An x-ray diffraction study of nonlinear polyethylene. I. Room-temperature observations

In this investigation on samples of high- and low-density polyethylene and ethylene-vinyl acetate copolymers, crystallinities ?_W and crystalline densities ρ_cW were obtained with the aid of wide-angle x-ray scattering (WAXS) methods. From small-angle x-ray scattering (SAXS) the following characteristics were obtained either directly or by combination with the WAXS data: values, or limiting values, of the crystallinity ?_S; crystal densities ρ_cS; thicknesses of the diffuse boundary layer; number-average thicknesses of the crystalline and amorphous layers; and both number and weight averages of the long periods. It was shown that a discrepancy between ?_S and ?_W cannot be attributed to the occurrence of large amorphous regions outside the regular stacks of lamellae; the data were reconciled by assuming that the WAXS crystallinities pertain to the cores of the crystalline lamellae, whereas part of the diffuse boundary layers is comprised in the values of ?_S. The ρ_cW and ρ_cS data of the nonlinear samples show systematic differences, which were attributed to partial incorporation of side groups in the crystalline regions at a concentration estimated to be of the order of 20–40% of the overall concentration. With increasing side-group concentration, the thickness of the core of the crystalline lamellae was found to approach the average length of the linear chain segments between side groups. On the basis of these observations a scheme for the crystallization of nonlinear polyethylene is proposed according to which a number of side groups is encapsulated by the growing crystal. The data can be explained by assuming that all chains, offered at a crystal face where growth takes place, crystallize directly, irrespective of whether the crystallizing stem carries a side group. Further crystallization would then proceed by chain folding at both ends of the first stem, until a noncrystallizable unit is met. In this scheme, allowance is made for about half the stems in the crystals to be connected by folds; this is required in view of the “overcrowding” effect. Finally, the effect of cooling rate and molecular weight on the thicknesses of the crystalline and amorphous layers is discussed, and differences between the amorphous densities of high-and low-density polyethylene are noted.     

Small-angle x-ray scattering studies of melting

The course of melting of melt-crystallized polyethylene fractions and of a poly(ethylene oxide)-polystyrene-poly(ethylene oxide) triblock copolymer has been followed by small-angle x-ray scattering (SAXS). Changes in the intensity and shape of the SAXS curves indicated that both surface melting and melting over the full crystallite thickness (full-strand melting) take place. Full strand melting is the final, irreversible process. Comparison with an analytical model indicates that in the earlier stages of the irreversible, full-strand process the crystallites melt out randomly throughout the bulk. Later stages may occur by the simultaneous melting of a larger stack of crystallites.     

Small-angle x-ray scattering studies of isothermally crystallized bulk polyethylene

A set of isothermally melt-crystallized polyethylene samples was examined using small-angle x-ray scattering (SAXS). Time and temperature of crystallization were the variable parameters used to create the set of samples. Following background subtraction, desmearing, and application of the Lorentz factor to the raw SAXS data it is possible to see many orders of reflection. This suggests that much higher degrees of order are present in isothermally melt-crystallized samples than had previously been thought possible. A combination of SAXS and DSC data indicates that there is no evidence for isothermal thickening in these samples. This study, coupled with data obtained from PE single crystals, produced information concerning the extrapolation of single-crystal data to fit bulk systems. In addition, the equilibrium melting point T determined is somewhat lower than previously claimed. This study also suggests that the surface energy of the mature crystals is always lower than that of the nucleated state and/or the nucleation factor Kσ_en increases with decreasing supercooling.     

Small-angle x-ray study of deformed bulk polyethylene

Scattering by extruded polyethylene films has been used to analyze in detail the small-angle scattering of x-rays by strained spherulitic samples. It is shown that the basic small-angle patterns in spherulitic specimens can be obtained by summing small-angle reflections from films drawn both parallel and normal to the direction of extrusion. The representation of a complex pattern by superposition of simpler reflection makes it possible to calculate local strains in various regions of spherulites in bulk specimens.     

