Chord length distributions and small-angle scattering

Chord length distributions describe size, shape and spatial arrangement of geometrical objects (particles). The chord length distribution is in principle proportional to the second derivative of the correlation function of small-angle scattering. It is calculable from a relative measurement of the scattering intensity I(h). In structure research, the characterization of numerous particle systems can be achieved by comparing experimental chord distributions with theoretical ones, provided the latter are available with sufficiently high precision for a lot of fundamental, universal shapes. Both sides of this concept are exemplified: – the step from a relative measurement of the scattering intensity of an isotropic two-phase sample to the chord length distribution (errors in and in , limited h-interval, corresponding to the region (1-2) nm < r in real space, must be observed); as well as the geometric matter of calculation of chord distributions as fingerprints for basic geometric figures, including the non-convex case. Received 15 March 1999 and Received in final form 26 April 2000     

A small-angle X-ray scattering study of the isotropic and nematic phases of V2O5 suspensions

We report small-angle X-ray scattering experiments performed in both the isotropic and nematic phases of aqueous V₂O₅ suspensions. We show that the scattering in the isotropic phase can be well described in the whole accessible q-range by only considering the form factor of non-interacting ribbons. We investigate the influence of concentration and pH on the dimensions of V₂O₅ ribbons and show that these parameters do not have any significant effect, as long as the system stays well within the chemical stability domain of the ribbons. We then show that nematic single domains display an anisotropic small-angle scattering pattern, even at scattering vectors small compared to that at which a characteristic correlation peak is observed. This feature is expected for a nematic phase, but was rarely observed. We finally try to describe this scattering within the framework of theories developed for the structure factor of a nematic polymer, and we reach the conclusion that chain ends are certainly important to understand this pattern. Received 21 July 1999 and Received in final form 17 December 1999     

The microstructure of hydrogen- and deuterium-doped nanocrystalline palladium studied by small-angle neutron scattering

By means of small-angle neutron scattering the microstructure of two nanocrystalline Pd samples (prepared by inert gas condensation) has been studied at room temperature in a Q-range from [0pt] to [0pt] . An additional subsequent doping of the two samples with H as well as with D (concentrations < 4 at%) caused contrast variations that provided more detailed structural information. The measured scattering intensity was modeled by a Porod contribution from large heterogenities (e.g. pores) and a contribution from spherical grains with a log-normal distribution of their radii. To account for the presence of grain boundaries, the grains were considered to be surrounded by a shell with a reduced Pd density and a thickness half as large as the thickness of the grain boundaries. For the above model, the data of the H-doped, D-doped and undoped sample were simultaneously fitted with one single set of adjustable parameters. The fits yielded for the two samples volume-weighted mean grain radii of 10 nm and 13 nm. The values for the grain boundary thickness lie between 0.2 and 0.8 nm. Almost all of the H- and D-atoms are, at low hydrogen concentrations, located in the grain boundaries. Received 1 May 2000     

Spontaneous condensation in DNA-polystyrene- b-poly(l-lysine) polyelectrolyte block copolymer mixtures

We investigated the condensation of calf thymus DNA by amphiphilic polystyrene_m-b-poly(l-lysine)_n block copolymers ( PS_m-b- PLys_n, m, n = degree of polymerization), using small-angle X-ray scattering, polarized optical microscopy and laser scanning confocal microscopy. Microscopy studies showed that the DNA condenses in the form of fibrillar precipitates, with an irregular structure, due to electrostatic interactions between PLys and DNA. This is not modified by the presence of hydrophobic PS block. Scattering experiments show that the structure of the polyplexes corresponds to a local order of DNA rods which becomes more compact upon increasing n. It can be concluded that for DNA/ PS_m-b- PLys_n polyplexes, the balance between the PLys block length and the excess charge in the system plays an essential role in the formation of a liquid crystalline phase.     

