Scattering of light by disordered spherulites

The scattering of light by a two-dimensional spherulite of radius R is calculated when there is disorder of optic axis orientation with respect to the radius. Special cases are considered when (1) the disorder occurs in the radial direction only, (2) the disorder occurs in the angular direction only, (3) there is combined radial and angular disorder, and (4) the optic axis makes a constant angle with the radius but there is disorder in the twist angle about the axis. In all of these calculations, a correlation function for disorder is defined and the scattering pattern depends on the ratio of the associated correlation distance to the size of the spherulite. With decreasing correlation distance, the azimuthal dependence of the scattering becomes less and there is a change in the variation of scattered intensity with scattering angles in a manner dependent upon the type of disorder.     

Scattering of light by deformed three-dimensional spherulites

A theoretical calculation of the H_v light-scattering patterns for deformed three-dimensional spherulites is presented. Affine deformation is assumed. The optic axis of the scattering element is allowed to lie at an arbitrary angle ß to the radius which is permitted to change in the course of the deformation in a manner that may depend upon the angular location in the spherulite. The consequences of twisting of the optic axis about the spherulite radius are also explored.     

Scattering of light by deformed disordered spherulites

The change in the light-scattering patterns upon deforming two-dimensional disordered spherulites is shown to arise from four effects occurring upon stretching: (1) the change in shape of the spherulite, (2) the change in average orientation of the optic axes of the scattering volume elements, (3) the change in deviation of the optic axis orientation angle from its average value, and (4) the change in the distance over which this deviation is correlated. The effects of these contributions upon the experimental scattering patterns are analyzed.     

Multiple-order light scattering from ringed spherulites

Light scattering from polymer films containing ringed spherulites may show multiple-order intensity maxima directly related to the period of the banded structure. Calculations based on spherulite models where the angle of twist of lamellae ω varies linearly with radial distance predict only first-order peaks. If the variation of ω is nonlinear, even though the ring spacing remains constant, higher-order intensity maxima will result. Other sources of multiple-order scattering are considered. It is concluded that for polyethylene the presence of multiple-order scattering is due to non-uniform twisting of the lamellae.     

Light scattering from assemblies of spherulites

A general equation describing the small-angle H_v light-scattering intensity for a system of N undeformed spherulites located at random within the sample and taking into account the truncation and interference effects is given. Scattering contour plots or radial scans are reported for various arrangements of the N spherulites. The results show that the interference effect may explain the speckled appearance of the experimental patterns. Moreover, the interference and truncation effects (for the special cases where truncation is considered here) do not seem to shift the position of the maximum scattering angle of the cloverleaf pattern as calculated from the single spherulite theory. Finally, the calculations show that the truncation effect increases the relative intensity of the pattern at large and low scattering angles and at azimuthal angles 0 and 90°C, as compared with the intensity at the position of maximum scattering angle.     

Light scattering from tilted two-dimensional spherulites

The theory of small-angle light scattering was developed for oblique incidence of the light beam on the surface of a two-dimensional spherulite. Results of the theory were compared with previously reported results of light scattering from two-dimensional and three-dimensional spherulites for normal incidence, and with some experimental patterns. The comparisons suggest that the scattering intensity distributions of two-dimensional spherulites deviate from those of three-dimensional spherulites when the sample surface is tilted with respect to the propagation direction of the incident beam, although they are almost identical when the sample surface is normal to the incident beam. Observation of the change of scattered intensity distributions upon tilting the samples thus provides a method of distinguishing between two-dimensional and three-dimensional spherulites. Moreover, this observation makes it possible to determine the degree of planar orientation of the optic axes of optically anisotropic scattering elements within two-dimensional spherulites. The calculations were carried out for special cases of two-dimensional spherulites with the optic axis orientation confined to the two-dimensional plane and randomly or helicoidally rotated around the spherulite radii.     

A lattice theory of light scattering from disordered spherulites

A lattice theory of orientational disorder in two-dimensional spherulites is developed in which the orientation direction of the optic axis in lattice cells is allowed statistically to deviate from its mean value in a manner correlated with the orientation in neighboring cells. The H_v light scattering patterns arising from such disordered spherulites deviate from the patterns for perfect spherulites in that there is excess intensity at both small and large scattering angles and the intensity at the maximum is lower. A comparison of the calculated scattering angular dependence with that which is experimentally measured permits assignment of values of correlation parameters. A consequence of this disorder is that the spherulite birefringence is reduced below that calculated on the basis of perfect crystalline orientation in agreement with experiment.     

