Otto Glatter

A new method for the evaluation of small-angle scattering data

A new numerical method is presented for simultaneous smoothing, desmearing and Fourier transformation of X-ray and neutron small-angle scattering data. The method can only be applied to scattering curves from dilute particle systems, i.e. for scattering media whose distance distributions are zero beyond a certain value. The distance distribution of the scattering medium is approximated by a linear combination of about 20 to 30 cubic B-splines. These spline functions have a restricted extension in real space. Their coefficients are adjusted by a weighted least-squares operation so that the series, after being Fourier transformed and smeared according to the geometry and wavelength distribution, represents an optimum smoothed approximation of the experimental data. Tendencies towards oscillations in the least-squares operation are suppressed by a new stabilization routine. The method offers a new possibility for the estimation of the radius of gyration, which is generally superior to the Guinier approximation.

SAXS experiments on absolute scale with Kratky systems using water as a secondary standard

For small-angle scattering, of X-rays (SAXS) and neutrons (SANS), the importance of absolute calibration has been recognized since the inception of the technique. The work reported here focuses on SAXS measurements using Kratky slit systems. In former days, only molecular weights or scattering per particle were determined, but today absolute calibration implies the use of the unit of cm−1 for the scattering curve. It is necessary to measure the so-called absolute intensity, which is the ratio of the scattering intensity to the primary intensity P0. Basically there are two possible ways to determine the absolute intensity. The first one is the direct method, which involves the mechanical attenuation of the primary beam by a rotating disc or a moving slit. The second is the indirect method, using secondary standards. Water is well suited as a calibration standard because of the angle-independent scattering. The essential advantage is that the scattering of water only depends on the physical property of isothermal compressibility. Before presenting an example of the practical performance of this method, the most important theoretical equations for an SAS experiment on the absolute scale are summarized. With the slit collimation system, the scattering curve of water can be measured with high enough statistical accuracy. As a first example, the scattering curve of the protein lysozyme on the absolute scale is presented. The second example is the determination of the aggregation number of a triblock copolymer P94 (EO17–PO42–EO17). Taking into account that at least 10% of the polymer sample consists of diblocks, the accuracy of around 10% for the determined aggregation number is rather good. The data of P94 are also considered on the particle scale in order to obtain the radial scattering-length density distribution.

The Interpretation of Real-Space Information from Small-Angle Scattering Experiments

The distance distribution function contains all the accessible information about the scattering medium. It is possible to determine the following particle parameters from this function: maximum dimension of the particle, radius of gyration, and zero-angle intensity (molecular weight). The shape of the distance distribution function enables one to distinguish and to recognize directly and rationally the following types of particles: compact globular particles; particles elongated in one dimension, with constant and with variable cross-section; spherical vesicles. The thickness of flat particles as well as the inner and outer diameter of spherical vesicles can be determined from the distance distribution function. Inhomogeneous particles with at least two regions of electron densities differing in their signs show typical features in the distance distribution. These characteristics can be found without further assumptions, independently of the shape of the particle. For concentric shells with different electron densities it is possible to obtain some general information about their structure. Residual concentration effects cannot be overlooked in real space. The formation of dimers can be analyzed with the aid of the distance distribution function.

Characterization and potential applications of nanostructured aqueous dispersions

The present article highlights recent advances and current status in the characterization and the utilization of nanostructured aqueous dispersions in which the submicron-sized dispersed particles envelope a distinctive well-defined self-assembled interior. The scope of this review covers dispersions of both inverted-type liquid-crystalline particles (cubosomes, hexosomes, micellar cubosomes, and sponge phases), and microemulsion droplets (emulsified microemulsions, EMEs). Recent investigations that have attempted to shed light on the characterization and the control of confined nanostructures of aqueous dispersions are surveyed, as these nanoobjects are attractive for various pharmaceutical and food applications. The focus has been placed on three main subjects: (1) our findings on the formation of EMEs and the modulation of the internal nanostructure, exploring how variations in temperature, oil content, and lipid composition significantly affect the confined nanostructures; (2) recent developments in the field of electron microscopy: using the tilt-angle cryo-TEM method or cryo-field emission scanning electron microscopy (cryo-FESEM) for observing the three dimensional (3D) morphology of non-lamellar liquid-crystalline nanostructured particles (cubosome and hexosome particles); and (3) recent studies on the utilization of nanostructured dispersions as drug nanocarriers.

