Small angle x-ray scattering (SAXS) can be considered as non-crystalline diffraction which provides important information about the dynamics and the structure of molecules; from biological molecules to more complex materials such as polymers and colloidal particles8.
SAXS is the inverse relationship between the particle size and scattering angle. When there are inhomogeneous electron density of colloidal size in a sample, then x-ray scattering will be observed at low angles (~ 0.1 - 10°). Only coherent scattering is considered since incoherent scattering is negligible. At these angles information regarding the shape and size of macromolecules, the pore size and distances of ordered constituents can be determined2.
As x-rays pass through a sample the electrons will resonate with the same frequency and coherent, secondary waves are emitted. At small scattering angles the phase difference between the waves are small and constructive interference occurs. So, at zero angle, all the waves are in phase so that the scattering is at a maximum1.
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Colloidal dimensions are large compared to the Cu Kα x-ray wavelength which is commonly used. For this reason the observed scattering is at small angles.
The particulate scattering curve can be calculated for any shape of a given individual particle. For particles where the properties are direction dependent, the scattering needs to be computed for all directions and the average taken1.
The only scattering comes from the interface between two phases in the transitional range at the high end of the resolution of a SAXS pattern. As a result, any scattering from other structural features are so random that their contribution is insignificant. The surface area (S) of the particles can then be determined from Porod's law2:
One way of computing the scattering curves is by making use of the distance distribution function of electrons - p(r). The scattering curve I is the Fourier inversion of p(r)1:
The scattering pattern is represented as the scattered intensity as a function of the magnitude of the scattering vector, h, in the above equation2. (h from the above equation is the same as q in the Porod's law equation)
The small angle scattering curve will remain virtually the same if the electrons are moved small distances compared to the size of the particle. For this reason the particle can be treated as having a uniform density distribution.
If the particles were in a vacuum, the scattered amplitude would be proportional to the electron density, ρ (number of moles of electrons per unit volume). The electron density difference applies to the dissolved particles where ρsolute - ρsolvent. The scattered amplitude is then proportional to ρsolute - ρsolvent and the intensity to (ρsolute - ρsolvent)2. Thus, if the electron densities are the same, the scattering is 'cancelled out' and virtually 'no particles' are detected. Electron density is a term used for electron density differences or contrast.
For dilute, homogeneous solutions, the scattered intensities of all the particles can be summarised. The challenge is to determine the size, shape, mass and electron density distribution from the scattering curve. The scattering curve is then compared to a reference particle. A series of modelling and approximation follow to evaluate the particle based on the experimental curve.
A number of parameters can be determined directly from the scattering curve1:
Radius of gyration, R: root mean square of the distances of all electrons from their centre of gravity i.e. expansion of the particle:
ln I vs (2θ)2 → Guinier plot
The cross-section factor for rod-like particles is given by the scattering curve multiplied by 2θ i.e. the curve only depends on the size, shape and electron density distribution of the cross-section.
The thickness factor for plate-like particles is determined by multiplying the scattering factor by (2θ)2 i.e. the curve depends on the particle thickness and the electron density distribution perpendicular to the plane of the particle.
Volume can be determined by using the following formula1:
Where a is the distance between the sample and the detector.
The integral is known as the variant and for a given concentration, this value is independent of the homogeneity of the solution.
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Particle molecular weight can be determined by dividing the scatter intensity by the primary intensity. The scattering of one electron is equal to 7.9 x 10-26 multiplied by the primary intensity per cm2 at a distance of 1 cm for the scattering electron.
One can also determine the mass per unit area for plate-like particles and the mass per unit length for rod-like particles.
The distance distribution function p(r) is the Fourier inversion of the scattering curve1:
To determine the shape of a particle the theoretical scattering curves can be calculated for a model of a specific shape.
By using the abovementioned parameters, good approximations of the particle shape and size can be obtained for biological macromolecules. For complex particle shapes, it's better to simplify the particle by comparing it to a large number of closely packed spheres.
Instead of using the scattering curves the theoretical and experimental p(r) functions can be used to determine the shape and size of molecules because this is close to how a human would interpret the information. The largest particle size can then be found where the r-value is zero.
The manner in which subunits are arranged can be determined by the labelling of heavy atoms or groups thereof. The scattering curves of the labelled and the unlabelled materials are determined and from that the differences between the distances of labelled materials can be determined from some difference curves. By using neutron scattering this technique is greatly improved because the labelling consists of replacing hydrogen with deuterium.
To refine the SAXS model further the inhomogeneity of the electron density of the particle is taken into account where the particle consists of different chemical compounds. One way of determining this is if the particles are spherically symmetrical, the radial electron density distribution is used by Fourier inversion of the scattering amplitude, where the amplitude of the square root of the intensity. The problem of assigning the correct sign for the phase is avoided by calculating the radial electron density from the distance distribution function. The electron densities of the chemical constituents are available and for this reason one can determine the relative arrangements of the different constituents from the radial electron density.
Another way of determining the electron density inhomogeneity is by using the phase contrast. The principle is that if the electron density of the solvent is changed, the scattering changes because the contrast between the particle and the solvent has changed.
