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 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°.     

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.     

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.     

Scattering from truncated spherulites

A general two-dimensional theory is derived to explain the light scattering from truncated spherulites. The severity of the truncation is expressed by a statistical parameter σ²/ā² which is the ratio of the variance σ² of the size of the spherulite to the square of its average size ā. The H_v light-scattering patterns are calculated for different values of the truncation parameter. It is observed that the truncation decreases the position of maximum scattering intensity of the pattern. It also increases the scattering intensity at small and large angles, but reduces it at intermediate angles. For a spherulitic polyethylene sample, the truncation parameter is found to equal 0.100 ± 0.030 as measured microscopically. The theory can also be used to calculate light-scattering patterns from row-nucleated spherulites. If it is assumed that the interference effect averages out to zero when a large number of spherulites is involved, a single “sliced” spherulite model can be used. Then, the scattering intensity per unit area decreases as the “slice” becomes very thin.     

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 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.     

Scattering of light from deformed regions surrounding voids and inclusions in high polymers

The H_v cloverleaf scattering pattern sometimes found for amorphous polymers can arise from the birefringent region caused by the strain surrounding a void or an inclusion in a polymer. The form of such a pattern may be theoretically explained by extension of a theory due to Goldstein.     

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.     

Scattering of two-dimensional analogs of disordered lamellae

Optical Fourier transformation is used to study some two-dimensional (2D) analogs of disordered lamellae microstructure. A connected network of thick lines is used as the 2D representation of the disordered lamellar structures which have been observed in surfactant solutions. It is shown that, in these amorphous structures, the position of the scattering peak is still given by the underlying lattice and the observed scattering can be decomposed as a fair approximation, as a product of a local form factor and a structure factor due to the correlations of the underlying lattice.     

Small-angle light scattering by random assemblies of incomplete spherulites. I. Sheaflike textures

The calculation of the scattering from a sheaflike sector of a two-dimensional spherulite has been carried out as a function of the apex angle of the sector. It is found that while for a complete spherulite the H_v scattered intensity is zero at zero scattering angle, there is an increasing intensity of scattering at 0° as the sector angle narrows. For very small values of the sector angle, the scattering becomes similar to that of a rod, with the exception that a scattering maximum is still seen at an angle close to that at which the spherulite scattering maximum occurs. The predictions of the model compare favorably with the scattering patterns observed for polymers in early stages of spherulitic growth.     

共振光散射光谱法测定痕量氯离子

定量测定氯离子的传统重量法、滴定法、硫氰酸汞比色法、比浊法的灵敏度和精密度均不够理想。共振光散射光谱法(Resonance Light Scattering RlS)是一种新的分子光谱技术,灵敏度高,可用普通的荧光分光光度计测量。本文用它测定痕量氯离子,结果令人满意。     

Small-angle light scattering by random assemblies of incomplete spherulites. Systems with partial degree of volume filling

A theory is presented to account for the effect of the impingement of growing spherulites on their H_v small-angle light scattering patterns. The theory is developed on the basis of results of computer-simulated two-dimensional spherulite growth and calculated scattered intensities. The impingement produces a lowering of the intensity of the scattering maximum and the diminishing of the overall sharpness of the scattering peak. The extent of these effects increases with area fraction of spherulites. A procedure is suggested for determining correction factors that may be applied to intensity data obtained during the course of spherulite crystallization. An interpretation is made of the type of average spherulite size determined from the scattering angle of maximum intensity.     

Scattering of light by films having nonrandom orientation fluctuations. II

The theory for scattering of light from films in which the orientation correlation between two scattering elements depends upon the angle β between the optic axes and the vector connecting the two elements is extended. A delta-function type dependence is assumed in which the correlation is strongest when β equals some preferred value β₀. Calculated results of scattered intensities are shown to be similar to experimental observations.     

Scattering of light from assemblies of oriented rods

The scattering patterns are calculated from anisotropic rods having an arbitrary angle of the polarizability axis with respect to the rod axis. The effect of a distribution of orientation of the rods is explored. This leads to a change in scattering patterns produced by orienting the rods upon stretching the sample. The scattering patterns are affected by the relationship between the refractive indexes of the rods and that of the surroundings. The influence of the optic axis orientation angle changing upon orienting the rods is explored. Theoretical patterns are compared with experimental ones obtained upon stretching polytetrafluoroethylene (PTFE) films.     

