Guinier Regime Research Articles

The dispersion of carbon nanotubes in composite materials studied by computer simulation of Small Angle Scattering

Although numerous efforts have been made to reinforce ceramic materials by adding a nanostructured phase like carbon nanotubes (CNT), the appearance of aggregates during the manufacturing processes continues to be a problem. Given the size of the CNT (nm–μm), techniques such as Small Angle Scattering (SAS) can be a useful tool to study the formation of the aggregates and to quantify the degree of homogenization of the nanophase in the matrix. In this work, systems with different concentrations of CNT have been simulated in different states of aggregation, starting from a perfectly homogeneous dispersion of individualized CNT, and progressively aggregated. A Guinier regime appears in the scattering signal in the range of low values of the modulus of the scattering vector, q, as the aggregation occurs. Two parameters from the intensity curve are proposed to quantify the quality of the dispersion of the nanophase in the ceramic matrix.

Light scattering study of highly absorptive, non-fractal, hematite aggregates

We present measurements of the scattered light intensity by aerosolized hematite aggregate particles. The measurements were made at a wavelength of 532 nm in the scattering angle range from 0.32° to 157°. Hematite has high values of the real and imaginary parts of the refractive index m = n + iκ = 3 + i0.5 at the studied wavelength. Scanning electron micrographs (SEM) indicated that the particles were aggregates whereas the optical microscope pictures showed that the aerosol had a bimodal distribution with effective mean diameters of roughly 1 and 10 µm. This is consistent with the light scattering results which displayed two Guinier regimes. The aggregates were composed of smaller grains with an approximate size of 200 nm. Ultra Small-Angle X-ray Scattering (USAXS) indicate that the aggregates were uniform and non-fractal. Mie calculations for a sphere equivalent to the aggregate size were compared to the experimentally observed results. The observed results showed an enhanced backscattering, whereas the Mie calculations did not due to the large imaginary part of the refractive index. Hematite aggregates were simulated by assuming they were composed of spherical monomers inside a spherical volume. Then the light scattering was calculated using the T-matrix method for these simulated aggregates. The calculated results show an enhanced backscattering. We present a dimensional analysis to estimate the extent of multiple scattering within the aggregate and find a correlation between the average number of scattering events within the aggregate and the enhancement in the backscattering.

Application of the scaling approach to particles having simple, fundamental shapes, in the Rayleigh–Debye–Gans diffraction limit

Presented is a scaling approach for understanding features, such as power laws and crossover points, of the light scattered in the m → 1, ρ = 2kRveq|m − 1| < 1, Rayleigh–Debye–Gans diffraction limit. The scaling approach is based on comparison of the length scale of the scattering, which is the inverse of the scattering wave vector, and the various length scales of the scattering entity. It will be shown that the scaling approach correctly predicts the exponents of the power law regions and the locations of the first and second Guinier regimes which define the boundaries of the power laws. Furthermore, the scaling approach yields a semi-quantitative prediction of the coefficients of the power laws. These Guinier boundaries and power law coefficients are described by a single parameter, the aspect ratio of the scattering object.

Soot with 1013 cm−3 high concentration and 25 Å radius of gyration as detected by small-angle X-ray scattering in a premixed ethylene-air flame at sooting threshold

SAXS experiments on a premixed ethylene-air flame burning at the so-called sooting threshold, i.e., a flame with Carbon-to-Oxygen molecular ratio between fuel C2H4 and air equal to [C]/[O] = 0.65 (ΦAir=25cm3/min, ΦC2H4=11.7cm3/min) is the object of this work. Measurements are performed at synchrotron SOLEIL / beamline SWING under the approved proposal #20130463. The height above burner considered for this case study is z=21mm (τres about 10ms). The background is made by considering an identical lean flame with [C]/[O]=0.19. The data processing approach consisting in subtracting to the signal S, the X-ray scattering from the air molecules surrounding the flame as background BCKG, is here revealed to be incorrect, owing to the fact that sometimes the S-BCKG evaluated as above is a negative quantity. This is caused by the much higher cool gas molecule number density at ambient conditions (300K) with respect to the lower gas molecule density inside the flame at hot temperatures (about 1800–2100K). Vice versa, by using a lean flame one obtains as BCKG the scattering contribute from the gas molecules alone in the absence of soot at about the identical premixed flame under test, which results in appreciating even the low signal from ultrafine incipient soot nanoparticles. The main results from the fit of SAXS data at larger q's are two: first, the demonstration of ultrafine nanoparticles, with gyration radius Rg=25±10Å, as inferred by the Guinier regime fit, and overall size D=45±10Å, as evaluated through Kratky plots; second, the fact that these entities are present in the premixed flame with a huge concentration, about N=1013cm−3. Such nanoparticles with D=45Å (Rg=25Å), do coexist contemporarily to isolated 20nm primaries and 160nm nanoparticle aggregates, which are observed with a number concentrations 6 and 8 order of magnitudes smaller with respect to the one of ultrafine nanoparticles.Our SAXS results do accord very well previous findings on size of premixed flame soot nanoparticles reported both by Wersborg et al. (1973) from molecular beam sampling and TEM analysis and, with concern to the number concentration, the results by Minutolo et al. (1998) and Sgro et al. (2009).

