Porod Region Research Articles

A new small-angle X-ray scattering model for polymer spherulites with a limited lateral size of the lamellar crystals.

As is well known, polymers commonly form lamellar crystals, and these assemble further into lamellar stacks and spherulites during quiescent crystallization. Fifty years ago, Vonk and Kortleve constructed the classical small-angle X-ray scattering theory (SAXS) for a lamellar system, in which it was assumed that the lamellar stack had an infinite lateral size [Vonk & Kortleve (1967 ▸), Kolloid Z. Z. Polym. 220, 19-24]. Under this assumption, only crystal planes satisfying the Bragg condition can form strong scattering, and the scattering from the lamellar stack arises from the difference between the scattering intensities in the amorphous and crystalline layers, induced by the incident X-ray beam. This assumption is now deemed unreasonable. In a real polymer spherulite, the lamellar crystal commonly has dimensions of only a few hundred nanometres. At such a limited lateral size, lamellar stacks in a broad orientation have similar scattering, so interference between these lamellar stacks must be considered. Scattering from lamellar stacks parallel to the incident X-ray beam also needs to be considered when total reflection occurs. In this study, various scattering contributions from lamellar stacks in a spherulite are determined. It is found that, for a limited lateral size, the scattering induced by the incident X-ray beam is not the main origin of SAXS. It forms double peaks, which are not observed in real scattering because of destructive interference between the lamellar stacks. The scattering induced by the evanescent wave is the main origin. It can form a similar interference pattern to that observed in a real SAXS measurement: a Guinier region in the small-q range, a signal region in the intermediate-q range and a Porod region in the high-q range. It is estimated that, to avoid destructive interference, the lateral size needs to be greater than 11 µm, which cannot be satisfied in a real lamellar system. Therefore, SAXS in a real polymer system arises largely from the scattering induced by the evanescent wave. Evidence for the existence of the evanescent wave was identified in the scattering of isotactic polypropyl-ene. This study corrects a long-term misunderstanding of SAXS in a polymer lamellar system.

Structural evolution of photocrosslinked silk fibroin and silk fibroin-based hybrid hydrogels: A small angle and ultra-small angle scattering investigation

Regenerated Bombyx mori silk fibroin (RSF) is a widely recognized protein for biomedical applications; however, its hierarchical gel structure is poorly understood. In this paper, the hierarchical structure of photocrosslinked RSF and RSF-based hybrid hydrogel systems: (i) RSF/Rec1-resilin and (ii) RSF/poly(N-vinylcaprolactam (PVCL) is reported for the first time using small-angle scattering (SAS) techniques. The structure of RSF in dilute to concentrated solution to fabricated hydrogels were characterized using small angle X-ray scattering (SAXS), small angle neutron scattering (SANS) and ultra-small angle neutron scattering (USANS) techniques. The RSF hydrogel exhibited three distinctive structural characteristics: (i) a Porod region in the length scale of 2 to 3nm due to hydrophobic domains (containing β-sheets) which exhibits sharp interfaces with the amorphous matrix of the hydrogel and the solvent, (ii) a Guinier region in the length scale of 4 to 20nm due to hydrophilic domains (containing turns and random coil), and (iii) a Porod-like region in the length scale of few micrometers due to water pores/channels exhibiting fractal-like characteristics. Addition of Rec1-resilin or PVCL to RSF and subsequent crosslinking systematically increased the nanoscale size of hydrophobic and hydrophilic domains, whereas decreased the homogeneity of pore size distribution in the microscale. The presented results have implications on the fundamental understanding of the structure–property relationship of RSF-based hydrogels.

SANS Evidence of Liquid–Liquid Phase Separation Leading to Inversion of Crystallization Rate of Broadly Distributed Random Ethylene Copolymers

Aiming to understand the inversion of crystallization kinetics observed by DSC, detailed SANS investigations of the melt structure of a broadly distributed ethylene–1-hexene copolymer have been undertaken in a wide range of temperatures that were reached either by heating the solid or cooling from the homogeneous melt state. In both cases, the observed SANS signal transitions from a scattering cross section consistent with a homogeneous melt state (high temperature range) to an intensity that in the low Q region displays the characteristics of the Porod region for particles dispersed in a homogeneous matrix (low melt temperature range). The latter structure is consistent with demixing of the highly branched molecules and corroborates the postulated liquid–liquid phase separation (LLPS) as an explanation for the peculiar crystallization kinetics observed by DSC. The solution temperature is found at 160 °C by heating the solid and at 150 °C when cooling from the one-phase melt, thus denoting the effect of c...