Atomic Rayleigh scattering cross-sections and the associated anomalous dispersion in the X-ray regions

Elastic scattering cross-sections for Pd, Ag, Cd, In, Sn, Sb, Pt, Au and Pb are measured at an angle of 90 in the X-ray region 5.41 keV. These energies fall between the high-energy side of the L- and M-shell absorption edges of the atoms considered. The present atomic region is significant for solid X-rays to assess the contribution of resonance and solid-state environmental effects. Also it is the anomalous scattering region for many of the atoms of the periodic table. Experimental results are compared with theoretical calculations based on form factor formalisms including the anomalous corrections and available recent S-matrix values. Based on the experimental evidence, the present results indicate the influence of solid-state environmental effects, the importance of anomalous corrections nearer to absorption edges, the correctness of revised high-energy limit values, the superiority of S-matrix predictions over form factor values on measured elastic scattering cross-sections in the X-ray regime and also show the resonance behavior around K, L and M absorption edges. Received: 27 January 1998 / Received in final form: 4 January 1999     

Sol-gel transition in worm-like micelles

Summary The statical and dynamical properties of lecithin/H₂O/cyclohexane cylindrical reverse micelles are investigated as a function of lecithin volume fraction, ϕ, and temperature at fixed water/lecithin ratio,w ₀. The viscosity data are well fitted by the Cates model when the breathing mode of micelles is taken into consideration, overlapping with the breaking and reforming mechanisms. We present some results from Brillouin-scattering experiment, performed across the sol-gel transition. In order to explain the experimentally observed ϕ-dependence of the hyperacoustic parameters, a mechanical model was developed from which the ϕ-dependence of the micelle size distribution was obtained. From a comparison with the viscosity data the entanglement length was estimated. Furthermore some new results from an incoherent quasi-elastic neutron scattering experiment are presented. The whole body of the experimental results suggests for a sol-gel transition triggered by topologic phenomena. When the lecithin volume fraction increases, the kinetic equilibrium between the breaking and reforming mechanisms of the micelles shifts the mean micelle length towards higher values and the entanglement of the micelles becomes highly favourable. The obtained results are discussed and compared with other findings in literature. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Momentum distributions in fluids determined by neutron compton scattering

Summary An introduction is given to the measurement of atomic momentum distribution by scattering of high-energy neutrons. Examples of measurements on liquid hydrogen, solid lithium, quantum fluids Ne and³He-⁴He mixtures are given. Future prospects for this new technique are discussed. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Nanoparticle chain-like formation in electrical double-layered magnetic fluids evidenced by small-angle X-ray scattering

Small-angle X-ray scattering (SAXS) was performed on a series of Electric Double-Layered Magnetic Fluids (EDL-MF) composed of ferrite type-- CoFe₂O₄, MnFe₂O₄, ZnFe₂O₄, NiFe₂O₄ and CuFe₂O₄--nanoparticles of different crystalline sizes ( D _XR ranging from 40 to 139?, as determined by X-ray diffraction). The information concerning the scattering objects was obtained through the analysis of the distance distribution function p(r) and of the size distribution function D(R), both retrieved from SAXS data. The results show that EDL-MF, in the absence of an applied magnetic field, are composed of small magnetic particle aggregates in solution. These agglomerates are elongated in one direction (chain-like) with the longest dimension varying from 240 to 330?. The cross-section size is of the order of D _XR. The data also demonstrate that the maximum dimension of these aggregates is independent of the ferrite type. On the other hand, the number of aggregated magnetic particles is nanoparticle-size-dependent. Accordingly, larger ferrite-type nanoparticles as those with D _XR = 139? form aggregates composed of 2-3 magnetic particles, whereas smaller ones with D _XR 40? form agglomerates of about 6 magnetic particles in solution. As the nanoparticle size is reduced, it might increase the particle surface defects. Such occurrence would affect the particle surface charge density, which could reduce the electrostatic screening, favoring the agglomeration phenomenon. Received 28 February 2000 and Received in final form 28 August 2000     

