S MALL ANGLE X- RAY ( MICRO ) DIFFRACTION

Scattering of X-rays at small angles close to the primary beam yields information on sizes, shapes and the internal structure of particles.⁶⁹ Apart from structural analysis of single particles, the method is suitable to study spatial correlations of particles.

Covering relevant length scales from one to a few hundred nanometer, small-angle X-ray scattering (SAXS) is a fundamental tool in the study of biological macromolecules.

The major advantage of the method lies in its ability to provide structural information about disordered systems, i.e. about macromolecules in solution. Solution scattering studies are in particular important, given the fact that it is not possible to crystallize numerous macromolecules with high biological significance. Moreover, SAXS allows one to study the structure of native particles in near physiological environments and to analyze structural changes in response to variations of external conditions at relative low effort.^69-71

2. Methods

However, due to the low information content of small angle scattering data from solutions in the absence of crystalline order, there is only a limited number of independent parameters that can in principle be extracted from such scattering data.

According to Shannon’s sampling theorem,^72-74 the number of degrees of freedom associated with I(q) on an interval [qmin , qmax] is estimated as

with Dmax representing the maximum particle diameter. Due to the fact that small angle scattering curves decay rapidly with q, they are reliably registered only at low resolution and, in practice, NS does not exceed 10–15.⁷⁵ Therefore, small angle scattering is commonly considered to be not only a resolution but also a low-information technique. Here, the number of Shannon channels is NS ≈ 8 (estimated with the program GNOM⁷⁶). Fitting complex models, which contain a larger number of parameters, to recorded scattering data can not yield a stable solution without additional a priori information and results are expected to display a strong dependence on initial parameters. Accordingly, the model derived for describing the dendrimers in chapter 6.2 includes only 4 independent fit parameters.

Molecular sizes, shapes and inter-molecular interferences contribute to the scattering curves. SAXS experiments measure the Fourier transform of the electron density of objects in the sample.⁷¹ The electron density of biomolecular assemblies is a convolution of the electron density of the single scattering objects with the 2D or 3D lattice of delta-functions that define their arrangement in the complex. Thus, the scattering intensity I(q) consists of Bragg peaks, determined by the structure factor S(q), accounting for interference effects between particles, with peak heights modified by the smoothly varying form factor, i.e. the single particle scattering function, F(q):

)

N is the number of particles. q is the scattering vector defined by the irradiated wavelength λ and the scattering angle θ according to the following equation:

λ θ πsin

= 4

q . (2-9)

Bragg peak heights and shapes are determined by lattice vibrations, defects, finite-size effects, and other distortions, whereas the peak positions determine the symmetry and dimensions of the lattice.^{71, 77}

2. Methods

2.2.1. Experimental setup

Small angle X-ray scattering (SAXS) measurements are conducted at the beam-line ID10b at the European Synchrotron Radiation Facility (ESRF, Grenoble, France). The ID10b beam-line is a multi-purpose, high-brilliance undulator beam-line for high resolution X-ray scattering and surface diffraction on solids and liquids.⁶² The sample is mounted onto the ID10b goniometer. Beryllium compound refractive lenses (CRL)⁷⁸ with a focal distance f ≈ 1.3m are used to focus the synchrotron X-ray beam of 8keV (λ = 1.55Å) down to a spot of dS ≈ 20µm in diameter. Moving the microdevice along coordinates x, and y, the X-ray beam is positioned at the desired position. A CCD camera with fluorescent screen is used as a detector. Resulting 2D images of diffraction patterns cover a q-range of 0.25–3.50nm^-1. Due to the high-flux X-ray beam, short exposure times of 30-120s are possible.

In addition, experiments are performed using an in-house Bruker AXS Nanostar (Bruker AXS, Karlsruhe, Germany). This setup includes a rotating anode X-ray source for Cu-Kα radiation (1.54Å) at a generator power of 4.05kW. The X-ray beam is adjusted to a size of dN ≈ 100µm or 400µm by a set of three pinholes. 2D scattering data are recorded using a virtually noise-free, real-time 2D Hi-Star detector with photon counting ability. Providing the opportunity of varying the sample-detector distance between 6-120cm, in-house a total q-range of 0.1–9.0nm^-1 is covered. This allows for resolving of feature sizes ranging from 0.7–62.7nm. Due to the lower adsorption of injected fluids compared to material of the channel walls, the in-house setup enables real space imaging of microfluidic devices via lateral and vertical X-ray absorption scanning with a step size down to 50µm. This allows for precise selection of measurement positions. According to the significantly lower intensity of the in-house X-ray beam, exposure times of up to two hours per image are necessary. Performing measurements under continuous flow conditions is therefore indispensable to the retention of molecular integrity, especially when analyzing complex materials that can be easily destroyed due to the high X-ray energy (chapter 9.4.3).^59-61

2.2.2. Data treatment

SAXS data were acquired from complex materials either inside of microfluidic devices or from samples loaded in thin quartz capillaries with a wall thickness of 0.01mm and diameters varying between 1.3-1.6mm. All collected data are first corrected for detector sensitivity by using the Bruker recording software SAXS. Background intensities

2. Methods

Figure 2-3: Experimental synchrotron beam-line (a) and in-house (b) setup.

2. Methods

Figure 2-4: Converting 2D raw data images to 1D intensity plots. (a) I(q) and (b) Iqmax(χ).

caused by scattering from the X-ray windows of the detector vacuum chamber, glass capillaries or microchannel devices, the solvent, and the beam stop overspill are determined by separate reference measurements. In order to account for different capillary diameters and to determine absolute scattering intensities, each reference data is weighted by the ratio of overall scattering intensities IGC(sample)/IGC(reference) before subtracting from the respective sample data. IGC is acquired by inserting glassy carbon in the beam path at a fixed position closely after the sample and measuring for 100s.

A typical 2D X-ray image shows patterns of intensity I(q,χ) corresponding to features of the liquid-crystalline lattices that are probed (Figure 2-4). The real spacing d of these features is inversely proportional to the momentum transfer

q = 4π/λsinθ, (2-10)

where λ is the X-ray wavelength and 2θ is the angle between incident and scattered radiation. A plot of intensity I versus q is obtained by azimuthally integrating the image over 360º:

2. Methods around a peak position qmax is also a quantity of interest, in particular because an alignment of materials occurs in microflow.

( )

∫

( )

Materials investigated in this thesis are not single-crystalline, and their 2D scattering images reflect a powder average from multiple orientations. This results in an intensity distribution along the pattern at fixed radial position q. In Figure 2-4, the extraction of data from a 2D raw data image is demonstrated and resulting plots of intensity I versus q and I versus the azimuthal angle χ (χ = 0º–360º) along a fixed q region, respectively, are shown. In addition, the full width at half maximum of a peak of I(χ), ∆χ, serves for quantifying the extent of material orientation within microdevices.

For recording in-house X-ray data and for their evaluation, the program SAXS (Bruker GmbH, Karlsruhe, Germany) is used. In addition, X-ray data processing is performed using the program Fit2D by Andy Hammersley.⁷⁹ For the analysis of the interaction of PAMAM dendrimers generation 6 and DNA, presented in chapter 8.2, the programs GNOM⁷⁶, DAMMIN,⁸⁰ and CRYSOL⁸¹ by Svergun et al. are used.