Small-angle scattering from fat fractals

A number of experimental small-angle scattering (SAS) data are characterized by a succession of power-law decays with arbitrarily decreasing values of scattering exponents. To describe such data, here we develop a new theoretical model based on 3D fat fractals (sets with fractal structure, but nonzero volume) and show how one can extract structural information about the underlying fractal structure. We calculate analytically the monodisperse and polydisperse SAS intensity (fractal form factor and structure factor) of a newly introduced model of fat fractals and study its properties in momentum space. The system is a 3D deterministic mass fractal built on an extension of the well-known Cantor fractal. The model allows us to explain a succession of power-law decays and respectively, of generalized power-law decays (GPLD; superposition of maxima and minima on a power-law decay) with arbitrarily decreasing scattering exponents in the range from zero to three. We show that within the model, the present analysis allows us to obtain the edges of all the fractal regions in the momentum space, the number of fractal iteration and the fractal dimensions and scaling factors at each structural level in the fractal. We applied our model to calculate an analytical expression for the radius of gyration of the fractal. The obtained quantities characterizing the fat fractal are correlated to variation of scaling factor with the iteration number.     

Small-angle photon scattering on complex nuclei

Photons of 3 GeV and 5 GeV were scattered on 7 different elements, ranging from Be to Au, and detected with a pair spectrometer. The angular distributions show diffractive patterns consistent with known nuclear sizes. Forward cross sections are 20–30% lower than expected from an A² dependence. This shadowing effect is qualitatively explained by photon interactions via intermediate hadronic states.     

Small-angle neutron scattering from micellar solutions

Micellar solutions are the suspension of the colloidal aggregates of the sur-factant molecules in aqueous solutions. The structure (shape and size) and the interaction of these aggregates, referred to as micelles, depend on the molecular architecture of the surfactant molecule, presence of additives and the solution conditions such as temperature, concentration etc. This paper gives the usefulness of small-angle neutron scattering to the study of micellar solutions with some of our recent results.     

Small-angle X-Ray scattering from polyurethane elastomers

Small-angle X-ray scattering is shown to be an effective technique, complementary to mechanical and thermal methods, for the study of structural ordering in polyurethane elastomers. Wide variations in intensity were found for different sample compositions and are interpreted in terms of the degree of hard segment aggregation. The results support previous conclusions regarding the organization in these samples based on analysis by scanning thermal methods. The polyether polyurethanes studied showed a higher degree of organization than the equivalent polyesters, and longer urethane segments enhance the extent of domain structure. Studies of the temperature dependence of scattering strengthen the previous assignment of a transition at about 150°C to the disordering of domain structure.     

Small-angle neutron scattering at fractal objects

The calculation of the correlation function of an isotropic fractal particle with the finite size ξ and the dimension D is presented. It is shown that the correlation function γ(r) of volume and surface fractals is described by a generalized expression and is proportional to the Macdonald function (D–3)/2 of the second order multiplied by the power function r ^(D–3)/2. For volume and surface fractals, the asymptotics of the correlation function at the limit r/ξ < 1 coincides with the corresponding correlation functions of unlimited fractals. The one-dimensional correlation function G(z), which, for an isotropic fractal particle, is described by an analogous expression with a shift of the index of the Macdonald function and the exponent of the power function by 1/2, is measured using spin-echo small-angle neutron scattering. The boundary case of the transition from a volume to a surface fractal corresponding to the cubic dependence of the neutron scattering cross section Q ^?3 leads to an exact analytical expression for the one-dimensional correlation function G(z) = exp(?z/ξ), and the asymptotics of the correlation function in the range of fractal behavior for r/ξ < 1 is proportional to ln(ξ/r). This corresponds to a special type of self-similarity with the additive law of scaling rather than the multiplicative one, as in the case of a volume fractal.     

Small-angle elastic scattering of 14.8 mev neutrons

The 14.8 MeV neutron elastic scattering differential cross sections of U, Pb, Hg, Ta, Sn and Al have been measured for angles extending from 10° down to 1.8° and for U and Pb down to 0.35°. Nuclear cross sections extrapolated to 0° have been compared with the Wick limit. The U nuclear cross sections did not rise as sharply toward 0°, and were not in as serious disagreement with the spherical optical model, as in earlier measurements. The smallest-angle data verified the Schwinger prediction.     

Small-angle elastic scattering of 2.5 MeV neutrons

Differential cross sections and asymmetries of 2.50 MeV polarized neutrons elastically scattered by In, Ho, Hg, Bi and U have been determined for scattering angles from 2.1° to 9.1°. The measurements were carried out by the use of a neutron positional spectrometer. The results, when compared with the predictions of the electromagnetic interaction, indicate some deviation of 〈σ(θ)P(θ)〉 from the expected values. The observed cross sections are systematically greater than those evaluated within the framework of the optical model.     

