Form and Structure Factor

The interactions of particle in a dilute system containing a number of homogeneous particles per unit volume are very weak therefore the intensity I(q) for the most part depends only on the shape and size of the particles.

(q) 2

(q) N F

I  (5)

where F(q) is the coherent sum of the scattering amplitudes of the individual scattering centres within the particle given by the Fourier transform of the distribution of those scattering centres in space. The scattering is described as coherent to indicate the relationship between the phase of different scattered waves and their accumulated amplitudes [35]. The scattering centres are characterised by the scattering length, b, of the atoms in the particle volume.

Typically in small-angle scattering, the scattering length density, ρ, is substituted instead of the scattering length, ρ=bM/VM, where bMis the sum of b for all the atoms in a molecule and VM is the corresponding molar volume. For neutrons b varies by isotope, while for X-rays b = ne× = 2.8110^-13cm, where neis the number of electrons

in the atom. In scattering, the contrast or the relative scattering length density (Δ ρ= ρ -ρM) is the parameter that specifies the difference between the scattering power of the particle and the medium in which it is contained, Therefore a consistent spherical particle of radius, RS, and volume, VS, gives a scattering amplitude of:

  _{( , )}

where V the volume of the particle, N VS is the volume fraction of the particles and the shape of the particle is described by the Bessel function (the equation inside the brackets) so that P(q, RS) is the square of the scattering form factor for a sphere.

Pedersen listed comprehensive of P(q, R) scattering functions for different particle shapes typically found in soft matter systems.

In this thesis the main models used to fit the small-angle scattering was uniform ellipsoid with the Hayter Penfold mean spherical approximation for charged spheres as the structure factor [36].Calculates the form factor for a monodisperse ellipsoid (ellipsoid of revolution) with uniform scattering length density. The form factor is normalized by the particle volume such that P(q) = scale*<f*f>/Vol + bkg, where f is the scattering amplitude and the < > denote an average over all possible orientations of the ellipsoid. The function calculated for the prolate ellipsoid is [37]

bkg

The structure factor S(q) mainly describes correlation between three components, surfactants hydrocarbon chain, surfactant headgroup and a counter ion. The Haytere Penfold structure factor model describes a correlation between charged micelles which can be calculated from the interparticle potential that determines the equilibrium arrangement of charged micelles. The interaction between identical charged spherical micelles of radius R3can be described by a screened coulomb interaction potential U(r) as follows [36, 38, 39] : permittivity of free space, ε is the dielectric constant of the solvent, and kis the Debye-Huckel inverse screening length. The structure factor of charged spherical micelles is calculated for a given micellar charge and ionic screening using the mean spherical approximation given by Haytere-Penfold when a volume fraction of scattering objects is sufficiently large [40].

At high particle concentrations the scattered rays from different particles will interfere.

This interference term or the structure factor, S(q), is a complex function. It is an oscillatory function and it can be used to determine how closely the particles pack together. In a dilute system where there are no interacting particles, S(q) 1. For particles with spherical symmetry and a narrow size distribution, I(q) is given as

I(q) = N V²∆^{p2p(q) S(q) (12)}

Where S(q) is the scattered intensity from the microstructure described through the pair correlation function, g(r). The partial structure factors and scattering amplitudes depend on the orientation of the particles. In the case of size polydispersity for spherical particles with small anisotropies and low volume fractions, the interactions can be considered to be independent of the orientation which leads to an analogous decoupling approximation. Then the averaging is applied over the orientation distribution of anisotropic particles. For spherical particles, the effect is less pronounced in S(q) as compared to that in P(q) at low polydispersities [41-43].

2.3 Reflectivity

In recent years there has been growing interest in studying surfaces and interfaces with nanostructures. Neutron and x-ray reflection from surfaces provides an important means to investigate the nature of surfaces and interfaces, as these techniques offer the ability to see deep inside nanofilms and thin layers. Several books have covered the theory of reflectivity in detail [6, 44, 45] but a summary is given here to assist understanding of the work reported in this thesis.

The intensity of scattering in reflectivity, is similar to that in the in the method described above and follows the Porod law asymptotically as the scattering vector q is increased. The absolute value of the intensity is proportional to the total area of interfaces within the sample. In addition, study of any deviation of the observed intensity curve from the Porod law can evaluate the diffuseness of the interfaces. The method of reflectivity measurement can be evaluated as an extension of the Porod law method to surfaces that are fundamentally flat and interfaces that are close to exposed surfaces and parallel to them. Small-Angle scattering and reflectivity at small q are measured where the radiation hits the surface at a small glancing angle and is reflected (or scattered) from the surface [6].

Figure 2.3.1 Schematic to explain the surface scattering; the wave vectors K0, K, and K1

indicate incident, reflected and refracted rays respectively.

The vector perpendicular to the surface is designated as the z axis; the incidence plane is defined as the plane including Z and K0. In general K and K1 are not in the plane of incidence. Thus the angle Φ between the Z axis and the projection of K on the XY plane

is not necessarily equal to zero. The detector is located to measure the reflected beam in the direction of K, where the scattering vector q is equal to K - K0. In the plane (figure 2.3.1) where the angle of Φ equals zero and θ is equal to θ0, the reflection referred to as the specular reflection. The wave vectors K0 and K are then associated to each other since the reflected beam mirrors the incident beam of light.