Studying banded spherulites in HDPE by electron microscopy

HDPE is a semi-crystalline thermoplastic polymer, with remarkable physical properties such as high chemical resistivity, high impact strength, and high modulus. Compared to the other semi-crystalline polymers, HDPE mostly possesses a high crystallinity, due to which, it exhibits a unique combination of mechanical and chemical resistance properties. In the present work, we have characterized the crystalline spherulites of neat and formulated HDPE compositions thoroughly characterized by different electron microscopy techniques such as scanning electron microscopy (SEM) and transmission electron microscopy (TEM). One of the critical steps to obtaining well-resolved SEM images is the sample preparation that typically involves the etching process to elucidate the crystalline spherulites. Though such traditional methodology can effectively be used for neat HDPE, it leads to the creation of undesirable experimental artifacts when used to investigate formulated HDPE compositions. An alternate TEM-based method provides clear images without any artifacts, apart from being a direct and green method and taking relatively a lesser measurement time.     

Orientational dependence of linear dichroism exemplified by dyed spherulites

D-sorbitol forms so-called spherulites from under-cooled melts. These polycrystalline formations have optically uniaxial radii. Melts pressed between glasses crystallize as plane sections of spheres. Dyes that are soluble in molten sorbitol become oriented as the crystallization front passes through the melt so as to form disks with large linear dichroism in the absorption bands of the dyes. The dyeing of spherulites is thus a general method of solute alignment. The linear optical properties of sorbitol spherulites containing the azo dye amaranth were analyzed in detail so as to correct a persistent confusion in the literature regarding the orientational dependence of linear dichroism. In cases involving thin film dichroism of multilayered samples requiring many corrections of intensity data in non-normal incidence, some authors have taken transmittance and others absorbance as having a cosine-squared angular dependence on the plane of the electric vector of linearly polarized light. Plane sections of doped spherulites present all orientations of an electric dipole oscillator in spatially localized region in normal incidence. As such, the samples described herein are ideally suited to resolving this confusion. Images of transmittance of dyed spherulites in polarized light were recorded with a CCD camera and simulated under the assumption that both absorbance and transmittance show a cosine-squared angular dependence but with respect to different angles. Transmittance with a cosine-squared dependence follows azimuthal rotations of the spherulite radii around the wave vector, while absorbance with a cosine-squared dependence follows rotations about axes perpendicular to the wave vector, natural consequences of the properties of the optical indicatrix that are often overlooked. Spherulites obviate the substantial experimental complexities that are engendered in non-normal incidence by sample reorientation. Thus, the principles of anisotropic absorption are given in a complete and intuitive fashion.     

Effect of birefringence on light scattering from deformed spherulitic polymer films

Light scattering from oriented samples of crystalline polymers is affected by the birefringence of the sample. An extension of the theory for scattering from uniaxially deformed two-dimensional and three-dimensional spherulites is made so as to include the retardation of the incident and scattered beams in passing through the birefringent sample. Strain influences scattering, in that it changes the birefringence of the sample and it also changes the anisotropy and shape of the spherulites. Scattering intensities are calculated for both crossed and parallel polarizers as a function of Ω, χ, and Φ, where Ω is the angle between the stretching direction of the sample and the horizontal direction, and χ and Φ are the angles between the stretching direction and the polarization directions of the polarizer and analyzer, respectively. It is shown that for crossed polarizers with Φ = 45° and χ = 45° birefringence changes largely influence the results but that for the polarizers parallel at Φ = 0° and χ = 0° or crossed at Φ = 90° and χ = 0° the birefringence effect is minimized. The intensity distributions for crossed polarizers at Φ = 45° and χ = 45° from polyethylene films stretched to give retardations up to several wavelengths, are found to be in good agreement with the calculated results.     

Scattering of light from oriented polymer films. II

A statistical theory for the scattering of light from oriented polymer films is developed in terms of angularly dependent generalized correlation functions. Numerical calculations of scattering patterns are carried out for special cases. The scattering depends upon two types of distributions describing (1) the orientation distribution of optical axes of scattering elements and (2) the angular dependence of correlation in orientation between pairs of optic axes. These distributions are expanded in Fourier series (in a two-dimensional treatment), the coefficients of which are functions of elongation and describe the elongation dependence of the scattering patterns.