Scattering of light by disordered spherulites. II. Effect of disorder in the magnitude of the anisotropy

The contribution to the disorder scattering by imperfect spherulites resulting from fluctuations in the magnitude of the anisotropy is analyzed for two-dimensional spherulites. The fluctuations are described in terms of a parameter characterizing the meansquare amplitude of the fluctuation and a correlation function describing the distance over which the correlation occurs. Cases considered are those where the correlation depends on either the radial or the angular separation of the scattering volume elements. As with the case of disorder in orientation, one finds that disorder in anisotropy may result in a nonzero value of intensity at μ = 0° and 90°, a decrease in the higher-order variation of scattered intensity with θ, and an increase in the intensity of scattering at higher values of θ over that for a perfect spherulite. In addition, disorder in the angular direction leads to an increase in the scattered intensity at small values of θ as compared with the zero intensity of scattering from a perfect spherulite at θ = 0°.     

Orientation-independent charge transport in single spherulites from solution-processed organic semiconductors

Due to the rapidity of morphological development during deposition, solution-processed organic semiconductor thin films exist in semicrystalline or polycrystalline states, incorporating a high degree of local variations in molecular orientation compared to their single-crystal counterparts. Spherulites, a common crystalline superstructure found in these systems, for example, incorporate a large distribution of molecular orientations about the radial axis to maintain their space-filling growth habit. Here, we aim to determine how this distribution of molecular orientations influences charge transport by fabricating arrays of devices on single spherulites. Given that the orientation distribution that is present about the radial axis mandates the presence of low-angle grain boundaries within single spherulites, we find intraspherulitic charge transport to be independent of the general direction of π-stacking; organic field-effect transistors exhibit comparable mobilities regardless of how their channels are oriented with respect to the general π-stacking direction.     

Cracking in polylactide spherulites

A polylactide of high optical purity was crystallized between 100 and 140 °C, in‐between two glass slides, and its morphology was investigated by polarizing optical microscopy, scanning electron microscopy, and atomic force microscopy, during subsequent heating and cooling cycles between ?15 °C and the crystallization temperature. It was found that dark circular rings show up on cooling on top of the spherulites and represent cracks of about 300 nm in width. This phenomenon is completely reversible, and the heating–cooling curves are centered at about 56 °C, which coincide with the T_g of polylactide. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 3308–3315, 2005     

Scattering from Ferrogels

Summary: This paper introduces a simple mean‐field theory for gels with magnetic properties. The main emphasis in this paper is the discussion of the scattering properties. Therefore, a simple model is introduced, such that the chains carry magnetic moments along their main axis. Naturally these magnetic moments interact sensitively to theory orientation with each other. Thus a distinguished interplay between chain and dipole orientation induces strong coupling between the macroscopic deformation and magnetic properties. These effects can be experimentally demonstrated with scattering methods. Here some mean‐field expressions for the structure factors are predicted and discussed in some detail.

The configurations adopted by polymers possessing a magnetic moment and undergoing dipole‐dipole interactions.     

Small-angle light scattering by random assemblies of incomplete spherulites. II. Truncated spherulites

Previous theoretical calculations of the scattering from spherulites are for isolated complete spheres, whereas most spherulitic polymer samples contain truncated spherulites as a result of impingement by other spherulites. The effect of such truncations on the scattering patterns for two-dimensional spherulites is explored as a function of the size, number and location of the truncations. The scattering of severely truncated spherulites is modified, particularly with regard to the enhancement of the H_V scattering at small angles. However, reasonable amounts of truncation corresponding to experimentally observed structures do not produce appreciable modification of the pattern so that the neglect of truncation will not lead to appreciable error in the estimated spherulite size from light scattering.     

Deformation of elastomeric polyolefin spherulites

The ethylene‐octene block copolymers in this study consist of long crystallizable sequences with low comonomer content alternating with rubbery amorphous blocks with high comonomer content. The crystallizable blocks form lamellae that organize into space‐filling spherulites even when the fraction of crystallizable block is so low that the crystallinity is only 7%. These unusual spherulites are highly elastic and recover from strains as high as 300%. This new class of thermoplastic elastomers is fundamentally different from conventional elastomeric olefin copolymers that depend on isolated, fringed micellar‐like crystals to provide the junctions for the elastomeric network. The elastomeric block copolymers are shown to be unique in that a hierarchical organization of space‐filling lamellar spherulites provides the junctions for the elastomeric network. The deformation of the elastic spherulites is readily studied with small angle light scattering, wide angle X‐ray diffractograms, and atomic force microscopy. At strains in excess of 300%, the spherulites break up into a fibrillar structure following lamellar deformation processes that are similar to those established for high density ethylenic polymers. The crystalline transformation produces a stiffer elastomer that exhibits complete recovery on subsequent loadings. Similar experiments on elastomeric random ethylene‐octene copolymers where fringed micellar crystals provide the physical crosslinks that connect the rubbery, amorphous chain segments reveal significant differences. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 1313–1330, 2009     

