Non-destructive microstructural analysis with depth resolution

Nuclear Instruments and Methods in Physics Research B 200 (2003) 382–389 www.elsevier.com/locate/nimb

Non-destructive microstructural analysis with depth resolution E. Zolotoyabko b

a,*

, J.P. Quintana

b

a Department of Materials Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel DND-CAT Research Center, Northwestern University, APS/ANL Sector 5, Building 432A, 9700 South Cass Avenue, Argonne, IL 60439-4857, USA

Abstract A depth-sensitive X-ray diffraction technique has been developed with the aim of studying microstructural modifications in inhomogeneous polycrystalline materials. In that method, diffraction profiles are measured at different X-ray energies varied by small steps. X-rays at higher energies probe deeper layers of material. Depth-resolved structural information is retrieved by comparing energy-dependent diffraction profiles. The method provides nondestructive depth profiling of the preferred orientation, grain size, microstrain fluctuations and residual strains. This technique is applied to the characterization of seashells. Similarly, energy-variable X-ray diffraction can be used for the non-destructive characterization of different laminated structures and composite materials. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 61.10.Nz; 68.55.Jk; 81.10.Aj Keywords: X-ray diffraction; Microstructure; Seashells

1. Introduction The study of increasingly complex materials and multilayered structures requires the development of X-ray diffraction techniques able to determine the three-dimensional distribution of structural characteristics in inhomogeneous systems. Two-dimensional mapping is under intensive development on synchrotron beam lines through X-ray microscopy [1–4], which is currently achieving a 100 nm lateral resolution. Extracting structural information with high-depth resolution is more problematical due to the relatively large

*

Corresponding author. Tel.: +972-4-829-4545; fax: +972-4832-1978. E-mail address: [email protected] (E. Zolotoyabko).

X-ray penetration depth in materials. This difficulty is increased for hard X-rays, which are used to measure thicker samples [5]. As an example, one can consider energy-dispersive X-ray diffraction [6], which utilizes high-energy white radiation and data accumulation at fixed angles by means of an energy-sensitive detector. In that method, the diffracting lozenge in the direction of the incident beam can reach hundreds of microns. The problem is partially solved by using some kind of triangulation technique [7,8]. While it is possible to separate diffraction signals from individual grains, this approach is time-consuming and requires sophisticated set-ups and software. In order to achieve depth profiling within a polycrystalline sample, the effective X-ray penetration depth must be changed. At fixed energy, the X-ray penetration depth depends on the entrance

0168-583X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 1 7 2 7 - 5

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angle between the incident beam and the sample surface. This is used in the ‘‘scattering vector method’’ [9,10], in which an asymmetric reflection is measured after the stepwise rotation of the sample round the diffraction vector. During rotation, the entrance angle varies, which results in the desired changes of the X-ray penetration depth. Under glancing incidence conditions the depth can be in the sub-micron range. This method, though very effective in characterizing the near-surface layers, encounters difficulties when used to study deeper layers buried in the sample interior. Alternatively, the X-ray penetration depth can be varied by small steps by accurately tuning the X-ray energy [11,12]. In this paper the application of energy-variable diffraction to the study of the microstructure in polycrystalline materials with depth resolution is described. This technique is very suitable for synchrotron beam lines, because X-ray energies can be precisely tuned by means of a conventional double-crystal monochromator. It uses a standard experimental diffractometer set-up and permits routine measurements of a whole diffraction pattern as well as of selected diffraction peaks. Thus, in principle, the complete set of structural and microstructural characteristics can be extracted from the diffraction data.

