Indirect Fourier Transformation Method Research Articles

Structural Investigation of Diglycerol Monolaurate Reverse Micelles in Nonpolar Oils Cyclohexane and Octane

Structure of diglycerol monolaurate (abbreviated as C(12)G(2)) micelles in nonpolar oils cyclohexane and n-octane as a function of compositions, temperatures, and surfactant chain length has been investigated by small-angle X-ray scattering (SAXS). The SAXS data were evaluated by the generalized indirect Fourier transformation (GIFT) method and real-space structural information of particles was achieved. Conventional poly(oxyethylene) type nonionic surfactants do not form reverse micelles in oils unless a trace water is added. However, present surfactant C(12)G(2) formed reverse micelle (RM) in cyclohexane and n-octane without addition of water at normal room temperature. A clear signature of one dimensional (1-D) micellar growth was found with increasing C(12)G(2) concentration. On the other hand, increasing temperature or hydrocarbon chain length of surfactant shorten the length of RM, which is essentially a cylinder-to-sphere type transition in the aggregate structure. Drastic changes in the structure of RM, namely, transition of ellipsoidal prolate to long rod-like micelles was observed upon changing oil from cyclohexane to octane. All the microstructural transitions were explained in terms of critical packing parameter.

Contrast Variation SANS Investigation of Composition Distributions in Mixed Surfactant Micelles

Small angle neutron scattering measurements have been performed on three systems (HFDeP-d5-C (N-1(1,1,2,2-tetrahydroperfluorodecanoyl)pyridinium-d5 chloride)/C16PC in 63 mM NaCl; HFDeP-d5-C/C12PC in 200 mM NaCl, and as an example of an ideally mixed system, SDS/SDS-d25 in 200 mM NaCl) containing micelles formed in a binary mixture of surfactants, in order to investigate the composition distribution of the mixed micelles. The experimental data were collected varying the contrast between the average scattering length density of micelles and aqueous solvent by changing the H2O/D2O ratio. Analysis of data includes a model-independent approach--the indirect Fourier transformation method and direct modeling-simultaneous fit at all contrasts by the scattering from micelles of equal size and shape with composition distribution and an effective interaction. It has earlier been shown (Almgren, M.; Garamus, V. M. J. Phys. Chem. B 2005, 109, 11348) that for micelles of equal size, independent of the composition, and with negligible intermicellar interactions, the scattered intensity at zero angle varies quadratically with the contrast, with the minimum intensity at the nominal match point proportional to sigma2, the variance of the micelle composition distribution. Within the regular solution framework, the composition distribution and its variance are uniquely defined by the value of the interaction parameter and the micelle aggregation number. At 25 degrees C, the first system gave sigma = 0.37, corresponding to a broad, bimodal composition distribution, the second sigma = 0.22, a broad distribution with a shallow minimum at the midpoint. For SDS/SDS-d25, we found sigma = 0.006 +/- 0.030, which is a smaller value than that of the binominal composition distribution expected for an ideally mixed system.

Structure and interaction in dense colloidal systems: evaluation of scattering data by thegeneralized indirect Fourier transformation method

The generalized indirect Fourier transformation (GIFT) technique is a versatile tool for theevaluation of small angle scattering data. It does not depend on models for the size andshape of the particles and requires model assumptions only for the interactioneffects that are typically not as sensitive to the details of the assumptions. Wereview here the development of the technique from its inception, focusing on theincluded interaction models for hard, charged and attractive spheres, and lamellae. Aconsiderable number of applications has also been reported ranging from surfactants,emulsions, microemulsions, food science, and ceramics to melts and block-copolymers.

