# Illuminating tiny structures: An introduction to small-angle scattering

We are surrounded by phenomena caused by the scattering of light. When enjoying a sunny day at the seaside, like in the photo at the top, why is the sky blue? Blue light scatters more than red light. Why is milk opaque? Protein and fat particles scatter light. If you are reading this with blue eyes, your eye color is due to light scattering. Scientists use the same general scattering principle to study the structure of soft materials using the scattering of well-defined radiation. Scattering measurements reveal structures between an ångström and hundreds of nanometers, an important region for studying soft matter. Just as the color of the sky results from light scattered by air molecules, the scattering of X-rays and neutrons tells us about the size and shape of compounds in soft materials along with their interactions, and I will focus on these two types of radiation.

In a small-angle scattering experiment (SAXS when using X-rays and SANS when using neutrons), a sample (whether solution, dispersion, or solid) is placed between an incoming beam of radiation and a detector (see Figure 1) [1]. The detector measures the scattering intensity as a function of angle, which in turn can be related to the size and shape of the sample’s components.

Scattering intensity is quantified as a function of the momentum transfer vector (or scattering vector) $\mathbf{Q}$, which is simply the difference between the momentum of the incoming  beam ($\mathbf{k_i}$) and the scattered beam ($\mathbf{k_s}$). The magnitude of $\mathbf{Q}$ (equal to $(4 \pi \sin{\theta}) / \lambda$) depends on the scattering angle ($2 \theta$) and the wavelength of the radiation ($\lambda$) [2].

The relationship between the magnitude of $\mathbf{Q}$ and the length scale being investigated ($d$) is given by the equation $Q = 2 \pi / d$, and this inverse relationship to distance is why measurements as a function of $Q$ are said to be in reciprocal space. (This relationship comes from Bragg’s law for crystals [3].) In real space, the arrangement of objects is described by the distances between them. In reciprocal space, the same arrangement would be given by $Q$. The scattering process is essentially a Fourier transform [4], a mathematical procedure to convert waves to frequencies, making it possible to go between real and reciprocal spaces.

Now, having established some of the fundamentals of waves and scattering, we need to think about how radiation interacts with materials. This interaction determines the way that scattering data look and also what information you can obtain. Specifically, X-rays interact with electrons, and neutrons interact with nuclei. The magnitude of the interaction is quantified by the amount per volume (the “scattering length density”), and a difference in scattering length density between solutes and solvents results in detectable scattering. The scattering length density can be thought of as the “refractive index” for neutrons or X-rays. Figure 2 shows the general idea of contrast for any scattering experiment, with a comparison between homogeneous and heterogeneous solutes. When the color of the solvent and particle are the same shade of blue (meaning they have the same scattering length density), there is no scattering from that component. When the colors are different (meaning that they have different scattering length densities), there is now scattering.

Interactions between X-rays and materials are fixed by their composition. However, neutrons interact differently with different isotopes of the same element. One particularly useful difference, especially for soft organic materials, is for structures with protons (1H) and those where protons are exchanged for deuterons (2H or D) [5]. It is possible to detect scattering from specific parts of complex systems by tuning the contrast of different components using solvents with different contrasts, as shown in Figure 2. Figure 3 shows calculated SANS (left) and SAXS (right) curves from chemically identical but isotopically distinct microemulsions, which are nanometer-sized droplets of water that are surrounded by surfactants in oil [6]. For the SANS curves, the dark grey areas represent deuterated oil or water (called D2O or heavy water), and the light grey areas represent standard hydrogenous oil or water. Multiple contrasts are required for a heterogenous particle, because no single solvent can match the whole particle (Figure 2, right). The curves are all different, but the actual structure of the microemulsion is always the same. It is only the contrast that differs. A precise determination of the structural dimensions of the microemulsion can be determined by analyzing all the data together, which gives more certainty than considering any one alone.

Scattering data is analyzed by comparing experimental data to known equations for how different shapes should scatter radiation as a function of $Q$. Luckily, many models are already known [7] for various shapes. However, as structures are determined somewhat indirectly from the scattering data, complementary information from other techniques, such as microscopy, is frequently obtained as well. The equation relating the shape and scattering as a function of $Q$ is called the “form factor”, an intraparticle property. For example, the curves in Figure 3 were all calculated using a form factor for a particle consisting of a spherical core surrounded by a shell of another material. At the extremes of $Q$, approximations can be used to calculate the size (at low-$Q$) and the interfacial roughness (at high-$Q$). In addition, the size distribution usually must be considered. In the example in Figure 3, the radii of the cores do not have a single value. There is a size distribution of about 20%.

For more concentrated dispersions or for more strongly interacting systems, the interactions between the particles must be considered. In these cases, an interparticle property called the “structure factor” contributes to the shape of the scattering curves. It can be calculated for particles that are considered to interact as hard spheres, charged spheres, or attractive spheres. Often samples are studied dilute to avoid the structure factor contribution. However, for systems that are necessarily concentrated or charged, it must be accounted for.

In this post, I focussed on the fundamentals of scattering with examples from nanoparticle dispersions in dilute conditions. However, these are not the only kind of soft materials that can be studied. Polymer solutions and blends, complex fluids, liquid crystals, gels, and a variety of biological materials (such as proteins and nucleic acids in solution or lipid bilayers) can be studied using small-angle scattering. The properties of soft materials often emerge out of their structures. By characterizing them in a quantitative way, scientists can determine the relationships between structures and their functions. Using small-angle scattering, we can not only better understand materials but also better predict ways of improving them. Small-angle scattering should be one of the tools employed by everyone interested in soft materials.

The featured image at the top is a photo of Bellevue Beach in Klampenborg in Copenhagen, Denmark. (Image taken by the author.)

[1] A detailed discussion of detection and instrumentation is beyond the scope of this post, but the book chapter “Small-Angle Neutron Scattering and Applications in Soft Condensed Matter” is a great introduction if you are interested in learning more. ^

[2] The angles being scattered are truly small, typically on order of 1° or less. By using $Q$, it is both the wavelength and angle that are important, and conveniently measurements performed using different wavelengths can be directly compared. ^

[3] Bragg’s law gives the conditions that a wave is diffracted by a series of planes. In a crystal diffraction measurement, peaks are observed in the data when Bragg’s law conditions are met. In a scattering measurement, where Bragg peaks are seldom observed, the relationship between $Q$ and $d$ is useful as a ruler for the length scale that is being examined. ^

[4] A Fourier transform is a mathematical operation that turns a periodic function, like a wave, into a probability of different frequencies. If, for example, you had a wave with a single wavelength, its Fourier transform would give a 100% probability at that wavelength. ^

[5] Deuterium is the heavy isotope of hydrogen with one neutron and one proton. ^

[6] A microemulsion is not just a small emulsion. Although, both are dispersions of two immiscible liquids, typically stabilized by a surfactant. A microemulsion is thermodynamically-stable and, in the case of a water-in-oil rather than oil-in-water, can be thought of as water-swollen surfactant micelle. Data for Aerosol OT-stabilized water-in-oil microemulsions can be found in the PhD thesis “Phase behaviour and interfacial properties of double-chain anionic surfactants” by Sandine Nave (University of Bristol). ^

[7] The equation for the scattering of a sphere was first given in the early 20th century by Lord Rayleigh who published this in the paper “The incidence of light upon a transparent sphere of dimensions comparable with the wave-length” in Proceedings of the Royal Society of London A in 1910. ^