Small-angle light scattering from an anisotropic sphere: The mueller matrix approach

The explicit form of the Mueller scattering matrix, which characterizes the small-angle light scattering from an anisotropic sphere when the requirements of the Rayleigh-Gans-Debye (RGD) approximation are fulfilled, contains all information obtainable about the RGD scattering from an anisotropic sphere taken as a model for a spherulite. A comparison of angular dependences of single matrix elements for the Lorenz-Mie sphere, the Rayleigh particle, and the simplified form of the presented matrix (taking a sphere without inherent anisotropy, i.e., Δn = 0) shows very good agreement within the limits of RGD approximations. The polarized small-angle light scattering intensities Hυ and Vυ are combinations of the single matrix elements. Their explicit form is in accord with the expressions for Hυ and Vυ intensities recently rederived from a 2 × 2 amplitude scattering matrix. It has been shown that the angular dependence of matrix elements is determined by the (n? — 1)/Δn parameter, where n — is the mean refractive index andδ n is the anisotropy, both measured relative to the surrounding medium. The expressions for Hυ and Vυ intensities derived by Stein and Rhodes fail for a sphere without inherenet anisotropy ( δ n = 0); and the commonly used procedure of size determination from a maximum of Hυ intensity has limited validity (it holds only approximately under the condition of a small phase shift and small (n?-1)/ δ n). Further theoretical work must be done to understand and construct scattering models for situations where the RGD approximately is inappropriate.     

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.     

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.     

Small-angle scattering of polarized light. I. comparison of theoretical predictions for isotropic and anisotropic spheres

This paper describes small angle light scattering (SALS) calculations and compares the results for isotropic and anisotropic spheres. For isotropic spheres the exact Mie theory is compared with the Rayleigh-Gans-Debye (RGD) and the anomalous diffraction (AD) approximations in order to study the validity of the approximations. For anisotropic spheres the RGD is compared with the AD approximation. A study of SALS in the RGD approximation shows that the first-order intensity maximum corresponding to a lobe of the four-leaf clover pattern is not only a function of the sphere diameter but also depends on the isotropic and the anisotropic refractive index of the sphere. The practical implications of this are discussed.     

A scattering study on intermicellar interactions and structures of polymeric micelles

The micellar structure is usually considered to be composed of a hard sphere (liquid) core and a heavily solvated corona. Therefore, the correction for intermicellar interactions at finite concentrations can be relatively complicated. In this article, small-angle neutron scattering of a copolymer, known as Pluronic L64 (PEO₁₃PPO₃₀PEO₁₃), in o-xylene in the presence of D₂O is used to demonstrate that, based on the hard sphere approximation, intermicellar interactions can be corrected by representing the micellar size as having an equivalent hard sphere radius. The procedure remains valid even if the micellar shape becomes asymmetric, with axial ratios of 3 ? 4. For the present system, the equivalent hard sphere radius corresponds to the micellar core radius plus one-half of the micellar shell thickness. With the equivalent hard sphere approach, the scattering behavior of the micelle could be described by using a core-shell structure. © 1994 John Wiley & Sons, Inc.     

Statistical approach to light scattering from deformed textured heterogeneous polymer materials

A statistical theory of light scattering from deformed isotropic and textured heterogeneous polymer materials is formulated. Two types of textured structures are analyzed: assemblies of optically isotropic and anisotropic rods and a spatially anisotropic distribution of isotropic spherical inclusion centers. The small-angle H_v light-scattering patterns are calculated. The appearance of scattering from isotropic rods and spheres in deformed materials has been demonstrated. The changes of the H_v scattering patterns as a function of elongation and strucuture parameters are discussed. © 1993 John Wiley & Sons, Inc.     

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.     

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.     

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.     

Studies on biaxial stretching of polypropylene film. IV. Overall orientation during one-step biaxial stretching and its application to the estimation of the principal refractive indices of isotactic polypropylene