Small-Angle Scattering from Hexagonal Liquid Crystals

In this paper, we discuss the scattering behavior of two-dimensional hexagonal liquid crystals with micellar cylinders as a building unit. We treat the hexagonal phase as an accumulation of ordered domains of finite size that typically consists of one hundred parallel cylinders whose axes are perpendicular to the lattice plane. When we suppose that no specific orientation is preferred, the lengths of the cylinders are rather large compared to their diameter, and the polydispersity of the size of the cylinders is negligible; it is therefore possible to split the scattering intensity into a product of the so-called form factor and the structure factor. This product approximation is the basic condition for the use of the generalized indirect Fourier transformation (GIFT) method and the deconvolution (DECON) method to evaluate the small-angle scattering data of hexagonal phases. The GIFT method provides the parameters of the structure factor model and the pair distance distribution function of the cylinders. Via the DECON technique, we can calculate the radial contrast profile of the cylinders from the pair distance distribution function that is obtained by the GIFT method.

Ultra-small-angle neutron scattering from dense micrometre-sized colloidal systems: data evaluation and comparison with static light scattering

Ultra-small-angle neutron scattering (USANS) probes the sameqregime as static light scattering (LS), making USANS an additional tool for the study of structures between 100 nm and 10 µm. Dense oil-in-water emulsions, which have already been characterized intensively by light scattering, are investigated in this study as a model system using USANS. The two basic problems of such scattering studies are the following: on the one hand, one has to use different scattering theories for USANS and LS, and on the other hand, in both cases one has to deal not only with particle interactions but also with multiple-scattering effects. For neutron scattering it is always possible to use the simpler Rayleigh–Debye–Gans (RDG) theory instead of the Lorenz–Mie theory, which generally describes light scattering from micrometre-sized globular objects. The samples have different contrasts in neutron and light scattering, such that only low-contrast (close to index match) LS data can be interpreted by the RDG theory. The data evaluation is performed by means of the generalized indirect Fourier transformation (GIFT) method, which enables the simultaneous calculation of the form and structure factors. The results are discussed and compared with those from light scattering experiments, taking into account the advantages of both methods. The effect of multiple scattering and its influence on data evaluation is also examined. Data evaluation by applying the GIFT method works well for both neutron and light scattering data, with results of comparable quality. The advantages of light scattering are the fast data acquisition and the large number of data points. USANS, on the other hand, covers a widerqrange and the problem of multiple scattering is not as severe as for light scattering, but still must not be neglected.

Ternary systems of nonionic surfactant Brij 35, water and various simple alcohols: Structural investigations by small-angle X-ray scattering and dynamic light scattering

Structural properties of ternary systems composed of nonionic surfactant dodecyl-poly(ethylene oxide-23) ether (C 12E 23, commercial name: Brij 35), water and various alcohols from ethanol to 1-decanol have been investigated using small-angle X-ray scattering (SAXS) and dynamic light scattering (DLS) techniques. All measurements were performed at the temperature 25 °C. SAXS experimental data were put on absolute scale using water as a secondary standard. The data of water-rich mixtures at low to moderate surfactant concentrations were evaluated using the generalized indirect Fourier transformation method (GIFT), which is based on the simultaneous determination of the intra- and inter-particle scattering contributions. In this way, the size and the shape of interacting scattering particles in real space could be deduced. The systems with a relatively low surfactant concentration (5 mass%) were studied most extensively. In these cases, the water-rich regions of the phase diagrams could be investigated into more detail, since in the alcohol-rich regions problems with the GIFT evaluation of the SAXS data were encountered. The presented results demonstrate the level of structural details that can be obtained on the basis of scattering methods and point out the specific stages of data evaluation and interpretation where one must be extremely precautious. As such they reveal the inner structuration of the complex ternary systems of our present interest. In parallel, they also indicate that the longer chain alcohols actually behave as real oil phases in the studied systems, as one might expect, and also confirm the well-known properties of different short to medium chain alcohols that act as co-solvents and/or co-surfactants in microemulsion systems depending on their chain length.

