high temperature. However, the failure mechanism of thermal barrier coatings (TBCs) spallation is not clear understood, especially the initiation of crack nucleation or failure propagation in nano-scale. Although in micro-scale, the morphology, strain/stress distribution and the failure behavior of TBCs has been widely studied by many 35, 44-50 scholars using various methods , it is a bit far from drastically explaining the failure mechanism of the TBCs failure behavior. The failure behavior research in nano-scale is necessary and significant, especially the nano-interface in the TBCs structure. In this study, a cross-sectional TBCs specimen containing grain interface was successfully fabricated. Moiré pattern was directly observed near the interface. Inspired by the Moiré method and inversion method, an inverse TEM Moiré method, for the first time, is proposed and applied to the characterization of TEM Moiré patterns with only one direction lattice visible. The inversion process was introduced in detail. The invisible lattice near the top-coat/thermally grown oxide (TC/TGO) interface in TBCs was inversed and reconstructed. The uniqueness of the inversion result was systematically explained. The strain distribution in the principal direction of the inversed invisible lattice, visible lattice, and Moiré fringe was calculated using the subset geometric phase analysis (S-GPA) method. The possible failure mechanism of the TBCs spallation from nano-scale to microscale and the feasibility of the inverse TEM nano-Moiré method was discussed and prospected.

2. Experiment and proposed method 2.1 Sample preparations and experimental procedure The base material used in this investigation was a commercial Ni-based super alloy. Firstly, the substrate specimen's surface was blasted to improve the roughness. Then the TBCs were air plasma sprayed onto the Ni-based substrate. The bond coat and top coating consisted of 45.5% Ni, 23% Co, 25% Cr, 6% Al, 0.5% Y and 8 wt.% YSZ powder, respectively. The spraying process was performed at the Institute of Process of Chinese Academy of Sciences in Beijing. The produced TBCs samples were cyclic heated in a resistance furnace (type: KSL-1200X-J, HEFEI KEJING MATERIALS TECHNOLOGY CO., LTD). The heating process was in three steps: 0.5h heating up to 1150 ˚C, 10h thermal insulation at 1150 ˚C, natural cooling to room temperature. After 6 times cyclic oxidation, a TGO layer was observed using a SEM (type: FEI Quanta FEG 450) and a super-depth microscope (type: KEYENCE VHX-500FE), with the TGO thickness about 1~3um. A cross-sectional TEM specimen was prepared successfully. Thin slices (approx. 1 mm in thickness) were cut from the TBCs sample and glued with the ZrO2 TC

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face to face of each two slices. Afterwards, the glued sample was mechanically thinned as thin as possible (30~40 um) and carefully glued on a 3mm diameter Cu holder. Then the sample was thinned in an ion milling Gatan 691 using argon ions with an initial voltage of 8 kV and incident angle of 8˚ until cavity arising. Finally, the sample was further ion-milled at 3 kV with an incident angle of 3˚ for 30 min to reduce the thickness around the cavity. The specimen was examined using a FEI Tecnai Remote (Tecnai G2 F20) transmission electron microscope operating at 200kV employing a combination of bright field (BF), high-resolution electron microscopy (HREM) as well as selected area electron diffraction (SAED) pattern analysis. Energy dispersive spectrometer (EDS) microanalysis was also performed in this TEM. 2.2 Subset geometric phase analysis The geometric phase analysis method was independently 51 52, 53 introduced by Takeda and Hÿtch , which is successfully applied in the displacement/strain field analysis of crystal 54 structures in nano-scale. Recently, Liu improved this method, where the windowed Fourier transform was added in the transforming process. The improved method was named as SGPA. Details concerning the theoretical discussions of GPA 55 method can be found in previous literature and the core 54 theoretical formula in S-GPA is as follows. The 2-dimensional windowed Fourier transform (2D-WFT) can be expressed as: Q(µ ,υ , ξ ,η ) = ∫

