The evaluation of the small-angle scattering of lamellar two-phase systems by means of interface distribution functions

Summary The basic conditions for the interpretation of the small-angle scattering of lamellar two-phase systems using interface distribution functions are discussed. It is shown that the determination of interface distribution functions requires the existence of a region in the intensity distribution in which Porod's law is valid. The influence of errors due to systematic deviations from Porod's law, non-planar lamellae, multiple scattering etc. on the evaluation is discussed, from which one can conclude that the experimental requirements for the applicability of the method can be met by modern small-angle equipment.An evaluation method has been developed which combines the desmearing and the determination of the interface distribution function using Bessel transforms.

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Zusammenfassung Die Grundbedingungen für eine Auswertung der Kleinwinkelstreuung lamellarer Zweiphasensysteme mit Hilfe von Grenzflächenverteilungsfunktionen werden diskutiert. Es wird gezeigt, daß die Bestimmung von Grenzflächenverteilungsfunktionen die Existenz eines Bereiches in der Intensitätsverteilung voraussetzt, in der das Porodsche Gesetz gültig ist. Der Einfluß von Meßfehlern bedingt durch systematische Abweichungen vom Porodschen Gesetz, unebene Lamellen, Mehrfachstreuung usw. auf die Auswertung wird diskutiert; hieraus ergibt sich, daß die. experimentellen Bedingungen für die Anwendbarkeit der Methode durch moderne Kleinwinkelapparaturen erfüllt werden können. Es wurde eine Auswertungsmethode entwickelt, die Entschmierung und Bestimmung der Grenzflächenverteilungsfunktion mit Hilfe von Besseltransformierten kombiniert.

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With 6 figures     

The determination of the lamellar thickness in semi-crystalline polymers by small-angle scattering

Three small-angle scattering methods for determining the thickness of the crystalline lamellae in polymers are described. These are based on 1. observation of the identity period, 2. determination of the specific surface of the phase boundary, and 3. analysis of the shape of the peak at the origin. The conditions for application of these methods and the types of average obtained are discussed, and some results are presented.     

Application of interface distribution functions to the evaluation of the small angle x-ray scattering of multicomponent polymer systems

A method for correcting the small angle x-ray intensity of multi-component polymer systems to compute the interface distribution function is presented. The resolution of this function can be improved to a great extent by evaluating the interference function from the measured intensity corrected for fluctuation scattering by use of a function which is proportional to the gas scattering of the investigated system. The method is applied successfully to polymer blends composed of components having similar morphology.Presented at the Spring Meeting of the Deutsche Physikalische Gesellschaft, Hamburg, March 14–16, 1988.     

Determination of the width of the domain boundaries in polymer two-phase systems by X-ray small-angle scattering

An accurate determination of the width of domain boundaries presupposes an unambigous elimination of the intensity component due to density fluctuations within the domains. The theoretical aspects of this problem are discussed and an application is demonstrated in an SAXS study of a styrene-isoprene-styrene blockcopolymer. The widths of the domain boundaries depend on the history of the sample, its values are, in general, smaller than theoretically predicted. The domain sizes correspond to the theoretical values, its variance is of the same order as the variance of the molecular weight of the domain-building blocks.Dedicated to Prof. Dr. Dr. h. c. O. Kratky on the occasion of his 80th birthday.     

Determination of spherulite size distribution by small-angle light scattering

The theoretical backgrounds of the method of spherulite size distribution determination, on the basis of the small-angle light scattering technique, are presented. Special corrections of experimental data are introduced to achieve great accuracy of the determination of scattered light intensity distribution. Experiments have been done on low-density polyethylene samples obtained in various crystallization conditions. For these samples, the stepwise distributions of spherulite size have been determined.     

Shear-induced coalescence in two-phase polymeric systems. I. Determination from small-angle neutron scattering measurements

It has been found that flow-induced coalescence occurs at a very rapid rate during the mixing of polymeric fluids. Furthermore, the rheological properties of the dispersed and continuous phases, as well as the nature of the flow field used in their blending, can greatly influence this coalescence. The significance of these findings is that in the development of a mixing scheme to obtain a desired morphology with minimum expenditure of time or energy, attention is usually focused only on the particle breakup aspects of the blending process. The competing coalescence can, however, be accelerated by the same conditions often employed to facilitate particle breakup (e.g., higher shear rates, reduced dispersed-phase viscosity). A better understanding of the mechanism for coalescence of viscoelastic fluids is clearly required. In this manner optimal blending of immiscible polymers can be achieved with respect to both the nature of the final material and the ease with which it is obtained.     