Small-Angle X-ray Scattering, Light Scattering, and NMR Study of PEO−PPO−PEO Triblock Copolymer/Cationic Surfactant Complexes in Aqueous Solution

The formation of triblock copolymer/surfactant complexes upon mixing a nonionic Pluronic polymer (PEO-PPO-PEO) with a cationic surfactant, hexadecyltrimethylammonium chloride (CTAC), has been studied in dilute aqueous solutions using small-angle X-ray scattering, static and dynamic light scattering, and self-diffusion NMR. The studied copolymer (denoted P123, EO(20)PO(68)EO(20)) forms micelles with a radius of 10 nm and a molecular weight of 7.5 x 10(5), composed of a hydrophobic PPO-rich core of radius 4 nm and a water swollen PEO corona. The P123/CTAC system has been investigated between 1 and 5 wt % P123 and with varying surfactant concentration up to approximately 170 mM CTAC (or a molar ratio n(CTAC)/n(P123) = 19.3). When CTAC is mixed with micellar P123 solutions, two different types of complexes are observed at various CTAC concentrations. At low molar ratios (>/=0.5) a "P123 micelle-CTAC" complex is obtained as the CTAC monomers associate noncooperatively with the P123 micelle, forming a spherical complex. Here, an increased interaction between the complexes with increasing CTAC concentration is observed. The interaction has been investigated by determining the structure factor obtained by using the generalized indirect Fourier transformation (GIFT) method. The interaction between the P123 micelle-CTAC complexes was modeled using the Percus-Yevick closure. For the low molar ratios a small decrease in the apparent molecular weight of the complex was obtained, whereas the major effect was the increase in electrostatic repulsion between the complexes. Between molar ratios 1.9 and 9 two coexisting complexes were found, one P123 micelle-CTAC complex and one "CTAC-P123" complex. The latter one consists of one or a few P123 unimers and a few CTAC monomers. As the CTAC concentration increases above a molar ratio of 9, the P123 micelles are broken up and only the CTAC-P123 complex that is slightly smaller than a CTAC micelle exists. The interaction between the P123/CTAC complexes was modeled with the hypernetted-chain closure using a Yukawa type potential in the GIFT analysis, due to the stronger electrostatic repulsion.

Colloidal Systems with Attractive Interactions: Evaluation of Scattering Data Using the Generalized Indirect Fourier Transformation Method

Different attractive interacting colloidal systems are characterized by means of static light scattering. As most of these samples are rather concentrated, multiple scattering is suppressed by partial contrast match and the use of very a thin sample cell (13 μm). This is possible with the laboratory built flat cell light scattering instrument, which is also capable of time-resolved measurements. The systems under investigation are oil-in-water emulsions with added polymer micelles or latex spheres, which give rise to depletion interaction. A second set of samples consists of electrostatically stabilized dense silica suspensions, which are destabilized by ionic strength or pH shift initiated by an in-situ reaction. The potential change from repulsive to attractive is measured time-resolved in real time. These suspensions are model systems for the so-called direct coagulation casting (DCC) method. Scattering data are evaluated using the generalized indirect Fourier transformation (GIFT) method. We added some new routines for calculating structure factors for attractively interacting systems to the already existing software package. We now can choose between a depletion interaction and an attractive square well potential, which was used for the DCC samples. The obtained results are in good agreement with the values known from preparation for the depletion samples and those predicted from DLVO calculations for the DCC series, respectively.

Borate–sugar complexation-induced charging of n-octyl-β- d-glucopyranoside micelles : Investigation of structure alteration using small angle neutron scattering

In this work, n-octyl-β- d-glucopyranoside (OG) micelles were charged in situ by borate buffer. The alteration of their structure was investigated by small angle neutron scattering. The scattering data were analyzed by the indirect Fourier transformation method and model fitting (cylindrical model). It was found that the size of charged OG micelles is reduced in comparison to non-charged micelles and the degree of the ionization of OG molecules is low. The dependence of micelle size on pD of the solvent was also observed.

