# A Review on Bitumen Rejuvenation: Mechanisms, Materials, Methods and Perspectives
## 6 Forefront/Vanguard Techniques Facing Complexity in Bitumen
### Probing Complexity
#### 6.2.2 Scattering of Neutrons (SANS), X-rays (SAXS) and Light  
Scattering probes the statistical ensemble of the nano-structures and deals with the diffusion of electromagnetic (or particle) waves by heterogeneities in material systems [93]. At small angles 20, the (coherent) scattering intensity in the so-called \"static approximation\" is given by:  
\[\mathbf{I}(\mathbf{q})\propto\left[\sum_{i}\mathbf{b}_{i}\ e^{i\mathbf{q}\mathbf{R}}\right]^{2} \tag{2}\]  
where \(b_{i}\) is the scattering length of the particle (chemical species) that occupy the position \(\mathbf{R}\) in the material system. For SANS experiments the scattering length density \(\rho(\mathbf{r})\) of the sample is defined as \(\rho(\mathbf{r})=\sum_{i}n_{i}(\mathbf{r})b_{i}\), where \(b_{i}\) is the scattering length of the nucleus of type \(i\), while \(n_{i}(\mathbf{r})\) is the corresponding number density of such nuclei. For X-ray scattering \(\rho(\mathbf{r})=\left(\frac{e^{2}}{mc^{2}}\right)n_{el}(\mathbf{r})\), where \(\left(\frac{e^{2}}{mc^{2}}\right)\) is the Thompson scattering length of the electron, and \(n_{el}(\mathbf{r})\) is the electron number density. By replacing \(b_{i}\) by a locally averaged scattering length density \(\rho_{i}(\mathbf{r})\), (where \(r\) is a variable position vector), it is possible to perform an integration over the sample volume, V:  
\[I(\mathbf{q})\propto\left[\int_{V}\rho(\mathbf{r})e^{i\mathbf{q}\mathbf{R}}d^{3}\mathbf{r}\right]^ {2} \tag{3}\]  
If isotropic samples are considered (i.e., where the orientation effects are averaged, due to the radial symmetry), the scattering intensity \(I(q)\) can be expressed as:  
\[I(q)\propto\int_{0}^{\infty}(\rho(\mathbf{r}))^{2}\frac{\sin q\mathbf{r}}{q\mathbf{r}}4\pi r ^{2}\mathbf{dr} \tag{4}\]  
Very often, it is not immediate from the experimental Small Angle Scattering (SAS) intensity profile to obtaining direct information about the \(\rho(\mathbf{r})\) function by inverse transform methods. In this case, SAS interpretation is based on the choice of suitable models expressed in terms of specified functions, which are capable of furnishing information on specific parameters connected to particular properties of the material system under consideration. The name of the technique is then further characterized by the probe which is used: if the probe is made of X-rays, then SAXS is considered; if the probe is a flux of neutrons, then SANS holds, etc. If the source is light, then it will obviously deal with light scattering. However, it must be noticed that, when dealing with light scattering, another technique, called dynamic light scattering (DLS) also known as photon correlation spectroscopy (PCS), is usually referred to. In such a method, time fluctuations of the scattering intensity as a consequence of the Brownian motion of nano-scatters in a solution are recorded. Then, a time-dependent scattering function is derived to the diffusion coefficient of the particles (the scatters) dispersed in the liquid phase [94,95]. In contrast to this (more widely known) technique, the static light scattering (SLS) configuration resembles the typical scattering apparatus. In the case of wide-angle scattering, higher \(q\) values are considered, which means that shorter distances are explored. In a typical Wide-Angle Scattering experiment, which can use in principle the same sources as small-angle, usually, the scatters are the atoms themselves. Typical interatomic distances are, therefore, probed: in the case of pure crystals, where positional order is dominant, this gives the famous Bragg law:  
\[2d\ \mathbf{sin}\mathbf{\theta}=n\mathbf{\Lambda} \tag{5}\]  
where \(n\) is an integer and \(d\) is the characteristic distance of reticular planes. However, when the order is weak, the principle still holds, and wide-angle scattering can also be used for amorphous materials as in the case of bitumen.