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coefficient, l / E3 and the Bragg angle, HB . Using Þ ¼ 12:3984= BraggÕs law, 2d sin HB ¼ k, where kðA ½EðkeVÞ is the X-ray wavelength and d is the spacing between diffracting atomic planes, yields: K ¼ k=ð4ldÞ:

ð1Þ

2. Experimental details

The penetration depth, K / E2 , varies rather slowly with the X-ray energy, as compared to ð1=lÞ / E3 , and, correspondingly, the depth adjustment can be made with precision. In order to illustrate the capabilities of the energy-variable diffraction, we applied it to the characterization of seashells, keeping in mind that seashells possess complicated microstructures which are also depth-dependent [14]. We used samples of the Acanthocardia tuberculata, which belongs to the bivalvia subgroup and has a simple aragonite structure with four molecules of CaCO3 per orthorhombic unit cell (a ¼ 4:9623, ). Measurements were b ¼ 7:968 and c ¼ 5:7439 A made with nearly flat pieces of shells about 20 mm2 in area, which were cut from seashells with a diamond saw. We focused on the nacre layer of the shells, which extends for hundreds of microns from the inner shell surface (adjacent to the mollusk mantle). According to scanning electron microscopy (SEM), the nacre layer is composed of large c-oriented lamellas (see Fig. 1). Between the nacre layer and mollusk mantle, a very thin (10 lm) inner prismatic layer is located (see Fig. 2).

Diffraction measurements were carried out at the 5BMD beam line of the Advanced Photon Source (APS) at Argonne National Laboratory. We used monochromatic radiation with the energy, E, ranging regularly between 7.5 and 30 keV, and up to 60 keV in specific measurements. All diffraction patterns were taken in the Bragg (reflection) scattering geometry. The incident beam was restricted by slits down to 1 mm vertically and 3 mm horizontally. Energy-variable X-ray diffraction provides nondestructive profiling of structural parameters based on the relationship between the X-ray energy, E, and the X-ray penetration depth, K. Neglecting extinction effects in polycrystalline materials, the value of K in the Bragg diffraction geometry, K ¼ sin HB =2l, depends [13] on the linear absorption

Fig. 1. SEM micrograph of the nacre layer, showing [0 0 1]oriented lamellae.

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(006)

1000

Intensity

(014) (004) 100

(015)

(012)

10

1 10

20

30

40

50

60

70

80

90

2Θ (deg)

Fig. 3. Diffraction pattern taken from the shellÕs inner surface at E ¼ 10 keV, which shows the strong (0 0 1) preferred orientation of the nacre layer.

Fig. 2. SEM micrograph of the less ordered prismatic layer adjacent to the mollusk mantle.

The energy range, E ¼ 7:5–30 keV, which is regularly used in our X-ray measurements, being converted to the X-ray penetration depth in aragonite (by means of Eq. (1)) corresponds to an interval of 5–70 lm. Energy variation around E ¼ 10, 20 or 30 keV by equal steps of 50 eV (which can be easily done with the computercontrolled double-crystal monochromator) results in penetration depth changes by steps of 100, 200 or 300 nm, respectively.

3. Results and discussion Energy-variable diffraction, in principle, enables phase analysis with depth resolution. In our shells only the single aragonite phase was observed in the shell interior. No calcite phase was found in the investigated shells. However, aragonite blocks can be differently packed within sub-layers, producing a variety of microstructures and resulting in the redistribution of the diffraction intensity between diffraction peaks and in some modifications of the shapes of diffraction profiles. 3.1. Analysis of diffraction intensities: preferred orientation X-ray diffraction patterns taken from the nacre layer (see Fig. 3) exhibited greatly enhanced ð0 0 lÞ

diffraction lines with even indexes, thus confirming the high degree of the c-type preferred orientation visible in SEM cross-sections. The crystalline texture is a result of mollusk activity preparing an organic template for the oriented aragonite growth. Biological control of the crystal texture in seashells and its important role in adapting crystal properties to functions is under intensive investigation [15]. The preferred orientation with depth resolution can be deduced from the diffraction intensity ratios, g, measured with appropriate X-ray reflections as a function of the X-ray energy. We used the ratio g ¼ Ið0 0 2Þ=Ið0 1 2Þ, between the (0 0 2) (preferred orientation) and (0 1 2) diffraction intensities, which is very small for a random powder: gp ¼ 0:042. A typical gðEÞ-dependence is plotted in Fig. 4(a). It is of a bell-like shape with a maximum at a certain depth and a slowly decreasing right-hand tile. In the probed depth range, the measured g-values were always found to be much larger than gp . In some specimens the g-values reached g 150 3000gp . Note that the diffraction patterns taken for comparison from the powdered seashells at 8, 21 and 60 keV exhibited unchanged intensity ratios very close to gp . Hence, the variation of the g-ratio with energy in Fig. 4(a) reflects in-depth changes of the degree of preferred orientation. Uniaxial preferred orientation can be treated with the Dollace–March approach [16], in which the modification of the diffraction intensity Iðh k lÞ ¼ Ip ðh k lÞMh k l , as compared with the powder diffraction intensity Ip ðh k lÞ, is described by the March function