Solving the generalized indirect Fourier transformation (GIFT) by Boltzmann simplex simulated annealing (BSSA)

The structure of colloidal particles can be studied with small-angle X-ray and neutron scattering (SAXS and SANS). In the case of randomly oriented systems, the indirect Fourier transformation (IFT) is a well established technique for the calculation of model-free real-space information. Interaction leads to an overlap of inter- and intraparticle scattering effects, preventing most detailed interpretations. The recently developed generalized indirect Fourier transformation (GIFT) technique allows these effects to be separated by assuming various models for the interaction, i.e. the so-called structure factors. The different analytical behaviour of these structure factors from that of the form factors, describing the intraparticle scattering, allows this separation. The mean-deviation surface is defined by the quality of the fit for different parameter sets of the structure factor. Its global minimum represents the solution. The former non-linear least-squares approach has proved to be inefficient and not very reliable. In this paper, the incorporation of the completely different Boltzmann simplex simulated annealing (BSSA) algorithm for finding the global minimum of the hypersurface is presented. This new method increases not only the calculation speed but also the reliability of the evaluation.

Small-angle scattering of interacting particles. II. Generalized indirect Fourier transformation under consideration of the effective structure factor for polydisperse systems

The indirect Fourier transformation (IFT) is the method of choice for the model-free evaluation of small-angle scattering data. Unfortunately, this technique is only useful for dilute solutions because, for higher concentrations, particle interactions can no longer be neglected. Thus an advanced technique was developed as a generalized version, the so-called generalized indirect Fourier transformation (GIFT). It is based on the simultaneous determination of the form factor, representing the intraparticle contributions, and the structure factor, describing the interparticle contributions. The former can be determined absolutely free from model assumptions, whereas the latter has to be calculated according to an adequate model. In this paper, various models for the structure factor are compared, e.g. the effective structure factor for polydisperse hard spheres, the averaged structure factor, the local monodisperse approximation and the decoupling approximation. Furthermore, the structure factor for polydisperse rod-like particles is presented. As the model-free evaluation of small-angle scattering data is an essential point of the GIFT technique, the use of a structure factor without any influence of the form amplitude is advisable, at least during the first evaluation procedure. Therefore, a series of simulations are performed to check the possibility of the representation of various structure factors (such as the effective structure factor for hard spheres or the structure factor for rod-like particles) by the less exact but much simpler averaged structure factor. In all the observed cases, it was possible to recover the exact form factor with a free determined parameter set for the structure factor. The resulting parameters of the averaged structure factor have to be understood as apparent model parameters and therefore have only limited physical relevance. Thus the GIFT represents a technique for the model independent evaluation of scattering data with a minimum of a priori information.

Convolution square root of band-limited symmetrical functions and its application to small-angle scattering data

A method for the deconvolution of the convolution square of a symmetrical function with a limited range of definition is presented. The solution function is approximated by a number of equidistant step functions. This allows the analytical computation of the integrals of overlap in one-dimensional (lamellar) symmetry, in two-dimensional (cylindrical) symmetry and in three-dimensional (spherical) symmetry. A special iterative linearized weighted-least-squares technique solves the non-linear convolution square-root problem without any a priori information on the solution. As an application, the electron or scattering length density ρ(r) from the distance distribution function p(r) of small-angle scattering is computed as well as the propagation of the statistical error from the input. The influence of imperfect realization of the symmetry conditions is discussed. Numerical instabilities that appear under certain conditions can easily be removed by a stabilization procedure.

