Small Angle X-ray Scattering (SAXS)
Small Angle Neutron Scattering (SANS)
Monte Carlo simulations on Meso-scale Models
Integral equation theory for liquids, colloids and polymers
Small Angle X-ray Scattering (SAXS)
A technique for structural investigations on a nanometer length scale (1-300 nm). Typical samples consist of macromolecules or colloids in solution and measurements are performed in situ. The technique provides information on the size and shape of the objects in solution as well as their inter-particle interactions. There is no requirement for crystallinity of the samples. The x-rays interact with the electrons in the sample and the technique is therefore sensitive to electron density differences. There exists both laboratory instruments based on more conventional sources and synchrotron based instrumentation.
Links:
General introduction to small-angle scattering:
http://www.isis.rl.ac.uk/largescale/loq/documents/sans.htm
A powerful SAXS instrument is located at the Department of Chemistry. A modified version of a commercially available small-angle x-ray equipment (NanoStar), which is produced by Anton Paar, Graz, and distributed by Bruker AXS has been installed in September, 2001. The instrument consists of a powerful rotating anode x-ray source (Cu Ka , 0.3 x 0.3 mm2 source point, 6 kW power) and a pinhole camera with two Göbel mirrors for monochromatizing and focusing the beam. The data is collected efficiently by a two-dimensional position-sensitive gas detector (HiSTAR).
The instrument has originally been constructed for position-resolved studies and the commercially available instrument has a very small beam diameter of only 0.3 mm at the sample position. At the Department of Chemistry at the University of Aarhus, we have developed a high flux version of the camera in which the beam size is larger. The instrumental configuration has been optimised by numerical calculations based on phase-space analysis and Monte Carlo simulations. This leads to a setup in which the collimation part of the instrument is longer, so that divergence of the beam is similar to that of the original camera. An extra pinhole is included after the Göbel mirrors, so that the beam is well-defined after the mirrors and so that the camera has a genuine three pin-hole collimation. The parts for the modifications have been produced in the Mechanical Workshop at the Department of Chemistry. The instrument covers with one setting a scattering vector range of 0.01-0.35 Å-1. The combination of a rotating anode and Göbel mirrors gives a flux, which is about 50 times higher than that of the original version of the camera. The instrument is equipped with a thermostated glass capillary holder for solution studies. The capillary is placed directly in the vacuum and therefore additional windows are avoided leading to a very low background scattering. The high flux as well as the low background is unique for this instrument and has allowed studies of samples with fairly low concentration and scattering contrasts which cannot be studied by more conventional instruments.
During the first four weeks the instrument was in operation, test measurements on several different samples including some very weakly scattering ones have been performed. Micelles of a Brij surfactant at various concentrations and temperatures, SDS micelles, DTAB micelles, and AOT microemulsions have been measured. The instrument did in all cases perform well. It has a quite low background (100 cps for empty camera) which easily allows the use of water for absolute calibration. Water gives about 150 cps of which 6 cps is noise and 27 is from the capillary, so a 4 hours water spectrum has a quite good statistics. The count rates are for a generator power of 4.05 kW.
The high flux SAXS camera at the Department of Chemistry at the University of Aarhus:
Small Angle Neutron Scattering (SANS).
A technique for structural investigations on a nanometer length scale (1-300 nm). Typical samples consist of macromolecules or colloids in solution and measurements are performed in situ. The technique provides information on the size and shape of the objects in solution as well as their inter-particle interactions. There is no requirement for crystallinity of the samples. The neutrons interacts with the nuclei in the sample and the interaction depend on the actual isotope. Hydrogen and deuterium have very different scattering length and this can be exploited in contrast variation measurements in which it is usual that part of the molecule in solution is deuterated. It is also usual to change the scattering length density of the solvent by mixing deuterated and protonated solvents. The technique requires a neutron source, i.e. a nuclear reactor or an accelerator based spallation source and therefore the experiments are performed at large scale facilities.
Links:
General introduction to small-angle scattering:
http://www.isis.rl.ac.uk/largescale/loq/documents/sans.htm
Neutron scattering facilities:
Paul-Scherrer Institute, Switzerland: http://sinq.web.psi.ch/sinq/instruments.html
GKSS Research Center, Germany: http://www.gkss.de/Themen/W/WFS/W_style_D/WFSframe.html
Laboratoire Léon Brillouin, CEA/Saclay, France: http://www-llb.cea.fr/index_e.html
Rutherford-Appleton Laboratory, UK: http://www.isis.rl.ac.uk/
Institute Laue-Langevin, France: http://www.ill.fr
Forschungszentrum Jülich, Germany: http://www.fz-juelich.de/iff/Institute/ins/Broschuere_NSE/
Hahn-Meitner Institute, BENSC, Germany: http://www.hmi.de/bensc/instrumentation/instrumente/v4/v4.html
Oak Ridge National Laboratory, USA: http://www.ornl.gov/divisions/casd/polymer/
National Institute of Science and Technology, USA: http://www.ncnr.nist.gov/programs/sans/
For further information contact Jan Skov Pedersen
Monte Carlo simulations on Meso-scale Models.
Scattering techniques are ideal for investigating the structure and interactions in soft matter. However, it is non-trivial to relate the measurements to structural properties of the scattering objects and their interaction. It is therefore of crucial importance to obtain expressions for the scattering intensity (cross section) which can be used for interpreting experimental data, for example, obtained by SAXS, SANS and light scattering. Monte Carlo simulations are ideally suited for developing the models for systems with many degrees of freedom, as the structure can be simulated in real space and the corresponding scattering cross section can be calculated. As the structure and interactions are known in real space, the models can be developed by fitting to the simulated scattering data and by comparing the derived results to the know structure.