Neutron reflectivity study of adsorbed diblock copolymers

Summary In this paper, we summarize our cumulative work on neutron reflectivity studies of polystyrene-poly(vinyl-2-pyridine) (PS-PVP) and polystyrene-polyethylene oxide (PS-PEO) adsorbed at a quartz-solvent interface. Deuterated toluene was chosen as the solvent since it is a good solvent for PS and a poor one for either of the other two blocks. In this case, the polystyrene dangles into the solvent while the other block acts as an anchor. The neutron reflectivity studies reveal that the form of the polymer density profile normal to the substrate may be varied from an extended ?brush? to a condensed ?mushroom? conformation by manipulating the ratio of the molecular weights of the two blocks. In addition, we present new data on the PS-PEO system in a poor solvent, deuterated cyclohexane, under conditions of shear flow in Poiseuille geometry. We find that when the PS-PEO diblock is absorbed from cyclohexane and is allowed to relax, the PS chain takes on a ?mushroom? conformation. However, when the shear is applied, the layer shear thickens due to the PS chains extending to nearly twice their original lengths. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Influence of the nematic order on the rheology and conformation of stretched comb-like liquid crystalline polymers

We have studied the rheology and the conformation of stretched comb-like liquid-crystalline polymers. Both the influence of the comb-like structure and the specific effect of the nematic interaction on the dynamics are investigated. For this purpose, two isomers of a comb-like polymetacrylate polymer, of well-defined molecular weights, were synthesized: one displays a nematic phase over a wide range of temperature, the other one has only an isotropic phase. Even with high degrees of polymerization N, between 40 and 1000, the polymer chains studied were not entangled. The stress-strain curves during the stretching and relaxation processes show differences between the isotropic and nematic comb-like polymers. They suggest that, in the nematic phase, the chain dynamics is more cooperative than for a usual linear polymer. Small-angle neutron scattering has been used in order to determine the evolution of the chain conformation after stretching, as a function of the duration of relaxation t _r. The conformation can be described with two parameters only: , the global deformation of the polymer chain, and p, the number of statistical units of locally relaxed sub-chains. For the comb-like polymer, the chain deformation is pseudo-affine: is always smaller than (the deformation ratio of the whole sample). In the isotropic phase, has a constant value, while pincreases as tr. This latter behavior is not that expected for non-entangled chains, in which p varies as t _r ^1/2 (Rouse model). In the nematic phase, decreases as a stretched exponential function of t _r, while p remains constant. The dynamics of the comb-like polymers is discussed in terms of living clusters from which junctions are produced by interactions between side chains. The nematic interaction increases the lifetime of these junctions and, strikingly, the relaxation is the same at all scales of the whole polymer chain. Received 5 May 1999 and Received in final form 18 October 1999     

Chain length dependence of anomalous swelling in multilamellar lipid vesicles

Using small-angle X-ray scattering, the repeat distance vs. temperature is measured for a homologous series of multilamellar vesicles of lecithins with varying acyl chain length in excess water condition around the lipid main transition. A systematic chain length dependence is found which is in accordance with a bending rigidity renormalization and critical unbinding of the lamellae close to the transition, as previously suggested in H?nger et al. [Phys. Rev. Lett. 72, 3911 (1994)]. Received 13 January 1999 and Received in final form 6 September 1999     

Aggregating block copolymers as model systems to study polymer brush dynamics

Summary A-B block copolymers in a selective solvent—good for the B-species and bad for the A-species—form micellar aggregates with a compact A-core with a corona (brush) of B ?hairs? reaching into the solvent. Whereas polystyrene(PS)-polyisoprene(PI) in decane forms spherical micelles with a PS core of about 10 nm radius, polyethylene(PE)-polyethylenepropylene(PEP) forms micellar platelets, the shape of which is goverend by the habitus of PE crystallites forming the core. These planar aggregates have large (several hundred nanometers) lateral extension and a core thickness in the range of 10 nm. Both systems are model systems for polymer brushes, either on a spherical surface or planar. Neutron spin-echo experiments allow for the investigation of the dynamics of the brushes which reflects their viscoelastic properties. Results of neutron small-angle and spin-echo investigations are reported. The brush dynamics is explained using a model based on an idea of de Gennes describing the brush properties in terms of scaling relations for osmotic pressure and viscosity of a semi-dilute solution with inhomogeneous density. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