∞

∫

∞

−∞ −∞

q(x,y)g(x − µ ,y − υ )e{ − iξ x − iη y}dxdy

(1) where Q(µ ,υ ,ξ ,η) represents the windowed Fourier spectrum; ξ and η represent the frequency components in the x and y directions, respectively; g is a window function representing the reciprocal lattice vector of the lattice; ( µ ,υ ) is the coordinate of the center of the window and the target window changes with different pairs of µ and υ . The 2D inverse WFT can be performed as: 1 ∞ ∞ ∞ ∞ q$ (x,y) = 2 ∫ ∫ ∫ ∫ Q(µ ,υ ,ξ ,η )g(x − µ ,y − υ ) × e{iξ x + iη y}dξ dηd µ dυ 4π −∞ −∞ −∞ −∞

(2) with Q(µ ,υ ,ξ ,η ), Q(µ ,υ ,ξ ,η ) / max( Q(µ ,υ ,ξ ,η ) ) ≥ Thr Q(µ ,υ ,ξ ,η ) =  , Q(µ ,υ ,ξ ,η ) / max( Q(µ ,υ ,ξ ,η ) ) < Thr 0

(3) where Q(µ ,υ ,ξ ,η) represents the filtered frequency spectrum; Q(µ ,υ ,ξ ,η) represents the power of Q(µ ,υ ,ξ ,η ) ; the mostly used Gaussian window function g is divided by πσ xσ y for normalization; Thr is the preset threshold. The wrapped phase can be obtained through the simple arctangent function of q$ (x,y) . Using phase unwrapping technique, the original phase field and the phase difference can be obtained. The displacement/strain can be calculated through the following matrix form equation.

 ∂u(x)

 ε xx ε xy   ∂x  =  ε yx ε yy   ∂u(y)   ∂x

∂u(x)  g ∂y   = − 1  1x ∂u(y)  2π  g2 x ∂y 

 ∂Pg1 −1 g1 y   ∂x   g2 y   ∂Pg2   ∂x

∂Pg1   ∂y  ∂Pg2   ∂y 

(4) where the subscripts x and y represent the x and y directions, u(x) and u(y) are the displacements in the x and y directions, Pg is the phase in the image, ε xx and ε yy are the direct strains, ε xy and ε yx are the shear strains, respectively. 2.3 Principle of the proposed inverse TEM nano-Moiré method Generally, Moiré pattern originates from the interference between two gratings, called as reference grating and specimen grating separately (see Fig. 1). It is widely accepted that the Moiré pattern observed in HRTEM is the overlap between two crystals (atoms in two adjacent layers) with almost exactly equal lattice parameters. For convenience, lattice will be substitute for grating, crystal or atom here. In Moiré method, the overlapped two sets of lattices are always called as reference lattice and specimen lattice. The light intensity of reference lattice can be written as 2π Ir ( x , y) = C r 0 + C r 1 cos y pr (5) in which C r 0 and C r 1 are constants related to the mean background intensity and modulation amplitude, respectively, and pr is the pitch of the reference lattice. 56 Here, the fringe multiplication phenomenon is not considered. It is considered that the primary frequencies of the reference and specimen lattices are matched, with tiny difference. The intensity distribution of the periodic specimen lattice can be expressed as 2π Is (x , y) = C s 0 + C s1 cos y ps (6) where C s 0 is the mean background intensity, C s1 is the modulation amplitude, and ps is the pitch of the specimen lattice. Since the frequency of the two lattices is assumed similar, clear Moiré patterns will be formed by the superposition of the two lattices. The intensity distribution function can be calculated by 2π 2π I = Ir − Is = C r 0 − C s 0 + C r 1 cos y − C s1 cos y pr ps (7) In Eq. (7), the information of reference lattice and specimen lattice is clear and evident, but the Moiré information is implicit. Fortunately, in the traditional Moiré method, there is a classical relation among the reference spacing, specimen spacing and Moiré distance. 1 1 1 = ± D pr ps (8) herein, D stands for the Moiré spacing. Substituting Eq. (8) into Eq. (7), we get

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(9) in which C0 is constant related to the mean background intensity; C1 and C2 are constants related to the modulation amplitude. p is the pitch of a specific lattice in one direction. Actually, pr and ps are all defined artificially, which can change sides with each other. In other words, pr is the relatively uniform lattice, which is the visible lattice in this study, and ps is the lattice with deformation (to be calculated), which is the invisible lattice in this study. So, p in Eq. (9) is the reference lattice here. It is obvious from Eq. (9) that the information of Moiré and a specific lattice is explicit and the other lattice is implicit. In a certain position, the term in the cosine function represents the corresponding phase value. The full field phase distribution of the lattice or the Moiré fringe can be obtained from the captured image using Fourier analysis.

Fig. 1 Diagram of description of Moiré order equation, where mr, ms, and m are the orders of reference lattice, specimen lattice and Moiré fringe, respectively.

Usually, the Moiré fringes appear simultaneously with a sort of lattice lines in a HRTEM image. Other lattice lines with different orientation are not visible. Such a case accords well with Eq. (9). Here, the visible lattice and Moiré are defined as reference lattice and Moiré fringe, respectively. And the invisible lattice is named as specimen lattice. In Eq. (9), the parameters of one kind of lattice pitch and Moiré spacing are clear and acceptable, so do these parameters in a HRTEM image. Afterwards, fringe orders can be determined one by one. The basis of seeking the specimen lattice distribution through inverse TEM nano-Moiré method is the Moiré order equation m=mr±ms, which can be transformed to the following form: ms=mr±m (10) where mr, ms, and m are the orders of reference lattice, specimen lattice, and Moiré fringe, respectively. As shown in Fig. 1, an image coordinate system is set up corresponding to the MATLAB software. It is noticed that ms has two expressions through Eq. (10) standing for two specimen lattice formations. The uniqueness will be discussed in section 3.

Fig. 2 Illustration of inverse TEM nano-Moiré method, (a) HRTEM image, (b) wrapped phase of the reference lattice in the dotted square in (a), where pr stands for the pitch of reference lattice, (c) the phase distribution of the pink dotted line in (b), (d) and (e) ordered reference lattice and Moiré fringe, respectively, (f) discrete points of ordered specimen lattice, (g) inversed distribution of a specimen lattice from interpolating the discrete points in (f), with doubled frequency representing the phase jump lines and the centerlines of the lattice alternately, (h) and (i) reconstructed specimen lattice representing the two specimen lattices in Eq. (10). (f) to (h) stands for a complete inversion process in a specific direction (representing one case of Eq. (10)). For convenience, the discrete points and inversed distribution of the specimen lattice corresponding to (i) is not shown here.

As shown in Fig. 2(a), a HRTEM image simultaneously containing Moiré and single lattice was firstly observed. Then Fourier analysis was applied to the image, which is different to 7 the literature ， where the fringe-centerlines method was used for inversion. Fourier analysis consists of computing the Fourier transform of the HRTEM image, extracting the fundamental frequency, and obtaining the phase from an inverse Fourier transform. The specific position of Moiré was obtained using the fringe-centerlines method and fitting each 7 Moiré fringe using a polynomial . And the reference lattice was always known, as the scanning parameter was set before operating an experiment. Here, the exact orders of the Moiré and reference lattice were calculated through the wrapped phase acquired from Fourier analysis. Fig. 2(b) is the wrapped phase field of the reference lattice in the dotted square in Fig. 2(a). The distance of each two adjacent phase jump line represents the spacing of reference lattice or Moiré fringe. It can be seen from Fig. 2(c), a typical line in Fig. 2(b), that there are alternating sign jump between positive and negative values. As the dotted black line in Fig. 2(c), one is the phase jump point, and the other stands for the centerline of a lattice. By searching the sign jump in the wrapped phase field, the exact location of reference lattice with doubled frequency than that 7 in literature and Moiré fringe can be obtained. Owning to the variety of the slopes of the lattice and Moiré, the searching process was performed in the two directions simultaneously, as the two yellow arrows signed in Fig. 2(b). Using an 857 connected objects strategy, the orders of the lattice and Moiré will be obtained, as shown in Fig. 2(d) and (e), where the colored lines represent different fringe orders,