Polypropylene film was biaxially stretched in one step in air at 140°C or 152°C, and the deformation was studied optically. A linear relation held between Δn_ss and v_A^?½ for v_A > 10, at both temperatures, where Δn_ss is the birefringence with respect to the normal to the film and v_A is the degree of stretching expressed as the factor by which the area of the film is increased. Extrapolation of data in this linear region yielded a value of 20 × 10^?3 for ?Δn_ss at infinite v_A. Since it is presumed that the polypropylene molecules lie completely parallel to the film surface when the film is stretched infinitely, ?Δn_ss at v_A^?½ = 0 must be just half Δn°, the intrinsic birefringence in the case of completely parallel orientation. Thus, Δn° must be 40 × 10^?3. This value was obtained experimentally in uniaxial stretching when the birefringence with respect to the direction of drawing was extrapolated to infinite extension. Similar relations held between n_p, the average of the refractive indices in the two stretching directions, and v_A, and between n_ss, the index normal to the film, and v_A. By similar extrapolations, (1/2)(n′_γ + n′_β) and n′_β = n*_α′ were estimated, and thence nα′ was obtained. Here, n′_α and n′_β are the refractive indices along the c axis (molecular chain axis) and b axis. All these optical parameters refer to a density of 0.900 g/cm³. Hence by applying a density correction to those values, the principal refractive indices and the intrinsic birefringence of polypropylene crystal were evaluated as follows: n_α = 1.5522, n_β = n*_α = 1.5106 and Δn_c° = 4.16 × 10^?3, where n*_α is the refractive index prependicular to the b and c axes of the crystal.     

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.     

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.     

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.     

Optical anisotropy of polymeric films measured by waveguide propagation mode determination

Extruded thin films of a liquid-crystalline charge-conjugated rigid-rod polymer poly(p-phenylenebenzobisthiazole), PBT, and a semicrystalline thermoplastic polyethylene-terephthalate (Mylar) were fabricated and examined for film thickness, refractive index, and linear attenuation coefficient. Optical waveguide modes were successfully induced on the polymeric films using a prism coupler at λ = 633 and 1300 nm. Highly consistent thickness values were obtained for the polymeric films. In addition, the anisotropic nature of the optical properties was determined using TE and TM propagation modes. A refractive index as high as 2.3 was observed on PBT film. The refractive index data suggested that the PBT and Mylar films were optically anisotropic with refractive indices n? (out-of-plane) invariably smaller than n∥ (in-plane). Large anisotropy was also discovered in the linear attenuation coefficient α, with α_∥/α_? ≈ 50 for the Mylar films. Complementary polarimetric and spectroscopic interference measurements were also applied to investigate the optical anisotropy of the extruded polymeric films. © 1992 John Wiley & Sons, Inc.     

Block copolymers of poly(ethylene terephthalate)-poly(butylene terephthalate). II. Small-angle light-scattering studies

Small-angle light-scattering (SALS) studies were carried out on block copolymers of poly(ethylene terephthalate) (PET) and poly(butylene terephthalate) (PBT), the synthesis and characterization of which have been reported in an earlier paper. Samples were crystallized isothermally from the melt at 95°C for predetermined crystallization times in order to follow the formation and growth of crystalline superstructure. During the early stages of crystallization, the block copolymers showed unusual H_v patterns with the four lobes along the polarizer directions, while at later stages they showed the usual H_v patterns with the four lobes at 45° to the polarizer directions. The unusual patterns are characteristic of PBT superstructure, while the usual patterns are characteristic of PET superstructure. These results show that PBT, which is the faster-crystallizing component, crystallizes first and provides nucleation sites for the crystallization of PET, which crystallizes later. Similar behavior was not observed in PET homopolymer and random copolymers of equivalent compositions. In each case the spherulite size increased with the time of crystallization. The ultimate spherulite size decreased with increasing PBT content in the block copolymer, thus showing an increase in nucleation density. It was demonstrated that light scattering is a useful tool to characterize block copolymers of two crystalline components which have different types of superstructure.     

Light scattering from swollen textured polymer network composites

Light scattering from a swollen textured heterogeneous polymer network is discussed using example of a network composite filled by spherical inclusions anisotropically distributed in the matrix. The dependence of the scattering intensity on the Fourier transform of the correlation function dry-state shear moduli fluctuations has been established. H_v scattering patterns from uniaxial textures and their ratio with the composite anisotropy parameter are analyzed. The sensitivity of the H_v scattering pattern on restrictions during the swelling is emphasized. © 1993 John Wiley & Sons, Inc.     