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where M1 ¼ Mða ¼ 0Þ, M2 ¼ MðaÞ (see Eq. (2)) and d1;2 , HB1;2 are the d-spacings and Bragg angles of the two reflections used, respectively. Experimental data were fitted by substituting the trial rðzÞ-functions in the form:

1.6 1.4 1.2

Intensity ratio

385

1 0.8 0.6

rðzÞ ¼

0.4

A 2

1 þ ðz=L1 Þ

þC

C 1 þ ðz=L2 Þ

2

;

ð4Þ

0.2 0 0

5

10

15

(a)

20

25

30

35

Energy (keV)

0.8

r(z)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

25

50

(b)

75

100

125

150

175

200

Depth (microns)

Fig. 4. (a) Measured energy-dependent (0 0 2)/(0 1 2) intensity ratios, gðEÞ, (points) superimposed on the solid line simulated by means of Eq. (3); (b) Depth-dependent March parameter, rðzÞ, used in simulations of the gðEÞ-curve.

Mh k l


3=2 sin2 a 2 2 ¼ r cos a þ : r

ð2Þ

In turn, Mh k l depends on the March parameter, 0 6 r 6 1, and the angle, a, between the atomic planes (h k l) and the planes of preferred orientation, which in our case are the (0 0 1)-planes. A lower March parameter indicates a higher degree of preferred orientation and vice versa. The limiting values are those of the powder sample ðr ¼ 1Þ and of a single crystalline layer ðr ¼ 0Þ. In layered structures the March parameter, rðzÞ, is depth-dependent. Taking into account the in-depth attenuation of the primary beam, expðz=KÞ, allows us to rewrite the diffraction intensity in an integral form and to express the intensity ratio as [12]:   R1 2lz M ðzÞ exp  dz 1 0 sin HB1 d1   ; gðEÞ ¼ gp R 1 ð3Þ d2 M2 ðzÞ exp  2lz dz 0

sin HB2

into Eqs. (2) and (3). Functions (4) allowed us to produce a wide spectrum of the March parameter distributions. The upper integration limit in Eq. (3) was taken to be three times larger than the Xray penetration depth, Km , at the highest energy. The depth-dependent function, rðzÞ, which provides the best fit to experimental data in Fig. 4(a), is plotted in Fig. 4(b). The fitting parameters are: A ¼ 0:25, C ¼ 0:65, L1 ¼ 20:2 lm, L2 ¼ 99 lm. In the investigated samples, the rðzÞ-functions are characterized by a minimum value centered at a certain depth, a weak left side ascent, and a slowly increasing long-range right side tile (see Fig. 4(b)). The left side ascent reflects the presence of the less ordered prismatic layer, visible in SEM images (see Fig. 2). The thickness of this layer is roughly given by the parameter L1 . The right-side tile reflects the gradual degradation of the preferred orientation with depth, a characteristic length equal to L2 . The measured parameters, L1 , L2 , as well as the maximum and minimum r-values, can be used to compare different shell specimens [12]. 3.2. Line profile analysis: grain size and microstrain fluctuations Energy-dependent diffraction profiles can be used in order to extract the depth dependences of the grain size and microstrain fluctuations. For this purpose, the measured profiles were decomposed onto the Lorentzian and Gaussian contributions. Practically, the (0 0 2) diffraction profiles were fitted to Voigt function which is the convolution of Gaussian and Lorentzian functions (see e.g. [17]). In our case of the [0 0 1]-preferred orientation, the grain size is the thickness of lamellae. The thickness, L, of the [0 0 1]-oriented lamellae was extracted from the Lorentzian widths, WL (FWHM in the 2H-scale), using the expression:

