Size-distribution analysis of polymer latex systems by use of the extrema in the angular light-scattering pattern: II. The polarization ratio

Previously, the angular positions of the extrema of the polarization ratio have been utilized in light-scattering studies as a method of particle-size analysis under the assumption of monodisperse, spherical particles. Since the consequences of the existing finite polydispersity on this method of analysis was not assessed, the results are questionable. This work is concerned with (1) reporting the quantitative effects of a finite polydispersity on the method of analysis, (2) pointing out previous misuse of the method, and (3) revising the method of analysis to include polydispersity and the exact Mie calculations. The method will then permit the characterization of a scattering system in terms of a modal diameter and a distribution-width parameter by utilizing prepared diagrams for a particular relative refractive index.     

On the modeling of small-angle X-ray scattering from systems of oriented fibrils

A method is outlined for the simulation of the scattering pattern from systems of oriented fibrils using the radial distribution function of a hard-disc fluid obtained under the Percus–Yevick approximation. In this manner both the diameter and the volume fraction of the fibrils may be estimated directly from the scattering pattern. The effect of polydispersity in the distribution of fibril diameters on the position of the scattering peak is discussed, suggesting that an estimate of the volume fraction from the peak position alone, such as in a modified Bragg's law, may be inaccurate.     

Size distribution analysis of polymer latex systems by use of the extrema in the angular light scattering pattern. III. Unpolarized and horizontally polarized scattered light

The extrema in the unpolarized and the horizontally polarized angular scattering patterns are used as a basis for size-distribution analysis of polymer latex systems composed of single, spherical, optically isotropic particles. The method of analysis is similar to that recently proposed for vertically polarized scattered light and the polarization ratio, where the modal diameter and a distribution width parameter are determined from the experimental angular scattering pattern and prepared theoretical diagrams obtained by Mie theory calculations. The method of analysis, based on all four scattering functions, is illustrated by use of a Dow polystyrene latex sample.     

Particle size analysis of polymer latexes by light scattering. V. The significance of an assumed distribution in the analysis of angular scattering data

In the analysis of light scattering data from polymer latex systems or other systems of spherical particles, it is necessary to assume a particle size distribution function. Theoretical angular scattering functions based on the assumed distribution and representing a wide range of size distribution parameters are compared to experimental data in order to obtain a best fit. In previous work, it has been shown that as the polydispersity increases beyond certain limits the uncertainty in the assignment of the size distribution parameters (i.e., the best fit) increases. This report is concerned with the analysis of angular scattering from unimodal systems and simulated cases where theoretical scattering functions for wide, negatively skewed distributions are used as “experimental data,” are analyzed by utilizing four different distribution functions. These functions represent different degrees of skewness and include negatively, positively, and normally skewed distributions. The results from the use of the various distribution functions are discussed with respect to the uncertainly in the assignment of distribution parameters resulting from the loss of structure in the angular scattering pattern due to increased polydispersity. Scattering data from the bimodal distribution are analyzed by assuming a unimodel distribution, and the consequences of this assumption are assessed.     

Quasielastic light scattering as a method for determining the distribution of relaxation times in a polymer system

Amongst other techniques, dynamic light scattering may be used to obtain molar mass distributions. The first step in this process consists in the Laplace inversion of the time correlation function that was measured by dynamic light scattering. This inversion gives a distribution of diffusion coefficients. In order to convert this distribution into the corresponding molar mass distribution, a relationship between diffusion coefficient and molar mass of monodisperse fractions has to be known. Such a relationship can be derived for linear and star-branched macromolecules from measurements of polydisperse systems, since the polydispersity of the distributions does not change with the molar mass. The problem is more involved with randomly branched materials, since in these cases the polydispersity increases strongly as the point of gelation is approached. A procedure is suggested for deriving the diffusion-molar mass dependence of monodisperse samples from polydisperse systems. After an outline of this background the method is applied to the three selected systems (i) radically polymerized linear PMMA, (ii) a star-branched microgel where monodisperse arms are attached to a microgel center and (iii) a randomly branched poly(dicyanate) sample based on bisphenol A. The results are compared with the combined column chromatography SEC/LALLS/VISC. Good agreement was found up to molar masses of about 10 millions g/mol, but systematic deviations occured in the high molar mass region. These differences result from the limitations of size permeation chromatography. Finally it is shown that the size distribution can be determind by this method, even for associates.     