Cavitation in strained polyethylene/aluminium oxide nanocomposites

The incorporation of metal oxide (e.g. Al2O3) nanoparticles has a pronounced positive effect on low-density polyethylene (LDPE) as an insulating material for high-voltage direct-current (HVDC) cables, the electrical conductivity being decreased by one to two orders of magnitude and charge species being trapped by the nanoparticles. The risk of debonding between the nanoparticles and the polymer matrix leading to electrical treeing via electrical discharges in the formed cavities was the motivation for this study. Scanning electron microscope (SEM), small-angle X-ray scattering (SAXS) and X-ray ptychographic tomography were used to study a series of LDPE nanocomposites which contained Al2O3 nanoparticles treated with silanes having terminal alkyl groups of different lengths (methyl, octyl and octadecyl). When specimens were subjected to a tensile strain (a typical specimen stretched beyond the onset of necking consisted of three zones according to SEM of specimens that were studied after removal of the external force: an essentially cavitation-free zone with low local plastic strain, a transitional zone in which local plastic strain showed a marked increase and the revealed concentration of permanent cavities increased with increasing plastic strain and a highly strained zone with extensive cavitation), the cavitation occurred mainly at the polymer-nanoparticle interface according to SEM and X-ray ptychographic tomography and according to SEM progressed with increasing plastic strain through an initial phase with no detectable formation of permanent cavities to a period of very fast cavitation and finally almost an order of magnitude slower cavitation. The polymer/nanoparticle interface was fractal before deformation, as revealed by the profile of the Porod region in SAXS, presumably due to the existence of bound polymers at the nanoparticle surface. A pronounced decrease in the interface fractal dimension was observed when the strain exceeded a critical value; a phenomenon attributed to the stress-induced de-bonding of nanoparticles. The strain-dependence of the interface fractal dimension value at low strain levels between composites containing differently treated nanoparticles seems to be an indicator of the strength of the nanoparticle-polymer interface.

Hierarchical self-assembly, relaxations and ergodic–non-ergodic transition in laponite ionogels

We report on the formation of entropic ionogels (gels in IL solutions) of laponite platelets in ionic liquid (IL), 1-ethyl-3-methyl imidazolium chloride, solutions at room temperature (0<[IL]<0.05% (w/v)). Self-assembled colloidal networks were formed following a growth mechanism given by, ηr∼(1−t/tgvis)−p for t<tgvis; and G0∼(t/tgrheo−1)β for t>tgrheo where ηr, G0 and tgvis,rheo represent relative viscosity, low frequency storage modulus and gelation time (determined from viscosity and rheology) respectively. The exponent p was found to increase from 0.23 to 0.85 with increase in IL concentration implying slower network growth kinetics, which helped in the formation of stronger ionogels (β decreased from 2.40 to 0.83). It was found that tgvis>tgrheo which indicated that the network, after formation, required considerable amount of equilibration time (longer for low IL concentration) to reorganize and attain a solid-like structure capable of sustaining shear deformation. Reorganization of clay platelets caused reduction in the inter platelet repulsion due the IL-double layer screening and facilitated formation of stronger ionogels for samples with [IL]≤0.03% (w/v). Dynamic structure factor data revealed two primary relaxation modes: (i) the anomalous slow mode that was frozen at all IL concentrations, and (ii) the diffusive fast mode that contained two diffusion components both strongly dependent on IL concentration. Neutron scattering data could be split into two distinct scattering regions: (i) 0.001<q<q# Å−1, I(q)∼q−α with α=0.97–1.67 (Power–law region) and (iii) q#<q<0.35Å−1, I(q)∼e−(q2Rc2/2) (Guinier–Porod region) with Rc=4.6–5.0Å−1. This implied existence of rod-like structures in homogeneous and rarefied clusters in fragile gels. In summary, it is shown that laponite gels could be systematically customized in IL solutions to generate a range of soft materials with properties solely dependent on IL concentration which may not exceed 0.05% (w/v).