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I = C 0 + C1 cos(

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continuously. Then the discrete points of ordered specimen lattice can be obtained through Eq. (10), as shown in Fig. 2(f). The complete ordered specimen lattice can be acquired by extracting the discrete points with same value and interpolation (see Fig. 2(g)). Generally, the distribution of a lattice is always sinusoidal. The centerline and boundary of a single lattice can be obtained through the sign jump searching process. Then the distribution of specimen lattice can be obtained using interpolation strategy, as the results shown in Fig. 2(h) and (i). The interpolation used here is spline interpolation. As the absolute value of fringe orders did not affect the shape of reference lattice, specimen lattice and Moiré fringe, the fringe orders were all set as 1, 2, 3..., n serially in the calculating process. Compared with the fringecenterlines method7, the ordered lines in this inverse TEM nano-Moiré method has been doubled. Hence, the inversion accuracy would be at least two times higher than the fringecenterlines method. Using the self-compiled MATLAB code, the inversion process can automatically operate, without too much manual intervention. The procedure for lattice reconstruction using inverse TEM nano-Moiré method is shown in Fig. 3.

wrapped phases of the reference lattice and Moiré fringe in the HRTEM image were extracted. Step 3. The sign jump searching strategy was conducted on the wrapped phases of the reference lattice and Moiré fringes. Then, the separate lines of the two kinds of fringes were obtained. Step 4. The separate lines were ordered using the 857 connected objects strategy. By using Eq. (10), the discrete points of ordered specimen lattice can be obtained. Step 5. Through spline interpolation, the completely ordered single lines and final distribution of specimen lattice can be obtained. It should be noted that there is an artful process in step 5. The initial obtained specimen lattice was a series of discrete points. There is a possible condition that a single point exists which stands for a specific order at the corner of the image. In this case, the slope of the interpolated line next to the single point was used to fit the ordered lattice line.

3. Results and Discussion After more than 60h’s oxidation at 1150˚C, the TBCs sample was cut, polished and observed. Common oxidized form of bond coat layer appeared accompanied with irregular distributed TGO in a thickness about 1~3um. The crosssections of TBCs observed using an optical microscope and a SEM was shown in Fig. 4(a) and 4(b), respectively. Then TEM sample, containing TC/TGO and TGO/bond-coat (BC) interface was carefully prepared, measured and analyzed, as shown in Fig. 4(c). The TC/TGO and TGO/BC interface was approximately determined and depicted after EDS analysis of various points. It is demonstrated by EDS that point 1 is enriched in Al and O (Fig. 4(d)), and point 2 is enriched in Zr, O and Y (Fig. 4(e)). It can be concluded that a TC/TGO interface exists between point 1 and point 2.

Fig. 3 The flow chart of inverse TEM nano-Moiré method.

The detailed steps are as follows: Step 1. A TEM sample was carefully prepared and the TEM experiment was highly concentrative operated. A HRTEM image was observed, with clear reference lattice and Moiré fringes. Step 2. Fourier analysis was applied on the captured image. Fourier transform of the image was computed. The fundamental frequency was extracted from the diffraction spectrum. Wrapped phase field was obtained through an inverse Fourier transform. Consecutively, the

Fig. 4 Microanalysis of TBC sample; cross-sections of TBCs observed using (a) optical microscopy, (b) SEM and (c) TEM; EDS of the TEM sample, (d) point 1 and (e) point 2 in (c).

In order to roughly identify the material and interface location, detailed EDS analysis was performed. Fig. 5 illustrates the element maps at the TC/TGO interface. Fig. 5(a) is a STEM

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image containing the interface, where the points 1 and 2 in Fig. 4(c) are included. The observation direction of Fig. 5(a) is the same as that of Fig. 4(c) or Fig. 6. The distribution of Zr (Fig. 5(b)) and Y (Fig. 5(g)) are almost the same, which is conformant to the chemical composition of YSZ powders used here. The white dotted curves in fig. 5(b) and 5(c) are considered to be the TC/TGO interface. It can be concluded that the main elements are Al and O in the dotted green area from Fig. 5. Considering that the distribution of Zr and Y is conformal with the chemical composition of YSZ powers, the position of the interface is accurately confirmed. Moreover, there are relatively few element Ni, Co, Cr and Y in the dotted green area, while Al is rich. This agrees with the expression that O and Al highly concentrate in the TGO layer after oxidation in previous literature34.