Analysis of small angle neutron scattering data from strongly interacting polydisperse ionic micellar solutions

Small angle neutron scattering (SANS) intensity distributions from ionic micellar solutions without added salt generally show a prominent interaction peak at finite Q (magnitude of the wave vector transfer) related to a certain inter-micellar correlation distance. Analysis of this type of data requires, aside from the structural model of the micelle itself, a statistical mechanical theory for dealing with the inter-micellar correlations. We present a method for a consistent analysis of SANS data taking into account the surfactant chain packing, the aggregation number polydispersity, and the effective micellar charge. The micelle is modelled as a two-region spheroidal particle and the inter-micellar correlations calculated according to a generalized one-component macroion (GOCM) theory. GOCM uses an effective inter-micellar pair potential which is a finite concentration extension of the well-known Derjaguin-Landau-Verway-Overbeek (DLVO) double layer interaction potential. Two micellar solutions are treated as examples, namely that of sodium dodecyl sulfate (SDS) and sodium 1,2-bis(2-ethylhexyloxycarbonyl)ethanesulfonate (AOT). The effects of polydispersity are appreciable immediately above the critical micellar concentrations (CMC). Both the structural parameters of the micelle and the free energies of micellization and micellar growth can be extracted from SANS data.     

Utilising spatial frequency filtering to extract nanoscale layer structure information from isotropic small-angle X-ray scattering data

 A separation method by spatial frequency filtering of the diffuse background of small-angle X-ray scattering (SAXS) is transferred to the case of isotropically scattering samples of polymer materials. Analysis of the residual discrete SAXS is demonstrated. Evaluations of model scattering curves from lamellar two-phase systems show that this technique, in general, results in a good separation. If samples with pure particle scattering or such with rough domain surfaces are investigated, the separation of a suitable background is possible, but is prone to some uncertainty which is estimated. In the case of particle scattering from lamellae the problem is solved by fitting a model function considering polydispersity to the Lorentz-corrected scattering intensity. After background correction the residual information on the distorted nanostructure is collected in an interface distribution function, from which topological parameters can be recovered with high accuracy. These parameters comprise average layer thicknesses and parameters of particle polydispersity. Parameter recovery is achieved by nonlinear regression with model functions describing stacking statistics. Automated versions of the technique are suited to process and analyse series of polymers collected in time-resolved synchrotron radiation experiments. Received: 24 July 2001 Accepted: 12 September 2001     

Polydispersions radiation envelope: Influence of particle size and assumed distribution function

Summary The influence of polydispersity, particle size and of assumed distribution on angular dependence of theMie intensity functions in systems with relative refractive indexm = 1.10 was studied theoretically. Two distribution functions were used; Stevenson-Heller-Wallach (SHWD) and logarithmic distribution of negative order (NOLD). It was found, that for highly polydisperse systems the functional dependences obtain also undulative character. By means of NOLD the angular shift of extremes on the curves studied with increasing polydispersity was not observed. The application of this distribution in comparison with SHWD shows more structured angular scattering patterns with increasing particle size and this fact can also be employed in modelling and in distribution analysis of dispersion with higher degree of polydispersity.With 4 figures     

Small-angle scattering from hexagonal liquid crystals

In this paper, we discuss the scattering behavior of two-dimensional hexagonal liquid crystals with micellar cylinders as a building unit. We treat the hexagonal phase as an accumulation of ordered domains of finite size that typically consists of one hundred parallel cylinders whose axes are perpendicular to the lattice plane. When we suppose that no specific orientation is preferred, the lengths of the cylinders are rather large compared to their diameter, and the polydispersity of the size of the cylinders is negligible; it is therefore possible to split the scattering intensity into a product of the so-called form factor and the structure factor. This product approximation is the basic condition for the use of the generalized indirect Fourier transformation (GIFT) method and the deconvolution (DECON) method to evaluate the small-angle scattering data of hexagonal phases. The GIFT method provides the parameters of the structure factor model and the pair distance distribution function of the cylinders. Via the DECON technique, we can calculate the radial contrast profile of the cylinders from the pair distance distribution function that is obtained by the GIFT method.     

Small-angle neutron scattering by two-component particles: Application to block copolymers

General equations for small-angle neutron scattering are derived for block copolymers in dilute solution. The effects of excluded volume and polydispersity are examined. The applicability of the method of variable contrast to such systems is discussed.     