Example 1: Block copolymer micelles: (Carsten Svaneborg’s PhD thesis, 2001)
(see also: http://alphixw.mpip-mainz.mpg.de/~svanebor/gallery1.html)
A di-block copolymer consists of two chemically distinct linear polymer chains covalently linked together. When dissolved in a solvent, which is only a good solvent for one block, micelles form. The micelles can have a spherical core as shown in the picture below. The cores are surrounded by a corona of solvated chains. The micelles can be modelled as a dense core surrounded by a diffuse corona of chains interacting through hard sphere excluded volume interactions. Chains are furthermore excluded from the core region.
At a zero-level approximation one can neglect the fact that the corona consists of chains, and assume it is a just radial density profile decaying away from the micellar core. This level of approximation provides core-shell models, and one can get an estimate of the width of the profile using such a model. All chain information has been averaged out, and therefore no information is available on single chain properties. However, the simulation results show that adding an effective single chain scattering expression in the form of a RPA approximation to the core-shell model provides an expression in agreement with the simulated scattering data. This is in effect the scattering one would get from a semi-dilute solution with a radial concentration profile.
The new model provides a very accurate representation of the micellar scattering within the statistical errors for reduced surface coverages less than one, and a very good fits are obtained for all the simulated surface coverages, e.g. reduced surface coverages less than 5. The reduced surface overage is the analogy to c/c* of polymer solutions. This result is obtained and validated for micelles (with a spherical core). However, the instantaneous density distribution of any tethered chain structure can be regarded as an average density distribution, and an instantaneous density fluctuation distribution. Thus the scattering can be regarded as arising from the Fourier transform of these two distributions. A fluctuation-dissipation theorem relates the zero-mode Fourier component of the density fluctuation distribution to the osmotic compressibility of the system.
The new model when applied to experimental scattering data provides the radius of gyration, corona osmotic compressibility/corona apparent second osmotic virial coefficient, as well as information on the shape of the radial profile. The model should furthermore be easily generalizabled to any tethered chain structure, where chains are not stretched too much, and concentration is locally in the dilute semi-dilute regime such that an RPA approximation is valid.
For the figure below, a standard micelle was chosen as having 44 chains, 50 bonds per chain, and a core radius of 10/3 b, where b is the Kuhn length. A chain has 8.33 statistically independent segments, and the micelle a reduced surface coverage of about 4 (the analogy to c/c* of polymer solutions). The experimental range of diblock copolymers is a reduced surface coverage less than five. The spheres in the picture illustrates the range of the hardcore excluded volume interactions. The radius (e) of the excluded volume hard spheres was chosen as (e/b=0.1) in order to mimic polystyrene in a good solvent.
Example 2: Monte Carlo Simulations of Polymers and Polyelectrolytes (PhD thesis of L. Cannavacciuolo, ETH-Zürich, Switzerland, 2001)
(see also: http://www.unifr.ch/physics/mm/)
The effect of electrostatic interactions on the flexibility of polyelectrolytes has been the subject of intense experimental and theoretical investigations and resulted in highly controversial results. The persistence length of polymers and polyelectrolytes is often determined by scattering methods such as small-angle neutron (SANS) or x-ray (SAXS) scattering. The problem with conventional polymer polyelectrolytes is their rather weak scattering power when performing SANS or SAXS experiments, which makes it almost impossible to produce data with a sufficient quality over the required range of scattering vectors q at low concentrations, where single-coil properties still can be resolved. We have studied experimentally and by Monte Carlos simulations solutions of polymer-like micelles ‘doped’ with ionic surfactants, which can serve as a model system for polyelectrolytes and help to clarify some of the open questions in the literature. However, a major problem in this context is the missing theoretical background for a quantitative interpretation of scattering data from polyelectrolyte solutions.
We have performed an extensive Monte Carlo simulation in order to determine the full scattering function of semi-flexible polyelectrolytes as a function of contour length, charge density and ionic strength as well as concentration c of chains. In these simulations we use an extension of a discrete representation of a Kratky-Porod worm-like chain model in the pseudo-continuous limit, which we have modified in order to take into account excluded volume interactions as well as electrostatic interactions (through a screened Coulomb potential) between the chains. We have performed both strictly single chain (c=0) as well as many chain simulations (c finite).
The Figure snapshot of a many chain simulation of wormlike chains with excluded volume interactions
Integral equation theory for liquids, colloids and polymers.
(For futher information see, for example, the homepage of prof. Arun Yethiraj, University of Wisconsin, USA: http://www.chem.wisc.edu/~yethiraj/research/polyel.html)
Liquid state integral equation theory was originally developed for atomic and small molecule fluids, but has in the last decades found widespread applications in colloids science. The theory provides a (inter-particle) pair correlation function when the inter-particle potential is specified. The structure factor, which describes the inter-particle interaction effects in a scattering experiment can be obtained from the pair correlation function by a simple Fourier transformation.
J. G. Curro and K. S. Schweizer have developed a theoretical approach for describing the equilibrium structure and properties of polymers in the bulk and in solution. This theory, known as the Polymer Reference Interaction Site Model or PRISM theory, is based on an extension to polymers of liquid state methods originally developed for atomic and small molecule fluids. An approximation, in which all sites are assumed to have equivalent interactions, transforms the complex, multiple chain/particle problem into a much simpler single chain problem. We have written computer codes that solves the PRISM non-linear integral equations as well as the simpler equations, but still non-linear intergral equations for spherical particles. We have initiated several projects within this field and expect to publish the results in the near future.
For further information contact Jan Skov Pedersen