Light scattering by some nematic liquid crystals due to phase transitions

New computer modelling of light scattering and its propagation through liquid crystal has been presented using T-matrix method in the structural phase transition regions. Numerical aspects of light scattering process, which are based on numerically solving Maxwell's equations, were calculated for some nematic liquid crystals. Firstly, we described in detail T-matrix method for computing light scattering from nematic liquid crystals and presented results of benchmark computations for the considered model. We reported results of extensive calculations for polydisperse, randomly oriented rod-like multilayered systems (nematic liquid crystals). Our results are associated with light scattering by ferroelectric and ferroelastic materials.     

Multiple small-angle scattering—A review

Small-angle scattering (SAS) is a powerful experimental technique in condensed matter physics for studying structural features of inhomogeneities of colloidal dimensions. So far the technique has been largely exploited to study thin samples for which the single scattering approximation, for the radiation-matrix interaction, holds good. The single scattering approximation is invalid when the thickness of the sample exceeds the scattering mean free path. This situation calls for a guideline to analyse the scattering data having significant contribution from multiple scattering. Since multiple scattering broadens the scattering profile, the beam broadening nature of multiple scattering can also be exploited, by making the sample suitably thick, to study large size inhomogeneities which are otherwise inaccessible to a small-angle scattering set up because of its resolution constraints. The present article presents a review and extension of the theoretical basis for analysing multiple scattering data from the point of view of a recent formalism on multiple small-angle scattering. The formalism is valid for both monodisperse and polydisperse scattering media characterized by the presence of large size inhomogeneities in the matrix. It is shown that multiple scattering from a polydisperse sample can be described by a system of coupled integrodifferential equation. However, multiple scattering from a monodisperse sample can be described by a Fokker-Planck type of equation. These equations have been analysed with an emphasis laid on the nature of the structural information pertaining to the inhomogeneities which is extractable from the multiple scattering profile. When the linear dimension of inhomogeneities becomes comparable to the scattering mean free path of the radiation in the sample, the statistical nature of the medium becomes pronounced. The statistical nature of the medium modulates the scattering profile. The modulation effect could be broadening or narrowing of the profile depending upon the nature of the inhomogeneities and their population distribution. The limiting regimes of validity and the implications of various approximations, frequently used to analyse the scattering data, have been indicated.     

Light-scattering by optically soft randomly oriented spheroids

In the framework of the Rayleigh–Gans–Debye approximation and anomalous diffraction approaches, the light scattering characteristics of randomly oriented spheroids have been investigated. It has been proved that the system of randomly oriented spheroids is equivalent to the system of polydisperse spherical particles that have the same values of volume and surface area as nonspherical particles. The power law size distribution meeting these requirements has been obtained.     

Optical equivalence of isotropic ensembles of ellipsoidal particles in the Rayleigh-Gans-Debye and anomalous diffraction approximations and its consequences

Light scattering by isotropic ensembles of ellipsoidal particles is considered in the Rayleigh-Gans-Debye approximation. It is proved that randomly oriented ellipsoidal particles are optically equivalent to polydisperse randomly oriented spheroidal particles and polydisperse spherical particles. Density functions of the shape and size distributions for equivalent ensembles of spheroidal and spherical particles are presented. In the anomalous diffraction approximation, equivalent ensembles of particles are shown to also have equal extinction, scattering, and absorption coefficients. Consequences of optical equivalence are considered. The results are illustrated by numerical calculations of the angular dependence of the scattering phase function using the T-matrix method and the Mie theory.     

Single scattering profile from small-angle scattering data affected by multiple scattering: use of a basis function set

Present work involves extraction of single-scattering profile from small-angle scattering data, affected by multiple scattering, using an existing model-independent algorithm [Mazumder et al., Physica B 241–243, 1222 (1998), Mazumder et al., J. Appl. Crystallogr. 36, 840 (2003)]. It is shown that implementation of the algorithm becomes effective by representing the multiple scattering profiles in terms of a set of basis function whose form of Hankel transformed pair is known analytically. In the present paper, the methodology has been presented by introducing a Gaussian basis set and has been tested for various simulated profiles. The performance of this methodology has also been tested for scattering profiles having complex internal features and moderate statistical noise. Subsequently, the same has been applied for experimental small-angle neutron scattering data.     

