Porod Exponent Research Articles

Systematic Investigation of Solution-State Aggregation Effect on Electrical Conductivity in Doped Conjugated Polymers

Systematic Investigation of Solution-State Aggregation Effect on Electrical Conductivity in Doped Conjugated Polymers

Using the USAXS technique to reveal the fat globule and casein micelle structures of bovine dairy products.

Cows' milk is a commodity used worldwide to make many dairy products. We have used the ultra small angle X-ray scattering (USAXS) technique to reveal the fat globule and casein micelle structures of some dairy products. USAXS covers the q-range 5×10-4Å-1<q<10-1Å-1, thereby allowing the study of micron-scale structures present in those dairy products. We measured the USAXS intensity, Iq, as a function of the scattering vector magnitude, q, for samples of skim milk, non-homogenized whole milk, homogenized whole milk, half and half and heavy cream, at two temperatures, 7°C and 45°C. The data collected from the scattering experiments were fitted using the Unified fit model run under the IRENA software from the Advanced Photon Source, Argonne National Laboratory (Illinois, USA). The fittings were carried out when the data were plotted as log[I(q)] vs log[q]. We observed a combination of linear regions (LRs) and knees. Two parameters of interest were obtained from the fittings, a radius of gyration, Rg, and a Porod exponent, P. Unified fit allowed us to fit up to four structural levels. One of the knees was centered at q ≈ 8×10-3Å-1 for all samples measured at 7°C, but vanished at 45°C. Two LRs were identified as being either due to casein micelles (CMs) or to fat globules (FGs). The porod exponent obtained from these LRs allowed us to describe the surface morphology of CMs and FGs. Two of the Rg values gave a rough estimate of the FGs and CMs sizes. FGs were identified for samples of homogenized whole milk, half and half and heavy cream in the q-region 2×10-4<q<8×10-4Å-1. We found that, in the absence of chymosin, or changes in pH, CaCl2 concentration or temperature changes, skim milk and non-homogenized whole milk displayed a Porod exponent that indicated a behavior characteristic of aggregation. Using computer simulations, we found that, seemingly, bovine CMs spontaneously formed approximately 1-dimensional aggregates possibly analogous to swollen randomly branched polymers.

Effect of grain-boundary diffusion process on the geometry of the grain microstructure of Nd−Fe−B nanocrystalline magnets

Hot-deformed anisotropic Nd$-$Fe$-$B nanocrystalline magnets have been subjected to the grain-boundary diffusion process (GBDP) using a $\mathrm{Pr}_{70}\mathrm{Cu}_{30}$ eutectic alloy. The resulting grain microstructure, consisting of shape-anisotropic Nd$-$Fe$-$B nanocrystals surrounded by a Pr$-$Cu-rich intergranular grain-boundary phase, has been investigated using unpolarized small-angle neutron scattering (SANS) and very small-angle neutron scattering (VSANS). The neutron data have been analyzed using the generalized Guinier-Porod model and by computing model-independently the distance distribution function. We find that the GBDP results in a change of the geometry of the scattering particles:~In the small-$q$ regime the scattering from the as-prepared sample exhibits a slope of about $2$, which is characteristic for the scattering from two-dimensional platelet-shaped objects, while the GBDP sample manifests a slope of about $1$, which is the scattering signature of one-dimensional elongated objects. The evolution of the Porod exponent indicates the smoothing of the grain surfaces due to the GBDP, which is accompanied by an increase of the coercivity.

Structural analysis of dendrimers based on polyhedral oligomeric silsesquioxane and their assemblies in solution by small-angle neutron scattering: Fits to a modified two correlation lengths model

The small-angle neutron scattering (SANS) technique has been used to investigate the structure of dendrimers based on polyhedral oligomeric silsesquioxane (POSS) in solution. Three generations and two distinct terminal groups (either vinyl or polyethylene glycol (PEG) terminal groups) as well as four concentrations for each dendrimer have been measured in deuterated tetrahydrofuran. Two main scattering regimes, one with a length scale of nm and the other with a length scale of sub-micron, are found to contribute to the SANS signals. In order to fit the SANS data, a new modified two correlation lengths model has been developed. Fit parameters include scale factors, correlation lengths, Porod exponents and “particle” elongation parameters. Our fit results show that the local structure does not correspond to single dendrimer molecules, but rather to POSS-rich domains surrounded by POSS-poor domains. The large-scale structure represents the overall morphology of multi-dendrimer aggregates. The flexible PEG outer links help to enhance the stability of inorganic POSS cages, and allow the hybrid polymers to adjust their conformation in order to respond to their surroundings. With increasing concentration, the local structure becomes more compact as evidenced by the decrease of correlation length and concurrent increase of the Porod exponent. The deformation of local structure during the compaction process is also observed. The structure becomes elongated at intermediate concentrations whereas it tends to be more globular at low and high concentrations.