Morphology Changes of Co Catalyst Nanoparticles at the Onset of Fischer–Tropsch Synthesis

Cobalt nanoparticles play an important role as catalysts for the Fischer–Tropsch synthesis, which is an attractive route for production of synthetic fuels. It is of particular interest to understand the varying conversion rate during the first hours after introducing synthesis gas (H2 and CO) to the system. To this end, several in situ characterization studies have previously been done on both idealized model systems and commercially relevant catalyst nanoparticles, using bulk techniques, such as X-ray powder diffraction and X-ray absorption spectroscopy. Since catalysis takes place at the surface of the cobalt particles, it is important to develop methods to gain surface-specific structural information under realistic processing conditions. We addressed this challenge using small-angle X-ray scattering (SAXS), a technique exploiting the penetrating nature of X-rays to provide information about particle morphology during in situ experiments. Simultaneous wide-angle X-ray scattering was used for monitoring the reduction from oxide to catalytically active metal cobalt, and anomalous SAXS was used for distinguishing the cobalt particles from the other phases present. After introducing the synthesis gas, we found that the slope of the scattered intensity in the Porod region increased significantly, while the scattering invariant remained essentially constant, indicating a change in the shape or surface structure of the particles. Shape- and surface change models are discussed in light of the experimental results, leading to an improved understanding of catalytic nanoparticles.

Water–polymer interfacial area in Nafion: Comparison with structural models

The water–polymer interfacial area per volume, S/V, which is reflected in the Porod region of small-angle scattering data, is an important parameter of different models of the Nafion fuel cell membrane. Therefore, we have compared published experimental S/V data of Nafion over a wide range of hydration levels with various structural models featuring stiff polymer backbones, in particular the parallel water-channel and the polymer ribbon models. The S/V curve at intermediate hydration levels typical of fuel-cell conditions (ca. 20 vol% water) matches that of the parallel water-channel model with molecular corrugation. At higher hydration levels, i.e. for membranes soaked in water or autoclaved at elevated pressures, the polymer-ribbon model matches the decreasing S/V ratio with increasing water content, while the polymer-bundle model predicts a higher surface area. However, the ribbon or bundle models cannot apply at low hydration (<3 water molecules per Nafion side group), since we show that the interfacial area in this regime must increase strongly with hydration, being determined by the available surface area of the water molecules. The pronounced asymmetry of the plot of S/V vs. water volume fraction is explained in terms of the difference in the diameters of the water molecules and the polymer aggregates.

Analysis of the Beaucage model

The Beaucage model is used to analyze small-angle scattering (SAS) data from fractal and particulate systems. It models the Guinier and Porod regions with a smooth transition between them and yields a radius of gyration and a Porod exponent. This model is an approximate form of an earlier polymer fractal model that has been generalized to cover a wider scope. The practice of allowing both the Guinier and the Porod scale factors to vary independently during nonlinear least-squares fits introduces undesired artefacts in the fitting of SAS data to this model. Such artefacts as well as an error in the original formulation of the model are discussed. This model is compared with other published models.

Scattering from generalized Cantor fractals

A fractal with a variable fractal dimension, which is a generalization of the well known triadic Cantor set, is considered. In contrast with the usual Cantor set, the fractal dimension is controlled using a scaling factor, and can vary from zero to one in one dimension and from zero to three in three dimensions. The intensity profile of small-angle scattering from the generalized Cantor fractal in three dimensions is calculated. The system is generated by a set of iterative rules, each iteration corresponding to a certain fractal generation. Small-angle scattering is considered from monodispersive sets, which are randomly oriented and placed. The scattering intensities represent minima and maxima superimposed on a power law decay, with the exponent equal to the fractal dimension of the scatterer, but the minima and maxima are damped with increasing polydispersity of the fractal sets. It is shown that, for a finite generation of the fractal, the exponent changes at sufficiently large wave vectors from the fractal dimension to four, the value given by the usual Porod law. It is shown that the number of particles of which the fractal is composed can be estimated from the value of the boundary between the fractal and Porod regions. The radius of gyration of the fractal is calculated analytically.

