Techniques for the characterization of physical gels - Physical Gels from Biological and Synthetic Polymers - page 22

2.2.2 Scattering by a single particle

-

Measuring the intensity scattered by a single particle I 1 (q) in other words, the intensity

measured in very dilute solutions cation of the structural unit or the

block which builds the framework of the gel. The angular dependence of the scattered
radiation by particles having simple shapes with well-de

- allows the identi

ned geometries is well known,
for instance for spheres, hollow spheres, ellipses, discs, cylinders, hollow cylinders,
polymer random coils, semi-

exible polymers and micelles (Burchard, 1983 , 1994 ).
From the measurement of intensity versus scattering vector in dilute solutions we can
derive not only the radius of gyration of the particles but also further information such as
the thickness of a disc, the cross-section of a rod or the persistence length of a polymer
coil. The limit of very small scattering vector is called the Guinier range; the scattered
intensity then follows a simple law, whatever the shape of the particles:

!

q 2 R g
3

I 1 ð q Þ/ exp ; qR g 1 : ð 2 : 9 Þ

In principle, in dilute solutions, when the logarithm of the intensity is plotted versus q 2 ,
the initial slope gives

R g =

3, so the particle radius of gyration is derived from the
negative slope (Guinier approximation). For the case of X-rays the radius of gyration is
the mean square distance from the centre of gravity, where the role of mass is played by
electrons. The overall intensities I 1 (q) scattered by single particles are monotonically
decreasing functions of the scattering vector , as shown on a double logarithmic scale in

Figure 2.2 . This also shows the difference of the scattered intensity I 1 (q) for particles of

q

1

1 Rod
2 Gaussian coil
3 Sphere
4 Guinier plot

10 − 2

1

2

3

4

10 − 1
10 − 1

1 10

〈 R g 〉 q

Figure 2.2

Scattering of single particles of various shapes: rod, Gaussian coil and sphere reduced to the same
radius of gyration, in a double logarithmic plot. The Guinier plot ( 2.9 ) is shown as a dashed line
( ln I ð q Þ¼ 1 h R g i q 2 = 3 Þ ). It is valid for any particle shape when qR g <1.