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 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 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.     

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.     

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.     

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.     

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 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.     

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. 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.     

The effect of spherulitic truncation on small-angle light scattering

The effects of spherulitic truncation on the H_v small-angle light-scattering (SALS) patterns are determined by computer simulation of spherulite nucleation and growth. The simulation is carried out for simultaneous and sporadic nucleation of two-dimensional spherulites and simultaneous nucleation of three-dimensional spherulites. The scattered intensity differences between truncated spherulites and round spherulites are determined as functions of the type of growth and the volume (or area) fraction of spherulites. Methods for the determination of certain geometrical characteristics of spherulites systems by SALS are developed. These characteristics include the volume (or area). fraction of spherulites, the average spherulite radius, and the average spherulite volume (or area). The results of this study are essential in the quantitative analysis of H_v SALS from spherulitic systems. The simulation process is readily extendable to the examination of other morphological phenomena by SALS.     

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.     

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.     

The effect of interference between anisotropic scattering entities on the light scattering from polymer films. I. Case of no impingement

Equations are developed for describing the effect of interspherulitic interference on the scattering of light by anisotropic spherulites. These are used to account for the variation in V_V and H_H scattered intensity during the course of spherulitic crystallization. The scattering depends upon the number of spherulites, their size, their anisotropy, the difference between one of their polarizabilities and that of the surroundings, and the radial distribution function of spherulite centers.     

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.     

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.     

对影响光散射强度分布的几种因素的探讨——Ⅰ.截断效应及内部无序性对散射光强分布曲线线形的影响

本工作从实验上考察了球晶截断效应及内部无序性对聚乙烯光散射强度分布曲线线形的影响,并同用二维截断球模型计算出的结果进行了比较,表明截断效应及内部无序性均可使散射光强度分布曲线漫宽,其程度随这些参数的增加而增加;截断效应主要影响小角度区域的散射光强,内部无序性则对大角度区域有较大的影响,二者相比内部无序性的影响似平起主要作用。所得结果与文献报导一致。     

Computer simulation of two-dimensional spherulite growth

The two-dimensional growth of spherulites with impingement was simulated by a computer for athermal, thermal, and combined primary nucleation mechanisms. The simulation provided data on the spherulite size distributions and spherulite shapes. The results of the computer simulation were compared with experimental data on poly(methylene oxide) films.     

聚氧乙烯球晶上彩色环状条纹的形成

用自然白光代替偏振光在显微镜下观察聚氧乙烯球晶的生长过程, 可以更清晰地看到彩色环形条纹的形成. 当成长中的二维球晶相互碰撞后, 被挤出的熔体改变球晶的原有结晶方向, 流向饼形球晶中央而在其表面上逆向结晶, 由于被挤出的熔体数量有限, 晶层的厚度逐渐减小, 在盖玻片下方形成一个盘状的楔形真空间隙. 此间隙导致彩色环状干涉条纹的形成. 实验用肉眼直接观察到二维球晶在生长过程中结晶方式的变化, 为二次结晶理论的发展提供了实验依据.     

Rainbow scattering of CO and N2 from LiF(001)

The angular intensity distributions of CO and N(2) molecules scattered from a LiF(001) surface have been measured as functions of surface temperature, incident translational energy, and incident azimuthal direction affecting surface corrugation at a high resolution. Although both molecules have the same molecular mass and linear structure, only the CO molecule shows a rainbow feature in its scattering pattern, while the N(2) molecule shows a single peak distribution. From the comparisons of the obtained results with the calculated predictions based on the newly developed classical theory of the ellipsoid-washboard model, the differences in scattering distribution are attributed to the effects of molecular anisotropy and center-of-mass position. With an increase in the extent of the molecular anisotropy such as that of N(2) and CO as compared with rare-gas atoms, the summation of several scattering distributions depending on molecular orientation results in smearing the rainbow scattering on the corrugated surface. This smearing effect, however, attenuates when center-of-mass position deviates from the molecular center, as that for CO.