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 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.     

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.     

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.     

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.     

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.     

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.     

Correlation between melting behavior and ringed spherulites in poly(trimethylene terephthalate)

The lamellar types as revealed by the multiple melting peaks and possible mechanisms of ringed spherulites in poly(trimethylene terephthalate) (PTT) were analyzed with differential scanning calorimetry (DSC), optical microscopy, and scanning electron microscopy. Several interesting correlations were found. If PTT is melt‐crystallized in a certain temperature range, it shows multiple melting peaks and rings in PTT. Once rings are formed in the original melt‐crystallized PTT, they do not disappear but persist and become even more apparent upon postcrystallization annealing at higher temperatures. Furthermore, for PTT that is capable of exhibiting ringed spherulites, a temperature range exists where rings do not form. This behavior can be interpreted in relation with the demonstrated thermal behavior in PTT. Reorganization took place upon postcrystallization scanning or annealing to or at higher temperatures. A postulation was proposed and rigorously tested with evidence to correlate the ringed spherulites and melting behavior. Rings in PTT may be related to multiple lamellae in the spherulites. Consequently, if a temperature of crystallization is selected so that there is only one type of lamella in the spherulites, then there should be no rings. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 80–93, 2002     

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.     

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 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.     

Crystal orientation in a semicrystalline polymer in relation to deformation of polymer spherulites. III. Small-angle light scattering

Small-angle polarized light scattering from a deformed three-dimensional spherulite is formulated on the basis of the deformation model proposed in Part II of this series. The intensity distribution of scattered light is discussed chiefly for the cross-polarization condition, the so-called H_v polarization, as a function of elongation of the spherulite. In the undeformed state, the scattered intensity distribution forms the typical fourleaf clover pattern, and the intensity decreases with increasing fraction of crystals oriented randomly (type R crystals) within the crystal lamellae of the spherulites. In a system composed of type R crystals and folded-chain crystals (type B crystals) within the lamellae, the four-leaf pattern moves to the horizontal zone near the equator with increasing elongation of the spherulite, and, simultaneously, extends to some extent to the vertical zone near the meridional direction as a parameter measuring the ease of lamellar untwisting increases. In a system composed, in addition to type R and type B crystals, of crystals transformed from type B to type C_a and type C_r due to tilting and unfolding of polymer chains, respectively, within the crystal lamellae an eight-leaf pattern appears, even at small elongation up to about 30%. Each lobe of the eight-leaf pattern undergoes a characteristic change with increasing elongation. In both systems, the scattered intensity increases with sharpening of orientation distribution of crystals within the crystal lamellae.     

Dynamic light scattering from spherical particles

The dynamic light scattering behavior of poly(butylmethacrylate) PBMA microgels and of kappa-casein micelles is compared with that from hard sphere latex particles. The latex particles and the kappa-casein micelles exhibited a single exponential decay of the time correlation function (TCF). For the microgels, progessively stronger deviations from a single exponential were observed as the scattering angle was made larger. These deviations are interpreted as being the result of internal modes of motion. From measurement of the first cumulant of the TCF, extrapolated towards zero angle, the translational diffusion coefficients D were determined, and the hydrodynamically effective radii were calculated via the Stokes-Einstein relationship. The ratio of the radius of gyration to the hydrodynamic radius was found to be?=0.775+0.012 for the latex particles, in good agreement with theory. The microgels, however, exhibit much lower?-parameters of 0.49 to 0.58, while the kappa-casein micelles showed the opposite behavior with values between 1.1 and 2.5. The results are interpreted on the basis of the DebyeBueche and Deutsch-Felderhof theory for porous spheres.     

Theory of light scattering from vesicles

The scattering and depolarization of light from spherical membrane systems, vesicles, is calculated in a new succinct formalism appropriate to spherical systems and the rotations of anisotropic elements composing them. The results of Tinker (1972) and Mishima (1980, 1981) are readily derived within this formalism and there is a considerable saving in numerical computation. The problem of fluctuations in the molecular packing of the component phospholipid molecules is also addressed.     

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.     

Dynamic light scattering from concentrated colloidal dispersions

From the scattered intensity as a function of scattering wave number, local ordering in concentrated latex dispersions in benzene is found. From the combination of intensity and photon correlation experiments follows the existence of strong hydrodynamic interactions between the dispersed particles.     

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.     

Depolarized light scattering from critical polymer blends

Depolarized light scattering of binary polymer blends in disordered state near the demixing critical point is considered both theoretically and experimentally. It is shown that the depolarized scattering in such systems is predominantly due to double scattering processes induced by composition fluctuations. For long enough polymer chains, this scattering is stronger than the contribution from intrinsic anisotropy fluctuations. The general equation for the static and dynamic double scattering function is obtained in terms of the system structure factor. The scattering functions are calculated both analytically and numerically (dynamic part) for polymer blends. We found that the depolarized intensity depends on the polymerization degree N and the relative distance from the critical point τ = 1 – χ*/χ (where χ is the Flory‐Huggins interaction parameter and χ* its critical value) as I_vh ∼︁ N²/τ², which is in good agreement with the experimental data. It is also shown that the dynamic scattering function is decaying non‐exponentially. We calculate the relaxation rate and the non‐exponentiality parameter as functions of the scattering angle and τ. These theoretical predictions are compared with experimental data for three chemically different blends.