Synchrotron Radiation Studies of Additives in Combustion, III: Ferrocene

Small-angle X-ray scattering (SAXS) has been used to examine the growth of soot primary particles (∼15–20 nm) and aggregates of primaries (>30 nm) formed in an ethylene flame doped with ferrocene. It is found that, early in the flame, the size of the primary particles formed when ferrocene is added tracks that for an undoped flame. Higher up (at height z above the burner greater than 18 mm), both primary and aggregate sizes are significantly smaller with ferrocene doping compared to the undoped flame. Within experimental error, the number densities of primary (N1) particles in the ferrocene flame are about the same as for the pure ethylene flame at low heights and higher (factor of 10) at larger heights (above 18 mm). On the other hand, the number concentration of aggregate particles (N2) in the doped flame is always greater than in the pure ethylene flame regardless of the height but with drops and increases in the ferrocene-doped flame case that can be correlated with corresponding increases and decreases in aggregate size. SAXS data at sufficiently long residence times in the flame show, through the value of the Porod exponent, that the particle surfaces of primaries and aggregates are much rougher in the doped versus undoped flame, which we interpret as being the effect of iron oxide inclusion that produces enhanced oxidation inside the soot material.

Small‐Angle Scattering From Branched Polymers

AbstractThe single‐polymer form factor is determined for branched polymers using a scaling argument in order to recover the low‐Q Porod exponent characteristic of the overall structure. The high‐Q Porod exponent characterizes the local branching structure. An alternative approach based on a high‐Q expansion contains information about functionality, branch length and branch content. The specific case of a starburst dendrimer for which the form factor is known is discussed. The model predictions are compared to small‐angle neutron scattering data from a dilute solution of dendrimer in D2O.magnified image

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.

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.

Insight into chain dimensions in PEO/water solutions

AbstractSmall‐Angle Neutron Scattering has been performed from poly(ethylene oxide) in deuterated water at temperature ranging from 10 to 80 °C. A simple fitting model was used to obtain a correlation length and a Porod exponent. The correlation length L characterizes the average distance between entanglements in the semidilute region and is proportional to the individual coil sizes in the dilute region. L was found to increase with temperature in the semidilute region but it decreases with temperature in the dilute region. This decrease is the precursor to the single‐chain collapse which applies to very dilute polymer solutions. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 2196–2200, 2007

Small-angle neutron scattering investigations on fractal aggregation and sintering behavior of La 1− xCa xCrO 3 synthesized by a combustion process

La 1− x Ca x CrO 3 ( x = 0.3) powder has been synthesized by gel combustion process. Fractal behavior of the agglomerates and pore morphology of the sintered compacts have been studied by small-angle neutron scattering. At low scattering vector ( q) regime the ‘ q − D f ’ power law variation of intensity of the powder sample and the green compact indicates that the powder particles are agglomerates and are composed of fractal aggregates. The value of the mass fractal dimension D f of these aggregates was found to be ∼2.2. The Porod exponent of the scattering profiles of the sintered specimens shows deviation from its normal value of −4 to ∼−3. This has been attributed due to superposition of scattering intensities from fine intra-granular pores on the scattering intensity from inter-granular pores. Loss of fractal structure of the aggregates upon heat treatment of the compacts at higher temperature has been observed. The pore size distribution gradually shifts towards higher radius size with the increase in sintering temperature due to faster elimination of finer pores.

Precipitation in 18 wt% Ni maraging steel of grade 350

The evolution of precipitates in maraging steel of grade 350 was studied using the complementary techniques of small angle X-ray scattering (SAXS) and transmission electron microscopy (TEM). These investigations revealed that ageing the steel at 703 K involved a rhombohedral distortion of the supersaturated b.c.c. martensite accompanied by the appearance of diffuse ω-like structures. This was followed by the appearance of well-defined ω particles containing chemical order. At the ageing temperature of 783 K, Ni 3(Ti, Mo) precipitates were the first to appear with a growth exponent of 1/3. The values of the Porod exponent obtained from the SAXS profiles indicated that the ω particles, formed below 723 K, had diffuse interfaces up to an ageing time of 48 h. On the other hand, Ni 3(Ti, Mo) precipitates, formed above 723 K, developed sharp interfaces in just about an hour. Also, the steel exhibited scaling in phase separation both at 703 and 783 K, but only during the early stages. Through this study it was established that, at temperatures of ageing less than 723 K, evolution of ω particles takes place through the collapse of the unstable b.c.c. lattice and, at temperatures above 723 K, precipitation of A 3B type of phases through the mechanism of clustering and ordering of atomic species. Sharp interfaces develop rather quickly when the mechanism of precipitation involves development and amplification of a concentration wave alone as in the nucleation of Ni 3(Ti, Mo) at 783 K than when an interplay of both the displacement and concentration waves is required as in the evolution of ω at 703 K. These results indicate towards the possibility of existence of two separate time–temperature–transformation (TTT) curves, one for the evolution of ω-phase and another for nucleation and growth of Ni 3(Ti, Mo).

