Small-angle X-ray scattering

Johannes Allwang

1 Small-angle X-ray scattering (SAXS)

SAXS is a powerful and popular characterization technique at \(\sim1\) to 100 nm length scales. It provides structural information and has become a standard technique in a wide field of studies, including research on proteins and gels [1], Bernado2007Structural, Papagiannopoulos2019Reorganizations.

Fig. 1.1 shows a schematic representation of a SAXS setup. The incoming X-ray beam is scattered by the sample, in the case of this a NP dispersion. A detector at the sample-detector distance \(SDD\) detects the intensity of the scattered beam on a two-dimensional detector. The scattering vector and momentum transfer are defined as in the light scattering case (Eq. (1.1)) with \(n=1\). Isotropic and elastic scattering are assumed for NP dispersions, and smearing is neglected.

\[\begin{eqnarray} \mathbf{q} = \mathbf{k_f} - \mathbf{k_i}, && q = \left| \mathbf{q} \right| = \frac{4\pi n}{\lambda_l}\sin(\Theta) \tag{1.1} \end{eqnarray}\]

Figure 1.1: Schematic of a SAXS setup.

Azimuthal integration of the 2D detector images gives the \(q\)-dependent intensity curve \(I_{sample}(q)\). Subtraction of the signal of pure buffer leads to the final SAXS scattering curve \(I(q)\).

2 Scattering curve and form factor

The detected scattering intensity depends on the sample structure via the following equation [2]:

\[\begin{equation} I(q) = N_pV_p^2(\rho_p - \rho_s)^2 P(q) S(q) \tag{2.1} \end{equation}\]

\(N_p\) is the number of observed scatterers, \(V_p\) is the detection volume, and \(\rho_p\) and \(\rho_s\) are the scattering length densities of particles and solvent, respectively. \(P(q)\) is the form factor of the sample and depends solely on the structure of the scattering particles. If \(P(q)\) is fully known, the distance distribution function \(p(r)\) that describes the frequency of distances \(r\) within a particle can be calculated, by:

\[\begin{equation} p(r) = \int_0^\infty P(q)\frac{\sin qr}{qr}q^2\mathrm{d}q \tag{2.2} \end{equation}\]

Inverting Eq. (2.2) allows the calculation of the form factor from the sample geometry. Note, that \(p(r)\) can generally not be calculated if forward scattering is present, as is the case for all the scattering curves collected in this \(S(q)\) is the structure factor and depends on the spacing and the interactions between the particles. The particles studied in this work are NPs from proteins where, the determination of an exact \(SLD\) is difficult. Instead of interpreting the system as proteins in a complicated network, we describe the whole NPs as scatterers with a hierarchical internal structure. Therefore, Eq. (2.1) simplifies to:

\[\begin{equation} I(q) = A P(q) \tag{2.3} \end{equation}\]

The scale \(A\) is a constant that depends on the \(SLD\) values of the particles and the solvent, and \(P(q)\) contains structural information about the NPs. The following sections describe different approaches to model the form factor.

3 Model fitting and discrete scattering objects

Model fitting is common practice in SAXS analysis, as it allows a comparison between the \(I(q)\) data and mathematical models. It is also used to find structural parameters like length scales and fractal dimensions. For simple geometric shapes, like spheres, \(P(q)\) can be calculated, whereas for more complicated systems, empirical models are available.

One of the simplest geometrical shapes is the homogeneous sphere and the corresponding form factor can be calculated by finding the distance distribution function \(p(r)\) and then applying the inverse of Eq. (2.2). For spherical scatterers of radius \(R\), volume \(V_{sphere}\) and scattering length contrast to the solvent \(\Delta\rho = \rho_{sample} - \rho_{solvent}\) the form factor reads:

\[\begin{equation} P(q) = \left[ 3\Delta\rho V_{sphere} \times \frac{\sin(qR) - qR\cos(qr)}{(qR)^3}\right]^2 \tag{3.1} \end{equation}\]

Fig. 3.1 shows sphere form factors for spheres of different sizes. The parameters \(\Delta\rho\) and \(V_{sphere}\) are chosen so that \(I(0) = 1\). The resulting curves feature a plateau at low \(q\)-values and decaying fringes starting around \(qR\sim 1\).

Figure 3.1: Sphere form factor for different values of \(R\) (given in the legend). code

4 Guinier and Porod law

In real systems the exact shape of the scattering particles is often unknown or too complex to describe it analytically. In that case, different approximations as well as empirical and fractal models allow the characterization of such systems. The most often used approximations are known as the Guinier law and Porod law. For any shape, the Guinier law describes \(P(q)\) for \(qR_G \ll 1\) with a simple exponential decay:

\[\begin{equation} I(q) \propto \exp\left({\frac{-q^2R_G^2}{3}}\right) \tag{4.1} \end{equation}\]

The most straightforward application of the Guinier law is plotting \(\ln(I)\) vs \(q^2\) (the Guinier plot) and performing a linear fit. The \(R_G\)-value of the particles can then be calculated from the slope of the fit (\(R_G = \sqrt{3\cdot \mathrm{slope}}\)).