Modeling single-scattering properties of small cirrus particles by use of a size-shape distribution of ice spheroids and cylinders

In this study, we model single-scattering properties of small cirrus crystals using mixtures of polydisperse, randomly oriented spheroids and cylinders with varying aspect ratios and with a refractive index representative of water ice at a wavelength of 1.88 μm. The Stokes scattering matrix elements averaged over wide shape distributions of spheroids and cylinders are compared with those computed for polydisperse surface-equivalent spheres. The shape-averaged phase function for a mixture of oblate and prolate spheroids is smooth, featureless, and nearly flat at side-scattering angles and closely resembles those typically measured for cirrus. Compared with the ensemble-averaged phase function for spheroids, that for a shape distribution of cylinders shows a relatively deeper minimum at side-scattering angles. This may indicate that light scattering from realistic cirrus crystals can be better represented by a shape mixture of ice spheroids. Interestingly, the single-scattering properties of shape-averaged oblate and prolate cylinders are very similar to those of compact cylinders with a diameter-to-length ratio of unity. The differences in the optical cross sections, single-scattering albedo, and asymmetry parameter between the spherical and the nonspherical particles studied appear to be relatively small. This may suggest that for a given optical thickness, the influence of particle shape on the radiative forcing caused by a cloud composed of small ice crystals can be negligible.     

Small-angle neutron scattering contrast variation on magnetite-myristic acid-benzene magnetic fluid

A magnetic fluid with magnetite dispersed in D-benzene and stabilized by myristic acid is investigated using the contrast variation technique in small-angle neutron scattering. The results obtained are interpreted within a new approach to the basic functions for polydisperse multicomponent and superparamagnetic systems. Myristic acid is considered an alternative to oleic acid, which is commonly used in these fluids. The parameters characterizing the particle size distribution function and the thickness of the myristic acid layer are determined. The data obtained are in good agreement with the results derived from previous investigations with the use of polarized neutrons.     

Combined Scheme of Reconstruction of the Particle Size Distribution Function Using Small-Angle Scattering Data

JETP Letters - To analyze polydisperse systems of nanoparticles, the particle size distribution function can be determined from small-angle X-ray and neutron scattering data using some algorithms....     

On determination of the structural parameters of polydisperse magnetic fluids by small-angle neutron scattering

Factors affecting the determination of the structural parameters of polydisperse particles of typical magnetic fluids (MFs) from small-angle neutron scattering data (SANS), including the possible anisotropy of the particle shape and the residual incoherent background, are analyzed. It is shown that the effect of particle anisotropy on scattering must be considered in combination with the influence of their polydispersity. The relative shift in the MF’s structural parameters caused by particle anisotropy does not exceed 10–20% for particles with an anisotropy parameter which is less than four and becomes insignificant for more anisotropic particles. The main effect of the residual incoherent background of neutron scattering is related to determination of the thickness of the stabilizing shell and the generalized contrast parameter. It is shown that restrictions on the available maximum values of the scattering vector, measured in the experiment, lead to a relative shift in these parameters by up to 10%, while the other structural parameters (the average particle radius and the width of the size distribution) are determined with sufficiently high accuracy (the relative systematic error is 5% or less).     

Structural analysis of composite metal-insulator materials by small-angle x-ray scattering

A small-angle x-ray scattering study of the structure of Cu: SiO₂ composite films obtained by magnetron cosputtering is reported. The experimental spectra are analyzed by direct numerical simulation of scattering from a polydisperse system of spherical particles with a high volume concentration. The calculated scattering spectra were found to fit well to the experiment if a log-normal particle distribution in size is assumed, and the parameters of this distribution were determined.     

Study of quasi-two- and three-dimensional ordered porous structures by means of small-angle X-ray scattering in the grazing incidence geometry

The structure of surface layers of thin metal inverse opals has been studied first by the grazing-incidence small-angle X-ray scattering technique. Contributions of the form factor and structure factor to the small-angle diffraction pattern have been separated using a numerical model of the scattering process. The complementary use of the small-angle X-ray scattering and grazing-incidence small-angle X-ray scattering techniques has provided independent information about the bulk and surface properties of the samples and allowed a type of defect in the investigated structures to be determined. The measurement results have been verified by atomic force microscopy.

     

Revision of the model of a fibril with amorphous nodules for oriented soft-chain semicrystalline polymers

A critical analysis of literature data about calculations of small-angle x-ray scattering (SAXS) from a model of a fibril with amorphous nodules (FAN) is made. Changes in the SAXS patterns are reviewed for the effect of stretching the FAN. Possibilities of using the calculation results for interpretation of microdeformational behavior of oriented, flexible chain semicrystalline polymers are analyzed. Invoking the FAN model, a retrospective analysis of some literature data and also analysis of new experimental results obtained by SAXS and wide-angle x-ray scattering methods for oriented samples of cellulose triacetate, poly(vinylidene fluoride-co-hexafluoropropylene), and poly(vinyl alcohol) are carried out.     

Angular structure of radiation scattered by monolayer of polydisperse droplets of nematic liquid crystal

We considered light scattering by a polydisperse ensemble of droplets of a nematic liquid crystal. To model light scattering by a monolayer of polymer-dispersed spherical droplets of a nematic liquid crystal with cylindrical symmetry of its internal structure, we proposed a semianalytical modeling method. The method is based on interference approximation of the theory of multiple wave scattering, anomalous diffraction approximation, and effective-medium approximation. The method takes into account cooperative optical effects in concentrated, partially ordered layers and can be used to analyze the small-angle structure of the intensity of scattered radiation in relation to the concentration, size, polydispersity of liquid crystal droplets, orientation of their optical axes, and refractive indices of the liquid crystal and polymer. The obtained relations can be applied to solving direct and inverse problems of light scattering in composite liquid crystal materials using data of polarization measurements. We present graphical results of solving the direct problem for components of the polarization vector of scattered wave. These results illustrate the formation of an angular structure for monolayers with a high concentration of polydisperse droplets of the liquid crystal in the range of small scattering angles (0 < θ _s ≤ 8°).     

Development of methods for processing spectra of multiple small-angle neutron scattering

A method has been developed for processing spectra of multiple small-angle neutron scattering (MSANS) to obtain information about substance heterogeneities, namely, their size and concentration. A method has been proposed for constructing the dependence of the MSANS line width on the sample thickness starting from the angular neutron distribution measured for one sample. The standard method for processing this dependence has been improved for application at any scattering multiplicity and complete account for the instrumental line of the double-crystal spectrometer. The method has been tested for the MSANS spectra of the samples of the Fe-Ni ferromagnet, Al powders, and HTSC ceramics.     

Characterization of the structure of ultradispersed diamond using X-ray diffractometry and small-angle X-ray scattering

Two samples of detonation-synthesized ultradispersed diamond have been studied using X-ray diffractometry and small-angle X-ray scattering. It has been shown based on the X-ray diffractometry data that two samples contain regions with both the diamond and graphite-like lattices. Grain radii inside both samples are evaluated from the small-angle scattering data as 30–50 nm. The samples also contain low-dimensional components. A broad Bragg peak corresponding to a set of interplanar distances from 5 to 15 nm is revealed in the small-angle scattering curve of sample no. 1. A structural model of ultradispersed diamond particles, which represents the diamond core surrounded by concentric graphite shells similar to the onionskin, is confirmed.     