By knowing the scattering curve of different solvents more information can be obtained. A theoretical particle with a similar shape when compared to the molecule with homogeneous electron density, can be determined. By comparing this information to the theoretical scattering curves of homogeneous bodies one can determine the shape of the particle. And one can also determine the inhomogeneity of the electron density of the particle from the scattering where the electron density of solvent is equal to the electron density of the particle.
With phase contrast determination the challenge is to find solvents which do not alter the molecule and which have different electron densities when compared to the particles. Alternatively neutron scattering can be used where hydrogen is replaced by deuterium.
Since the 1990's the SAXS technique has provided more information on the structure and shape as well as the modelling of macromolecular complexes with the use of rigid body refinement6. The short response times make it ideal for studying biological processes as they occur7.
Parameters such as particle size, shape, distribution, or surface-to-volume ratios, can be used to determine the structure of micro- and nanoscale particle systems. The material to be studied can be either solid, liquid or gas phase and can be made up of the same elements or be a combination of different elements. This technique can also be used to study the structure of ordered systems (e.g. lamellae) because it is accurate, non-destructive and very little sample preparation is required1.
SAXS can be used to determine how external influences such as temperature, humidity, light, solvents etc. cause changes in the structure of the particle. With biological macromolecules one will use this technique to investigate evenly dispersed solutions with particles with identical shape and stable structure. However for natural or synthetic molecules that structure is constantly changing. The SAXS technique can then be employed to determine the molecular weight (i.e. weight average) and the degree of coiling (i.e. persistent length) of such a molecule. The persistent length can be obtained from the bend in the curve of the I(2θ)âˆ™(2θ)2 vs 2θ plot. The degree of coiling is averaged over the entire molecule and even branched molecules can be measured1.
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Colloidal particles in solution:
Dissolved particles can also be characterised by the SAXS technique and it doesn't matter whether it is low molecular weight or extremely high molecular weight molecules. This can be achieved by adjusting the width of the entering x-ray beam and by varying the analysis time1.
The properties of nanomaterials can be very different from the bulk material. For this reason the physical, chemical and biological properties of these materials are being investigated with the use of the SAXS technique. The x-rays penetrating the sample are scattered on the interfaces of the nanostructures and provide a specific pattern for the material. From this pattern the size and shape and the orientation of nanoparticles can be determined4.
Surfactant micelles are also successfully studied using the SAXS technique. Surfactants consist of two parts - a hydrophilic and a hydrophobic part. As soon as surfactants (such as detergents or soaps) are placed in water, the water molecules start to form structures which are called micelles. These micelles can arrange in spherical, cylindrical or lamellar shapes. With the SAXS technique the micelle size and shape, phase behaviour as well as the inner structure of the vesicle walls can be determined4.
In systems where two immiscible liquids are mixed (i.e. emulsions such as oil and water), the system can only be stable if a surfactant covers the interface between the two types of molecules. The SAXS technique is employed to determine the shape and size of the inner structure, the size distribution of the droplets in the system, the emulsion stability at different temperatures and the manner in which encapsulated agents are being transferred4.
The SAXS technique is recommended for research on biological materials because studies can be done under the biological conditions and structural changes can be monitored in vitro. By applying the SAXS technique the structural information of proteins can be obtained in terms of their shape and size. The inner structure, aggregation state and molecular weight can also be determined4.
In mixtures of particles with liquids, the stability of the system can be determined by looking at the size distribution of the dispersion. The following properties can be determined with SAXS:
Stability of the dispersion;
Nucleation of particles;
The state of aggregation; and
The shape and size distribution of dispersed particles4.
Natural and man-made fibres can be studied for their mechanical, optical and thermal properties. Especially with man-made fibres the production and processing can be monitored since these factors determine the fibre properties. With SAXS the internal structure can be determined, specific surface, crystallinity, orientation and the distribution of orientation4.
SAXS can be used to study the structure and performance of polymers and nanocomposites. The crystallinity, shape and inner structure, periodic nanostructures as well as orientation are properties which can be determined4.
Liquid crystals form a variety of liquid crystal phases depending on their concentrations in water. Their properties are between conventional liquids and solid crystals. There are a variety of man-made liquid crystals (e.g. liquid crystal displays) and biological liquid crystal systems such as cell membranes. In order to determine the particle size distribution and shape, crystallinity, ordering of aggregates and the orientation of particles, the SAXS technique is used4.
Initially it was quite challenging to apply particle scattering to closely packed particles. These systems only have about 1% open spaces per volume between particles when compared to dilute systems. A technique was invented to swell the system with air to more than its original volume in order to disperse the densely packed particles. The dimensions obtained from these air swollen systems compared well with the densely packed systems1.
There are different ways in which SAXS can be applied to metals, alloys, ceramics and glasses. Particle scattering can be applied given that the electron density contrast between the phase and the matrix is greater than the varying electron density in the matrix itself. Unfortunately, particle anisotropy and polydispersity cannot be completely separated on the scattering curve. In practice, one can just consider the spherical particles in inorganic systems which are homogeneously dispersed and approximate the scattering1.
