What is measured in a Small Angle X-ray Scattering (SAXS) ?

X-rays are used to investigate the structural properties of solids, liquids or gels. Photons interact with electrons, and provide information about the fluctuations of electronic densities in the matter. A typical experimental set-up is shown on Figure 1 : a monochromatic beam of incident wave vector k_i is selected and falls on the sample. The scattered intensity is collected as a function of the so-called scattering angle 2θ. Elastic interactions are characterised by zero energy transfers, such that the final wave vector k_f is equal in modulus to k_i. The relevant parameter to analyse the interaction is the momentum transfer or scattering vector q=k_i–k_f, defined by :

The scattered intensity I(q) is the Fourier Transform of g(r), the correlation function of the electronic density r(r), which corresponds to the probability to find a scatterer at position r in the sample if another scatterer is located at position 0 : elastic x-ray scattering experiments reveal the spatial correlations in the sample. Small angle scattering experiments are designed to measure I(q) at very small scattering vectors q»(4p/l)q, with 2q ranging from few micro-radians to a ten of radians, in order to investigate systems with characteristic sizes ranging from cristallographic distances (few Å) to colloidal sizes (up to few microns).

· Electronic contrast. The number of photons scattered by one sample is proportional to its total volume V and to its electronic contrast . In the simple case of a binary system for instance, like scattering objects of density r₁ embedded in a solvent of density r₂, the electronic contrast is Dr=r₁-r₂. The higher the contrast between particles and solvent, the greater the signal.

· Absolute intensity. The experimental intensity is usually fitted in order to determine the density r(r), the size, the shape and the internal structure of one elementary scatterer, as well as the structure and the interactions between scatterers. The determination of physical quantities, such as molecular weight, particle volume, specific surface or osmotic compressibility, is feasible only if I(q) is measured on absolute scale. As shown on figure 1, a part of the incident red beam is absorbed in the material. The number of photons scattered in the solid angle DW in the direction 2q have to be normalised with respect to the number of photons transmitted through the sample (green beam). This imply to control several parameters : the sample thickness e (cm) and transmission T, the incident flux of photons f₀(photons/s) and the solid angle of the experiment DW. The absolute intensity can then be measured in absolute units (cm^-1).

What does « q-range » mean ?

A typical small angle scattering intensity profile is shown on figure 2a. The intensity is plotted versus q, in the range qmin-qmax defined by the experimental set-up and usually fixed by geometric limitations

Intuitively, a measurement made at a given q₀ allows to investigate the density fluctuations in the sample on a distance scale D₀=2p/q₀. It is equivalent to observe the system through a 2p/q₀ diameter « window » in real space, as shown in figure 2b. The red circle is the observation window. A scattering signal is observed if the contrast Dr inside the circle is different from zero. To study objects much smaller or much larger than D₀=2p/q₀, another « window » has to be chosen. The smallest (largest) observation window is given by D_min(max₎=2p/_qmax(min) : it determines the size of the smaller (bigger) particles that can be observed with the instrument.

Figure 2 :

left: Example of scattering intensity profile measured between qmin and qmax.
Right: Binary sample and « q-window » corresponding to a measurement at a given q₀.

High q domain :

The window is very small : there is a contrast only at the interface between the two media. This domain, called the Porod’s region, gives information about the surfaces.

Intermediary zone :

The window is of the order of the elementary bricks in the systems. The form factor P(q) can be measured (size, shape and internal structure of one particle).

Low q domain :

When the observation window is very large, the structural order can be obtained : it is the so-called structure factor S(q), which allows to calculate the interactions in the system.

Porod’s law : specific surface and interface

When two media are separated by a sharp interface, the scattered intensity follows an asymptotic law in the high q region : I(q)=Aq-4+B. This law is called the Porod’s limit (and is not verified any more for more complicated interfaces). The asymptotic value, when the electronic contrast of the sample is known, and when the intensity is expressed in absolute scale, allows to calculate the specific surface S of the particles.

The q-range of validity of a Porod’s law can cover several decades, giving information about the particle sizes. When various types of particles, with different characteristic scales, exist in the sample, it is possible to measure different Porod’s law, as shown in the following example. Borosilicated glass are alterated and studied as a function of time, in order to investigate the alteration process. The initial glass before alteration can be represented as a two-media system, with large grains distributed in the solvent. The scattering signal shown in red is a Porod’s law resulting from the sharp interface between the solvent and the grain, ranging from 10^-4 to 0.1 Å^-1. When the sample is alterated a second Porod’s law appears at larger q : small pores are created in each grain. It is the sharp interface pore-grain that gives rise to that second law. As the pore’s size increases with time, the Porod’s law after 8 weeks is shifted to smaller q respect to the 2 weeks alteration glass signal. In this particular case, measuring the intensity on absolute scale is critical to calculate the pore’s surface (and radius, if a spherical shape is assumed), as well as the grain’s surface.

When two media are separated by a sharp interface, the scattered intensity follows an asymptotic law in the high q region : I(q)=Aq^-4+B. This law is called the Porod’s limit (and is not verified any more for more complicated interfaces). The asymptotic value, when the electronic contrast of the sample is known, and when the intensity is expressed in absolute scale, allows to calculate the specific surface S of the particles.