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E. Zolotoyabko, J.P. Quintana / Nucl. Instr. and Meth. in Phys. Res. B 200 (2003) 382–389

L ¼ 2d

tan HB : WL

ð5Þ

The lamellae thickness in the investigated samples is between 200 and 600 nm, which is in accordance with direct observations by SEM. It means that lamellae visible in SEM cross-sections (see Fig. 1) are rather thick single crystals. The lamellae thickness, LðKÞ, in one of the samples plotted in Fig. 5, as a function of the X-ray penetration depth, K. In the specific sample,the lamellae are around 300 nm thick. The LðKÞ-function has a gentle maximum near the inner shell surface, at a depth of 10–20 lm. This is reasonable if one takes into account the presence of the thin prismatic layer underneath the nacre. After reaching the maximum, the thickness of the lamellae was found to decrease slowly with the sample depth (see Fig. 5). It should be stressed that our method (like every X-ray diffraction method) provides the material parameters, P ðKÞ, which are, in fact, integrated over the X-ray penetration depth, K. For an infinitely thick sample (meaning physically that the sample thickness T K) R1 pðzÞ expðz=KÞ dz P ðKÞ ¼ 0 R 1 expðz=KÞ dz 0 Z 1 1 ¼ pðzÞ expðz=KÞ dz; ð6Þ K 0 where pðzÞ is the actual distribution of the material parameter, p, across the sample depth, z. The integration mentioned arises as a result of the

500

L(nm) 400 300 200 100 0 0

10

20

30

40

50

60

70

80

X-ray penetration depth (microns)

Fig. 5. The thickness, L, of the [0 0 1]-oriented lamellae in the nacre layer, as a function of the X-ray penetration depth, K.

exponential (in-depth) attenuation of the X-ray intensity. For slowly varying functions (as is the case with the measured dependence, LðKÞ) the distinction between P ðKÞ and pðzÞ is not very pronounced. In the extreme case, pðzÞ ¼ p0 ¼ const:, both functions are identical: P ðKÞ ¼ p0 . This is the reason why, when analyzing the lamellae thickness, L, and the averaged microstrain fluctuations, rm (see below), we have restricted ourselves to the integrated (over K) functions. The averaged microstrain fluctuations, rm , (dispersion of the Dd=d-distribution) were extracted from the Gaussian widths, WG (FWHM in the 2H-scale): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðWG2  WI2 Þ ffi rm ¼ pffiffiffiffiffiffiffiffiffiffi ; ð7Þ 4 2 ln 2 tan HB where WI is the instrumental width. For highly monochromatized and parallel synchrotron radiation, WI is determined by the transmission function of the detecting system which, in turn, depends on the slit configuration. The value of WI ¼ 0:058°, was extracted from the diffraction profiles (2H-scans) measured with a 150 lm thick single-crystal plate of quartz. The procedure of microstrain determination was tested with a 1 lm thick Al film deposited by the electron beam technique on the quartz crystal mentioned above. Due to the small thickness of the Al layer (as compared with the X-ray penetration depth at the lowest energy used, E ¼ 7:5 keV) no variations in the microstrain magnitude are expected. In fact, data collected with the Al film revealed practically constant rm 0:00045 over the entire energy range. The averaged microstrain fluctuations, rm in the investigated specimens were found at a level of 0.001 and slightly increased with depth. A typical rm ðKÞ-dependence is plotted in Fig. 6. The rm values are 2–2.5 times greater than that in the Al sample. Most probably, the microstrains in the investigated seashells are influenced by the inclusions of an organic phase (a few volume percent) entered the nacre layer during the bio-mineralization process (see e.g. [18]). According to [18], the organic phase is concentrated between lamellar sheets. We found that an increase in rm ðKÞ