Q-space analysis of light scattering by ice crystals

Q-space analysis is applied to extensive simulations of the single-scattering properties of ice crystals with various habits/shapes over a range of sizes. The analysis uncovers features common to all the shapes: a forward scattering regime with intensity quantitatively related to the Rayleigh scattering by the particle and the internal coupling parameter, followed by a Guinier regime dependent upon the particle size, a complex power law regime with incipient two dimensional diffraction effects, and, in some cases, an enhanced backscattering regime. The effects of significant absorption on the scattering profile are also studied. The overall features found for the ice crystals are similar to features in scattering from same sized spheres.

A light-scattering study of Al2O3 abrasives of various grit sizes

We report light scattering phase function measurements for irregularly shaped Al2O3 abrasive powders of various grit sizes. Q-space analysis is applied to the angular scattering to reveal a forward scattering regime, Guinier regime, power law regime with quantifiable exponents, and an enhanced backscattering regime. The exponents of the power laws for Al2O3 abrasives decrease with increasing internal coupling parameter ρ′, which is in agreement with previous observations for other irregular particles. Unlike other dust particles previously studied showing single power laws under Q-space analysis, the largest three abrasives, for which ρ′≳100, showed a kink in the power law, which is possibly due to the higher degree of symmetry for the abrasives than for all the particles studied previously. Direct comparison of the 1200, 1000, and 800 grit abrasive scattering to scattering by corresponding spheres shows that the scatterings approximately coincide at the spherical particle qR≃ρ′ crossover point. Furthermore, the scattering at the maximum qR=2kR by the irregularly shaped abrasives is close to the geometric centers of the glories of the spheres.

Light scattering Q‐space analysis of irregularly shaped particles

AbstractWe report Q‐space analysis of light scattering phase function data for irregularly shaped dust particles and of theoretical model output to describe them. This analysis involves plotting the scattered intensity versus the magnitude of the scattering wave vector q = (4π/λ)sin(θ/2), where λ is the optical wavelength and θ is the scattering angle, on a double‐logarithmic plot. In q‐space all the particle shapes studied display a scattering pattern which includes a q‐independent forward scattering regime; a crossover, Guinier regime when q is near the inverse size; a power law regime; and an enhanced backscattering regime. Power law exponents show a quasi‐universal functionality with the internal coupling parameter ρ′. The absolute value of the exponents start from 4 when ρ′ &lt; 1, the diffraction limit, and decreases as ρ′ increases until a constant 1.75 ± 0.25 when ρ′ ≳ 10. The diffraction limit exponent implies that despite their irregular structures, all the particles studied have mass and surface scaling dimensions of Dm = 3 and Ds = 2, respectively. This is different from fractal aggregates that have a power law equal to the fractal dimension Df because Df = Dm = Ds &lt; 3. Spheres have Dm = 3 and Ds = 2 but do not show a single power law nor the same functionality with ρ′. The results presented here imply that Q‐space analysis can differentiate between spheres and these two types of irregularly shaped particles. Furthermore, they are applicable to analysis of the contribution of aerosol radiative forcing to climate change and of aerosol remote sensing data.

