polymer hybrid structures
1.5.4 Small angle neutron scattering (SANS)
Scattering techniques are powerful tools for structure determination in micellar solutions. It includes scattering of visible light, neutrons and x-rays. Characteristic for each method is the covered range of the scattering vector q, which determines the length scale on which structural investigations can be carried out. The scattering vector depends on the wavelength of the scattered photons and is defined as:
4 sin q 2
Eq. 1.13
λ is the wavelength of the scattered waves and θ the scattering angle.
Light scattering and SANS/SAXS appear to be complementary since light covers a range of rather low q-values whereas x-rays and neutrons enable structural studies on a local length scale in the range of atomar or molecular distances. Consequently, it is a powerful tool for structural investigations of micellar solutions. The setup of a neutron scattering device is shown in Scheme 1.11.122
Scheme 1.11. Setup of a SANS experiment.122
For scattering purposes, cold neutrons are needed (E = 0.1-10 MeV, λ = 30-3 Å), which are commonly generated by a nuclear reactor and subsequently decelerated using moderating systems.123 The monochromator provides a neutron beam with a wavelength dispersity of Δλ/λ ≈ 10 % and the collimation setup focuses the beam before passing the sample to be investigated. Detection occurs via gaseous substances like BF3, emitting ionizing particles upon neutron impact.123 The detecting unit is a two-dimensional multi-detector which can be moved along the neutron flight direction in order to cover a q-range as broad as possible.
33 The primarily recorded data are two-dimensional scattering images. By a radial averaging of these images, one obtains the q-dependent intensity profile , which is defined as follows:
Eq. 1.14
Δρ is the difference in scattering length density between the scattering objects and the surrounding medium, N is the number of scattering particles, P(q) the form factor and S(q) the structure factor. For diluted samples where no interactions between the scattering particles take place, S(q) ≈ 1 applies and the intensity profile exclusively contains information about the particle dimensions. By using a variety of fit models for the form factor, one obtains useful information, such as shape, diameter and consistency of the scattering objects.124,125 In semi-diluted and concentrated solutions, the influence of the structure factor becomes important. At sufficiently high concentration, an intensity maximum occurs, whose position provides information about the center-to-center distance of the scattering objects. If ordered domains are present, additional higher order reflections are visible at defined q values. These values, normalized to the position of the structure factor maximum, provide information about the type of crystal lattice present in the sample (Table 1.1).
Table 1.1. Common crystal lattice types and their crystallographic specifications occurring in concentrated micellar solutions.126-128
When dealing with concentrated, ordered micellar solutions, the primarily recorded 2D diffraction patterns provide additional useful information. In principle, three kinds of patterns are observable (Fig. 1.13). A pattern can be fully isotropic, i.e. the intensity does not depend on the azimuthal angle (Fig. 1.13A), which points to the presence of randomly oriented domains with a local order, like in powder diffraction experiments. Smeared Bragg reflections, as shown in Figure 1.13B, point to the presence of randomly oriented, ordered
Introduction
34
domains with a weak preferred orientation. The presence of fully aligned monodomains (i.e.
macroscopically oriented sample), finally leads to patterns with sharp Bragg spots as shown in Figure 1.13C. Such a pattern either stands for one single macroscopic orientation (“single crystal”), or represents a superposition of patterns originating from a set of different macroscopic orientations. In general a higher degree of order in micellar solutions, allowing for a detailed investigation of packing characteristics, can be achieved by applying shear to the sample.129
Figure 1.13. Sketches of 2D-scattering patterns obtainable form SANS experiments: A) isotropic pattern; B) anisotropic pattern with smeared Bragg reflections; C) strongly anisotropic pattern with sharp Bragg spots.
A detailed interpretation of patterns from fully aligned domains, including indexation of the reflections, is commonly achieved with the help of reticular planes.128 These are planes at which an incident beam is refracted under the condition angle of incidence = angle of reflection. The orientation of these planes relative to the crystal lattice is expressed by so called Miller indices (hkl) (Fig. 1.14).
Figure 1.14. A) Schematic depiction of the definition of a crystallographic reticular plane; B) reticular planes and their corresponding Miller indices.
35 In general, only reflections are allowed, for which the Bragg relation is fulfilled:
2 hklsin
n d Eq. 1.15
dhkl is the distance between two particular {hkl} planes. For symmetry reasons, the Miller indices h,k,l, and their sum n, has to adopt characteristic values for each lattice. These values are summarized in Table 1.1. The different spots in the 2D pattern lie on concentric circles.
Those located on the same circle have the same scattering angle, which means that they originate from identical {hkl} planes. Reflections from planes of the same order n on the other hand lie on a vertical axis within the pattern starting with n = 0 in the center130. With these predictions, an easy indexation of the observed reflections can be carried out giving the opportunity to determine the lattice type of the sample. The positions of the indexed reflections allows for further conclusion about the preferred orientation of the crystal domains relative to the incident neutron beam. Finally, it is also possible to derive quantitative information from 2D patterns, in particular lattice spacings. For a simple cubic packing for instance the lattice spacing a can be calculated according to the relation: