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.     

Small-angle scattering of polarized light. IV. Isotropic to birefringent transition for spheres

The Rayleigh-Gans-Debye approximation is used to investigate the small-angle, cross-polarized light scattering by a sphere of radius a, birefringence Δμ and relative index μ. If θ_max is the polar scattering angle of the intensity maxima, the quantity Û_max = 4πa/λ × sin (θ_max/2) behaves in two different ways according to the signs of Δμ and μ ? 1. When Δμ > 0, μ > 1 or Δμ < 0, μ < 1, then Û_max varies from 2.8 to 4.1 as Δμ increases from zero. If Δμ < 0, μ > 1 or Δμ > 0, μ < 1, then Û_max goes from 2.8 to about 6, thereafter decreasing to 4.1. Another interesting result is that the value of Û_max for a highly briefringent sphere is 4.1 only for large diameters. It decreases to 4.0 when the diameter decreases.     

Analysis of lattice thermal conductivity and phonon-phonon scattering relaxation rate: Application to Ge

The three-phonon scattering relaxation rates and their temperature exponents have been analysed in the frame of Guthrie's classification of the phonon-phonon scattering events as class I and class II events and as a result of this, a new expressionτ _3ph ^-1 =(B _N,I+B _U,Ie^-θ/αT) g(w)T ^m I ^(T)+(B _N,II+B _U,IIe^-θ/αT)g(w)T ^mII^(T) for the three phonon scattering relaxation rates has been proposed for the first time to calculate the lattice thermal conductivity of a sample. Using the expression proposed above, the lattice thermal conductivity of Ge has been analysed in the temperature range 2–1000K and result obtained shows a very good agreement with the experimental data. The percentage contributions due to three-phonon normal and umklapp processes are also reported. The role of four phonon processes is also included at high temperatures. To estimate an approximate value of the scattering strength and the phonon conductivity, the analytical expression is also obtained in the frame of the expression proposed above forτ _3ph ^-1 .     

A study of diffractometer line shapes in melt crystallized polyethylene

Wide angle X-ray diffraction scans of isothermally crystallized samples of polyethylene containing known levels of chain defects have been investigated. An analysis of scattering curves using a powder diffractometer is reported. The analytical deconvolution of Lorentz and Lorentz squared profiles for the PE experimental scattering curve and the standard is performed to give the pure profile for the polymer. The separate contribution of paracrystalline lattice distortions and coherently diffracting domains, in different crystallographic directions, to the broadening of the diffraction profiles is discussed in the light of the level of defects present.     

Multiple collisions in molecular beam scattering experiments

Due to the forward peaked differential cross section for elastic atom—atom scattering the effect of multiple collisions has to be considered in the analysis of crossed beam measurements of the total cross section and especially of the small angle differential cross section at large values of the beam attenuation. At angles θ ≈ θ₀, with θ₀ the quantum mechanical scaling angle of the elastic differential cross section, the correction for the latter case amounts to 20% at beam attenuations I/I₀ = exp(?1). Firstly, a careful analysis of the probabilities for single and multiple scattering is given, resulting in an expression for the measured beam signals which is correct for all values of the beam attenuation. The probability for multiple scattering is then calculated for an inverse power potential V(r) = ?C_sr^?s, with s = 4 through s = 7, which include both the case of ion—atom scattering (s = 4) and atom—atom scattering (s = 6). The results are given as effective differential cross sections σ_n(θ) for n-fold scattering. They are described by a single, simple analytical function with four free parameters that have been determined for n = 2, 3 and 4 by a least squares method. The σ_n(θ) are normalised to the total cross section Q.     

Synthesis and Characterization of a Thermoresponsive Polyoxometalate–Polymer Hybrid

