Biomedical Engineering Reference
In-Depth Information
In this expression, the ODF is normalized to unity. If it is not, as is usually
the case when the distribution function is replaced by variation in X-ray dif-
fraction intensity around the azimuthal angle α,
I
( α), then
π
∫
() () (
)
I
i
P
d
P
P
(
)
d
α
)
(
(
)
n
0
[2.16]
P
=
π
∫
() ()
I
d
α
)
(
0
For
n
= 2, the Legendre polynomial reduces to Hermans' orientation func-
tion commonly referred to as
P
2:
1
2
(
)
2
[2.17]
P
P
3
2
1
=
3
P
2 is adequate for cases involving fi ber symmetry. IR dichroism and bire-
fringence (refractive index) measurement yield
P
2, Raman and fl uorescence
measurements yield
P
2 and
P
4. In contrast, XRD and NMR measurements
yield the complete ODF. In systems without cylindrical symmetry, the ODFs
can be expressed in terms of generalized spherical harmonics, of which
Legendre polynomials are a subset.
26
When semicrystalline polymers are deformed, both crystalline and the
amorphous domains get oriented. Thus, in addition to the crystalline ori-
entation, it is also necessary to determine the amorphous orientation. In
general, the degree of amorphous orientation is smaller than the crystalline
orientation. Amorphous orientation is especially important in fi bers and
oriented fi lms, more so than the crystalline orientation. Amorphous orienta-
tion contributes to the anisotropic properties such shrinkage upon exposure
to heat and moisture (solvent), and enhancement of the strength and mod-
ulus along the direction of orientation, and it affects the diffusion of host
molecules into the polymer. It has been shown that amorphous components
in polymers can be separated into isotropic (unoriented) and anisotropic
(oriented) components.
15-17,27
Optical birefringence, spectroscopy (NMR
and IR), and diffraction are commonly used to determine the amorphous
orientation.
X-ray diffraction is perhaps the only technique that can provide a com-
plete orientation distribution function (ODF) that can also be presented
in the form of a pole fi gure directly. Roe and Krigbaum have described a
method for calculating the crystallite orientation, that is, orientation of any
axis within the crystal, from the orientation distribution of the normals to
the different crystallographic planes (pole fi gures).
28-30
The detailed infor-
mation given by such a pole fi gure inversion method is seldom required to
understand basic material properties, and hence will not be discussed here.