Structure and order in lamellar phases determined by small-angle scattering

Multilamellar phases can be identified and characterized by small-angle scattering of X-rays (SAXS) or neutrons (SANS). Equidistant peaks are the typical signature and their spacing allows the fast determination of the repeat distance,i.e.the mean distance between the midplane of neighbouring bilayers. The scattering function can be described as the product of a structure factor and a form factor. The structure factor is related to the ordering of the bilayers and is responsible for the typical equidistant peaks, but it also contains information about the bilayer flexibility and the number of coherently scattering bilayers. The form factor depends on the thickness and the internal structure (scattering length density distribution) of a single bilayer. The recently developed generalized indirect Fourier transformation (GIFT) method is extended to such systems. This method allows the simultaneous determination of the structure factor and the form factor of the system, including the correction of instrumental broadening effects. A few-parameter model is used for the structure factor, while the determination of the form factor is completely model-free. The method has been tested successfully with simulated scattering data and by application to experimental data sets.

Nonionic Surfactant Brij 35 in Water and in Various Simple Alcohols: Structural Investigations by Small-Angle X-ray Scattering and Dynamic Light Scattering

Small-angle X-ray scattering (SAXS) and dynamic light scattering (DLS) techniques have been used to study the structural properties of nonionic surfactant dodecyl-poly(ethylene oxide-23) ether (C12E23, commercial name Brij 35) in water and in various alcohols from ethanol to 1-decanol. All measurements were performed at 25 °C. SAXS experimental data were put on absolute scale using water as a secondary standard and evaluated using the generalized indirect Fourier transformation method (GIFT), which is based on the simultaneous determination of the intra- and interparticle scattering contributions. In this way, the size and shape of interacting scattering particles in real space could be deduced. In the Brij 35/water binary system (surfactant concentrations from 0.5 to 25 mass %) globular micelles with maximum dimensions ranging from 9 to 10.5 nm and radii of the hydrophobic core around 1.7 nm were found. At low surfactant concentrations the micelles were more or less spherical, whereas slight elongation o...

Comparison of Small-Angle Scattering Methods for the Structural Analysis of Octyl-β-maltopyranoside Micelles

Small-angle neutron scattering (SANS) and small-angle X-ray scattering (SAXS) experiments were performed to investigate the micelle structure of n-octyl-β-maltopyranoside (OM). Density measurements were carried out to obtain the volumetric characteristics of OM for micelle structure analysis. Both temperature and concentration were varied in scattering experiments to study their effects on micelle size. The scattering data were analyzed by the indirect Fourier transformation method (IFT) and model fitting. The IFT method gave the radius of gyration of the micelles and their pair distance distribution function. It was found that the radii of gyration from SANS data were much smaller than those from SAXS data at similar solution conditions. Moreover, pair distance distribution functions from SANS and SAXS data were also different. Model fitting indicated that a spherical shell model can be used to describe both SANS and SAXS data using similar structure parameters. Comparison of SAXS data in D2O and H2O sho...

Analysis of neutron double-crystal diffractometer scattering curves including multiple scattering

Double-crystal diffractometers (DCD) of the Bonse–Hart type are used for small-angle neutron scattering (SANS) experiments with very high resolution. Scattering from inhomogeneities in the size range of typically 100 nm to 10 µm can be detected with a DCD. The evaluation of DCD scattering curves is more complicated than for conventional SANS instruments with point collimation for two reasons: first, a DCD operates with a long-slit geometry, and second, owing to the high scattering power of large inhomogeneities, multiple small-angle scattering is likely to play a significant role in many DCD experiments. A numerical procedure is proposed for the determination of particle size distributions from DCD scattering curves influenced by multiple scattering. This procedure is based on ann-fold convolution of single-scattering cross sections, wherendepends on the degree of multiple scattering. It makes use of the indirect Fourier transformation method and thus, firstly, size distributions of arbitrary shape can be determined, and secondly, the calculation is fast, allowing an interactive analysis. The procedure is demonstrated with scattering from latex particles with different degrees of multiple scattering. The results show that the size and volume fraction of the latex spheres can be determined with a reasonable error even from scattering curves that are strongly influenced by multiple small-angle scattering.

