2.2.1. Experimental Setup
XRD is widely used to measure the elastic strain in crystalline material [
8,
9]. The experimental geometry of the system is presented in
Figure 3. The method is used in reflexion. The samples of plane geometry are at a fixed distance of an area detector. The samples are cylinders of radius 6 mm and thickness 4 mm. The energy of 20 keV (optimum for photons flux on BM02) is well adapted to optimize the film response. A XRD diagram has been recorded using an area detector (FReLoN, ESRF, Grenoble, France), which is large enough to extract the stresses evolution in both the film and the metal, and fast enough to obtain data in real time, as in previous experiments [
4].
The beamline is equipped with a goniometer, a sample carrier, an area detector, and two photodiodes. The first photodiode is placed in front of the sample and makes it possible to know the flux of incoming photons. The second photodiode is placed behind the sample and along the beam axis to allow alignment of the sample surface with the beam, which is important for height adjustment.
An induction furnace (ESRF, Grenoble, France) was used to simultaneously carry out oxidations with in-situ measurements and especially strain measurements during this oxidation. The induction furnace was provided by the Sample Environment Support Service of ESRF. The system was driven by a high-power generator (maximum 3kW). The sample carrier is a cylindrical support, surrounded by the induction coil with 7 mm diameter and 4 mm depth. A thermocouple is placed at the bottom of this support to control and regulate the temperature (the temperature can reach a maximum value of 1600 °C). The temperature on the surface of the sample was measured by a pyrometer (FAR, model FMPI, Riverview, Michigan, USA). The speed of heating and cooling can vary between 1 °C/min and 500 °C/min with a temperature of accuracy +/-1 °C. The main advantage of this furnace is its ability to heat or cool very quickly (thermal characteristic time of the furnace with our metallic specimen is around 13 s), with a good control of the heating and cooling speeds, which is an advantage to develop particular experiments (thermal cycling or temperature jumps. However, even if the use of a thermocouple and a pyrometer allows a correct knowledge of the oxidation temperature, it cannot be ignored that there may be a temperature gradient between the top and bottom faces of the sample (in contact with the bottom thermocouple). This may generate non-homogeneous expansion of the samples and may disturb especially disturb the height adjustment. The induction furnace is used for the oxidation of samples in air.
The X-ray beam size is 1 mm × 1 mm and the angle of incidence is 5°. The detector is located at a distance of 22 cm from the center of the goniometer. This detector makes an angle of 18° with the incident beam.
The calibration of the detector FReLoN was carried out using an ideal silicon powder as reference (NIST SRM640), in order to adjust the correspondence between pixel and angular degrees. The calibrant powders and samples have been placed at point O using the absorption profile of the sample from the direct beam recorded by a photodiode placed behind the sample position. The calibration is a very important step because a poor positioning of the camera can induce an offset at the pixels and, subsequently, an offset on the positions of the Debye-Scherrer rings.
In order to make the sample surface parallel to the beam, adjustment steps are required before each experiment to have a better accuracy. The procedure for adjustment of the flatness of the sample is as follows.
Note the detected intensity of the uncut incident beam
Adjust the sample to partially cut off the incident beam, detected intensity is then less than the previous intensity
Rotate the sample around the axis to find the ideal position, for which the detected intensity is maximum. After this adjustment, the surface of the sample is then parallel to the incident beam
Adjust precisely the height of the sample so that the detected intensity is equal to the ideal position
Because of all these complex steps and even with all calibration and adjustment processes, it is necessary, to obtain a correct stress value, to propose a correction method for possible errors or systematic bias such as beam deviation. It is explained in
Section 2.3.
The associated stress can be determined thanks to the sin
2 method, and using the radiocrystallographic elastic coefficients. Because the material consists of randomly oriented crystallites, Bragg’s law is applied to evaluate the changes of distance between crystallographic planes and taking into account the diffraction geometry (
Figure 3). Then, rings are obtained with the help of an area detector. From [
9], a relation is given to transform ψ as a function of θ and γ.
ψ is the angle between the normal to the surface and the normal to the diffracting planes, 2θ is the angle between the incident beam and the diffracted beam, ω is the incident angle, and γ is the angle between the vertical passing through the position of the direct beam and a given position on the diffraction ring.
Thus, each gamma value is related to ψ and the diffraction patterns are fitted for each ψ value, to obtain the corresponding d-spacing. Afterwards, the sin2ψ method is used to determine the associated stress. In the present study, the measured strain corresponds to the average through a depth of a few microns below the surface of the metallic sample.
2.2.3. Data Processing
Raw data (see example in
Figure 5) are images of the diffracted rings. It has to be analyzed to get the stress in the oxide layers and in the metal. For a stress-free material, the data must draw perfect arcs of circles.
Figure 5 shows an example recorded for the NiCr/Cr
2O
3-system. As we can see in
Figure 5, the intensity for the metal rings are not as homogenous as for the oxide rings. This is mainly due to a grain size effect, which is bigger in the metal (around 400 µm [
13]) than in the oxide (less than 1 µm [
14]) when compared to the beam size. With time, oxide rings become more continuous. Except for the very beginning of the oxidation, we are easily able to fit the peaks because of the good resolution of the area detector and it is sufficient for sin
2ψ method.