Small angle neutron scattering

Small Angle Neutron Scattering (SANS) is a technique that enables to probe the 3-D structure of materials on a typical size range lying from ∼ 1 nm up to ∼ a few 100 nm, the obtained information being statistically averaged on a sample whose volume is ∼ 1 cm3. This very rich technique enables to make a full structural characterization of a given object of nanometric dimensions (radius of gyration, shape, volume or mass, fractal dimension, specific area…) through the determination of the form factor as well as the determination of the way objects are organized within in a continuous media, and therefore to describe interactions between them, through the determination of the structure factor. The specific properties of neutrons (possibility of tuning the scattering intensity by using the isotopic substitution, sensitivity to magnetism, negligible absorption, low energy of the incident neutrons) make it particularly interesting in the fields of soft matter, biophysics, magnetic materials and metallurgy. In particular, the contrast variation methods allow to extract some informations that cannot be obtained by any other experimental techniques. This course is divided in two parts. The first one is devoted to the description of the principle of SANS: basics (formalism, coherent scattering/incoherent scattering, notion of elementary scatterer), form factor analysis (I(q→0), Guinier regime, intermediate regime, Porod regime, polydisperse system), structure factor analysis (2nd Virial coefficient, integral equations, characterization of aggregates), and contrast variation methods (how to create contrast in an homogeneous system, matching in ternary systems, extrapolation to zero concentration, Zero Averaged Contrast). It is illustrated by some representative examples. The second one describes the experimental aspects of SANS to guide user in its future experiments: description of SANS spectrometer, resolution of the spectrometer, optimization of spectrometer configurations, optimization of sample characteristics prior to measurements (thickness, volume, hydrogen content…), standards measurements to be made and principle of data reduction.

Upgrade of MacCHESS facility for X-ray scattering of biological macromolecules in solution.

X-ray scattering of biological macromolecules in solution is an increasingly popular tool for structural biology and benefits greatly from modern high-brightness synchrotron sources. The upgraded MacCHESS BioSAXS station is now located at the 49-pole wiggler beamline G1. The 20-fold improved flux over the previous beamline F2 provides higher sample throughput and autonomous X-ray scattering data collection using a unique SAXS/WAXS dual detectors configuration. This setup achieves a combined q-range from 0.007 to 0.7 Å(-1), enabling better characterization of smaller molecules, while opening opportunities for emerging wide-angle scattering methods. In addition, a facility upgrade of the positron storage ring to continuous top-up mode has improved beam stability and eliminated beam drift over the course of typical BioSAXS experiments. Single exposure times have been reduced to 2 s for 3.560 mg ml(-1) lysozyme with an average quality factor I/σ of 20 in the Guinier region. A novel disposable plastic sample cell design that incorporates lower background X-ray window material provides users with a more pristine sample environment than previously available. Systematic comparisons of common X-ray window materials bonded to the cell have also been extended to the wide-angle regime, offering new insight into best choices for various q-space ranges. In addition, a quantitative assessment of signal-to-noise levels has been performed on the station to allow users to estimate necessary exposure times for obtaining usable signals in the Guinier regime. Users also have access to a new BioSAXS sample preparation laboratory which houses essential wet-chemistry equipment and biophysical instrumentation. User experiments at the upgraded BioSAXS station have been on-going since commissioning of the beamline in Summer 2013. A planned upgrade of the G1 insertion device to an undulator for the Winter 2014 cycle is expected to further improve flux by an order of magnitude.

Internal structures of agar-gelatin co-hydrogels by light scattering, small-angle neutron scattering and rheology

Internal structures of agar-gelatin co-hydrogels were investigated as a function of their volumetric mixing ratio, [Formula: see text] , 1.0 and 2.0 using dynamic light scattering (DLS), small-angle neutron scattering (SANS) and rheology. The degree of non-ergodicity ( X = 0.2 ± 0.02) , which was extracted as a heterodyne contribution from the measured dynamic structure factor data remained less than that of homogeneous solutions where ergodicity is expected (X = 10. The static structure factor, I(q) , results obtained from SANS were interpreted in the Guinier regime (low-q , which implied the existence of ≈ 250 nm long rod-like structures (double-helix bundles), and the power law (intermediate-q regions) yielded I (q) ~ q(−α) with α = 2.3 , 1.8 and 1.6 for r = 0.5 , 1.0 and 2.0. This is indicative of the presence of Gaussian chains at low r , while at r = 2 there was a propensity of rod-shaped structures. The gel strength and transition temperatures measured from frequency sweep and temperature ramp studies were suggestive of the presence of a stronger association between the two biopolymer networks at higher r . The results indicate that the internal structures of agar-gelatin co-hydrogels were highly dependent on the volumetric mixing ratio.