Small-angle X-ray scattering from styrene-butadiene-styrene block copolymers

Abstract

A new type of time-dependent and strain-history-dependent viscoelasticity was discovered in semidilute polymer solutions. Dynamic moduli G′ and G″ of 20% and 10% nitrile butadiene rubber (NBR) solutions were recorded as a function of time while oscillatory shear deformations were maintained. The moduli decrease with time was observed only at lower frequencies. The time dependence of G′ was more pronounced than that of G″. At a higher temperature, the time dependence was extended toward higher frequencies also, and the time dependence became stronger. Lowering the concentration of solution gave a similar effect as increasing temperature. After the cessation of oscillations, a slow recovery was observed. The recovery was somewhat faster at the higher temperature. The time-dependent moduli and their recovery were explained by the change and recovery of structures associated with long branches and gels in the NBR. The structure change occurred at higher frequencies also, but it was not observed during the application of oscillation. Only in subsequent measurements at lower frequencies could the structure change be detected. Thus, the change may be regarded as strain history dependent. The mechanism of the structural change was explained with either the entanglement or osmotic pressure models, depending on concentration.     

Small-angle neutron spin-flip scattering in ferromagnetic films

This paper reports on the results of reflectometric measurements of anisotropic (Co₆₇Fe₃₁V₂) and almost isotropic (Fe) films prepared by magnetron sputtering. Nonspecular reflections and the corresponding peaks of the intensities of refracted neutrons have been observed for the alloy samples in magnetic fields H ≤ 7 Oe applied in the film plane along the easy magnetization axis. For iron films, angular splitting of the reflected neutron beam becomes observable only at H > 100 Oe and increases with an increase in the magnetic field. A general scheme has been proposed for this small-angle scattering with allowance made for different variants of changes in the Zeeman energy of neutrons. This scheme has allowed us to identify the magnetic structures of Co-Fe films. The magnetization of 0.15-μm-thick films with uniaxial and unidirectional textures leads to the formation of unidirectional textures characterized by different intensity distributions, for which the qualitative differences are retained with an increase in the magnetic field from 7 to 800 Oe. It has been revealed that, for 2.5-μm-thick films with the initial unidirectional texture, the oppositely magnetized states are nonequivalent.     

Small-angle light scattering from polymer-dispersed liquid-crystal films

A method is developed for modeling and computing the angular distribution of light scattered forward from a single-layer polymer-dispersed liquid-crystal (PDLC) film. The method is based on effective-medium approximation, anomalous diffraction approximation, and far-field single-scattering approximation. The angular distribution of forward-scattered light is analyzed for PDLC films with droplet size larger than the optical wavelength. The method can be used to study field-and temperature-induced phase transitions in LC droplets with cylindrical symmetry by measuring polarized scattered light intensity.     

Small-angle scattering of X-rays by electron-irradiated copper

Small-angle scattering of annealed electron-irradiated copper shows that isolated monovacancies persist through recovery stage III. The scattering appearing at small angles after annealing between 260 K and 300 K can be accounted for by the condensation of self-interstitials to form small dislocation loops.     

Small-angle neutron scattering study of dynamically polarized polyethylenes

We carried out a small-angle neutron scattering (SANS) study of dynamically polarized polyethylene (PE) samples doped with 2,2,6,6-tetramethylpiperidine 1-oxyl (TEMPO). The transmission of the PE with almost fully polarized neutrons (98.5%) increased with increasing the proton polarization, P. The incoherent scattering cross section decreased with increasing P. The effect of P on the polarized neutrons’ transmission and the incoherent scattering cross section agreed well with the theory. The q-dependence of the coherent scattering, which reflects a two-phase structure of PE composed of crystalline and amorphous domains, was kept unchanged by the proton polarization, but the intensity increased by a factor of 3 and 6 for P=+23% and −23%, respectively. The results mean that the contrast between the two phases was successfully enhanced by a dynamic nuclear polarization (DNP) technique. However, the enhancement is only 1/13–1/16 of the enhancement calculated by assuming a homogeneous polarization through the PE sample. The discrepancy suggests that P in amorphous domains (25%) should be higher than that in crystalline domains (22%) by 3%, which in turn may suggest the partial depolarization of proton spins on the way of the spin diffusion from amorphous domains, where TEMPO radicals localize, to crystalline domains.     

Small-angle light scattering studies of dense AOT-water-decane microemulsions

Summary We have performed extensive studies of a three-component microemulsion system composed of AOT-water-decane (AOT=sodium-bis-ethylhexyl-sulfosuccinate is an ionic surfactant) using small-angle light scattering (SALS). The small-angle scattering intensities are measured in the angular interval 0.001–0.1 radians, corresponding to a Bragg wave number range of 0.14 μm⁻¹<Q<<1.4 μm⁻¹. The measurements were made by changing temperature and volume fraction ϕ of the dispersed phase (water + AOT) in the range 0.05<ϕ<0.75. All samples have a fixed water-to-AOT molar ratio,w=[water]/[AOT]=40.8, in order to keep the same average droplet size in the stable one-phase region. With the SALS technique, we have been able to observe all the phase boundaries of a very complex phase diagram with a percolation line and many structural organizations within it. We observe at the percolation transition threshold, a scaling behavior of the intensity data. This behavior is a consequence of a clustering among microemulsion droplets near the percolation threshold. In addition, we describe in detail a structural transition from a droplet microemulsion to a bicontinuous one as suggested by a recent small-angle neutron scattering experiment. The loci of this transition are located several degrees above the percolation temperatures and are coincident with the maxima previously observed in shear viscosity. From the data analysis, we show that both the percolation phenomenon and this novel structural transition are derived from a large-scale aggregation between microemulsion droplets.     