Evaluation of double-crystal SANS data influenced by multiple scattering

Evaluation of small-angle neutron scattering (SANS) data is often complicated by multiple scattering effects if large particles of relatively high volume fraction have to be studied and dilution or contrast reduction is impossible. The use of pin-hole SANS instruments is often limited due to the contradictory requirements of high resolution and short wavelength needed to keep scattering contrast as low as possible. Double crystal (DC) SANS diffractometers of Bonse-Hart and bent-crystal type are useful alternatives in such cases, as they permit reaching very high resolution with thermal neutrons. A method for SANS data evaluation suited to DC instruments is presented. It includes the common scheme of the indirect Fourier transformation method, but takes multiple scattering into account. The scattering medium is described by the frequency function g(x) defined as the cosine Fourier transform of slit-smeared data. Although a simplistic model of polydisperse spheres is used to represent g(x), resulting g(x) function and some integral parameters are independent of this model. Tests on simulated data show, that the method reproduce well true values of microstructural parameters, though systematic errors are observed in the cases when the unscattered part of incident beam completely disappears. If the scattering power <N> is known and kept fixed during fitting, then other parameters are reproduced well also in the regime of strong multiple scattering. The evaluation procedure permits simultaneously fitting to several sets of data measured for different Q-regions, resolutions and sample thicknesses. It has proved to provide reliable results for particle sizes ranging from about 100 Å to several microns and <N> < 10.

Small-Angle Scattering of Interacting Particles. I. Basic Principles of a Global Evaluation Technique

Indirect Fourier transformation is a well established method for the evaluation of small-angle scattering data. This technique, however, is restricted to dilute solutions, as for higher concentrations particle interactions can no longer be neglected. As the scattering intensity contains intra- and interparticle scattering contributions, the evaluation of scattering data is no longer just the solution of a linear weighted least-squares problem because the scattering intensity can, under certain conditions, be written as the product of the particle form factor and the so-called structure factor, which leads to a highly non-linear problem. In this paper a global evaluation technique including the structure factor is presented so that it is possible to determine the form factor and the structure factor simultaneously. This technique can be understood as a generalized version of the indirect Fourier transformation method. Like in the indirect Fourier transformation, there are no models or no analytical restrictions used for the form factor, and the structure factor is parameterized with up to four parameters for a given interaction model. A simultaneous determination of these two functions is possible due to the different analytical behavior of these functions, which also leads in most cases to the existence of a global minimum in the parameter surface. An algorithm to solve this nonlinear least-squares problem has been developed and applied to simulated data for a variety of different uncharged systems.

Cross-Section Structure of Cylindrical and Polymer-Like Micelles from Small-Angle Scattering Data. I. Test of Analysis Methods

The application of the indirect Fourier transformation method and the square-root deconvolution procedure of Glatter [Glatter (1977). J. Appl. Cryst. 10, 415–421; Glatter (1981). J. Appl. Cryst. 14, 101–108] for a determination of the cross-section structure of cylindrical and polymer-like micelles is investigated. The data analysis methods assume: (i) the rod-like structures are infinitely long and straight; (ii) the particles have cylindrical symmetry; (iii) the particles are monodisperse in the cross-section size and have no fluctuations in the cross-section structure. The influence of the violations of these assumptions on the applicability of the methods and the limitations have been investigated by analyzing series of simulated scattering data. In addition, the influence of finite range of scattering vectors for the data and the presence of diffuse correlations along the rod-like particles have been determined.

Cross-Section Structure of Cylindrical and Polymer-like Micelles from Small-Angle Scattering Data. 2. Experimental Results

We report a small-angle neutron scattering (SANS) study of the cross-section structure of polymer-like lecithin reverse micelles in deuterated cyclohexane. We demonstrate that the application of the indirect Fourier transformation and square-root deconvolution methods to data from SANS measurements with cylindrical polymer-like micelles allows for a direct verification of the previously postulated geometrical model of flexible tubular structures with a well defined water core and a surfactant shell. By combining contrast variation experiments and data analysis performed on an absolute scale, we quantitatively deduce information on properties such as the extension of the aqueous core and the degree of water penetration into the headgroup and solvent penetration into the tail region.

