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

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