Fig. 5 STEM-EDS element maps of the TC/TGO interface.

Before the observation of a higher magnification, SAED analysis was performed, shown as the red circled area in Fig. 6. The map on the right is the electron diffraction pattern, which indicates that the main material in this area is c-ZrO2 and the lattice parameter in [200] direction is 0.265nm. Although the diffraction point of other composition is blurry, α-Al2O3 can still be found and the lattice parameter in [104] direction is about 0.255nm by calculating, which agrees with the PDF card PDF#88-0826. The white dotted square area is the same area in Fig. 5.

Fig. 6 BF image of TBCs sample with an SAED map.

TEM analysis in a higher magnification was conducted in the white dotted area of Fig. 6. The details are shown in Fig. 7. It is obvious that Moiré pattern has been formed in region III and there is also visible Moiré in region II, even though it is a bit obscure. So the image contrast was partly adjusted. Apparently, the directions of the Moiré patterns in the two areas are different. So, we can conclude that there must be an interface between the two areas. After an exhaustive analysis, it is confirmed that the visible lattice in region II and region III is α-Al2O3 in [104] direction and c-ZrO2 in [200] direction, separately. Hence, the interface in the green area is TC/TGO interface. The insert map is the FFT of the HRTEM image, where the c-ZrO2 and α-Al2O3 are shown directly. Table 1, table 2 and table 3 show the angles between the principal direction of lattice line and the horizontal direction with the lattice parameter (lattice spacing) of the three regions marked in Fig. 7. The principal direction of a lattice line is defined as the direction perpendicular to the lattice line. The results in the three tables are calculated using S-GPA method. The different columns in these tables stand for various square areas in the corresponding region. The exact value of the angle and spacing is an average effect of the region selected and calculated. ‘AV’ stands for average value in table 1 to 3. It is obvious that the values of the angle and lattice spacing change with different selected areas in each region, which means that there are deformations and stress in these areas. Particularly, both visible lattices in region I and region II are α-Al2O3, but the angle and spacing is a little different from each other. The angle is in a difference less than 1 degree and a few percent of the lattice parameter. The inner mismatch in the TBC structure is more likely to come from the high temperature oxidation and its evolution. Thermal growth stress usually exists in these regions. The mismatch of the lattice spacing may lead to dislocation even crack nucleation. Especially, the mismatch is more evident near the interface, which means that the deformation and stress here are larger.

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Fig. 7 HRTEM image of TC/TGO interface. Table. 1 The angle between the principal direction of lattice line and the horizontal direction with the lattice parameter (lattice spacing) of region I in Fig. 7 calculated using S-GPA.

I Angle (˚) Spacing(nm)

Calculated results with different reference areas

AV

99.874

100.177

101.172

100.554

99.488

100.272

100.386

100.65

100.141

100.354

100.307

0.2443

0.2529

0.2555

0.2564

0.2345

0.2582

0.2547

0.2533

0.2532

0.2432

0.2506

Table. 2 The angle between the principal direction of lattice line and the horizontal direction with the lattice parameter (lattice spacing) of region II in Fig. 7 calculated using S-GPA.

II Angle (˚) Spacing(nm)

101.146

101.27

Calculated results with different reference areas 100.746 100.88 101.727 100.927 101.18 101.28

101.507

100.793

AV 101.146

0.2519

0.2524

0.2520

0.2527

0.2522

0.2524

0.2519

0.2559

0.2517

0.2518

0.2523

Table. 3 The angle between the principal direction of lattice line and the horizontal direction with the lattice parameter (lattice spacing) of region III in Fig. 7 calculated using S-GPA.

