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.     

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.     

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.     

Moderately concentrated solutions of polystyrene. II. Integrated-intensity light scattering as a function of concentration,temperature, and molecular weight

Integrated-intensity light scattering data are reported for moderately concentrated solutions of polystyrene in benzene and in cyclopentane. The benzene system is one for which the second virial coefficient A₂ is large; data obtained over the range 0.5 < A₂Mc < 30, with c the polymer concentration, are analyzed in terms of the (extrapolated) intensity at zero angle and the angular dependence of the intensity. The former is discussed in terms of power law representations based on scaling relations, which are found to represent the data. The latter is discussed in terms of the dependence of the chain dimensions on concentration. With cyclopentane, the behavior is similar for temperatures for which A₂ is near its maximum, but for T near either Θ_U or Θ_L, for which A₂ is zero or small, the angular dependence of the scattering is distinctly different, with the intensity exhibiting a maximum as a function of scattering angle.     

Small-Angle light scattering from an anisotropic sphere: Anisotropy and size effects

The recently developed theory of light scattering from an anisotropic sphere (taken as a model for a spherulite) summarizes all information obtainable from such scattering system when the Rayleigh-Gans-Debye approximation is appropriate. The commonly used procedure of size determination from the maximum of the H_v intensity is tested for a number of models differing in radius size, inherent anisotropy δn and mean refractive index ¯n. It has been confirmed, that the parameter (¯n - 1)/δn plays the decisive role in the angular dependence of the H_v intensity and the influence of that parameter on the size determination and resulting accuracy is specified. The results differ for positive and negative spherulites and the stabilizing role of increasing spherulite size in accuracy of procedure is demonstrated. It has been shown that the RGD approximation is appropriate especially for the case of homogeneous spherulite texture (thus implying, that the refractive index of the surrounding is close to the average refractive index of a single spherulite). The results extrapolated outside of the RGD approximation display invariable U_max position for isotropic contribution with increasing sphere size in contrast to the oscillatory character of U_max predicted by the LM solution. It is proposed to construct the anisotropic solution under an approximation recently suggested for thin anisotropic shell. © 1994 John Wiley & Sons, Inc.     

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 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 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.     

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.     

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.     

Small-angle x-ray scattering of semicrystalline polymers. I. Review of existing models

The effect of previously proposed distributions of particle size and interparticle “gap” lengths on the small-angle x-ray scattering of a paracrystalline one-dimensional macrolattice has been examined. It was concluded that the general paracrystalline model, in which the fluctuations of crystalline and amorphous thickness both contribute to the destruction of long-range order, best describes the structure of lamellar aggregates in semicrystalline polymers. By using this model, the influence of symmetric and asymmetric lattice statistics on the positions of the scattering maxima were investigated. It was found that positively skewed thickness distributions result in the second-order maximum occurring at an angle greater than twice that of the first-order maximum (sx?₂/sx?₁ > 2.0); the position of the first-order maximum is generally greater than the Bragg angle of the structure. With negatively skewed distributions, the ratio of the scattering angles, sx?₂/sx?₁, is less than 2.0, and the first maximum is displaced below the Bragg angle. Qualitatively similar behavior is found with lattices characterized by symmetric lattice statistics, though these deviations from the Bragg conditions are smaller than in the case of negatively skewed distributions. The ratio of the scattering angles of the second and first maxima best reflects the general shape of the lattice statistics in a paracrystalline lattice. The effect of a transition zone, having properties intermediate between those of the crystalline and amorphous regions, was also considered. While the intensity of the higher-order maxima is decreased, no significant shift of the scattering angles results from the incorporation of such a transition zone.     

Light scattering from spherulitic materials

The azimuthal angular dependence of the depolarized component of the light scattered from spherulitic materials is derived by an algebraic method that avoids the difficult angular integrations of the usual approach. The result appears as a sum of products of two factors, a molecular factor, that depends only on the structure and the scattering angle θ, and a geometrical factor that depends only on the azimuthal angle ? and the scattering angle θ. The molecular factors are evaluated for models of spherulitic structure that assume a constant tilt of the optical polarizability tensor. The radial distribution, in principle, is arbitrary, and an evaluation for the layered spherulite is made. If the tilt angle is ω when the azimuthal patterns depend only on a linear combination of P₂(cos ω) and P₄(cos ω), where P_n(x) is the Legendre polynomial of order n. In our theory the V_H scattering pattern is a four-leaf clover whose axes are restricted by the theory to be at either 0 or 45° to the polarization directions.     

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°.     

Effect of interference between anisotropic scattering entities on light scattering from polymer films. II. The correlation function approach

A theory is developed by use of the correlation function approach for calculating both the H_v and V_v intensity of scattered light for a concentrated assembly of spherulities. The scattering becomes a function of the radial and tangential polarizabilities of the spherulite α_r and α_t, the polarizability α_m of the medium, surrounding the spherulites, and the volume fraction ?_s of spherulites. The “effective polarizability of the surroundings” α_s, which appeared in previous theories, becomes function of these variables. The theory can explain, for example, why the V_v scattered intensity passes through a maximum during the course of crystallization.     

Small-angle neutron scattering study of homopolymer blends and blockcopolymers of polystyrene/polyparamethylstyrene

Small angle neutron scattering (SANS) experiments were carried out at one mixture and two block copolymers of polystrene (PS) and poly(p-methylstyrene)(PpMS) at different temperatures ranging from 107 to 295°C. Both block copolymers show a maximum in scattering intensity, which increases with decreasing temperature approaching the spinodal point. Theoretical curves from Leibler's mean field theory agree very well with the experimental points with (XN) as the only fitting parameter, where χ is the Flory-Huggins interaction parameter and N is the degree of polymerization. The reciprocal value of I(q_m)⁻¹ of the maximum intensity for the block copolymers as well as the reciprocal intensity at zero scattering vector (I(q=0)⁻¹) for the mixture obey well the ansatz I = A + B/T within the experimental temperature range. The spinodal values of (X^N)_S are in good agreement with the theoretical values from Leibler.     

Light scattering from crystalline superstructure in tubular-extruded polybutene-1 films

Light scattering from polybutene-1 films prepared by tubular extrusion was studied in order to investigate its crystalline superstructure and the deformation mechanism. Analysis of the light-scattering patterns together with electron micrographs, indicate the existence of sheaflike crystalline superstructures. The sheaves are aligned nearly side by side with their axes preferentially oriented perpendicular to the machine direction. The H_v scattering patterns exhibit a “butterfly” appearance. Analysis of the patterns in terms of scattering and azimuthal angles at which the scattering intensity is a maximum yields information on the size and shape of the sheaflike texture. The information should be of importance in studies of anisotropic crystal growth caused by molecular orientation in the melt and of the deformation mechanism of the texture. The deformation behavior should be representative of that of different parts of spherulites, at least qualitatively; the deformation of the texture along the machine and transverse directions corresponds to deformation of equatorial and meridional regions of a spherulite, respectively.     

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.     

Influence of crystallization temperature,molecular weight,and additives on the crystallization kinetics of poly(ethylene terephthalate)

The spherulite growth rate, the maximum spherulite radius, and the overall rate of crystallization of poly(ethylene terephthalate) (PETP) were measured by means of scattering and transmission of depolarized light. The influence of crystallization temperature, molecular weight, and additives on the above-mentioned quantities was investigated. An expression has been derived for the spherulite growth rate of PETP as a function of crystallization temperature and the number-average molecular weight for M?_n in the range of 19,000 to 39,000.     

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.