When the surface is completely flat and there is no difference in the scattering length density in the X and Y directions in the layer of material below the surface, only specular reflection is observed. The ratio of the reflected beam energy to the incident beam energy (reflectivity R), is then measured as a function of the magnitude of q while its direction is maintained normal to the surface (ie qz). Changes in q can be accomplished either by varying the angles θ0and θ (=θ0) ie of incidence and reflection at the same time or by changing the wavelength λ while keeping the geometry of the scattering fixed. The result of measurement of R as a function of qz is then analysed to find information about the variation in the scattering length density, p(z), in the material as a function of depth Z from the surface. In general most studies of X- ray and neutron reflectivity are focused on the measurement of such specular reflectivity [6, 24].

Specular reflectivity data of this type is normally analysed using a layer (or slab) model consisting of defined layers with different scattering length densities arranged parallel to the surface. However this type of scattering was not analysed in this thesis so no further detail is given here.

The off-specular or diffuse scattering occurs when the surface is not perfectly flat or when the material near the surface contains some inhomogeneities in scattering length density in the direction parallel to the surface. The scattering vector q in such a diffuse scattering measurement also contains a qx or qy component. The scattering is then analysed to get information about the surface topology or scattering length density inhomogeneities in the x or y direction [6].

Solutions of surfactants or polymer/surfactants systems demonstrate a range of one and two dimensionally ordered structures such as lamellar and hexagonal lyotropic phases.

Reflectivity measurements can be used to characterize the structure and the degree of order in these complex systems near interfaces, by considering the position and intensities of diffraction peaks in R(q). From R(q) intensities and the position relationship of Bragg peaks it is possible to determine the crystallographic symmetries of these ordered system. The assignment of structures using Bragg peaks is discussed in more detail below in section 2.4.

2.3.1 Neutron Reflectivity Experimental Measurements

Neutron reflectivity measurements of the films grown at the air/water interface were performed on SURF and CRISP instruments (Target Station I) at the ISIS Pulsed Neutron Source facility within the Rutherford Appleton Laboratories. Both SURF and CRISP instruments have been established for the study of surfaces using specular neutron reflectivity. The neutrons wavelengths used were 0.55 - 6.8 Å in pulses at 50 Hz to give a q range of 0.048 - 1.1 Å^-1. A hydrogen moderator at 25 K is used to cool the neutrons in each pulse prior to travel down a flight path with a series of four slits, two before and two after the sample. An optical laser is used to aid alignment of the sample height that is controlled on a sample stage with 0.05 mm accuracy. The incident angles used for the reflectivity experiments described here was 1.5^oand 0.5^o. The detector is a He³ gas detector [18]. The solvent for all neutron reflectivity experiments was D²O. The samples were poured in to sample holders (Teflon troughs) with dimensions of 152mm, 42mm, 3mm which hold 20 - 30ml of solution with a meniscus well above the side of the trough. These sample holders sit within the facilities own heat controlled sample holders allowing temperature control of the solution during each experiment. The temperature was maintained at 35^oC for the experiments described in this work, to ensure the surfactants remained in solution.

2.3.2 X-ray Reflectivity Experimental Measurements

X-ray reflectometry experiments were performed in the I07 beamline at the Diamond Light Source in Oxford UK, and on ID10B Troika II beamline at the European Synchrotron Radiation Facility in Grenoble, France.The solvent for all x-ray reflectivity experiments was H2O. Thefilms grown at the air/water interfacewere poured in to sample holders (Teflon troughs) with dimensions of 152mm, 42mm, 3mm and hold 20- 30ml of solution with a meniscus well above the side of the trough. All samples measurements were recorded at 35ºC.

The ID10B beamline is a high-brilliance undulator beamline at the ESRF designed for high resolution X-ray scattering and surface diffraction [46]. ID10B used photons with an energy range of 8 keV provided by a diamond double-crystal monochromator.

Studies of film formation at the air-liquid interface were performed using time-resolved X-ray reflectometry as the flux of the synchrotron radiation was sufficient to allow short scans to be collected. Time-resolved experiments were able to cover a Qzrange of 0.085

– 0.25 A^-1 by using a linear detector with a vertical orientation to collect data over a range of reflected angles [46]. X-ray reflectometry and grazing-incidence diffraction experiments were performed after film formation. X-ray reflectometry patterns were collected over a Qz range of 0.0014 – 0.3535 A^-1 and grazing-incidence diffraction patterns were collected over a Qzrange of 0.0014 – 0.89 A^-1and a Qxyrange of 0.0 – 0.2 A^-1. A linear detector with 1024 channels and with a vertical alignment was used to collect data during grazing-incidence diffraction experiments and was rotated incrementally in XY plane. Grazing incidence diffraction patterns were calibrated using the known straight through beam position, measured number of pixels per degree and the motor positions during the measurements.

Diamond Light Source beamline I07 is a high-resolution X-ray diffraction beamline for investigating the structure of surfaces and interfaces of the films grown at the air/water interface and on silicon wafers.

Grazing indicence diffraction patterns were collected with a Qzrange of 0.0 - 0.55 Å^-1 and Qxyrange of 0 - 0.25 Å^-1 at an incident angle of 0.25 º, using an energy of 12.5 or 12.2 keV. X-rays were detected using a two-dimensional pixel detector (Pilatus 2M, 195 X 487 pixels with a size of 172 X 172 µm each) [26]. Sample-to-detector distance was calculated using the image of silver behenate powders spread on a silicon wafer, taken in reflection mode. The peak positions were well described the expected diffraction pattern of the silver behenate lamellar structure.