Antishadow scattering and diffraction dissociation

Zeitschrift für Physik C Particles and Fields - We consider diffractive dissociation processes at high energies in the framework of chiral quark model with unitarization via generalized...     

Vacuum structure and diffraction scattering

A simple model of the vacuum is developed in order to explain the properties of the pomeron observed in experiments. In the model, pomeron exchange corresponds to two-gluon exchange. We explain how it can be that when the pomeron couples to a hadron, the two gluons couple predominantly to the same quark in the hadron, and how together their coupling is similar to that of aC=1 isoscalar photon.     

Small-angle neutron and dynamic light scattering study of gelatin coacervates

The state of intermolecular aggregates and that of folded gelatin molecules could be characterized by dynamic laser light and small-angle neutron scattering experiments, which implied spontaneous segregation of particle sizes preceding coacervation, which is a liquid-liquid phase transition phenomenon. Dynamic light scattering (DLS) data analysis revealed two particle sizes until precipitation was reached. The smaller particles having a diameter of ～50 nm (stable nanoparticles prepared by coacervation method) were detected in the supernatant, whereas the inter-molecular aggregates having a diameter of ～400 nm gave rise to coacervation. Small-angle neutron scattering (SANS) experiments revealed that typical mesh size of the networks exist in polymer dense phase (coacervates) [1]. Analysis of the SANS structure factor showed the presence of two length scales associated with this system that were identified as the correlation length or mesh size, ξ = 10.6 Å of the network and the other is the size of inhomogeneities = 21.4 Å. Observations were discussed based on the results obtained from SANS experiments performed in 5% (w/v) gelatin solution at 60°C (ξ = 50 Å, ζ = 113 Å) and 5% (w/v) gel at 28°C (ξ = 47 Å, ζ = 115 Å) in aqueous phase [2] indicating smaller length scales in coacervate as compared to sol and gel.     

Arrays of interacting ferromagnetic nanofilaments: Small-angle neutron diffraction study

Magnetic properties of spatially ordered arrays of interacting nanofilaments have been studied by means of small-angle diffraction of polarized neutrons. Several diffraction maxima or rings that correspond to the scattering of the highly ordered structure of pores/filaments with hexagonal packing have been observed in neutron scattering intensity maps. The interference (nuclear-magnetic) and pure magnetic contributions to the scattering have been analyzed during the magnetic reversal of the nanofilament array in a field applied perpendicular to the nanofilament axis. The average magnetization and the interference contribution proportional to it increase with the field and are saturated at H = H _S . The magnetic reversal process occurs almost without hysteresis. The intensity of the magnetic contribution has hysteresis behavior in the magnetic reversal process for both the positive and negative fields that form the field dependence of the intensity in a butterfly shape. It has been shown that this dependence is due to the magnetostatic interaction between the filaments in the field range of H ≤ H _S . A theory for describing the magnetic properties of the arrays of interacting ferromagnetic nanofilaments in the magnetic field has been proposed.     

Small-angle X-ray scattering from salt-free solutions of star-branched polyelectrolytes

The dispersion state of sodium-sulphonated polystyrene ( NaPSS) star-branched polyelectrolytes was investigated in salt-free aqueous solutions, by use of the small-angle X-ray scattering technique. With respect to polystyrene (PS) star-branched polymers of identical functionality, the ordering phenomenon occurring in the neighborhood of the overlap concentration c^* is reinforced and observed in a larger range of concentrations. Moreover, the degree of order is no longer maximum at c^* and is improved as the concentration decreases. The dispersion state is then mainly controlled by the electrostatic interaction. A crystalline order should therefore be achieved with stars of lower functionality, provided the electrostatic interaction is added to the osmotic repulsion. On the other hand, unusual scattering patterns are measured for aqueous solutions of NaPSS star polyelectrolytes. Indeed, a diffuse scattering is revealed at high angles, in addition to the regular diffraction rings related to preferred interstar distances. It is similar to the broad scattering peak produced by semidilute solutions of NaPSS linear polyelectrolytes and associated to the electrostatic correlation hole within the isotropic model. In the dilute regime (c < c ^*), it is just an intramolecular characteristic and represents the electrostatic repulsion between arms belonging to the same star. In the semidilute regime (c > c ^*), it also reflects the electrostatic repulsion between arms of distinct stars. So, as the concentration increases, it is mainly caused by the interpenetration of NaPSS stars. Such an observation is in agreement with the composite structure earlier proposed by Daoud and Cotton for star semidilute solutions. For c > c ^*, NaPSS star aqueous solutions can therefore be pictured as effective stars immersed in a matrix formed by the overlap of the arm ends. With respect to the dilute regime, the effective stars are smaller; the higher the concentration the smaller the size. Received 14 May 1999 and Received in final form 15 March 2000