Solution characterization of the novel organometallic polymer poly(ferrocenyldimethylsilane)

A series of four well‐defined poly(ferrocenyldimethylsilane) (PFS) samples spanning a molecular weight range of approximately 10,000–100,000 g mol⁻¹ was synthesized by the living anionic polymerization of dimethyl[1]silaferrocenophane initiated with n‐BuLi. The polymers possessed narrow polydispersities and were used to characterize the solution behavior of PFS in tetrahydrofuran (THF). The weight‐average molecular weights (M_w ) of the polymers were determined by low‐angle laser light scattering (LALLS), conventional gel permeation chromatography (GPC), and GPC equipped with a triple detector (refractive index, light scattering, and viscosity). The molecular weight calculated by conventional GPC, with polystyrene standards, underestimated the true value in comparison with LALLS and GPC with the triple detection system. The Mark–Houwink parameter a for PFS in THF was 0.62 (k = 2.5 × 10⁻⁴), which is indicative of fairly marginal polymer–solvent interactions. The scaling exponent between the radius of gyration and M_w was 0.54, also consistent with marginal polymer–solvent interactions for PFS in THF. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 3032–3041, 2000     

Effects of amorphous poly(vinyl acetate) on crystalline morphology of poly(3-hydroxybutyric acid-co-3-hydroxyvaleric acid)

The amorphous and crystalline phase behavior, spherulite morphology, and interactions between amorphous poly(vinyl acetate) (PVAc) and poly(3-hydroxybutyric acid-co-3-hydroxyvaleric acid) (PHBV) were examined using differential scanning calorimetry, polarized-light optical and scanning electron, atomic-force microscopy (DSC, POM, SEM, AFM), and small-angle X-ray scattering (SAXS). The PHBV/PVAc blend was found to be miscible with an almost linear T _g-composition relationship, indicating perfect homogeneity. Interaction parameter by melting point depression is a negative value of χ = −0.32, suggesting quite favorable interaction strength. With the intimate interaction between the amorphous PVAc and crystalline PHBV polymers, effects of PVAc on the spherulitic morphology of PHBV are quite significant. Owing to the higher T _g of PVAc (than that of PHBV), the spherulite growth rate of PHBV was depressed by increasing PVAc content in blends. Neat PHBV exhibits ring-banded spherulites when crystallized at T_c = 60 ~ 110^° C {T_{\rm{c}}} = {6}0\sim {11}0^\circ {\hbox{C}} ; however, with increasing PVAc content in the blends, the temperature range at which the PHBV/PVAc blends exhibit ring-banded spherulites remains similar but the regularity increases, and the inter-ring spacing significantly decreases. In addition, the spherulite size and ring-band patterns therein are strongly dependent on T _max (190 vs. 220 °C, respectively, for erasing prior nuclei), from which the blends were quenched to a T _c (60–110 °C) for crystallization. For PHBV/PVAc blends crystallized at the same T _c from different T _max, higher T _max tends to erase nuclei, leading to larger spherulites. However, such larger spherulites owing to higher T _max are not necessarily packed with thicker lamellae.     

The influence of hydrogen chloride doping level on the complex refractive indices of anisotropic polyaniline film: Application of a new internal reflection waveguide coupling technique

Nondestructive three‐dimensional refractive‐index measurements are used for the determination of both crystallinity and orientation in thin polymer films. The prism waveguide coupler is particularly suitable for three‐dimensional isotropic and anisotropic thin‐film studies because of the quantitative character of the information obtained and the ease of data acquisition. It has been limited, however, to measuring the refractive index of optically transparent or weakly absorbing films. This study shows that a modified prism waveguide coupler can be used to determine the complex refractive index over a range from transparent to highly absorbing films from the internally reflected light intensity. Thus, both the refractive index, n, and the extinction coefficient, k, can be obtained. This method is used to determine the anisotropic three‐dimensional n and k values of spin‐coated emeraldine base and hydrogen chloride doped emeraldine salt polyaniline films. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2481–2490, 2001     

Characterization of poly(methyl methacrylate) nanoparticles prepared by nanoprecipitation using analytical ultracentrifugation,dynamic light scattering,and scanning electron microscopy

Nanoprecipitation represents an effective method for the production of polymeric nanoparticles. This technique was used to prepare nanoparticles from solutions of poly(methyl methacrylate) and its copolymers. Since the regulation of main parameters like particle size, particle size distribution, and molar particle mass is very important for future applications, the stable nanoparticle dispersions were examined by scanning electron microscopy, velocity sedimentation, and dynamic light scattering, whereby advantages and disadvantages of each characterization techniques are discussed. Polydispersities of particle size distributions are determined by the ratio of d_w/d_n, where d_w and d_n are weight‐ and number‐average diameters, respectively. The particle characteristics strongly depend on the chemical structure of the polymers and the way of preparation and, therefore, vary in the studied cases in the range of 6 < d_w < 680 nm, whereas the polydispersity index d_w/d_n changes in the range of 1.02 to 1.40. It is shown that nanoparticles in a desirable size range can be prepared by solvent–nonsolvent methods (dialysis technique or dropping technique). © 2010 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 48: 3924–3931, 2010