A comprehensive,time-resolved SANS investigation of temperature-change-induced sponge-to-lamellar and lamellar-to-sponge phase transformations in comparison with <Superscript>2</Superscript>H -NMR results

Time-resolved small-angle neutron scattering (TR-SANS) was employed to observe temperature-induced phase transitions from the sponge (L ₃ to the lamellar ( L _α phase, and vice versa, in the water-oil (n -decane)-non-ionic surfactant ( C₁₂E₅ system using both bulk and film contrast. Samples of different bilayer volume fractions φ and solvent viscosities η were investigated applying various amplitudes of temperature jump ΔT . The findings of a previous ²H -NMR study could be confirmed, where the lamellar phase formation was determined to occur through a nucleation and growth process, while it was concluded that the L ₃ -phase develops in a mechanistically different and more rapid manner involving uncorrelated passage formation. Likewise, the kinetic trends of the nucleation and growth transition (decreased transition time with increase of φ and ΔT were witnessed once again. Additionally, NMR and SANS data that demonstrate a strong dependency of that process on solvent viscosity η are presented. Contrariwise, it is made evident via both SANS and NMR results that the L _α -to-L ₃ transition time is independent (within experimental sensitivity) of the varied parameters (φ , ΔT , η . Unusual scattering evolution in one experiment, originating from a highly ordered lamellar phase, intriguingly hints that a major rate determining factor is the disruption of long-range order. Furthermore, the bulk contrast investigations give insight into structure peak shifts/development during the transitions, while the film contrast experiments prove the bilayer thickness to be constant throughout the phase transitions and show that there is no evidence for a change in the short-range order of the bilayer structure. The latter was considered possible, due to the different topology of the L ₃ and L _α phases. Lastly, an unexpected yet consistent appearance of anisotropic scattering is detected in the L ₃ -to- L _α transitions.     

Structure and properties of some glass-forming liquid alloys

Some physical properties (kinematic viscosity, surface tension and magnetic susceptibility) of some Fe-based metallic melts of easily glass-forming alloys have been measured during heating and subsequent cooling. The results indicate that molten liquid metallic alloys undergo a number of structural transformations ranging from the initial microheterogeneous state formed after melting up to the true solution state. Studies by small angle neutron scattering on a eutectic SnPb melt confirm this conclusion in that two families of different sizes have been seen, one in the range 1 to 2 nm and one of size larger than 100 nm. Both kind of particles have relatively sharp interfaces and the size of the smaller particles is found to depend on temperature. Received 25 February 1999 and Received in final form 28 October 1999     

Structural characterization of Rh/pumice SMAD catalysts

The structure of Rh/pumice catalysts prepared by the SMAD (Solvated Metal Atoms Dispersion) technique at different metal loadings has been investigated by EXAFS (Extended X-ray Absorption Fine Structure Spectroscopy), XPS (X-ray Photoelectron Spectroscopy), SAXS (Small-Angle X-ray Scattering), WAXS (Wide-Angle X-ray Scattering) and TEM (Transmission Electron Microscopy). According to EXAFS and XPS, a fraction of the Rh atoms is oxidised, but a noticeable part is also present as Rh ⁰. The Rh oxidation is attributed to the interaction of the Rh atoms with the hydroxyls of the support; after the formation of the oxide, the nucleation of metallic rhodium becomes possible. The WAXS data do not show evidence of rhodium fcc crystallites; the metal-bearing particles are probably amorphous and/or very small, as results from the SAXS and TEM data analysis. The disagreement between the latter two techniques, resulting in a small-angle determination of the average size of the particles that is about half that of TEM in the catalyst with the higher Rh loading, is acknowledged and discussed. Preliminary catalytic tests are described, demonstrating the suitability of using a low surface area support for the preparation of SMAD catalysts. Received 2 February 1999     

Self-organized criticality on quasiperiodic graphs

It is shown that quasielastic scattering of ultracold neutrons due to diffusive motion of scatterers at the surface of liquid polymer (fomblin) or adsorbed hydrogenous contaminations of the surface of neutron traps may be possible reason of their energy spreading during long storage time in closed traps, which was observed in the recent experiments. Received 8 July 1998     