Improvements in real‐space deconvolution of small‐angle scattering data

In the cases of spherical, cylindrical or lamellar symmetry it is possible to calculate the radial density distribution from small-angle scattering intensities via indirect Fourier transformation and a convolution square-root technique avoiding the phase problem. The density function is expressed in terms of step functions. All this is accomplished by a method described previously. Firstly, the necessary modifications of the method allowing for arbitrary step widths are discussed. Additional background to the scattering intensities is an important problem in practical application, so it is shown that remaining background scattering is mainly eliminated automatically and does not essentially influence the results in the convolution square-root technique. Finally, the introduction of a least-squares variation algorithm allows for the optimization of simple step models.

Self-assembled structures and pKa value of oleic acid in systems of biological relevance.

In the human digestion process, triglycerides are hydrolyzed by lipases to monoglycerides and the corresponding fatty acids. Here we report the self-assembly of structures in biologically relevant, emulsified oleic acid-monoolein mixtures at various pH values and oleic acid concentrations. Small-angle X-ray scattering, cryogenic transmission electron microscopy, and dynamic light scattering were used to investigate the structures formed, and to follow their transitions while these factors were varied. The addition of oleic acid to monoolein-based cubosomes was found to increase the critical packing parameter in the system. Structural transitions from bicontinuous cubosomes through hexosomes and micellar cubosomes (Fd3m symmetry) to emulsified microemulsions occur with increasing oleic acid concentration. At sufficiently high oleic acid concentration, the internal particle structure was also found to strongly depend on the pH of the aqueous phase: transformations from emulsified microemulsion through micellar cubosomes, hexosomes, and bicontinuous cubosomes to vesicles can be observed as a function of increasing pH. The reversible transition from liquid crystals to vesicles occurs at intestinal pH values (between pH 7 and 8). The hydrodynamic radius of the particles decreases from around 120 nm for internally structured particles to around 60 nm for vesicles. All transitions with pH are reversible. Finally, the apparent pK(a) for oleic acid in monoolein could be determined from the change of structure with pH. This value is within the physiological pH range of the intestine and depends somewhat on composition.

Effects of Reagent and Enzymatically Generated Hypochlorite on Physicochemical and Metabolic Properties of High Density Lipoproteins

Myeloperoxidase (MPO), a protein secreted by activated phagocytes, may be a potential candidate for the generation of modified/oxidized lipoproteins in vivo via intermediate formation of HOCl, a powerful oxidant. During the present study, the effects of reagent NaOCl and OCl− generated by the MPO/H2O2/Cl− system on physicochemical and metabolic properties of high density lipoprotein (HDL) subclass 3 (HDL3) were investigated. Up to a molar oxidant:lipoprotein ratio of approximately 30:1, apolipoprotein A-I (apoA-I), the major HDL3 apolipoprotein component, represented the preferential target for OCl− attack (consuming 35–76% of the oxidant), thereby protecting HDL3 fatty acids (consuming between 17 and 30% of the oxidant) against OCl−-mediated modification. At molar oxidant:HDL3 ratios ≥ 60:1, we have observed pronounced consumption of HDL3 unsaturated fatty acids with concomitant formation of fatty acid chlorohydrins. Modification of HDL3 in the presence of the MPO/H2O2/Cl− system resulted in amino acid oxidation in a manner comparable with that found with reagent NaOCl only. Treatment of HDL3 with reagent NaOCl as well as modification by the MPO/H2O2/Cl− system resulted in significantly enhanced turnover rates of HDL3 by mouse peritoneal macrophages, an effect that was not a result of HDL3 aggregation as judged by dynamic and static light-scattering experiments. In comparison with native HDL3, the degradation by macrophages was enhanced by 4- and 15-fold when HDL3 was modified with reagent NaOCl or the MPO/H2O2/Cl− system. Finally, the ability of HDL3 to promote cellular cholesterol efflux from macrophages was significantly diminished after modification with reagent NaOCl. Collectively, these results demonstrate that the modification of HDL3 by hypochlorite (added as reagent or generated by the MPO/H2O2/Cl−system) transformed an antiatherogenic lipoprotein particle into a modified lipoprotein with characteristics similar to lipoproteins commonly thought to initiate foam cell formation in vivo.