Small-Angle X-ray Scattering by Nanostructured Materials

Abstract

This chapter contains the basic theory of small-angle X-ray scattering (SAXS) and its applications to low-resolution studies of nanostructured materials. The primary purpose is to explain how to obtain structural information from simple systems whose low-resolution structure can be described by a two-electron density model, consisting of either homogeneous nanoparticles embedded in a (solid or liquid) medium with constant electron density or two-phase bicontinuous systems. The presented SAXS theory and the examples of applications refer to different procedures for determinations of geometrical parameters associated to nanoparticles or clusters in dilute solution, spatially correlated nanoparticles, and more general two-phase systems, namely, particle radius of gyration, interface area, size distribution, fractal dimension, and interparticle average distance. Other described applications are in situ SAXS studies of mechanisms involved in transformation processes leading to nanostructured materials such as those occurring in nanophase separation and along the successive steps of sol–gel routes. One section is dedicated to present the basic concepts and describes an application of grazing incidence small-angle scattering (GISAXS), which allows for studying nanostructured thin films and thin layers located close to the external surface of solid substrates. Most of the reported applications refer to nanostructured materials obtained by sol–gel processing and are based on experimental results published by the author and collaborators.

Appendix: Experimental Issues

Basic Comments

Monochromatic X-ray beams are characterized by their photon energy E or wavelength λ, both related by λ = hc/Ε, where h is the Plank constant and c is the speed of light in vacuum, i.e., λ(Å) = 12.398/E(KeV). The wavelengths of typical monochromatic beams used in SAXS experiments are within the range 0.6–2.0 Å circa (i.e., photon energies ranging from ~6 to ~20 KeV). The X-ray beams produced by synchrotrons or typical commercial sources are usually monochromatized by quartz, germanium, or silicon single crystals, which yield incident beams with very narrow pass-bands (Δλ/λ < 10⁻³).

Considering, for example, a typical SAXS experiment with an X-ray wavelength λ = 1.542 Å (λCuKα), a sample-to-detector distance D = 1 m, a beam-stopper with a diameter ϕ ₁ = 5 mm and a circular 2D detector with a diameter ϕ ₂ = 150 mm, and remembering that $q=\left(4\pi /\lambda \right) \sin \theta \approx \left(2\pi /\lambda \right).2\theta$ for low q, the range of scattering angles to be covered results 0.14° < 2θ < 4.3°, and the corresponding minimum and maximum q values are 0.01 Å⁻¹ and 0.30 Å⁻¹, respectively. Different lower and upper q limits can be reached by selecting adequate beam collimation, sample-to-detector distances and/or X-ray wavelengths. The choice of the experimental q range depends on the sizes of the nanoparticles to be studied.

X-ray beams for SAXS experiments are produced by classical sealed X-ray tubes, rotating anode X-ray generators and synchrotron sources. Synchrotron radiation sources are often preferred because they provide powerful, continuously tunable and well collimated X-ray beams. Another closely related experimental technique often used for same or similar purposes is small-angle neutron scattering (SANS), its basic theory being essentially the same as that developed for the SAXS technique.

Choice of Sample Thickness

Classical SAXS experiments are performed in transmission mode and usually under normal incidence. The first step for planning SAXS experiments is to determine the sample thickness that maximizes the scattering intensity for a given material and photon energy. The SAXS intensity produced by any material with arbitrary structure, as a function of sample thickness t, is given by

$$ I(t)\propto t{e}^{-\mu \rho t} $$

(48)

where ρ is the mass density and μ the mass X-ray absorption coefficient, which is a function of chemical composition of the material and photon energy. The absorption coefficient can be obtained from tables published by Henke et al. (1993) or by using an online program accessible in their web page.

Examples of the function defined by Eq. 48 are plotted in Fig. 17 for three different materials. The optimum thickness t_max corresponding to the maximum of the I(t) function is

Fig. 17

(a) Examples of scattering intensities in arbitrary units as functions of sample thickness for an incident X-ray beam with a wavelength λCuKα = 1.542 Å, corresponding to different selected materials: Cu, SiO₂, and H₂O, whose optimum thicknesses t _max are 22 μm, 132 μm, and 1.00 mm, respectively

$$ {t}_{\max }={\left(\rho \mu \right)}^{-1} $$

(49)

This implies that the transmittance of samples with optimum thickness is

$$ T=\left({I}_{\mathrm{transmitted}}/{I}_{\mathrm{incident}}\right)={e}^{-1}=0.37. $$

(50)

Notice that the t _max values determined by Eq. 49 are just a guide for a convenient choice of sample thickness. However, it is always advisable to avoid the use of very thick or very thin samples which would lead to high absorption and low probed volumes, respectively, both yielding weak scattering intensities.