A new Guinier–Porod model

Small-angle scattering (SAS) curves are characterized by two main features: the Guinier region and the Porod region. Standard linear plots are available to fit SAS data and obtain a radius of gyration and a Porod exponent. A new Guinier–Porod empirical model is introduced to fit SAS data from spherical as well as nonspherical objects such as rods or platelets. It also applies to shapes intermediate between spheres and rods or between rods and platelets. The new model is used to fit SAS data from a Pluronic solution that sequentially forms unimers, then spherical micelles, then cylindrical micelles, then lamellar micelles upon heating. This single model can fit structures associated with all four phases as well as the intermediate structures.

Small-Angle Neutron Scattering of Percolative Perfluoropolyether Water in Oil Microemulsions

A water in oil microemulsion system composed of water, surfactant, and oil, the latter two components of perfluoropolyether (PFPE) type, has been studied by small-angle neutron scattering (SANS) with the aim of knowing the microstructure of the system and to have an insight on the connection between microstructure characterization and percolation behavior. In fact, along the dilution line W/S = 11 of the phase diagram, dielectric spectroscopy and conductivity studies revealed a dynamic percolation process taking place approaching and above the dynamic percolation threshold, leading to a system composed of droplet clusters with percolation thresholds varying with temperature from a 0.501 volume fraction of the dispersed phase at 9.3 degrees C to 0.205 at 32.5 degrees C. The SANS experimental spectra of this work have been studied by modeling the microemulsion droplets as adhesive hard spheres. For all of the samples, the surfactant area per polar head has been also measured in the Porod region of the SANS spectra. Geometric and potential parameters as well as the osmotic pressure, the second virial coefficient, and the distance between droplets have been extracted from data as a function of droplets concentration. At low concentration, that is, below percolation thresholds, the droplets behave as hard spheres, whereas at threshold and above, adhesion changes significantly the samples. In fact, for each temperature, the measured size increases versus concentration from 30 to 50 A, and the area per polar head decreases correspondingly, suggesting that a process of dynamic fusion of droplets occurs in the system above threshold, that is, couples of droplets stick and unstick continuously with interdigitation of the surfactant tails.

Effects of Nitrogen Concentration and Plastic Deformation on Small Angle Neutron Scattering for Austenitic Stainless Steels

The small angle neutron scattering (SANS) profile has been found to be influenced by nitrogen addition and plastic deformation for austenitic stainless steels. In solution-treated specimens, the slope of plots for SANS intensity versus the magnitude of scattering vector, q, becomes larger than −4 in the Porod region, suggesting the influence of short range N ordering. The change in scattering intensity profile with tensile deformation was in situ measured and it is found that SANS intensity becomes larger with increasing of plastic strain except a q region around 0.6 nm−1 for N bearing steel, where the intensity decreases in the beginning and then increases with increasing of plastic strain. This temporary decrease in SANS intensity is confirmed for cold rolled specimens. Severe deformation given by swaging to a N free steel brings larger increase in SANS intensity than a N bearing steel. The influence of plastic deformation on SANS intensity is larger in austenitic steels than in ferritic steels.

Surface aggregate structure of nonionic surfactants on silica nanoparticles

The self-assembly of two nonionic surfactants, pentaethylene glycol monododecyl ether (C12E5) and n-dodecyl-{\ss}-maltoside ({\ss}-C12G2), in the presence of a purpose-synthesized silica sol of uniform particle size (diameter 16 nm) has been studied by adsorption measurements, dynamic light scattering and small-angle neutron scattering (SANS) using a H2O/D2O mixture matching the silica, in order to highlight the structure of the surfactant aggregates. For C12E5 strong aggregative adsorption onto the silica beads, with a high plateau value of the adsorption isotherm above the CMC was found. SANS measurements were made at a series of loadings, from zero surfactant up to maximum surface coverage. It is found that the spherical core-shell model nicely reproduces the SANS data up to and including the local maximum at q = 0.42 nm-1 but not in the Porod region of high q, indicating that the surface area of the adsorbed surfactant is underestimated by the model of a uniform adsorbed layer. A satisfactory representation of the entire scattering profiles is obtained with the model of micelle-decorated silica beads, indicating that C12E5 is adsorbed as spherical micellar aggregates. This behaviour is attributed to the high surface curvature of the silica which prevents an effective packing of the hydrophobic chains of the amphiphile in a bilayer configuration. For the maltoside surfactant {\ss}-C12G2 very weak adsorption on the silica beads was found. The SANS profile indicates that this surfactant forms oblate ellipsoidal micelles in the silica dispersion, as in the absence of the silica beads.