A lot of inhomogeneities in the system will lower the intensities of the scattering at low angles but even small interparticle interferences can be detected by the p(r) function1.
However, some parameters can be directly determined from the scattering theory:
The mean square fluctuation of electron density can be determined using the following equation1:
The volume fractions of a two-phase system with know electron density difference can be determined from the following equation1:
According to Porod's Law the scattering curve at the tail end can be explained by1:
Where the inner surface of the sample can be related to K (Å2/ Å3 of the sample) by using the following equation1:
The quantities of XXXX and O/V can be used to evaluate the accuracy any model.
When monochromatic beams of all x-rays interact with the sample, some of the x-rays simply pass through the sample while other x-rays are scattered. The scattering pattern contains the information required and is detected by the detector. The biggest challenge is to separate the scattered intensities from the high intensity produced by the main beam. In order to achieve this separation the x-ray beam needs to be focused or collimated2.
Figure 1 shows a simplified diagram of the SAXS instrumental design3
Conventional laboratory equipment makes use of two-types of collimation2:
Point collimation: the beam is focused into a small spot which then illuminates the sample. The scattering pattern can be seen as a set of rings around the primary beam. Only a small portion of the sample is illuminated at a time and unless focusing optics (i.e. bent mirrors, collimating optics) are used, the measuring time will be in terms of hours or days. This experimental mode is preferred for oriented nanostructures.
Line collimation: the x-ray beam line is a long, narrow line so that a larger volume of the sample is illuminated. For this reason the scattered intensity is larger than with point collimation when using the same flux density. The measurements are fast and the resolution and data quality is very good.
One disadvantage is that the recorded pattern is a combination of pinhole patterns which leads to a smeared pattern. When dealing with an isotropic system (i.e. random distribution) the smearing is removed by using deconvolution methods which are based on Fourier transformation (i.e. in isotropic nanostructured materials)1.
The SAXS technique is regarded as a high precision technique because of the good quality scattering results that can be obtained. Commercially available instrumentation can be used for routine and research and development work.
But for highly specialised experiments, synchrotron radiation can be used because x-ray intensities are by far higher than those produced by conventional x-ray generators. Problems such as the determination of density inhomogeneities in a particle can be solved with the use of neutron diffraction because of the high neutron flux created by these reactors1.
Experimental work and advances in SAXS:
Protein crystallographers are increasingly making use of SAXS to study biomacromolecular complexes and even structural biologists have found increased interest in using SAXS. The reason is that the SAXS technologies, instrumentation and algorithms to retrieve better structural data have increased over the past few years. The SAXS small interest group of the American Crystallography Association are having more sessions focused on nanomaterials and structural biology5.
SAXIER is a European project which aims to explore innovative applications of SAXS at beamlines at synchrotron facilities. The aim of the initiative was to develop automated data analysis software and hardware developments of the optical elements. Structural biologists will find this of particular interest in their studies of proteins and biomacromolecular complexes. Other fields that will benefit from these developments are advanced nanomaterials and surface chemistry7.
The five year SAXIER project was successfully completed at the end of last year and some of the highlights include7:
X-ray optics which can focus the beam down to 100 nm and nanomanipulation apparatus for the analysis of minute samples, were developed at ESRF (France). X-ray fluorescence, Raman spectroscopy and optical microscopy combined with SAX for in situ analysis are novel developments which are now available.
Innovative microfluidic sample environments were developed through a collaboration of ESRF, Elettra (Italy) and EMBL (Germany). These environments will be used to describe macromolecular solutions, chemical and biological reactions and nanofabricated functional surfaces. By making use of an innovative continuous flow jet-mixer, time resolution below 100 μs can be obtained in studies of fast reactions.
Two other innovative developments were completed by EMBL and ULIV. On gas phases the spurt of macromolecules coming out of a mass spectrometer can be analysed and in cryo-frozen solution the reactions can be halted for analysis.
EMBL and SOLEIL (France) took on the enormous task of automating SAXS experiments and especially, the data analysis.
Another development by the two abovementioned institutions was the coupling of gel-filtration chromatography (GF-HPLC) with SAXS in order to purify macromolecular solutions as they come through the system and to study any momentary occurrences.
In order to perform SAXS experiments remotely, a robotic sample changer was developed for one of the EMBL beamlines and connected to an automated channel for the analysis of macromolecular solutions. In May 2009, the first remote SAXS experiment was conducted. The protein sample was sent to EMBL where it was placed in the instrument and the research team in Singapore could control the experiment from where they were. The team in Singapore could load the sample, acquire the data, conduct automated data analysis and build a three dimensional model while it was watched by students, professors and other interested parties! Within a couple of minutes after the start of the experiment, structural evidence and the shape of a subunit were acquired.
The manner in which SAXS differs from conventional x-ray scattering techniques is that a crystalline sample is not required. The scattering curves provide information about the dimensions, size and shape of the molecules.
SAXS is an ideal technique to unravel the mysteries of biological macromolecules and nanomaterials. Each day new applications are found or more efficient ways of resolving the data.