To perform the analysis, the rings were subdivided into 128 sectors γ whose values are between {−61.5 ° and +65.5 °} with a step of 1° (
Figure 6).
From these data, it is possible to plot the intensities of each sector γ as a function of 2θ, giving a diffraction diagram. The peaks studied for the chromium oxide are (104), (110), (116), and the peak for the nickel-chromium alloy is (111). The 2θ free of stress reference positions are summarized in
Table 3.
Each diffraction diagram has to be simulated by using different functions and after background subtraction. Indeed, it is necessary to minimize the influence of the background. Four main distribution functions have been tested to simulate the peaks profile, which are Gauss, Lorentz, Pearson 7, and Pseudo-Voigt. Concerning the background, we choose a polynomial relationship. In addition, the order of the polynomial relationship is also tested. In the computing procedure, we can simulate the peak distribution and the background noise together or separately.
Figure 7 shows an example, which simulates the peak shape and the background separately. It shows four curves, which are initial experimental peak, simulated peak with initial parameters, experimental peak after removing the background, and simulated peak with final parameters. Initial experimental peak corresponds to the initial experimental data. Simulated peak with initial parameters is obtained with the initial set of parameters used in the distribution function. The position, width and intensity of peaks are close to the simulation peak with these initial parameters. Then by applying an optimization method (non-linear fitting function in Matlab software), we can simulate the peak more correctly and find the final parameters (position, width, and intensity). Moreover, thresholds enable obtaining a good quality for fitting process. For example, we have a threshold for intensity: an intensity value below this threshold leads to consider the fitting of the peak as not good, etc. To obtain the peak after removing the background, we simulate the background by using a polynomial relationship and remove the simulated results from the initial experimental data. Finally, the peak is simulated with one of the distribution functions mentioned before, by using the final parameters.
Figure 8 shows the results that were obtained for the different distribution functions. The simulation results are quite similar for the different output parameters obtained, such as intensity, position, and width of the peak. Finally, to choose the best function, we need to test systematically this procedure by considering a significant number of experimental data.
After comparing different quality indicators (such as slope, average value, average absolute difference and difference between maximum and minimum), we consider that a Pearson 7 distribution function with a simultaneous background simulated by an order 3 polynomial function lead to a maximization of the fitting quality. Therefore, these parameters have been considered for the simulation of all the data.
To summarize, the first two peaks are fitted together and the third peak is fitted by itself. Peaks are simulated first to hang the optimization of parameters with some criteria/error to stop the optimization. Then a simulation in return is performed with the optimized parameters to check visually the calculation. For the background noise, it is typically a cubic power law used for the fitting. Even with the procedure of calibration of the detector, it is not possible to remove all of the background noise. All of this procedure has been automated in a Matlab software. Moreover, we randomly selected 100 pictures to apply this procedure for different simulation models, which correspond to a relatively stable period.
Afterwards, to visualize the stress influence, the 2θ position can be plotted as a function of γ for a given peak (
Figure 9).
Figure 9 shows an example of 2θ versus γ for the peak (116) in the oxide. A symmetrical variation of the positions 2θ is observed, which corresponds to a stressed material. The same trend (not shown) is observed in the metal.
To draw the curves of sin2ψ, different other quality criteria have also been defined in the computing process. For example, a criterion corresponds to the confidence interval around the linear fit that control the removal of abnormal points. In addition, the correlation coefficient of the fit and the difference between the first point and the last point of the linear fit are also used to assess the quality of the fit. If all of these criteria are satisfied, we can consider that the fitted curves are correct.
Figure 10 gives an example of result obtained for ψ < 0. The trend for ψ > 0 is quite similar (but is not shown in this article).
In order to more legibly display the results, we calculate an average for every five data, which are indicated by the black points in
Figure 10. The “error” bar corresponds here to the range for which we can find the other points. It is calculated from the difference value between every five data. The slope of this line corresponds to
and the intercept to
. The range of variation of the values
is 7.2 × 10
−3. The error bar is in the range 2.3 × 10
−4 to 9.8 × 10
−4, which is small when compared to the variation of the values for
. The relationship between
and
is well linear, which corresponds to equation (12). The stress value corresponds to only one point in the stress-time curves evolution. After repeating this procedure for all the experimental pictures/times, we can finally get the stress-time curve for the different (hkl) oxide families.
The same analysis has been undertaken for the metal. For a better comparison of the results between oxide and metal, the same kind of display has been adopted.
Figure 11 shows an example of the results obtained. The error bar is from 3.5 × 10
−4 to 2.7 × 10
−3, which is higher than the error bar in the oxide. However, we can again consider with confidence that the relationship between
and
is linear, as in the oxide layer. Therefore, after repeating this procedure for all of the experimental data, the stress-time curve for the (111) metal peak can be also obtained.
Table 4 shows the data quality comparison for oxide and metal.
As written before, the error bar is calculated from the difference between five data. By comparing the values of this error bar, we can check the data quality of the results in oxide and metal. The proportion of average error bar is the proportion of average error bar in the variation of . The variation range of is very similar in oxide and in metal. The maximum error bar and the average error bar are larger in metal than in oxide. The proportion of average error bar is also larger in metal than in oxide. That is to say, from these criteria, the data quality of the results in oxide is a little bit better than in metal.