We report the synthesis of the first organo‐POM with thermoresponsive properties. Our concept will provide chemists with a new tool to design POMs whose solubility is reversibly controllable through an external stimulus. POM–polymer TBA₇[POM]‐poly(N,N‐diethylacrylamide) (POM–PDEAAm), was prepared by grafting PDEAAm‐NH₂ (obtained by RAFT polymerization) onto the activated Dawson acyl‐POM, α₂‐[P₂W₁₇O₆₁SnCH₂CH₂C(?O)]^6?. Extensive MS analysis was used to monitor the chain‐functionalization steps and to confirm the formation of the hybrid. Aqueous solutions of the (NH₄)₇[POM–PDEAAm] exhibited a LCST of 38 °C. Thus, the solubility/aggregation of the hybrid was reversibly controlled by changing the temperature. Above 38 °C, the solution became cloudy, and cleared again upon cooling. Dynamic light scattering (DLS) revealed the formation of small aggregates in the range 100 nm. We assumed that the charged POM head units prevented the formation of the larger‐scattering aggregates that are usually observed for PDEAAm, and promoted the formation of micelle‐like structures. The conjugate exhibited a temperature transition, which was different from that of the polymer and depended on the counterions associated with the POM. This result demonstrates the potential for merging organic (in this case, polymer) and inorganic structures to afford materials that exhibit new properties.     

Microfibril diameter in celery collenchyma cellulose: X-ray scattering and NMR evidence

Cellulose isolated from celery collenchyma is typical of the low-crystallinity celluloses that can be isolated from primary cell-walls of higher plants, except that it is oriented with high uniformity. The diameter of the microfibrils of celery collenchyma cellulose was estimated by three separate approaches: ¹³C NMR measurement of the ratio of surface to interior chains; estimation of the dimensions of the crystalline lattice from wide angle X-ray scattering (WAXS) measurements using the Scherrer equation; and the observation that microfibrils of this form of cellulose have the unusual property of packing into an irregular array from which small angle X-ray scattering (SAXS) shows features of both form and interference functions. The interference function contributing to the SAXS pattern implied a mean microfibril centre-to-centre distance of 3.6 nm, providing an upper limit for the diameter. However modelling of the scattering pattern from an irregular array of microfibrils showed that the observed scattering curve could be matched at a range of diameters down to 2.4 nm, with the intervening space more or less sparsely occupied by hemicellulose chains. The lateral extent of the crystalline lattice normal to the 200 plane was estimated as a minimum of 2.4 nm by WAXS through the Scherrer equation, and a diameter of 2.6 nm was implied by the surface: volume ratio determined by ¹³C NMR. The WAXS and NMR measurements both depended on the assumption that the surface chains were positioned within an extension of the crystalline lattice. The reliability of this assumption is uncertain. If the surface chains deviated from the lattice, both the WAXS and the NMR data would imply larger microfibril diameters within the range consistent with the SAXS pattern. The evidence presented is therefore all consistent with microfibril diameters from about 2.4 to 3.6 nm, larger than has previously been suggested for primary-wall cellulose. Some degree of aggregation may have occurred during the isolation of the cellulose, but the larger microfibril diameters within the range proposed are a consequence of the novel interpretation of the experimental data from WAXS and NMR and are consistent with previously published data if these are similarly interpreted.     

A structural development of an organogel explored by synchrotron time-resolved small-angle X-ray scattering

We investigated the gelation process of 12-hydroxystearic acid (12-HSA)/dodecane gel (turbid gel), 12-HSA/xylene and 12-HSA/toluene gels (transparent gels) by using time-resolved synchrotron small-angle X-ray scattering technique. The fibers were developed via nucleation and growth mechanism, and the induction period was longer at higher temperatures. After the induction period, the fiber growth can be divided into two regimes. In the first stage, the scattering intensity increased with time without appearance of any crystalline peak, and in the second stage (001) peak appeared due to crystalline nucleation. In the former regime, the fibers grew with increasing the cross-sectional size, while in the latter regime, they grew with keeping it almost constant, i.e., with keeping the fiber thickness constant. The fiber thickness for the turbid gel (radius of the gyration R _c of ～100 Å) was thicker than that for the transparent gel (R _c of ～82 Å). The fractal dimensions of the fibrillar aggregates for the turbid gels (2.0–2.3) at various temperatures were larger than those for the transparent gels (1.4–1.6), suggesting that the structure of the former gels is more branching or more compact in comparison with the latter.     

Analysis of the small-angle X-ray scattering of linear polyethylene by a paracrystalline lattice model

Three paracrystalline lattice models for the interpretation of the small angle scattering of polyethylene are discussed: The “lattice model”, the “stapel model” (often referred to as the lamellar stack model) and the “proportional model”. While the applicability of the first model is restricted, the latter models differ in the statistical assumptions of lamellar and interlamellar thickness distributions. The principal advantage of the proportional model over the stapel model is its applicability through the adjustment of only three parameters: long period, crystallinity and one statistical parameter. Small angle X-ray curves of linear polyethylene are interpreted by the proportional model. The results are in good agreement with stapel model calculations.     