Effect of macromolecular polydispersity on diffusion coefficients measured by Rayleigh interferometry

Rayleigh interferometry has been extensively used for the precise determination of diffusion coefficients for binary and ternary liquid mixtures. For ternary mixtures, the 2x2 matrix of multicomponent diffusion coefficients is obtained. Polydispersity adds complexity to the meaning of these measured diffusion coefficients. Here we discuss three important issues of polydispersity regarding the diffusion measurements extracted from this interferometric technique. First, we report novel equations for the extraction of diffusion moments from the Rayleigh interferometric pattern. These moments are used to define polydispersity parameters for macromolecular systems. We have experimentally determined mean diffusion coefficients and polydispersity parameters for aqueous solutions of poly(ethylene glycol) and poly(vinyl alcohol) at 25 degrees C. Aqueous solutions of poly(ethylene glycol) mixtures were used to examine the accuracy of the polydispersity parameters. Second, we compare Rayleigh interferometry to dynamic light scattering. Specifically, we have performed diffusion measurements on the same system using both techniques. To our knowledge, no direct experimental comparison between dynamic light scattering and classical methods for the measurements of diffusion coefficients has been previously reported in relation to polydispersity. We find that substantial discrepancies (i.e., 1 order of magnitude) between the mean diffusion coefficients obtained from these two different techniques can be observed when polydispersity is large. Third, for two-solute mixtures with one polydisperse solute, we report a novel corrective procedure for extracting accurate ternary diffusion coefficients from Rayleigh interferometry. Computer simulations were used to examine the accuracy of the extracted ternary diffusion coefficients.     

Classical light scattering studies on structure formation in polymer solutions

The scattered light is very sensitive to the presence of supermolecular structures in polymer solutions, offering the possibility to study structure formation in situ by classical light scattering and to characterize the generated particles. An improved analysis of light scattering data yields information on the structure type, polydispersity of the systems, particle mass and size as well as the degree of swelling. The flocculation process of ampholytic modified poly(acrylonitrile) was investigated in dependence on the pH-value and the content of different salts in the solvent. Results obtained give a deeper insight into the flocculation mechanism.     

Supermolecular structures in polymer solutions interpretation of static light scattering data

The characterization of supermolecular structures in polymer solutions by static light scattering requires an improved analysis of the scattering curves. A comparison of the experimental curves with model calculations for various basic structure types in a scaled plot offers the possibility to extend the range of a reliable determination of the structural parameters to particle sizes up to 500 nm. The influence of the polydispersity and problems of an unambiguous interpretation are discussed. Especially, a closed analytical expression for the scattering function of polydisperse systems of spheres with a Schulz-Zimm distribution of radii was derived.     

The self-preserving size distribution theory. I. Effects of the Knudsen number on aerosol agglomerate growth

Gas-phase synthesis of fine solid particles leads to fractal-like structures whose transport and light scattering properties differ from those of their spherical counterparts. Self-preserving size distribution theory provides a useful methodology for analyzing the asymptotic behavior of such systems. Apparent inconsistencies in previous treatments of the self-preserving size distributions in the free molecule regime are resolved. Integro-differential equations for fractal-like particles in the continuum and near continuum regimes are derived and used to calculate the self-preserving and quasi-self-preserving size distributions for agglomerates formed by Brownian coagulation. The results for the limiting case (the continuum regime) were compared with the results of other authors. For these cases the finite difference method was in good in agreement with previous calculations in the continuum regime. A new analysis of aerosol agglomeration for the entire Knudsen number range was developed and compared with a monodisperse model; Higher agglomeration rates were found for lower fractal dimensions, as expected from previous studies. Effects of fractal dimension, pressure, volume loading and temperature on agglomerate growth were investigated. The agglomeration rate can be reduced by decreasing volumetric loading or by increasing the pressure. In laminar flow, an increase in pressure can be used to control particle growth and polydispersity. For D(f)=2, an increase in pressure from 1 to 4 bar reduces the collision radius by about 30%. Varying the temperature has a much smaller effect on agglomerate coagulation.     

Interpretation of light scattering from supermolecular structures in liquid systems by master curves

By a detailed analysis of light scattering curves, a more complete characterization of colloidal particles may be achieved. For this purpose, a fitting procedure based on theoretical master curves of models of polydisperse systems of homogeneous spheres and Gaussian coils is presented. The use of a suitable logarithmic distribution function makes it possible to separate the influence of polydispersity from that of particle size on the shape of the scattering curve. A double logarithmic plot of master curves reduces the fitting procedure to translations of the experimental curves. The reliability and accuracy of this procedure are demonstrated by light scattering results on solutions of a polyelectrolyte complex with variation of salt content.     