Small-angle x-ray scattering study of ionomer deformation

The small-angle x-ray scattering (SAXS) pattern from a cesium salt of a 6.1 mole % ethylenemethacrylic acid (E-MAA) copolymer is shown to become azimuthally dependent on sample elongation. SAXS was measured using the Oak Ridge National Laboratory (ORNL) spectrometer with pinhole collimation and a two-dimensional position-sensitive detector. The sample was quenched prior to deformation to avoid crystallization of the ethylene unit which would complicate the interpretation of scattering. The observed SAXS patterns are interpreted in terms of several proposed models for the structure of ionomers. A model in which ionic aggregates are arranged on a paracrystalline lattice is found to be largely in disagreement with the results for undeformed and deformed samples. Spherical and lamellar models incorporating local structure around a central ionic core are capable of predicting the observed SAXS for the undeformed sample. A model of ellipsoidal deformation of the spherical shell-core model fails to predict the correct azimuthal dependence of scattering. However, a deformation scheme involving rotation of the lamellar model is more satisfactory.     

Small-angle x-ray scattering of semicrystalline polymers. II. Analysis of experimental scattering curves

The small-angle x-ray scattering (SAXS) patterns of a number of linear polyethylene (PE) and polyoxymethylene (POM) samples have been measured and compared to the intensity functions of one-dimensional paracrystalline lattices. It was found that the ratio of the angular positions of the second and first scattering maxima (θ₂/θ₁) is generally less than or equal to 2.0, implying that the paracrystalline lattice statistics are symmetric or moderately skewed to larger periods. The Bragg spacing (“long period”) of such samples is within 3% of the identity period of the macrolattice. With quenched POM the ratio θ₂/θ₁ is substantially larger than 2.0, which indicates either extremely asymmetric lattice statistics or coexisting structures within the material. From consideration of the reduced widths of the first scattering maxima, it was found that some broadening is present in addition to that from the paracrystallinity. This excess broadening could result from a finite lattice length of ～1000 Å. The need for careful experimental technique for obtaining the actual position of the scattering maximum is emphasized. In addition, it is demonstrated that the scattering curve and the correlation function of the system yield essentially the same apparent structural periods.     

On the structure of glassy polymers. III. Small-angle x-ray scattering from amorphous polyethylene terephthalate

The small-angle x-ray scattering (SAXS) from glassy polyethylene terephthalate has been measured using a Bonse–Hart system. The data cover the angular range (2θ) between 20 sec and 2 deg. After correcting for absorption, background, and beam divergence, the data have been placed on an absolute basis by comparison with the scattering from a standard silica suspension. The corrected absolute intensity decreases strongly with increasing angle over the range between 20 sec and 15 min, decreases more gradually in the range between 15 min and 45 min, and reaches a nearly constant asymptotic value over the range between 45 min and 2 deg. The magnitude of the scattering in the constant range, about 0.4 (electrons)² Å^?3, is very close to the value predicted by the thermodynamic fluctuation theory for fluids applied at the glass-transition temperature [0.34 (electrons)² Å^?3]. The increase in intensity at angles smaller than about 45 min cannot be described by structures on the scale and volume fraction of the nodules reported in amorphous PET (50–100 Å), but can be well represented by small concentrations of heterogeneities, ranging in size from 100 to 2000 Angstroms, superimposed on the thermal density fluctuations frozen-in at the glass transition. The bulk structure of this material seems well described as a random amorphous solid, containing simple thermal fluctuations and a small concentration (<1 vol-%) of heterogeneities covering a range of sizes. The heterogeneities in the small end of the range may well be crystallites which formed on cooling.     

Small-angle x-ray scattering of isotactic polystyrene

Isothermal crystallization process of isotactic polystyrene at 167°C has been studied by smallangle x-ray scattering. The observed SAXS intensities consist of the twophase lamellar structure component, the density fluctuation, and the foreign particle components. The profile of lamellar structure component remains unchanged during crystallization while its intensity increases with crystallization. The lamellar structure of isotactic polystyrene is investigated on the basis of the interface distribution function. An interface distribution function is obtained from the lamellar structure component after correcting the effect of the finite thickness of boundary regions between crystalline and amorphous phases. In order to obtain the structure parameters, the Gaussian correlation model is used, in which the correlation between the distributions of neighboring crystal and amorphous thicknesses is taken into account. Agreement is satisfactory between the experimental results and the calculations. The structure parameters of isotactic polystyrene are determined for isothermal crystallization at 167°C as follows: the average and the standard deviation of crystal thickness are 40 A and 10 A, respectively, those of amorphous thickness are 70 A and 23 A, and the standard deviation of long period is 31 A.     