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 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.     

SAXS determination of the amorphous surface layer thickness of polymer single crystals

Polyethylene single crystals were grown from 0.1% solutions in xylene at 80 and 87°C. Oriented mats were made from each preparation and the small-angle x-ray scattering (SAXS) profiles obtained. Following treatment of the raw data for main-beam position and width, background scatter, and the Lorentz factor, five Bragg reflections were resolved. A one-dimensional lattice was used as a model for the oriented mats of single crystals. This model contains three parameters. An additional parameter G_x was also introduced to demonstrate the general effect of a broadening factor on the model. The effect of each parameter on the calculated diffraction pattern was examined. From this examination it was found that by assuming that the broadening functions are zero we can determine directly from the number of observable peaks the maximum possible thickness of the amorphous surface. Further, we find that the thickness of the amorphous layer must be less than the maximum value calculated if G_x is assigned values greater than zero. A “best-fit” diffraction pattern was generated in order to estimate how much smaller the surface thickness can be such that one can still resolve five diffraction maxima. The range of amorphous surface thicknesses found from the calculated diffraction profile is 12–20 Å. This is in good agreement with complementary studies performed on the same crystal preparations.     

Quantitative analysis of calcium and fluorine by high-sensitivity low-energy ion scattering: Calcium fluoride

Low-energy ion scattering (LEIS) probes the atomic composition of the outer surface. Well-defined reference samples are used for the quantitation. For elements like fluorine and calcium, it is not easy to find suitable, clean, and homogeneous references, since fluorine is a gas and calcium is a very reactive metal. In contrast to surface analytic techniques such as XPS, the extreme surface sensitivity of LEIS makes it difficult to use stable compounds like CaF₂ as reference, since these compounds are not homogeneous at the atomic scale. With LEIS, CaF₂ is not expected to show an atomic ratio F/Ca = 2.0. Thus, before CaF₂ can be used as reference, its atomic surface concentrations have to be determined. Here, 3-keV He⁺ scattering by a LiF(001) single crystal, an evaporated layer of Ca, and a Cu foil are used as basic references. High-purity CaF₂ is available in two forms: a single crystal and a powder. For a practical reference, powders are preferred, since if bulk impurities segregate to the surface, they will be dispersed over a large surface area. It is found that both CaF₂ (111) and powder have similar F/Ca atomic ratios. This confirms the F termination for both samples. For the powder, the F and Ca signals are reduced by 0.77 ± 0.03 in comparison with those for the single crystal. The atomic sensitivity factors and relative sensitivity factors have been determined for F, Ca, and Cu.     

Light scattering and dielectric relaxation studies of glass forming liquids

Extended Abstract: Glass forming organic liquids and polymers exhibit long range density fluctuations with correlation length ξ in the range of 10–300 nm at temperatures above T_g (1 - 6). This follows from dynamic and static light scattering experiments revealing some unexpected features, which cannot be explained on the basis of conventional liquid state theories: (i) In static light scattering the intensity I(q → 0) is no longer proportional to the isothermal compressibility, (ii) This excess scattering I_exc shows a strong q-dependence (q = (4π/Λ.)sin(θ/2)) corresponding to a correlation length ξ in the above mentioned range, (iii) The Landau-Placzek ratio I_Rayleigh/2I_Brillouin is much too high compared with the results of light scattering theories, (iv) In photon correlation spectroscopy a new ultraslow hydrodynamic mode (Γ ˜ q²) is detected with relaxation rates Γ about 10⁻⁶ to 10⁻⁹ lower than those of the α-process at a given temperature. In order to explain these observations, a two-state fluid model is proposed, which starts from the coexistence of “liquid-like” and “aperiodic solid-like” regions within the liquids. Such ideas have been discussed many times before, so for example A.R. Ubbelohde (7) speculates about “anticrystalline” clusters in liquids. Molecular dynamics simulations of atomic liquids showed that long range orientational fluctuations appear upon supercooling (8). A preferred icosahedral ordering is observed (9) and the number of icosahedral clusters increases with decreasing temperature (10). In connection with the interpretation of the dynamics of supercooled liquids different “two-state” models have been proposed (11 - 15). For the explanation of the light scattering results we propose that the molecules in the different dynamic states (“liquid” or “solid”) aggregate during annealing of the liquid at temperatures above T_g. Experiments showed that the equilibration times can be rather long (3 - 5), but nevertheless the liquids exhibiting long range density fluctuations are in the state of lowest free energy. We claim that our observations are the first experimental proof of the existence of such different dynamic states, which have been discussed many times before. The extended secondary clusters can also be detected by ultra small angle X-ray scattering.     