The Porod law describes the scattering at \(qR_G \gg 1\) by a power law:

\[\begin{equation} I(q) \propto q^{-n_P} \tag{4.2} \end{equation}\]

In the original Porod law, scatterers with smooth surfaces were considered and the Porod exponent is \(n_P = 4\). Later, the theory was expanded to surface fractals (\(3<n_P<4\), depending on the roughness of the surface), mass fractals (\(1<n_P<3\) [3]), including polymer chains (\(5/3<n_P<3\) [4]). The corresponding Porod exponents are summarized in the following table.

5 Kratky plot

The Kratky plot \(q^2\times I(q)\) vs \(q\) is a commonly used technique based on the porod law [TODO, check this]. It can qualitatively reveal the degree of unfolding in a protein dispersion (Fig. 5.1 (a)). Fully folded proteins have a peak [TODO, fix] while fully unfolded proteins have a plateau or diverge. If the Kratky plot does not go to 0, there are unfolded parts. For polymer chains (Fig. 5.1 (b)) the Kratky plot can reveal the solvent quality as polymers will occupy a more compact structure in worse solvent.

Figure 5.1: Schematic Kratky plots for (a) globular proteins in different conformations and (b) polymers in solution of different quality. code

6 Beaucage form factor

A number of empirical models based on the Guinier and Porod models have been proposed to allow the description of scattering data where the exact microscopic model is unknown. Generally, they reproduce the Guinier law at low \(q\) and the Porod law at large \(q\) with a continuous intermediate region. Those models can be used to fit SAS data and obtain \(R_G\) and \(n_P\) even when their shape and internal structure are unknown. The original Beaucage model was introduced some 27 years ago. However, the practice of allowing both Guinier and Porod prefactors to vary independently in a non-linear least-squares fit gives rise to undesired artifacts. A corrected version of the Beaucage model was introduced in 2010 by Hammouda [3]. The form factor reads:

\[\begin{eqnarray} I_{Be}(q) &= G \exp\left({\frac{-q^2R_G^2}{3}}\right) + \frac{C}{q^{n_P}} \left[\mathrm{erf}\left(\frac{qR_G}{\sqrt{6}}\right)\right]^{3n_P}\\ C &= \frac{Gn_P}{R_G^{n_P}}\left[\frac{6n_P^2}{(2+n_P)(2+2n_P)}\right]^{n_P/2} \cdot \Gamma\left(n_P/2\right) \tag{6.1} \end{eqnarray}\]

The Beaucage model can be used to illustrate how scattering curves depend on the length scale of the studied particles and their fractal dimension. Fig. 6.1 shows the Beaucage form factor for different structural parameters. It features a shoulder roughly at \(q\sim 1/R_G\) that corresponds to the Guinier law (Eq. (4.1)) and a power law decay with exponent \(n_p\).

Figure 6.1: Beaucage form factor for (a) \(n_P=4\) and different \(R_G\)-values (given in the legend) and (b) \(R_G=100 m\) and differernt \(n_P\) values (given in the legend). code

In this , the interior of the NPs can form a gel when the proteins unfold and bond with each other. The Beaucage form factor can still be used to model the data, as it can describe fractal systems as well as compact particles. Using this empirical form factor, therefore, allows an unbiased observation of the internal structure of protein-polysaccharide NPs in the two qualitatively different cases of folded proteins and unfolded proteins as well as partially folded proteins. It can further be used to describe larger structures within the NPs that consist of several proteins and PS molecules. The only situations when this uses a different form factor is when a secondary maximum can be observed, hinting at a compact shape with a smooth surface. In this case a polydisperse sphere form factor is used.

7 Gels

This section is based on Shibayama at al [5].

So far, only the scattering from separate particles has been considered. To describe the scattering intensity of polymer networks in the semidilute regime empirical functions of the following form are used [5]:

\[\begin{equation} I(q) = \frac{I_L(0)}{\left(1 + (q\xi)^m\right)} \tag{7.1} \end{equation}\]

\(\xi\) is the correlation length of local fluctuations and indicates the size of blobs in the blob model. \(m\) is an exponent analogous to the Porod exponent and depends on the polymer solvent interaction. For simple polymer networks with no solvent interaction \(m=2\), for polymers in a good solvent \(m=5/3\) and in a bad solvent \(m=3\). Note, that Eq. (7.1) is simply an empirical function that contains the correct length scale \(\xi\) and exponent \(m\). In this , a Beaucage form factor is used instead because it is more general.

In more densely packed polymer networks, there are long range static like density fluctuations in addition to the local ones. The mean size of these nonformities is called \(\Xi\). The scattering can be described by adding another term to Eq. (7.1).

\[\begin{equation} I(q) = I_G(0)e^{-(\Xi q)^2} + \frac{I_L(0)}{\left(1 + (q\xi)^m\right)} \tag{7.1} \end{equation}\]

The new term is equivalent to a Guinier approximation and could also be exchanged by different form factors, depending on the shape of the unformities.

References

[1] A. G. Kikhney, D. I. Svergun, FEBS Letters 2015, 589, 2570.

[2] L. A. Feigin, D. I. Svergun, Structure Analysis by Small-Angle X-Ray and Neutron Scattering, Springer US, 1987.

[3] B. Hammouda, Journal of Applied Crystallography 2010, 43, 1474.

[4] M. Rubinstein, R. H. Colby, Polymer Physics (Chemistry), Oxford University Press, USA, 2003.

[5] M. Shibayama, T. Tanaka, C. C. Han, 1992, 97, 6829.