Small-angle neutron and x-ray scattering study of swollen nylon-6

Results of swelling and small-angle scattering experiments on samples of nylon-6 swollen with heavy water are discussed on the basis of the lamellar and switchboard models. The small-angle neutron scattering (SANS) intensity is very sensitive to the distribution of water in swollen samples, while the small-angle x-ray scattering (SAXS) data characterize the dry samples. The observed values of the mean-square fluctuation of scattering-density can be explained by a model with assumed inhomogeneous swelling of the amorphous phase.     

Small-angle x-ray diffraction studies of plastically deformed polyethylene. II. Influence of draw temperature,draw ratio,annealing temperature,and time

The angular dependence of scattering intensity of drawn polyethylene (PE) was investigated with a small-angle Kratky camera. At constant drawing temperature the intensity drops drastically with increasing draw ratio; however, the position and the half-width of the first maximum remain nearly unchanged. The drop in intensity can be explained only by a reduction of effective electron density difference between amorphous and crystalline components. The latter contains more vacancies, and the former contains more and better packed tie molecules. This increases the average density of the amorphous layer and decreases that of the crystalline component. As the temperature of the drawing increases, the draw ratio attainable at the applied draw rate drops and the intensity of scattering and the long period rapidly increase. In addition, a second-order maximum appears, indicating a better order of lamellar stacking, in good agreement with electron microscopy. The first annealing effect is an extremely rapid increase in scattering intensity and long period. The subsequent increase is rather slow and proportional to the logarithm of annealing time. The long period in such an experiment is independent of the draw ratio; however, the scattering intensity depends on it quite strongly even after prolonged annealing.     

Small-angle x-ray scattering of distorted lamellar structures

Small-angle x-ray scattering (SAXS) intensity for the lamellar structure of polymeric materials has been formulated with consideration of structural defects such as the finiteness of the lamellar stack, the lamellar bend, and the paracrystalline distortions. In particular, the effects of the lamellar bend on the SAXS profile have been elucidated on the basis of Vonk'xss formula γ₁(x) – γ(x)exp(?2x/d). Here, the scattering profile due to the lamellar bend is shown to be expressed by a Cauchy function. The integral breadth is equal to 2π/d, being independent of the order of scattering. As an example of the SAXS analysis based on the theory, the characterization of the lamellar structure in the “hard” elastic polypropylene films is reported. The long period and the lamellar thickness are evaluated from the correlation function, and the distortion length and Hosemann's g factor are estimated according to the procedure presented here. On the basis of these structural parameters, the relationship between the manufacturing process and the lamellar structure of the polypropylene films is discussed. © 1993 John Wiley & Sons, Inc.     

Small-angle x-ray scattering study of density fluctuation in pressure-densified polystyrene glasses

Densified polystyrene glasses, prepared by cooling from the liquid state under elevated pressure, were studied by small-angle x-ray scattering at ambient pressure. The density fluctuation, determined from the x-ray data, showed a decrease with increasing pressure up to about 1.5 kbar, and then leveled off to a fairly constant value. The reduction in the density fluctuation produced by the pressure is much greater than the associated decrease in the specific volume. The observed change in density fluctuation is consistent with the view that the density fluctuation in glassy polymers consists of dynamic and quasistatic components and that the first of these can be correlated with the compressibility of the glass. The present data on the density fluctuation, in conjunction with the available data on volume and enthalpy, can be interpreted to mean that in pressure-densified glasses unfavorable chain configurations are trapped in local energy minima, and the strain energy thus stored can promote segmental motion leading to volume expansion at temperatures far below T_g. Some preliminary evidence indicating the formation of microcavities in these pressure-densified glasses is also presented.     

Small-angle x-ray scattering studies of crazed glassy polymers

A small-angle x-ray scattering (SAXS) study of the relaxed craze structure in polystyrene was performed using the Oak Ridge National Laboratory 10-m SAXS facility. Coupled with known results from transmission electron microscopy studies, the SAXS patterns can be interpreted as scattering from an open-cell foam with void spaces interspersed among the fibrils. Results have shown the scattering centers in crazed polystyrene can be modeled as cylinders the axes of symmetry of which are parallel to the tensile axes. Scattering centers are bimodal in their size distribution, with aspect ratios of 1.0 and 2.6. Crazes in lower-molecular-weight polystyrene have more and larger scattering centers than crazes in higher-molecular-weight polystyrene, while variations in strain rate and test temperature during craze formation have no effect on the relaxed craze morphology. A comparison of SAXS patterns from polystyrene and polycarbonate indicates that the morphologies of their respective crazes are significantly different.     