A simple representation of small-angle light scattering from an anisotropic sphere

An explicit form of Mueller's phenomenological matrix has been derived which characterizes the small-angle light scattering from an anisotropic sphere when the requirements of the Rayleigh-Gans approximation are fulfilled.     

The influence of crystallinity distribution on small-angle x-ray scattering from semicrystalline polymers

Theoretical x-ray scattering curves have been developed for the lamellar structure in semicrystalline polymers in which there are present distributions of lamellar thickness and crystallinity. The models have been tested against samples of linear low-density, low-density and crosslinked polyethylenes. When variation in crystallinity is present in a material, a major effect is an increase in the magnitude of near-zero angle scattering. The Bragg maximum can appear as an ill-defined hump on an apparently high level of background scattering. The shape of the Bragg peak is influenced more by crystallinity distribution than by lamellar thickness distribution. Of the polymers we have studied so far only linear low density polyethylene shows significant crystallinity distribution effects. A “rule-of-thumb” method for rapid estimation of crystallinity distribution effects has been developed, obviating the need for lengthy simulation.     

A small-angle neutron scattering study of cholic acid-based organogel systems

Small-angle neutron scattering measurements were performed on some cholic acid-based gel systems in order to gain detailed information about the network structure. The presence of thin fibers with a radius of about 10-20 A was found for various gelators. Two types of interaction between different sorts of fibers were demonstrated, depending on the molecular structure of the gelator. The first type involves the presence of microcrystalline knots with a dimension of about 100-200 A between the fibers. Upon heating, this network gradually disintegrates. The second type involves loose entanglements between flattened fibers. The occurrence of these types of interaction is related to the length of the alkyl tail attached to cholic acid.     

Protein diffusion across the interface in aqueous two-phase systems

We present a detailed study of the diffusive transport of proteins across a fluid phase boundary within aqueous two-phase systems. The aim of the work is to investigate whether local effects at the phase boundary cause a retardation of the diffusive transport between the phases. Possible modifications of interfacial mass transfer could be due to protein adsorption at the phase boundary or local electric fields from electric double layers. Experiments with a microfluidic system have been performed in which protein diffusion (bovine serum albumin and ovalbumin) within a bilaminated configuration of two phases containing polyethylene glycol and dextran is analyzed. A one-dimensional model incorporating phase-specific diffusion constants and the difference in chemical potential between the phases has been formulated. A comparison of experimental and simulation data shows a good overall agreement and suggests that a potential local influence of the phase boundary on protein transport is insignificant for the systems under investigation.     

On the modeling of small-angle X-ray scattering from systems of oriented fibrils

A method is outlined for the simulation of the scattering pattern from systems of oriented fibrils using the radial distribution function of a hard-disc fluid obtained under the Percus–Yevick approximation. In this manner both the diameter and the volume fraction of the fibrils may be estimated directly from the scattering pattern. The effect of polydispersity in the distribution of fibril diameters on the position of the scattering peak is discussed, suggesting that an estimate of the volume fraction from the peak position alone, such as in a modified Bragg's law, may be inaccurate.     

Calculating atomic fast-electron scattering amplitudes by means of electron wave functions

Conclusions It is very important to reduce the errors in the theoretical approximation associated with inaccuracy in calculating the atomic-scattering amplitudes for structural electron diffraction.Vapor electron diffraction is advanced by calculating the atomic scattering amplitudes by means of a method that gives them directly from the electron wave functions for the atoms, in conjunction with the software suite written for the purpose.High-Temperatures Institute, USSR Academyof Sciences. Translated from Zhurnal Strukturnoi Khimii, Vol. 31, No. 1, pp. 41–48, January–February, 1990.     