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0.04

0.002

σm

∆2 Θ (deg) 0.03

0.001

0.02

0.01

0 0

10

20

30

40

50

60

70

80

X-ray penetration depth (microns)

Fig. 6. The averaged microstrain fluctuations, rm , in the nacre layer, as a function of the X-ray penetration depth, K.

correlates with a decrease in LðKÞ (compare Figs. 5 and 6). This correlation is reasonable, since decreasing the lamellae thickness results in an increasing number of inter-lamellar boundaries where non-homogeneous deformation fields are expected. 3.3. Diffraction peak position: residual strains Important information on the shellÕs strain (stress) state can be obtained by measuring the diffraction peak positions ð2HÞ as a function of energy. In principle, the experimental data taken in symmetric Bragg scattering geometry allow us to deduce the depth dependence of the d-spacings perpendicular to the sample surface. Using strainfree aragonite d-spacings, d0 from the X-ray powder diffraction file (JCPDS 41-1475), the relative difference: e ¼ ðd  d0 Þ=d0 can be calculated. However, this is the most problematic part of the technique. In fact, the measured peak positions, 2Hm , are influenced by several factors that must be taken into account and the peak positions properly corrected. The first correction, D2H1 , takes into account the ‘‘chromatic aberration’’ of the instrument related to the subtle energy-dependent alterations of the incident beam trajectory. In order to find the D2H1 ðEÞ-function we performed energy-variable diffraction measurements with a powdered Al2 O3 (alumina) standard obtained from NIST. The diffractometer was aligned at E ¼ 7:5 keV. The difference between measured and calculated (1 0 4)Al2 O3 peak positions as a function of the

0 0

5

10

15

20

25

30

35

40

45

Energy (keV)

Fig. 7. Data taken with the powder Al2 O3 standard (alumina), which illustrate the ‘‘chromatic aberration’’ of the experimental set-up. The points represent the difference (in degrees) between measured and calculated (1 0 4)Al2 O3 diffraction peak positions as a function of the X-ray energy, E.

X-ray energy, is plotted in Fig. 7. It is seen that at all energies, E > 7:5 keV, the measured Bragg angle is larger than the calculated one, the difference reaching the maximum value D2H1 0:03° between 20 and 25 keV. The D2H1 ðEÞ-values thus obtained were subtracted from the diffraction angles measured with seashell specimens. The second correction, D2H2 , arises due to the fact that at higher energies the diffraction signal is coming from deeper layers. This geometrical factor leads to the well-known aberration in powder diffraction, viz. the detector is ‘‘seeing’’ the maximum of the diffraction intensity at angles lower than those theoretically expected. An analysis of our scattering geometry shows that this correction is: D2H2 ðEÞ ¼ 2K cos HB =H , where H is the distance between the sample and the slit in the front of the detector. The D2H2 values should be added to the measured 2Hm values. A similar correction should be applied to the alumina data, but using respective penetration depths that are larger than in aragonite. The third correction, D2H3 , is related to the possible curvature of the samples (bowl-shaped) under investigation, which also leads to the reduction of the measured 2Hm values, depending on the maximum bow value, B. Assuming that the size of the sample is much smaller than the radius of curvature and averaging over the bow values, yields D2H3 ¼ B cos HB =H . The magnitudes of the bows in the investigated samples were measured by

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0.002

ε

0.0015 0.001 0.0005 0 0

10

20

30

40

50

60

70

80

X-ray penetration depth (microns)

Fig. 8. The (0 0 2) d-spacing variation, e ¼ Dd=d, as a function of the X-ray penetration depth, K.

optical microscope. Finally, the corrected peak position, 2Hc , 2Hc ¼ 2Hm  D2H1 þ D2H2 ðsampleÞ  D2H2 ðstandardÞ þ D2H3

ð8Þ

was used in calculating the d-spacings in seashell samples. An example of the dependence thus obtained, eðKÞ ¼ ½ðd  d0 Þ=d0 , on the X-ray penetration depth, K, for the (0 0 2) reflection (preferred orientation) is shown in Fig. 8. We stress that eðKÞ 103 > 0, i.e. d > d0 All investigated samples revealed similar behavior of eðKÞ in the depth range probed. Note that the increased d-spacings (on the same scale of eðKÞ 103 ) were also found in the powdered seashell specimens, which are not affected by the curvature effects. In our opinion, the obtained swelling of the aragonite lattice indicates the presence of an organic phase within the lamellae. This finding may be very important to the bio-mineralization problem.