Shrinking of anionic polyacrylate coils induced by Ca2+, Sr2+ and Ba2+: A combined light scattering and ASAXS study

Anionic polyacrylate chains (NaPA) form precipitates if alkaline earth cations are added in stoichiometric amounts. Accordingly, precipitation thresholds were established for three different alkaline earth cations Ca²⁺, Sr²⁺ and Ba²⁺. Close to the precipitation threshold, the NaPA chains significantly decrease in size. This shrinking process was followed by means of combined static and dynamic light scattering. Intermediates were generated by varying the ratio [MCl₂]/[NaPA] with M denoting the respective alkaline earth cation. All experiments were performed at an inert salt level of 0.01M NaCl. Similar coil-to-sphere transitions could be observed with all three alkaline earth cations Ca²⁺, Sr²⁺ and Ba²⁺. Based on these findings, supplementary conventional and anomalous small-angle X-ray scattering experiments using selected intermediates close to the precipitation threshold of SrPA were performed. The distribution of Sr counterions around the polyacrylate chains in aqueous solution provided the desired scattering contrast. Energy-dependent scattering experiments enabled successful separation of the pure-resonant terms, which solely stem from the counterions. The Sr²⁺ scattering roughly reflects the monomer distribution of the polyacrylate chains. Different ratios of the concentrations of [ SrCl₂]/[NaPA] revealed dramatic changes in the scattering curves. The scattering curve at the lowest ratio indicated an almost coil-like behaviour, while at the higher ratios the scattering curves supported the model of highly contracted polymer chains. Most of X-ray scattering experiments on intermediate states revealed compact structural elements which were significantly smaller than the respective overall size of the NaPA particles.     

Neutron scattering from equilibrium-swollen networks

Small-angle neutron scattering measurements were performed on end-linked poly (dimethylsiloxane) (PDMS) networks swollen to equilibrium with d-benzene. Comparison was made with equivalent concentration PDMS solutions. Equilibrium-swollen networks consistently displayed a linear scattering regime at low q followed by a good-solvent-like scaling regime at high q in agreement with the predictions of the Gel Tensile Blob (GTB) model. Data are fit using the unified function modified for the GTB model (3-parameter fit). Equilibrium-swollen networks display a base structural size, the gel tensile-blob size, ξ, that was found to be independent of the molecular weight between crosslinks for the series of molecular weights studied, consistent with the predictions of the model. The length of the extended tensile structure, L, can be larger than the length of the fully extended network strand. The predicted scaling relationship for L, L ∼ Q^1/2N_avg, where N_avg = (1/fN_c²+1/4N_e² , Q is the equilibrium swelling ratio, N_c is the molecular weight between crosslinks, N_e is the entanglement molecular weight and f is the crosslink functionality is in agreement with experimental results for the networks studied.     

Morphology of Polystyrene Core–dPoly(ethylene oxide) Shell Arborescent Copolymer Micelles from Small-Angle Neutron Scattering Analysis

Arborescent (dendrigraft) copolymers with a branched polystyrene (PS) core grafted at the chain termini with deuterated poly(ethylene oxide) segments (PS-dPEO) were characterized in benzene and acetone by small-angle neutron scattering measurements using the contrast matching technique. While copolymers incorporating a G1 (twice-grafted) PS core aggregated to some extent, the portion of the scattering curve corresponding to non-aggregated copolymer molecules could still be analyzed to determine the shape and segment radial density profile for core and shell. These were derived from the pair distance distribution function P(r) and the scattering length density contrast profile Δρ(r) = ρ(r) ? ρ(solvent), obtained by the indirect Fourier transformation and deconvolution methods. The profiles obtained for the G1 copolymer are consistent with a well-defined PS core–dPEO shell morphology, only observed previously for upper generation (G4) polymers with deuterated PS (dPS) chains grafted randomly on arborescent PS substrates. Detailed morphological analysis could not be carried out for an analogous G3 arborescent PS copolymer terminally grafted with dPEO segments due to extensive aggregation in both solvents.