Interaction of blockcopolymer micelles as probed by small-angle neutron and by small-angle X-ray scattering

 The analysis of the interaction of micelles formed by a blockcopolymer is given by means of small-angle X-ray (SAXS) and small-angle neutron scattering (SANS). The blockcopolymer consists of poly(styrene) and poly(ethylene oxide) (molecular weight of each block: 1000 g/mol) and forms well-defined micelles (weight-association number: 400, weight-average diameter: 15.4 nm) in water. The internal structure has been studied previously (Macromolecules 29:4006 (1996)) by SAXS. There it has been shown that the micelles are spherical objects. The structure factor S(q) as a function of the scattering vector q (q=(4π/λ) sin (θ/2); λ: wavelength of the radiation in the medium; θ: scattering angle) can be extracted from both sets of small-angle scattering data (SANS: q≤0.4 nm^-1; SAXS: q≤0.6 nm^-1). It is shown that particle interaction in the present system can be described by assuming soft interaction which is modeled by a square-step potential. Received: 12 May 1997 Accepted: 9 July 1997     

Investigation of humate-cetyltrimethylammonium complexes by small-angle X-ray scattering

The structural examination of the complexes formed between humic acid and cationic surfactants has environmental implications. A humic acid (HA) dissolved in 0.1 M NaOH (5 g/L) was reacted with a cationic surfactant (hexadecyltrimethylammonium bromide or CTAB) at initial solution concentrations of 1, 5, 10, 20, 30, 40 and 50 mM. The HA precipitated at CTAB concentrations of 20, 30, and 50 mM but the complexes were soluble at 40 mM and below 20 mM. The charge neutralization between humic acid anions and CTAB micelles and the subsequent charge reversal due to hydrophobic interactions explain the behavior of the HA-CTAB complexes. The HA solution (5 g/L), reaction products (supernatants and precipitates), and pure cationic surfactant solutions were studied by the small-angle X-ray scattering (SAXS) technique in order to determine the structure of HA-CTAB complexes. The scattering intensity (I(q)) of various HA-CTAB systems were recorded over a range of scattering vectors (q=0.053-4.0 nm(-1)). HA forms networks in an alkaline solution with a characterization length of 7.8 nm or greater. The HA-CTAB precipitates and the 50-mM CTAB solution gave d(100) and d(110) reflections of a hexagonal structure. The hexagonal array of cylindrical CTAB micelles has a lattice parameter of 5.01 nm in pure solution, and the parameter decreases in the order: 4.96, 4.91, and 4.85 nm for the precipitates of HA-CTAB (50, 30, and 20 mM, respectively), indicating that the structure of CTAB micelles was disturbed by the addition of HA. The molecular properties and behavior of HA in solution were discussed.     

Characterization of polystyrene networks by small-angle neutron scattering

Polystyrene networks prepared by anionic polymerization have been characterized by small-angle neutron scattering. Two kinds of systems have been examined: (A) networks with labelled branch points allowing characterization of the spatial distribution of crosslinking points; (B) networks containing a low proportion of chains labelled with perdeuterated polystyrene in order to characterize the conformation of individual elastic chains of the polymeric network. The dependence of the results on swelling and uniaxial extension is discussed.     

Scattering structure factor of colloidal gels characterized by static light scattering, small-angle light scattering, and small-angle neutron scattering measurements

The scattering structure factor of a colloidal gel in a q range of 5 orders of magnitude has been determined by combining static light scattering, small-angle light scattering, and neutron scattering measurements. It exhibits simultaneously two types of structure information: a mass fractal scaling within the clusters that constitute the gel and a surface fractal scaling for length scales larger than that of the clusters. Such scattering behavior can be well interpreted by the pair-correlation function proposed in the literature to model an ideal structure constituted of mass fractal objects inside surface fractal objects.     

Analysis of the conformation of wormlike chains by combination of small-angle x-ray and small-angle neutron scattering

An analysis of the conformation of a wormlike polymer by small-angle scattering is presented. By a combined investigation of small-angle X-ray and of small-angle neutron scattering the effect of the finite size of the repeating units can be eliminated. The procedure suggested herein therefore allows to obtain the scattering function for a respective infinitely thin chain. The latter quantity is compared to current models of the scattering function of wormlike chains.     

The interface between immiscible polymers in composite latexes: a small-angle x-ray scattering study employing contrast variation

An investigation of the radial structure of composite latex particles by small-angle x-ray scattering (SAXS) is given. Measurements at different contrasts were done by addition of sucrose to the dispersion medium water. The latex particles investigated here consist of a poly(styrene) core and a shell of poly(methylmethacrylate) and were prepared by seeded emulsion polymerization. Since the electron density of both polymers can be easily matched by concentrated sucrose solution, a full analysis of the radial electron density by contrast variation can be given. Depending on the mode of monomer addition during the second polymerization step a very sharp or a diffuse interface between the two incompatible polymers may result.Respectfully dedicated to Prof. E. W. Fischer on the occasion of his 65th birthday