The structure of casein aggregates during renneting studied by indirect Fourier transformation and inverse Laplace transformation of static and dynamic light scattering data, respectively

Aggregation of casein micelles after addition of the proteolytic enzyme chymosin has been studied by static and dynamic light scattering at three different concentrations of casein corresponding to dilutions 1:100, 1:500, and 1:1000 of native milk. The static light scattering data have been analyzed by an indirect Fourier transformation method which gives the distance distributions as a function of time. From these curves radius of gyration and an average number of casein micelles in the aggregates have been derived as a function of time. The dynamic light scattering experiments give the hydrodynamic radius as a function of time after the addition of rennet. The initial radius of gyration for the intact casein micelles is 140 nm. The corresponding hydrodynamic radius is also 140 nm. This shows that the casein micelles are not solid spheres. Inspection of a plot of relative mass versus radius of gyration for the aggregates appearing after the addition of chymosin shows that two processes take place. First extended linear aggregates are built up to a relative mass of the aggregates of about 10 and then restructuring of aggregates occurs such that increasingly compact objects are formed. Whereas the first process exhibits a relatively fast growth in size, the aggregates grow slowly in size during the second process. Further evidence of the formation of linear aggregates followed by more dense aggregates was obtained by forming the ratio between the radius of gyration and the hydrodynamic radius. This ratio increases to values of about 2.5 (indicating that linearly extended molecules are present followed) by a decrease to about 1. The log–log plot of mass versus radius of gyration is linear up to relative masses of about 10 with a slope of about 2. This extends up to sizes of 1 μm in diameter. The slope then increases to values indicating branching and thereby the formation of more compact aggregates. For relative masses below 10 and sizes below 1 μm sedimentation is unlikely to occur and information about the mechanism of aggregation can be obtained. The aggregation number as a function of time has been analyzed in terms of Smoluchowski’s equations with a rate constant including both functionality and a changing barrier height as a function of the extent of proteolysis. The functionality obtained from Smoluchowski’s equations is about 2.1.

Small-angle X-ray scattering study of Zr 4(Fe,Cr) precipitate in beta-quenched zircaloy-4

Zircaloy-4 through beta quenching treatment has been extensively used as cladding and core structure material in water-cooled nuclear reactors, due to its high nodular corrosion resistance and mechanical strength. Research shows that corrosion resistance is highly related to the sites and size distribution of the second phase precipitate particles after quenching. The nodular corrosion resistance decreases as particle size increases, while uniform corrosion resistance increases. The existence of a large amount of quenched-in vacancies enhances the nucleation and growth of precipitates during the quench process. The microstructure of zircaloy-4 transforms from beta phase (bcc) to Widmanstaetten alpha phase (hcp) with a lathe structure by martensitic transformation. The main intermetallic precipitates, Zr{sub 4}(Fe,Cr), with a fcc phase mostly precipitated at the lathe boundaries, having sizes from 4 to 35 nm. These particles can be considered as a polydispersive system. SAXS (Small-Angle X-ray Scattering) technique, which is used in this study, is a powerful tool to determine the size distribution of fine particles. The IFT (Indirect Fourier Transformation) method is used in this study to analyze the measured scattering intensity distribution which comes from the scattering of all particles (below 50 nm) within the specimen. This work attempts to more accurately reconstructmore » or determine the size distribution of the second phase particles in beta-quenched zircaloy-4 without prior assumption of the shape of size distribution. The results can be used in future work to explore the mechanism of growth kinetics of Zr{sub 4}(Fe,Cr) precipitate.« less

A treatment of instrumental smearing effects in circularly symmetric small-angle scattering

Instrumental smearing effects can result in a significant deviation of measured small-angle scattering data from the ideal cross section. The effect of various contributions to smearing is described. The case of circularly symmetric scattering measured on an instrument collimated by circular apertures is treated in detail. For such measurements, it is shown how known scattering functions can be smeared to allow estimation of parameters by a fit to the observed data, and also how the indirect Fourier transformation method can be used to desmear the observed scattered intensity.