4. Summary We have developed energy-variable X-ray diffraction in order to non-destructively measure the spatial distributions of microstructural characteristics in inhomogeneous polycrystalline materials. It was shown that this approach is well suited for synchrotron beam lines and helps to extract depth-resolved structural parameters from energy-dependent X-ray diffraction data. By using synchrotron radiation from third-generation synchrotron sources the depth profiling can be per-

formed to a depth of hundreds of microns. The minimal step depends on the material investigated and, from the energy point of view, can be reduced below 100 nm. We have shown that the diffraction intensities measured allow us to extract the depth-dependent preferred orientation, while the shapes of the diffraction profiles provide information on the depthdependent grain size and averaged microstrain fluctuations. Besides that, the diffraction peak positions are influenced by depth-dependent residual strains. Application of this technique to seashells allowed us to characterize the microstructural evolution in the nacre layer. Individual diffraction profiles were treated in the course of the line profile analysis by utilizing Voigt functions. We found that the nacre layer in the investigated specimens of the Acanthocardia tuberculata shells consists of [0 0 1]-oriented single-crystal lamellae, hundreds of nm thick, which are packed nearly parallel to the inner surface of the shell. The lamellae thickness, L, has a maximum at a depth of 10–20 lm, which is where the nacre layer begins. We remind that underneath the nacre layer the inner prismatic layer is located. The thickness of the lamellae slowly decreases in the deeper nacre layers. That reduction of L correlates with the observed slow in-depth growth of the averaged microstrain fluctuations. In our view, the latter is influenced by an organic phase concentrated within the inter-lamellar boundaries. The presence of thick well-ordered lamellae leads to the enhancement of the (0 0 l) diffraction intensity, which can be used to characterize the degree of preferred orientation in the nacre layer. In the case of uniaxial preferred orientation an analytical algorithm has been developed based on March functions. The texture distribution is characterized quantitatively by depth-dependent March parameters, which allows us to compare samples taken from different shells. It was found that the preferred orientation, like the lamellar thickness, has a maximum at a certain depth, viz. where the nacre layer starts to grow on top of the inner prismatic layer. After reaching the maximum, the degree of preferred orientation gradually decreases in the depth of the seashell.

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Information on depth-resolved residual strains (stresses) can be retrieved from the energy-dependent diffraction peak positions. We indicate that this treatment requires a proper correction procedure that takes account of the experimental geometry and sample curvature. The d-spacings determined were found to be systematically larger that those expected for the non-distorted aragonite lattice. Most probably the observed swelling is due to the presence of the organic phase within singlecrystal lamellae. This finding may be of great importance to the bio-mineralization problem and deserves further investigation.

Acknowledgements This work was performed at the DuPontNorthwestern-Dow Collaborative Access Team (DND-CAT) Synchrotron Research Center located at Sector 5 of the Advanced Photon Source. DND-CAT is supported by the E.I. DuPont de Nemours & Co., the Dow Chemical Company, the US National Science Foundation through grant DMR-9304725, and the State of Illinois through the Department of Commerce and the Board of Higher Education grant IBHE HECA NWU 96. Use of the Advanced Photon Source was supported by the US Department of Energy, Basic Energy Sciences, Office of Energy Research, under contract no. W-31-102-Eng-38. The technical support of J. Kulpin and A. Philippides (DND-CAT) is gratefully acknowledged. We thank D. Shaham (Technion) for taking the SEM micrographs of the seashell specimens, and B. Pokroy (Technion) for his help in part of synchrotron measurements. One

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of the authors (E.Z.) would like to thank the Fund for the Promotion of Research at Technion for partial financial support.

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