Biomedical Engineering Reference
In-Depth Information
2.6
Small-angle X-ray scattering (SAXS)
The contrast in X-ray scattering arises for the electron density differences.
Therefore, SAXS occurs whenever there are density heterogeneities or fl uc-
tuations at the nanoscale. Thus, SAXS can be used to characterize those
heterogeneities that are present in all biomaterials (polymers, metals and
ceramics) in the form of organized assemblies of molecules, particles, voids
and pores. When examined at large length scales, ~µm, the crystalline and
amorphous regions in semicrystalline polymers are homogeneously distrib-
uted. But, at smaller length scales, 10-100 nm, the crystalline and amorphous
regions are clearly segregated into separate domains. Such phase separa-
tion gives rise to fringed micelles in cellulose, fi brillar morphology in silk,
and lamellar structure in many semicrystalline polymers. The nature of this
domain formation, how the phase segregation depends on processing condi-
tions and how it affects the properties, can be investigated by SAXS. Typical
applications of SAXS in biomaterials include the determination of the size
in solution of macromolecules, nanoparticles, vesicles and polymersomes,
organization of the crystalline or organized assembly of polymer chains in
solid materials, and in general the study of phase separation characteristics.
SAXS requires a different arrangement than for the wide-angle X-ray
diffraction to permit the data to be collected close to the primary beam.
SAXS requires a highly collimated beam, a long distance between the sam-
ple and the detector, and evacuation of the beam path from the sample to
the detector. An example of a SAXS apparatus is shown in Fig. 2.4b.
2.6.1 Size of proteins and other particles in solution
The use of SAXS to analyze polymers in solution is in some ways similar to
light scattering. Figure 2.14 shows an example of the SAXS data that can be
obtained from a solution of proteins. These data can be used to calculate the
radius of gyration (
R
g
) from the observed intensity
I
(
Q
),
Q
being the scat-
tering vector, (4
π
sin
θ
)/
λ
, using Guinier approximation
22
R
Ie
QR
g
/
3
I
[2.22]
()
()
Q
−
=
R
g
is a measure of the size of the molecule. It is the root mean square dis-
tance within the molecule from its center of mass:
∫
∫
∑
∑
2
dv
2
2
rdv
r dv
fr
f
ρ
()
r
r
kk
ffr
2
=
[2.23]
R
=
=
g
∫
∫
dv
()
r
dv
ρ
k
where the fi rst expression is calculated for discrete atoms of atomic num-
ber
f
k
at distances
r
k
from the origin, the second for homogeneous particle