Preliminary SANS studies of the structure of nickel powders on the nanoscale

Nickel powders produced by hydrogen reduction of nickel ammine/ammonium sulphate solution were examined for their structure on the nanoscale using the small-angle neutron scattering (SANS) technique. It is known that the particle size and density of the powder depends on the number of times the nickel solution bath was subjected to hydrogen reduction, however the porosity on the nanoscale is open to question. Porosity in commercial nickel powders is a concern since these nanopores can trap sulphur impurities from solution and lead to unacceptably high levels of sulphur in the final nickel product. SANS data over the q-range (0.002–0.18 Å −1) was collected on nickel powders that had been exposed to 0, 1, 2, 40, and 60 cycles of hydrogenation in an industrial refining process. Results from this study indicate general trends in the nanostructure consistent with a roughening of the surface with the power law exponent in the Porod region consistently exceeding 4. Further experiments are required to clarify the overall porosity.

A review of small-angle scattering models for random segmented poly(ether-urethane) copolymers

Small-angle X-ray scattering data from several experimental and commercial poly(ether urethane) formulations was used to test various scattering models based on different morphologies. The best fits were generally found with ‘globular’ scattering models based on a distorted one-dimensional lattice or the Percus–Yevick model of liquid structure. Whilst this is not conclusive proof of the morphologies exhibited by the materials studied, these scattering models are roughly consistent with many AFM and TEM studies, which indicated discrete globular or elongated cylindrical microdomains. Moreover, reasons why meandering elongated or finite lamellar microdomains may behave more like globular scattering bodies are discussed. Hence, these models are proposed as a basis for interpreting scattering data from polyurethanes. Other models were unable to fit the observed scattering data adequately. A model based on the weak segregation of copolymers was rejected on the basis of deviations from the observed scattering behaviour in the Porod region. A model based on stacks of ideal (infinite, parallel, flat) lamellae was ruled out, since it was unable to reproduce the observed peak width. The Teubner–Strey model, based on microemulsion structure, was found to be incapable of fitting the data at low q, particularly for strained samples. However, this model appeared to be more successful at reproducing the scattering observed at elevated temperatures. One possible inference is that the morphologies became more akin to microemulsions during heating.

Small angle scattering in the Porod region from hydrated paper sheets at varying humidities

Abstract The microscopic changes in the structure of paper at different equilibrium moisture contents were examined using the novel technique of contrast variation small angle neutron scattering (SANS). Contrast variation, by appropriate selection of the H2O:D2O ratio, was used to simplify the scattering from hydrated paper to a two-phase approximation. The two phases were paper polymers (cellulose, lignin and hemicelluloses) and a phase consisting of water and voids. The two-phase approximation is justified by consideration of the chemistry and density of each of the polymer species belonging to paper, and by matching the scattering length density of the sorbed moisture to air. The neutron intensity at the higher scattering angles examined was successfully fitted using the Porod law, which is applicable to scattering from a two-phase system at the high scattering angle limit. This limit is related to the specific surface area between the two phases. The limiting scattered intensity of neutrons was found to increase with increasing moisture content for a range of paper materials. It is concluded that the sorption of water increased the limiting intensity because water disrupted polymer-polymer bonding. The Porod law also predicts the slope of the intensity/scattering angle curve at the high scattering angle limit.

Structure and Phase Transitions of Poly(heptamethylene p,p‘-bibenzoate) As Studied by DSC and Real-Time SAXS/MAXS Employing Synchrotron Radiation

The structure and phase transitions of liquid-crystalline poly(heptamethylene p,p'bibenzoate) have been studied by DSC and variable-temperature real-time SAXS/MAXS employing synchrotron radiation. It was suggested that three-dimensional crystal order appeared within the smectic domains during cooling the sample because the crystal correlation length is nearly the same as that of the smectic domains. These estimated domain sizes are of the order of the average macromolecular chain length, so that it seems that no chain folding exists in the present case. It was found that SAXS intensity in Porod's region obeys the power law, and it deviates significantly from Porod's law. These deviations were explained by an assumption of anisotropic density fluctuations within the phases. It was supposed that the melting process is ruled by an increasing quantity of anisotropic fluctuations within the crystallites but not by surface melting.

