Small-angle polarized light scattering from amorphous spheres

Small-angle light-scattering measurements have been made using micron-diameter isotropic and spherical polymer latex particles placed between crossed polarizers. Four-leaf clover patterns are obtained, reminiscent of those commonly found for spherical birefringent scatterers. The experimental results compare closely with predictions of Mie scattering theory for isotropic spheres.     

Small-angle scattering of polarized light. I. comparison of theoretical predictions for isotropic and anisotropic spheres

This paper describes small angle light scattering (SALS) calculations and compares the results for isotropic and anisotropic spheres. For isotropic spheres the exact Mie theory is compared with the Rayleigh-Gans-Debye (RGD) and the anomalous diffraction (AD) approximations in order to study the validity of the approximations. For anisotropic spheres the RGD is compared with the AD approximation. A study of SALS in the RGD approximation shows that the first-order intensity maximum corresponding to a lobe of the four-leaf clover pattern is not only a function of the sphere diameter but also depends on the isotropic and the anisotropic refractive index of the sphere. The practical implications of this are discussed.     

Small-angle scattering of circularly polarized light from an anisotropic sphere

A theory of the small-angle scattering of circularly polarized light from an anisotropic sphere has been derived. The validity of the theory has been verified, and a relationship between the structural information thus obtained and the structural information obtained with linearly polarized light has been demonstrated.     

Small-angle scattering of polarized light. II. experiments with isotropic spheres

New experimental measurements are reported of small-angle polarized and depolarized light scattering from almost monodisperse isotropic, spherical particle, polystyrene latexes. The shape and intensity distribution of the scattering patterns is shown to compare closely with the calculated patterns based on the exact Mie scattering theory from a single sphere.     

Small-angle light scattering from an anisotropic sphere: The mueller matrix approach

The explicit form of the Mueller scattering matrix, which characterizes the small-angle light scattering from an anisotropic sphere when the requirements of the Rayleigh-Gans-Debye (RGD) approximation are fulfilled, contains all information obtainable about the RGD scattering from an anisotropic sphere taken as a model for a spherulite. A comparison of angular dependences of single matrix elements for the Lorenz-Mie sphere, the Rayleigh particle, and the simplified form of the presented matrix (taking a sphere without inherent anisotropy, i.e., Δn = 0) shows very good agreement within the limits of RGD approximations. The polarized small-angle light scattering intensities Hυ and Vυ are combinations of the single matrix elements. Their explicit form is in accord with the expressions for Hυ and Vυ intensities recently rederived from a 2 × 2 amplitude scattering matrix. It has been shown that the angular dependence of matrix elements is determined by the (n? — 1)/Δn parameter, where n — is the mean refractive index andδ n is the anisotropy, both measured relative to the surrounding medium. The expressions for Hυ and Vυ intensities derived by Stein and Rhodes fail for a sphere without inherenet anisotropy ( δ n = 0); and the commonly used procedure of size determination from a maximum of Hυ intensity has limited validity (it holds only approximately under the condition of a small phase shift and small (n?-1)/ δ n). Further theoretical work must be done to understand and construct scattering models for situations where the RGD approximately is inappropriate.     

Small-angle scattering of polarized light. IV. Isotropic to birefringent transition for spheres

The Rayleigh-Gans-Debye approximation is used to investigate the small-angle, cross-polarized light scattering by a sphere of radius a, birefringence Δμ and relative index μ. If θ_max is the polar scattering angle of the intensity maxima, the quantity Û_max = 4πa/λ × sin (θ_max/2) behaves in two different ways according to the signs of Δμ and μ ? 1. When Δμ > 0, μ > 1 or Δμ < 0, μ < 1, then Û_max varies from 2.8 to 4.1 as Δμ increases from zero. If Δμ < 0, μ > 1 or Δμ > 0, μ < 1, then Û_max goes from 2.8 to about 6, thereafter decreasing to 4.1. Another interesting result is that the value of Û_max for a highly briefringent sphere is 4.1 only for large diameters. It decreases to 4.0 when the diameter decreases.     