Formation of ringed spherulites in polyethylenes

The effects of molecular weight, molecular weight distribution, crystallization temperature, quenching medium, and sample preparation on the formation of ringed spherulites in linear polyethylenes were studied by polarized light microscopy and small-angle light scattering. When the samples were crystallized at a predetermined temperature, ringed spherulites were formed over a narrow range of temperature and molecular weight with both fractionated and unfractionated polymer samples. Quenching the samples in air at room temperature considerably extended the range of molecular weights for the formation of ringed spherulites. Minor modification of an airquench method further extended the range and yielded better-defined structures. The results are interpreted in terms of the anisotropy of the melt, the thermal conductivity of the quenching medium, and the shear stress applied during the crystallization process. That highly specific conditions are necessary for spherulite formation, of both the conventional and ringed type, is a major conclusion of this study.     

Deformation of polybutene-1 spherulites

The deformation of fresh and aged polybutene-1 spherulitic samples has been investigated by microscopic observation, interferometry, studying macroscopic and spherulitic birefringence changes, and study of light-scattering patterns. The spherulite deformation is not affine, the microscopic deformation ratio being less than the macroscopic deformation ratio of the sample and greater in the equatorial regions of the spherulite than in the polar regions. The deviation from affine deformation is less for fresh spherulites than for the aged, where void formation occurs in the equatorial part of the spherulite. This gives rise to large scattering by this part of the spherulite and to form birefringence. The spherulite birefringence and its change with elongation is dependent upon the degree of aging of the sample. The spherulite birefringence is more negative for the aged sample. In the polar regions of the spherulite, this negative birefringence decreases and turns positive at higher elongations, characteristic of a reorientation of the crystals with their optic axes turning from being perpendicular to parallel to the spherulite radius. The spherulite birefringence in the equatorial direction becomes somewhat more negative on stretching a fresh sample but less negative on stretching an aged one. Spherulite distortion and orientation changes are apparent from the light-scattering patterns of films possessing small spherulites. The changes in V_v and H_v scattering patterns upon stretch are different for the fresh and aged samples. The V_v patterns of the fresh samples decrease in intensity with time after stretching a fresh sample with the H_v patterns do not.     

Scattering from polymer-like micelles

Polymer-like micelles are analogs to polymer solutions and provide an exciting class of materials for both applications and fundamental understanding of polyelectrolyte systems. Small angle neutron and X-ray scattering have been key to the characterization of these materials from the first observations of linear micelle growth. As new materials are developed, these techniques continue to be utilized and combined with other analytical tools to characterize the length and time scales of polymer-like micelle behavior. Recent reports on the use of small-angle scattering to characterize polymer-like and wormlike micelles are reviewed, with focus on new materials, improvements in analytical approaches and anisotropic structures.     

Mechanism for twisting in polyethylene spherulites

A model is proposed to account for the regular twisting of radial ribbons in certain polymer spherulites. The model assumes that the ribbons have crystallographically regular fold surfaces and that growth of the ribbons is nucleation-controlled. The model leads directly to a possible mechanism of spherulitic growth in some polymers.     

Terraces on banded spherulites of polyhydroxyalkanoates

On banded spherulites of poly(3‐hydroxybutyrate) and its copolymers, fine circles were observed between either cross‐polarizers or without polarizers. Atomic force microscopy was applied to confirm that these circles are terraces with heights up to several hundred nanometers rather than cracks. Real‐time observation demonstrated that the terrace forms at the front of the growing spherulites just before or exactly when two spherulites impinge on each other. Terraces were observed on the spherulites crystallized from melt confined between glass or polyimide slides rather than poly(ethylene terephthalate) slides. The formation of the terraces may have resulted from instability of the moving boundary of the melt film confined between the spherulite surface and cover slide. Wettability of the substrate played an important role in the formation of the terraces. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 2128–2134, 2003     

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.