Kinetics of phase separation in the driven lattice gas: Self-similar pattern growth under anisotropic nonequilibrium conditions

The driven lattice gas (DLG) evolving at low temperature helps understanding the kinetics of pattern formation in unstable mixtures under anisotropic conditions. We here develop a simple theoretical description of kinetics in Monte Carlo simulations of the DLG. A Langevin continuum analog is also studied which is shown to exhibit the same behavior. We demonstrate that pattern growth is mainly a consequence of single-particle processes and that, after a short transient time, in which a surface evaporation/condensation mechanism is important, hole diffusion in the bulk becomes dominant. Consequently, there is a unique relevant length that behaves $l(t) \sim t^{1/3}$ for macroscopic systems except at some very early (perhaps unobservable) time. This implies sort of self-similarity, namely, the spatial pattern looks alike, but for a (non-trivial) change of scale at different times. We also characterize the structure factor, in which we identify Guinier and Porod regions, and its scaling form with both time and size. The underlying anisotropy turns out to be essential in determining the macroscopically-emergent peculiar behavior.

Small‐Angle X‐Ray Scattering Study of the Quasi‐Crystal Structure of Montmorillonite‐CTAB in Suspension

The quasi‐crystals of hexadecyltrimethylammonium (CTA+)‐montmorillonite formed in suspension were investigated by small‐angle x‐ray scattering (SAXS). The d‐spacing of the quasi‐crystals increased from 17.1 to 21.7 Å, corresponding to an increasing surfactant loading from 20 to 250% of the clay's cation exchange capacity (CEC). One‐dimensional grating theory was used to describe the scattering in the Porod region so that the number of clay layers and crystal size were obtained. The quasi‐crystal reaches maximum size at a surfactant loading equal to the clay's CEC. The scattering peaks in the Porod region are due solely to quasi‐crystals. Small‐angle x‐ray scattering avoids intermediate peaks resulting from the artificial interstratification of free and complexed layers in conventional x‐ray diffraction analysis.

Nanoscale uniformity of pore architecture in diatomaceous silica: a combined small and wide angle x‐ray scattering study

Combined small and wide angle X‐ray scattering (SAXS and WAXS) analysis was applied to purified biogenic silica of cultured diatom frustules and of natural populations sampled on marine tidal flats. The overall WAXS patterns did not reveal crystalline phases (WAXS domain between 0.07 to 0.5 nm) in this biogenic silica, which is in line with previous reports on the amorphous character of the SiO2 matrix of diatom frustules. One exception was the silica of the pennate species Cylindrotheca fusiformis Reimann et Lewin, which revealed wide peaks in the WAXS spectra. These peaks either indicate the presence of a yet unknown crystalline phase with a repetitive distance (d‐value ≈0.06 nm) or are caused by the ordering of the fibrous silica fragments; numerous girdle bands. The SAXS spectra revealed the size range of pores (diameter d between 3.0 and 65 nm), the presence of distinct pores (slope transitions), and structure factors (oscillation of the spectra). All slopes varied in the range of −4.0 to −2.5, with two clear common regions among species: d &lt; 10 nm (slopes –4, denoted as region I and also called the Porod region), and 10.0 &lt; d &lt; 40.0 nm (slopes −2.9 to −3.8, denoted as region II). The existence of these common regions suggests the presence of comparable form (region I) and structure (region II) factors, respectively the shape of the primary building units of the silica and the geometry of the pores. Contrast variation experiments using dibromomethane to fill pores in the SiO2 matrix showed that scattering was caused by pores rather than silica particles. Electron microscopic analysis confirmed the presence of circular, elliptical, and rectangular pores ranging in size from 3 to 65 nm, determining the structure factor. The fine architecture (length/width ratio of pore diameters) and distribution of the pores, however, seemed to be influenced by environmental factors, such as the salinity of and additions of AlCl3 to the growth medium. The results indicate that diatoms deposit silica with pores &lt;50 nm in size and are highly homologous with respect to geometry. Consequently, it is suggested that in diatoms, whether pennate or centric, the formation of silica at a nanoscale level is a uniform process.