Medium range order of bulk metallic glasses determined by variable ...

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830 J.W. Deng et al. / Micron 43 (2012) 827–831 Fig. 5. Summary of the peak variance V(k) values obtained using different nominal resolutions R for all tested amorphous materials. Table 2 Correlation lengths and characteristic widths W of the amorphous materials. The mean atomic radii, r, of the materials are listed, as well as ratios /r between the correlation lengths and the mean atomic radii. (nm) W (nm) r (nm) /r a-Si 3N 4 0.69 ± 0.055 2.2 0.096 7.2 Zr-S2 0.81 ± 0.35 2.6 0.15 5.4 Zr-Vit105 0.85 ± 0.34 2.7 0.15 5.7 Fe-BMG 0.88 ± 0.11 2.8 0.13 6.8 amorphous materials, and the greatest V(k) were obtained with the 0.6 nm resolution. 4. Discussion 4.1. MRO length of the amorphous materials The major advantage of variable resolution FEM is that it semiquantitatively determines MRO length of amorphous materials with no needs of prior knowledge on the material structure. By the use of pair persistence model (Gibson et al., 2000), the correlation lengths of the amorphous materials can be determined from the variable resolution FEM results. By assuming that the four-body atom correlation function g 4 has a Gaussian decay envelope, the variance V(k,Q) is transformed into following expression, ( Q 2 V(k, Q ) = 1 3 P(k) ) + ( 4 2 P(k) ) Q 2 . (2) is correlation decay length, which is closely related with the MRO length. The structure information independent on MRO length is separated into P(k), which is a pair persistence function. Accordingly, a linear correlation can be derived between Q 2 /V(k,Q) and Q 2 in low resolution (i.e. comparable to the MRO length scale of amorphous materials), and the correlation length is obtained as = 1 √ m 2 c , (3) where m = 4 2 /P(k) and c = 1/ 3 P(k) are the slope and intercept of the linear fitting of Q 2 /V(k,Q) against Q 2 (Fig. 6). Correlation lengths of the amorphous materials are presented in Table 2, where correlation lengths of BMGs are between 0.81 nm and 0.88 nm, which are considerably larger than that of a-Si 3 N 4 (∼0.69 nm). The measured correlation lengths are commonly significant smaller than the acknowledged MRO lengths in amorphous materials. A characteristic width W for MRO was proposed as W = √ 10 Fig. 6. Plot of the calculated Q 2 /V(k) against Q 2 values for all the amorphous materials according to the variable resolution FEM results. Here the V(k) values were chosen from Fig. 5. The corresponding Q values were listed in Table 1. Straight lines in different colors represent linear fitting to the data points. Using the slope and intercept of each fitting line, the correlation lengths of tested amorphous materials were extracted. by assuming as a radius of gyration for the correlated spheroidal ordered region (Gibson et al., 2000). In this work, W would be between 2.6 and 2.8 nm for BMGs, and ∼2.2 nm for a-Si 3 N 4 , which lies in the reported MRO range (0.5–3 nm) (Table 2). It is noteworthy that the pair persistence model predicts a monotonically decreasing V with the increase of R, which is inconsistent with our experimental results. Nevertheless, in order to compare with previous reported FEM results, we still used this method for the calculation of correlation lengths in this work. The obtained correlation lengths of a-Si 3 N 4 (0.69 nm) and BMGs (0.81–0.88 nm) are consistent with correlation lengths of amorphous materials determined in previous researches (e.g. 0.3–0.6 nm for a-Si (Bogle et al., 2010), 0.6–0.9 for CuZr (Hwang and Voyles, 2011)). Therefore, the calculation is reliable, although better understanding of the experimental results would need more advanced computation models as suggested by Stratton and Voyles (2008). 4.2. Size effect of objective aperture on the normalized intensity variance of FEM images As presented in Fig. 5, the highest variance V(k) peaks were all obtained when 0.6 nm resolution was employed (corresponding to 10 m objective aperture) for the amorphous samples in this work. This is probably due to the following reasons. In dark-field TEM imaging, the bright speckles arise from local ordered regions where coherent diffractions take place between structurally correlated atoms. In variable resolution FEM, the size of objective aperture determines the microscopy resolution in real space (proportional to 1/Q), which sets a lateral limit that the structurally ordered atoms can interfere coherently within. In this way, the size of objective aperture changes the sampling volume of dark-field imaging (Treacy and Gibson, 1996; Treacy et al., 2005). When the aperture size is appropriate, the obtained intensity variance value will reach a maximum because the sampling volume is comparable to the size of ordered region so that all structurally correlated atoms are contained in the sampling volume. A larger sampling volume (corresponding to smaller aperture) will include atoms outside an ordered region, thus decrease the obtained variance due to their incoherent interference. In contrast, a smaller sampling volume (larger aperture) will probe each ordered region multiple times, which also diminishes the variance value according to the