Dynamic light scattering by optically anisotropic colloidal particles in polyacrylamide gels

The translational and rotational motions of optically anisotropic spherical particles embedded in cross-linked polyacrylamide gels is studied by dynamic light scattering. The particles are liquid crystal droplets solidified in the nematic phase. The amount of cross linkers is varied to cross the sol-gel transition where the system becomes nonergodic for both translational and rotational diffusion modes of the probes. The translational and rotational dynamic correlation functions are obtained by measuring the intensity correlation function between crossed polarizers in the parallel and perpendicular geometries. Data from nonergodic systems is analyzed using an extension, to include rotations, of the method of Pusey and van Megen [Physica A 157, 705 (1989)]. Both diffusion modes are observed to be arrested as the rigidity of the gel increases.     

Small-angle light scattering from a distribution of spherulite sizes

Depolarized small-angle light scattering from spherulites in semicrystalline polymers gives rise to a characteristic cloverleaf pattern. For scattering from a single spherulite, the position of the maximum in scattered intensity is readily related to the spherulite radius. For a distribution of spherulites, the maximum should be related to some characteristic measure of the distribution. It is shown for a wide variety of distributions that this characteristic radius is a ratio of high moments of the size distribution, specifically R* ≈ 〈R⁷〉/〈R⁶〉. The shape of the light-scattering profile should in principal be related to the nature of the spherulite distribution. Calculations of scattering profiles from a variety of distributions fail to demonstrate this, owing to the strong dependence of scattering power on spherulite size. Exceptions are noted for the case of certain bimodal distributions.     

Small-angle polarized light scattering by spherulites

The light-scattering matrix for a three-dimensional spherulite is derived within the Ray-leigh-Gans-Debye light-scattering approximation. New expressions for the polarized, small-angle light-scattering intensities I_VV and I_VH are derived from the scattering matrix. These expressions are compared with the I_VV and I_VH expressions derived for a spherulite by Stein and Rhodes. For the case of a weakly anisotropic spherulite having an average refractive index mismatch with its surroundings, the two sets of expressions predict different I_VV and I_VH intensities. In particular, our expressions show that the I_VVand I_VH patterns usually attributed to the spherulitic anisotropy and crystallinity are also predicted for an isotropic sphere. This is in accord with recent experiments.     

Small-angle light scattering from hedrites,ovoids, and spiral ovoids. II. Experimental

Procedures for obtaining hedrites, ovoids, and spiral ovoids in polyoxymethylene are described. Results of morphological studies on these structures by optical and electron microscopy are presented. Small-angle light scattering studies on single isolated supermolecular structures are described. The V_v and H_v diffraction patterns obtained experimentally are discussed in the light of theoretical results of Part I of this study.     

Small-angle light scattering from thin polymer films: Multiple scattering from samples with rodlike morphology

Hartel's theory for multiple scattering has been generalized to the case of small-angle light scattering (SALS) by polymers having a roadlike morphology. It is shown that multiple scattering tends to make the scattering tends to make the scattering patterns more diffuse and leads to an underestimation of the size of the units (rods) measured from such patterns. The error induced by neglecting multiple scattering has been estimated at 10% for a transmittance of 75% and at 22% for a transmittance of 50%. A correction method based on Hartel's procedure is suggested.     

Small-angle light scattering from hedrites,ovoids, and spiral ovoids. I. Theoretical considerations

Taking into consideration results of our microscopy studies and data given in the literature on the structure of hedrites, ovoids, and spiral ovoids, we have developed optical models for these morphological entities. These models are applied to calculations of small-angle scattered light intensity distributions for these structural elements. Analytic formulae computed for the V_v and H_v scattering patterns contain terms dependent only on azimuthal angle μ and vertical angle θ. For hedrites and ovoids, equations are derived relating the characteristic points of scattered light intensity patterns to structural parameters. The corresponding diagrams of light intensity distributions are given.     