Phase-separation kinetics of a multicomponent alloy

Phase separation kinetics of multicomponent 350-grade maraging steel has been investigated on both recrystallized as well as cold-worked specimens by small-angle x-ray scattering, wide angle x-ray scattering, and transmission electron microscopy at two different temperatures, viz. 430 \ifmmode^\circ\else\textdegree\fi{}C and 510 \ifmmode^\circ\else\textdegree\fi{}C, for different aging times. Unlike previous observations, at both the temperatures, dynamical scaling behavior is observed at the early stages of phase separation accompanied by diffuse interface of the secondary phases. Porod exponents have been found to be greater than 4. At late stages, the precipitate-matrix interface becomes sharp --- the Porod exponent is close to 4 but clear deviation from the dynamical scaling behavior is evident. At $430\ifmmode^\circ\else\textdegree\fi{}\mathrm{C},$ the phase separation is attributed to the formation of an ordered $\ensuremath{\omega}$ phase through a mechanism involving chemical ordering and the $\ensuremath{\omega}$-like lattice collapse in the bcc structure. Time (t) dependent population averaged precipitate radius follows ${t}^{1/5}$ power law indicating cluster diffusion mechanism of Binder-Stauffer type for the entire range, 30 min--72 h, of aging time. At $510\ifmmode^\circ\else\textdegree\fi{}\mathrm{C},$the phase separation is attributed to the formation of ${\mathrm{Ni}}_{3}(\mathrm{T}\mathrm{i},\mathrm{M}\mathrm{o})$ with ${\mathrm{DO}}_{24}$ structure through the process of nucleation and growth. Average precipitate radius follows ${t}^{1/3}$ Lifshitz-Slyozov power law for the entire range, 5 min--18 h, of aging time. The system, despite being multicomponent and complex, appears to follow two distinct time-temperature-transformation curves. As far as the effect of cold work on phase-separation behavior is concerned, it has been found that cold work facilitates the growth of the precipitates. Also, it narrows down the size distribution and enforces strong spatial correlation of the precipitates. Cold working the material is found to be detrimental to the dynamical scaling behavior.

A small-angle x-ray scattering study of microstructure evolution during sintering of sol-gel-derived porous nanophase titania

The evolution of microstructure as a function of firing temperature in sol-gel derived porous titania xerogels was investigated by small-angle x-ray scattering (SAXS). SAXS curves for xerogels fired below 550 °C exhibit a well-defined structure peak. This peak indicates the presence of a high degree of order in the electron density correlations associated with the interparticle structure factor. Results from electron microscopy and SAXS give a primary particle mean diameter of 50 Å, while scaling analysis of the scattered intensity at large momentum transfer values yields the Porod exponent 4, indicating a sharp transition zone between the solid and void phases. The pore volume fraction of the unfired xerogel is consistent with random close-packing of spheres. The internal surface area decreases almost linearly with increasing firing temperature. Rapid grain growth and pore coarsening begin near 90% of theoretical density, and lead to a breakdown in pore interconnectedness and the development of isolated pores. Observed enhanced sintering properties may be attributed primarily to the large surface-to-mass ratio of the sol-gel particles. The SAXS curves were adequately fit using a bicontinuous phase model developed for late-stage spinodal decomposition structures. Alternatively, the microstructure can be described by a hierarchical close-packing of spheres model. SAXS results are compared with data from gas adsorption and electron microscopy.

Small-angle scattering of shaley rocks with fractal pore interfaces

Differences have been found in the outer part of the neutron and x-ray small-angle scattering curves for various porous shaly rocks, and are analyzed in terms of fractal interfaces. Neutrons are relatively insensitive to the presence of either hydrocarbons or water in the pores, and always give a smaller Porod exponent than that for x rays, indicating that the hydrocarbon and/or water tend to reside in the small re-entrant parts of the mineral-pore interfaces