J.W. Deng et al. / Micron 43 (2012) 827–831 831 statistical feature of variance (Treacy, 2007; Treacy et al., 2005; Voyles and Abelson, 2003). Small objective apertures, which bring the nominal resolution R to the MRO length scale (0.5–3 nm), were suggested for FEM experiments to obtain reliable results (Treacy, 2007; Treacy et al., 2005). In this work, the correlation length is between 0.69 and 0.88 nm, while the characteristic width W is between 2.2 and 2.8 nm. This indicates that the maximum variance is obtained when the nominal resolution R approaches the correlation length rather than the characteristic width of MRO. This consists with recent variable resolution FEM experiment on Cu–Zr metallic glass in the STEM nanodiffraction mode (Hwang and Voyles, 2011). By changing the probe size, FEM resolution was adjusted in STEM. Their results show that the maximum variance values increase monotonically as the resolution goes down to 0.8 nm, which is significantly smaller than the MRO length usually accepted but close to the correlation length of their samples (0.68–0.89 nm). Unfortunately, since there is no experiment results for even smaller probe size, the downward trend of variance V(k) is not available for further decrease of the resolution in STEM. Compared with previous FEM works on a-Si, a-Ge, etc., much smaller resolutions have been used for the BMG and a-Si 3 N 4 samples in this study and these resolutions work adequately for the samples. Nevertheless, since different amorphous materials may have different MRO length scales, the appropriate resolution for samples in this work might not be the optimum condition for other materials. Hence, it is necessary to resolve the appropriate resolution for studied materials prior to conducting FEM experiments in TEM as suggested by Treacy (2007). This is in fact also a motivation for us to conduct the variable resolution FEM study here. 4.3. Comparison between MROs of metallic and covalent bond amorphous materials Although quantitative comparison of FEM results between different materials is hard due to influences from such as sample thickness, imaging conditions (Daulton et al., 2010; Yi and Voyles, 2011), a preliminary comparison of MRO lengths and magnitudes between covalent bond a-Si 3 N 4 and metallic bond BMGs is still possible especially when experiment configuration was kept consistent. Firstly, the variance peaks of BMGs are greater than that of a-Si 3 N 4 as shown in Fig. 5. Since the variance value corresponds to the extent of structural ordering, the greater variance indicates a higher ordering or an inhomogeneity in the BMG samples. Secondly, the correlation lengths of BMGs (0.81–0.88 nm) are larger than that of a-Si 3 N 4 (0.69 nm). However, in order to compare the MRO length in a relative scale, each correlation length was divided by the mean atomic radius r of the amorphous material (Table 2). Here, the mean atomic radius was calculated from Goldschmidt atomic radii of elements and chemical composition of each material (Gale and Totemeier, 2004). Although a-Si 3 N 4 has a much smaller correlation length, its /r value is in fact similar to or even slightly larger than that of BMGs. The spatial heterogeneity/homogeneity of BMGs on the nanoscale is also an interesting research topic. Researchers have suggested that the heterogeneity in BMG can play an important role during the initiation of shear bands and the subsequently plastic deformation process (Murali et al., 2011; Wang et al., 2012). However, the main focus of present work is to determine the length scale of medium range order in different amorphous materials. In order to reveal the structure character of heterogeneity in BMGs, which is beyond the scope of this work, more experiments and analysis would be needed. 5. Conclusions Variable resolution FEM has been successfully performed in TEM by varying the objective aperture size in a hollow-cone darkfield imaging mode on metallic and covalent bond amorphous materials. Correlation lengths of = 0.69 nm for covalently bonded a-Si 3 N 4 and = 0.81–0.88 nm for BMGs were extracted by fitting the variance values at different resolution situations. The experiment reveals that maximum intensity variance was obtained when the nominal resolution of FEM images approaches the correlation length of the studied amorphous materials. Acknowledgments The authors thank B. Wu for technical supports and F.F. Wu, Q.P. Cao and J.Z. Jiang for sample preparation. The authors would also like to thank the Hundred Talents Project of Chinese Academy of Sciences, the Cheung Kong Scholars Programme of China, the Natural Sciences Foundation of China (Grant Nos. 50671104 and 50921004) and the Special Funds for the Major State Basic Research Projects of China (Grant No. 2009CB623705) for the financial support. References Alamgir, F.A., Jain, H., Williams, D.B., Schwarz, R.B., 2003. Micron 34, 433. Bogle, S.N., Nittala, L.N., Twesten, R.D., Voyles, P.M., Abelson, J.R., 2010. Ultramicroscopy 110, 1273. Cheng, Y.Q., Ma, E., 2011. Prog. Mater. Sci. 56, 379. Cockayne, D.J.H., 2007. Annu. Rev. Mater. Res. 37, 159. Cowley, J.M., 2004. Micron 35, 345. Daulton, T.L., Bondi, K.S., Kelton, K.F., 2010. Ultramicroscopy 110, 1279. Egerton, R.F., 1996. Electron Energy-Loss Spectroscopy in the Electron Microscope, 2nd ed. Plenum Press, New York. Elliott, S.R., 1991. Nature 354, 445. 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