Page 56 - Handbook of Properties of Textile and Technical Fibres

Testing and characterization of fibers 37

frequencies. A compressive stress produces a shift in the opposite direction. The strain-
induced Raman wave number shift (Dn) is linearly related to the tensile strain (Dε)so
that we can write the empirical relationship (Colomban, 2002):

ε
Dn ¼ S Dε (2.11)
and the above equation is microscopically analogous to Hooke’s law

Ds ¼ E Dε (2.12)
Young’s modulus, E, is indeed the result, at the macroscopic scale, of the force con-
stant of the various chemical bonds. Consequently we can write:

ε
ε
Dn ¼ S Dε ¼ S Ds (2.13)
In this way it can be seen from Eq. (2.13) that a wave number shift due to an
imposed macroscopic strain can be used to calculate internal stress states of molecular
species. This also means that a wave number shift can be used to calculate internal
stress states. Thus Fig. 2.10(b) shows how scanning across a 26-mm-diameter PA66
fiber reveals a decrease in wave number, and therefore a fall in frequency toward
the center of the fiber. This directly demonstrates that the surface of the fiber is in
compression, with respect to its core, due to the effects of cooling during manufacture.
The value of the compressive residual stresses can be evaluated at about 200 MPa from
the calibration plot.

2.3.5 X-ray diffraction and related scattering techniques

Diffraction is the scattering of waves from a regular array with distances between
layers in the structure similar to the lengths of the incident wave form. At particular
angles the waves scattered from different rows or planes in the material are in phase
and interfere constructively. At other angles the interference leads to a reduction in
intensity so that peaks in intensity are observed at angles for which the scattered waves
are in phase. Fig. 2.11 shows this concept, which is known as Bragg diffraction and
leads to the relationship:

2d sinðqÞ¼ nl (2.14)

This relationship can be understood by considering the geometry of the layers
shown in Fig. 2.11, noting that the incident angle is q, the regular distance between
the structural layers is d, and the wavelength of the incident rays is l.
Wide angle X-ray scattering is the most commonly used technique in which the
specimen is impinged by a monochromatic X-ray beam at q angles usually in the range
3e45 degrees as shown in Fig. 2.11. The diffraction pattern generated allows the
chemical or phase composition of materials to be determined and dimensions of the
atomic structure to be obtained. Small angle X-ray scattering (SAXS), with scattering