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W. Qiu and X.-Y. Liu
(120), and (300) Bragg reflections and an amorphous halo. The meridian data, on the
other hand, is deconvoluted into two more crystalline peaks corresponding to (002)
and (102) as well as an amorphous halo. The crystallinity is determined by the ratio
of the area under the crystalline peaks in the equatorial data (i.e., the (100), (200),
(120), and (300) peaks) to that of the total reflection patterns. According to Scherer’s
formula, the crystalline size in one dimension is equal to
0.9λ
FWHMCOSθ
, where FWHM
represents the full width at half-maximum of the peak at the diffraction angle θ, and
the wavelength of the incident ray λ is 0.15418 nm. The crystallite sizes along the a,
b, and c directions are determined by the position and the full width at half maximum
(FWHM) of the (200), (120), and (002) peaks, respectively [8]. For spider dragline
silk fibers, the crystallite size is measured to be a = 2.1 nm, b = 2.7 nm, and c =
6.5 nm. The dimensions of silkworm silk fibers are relatively larger, i.e. a = 2.3 nm,
b = 4.1 nm, and c = 10.3 nm.
Orientation Function f : Experimentally, information about the orientation of crystallites can be measured via the WAXS intensity integration as a function of the azimuth
angle at the radial position of the equatorial (120) and (200) peaks (Fig. 6.20c). Here,
the orientation function f is defined by the Hermans orientation function, as Eq. 6.1,
f =
3 < cos
2
φ > −1
/2
(6.1)
where φ is the angle between the c-axis of the crystallites and the fiber axis. For
the two reflections, (200) and (120), which are not orthogonal but have a known
geometry in the equatorial plane, the expression of
φ> is determined using the
Eq. 6.2
< cos
2
φ >= 1 − 0.8 < cos
2
φ 200 > −1.2 < cos
2
φ 120 >
(6.2)
The FWHM values of the (200) and (120) peaks were measured in a direction
perpendicular to that of the fiber axis using the following Eq. 6.3,
< cos
2
φ 200 >= 1 − [cos(0.4FWHM 200 )]
2
(6.3)
Thus, the FWHM data can be applied to calculate the orientation function. If f =
1, the β-crystallites are oriented in a direction that is completely parallel to the fiber
axis. However, if f = 0, the β-crystallites are oriented randomly [8].
Inter-crystalline distance: As described by the molecular crystal network structure,
crystallites, which scatter the incident X-rays, are embedded randomly into amorphous regions and have a cylindrical symmetry along the fiber axis. In this regard,
the SAXS intensity in the equatorial direction can be determined using the Eq. 6.4
I =
Kl
2
c
1 + l 2
c q 2
3/2
(6.4)
W. Qiu and X.-Y. Liu
(120), and (300) Bragg reflections and an amorphous halo. The meridian data, on the
other hand, is deconvoluted into two more crystalline peaks corresponding to (002)
and (102) as well as an amorphous halo. The crystallinity is determined by the ratio
of the area under the crystalline peaks in the equatorial data (i.e., the (100), (200),
(120), and (300) peaks) to that of the total reflection patterns. According to Scherer’s
formula, the crystalline size in one dimension is equal to
0.9λ
FWHMCOSθ
, where FWHM
represents the full width at half-maximum of the peak at the diffraction angle θ, and
the wavelength of the incident ray λ is 0.15418 nm. The crystallite sizes along the a,
b, and c directions are determined by the position and the full width at half maximum
(FWHM) of the (200), (120), and (002) peaks, respectively [8]. For spider dragline
silk fibers, the crystallite size is measured to be a = 2.1 nm, b = 2.7 nm, and c =
6.5 nm. The dimensions of silkworm silk fibers are relatively larger, i.e. a = 2.3 nm,
b = 4.1 nm, and c = 10.3 nm.
Orientation Function f : Experimentally, information about the orientation of crystallites can be measured via the WAXS intensity integration as a function of the azimuth
angle at the radial position of the equatorial (120) and (200) peaks (Fig. 6.20c). Here,
the orientation function f is defined by the Hermans orientation function, as Eq. 6.1,
f =
3 < cos
2
φ > −1
/2
(6.1)
where φ is the angle between the c-axis of the crystallites and the fiber axis. For
the two reflections, (200) and (120), which are not orthogonal but have a known
geometry in the equatorial plane, the expression of
φ> is determined using the
Eq. 6.2
< cos
2
φ >= 1 − 0.8 < cos
2
φ 200 > −1.2 < cos
2
φ 120 >
(6.2)
The FWHM values of the (200) and (120) peaks were measured in a direction
perpendicular to that of the fiber axis using the following Eq. 6.3,
< cos
2
φ 200 >= 1 − [cos(0.4FWHM 200 )]
2
(6.3)
Thus, the FWHM data can be applied to calculate the orientation function. If f =
1, the β-crystallites are oriented in a direction that is completely parallel to the fiber
axis. However, if f = 0, the β-crystallites are oriented randomly [8].
Inter-crystalline distance: As described by the molecular crystal network structure,
crystallites, which scatter the incident X-rays, are embedded randomly into amorphous regions and have a cylindrical symmetry along the fiber axis. In this regard,
the SAXS intensity in the equatorial direction can be determined using the Eq. 6.4
I =
Kl
2
c
1 + l 2
c q 2
3/2
(6.4)