Anomalous diffraction of light with geometrical path statistics of rays and a Gaussian ray approximation

The anomalous-diffraction theory (ADT) of extinction of light by soft particles is shown to be determined by a statistical distribution of the geometrical paths of individual rays inside the particles. Light extinction depends on the mean and the mean-squared geometrical paths of the rays. Analytical formulas for optical efficiencies from a Gaussian distribution of the geometrical paths of rays are derived. This Gaussian ray approximation reduces to the exact ADT in the intermediate case of light scattering for an arbitrary soft particle and describes well the extinction of light from a system of randomly oriented and (or) polydisperse particles. The implications for probing of the sizes and shapes of particles by light extinction are discussed.     

Study of dispersion and absorption of an ensemble of spherical particles inside an optical cavity and new possibilities of predicting the optical characteristics of biological media by intracavity laser spectroscopy

Optical characteristics, namely, dispersion and absorption spectra of an ensemble of spherical particles randomly oriented inside an optical cavity are investigated. The study is based on the self-consistent matching of new data from the inhomogeneous optical cavity with data from the scattering of an ensemble of spherical particles of different size, randomly oriented in free space. As a result, a new model, which self-consistently accounts for multiple scattering in the optical cavity, has been developed to predict absorption and dispersion of ensembles of spherical particles. This model is supposed to enhance potentiality of the intracavity method for plotting wavelength dependences of optical characteristics of media. A specific calculation of dispersion and absorption dependences on the wavelength shows that this method can be used for investigation of biological media consisting of spherical particles, in particular, erythrocyte suspension.     

Theoretical and experimental study of blurring of a femtosecond laser pulse in a turbid medium

We propose an analytical model of the spatio-temporal structure of a short laser pulse transmitted through a layer of an optically inhomogeneous medium with high anisotropy of scattering. The light-field brightness in the medium is represented as a finite series in terms of multiplicities of the small-angle scattering, while the contribution from the higher-order scattering is allowed for as a quasi-diffuse component. The scattered-pulse structure is calculated on the basis of solving the radiative-transfer equation in the small-angle approximation with allowance for the effect of multipath light propagation. Compared with the first approximation of the multiple-scattering theory (attenuated nonscattered light plus the diffuse component), this approach makes it possible to describe more correctly the transformation of the spatio-angular distribution of light in the medium when passing from the single-scattering to multiple-scattering regime, as well as specify the temporal profile of the scattered pulse. The temporal profile of the femtosecond pulse transmitted through a layer of model scattering medium with various concentrations of scatterers is studied experimentally. The blurred-pulse structure is studied with the help of nonlinear optical gating in the case of noncollinear generation of the second harmonic. Good agreement between the theoretical and experimental time profiles of the scattered pulse is shown for the optical-thickness intervals corresponding to both the predominantly low multiplicity scattering and multiple small-angle scattering, which allows us to use the proposed analytical model for solving the inverse problem of the pulse sounding of a homogeneous turbid medium. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 51, No. 4, pp. 333–348, April 2008.     

Structure determination of Pt‐coated Au dumbbells via fluctuation X‐ray scattering

A fluctuation X‐ray scattering experiment has been carried out on platinum‐coated gold nanoparticles randomly oriented on a substrate. A complete algorithm for determining the electron density of an individual particle from diffraction patterns of many particles randomly oriented about a single axis is demonstrated. This algorithm operates on angular correlations among the measured intensity distributions and recovers the angular correlation functions of a single particle from measured diffraction patterns. Taking advantage of the cylindrical symmetry of the nanoparticles, a cylindrical slice model is proposed to reconstruct the structure of the nanoparticles by fitting the experimental ring angular auto‐correlation and small‐angle scattering data obtained from many scattering patterns. The physical meaning of the refined structure is discussed in terms of their statistical distributions of the shape and electron density profile.     

Multiple small-angle neutron scattering in media with a high concentration of inhomogeneities

Multiple small-angle neutron scattering (MSANS) on a polydisperse system of scatterers was considered taking into account interparticle interference. The effect of the scatterer’s size distribution on the MSANS spectrum was studied. The exponential and log-normal distributions were analyzed. It was shown that the MSANS linewidth decreases at high scatterer concentrations due to the contribution of interparticle interference. The dependence of this effect on the scattering multiplicity and scatterer’s size dispersion was studied in detail.