Determination of spherulite size distribution by small-angle light scattering

The theoretical backgrounds of the method of spherulite size distribution determination, on the basis of the small-angle light scattering technique, are presented. Special corrections of experimental data are introduced to achieve great accuracy of the determination of scattered light intensity distribution. Experiments have been done on low-density polyethylene samples obtained in various crystallization conditions. For these samples, the stepwise distributions of spherulite size have been determined.     

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 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 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 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 x-ray scattering from solution of rodlike particles with length distribution

The scattering function of rods with a constant radius, 8 Å, and a length distribution of the Schulz–Zimm type was calculated on a computer as a model of helical synthetic polypeptide. The influence of length and length distribution on the scattered intensity of small-angle x-ray scattering was clarified. As length grows and length distribution broadens it is difficult to obtain reliable values of molecular weight and radius of gyration from a Zimm plot. The influence of length distribution on the mass per unit length, M_q, and the radius of gyration of the cross section, 〈S〉^1/2, from the Guinier plot of the cross-sectional factor decreases as the length increases, and reliable values of M_q and 〈S〉^1/2 can be obtained even for rods with wide distribution for rods more than 600 Å long. In particular, it is pointed out that the value of 〈S〉^1/2 is little influenced by length and length distribution.     

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.     

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 Hv light scattering from deformed spherulites with orientational fluctuation of optical axes

Mathematical evaluation was done for small-angle light scattering from disordered spherulites under Hv polarization conditions. The calculation was carried out for a two-dimensional deformed spherulite whose major optical axes are oriented at 0 or 45° with respect to the radial direction. The calculated results were compared with the scattering patterns observed for polypropylene (PP) spherulites, whose optical axes are oriented parallel to the radial direction, and poly(butylene terephthalate) (PBT) spherulites, whose optical axes are oriented at 45° with respect to the radial direction. The degree of disorder for PBT was much larger than that for PP. By selecting a parameter associated with the degree of disorder of the optical axes with respect to the radial direction, the patterns calculated as a function of draw ratios were in good agreement with the observed patterns, which changed from four leaves to streaks extended in the horizontal direction. Through a series of observed and calculated patterns, it turns out that an increase in the disorder under the deformation process occurs drastically even for perfect spherulites in an undeformed state.     

Effect of disorder in the twist angle on light scattering by a three-dimensional spherulite

Light scattering patterns are calculated for imperfect three-dimensional spherulites with fluctuations in the twist angle. The fluctuations are described in terms of a parameter characterizing the distance correlation function. Cases are considered in which (i) the principal axis of the scattering element makes a constant angle with the radius but there is disorder in the twist angle about the axis, and (ii) there is combined twist disorder and orientation disorder of the scattering elements. Calculations suggest that the disorder in the twist angle may lead to a decrease in the higher-order variation of scattered intensity with scattering angle and deviation from the four-leaf-clover-type scattering characteristic of a perfect spherulite at lower scattering angles. On the other hand, disorder in orientation has little effect on the scattering pattern.     

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 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 light scattering study of quasi-two-dimensional structures of magnetic particles

We studied quasi-two-dimensional magnetic domain structures using a small-angle light scattering technique. The structures are formed when magnetic particles in an aqueous suspension between two parallel glass plates are assembled and oriented in a magnetic field H perpendicular to the plates. Three different structure regions can be identified, and they are bounded by two critical field lines H_c1 (φ) and H_c2(φ). Below H_c1 (φ) the system consists of single particles; between H_c1 (φ) and H_C2 (φ), the particles assemble into elongated disordered domains; and above H_c2 (φ), the domains develop a strong in-plane correlation. In the strong field limit H H_c2, the structure factor of the magnetic fluid resembles that of a two-dimensional liquid. However, long lasting metastability and hysteresis observed in the experiment suggest that the system is essentially locked in a glassy state.     

Small-angle neutron scattering from a hexagonal phase under shear

Summary The shear orientation of a micellar hexagonal liquid crystalline phase was investigated by small-angle neutron scattering. The hexagonal phase in the quiescent state showed a symmetrical scattering pattern typical of a polydomain structure. Enhanced scattering along the flow direction was observed during shear and the anisotropy of scattering intensity became stronger with increasing shear rate. The anisotropic scattering pattern corresponds to an orientation perpendicular to the flow direction and can be interpreted as a log-rolling state. The oriented sample did not relax after cessation of shear. The results from small-angle neutron scattering confirm data obtained previously from rheo-small angle light scattering measurements and are discussed in comparison to shear alignment of lyotropic liquid crystalline polymer solutions.     

Small-angle light scattering and crystallization processes in solid polymer films

Information, not previously measurable, about the internal crystallization processes occurring within spherulitic films can now be obtained by combining refractive index, birefringence, and small-angle light-scattering measurements. The surroundings of a spherulite in a solid film are composed of both the adjacent spherulites and their interstices (boundaries), and it is shown that different crystallization processes can occur in each of these regions. Upon annealing a quenched isotactic polypropylene film, the rate of crystallization within the interstices of the space-filling spherulites is observed to be greater than the rate of crystallization within the spherulites themselves.     

Small-angle light scattering from optically anisotropic spheres and disks. Theory and experimental verification

The small-angle light scattering (SALS) theory for optically anisotropic spheres and disks is examined in depth. An error is found in the existing sphere equations. The correct form of the equations is identified and then experimentally verified for dilute starch suspensions. Increased concentrations and solid films of starch granules are used to identify the effect of concentration on the scattering envelope. Spherulitic films of isotactic polypropylene, isotactic polystyrene, nylon 610, PET, and nylon 66 are then used to examine different aspects of the SALS theory. Experimental observations are found to agree with the predictions of the correct SALS equations. Disk theory is interrogated and correlated with predictions for spheres. It is found that the predicted patterns from spheres and disks are very similar under identical optical conditions, in contradiction to earlier predictions. A method is developed for identifying the optical sign of spherulites too small to be seen in the optical microscope. This study constitutes a comprehensive examination of SALS theory and includes many other aspects of the phenomena. A catalogue of theoretical V_v SALS patterns from spheres and disks is also included.     

Small-angle neutron scattering from two-phase polymeric systems

A small-angle neutron scattering method is presented for the determination of single-chain scattering functions in heterogeneous two-phase polymeric materials such as incompatible polymer blends and microphase-separated diblock copolymers. Development of the method makes use of both the discrete and continuum approaches to scattering theory. In this fashion it is shown that extraction of the single-chain scattering function requires only two experimental observations, regardless of the compressibility of the system. These experiments consist of small-angle neutron scattering profiles from two samples: an unlabeled two-phase material and a similar two-phase material in which a portion of the molecules in one phase have been replaced by molecules which are contrast labeled by isotopic deuterium substitution. Anticipated difficulties associated with actual application of the procedure and requirements of the necessary experimental samples are discussed.