III Angle (˚) Spacing(nm )

118.03 2 0.2663

118.3 7 0.266 5

Calculated results with different reference areas 118.76 117.40 117.82 118.06 117.77 118.27 5 5 2 8 5 2 0.2639 0.2629 0.2660 0.2662 0.2657 0.2694

By using the developed inverse TEM nano-Moiré method, the inversed sub layer invisible lattice was obtained. According to Eq. (10), there should be two different results for each pair of visible lattice and Moiré, as indicated in table 4. For convenience, the inversion results are written as ms=mr-m for minus value and mr+m for plus value, respectively. Coincidently, the inversed minus value in region II is very close to that of reference lattice in region III, and specifically, the principal direction is in a difference of about 1 degree and the lattice parameter differs about 0.1 angstrom. This implies that the lattice lines connect well with each other at a certain distance regardless of the spacing difference. So it is considered that the upper visible layer in region III and the lower invisible layer in region II connect with each other, meaning that the two grains partly contacting in 3-dimentional

117.67 7 0.2653

117.98 8 0.2656

AV 118.01 7 0.2658

(3D) space. Then dislocations may occur, because the mismatch will accumulate to be bigger and bigger as the connecting points prolonging. For the plus value of region II, there is not exactly the same spacing value but two similar spacing representing other crystal oritation in PDF#88-0826. Considering the continuity of the lattice line, the plus value is still excluded. There are blurry lattice lines in the upper left parts of region II, the principal direction of which is similar to that of the reference lattice in region III. It is considered that the minus value represents the invisible weak lattice in region II. So it is confirmed that the sub layer invisible lattice in region II is α-Al2O3 with lattice constant 0.2552nm. In PDF#49-1642 of c-ZrO2, there is no lattice constant close to 0.2164nm, which indicates that the plus value in region III is also excluded. For the minus value in region III, the lattice spacing 0.2972nm

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stands for the crystallographic direction families. It comes to a conclusion that the sub layer invisible lattice in region III is c-ZrO2. α-Al2O3 crystal and c-ZrO2 crystal are in region II and region III in the 3D space, respectively. Furthermore, it can be concluded that there are interfaces in the region II and region III, since the orientations of the upper layer and sub layer are different. Then a 3D interface is likely to be formed in the green area marked in Fig. 7 and Fig. 9, which is complicated and interesting. Table. 4 The inversed result of specimen lattice of region II and region III in Fig. 7.

--Angle(˚) Spacing(nm)

Region II mr-m mr+m 119.025 85.174 0.2552 0.2226

Region III mr-m mr+m 139.833 103.177 0.2972 0.2164

As shown in Fig. 8, the visible layer lattice in region III and the sub layer invisible lattice in region II are all displayed. The color of the overlap part of the two lattices is lighter. In a relatively large range, the lattice lines connect well with each other. And dislocations appear at the edge of the two crystals, as shown in the green dotted region A. It is also found that dislocation exactly appears in the Moiré fringe at the same location of the original HRTEM image marked as B with a specific rotation angle, which is a powerful and effective certification of the inversion method. There will be big strain and stress around a dislocation and crack nucleation is likely to develop at this place. There is reason to believe that the position of initial contact between different crystals are weak. Especially in the interface of TC/TGO, cracks may have a great possibility originate from the dislocation. Cracks may form, propagate, and finally led to spallation failure at these weak points near the interface.

Fig. 8 The inversion result of TC/TGO interface using TEM nano-Moiré method.