Structural and surface plasmon behavior of Cu nanoparticles using different stabilizers

Copper nanoparticles (CuNPs) are prepared in aqueous medium with hydrazine hydrate as the reductant. Different stabilizers PVA, PVP and starch are considered. For each stabilizing agent, the effect of concentration of NaOH on the synthesis of CuNPs is investigated. The structural characterization of CuNPs dispersed in aqueous medium has been done with the help of small angle X-ray scattering (SAXS). The SAXS studies demonstrate the formation of spherical NPs with bimodal size distribution. The primary (smaller) particles/clusters aggregate to form the larger nanoparticles. The scattered intensity after Guinier regime displays fractal behavior. An attempt has been made to understand the effect of stabilizer and NaOH concentration on the formation of such fractal aggregates. Correlations have been drawn between the particles size distribution in the sol and the localized surface plasmon resonance (LSPR) behavior of such CuNPs in aqueous solution. Dominance of smaller nanoparticles (∼5 nm or less) in the sol leads to suppression of the LSPR.

Clathrin Triskelia Show Evidence of Molecular Flexibility

The clathrin triskelion, which is a three-legged pinwheel-shaped heteropolymer, is a major component in the protein coats of certain post-Golgi and endocytic vesicles. At low pH, or at physiological pH in the presence of assembly proteins, triskelia will self-assemble to form a closed clathrin cage, or “basket”. Recent static light scattering and dynamic light scattering studies of triskelia in solution showed that an individual triskelion has an intrinsic pucker similar to, but differing from, that inferred from a high resolution cryoEM structure of a triskelion in a clathrin basket. We extend the earlier solution studies by performing small-angle neutron scattering (SANS) experiments on isolated triskelia, allowing us to examine a higher q range than that probed by static light scattering. Results of the SANS measurements are consistent with the light scattering measurements, but show a shoulder in the scattering function at intermediate q values (0.016 Å −1), just beyond the Guinier regime. This feature can be accounted for by Brownian dynamics simulations based on flexible bead-spring models of a triskelion, which generate time-averaged scattering functions. Calculated scattering profiles are in good agreement with the experimental SANS profiles when the persistence length of the assumed semiflexible triskelion is close to that previously estimated from the analysis of electron micrographs.

Universal Growth of Microdomains and Gelation Transition in Agar Hydrogels

Investigations were carried out on aqueous sols and gels of agar (extracted from red seaweed Gelidiella acerosa) to explore the growth of microdomains en route to gelation. Isothermal frequency sweep studies on gel samples revealed master plots showing power-law dependence of gel elastic modulus, |G*|, on oscillation frequency, omega as |G*| approximately omegan, independent of temperature, with 0.5<or=n<or=1.4. Dynamic structure factor data from sol samples comprised of a double-exponential relaxation function, S(q,t)=A exp(-DSq2t)+B exp(-DLq2t) where DS and DL are the two translational diffusion coefficients and q is the scattering wave vector. This yielded hydrodynamic radii (from DS), with RS varying from approximately 20 nm (for sol) to 250 nm (at gelation point). The second hydrodynamic radius (from DL) obtained was RL in the range of approximately 200-500 nm (for sol) to approximately 1000 nm (at 38 degrees C, gelation point). These data could be universally fitted to RS approximately epsilon(-3/5) and RL approximately epsilon-1/3 (epsilon=(T/Tg-1), T>Tg). The S(q,t) behavior close to the gel transition point (Tg approximately (38+/-3 degrees C determined from rheology) followed a stretched exponential function: S(t)=A exp(-t/ts)beta. The beta factor increased from 0.25 to 1 as the gel temperature approached 25 degrees C from Tg, and relaxation time, ts, showed a peak at T approximately 30 degrees C. The SLS data (in the sol state) suggested the scaling of scattered intensity, Is(q) approximately epsilon(-gamma) (epsilon=(T/Tg-1), T>Tg) with gamma=0.13+/-0.03, and the presence of two distinct domains characterized by a Guinier regime (low q) and a power-law regime (high q). Close to and above Tg (+2 degrees C), IS(q) scaled with q as Is(q) approximately q(-alpha) with alpha=2.2+/-0.2, which decreased to 1.4+/-1 just below Tg (-2 degrees C), implying a coil-helix transition for 0.2% (w/v) and 0.3% (w/v) samples. For a 0.01% sample, alpha=3.5+/-0.5 which indicated the presence of spherical microgels.