For samples containing large fractions of high Z atoms the optimum thicknesses could be extremely low using CuλKα photons (E = 8.04 KeV). For these materials X-ray beams with higher photon energy should be employed. On the other hand, in order to minimize fluorescence effects, the use of beams with photon energies above and close to absorption edges of sample elements should be avoided.

Subtraction of Parasitic Scattering

Before further analysis of experimental SAXS results, a pretreatment of rough data is required. For anisotropic 2D SAXS patterns, the vector $\overrightarrow{q}$ associated to each detector pixel is calculated. For isotropic 2D SAXS patterns, the scattering intensity is defined as a function of the modulus of the scattering vector, which is determined by circular averaging.

In order to subtract the parasitic scattering intensity produced by slits, cell windows, and air, two SAXS patterns should be recorded: (i) the total scattering intensity (from sample plus parasitic scattering) defined by the counting rate ${R}_T\left(\overrightarrow{q}\right)$ and (ii) the parasitic scattering intensity given by the counting rate ${R}_P\left(\overrightarrow{q}\right)$ recorded under same experimental conditions but without sample. The scattering intensity exclusively related to the sample is given by

$$ R\left(\overrightarrow{q}\right)=\left[\left({R}_T\left(\overrightarrow{q}\right)-{R}_D\right)/{T}_{S+W}\right]-\left[\left({R}_P\left(\overrightarrow{q}\right)-{R}_D\right)/{T}_W\right] $$

(51)

where R_D is the counting rate associated to the detector noise, T_S+W is the transmittance of the sample and cell thin windows, and T_W is the transmittance of the empty sample cell. For solid samples placed in a windowless holder, we have T _W = 1. Often the counting rate associated to parasitic scattering for macromolecules in dilute solution is determined with the sample cell filled with same buffer, thus under this condition the scattering intensity due to statistical density fluctuations in the solvent is also subtracted.

When SAXS experiments are conducted using synchrotron beam lines with continuously decreasing electronic current, the effects of time variation of the intensity of the incident X-ray beam should be properly accounted for.

C orrection of Smearing Effects

The use of X-ray incident beam with rather large cross-section and/or X-ray detectors with large pixel size may produce serious smearing effects on the SAXS curves. However, most of the modern commercial setups and synchrotron beam lines provide an incident beam with pinhole-like cross-section and use X-ray detectors with very small pixel size, thus often making mathematical desmearing procedures unnecessary.

When using commercial setups yielding an incident X-ray beam with large cross section (for example a beam with linear cross-section), two approaches can be applied for quantitative analyses: (i) fitting the theoretical model of SAXS curve to previously dismeared experimental functions or (ii) fitting the previously smeared theoretical model of SAXS curve to the experimental function. Since mathematical desmearing of experimental SAXS patterns leads to results with rather high statistical noise, the second procedure is generally preferred.

Determinations of SAXS Intensity in Relative and Absolute Units

For pin-hole collimation of the incident beam, the counting rate $R\left(\overrightarrow{q}\right)$ corresponding to the X-ray photons scattered by the sample is proportional to the $I\left(\overrightarrow{q}\right)$ function used along this chapter. Thus Eq. 51 directly yields the scattering intensity in relative scale or arbitrary units, to which model functions are fitted after adequate scaling. However, the SAXS intensity given in absolute scale provides additional information that is often useful for detailed structural characterization.

The typical scattering intensity function in absolute scale is the differential scattering cross-section per unit volume (dΣ/dΩ). This function is related to the SAXS intensity $I\left(\overrightarrow{q}\right)$, which was used along this chapter, by $\left(d\Sigma /d\Omega \right)\left(\overrightarrow{q}\right)=I\left(\overrightarrow{q}\right).{r}_e^2/V$.