Figure 3 : USAXS on borosilicated glasses.

Form factor P(q) : size and shape of particles

Generally, the size and the shape of the particles is of fundamental interest to understand a material. X-ray small angle scattering gives valuable information, providing that the contrast is sufficient. If the particles are of the order of one micron, or larger, the results can be checked by light scattering. If the system is complex (three different media for instance), it can be completed by neutron scattering, which offers the possibility of contrast variation methods.

Figure 4 : USAXS determination of the form factor of silica spheres.

Experimentally, form factors can only be measured in the dilute regime where particles can be considered as independent scatterers without any interactions. In this case, the intensity is directly proportional to :

the contrast Dr.
the volume fraction f and the volume of one particle V_P.
the form factor of a single particle P(q).

· Elementary shapes. Form factors are easily calculated for spheres, cylinders, disks, rods, micelles, lamellas or Gaussian polymers, which are the basic shapes encountered in soft condensed matter. However, the only analytical expression is the sphere form factor that is represented on Figure 5.

· Example. The experimental intensity scattered by a diluted solution of silica spheres of radius R=3000 Å is shown above on figure 4. The experiment has been performed on our laboratory Bonse/Hart camera. The data are compared to the theoretical form factor P(Q). Experimental data are smoothed compared to the model because of the instrument resolution (and some polydispersity). The location of the first minimum gives the radius of the particles : qR=4.5. A very good fit is obtained in this case with R=3000 Å. The extrapolation of the absolute intensity at q=0 allows to calculate the volume fraction of particles in the solvent.

Structure factor S(q) : interactions between particles

Complex systems are described through interaction potentials. Are the interactions attractive or repulsive, electrostatic or not ? is there any long range order in the system ? What is the effect of temperature, salt or pressure on the equilibrium ? SAXS experiments combined with adapted models bring some answers, provided the structure factor S(q) can be extracted from the scattered intensity I(q).

For instance, in the case of centrosymmetric identical particles in solution, the signal due to one single particle can be dissociated from the signal arising from the interactions between particles. The intensity is then proportionnal to the product P(q)*S(q).

The limit S(q=0) gives the osmotic compressibility of the sample, but this quantity is only accessible if absolute intensities can be measured. When there is no interaction, as in the previous example of Silica spheres in diluted regime, the structure factor is equal to 1. Repulsive electrostatic nteractions appear between these charged silica spheres when one concentrates the sample. The experimental signal is reported in Figure 5: at large q, where the « observation window » mainly shows the particle itself, the form factor of a sphere is predominant. The intensity profile in this region is identical to the one mesaured in diluted regime. However, an interaction peak appears at small q, the position of which is related to the main first neighbour distance D between the silica spheres. At low q the signal results from the competition of P(q) and S(q).

By dividing I(q) by P(q), the experimental structure factor is obtained. A model taking into account electrostatic repulsions and Van der Waals attractive potential is shown in blue. Fitting the data with the model allow to quantify the interaction potential.

Technical potential: a view of our possibilities

Figure 6 : Intensity versus scattering vector : the technical platform, combining three different cameras, allow to measure signals from 3.10^-4 Å^-1 up to 6 Å^-1, and intensities varying from few 10^-2 to 10⁴ cm^-1. The D-scale gives the corresponding direct measure of chracteristic sizes in the system : from few Å to few microns.

The three different (I,q) regions accessible with our cameras are drawn in pastel colors on figure 6, and compared to the USAXS performances at the ESRF. There is an overlap between the 2D cameras and the Bonse/Hart camera, which allows a good recovering of data on four decades in q and nine decades in intensity. The combination of the three instruments constitute a powerful tecnical platform, with a high sensitivity and complementarity.

Some typical materials studied in soft condensed matter are located on the same (I,q) diagram: we are able to measure scattering from colloidal crystals at very small q, mesoporous materials at large q, or water and molecular liquids at very small intensities.

Conclusion : absolute intensity

Absolute intensities are experimentally very difficult to obtain, but crucial to quantify a system. We have shown how specific surfaces, for instance, can be extracted from Porod’s law only if the intensity is given in absolute units (cm^-1). Figure 7 illustrates for instance the case of TiO₂ powders used in the SYNROC australian program. These materials are constituted by an assembly of hierarchical aggregates : the elementary bricks are 8 nm grains, which form aggregates of 100 nm, themselves contained in larger aggregates. Experimentally we have observed two Porod’s limits corresponding to the two smallest sizes in the system. The use of two different cameras was necessary to cover the whole interesting q-range. We are showing here the scattering in absolute units measured on the same sample, on the three cameras available at the laboratory. The measurements have been made completely independently one from the other: an excellent connection is observed between the three data sets. The shape of the signal, as well as the intensities, are identical. The first cristallographic Ti-Ti peak is obtained at large q.

Figure 7 : Superposition of scattering data obtained on the three cameras on TiO₂ powders. A schematic representation of the system is represented on the right. The red grains give rise to a Porod's law at large q. The corresponding specific surface is calculated.

These results confirm the complementarity of the three machines, and the coherence of the measurements : absolute intensities are measured with a very good reliability on a large q-scale. As shown on figure 7, where the first Ti-Ti peak is evidenced at high q, cristallographic peaks can also be measured.