On the structure of glassy polymers. II. Small-angle x-ray scattering from poly(methyl methacrylate)

The small-angle x-ray scattering (SAXS) from glassy poly(methyl methacrylate) has been measured using a Bonse–Hart system. The data cover the angular range (2θ) between 20 sec and 2 deg. After correcting for absorption, background, and beam divergence, the data have been placed on an absolute basis by comparison with the scattering from a standard silica suspension. The corrected absolute intensity decreases strongly with increasing angle over the range between 20 sec and 30 min, and is nearly constant between 30 min and 2 deg. The magnitude of the scattering in the constant range, 0.6 (electrons)² Å^?3, is within a factor of 1.5 of the value predicted by the thermodynamic fluctuation theory for fluids applied at the glass transition temperature. The increase in intensity at smaller angles cannot be described by structures on the scale of the nodules reported in highly isotactic PMMA (150–200 Å), but can be well represented by small concentrations of heterogeneities, several thousand angstrom units in size, superimposed on the thermal density fluctuations frozen-in at the glass transition. The bulk structure of this material is well described as a random amorphous solid, containing simple thermal fluctuations and a small concentration of relatively large heterogeneities.     

Structure of glassy polymers. V. Small-angle x-ray scattering from polystyrene

Small-angle x-ray scattering (SAXS) from glassy atactic polystyrene has been measured using a Bonse–Hart system. After correcting for absorption, background, and beam divergence, the scattering has been placed on an absolute basis using a standard silica suspension as a reference.The desmeared absolute intensity decreases strongly with increasing scattering angle over the range between 20 sec and 20 min. At larger angles, the intensity decreases much more slowly with increasing angle and approaches the value expected for density fluctuations frozen-in at the glass transition. The angular variation of intensity is well described by the scattering from heterogeneities of various sizes and concentrations superimposed on the scattering from thermal density fluctuations. These heterogeneities range in radius from 10 to 4000 Å. The present SAXS results on glassy polystyrene seem inconsistent with the presence of nodular features as representative of the bulk polymer.     

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.     

Dynamic x-ray diffraction studies of high-density polyethylene

Dynamic x-ray diffraction is conducted to explore the structural origin of the α and β mechanical dispersions of a melt-crystallized high-density polyethylene. It is shown that the real component of the strain orientation coefficient for the crystal c axis C decreases with increasing frequency at a rate which decreases with decreasing temperature. Values of C for the c axis are positive, C for the a axis negative, and C for the b axis close to zero, suggesting that the predominant relaxation process is crystal rotation about the b axis. The activation energy found from Arrhenius plots of C corresponds to that of the α₁ mechanical dispersion. The dynamic birefringence in this region is dominated by the contribution from crystal orientation changes. At low temperatures, the imaginary component KC of the strain-optical coefficient of the crystal phase approaches zero, while KC of the amorphous phase exhibits a somewhat broad dispersion peak corresponding to the β birefringence dispersion. This suggests that the principal contribution to the β birefringence dispersion arises from the amorphous phase, probably owing to the amorphous orientation process. Contrary to the case of low-density polyethylene, the dynamic crystal lattice deformation and compliance functions reveal distinct frequency dispersions corresponding to the α₁ and α₂ mechanical processes. The α₁ lattice dispersion is thought to be associated with the α₁ crystal orientation dispersion, while the α₂ lattice dispersion is believed to be the inherent one arising from the onset of intracrystalline chain motions.     

Small-angle x-ray scattering experiments for investigating the validity of the two-phase model

A New method for evaluating SAXS curves of polymer samples with lamellar structure is applied to two typical scattering curves measured with a solution-crystallized linear polyethylene and a melt-crystallized branched polyethylene respectively. The method permits a rigorous check of the validity of the two-phase model and yields, without additional measurement, the volume fractions of the two phases and the difference in their densities. The densities can than be obtained by measuring the overall density of the sample. The results are: ρ_c = 0.996 g/cm³,ρ_a = 0.854 g/cm³, w_a = 0.20 for the solution-crystallized sample; ρ_c = 0.967 g/cm³,ρ_a = 0.850 g/cm³, w_a = 0.36 for the melt-crystallized sample.