For SAXS measurements using pin-hole collimation (i.e., with a point-like incident beam cross-section), the differential scattering cross-section per unit volume is given by

$$ \frac{d\Sigma}{d\Omega}\left(\overrightarrow{q}\right)=\frac{R\left(\overrightarrow{q}\right)/\eta }{I_0V.\Delta \Omega} $$

(52)

where $R\left(\overrightarrow{q}\right)$ is the photon counting rate, η is the detector efficiency, I₀ is the photon flux of the incident X-ray beam (number of photons per unit cross-section.second), V is the probed sample volume, and ΔΩ is the solid angle associated to the surface area of the detector pixel. The usual unit for the differential scattering cross-section per unit volume is cm⁻¹. Equation 52 can also be written as

$$ \frac{d\Sigma}{d\Omega}\left(\overrightarrow{q}\right)=\frac{R\left(\overrightarrow{q}\right).{L}^2}{R_0{t}_s\Delta a} $$

(53)

where R₀ is the counting rate (number of photons/second) corresponding to the total incident beam, t _s is the sample thickness, Δa is the surface area of the detector pixel, and L is the sample-to-detector distance. It is assumed in Eq. 53 that the efficiency of the detectors that records $R\left(\overrightarrow{q}\right)$ and R_o are identical. When different detectors are used for the measurements $R\left(\overrightarrow{q}\right)$ and R_o, the counting rates should be properly normalized to equivalent efficiencies.

Equation 53 is usually applied to plate-shaped solid samples or to liquids contained in cells with parallel thin windows for entrance of the incident X-ray beam and exit of the scattered photons. Determinations of SAXS intensity in absolute units associated to powdered samples or liquid samples contained in cylindrical capillaries are also possible but their evaluation is less precise (Fan et al. 2010).

Since the measurement of R_o is in practice difficult using standard detectors, the differential scattering cross-section per unit volume of solid materials is generally determined by means of an independently calibrated sample, such as Lupolen or glassy carbon (Fan et al. 2010).

In order to determine the differential scattering cross-section per unit volume associated to colloidal particles embedded in a liquid medium, it is also recorded – under the same experimental conditions – the SAXS intensity produced by statistical density fluctuations in water. The differential scattering cross-section per unit volume of water ${\left(d\Sigma /d\Omega \right)}_{{\mathrm{H}}_2\mathrm{O}}$ – which is a isotropic and constant function at small q – is given by (Guinier and Fournet 1955):

$$ {\left(\frac{d\Sigma}{d\Omega}\right)}_{{\mathrm{H}}_2\mathrm{O}}\left(q\to 0\right)={\left({N}_{{\mathrm{H}}_2\mathrm{O}}{n}_e{r}_e\right)}^2kT\beta $$

(54)

where ${N}_{{\mathrm{H}}_2\mathrm{O}}$ is the number of water molecules per unit volume, n _e is the number of electrons per water molecule, k is the Boltzmann constant, T is the absolute temperature, and β is the isothermal compressibility of water at room temperature. Since all parameters in Eq. 54 are known, the differential scattering cross-section per unit volume of water can be written as

$$ {\left(\frac{d\Sigma}{d\Omega}\right)}_{{\mathrm{H}}_2\mathrm{O}}\left(q\to 0\right)={\mathrm{1.65.10}}^{-2}{\mathrm{cm}}^{-1} $$

(55)

If the counting rate associated to a isotropic liquid sample (for example proteins in liquid buffer), [R(q)]_sample, and that corresponding to water, $R{(q)}_{{\mathrm{H}}_2\mathrm{O}}$, are determined under same experimental conditions, the differential scattering cross section per unit volume of the studied sample is given by

$$ \frac{d\Sigma}{d\Omega}(q)={\mathrm{1.65.10}}^{-2}\frac{{\left[R(q)\right]}_{\mathrm{sample}}}{<R{(q)}_{{\mathrm{H}}_2\mathrm{O}}>}{\mathrm{cm}}^{-1} $$

(56)

where $<R{(q)}_{{\mathrm{H}}_2\mathrm{O}}>$ is an average value taken within the small q range over which the counting rate is approximately constant. If SAXS measurements corresponding to sample and water are conducted under different experimental conditions, adequate corrections should be applied. Additional details on this matter were reported by Fan et al. (2010).