The positive and negative signs in the inversion formula are determined by rotating the specimen lattice or the combination judgment of the relationship between the included angle between Moiré fringe and the horizontal direction and the included angle between specimen lattice 7 lines and the horizontal direction . However, rotating specimen lattice or reference lattice is not practical here in a TEM sample. Three rules are put forward to ensure the uniqueness of the inversed lattice. Maybe it is not strict enough, but effective and reasonable to some degree. 1). Find out the PDF card and compare the lattice constant in different orientation. 2). Considering the continuity of materials with different element type, compare the inversed lattice with the local structure in the inversed area or make the analysis through the diffraction spectrum in a local area. If there is a diffraction point in the corresponding direction, the inversed result will be more credible. 3). Principle of proximity. Consider the rationality of the inversed lattice constant and angle. Since the principal direction of the inversed sub layer lattice in region II and the visible lattice in region III are similar with each other, it is considered that part of the upper layer lattice in region III and the sub layer lattice in region II are connected in the same plane. Based on the inversed result in table 4, a three layers (or more) 3D lattice structure is formed. The first layer is the visible α-Al2O3 lattice in region I and region II. The second layer is the inversed invisible α-Al2O3 lattice in region II and the visible c-ZrO2 lattice in region III. The third layer is the inversed invisible c-ZrO2 lattice in region III. Even though a TEM sample is very thin, with dozens of nanometers, it is essentially a 3D object and the 3D information can be partly reconstructed using the developed inverse TEM nanoMoiré method. Therefore, the developed inverse TEM nanoMoiré method provides a possible way for 3D reconstruction. Particularly, it should be noted that relatively less information was needed in the reconstruction process. Based on the inversion result, the possible 3D interface distribution of the grains in the TBCs structure is illustrated in Fig. 9. The colored boxes indicate the single grains, which are identified by the inverse TEM nano-Moiré analysis. In fact, the width of interface area marked green in Fig. 7 is only a few nanometers, which means that the grain boundary is almost perpendicular to the thin section. The interface in Fig. 7 is considered to be the interface 2 in Fig. 9. The other two interfaces 1 and 3 are invisible in Fig. 7 and they are considered to be parallel to the thin section (see Fig. 9). Since the invisible grains (ZrO2 grain 2 and Al2O3 grain 2) are likely to be very thin in the observed thin section, more information is needed to determine the angle between the grains for further analysis. A promising way is presented here. By rotating the specimen in the out-of-plane direction in a slight angle, the Moiré pattern will change due to the shape and thickness of the invisible grain, so does as the visible lattice. Through analyzing the Moiré patterns, the 3D structure could be further ascertained.

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Fig. 9 Schematic illustration of the 3D interface reconstructed.

Fig. 10(a) and 10(b) illustrate the inversed sub layer lattice and the extracted visible lattice and their real strain distribution. Fig. 10(c) show the Moiré fringes and its corresponding relative strain. The two regions numbered as II and III are overlapped with each other, and the region II is moved to the right with some distance for convenience. The superposed area was marked with dotted pink line and black line in the two areas, separately. The strain field was 54 calculated using the S-GPA method aforementioned. The strain field in different regions was calculated separately. The dotted white square shown in Fig. 10(a), 10(b) and 10(c) is the one used for reference phase reconstruction in S-GPA method. The strain fields of the two areas were calculated independently. The reference phase reconstruction area in the region II and region III is set away from the interface. Strain in a lattice away from the interface or defects are very small, and can be neglected. So the calculated strain in these two areas in Fig. 10(a) and 10(b) are real strain. Strain value is comparatively small and uniform in the middle of the strain field. However, the strain value is larger and nonuniform at the boundary areas, especially around the interface of the two dotted closed irregular areas. The lattice structure and the Moiré fringes are extracted using Fourier analysis shown in Fig. 10(b) and 10(c), respectively. The Moiré fringes in the c-ZrO2 area are straight and uniform with eyesight. In the α-Al2O3 region, there are

evident visible deformations in the Moiré fringes, meaning that relatively large deformations are between the upper and lower lattice. There may be mismatches between the two αAl2O3 crystals. Mismatches may come from some impurity atoms mixed in the interface or some direct misfit between different crystals. No matter what the mismatch is, it is the possible and important reason to induce crack nucleation or dislocations. The relative strain is also calculated and showed in Fig. 10(c). The Moiré pattern is an interference pattern generated by the overlaid lattices and each layer of the lattice is not perfectly uniform. It is hard to confirm where of the Moiré pattern is completely free of distortion, so the strain calculated from the Moiré fringe is defined as relative strain. Similarly, the strain distributed around the interface is relatively large and nonuniform. In the red dotted circle, the strain in this area is not uniform and large. Maybe impurity atoms exist in this area and the impurity atoms are all in the sub layer invisible lattice based on the real strain field in Fig. 10(a) and 10(b). It is clear that the strain in the visible layer lattice is relatively uniform than that in the inversed invisible layer lattice. The strain distribution of the same material in different crystal may be drastically different from each other. The crystal (or interface) with nonuniform strain distribution may be weak and failure may form or propagate in the fragile crystal (or interface). In region III, the upper and lower lattices are all c-ZrO2, meaning that two ZrO2 crystals are connected with each other. The ZrO2 is a kind of brittle material. Therefore, the strain field is relatively uniform in this area. If large strain exists, cracks maybe occur. In reality, ZrO2 is stable and cracks always arise in the TGO layer but not the ceramic layer. For TGO structure (mainly consists of α-Al2O3), TGO is always growing and changing during the oxidation process. Compared with ZrO2, the TGO structure is unstable and defects may form in TGO region. The defects, like dislocations, maybe the core factor to form crack nucleation.

