For ribbon A with the composition of Zr65Cu27.5Al7.5 we analyzed the crystallization behavior further. For the calculation of the crystallized volume fraction we are restricted to the q-range from 1 to 6 Å−1, because, to our knowledge, there was no literature data of the Bragg peak positions for primary crystalline phase Zr2(Cu,Al) available for higher values. In order to determine the crystallized volume fraction, it is most crucial to seperate the crystalline from the amorphous contribution in the diffraction profiles [60]. For this purpose we fitted the total
Chapter 4. Results
diffraction intensity with a single function. In this function we attribute a multiple Gaussian fit to the Bragg peaks of the crystalline phase, a Pseudo-Voigt function [60] in order to describe the amorphous maxima and the background was fitted with polynom of second order. Before we analyzed a diffraction pattern during the crystallization process we fitted the first three maxima of a purely amorphous spectrum with a multiple Pseudo-Voigt function (see Fig. 4.35).
Iam(q) = The result of this fit, particularly the full widths at half maximum (FWsHM) βam,1 = 0.37776, βam,2 = 0.7 and βam,3 = 0.7, the absolute heights of the maxima Aam,1 = 60.23,Aam,2 = 9.0481 andAam,3 = 3.5676 and the Gaussian content η1 = 0.16354,η2 = 0 and η3 = 1 is kept constant and is used for the next fit of the partially crystallized diffraction profiles. The positions of the amorphous maxima qam,max,i still remain variable in order to allow shifts during thermal expansion. It is visible that the second amorphous maximum can be modeled only with a Lorentzian contribution and the third amorphous maximum requires only a Gaussian part to describe the curve. The FWHM of the second and the third peak were partly varied manually because the fitting program tends to overemphasize one of the peaks.
For the fit of the entire, partially crystallized diffraction pattern we used the following function,
Figure 4.35: Multiple Pseudo-Voigt Fit of the purely amorphous diffraction patterns before the heating starts.
where the Bragg peaks are described by a Multiple Gaussian function [60]:
Chapter 4. Results The line 4.6 corresponds to the Gaussian fits of the Bragg peaks with the height of the j-th Bragg peakacr,j, its position qcr,max,j and its FWHM βcr,j. We allowed a maximum number of 10 crystalline peaks which form during crystallization because we found 10 crystalline peaks which belong to the dominant Zr2(Cu,Al) phase found in section 4.9 in the lines 4.7-4.10 are the fits of the amorphous maxima which include the results for the FWHM, the heights of the peaks and the Gaussian content from the previous fit of the purely amorphous profile. The parameter Aam,rel is the peak height of the amorphous contribution in the partially crystallized profile relative to the peak height of the purely amorphous profile and therefore it varies between 0 and 1. The quantitiesbiof the polynomial contribution model the remaining background which was not entirely removed by the procedure described in section 3.14. The fit of the partially crystallized profile is shown in Fig. 4.36 (blue curve). The difference between the data points and the fit verifies a good agreement (see Fig. 4.36 bottom panel).
With the information from the fits it is possible to separate the crystalline from the amor-phous contribution in the partially crystallized diffraction pattern and therefore we can make a statement about the crystallized volume fraction according to equation 2.10. As expected, the crystallized volume fractionXcryst versus the scattering vector q of the measurement shown in Fig. 4.36 increases at the positions of the Bragg peaks and decreases in between (Fig. 4.37).
For the value of the crystallized volume fraction of the entire diffraction pattern we used the last analyzed value Xcryst(q= 6.0).
We analyzed the crystallized volume fraction of the samples of ribbon A which crystallize into the Zr2(Cu,Al) phase and connected the value Xcryst(q= 6.0) of each frame to the time where
Chapter 4. Results
Figure 4.36: The top panel shows an exemplary, partially crystallized diffraction profile of a sample made of Zr65Cu27.5Al7.5 (red curve) and a multiple Gaussian fit (blue curve). The bottom panel shows the difference between the data points of the diffraction profile and the fit.
Figure 4.37: Exemplary crystallized volume fraction as a function of scattering vector in the q-range of 1 to 6Å−1.
the illumination of the frame ends (Fig. 4.38). In all the curves the increase in the crystallized volume fraction ends suddenly at a certain value between 30 and 50 %. It is also noticeable that the slope of the Xcryst-increase rises from sample A1 (black curve) to sample A4 which exhibits the highest slope. This is not correlated to the crystallization temperature which is the highest for the sample A9 (dark green curve). The increase of the crystallized volume fraction can be well approximated by the Johnson-Mehl-Avrami (JMA) equation 2.9. The fitting parameter k
Chapter 4. Results
represents the aforementioned change in slope ofXcrystwith the temperature where the sample crystallizes (Fig. 4.39).
Despite the fact that the JMA model fits the data well, the interpretation of the resulting
Figure 4.38: Crystallized volume fraction for the samples of ribbon A which crystallize into the Zr2(Cu,Al)-phase. The solid lines are JMA-fits. The inset shows the temperature versus time curves of the samples of ribbon A during rapid heating which is copied from Fig. 4.19.
parameters can be misleading when the application of the model is not verified. First, the JMA model was designed for isothermal crystallization behavior and for non-isothermal experiments it can only applied in a very limited number of cases [57]. Particularly, for non-isothermal conditions it needs to be verified that the nucleation process is already finished when the crys-tal growth process starts [57] so that the number of nuclei does not change any more during the crystallization process. A common test of the applicability of the JMA model is to test the linearity of the plot ln(−ln(1−α)) versus the inverse temperature. However, even if the resulting is curve is linear the application of the model is not valid and only a SEM study of the crystallization behavior can bring a conclusive result [1].
Chapter 4. Results
Figure 4.39: Graph of the fitting parameter k of the VFT fit as a function of the temperature where the sample crystallizes.