Analysis of small-angle scattering data from micelles and microemulsions: free-form approaches and model fitting

The free-form methods for analyzing small-angle scattering data have, during the last years, found more widespread use for micelles and microemulsions. Recent developments have made them applicable also to systems with size polydispersity and particle correlations, however, model fitting still constitutes a very important and partly complementary analysis tool.     

Observation of two diffusive relaxation modes in microemulsions by dynamic light scattering

Water-in-oil microemulsions stabilized by AOT and dispersed in n-alkane oils with a constant molar water-to-surfactant ratio were studied by dynamic light scattering. A dilution series (in the range of volume fraction of water plus surfactant, phi approximately 0.02-0.52) was used, which allowed us to extract information about droplet sizes, diffusion coefficients, interactions, and polydispersity from experimental data. We report the observation of two diffusive relaxation modes in a concentrated microemulsion (0.20 < phi < 0.5) due to density (collective diffusion) and concentration or polydispersity (self-diffusion) fluctuations. Below this concentration it was difficult to resolve two exponentials unambiguously, and in this case one apparent relaxation mode was observed. It was found that for a given composition self-diffusion is more pronounced in apparent relaxation mode for a shorter chain length alkane. The concentration dependence of these diffusion coefficients reflects the effect of hard sphere and the supplementary attractive interactions. It was observed that the attractive part becomes more pronounced in the case of a large alkane chain oil at a given temperature. This explains the shift of the region of microemulsion stability to lower temperatures for higher chain length alkanes. Increase in hydrodynamic radius, Rh, obtained from the diffusion coefficient extrapolated to infinite dilution was observed with increase of alkane chain length. The polydispersity in microemulsion systems is dynamic in origin. Results indicate that the time scale for local polydispersity fluctuations is at least 3 orders of magnitude longer than the estimated time between droplet collisions.     

Small angle X-ray scattering investigations on vinylpyridine-butadiene rubbers

Methylvinylpyridine-butadiene copolymers vulcanized by various α, ω dibromo-alkanes were investigated by X-ray small angle scattering. In these systems, employing the Hosemann-Joerchel analysis, only one cluster dimension of about 45 Å has been found with a polydispersity not exceeding 35 per cent. Stretched rubbers show interference effects both in the meridional and equatorial directions.     

Synthesis and characterization of non-spherical gold colloids in block-copolymer micelles

Gold colloids of well-defined shape, size and polydispersity are synthesized by heterogeneous reduction of gold salts in amphiphilic block-copolymer micelles. The resulting hybride systems consisting of the noble metal colloid and the stabilizing shell of block-copolymers are characterized by electron microscopy and small-angle x-ray scattering. Opposite to classical noble metal colloids, the block-copolymer stabilized systems exhibit an extra-ordinary high colloidal stability which makes analysis and sample handling very simple.It turned out that non-spherical metal colloids are formed in the early stages of the reduction process where the amount of gold exceeds the reduction agent. At later stages, the colloids break up to globular subunits again. The reaction period of colloidal anisometry is reflected in a violet color of the reaction solution, which is caused by a typical structured, double-resonance plasmon band.Small-and wide-angle x-ray scattering reveal a quantitative measure for the size, shape, anisometry and local order in each step of the reduction process. It is suggested that the primary aubergine-shaped particles consist of globular primary particles which are glued together via non reacted gold salt. This observation gives rise to some interesting possibilities of the supramolecular handling of colloids.     

Small angle scattering and zeta potential of liposomes loaded with octa(carboranyl)porphyrazine

In this work, the physicochemical characterization of liposomes loaded with a newly synthesized carboranyl porphyrazine (H2HECASPz) is described. This molecule represents a potential drug for different anticancer therapies, such as boron neutron capture therapy and for photodynamic therapy or photothermal therapy. Different loading methods and different lipid mixtures were tested. The corresponding loaded vectors were studied by small angle scattering, light scattering, and zeta potential. The combined analysis of structural data at various lengths of scales and the measurement of the surface charge allowed us to obtain a detailed characterization of the investigated systems. The mechanisms underlying the onset of differences in relevant physicochemical parameters (size, polydispersity, and charge) were also critically discussed.