Thermotropic Ionic Liquid Crystals via Self-Assembly of Cationic Hyperbranched Polypeptides and Anionic Surfactants

This work describes the dilute solution and solid-state structure of polyelectrolyte complexes generated from hyperbranched polylysine (HBPL) and various anionic, sodium alkyl sulfate surfactants. In dilute solution, the radius of gyration (Rg) of the HBPL and HBPL-surfactant complexes was determined in the Guinier regime in good solvent conditions by means of small-angle X-ray scattering (SAXS). With increasing molecular weight of the HBPL, an increase in Rg up to a maximum of 3.7 and 4.2 nm was observed for the HBPLs and HBPL-surfactant complexes, respectively. In the solid state, HBPL-surfactant complexes were found to form liquid crystalline (LC) phases, whose thermal stability and structure depended both on the molecular weight of the HBPL as well as on the nature of the anionic surfactant. Depending on the surfactant alkyl chain length, liquid crystalline phases with short range liquid-like order, columnar hexagonal packing or lamellar ordering were observed. By combination of small-angle X-ray scattering, differential scanning calorimetry (DSC), and cross-polarized light optical microscopy (CPOM), the exact structure of the LC phases, as well as their region of thermal stability, could be identified. HBPL-sodium dodecyl sulfate LC phases showed thermotropic behavior and underwent two transitions with increasing temperature. First, at lower temperatures, an order-nematic transition was observed. Upon further temperature increase, a second transition from a nematic to an isotropic phase was observed. Structural models for these different LC phases are proposed. © 2007 American Chemical Society.

Multiple-length-scale small-angle X-ray scattering analysis on maghemite nanocomposites

Small-angle X-ray scattering (SAXS) analysis has been performed on maghemite–poly(4-vinylpyridine) nanocomposites prepared by in situ precipitation from iron–polymer coordination compounds. According to electron microscopy observations, the nanocomposites contain isolated spherical particles with a narrow size distribution, uniformly distributed throughout the polymer matrix. The scattering intensity of nanocomposites has relevant contributions from both the polymer and the nanocomposites, showing features characteristic of multiscale structured systems, namely two power laws and a Guinier regime. The data have been analysed in terms of Beaucage's unified approach and it is found that the maghemite particle size increases with the iron/polymer weight ratio used in the preparation of the nanocomposites. SAXS curves also feature a bump that was analysed as arising from a second particle population or from interactions. Magnetization and transmission electron microscopy results give arguments favouring the latter interpretation. It is found that the maghemite particle sizes vary linearly with the iron weight ratio used in the preparation of the nanocomposites.

Conformational modification of conducting polymer chains by solvents: Small-angle X-ray scattering study

The present small-angle X-ray scattering (SAXS) investigations reveal solvent-assisted conformational modifications in the conducting polymer polyethylene dioxythiophene (PEDOT) doped with polystyrene sulfonate (PSS). The dimensionality of PEDOT-PSS chains in water, glycerol and dimethyl sulfoxide (DMSO) solutions, obtained from SAXS profiles, are 2.68–2.3. Radius of gyration values from pair distribution functions analysis are used to identify the Guinier regime, and give evidence for chain expansion in glycerol and DMSO solutions, with improved assembly of chains. Conformational modifications are suggested to enhance the conductivity of PEDOT-PSS films in glycerol/DMSO solutions.

Structure development in aerogel-processed nanocrystalline alkaline earth oxides as revealed by SANS

Nanocrystalline MgO, CaO and SrO were prepared according to a modified aerogel process (AP). Small-angle neutron scattering (SANS) was used to probe the nanoscale structural features of these materials after each stage of the synthetic process, including hydrolysis, supercritical drying and calcining. SANS data were interpreted using a classical analysis involving power-law and Guinier regimes, and by application of the maximum entropy method. Results are compared with previously published structural data based on X-ray diffraction, electron microscopy and gas adsorption. It is found that the gel hydrolysis product suspended in methanol and toluene exhibits rod-like scattering at small length scales. This is consistent with a filiform morphology previously reported for air-dried Mg(OH)2alcogel, yet SANS data for air-dried alcogels tested in this study indicate no evidence for low-dimensional structure on any length scale. A previous assertion of mass fractal structure in the AP aerogels and oxides was not confirmed by the present data. Instead, surface fractal scattering was found to be the most dominant characteristic feature associated with the SANS data for all AP powders examined. Additionally, MgO and CaO exhibited a correlation peak that corresponds to liquid-like ordering at Bragg length scales of 5.9 nm and 20.3 nm, respectively. These values are roughly consistent with previous independent estimates of primary particle size, suggesting that local packing of primary crystallites is facilitated by the calcination/dehydration process. An alternative interpretation treats these features as Guinier scattering regions. Fitting of results using the unified Guinier/power-law equation yields sphere-equivalent radii for the primary particles that are nearly identical to the Bragg lengths calculated from the positions of the maxima. Air-dried alcogels produced very weak maxima that could be interpreted either as correlation peaks or as Guinier regions. No maxima were observed for aerogel samples. Maximum entropy analysis using a spherical shape factor produced interesting but complex results for the calculated volume size distributions of these materials. Overall, the observed trend shows an increase in structural feature size with increasing metal cation size.