Small-angle x-ray scattering of semicrystalline polymers. II. Analysis of experimental scattering curves

The small-angle x-ray scattering (SAXS) patterns of a number of linear polyethylene (PE) and polyoxymethylene (POM) samples have been measured and compared to the intensity functions of one-dimensional paracrystalline lattices. It was found that the ratio of the angular positions of the second and first scattering maxima (θ₂/θ₁) is generally less than or equal to 2.0, implying that the paracrystalline lattice statistics are symmetric or moderately skewed to larger periods. The Bragg spacing (“long period”) of such samples is within 3% of the identity period of the macrolattice. With quenched POM the ratio θ₂/θ₁ is substantially larger than 2.0, which indicates either extremely asymmetric lattice statistics or coexisting structures within the material. From consideration of the reduced widths of the first scattering maxima, it was found that some broadening is present in addition to that from the paracrystallinity. This excess broadening could result from a finite lattice length of ～1000 Å. The need for careful experimental technique for obtaining the actual position of the scattering maximum is emphasized. In addition, it is demonstrated that the scattering curve and the correlation function of the system yield essentially the same apparent structural periods.     

Light scattering from polymer films having optically anisotropic rodlike texture. III. Inter-rod interference

Effects of interparticle interference on depolarized light scattering from polymer films having nonspherulitic (rodlike, disklike, or fibrous) crystalline superstructures are explored, both theoretically and experimentally. Discrepancies between the observed “rodlike scattering” and the predicted scattering based on “single-rod” theory are pointed out and accounted for in terms of various types of aggregation models for polymeric solids and solutions.     

Small-angle X-ray scattering and light scattering on lysozyme and sodium glycocholate micelles

Small-angle X-ray scattering (SAXS) together with static (SLS) and dynamic light scattering (DLS) measurements were carried out on aqueous solutions of lysozyme (LY) and of the ionic biological detergent sodium glycocholate (NaGC). Apparent diffusion coefficients (D app), excess Rayleigh ratio, and SAXS spectra were measured for 0.1 M NaGC solutions at different ionic strengths (0.05-0.30 M NaCl). The same data were collected for LY in sodium acetate buffer 50 mM without and with 92 mM NaCl as a function of protein concentration (10-80 g L(-1)). A correlated analysis of SLS data and SAXS spectra was first tested on the LY samples and then extended to the interpretation of the NaGC data to infer information on particle structure and interaction potential. A hard-core (HC) interaction shell of uniform thickness, a screened Coulomb potential of the electric double layer (EDL) or the complete DLVO potential were alternatively used to represent the long-range tail of the interaction potential. Whenever an essentially repulsive tail is expected, all the representations give reasonable results, but the data analysis does not allow the discrimination between the oblate and the prolate symmetries of the NaGC aggregates. The DLVO model allows the interpretation of the data even when the attractive component determines the tail character. With this model an overall fit of the micelle data at all the NaCl concentrations was successfully performed by assuming a simple spherical symmetry of the micelles and invariant values of their ionization degree and Hamaker constant, thus considering just the screening effect of the added electrolyte. Whatever model is used, the results point out that the aggregates are quite hydrated (26-38 water molecules per monomer) and very slightly grow by increasing the NaCl concentration. When spherical symmetry is assumed the aggregate radii for all the samples fall in the range 15-16 A. From the SAXS and SLS, best fitting geometrical parameters, and interparticle structure factor, a D app value was calculated for each sample. An excellent consistence is achieved for LY results. On the contrary, calculated D app values systematically lower than the experimental values are always obtained for the NaGC micelles. Micelle polydispersity and internal dynamics seem to be the most probable reasons of the bad agreement.     

Small-angle light scattering from thin polymer films: Multiple scattering from Debye–Bueche scatterers

An extension of Hartel's theory for multiple scattering has been applied to the case of small-angle light scattering from polymer films with random two-phase morphology. The scattering is treated in terms of Debye–Bueche theory from which values of the correlation length, an average phase size, and mean-square fluctuation in polarizability are determined. It is shown that multiple scattering leads to a reduction in the angular dependence of scattering due to an enhancement of scattering at high angles. This leads to an underestimate of the correlation length and an overestimate of the magnitude of the mean-square polarizability fluctuation.