Small-angle neutron scattering of a model crystallizable polymer: Crystallization and size-mismatch effects

Small-angle neutron scattering (SANS) experiments have been conducted on mixtures of nearly monodisperse hydrogenated polybutadiene (HPB) and its deuterated counterpart (DPB) in molten and semicrystalline states. A direct comparison of scattering patterns from molten and quench-crystallized samples shows unambiguously that chain conformation is unchanged by crystallization over large and intermediate size scales. The HPB's are shown to have slightly larger coil dimensions than linear polyethylenes of the same molecular weight. The scattering law for mixtures of different chain lengths was also investigated. Mismatched mixtures of monodisperse components give rise to scattering with an angle dependence well described by the Debye function for a Gaussian chain, even in the 50/50 composition range. This permits evaluation of SANS parameters [apparent radius gyration (R_g)_e and apparent degree of polymerization N_e] by analysis of the full scattering curve, removing the restrictions that the Guinier condition (qR_g < 1) be satisfied. The values of (R_g)_e and N_e obtained from experiment are in good agreement with predictions of the scattering law for ideal mixtures.     

mj Rainbows in molecule-surface scattering

A quasiclassical trajectory study of N₂ scattering from a rigid corrugated crystal surface was carried out. Rainbow maxima were seen in the m_j distribution. The maximum value of product j at which m_j, rainbows occur is found to increase linearly with the corrugation and decrease linearly with the hardness of the interaction potential.     

Watching Nanoparticles Form: An In Situ (Small‐/Wide‐Angle X‐ray Scattering/Total Scattering) Study of the Growth of Yttria‐Stabilised Zirconia in Supercritical Fluids

Understanding nanoparticle‐formation reactions requires multi‐technique in situ characterisation, since no single characterisation technique provides adequate information. Here, the first combined small‐angle X‐ray scattering (SAXS)/wide‐angle X‐ray scattering (WAXS)/total‐scattering study of nanoparticle formation is presented. We report on the formation and growth of yttria‐stabilised zirconia (YSZ) under the extreme conditions of supercritical methanol for particles with Y₂O₃ equivalent molar fractions of 0, 4, 8, 12 and 25 %. Simultaneous in situ SAXS and WAXS reveals a quick formation (seconds) of sub‐nanometre amorphous material forming larger agglomerates with subsequent slow crystallisation (minutes) into nanocrystallites. The amount of yttria dopant is shown to strongly affect the crystallite size and unit‐cell dimensions. At yttria‐doping levels larger than 8 %, which is known to be the stoichiometry with maximum ionic conductivity, the strain on the crystal lattice is significantly increased. Time‐resolved nanoparticle size distributions are calculated based on whole‐powder‐pattern modelling of the WAXS data, which reveals that concurrent with increasing average particle sizes, a broadening of the particle‐size distributions occur. In situ total scattering provides structural insight into the sub‐nanometre amorphous phase prior to crystallite growth, and the data reveal an atomic rearrangement from six‐coordinated zirconium atoms in the initial amorphous clusters to eight‐coordinated zirconia atoms in stable crystallites. Representative samples prepared ex situ and investigated by transmission electron microscopy confirm a transformation from an amorphous material to crystalline nanoparticles upon increased synthesis duration.     

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.     

方差方法用于高聚物X光衍射峰形的分析

假设PET纤维中的晶格畸变可用次晶模型描述, 用其x光衍射105的方差-区间函数测定出一组热处理样品的晶粒尺寸和晶格畸变因子。说明了此法所得晶粒尺寸为“数均”尺寸。发现样品中晶区的完善性随其热处理温度的升高而改善; 晶粒尺寸与畸变因子之间的关系符合Hosemann的a~*定则。