Fig. 10 Strain distribution of (a) real value of the inversed invisible layer lattice, (b) real value of the extracted visible layer lattice and (c) relative value calculated from the Moiré fringe.

Generally, the Moiré fringe and lattice structure always coexist in a unilateral directivity in the HRTEM image. The proposed inverse TEM nano-Moiré method can conveniently

be used. Additionally, the inverse TEM nano-Moiré method can be easily used for multi-direction Moiré inversion and more information can be obtained to analyze the lattice

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structure. The details of the atom distribution can be accurately got and it will be helpful to quantize the material structure. However, owning to the formation of Eq. (10), the inversion results will be much more complicated than those in this study. Various combinations can be made. Then the three suggestions aforementioned are strongly recommended for concrete analysis. It is accessible that the elastic theory of continuum mechanics can be applied in the stress measurement of material in a range of tens of nanometers. However, the observed Moiré or lattice is all in a unilateral directivity here. Deformation information in the principal direction of the visible Moiré or lattice is obtainable, but the information in another direction is not available. Hence, the real stress distribution cannot be obtained using strain-stress relationship. Nevertheless, the stress distribution can be qualitatively evaluated using the strain value. The real strain value near the interface is large and nonuniform, so should be the stress. Then the stress at the interface is more likely to surpass the tolerance of the structure. Especially there are defects, like dislocation formed in the interface. The larger stress at the interface may cause crack nucleation. After periods of oxidation, the crack nucleation may propagate along the weak point (defects) at the TC/TGO interface and develop into microlevel even macroscopic scale. Afterwards, ceramic spallation may occur due to the crack propagating. Actually, it has been reported that the residual stress near the TC/TGO or 44 TGO/BC interface is large and damage initiation and progression in the form of microcracks can occur in many 46 different ways in TBCs . The stresses at the TC/TGO interface are tensile at the undulation crests and compressive at the 44 troughs and the sign of the stress may change opposite after 46 heat treatment . In our study, it can be concluded that the symbol of stress at the interface would be alternating in positive or negative. Stress concentration may occur at different locations, crests or troughs, and even the transition 44, 46 . Once the stress at these points surpasses the part tolerance of the structure, dislocations or cracks will form, propagate, and develop. Finally, these cracks will result in the ultimate spallation failure of the TBC.

the weak point that may lead to the final failure of the TBCs spallation. Also, the possible 3D structure near the interface is reconstructed using relatively less information. Due to the lack of 3D information, the 3D inversion result could be nonuniqueness. It seems that the proposed method is a semi quantitative method, since the real 3D structure cannot be entirely determined. A possible way is put forward to ascertain the 3D structure. In addition, the proposed inverse TEM nanoMoiré method is considered to be a promising technique in other condition, like a Moiré pattern in a larger field of view without any visible lattice. In this condition, the reference lattice may come from the SAED pattern or a larger magnification with visible lattice fringe.

Acknowledgements This work was financially supported by the National Natural Science Foundation of China [grant numbers 11372037, 11572041, and 11232008].

Notes and references 1. 2. 3. 4. 5. 6.

7. 8. 9. 10.

4. Summary

11.

In this study, an inverse TEM nano-Moiré method, for the first time, is proposed and applied to the characterization of TEM Moiré patterns with only one direction lattice visible. Compared with the existing method, the newly developed method can reconstruct the invisible information (lattice or structure) directly from the coexisting lattice and Moiré patterns. The uniqueness of the inversion result can be guaranteed by the diffraction analysis or elemental analysis. Veiled information near the TC/TGO interface in TBCs was successfully visualized by the inversion method. The strain distribution in the principal direction of the invisible lattice, TEM Moiré and visible lattice was calculated simultaneously using the S-GPA method for the first time. The contact area, especially the contact edge of the grain interface is possibly

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