Structure and properties of poly(vinyl alcohol)/tungsten trioxide hybrids

PVA/WO 3 hybrids were produced by using ion exchange process of Na 2WO 4·2H 2O. Microstructures of this hybrid was estimated by SAXS measurement as follows; the WO 3 rich domains having radius of gyration of about 5.7 nm (estimated in Guinier regime), in which PVA chains were partly incorporated, were dispersed evenly in PVA matrix with the correlation length of 20–30 nm (approximated by Bragg equation). The electron density difference between WO 3 and matrix, Δ ρ, was rather high and sharp, and this indicates that the domains dispersed in PVA matrix is rich in WO 3 component. The physical properties of the hybrids were affected by the morphology. The mechanical properties of the hybrid were improved markedly, e.g. the tensile strength increased from 35 MPa (PVA) to 55 MPa at 27 wt% of WO 3 content, and the modulus also increased above T g with increasing amount of WO 3. A tan δ peak appeared at 75 °C shifted to the higher temperature side with WO 3 content. These were caused by strong interaction between PVA chains and WO 3 domains, which was formed by incorporation of PVA chain into WO 3 domains having hydrogen bonding. This hybrid was transparent and showed the photochromism especially at low concentration of WO 3. When UV–visible light of wavelength above 350 nm was irradiated on the hybrid, a minimum occurred at 630 nm in the UV–visible spectrum, and color changed from yellow to blue. Bleaching occurred in the dark in 100% relative humidity. Consequently, the ion exchange process was suitable to obtain PVA/WO 3 hybrid, since this procedure was simple and economical, and hybrid thus produced had excellent mechanical and photochromism properties.

Small-angle X-ray scattering study of the smart thermo-optical behavior of zirconyl aqueous colloids

The smart thermo-optical systems studied here are based on the unusual thermoreversible sol-gel transition of zirconyl chloride aqueous solution modified by sulfuric acid in the molar ratio Zr/SO4:3/1. The transparency to the visible light changes during heating due to light scattering. This feature is related to the aggregates growth that occurs during gelation. These reversible changes can be controlled by the amount of chloride ions in solution. The thermoreversible sol-gel transition temperature increases from 323 to 343 K by decreasing the molar ratio Cl/Zr from 7.0 to 1.3. In this work the effect of the concentration of chloride ions on the structural characteristics of the system has been analyzed by in situ SAXS measurements during the sol-gel transition carried out at 323 and 333 K. The experimental SAXS curves of sols exhibit three regions at small, medium and high scattering vectors characteristics of Guinier, fractal and Porod regimes, respectively. The radius of primary particles, obtained from the crossover between the fractal and Porod regimes, remains almost invariable with the chloride concentration, and the value (4 Å) is consistent with the size of the molecular precursor. During the sol-gel transition the aggregates grow with a fractal structure and the fractal dimensionality decreases from 2.4 to 1.8. This last value is characteristic of a cluster-cluster aggregation controlled by a diffusion process. Furthermore, the time exponent of aggregate growth presents values of 0.33 and 1, typical of diffusional and hydrodynamic motions. A crossover between these two regimes is observed.

SCATTERED INTENSITY FROM HIGHER CONNECTED POLYMERS IN SOLUTION

The aim of this work is to compute the structure factor of any arbitrary D-dimensionally-connected polymers (1≤D&lt;2) in dilute solution. We use the standard cut-off function method, which is successfully applied to liquid systems and to fractal aggregates. For monodisperse systems, we find that the corresponding structure factor is simply given by the Gauss hypergeometric function 2F1, the three parameters of which depend explicitly on the fractal and the Euclidean dimensions. This function reproduces the two limiting behaviors, in the Guinier and intermediate regimes. This result is applied to several systems, namely, compact and convex polymeric objects, rod-like, linear and branched polymers. We then extend this result to polydisperse branched polymers. We show that polydispersity induces a change in the structure factor, from the simple hypergeometric function 2F1 to the generalized one 2F1, with four parameters that also depend on both fractal and Euclidean dimensions.