Quasi-crystalline Approximation Research Articles

Detailed comparison of recent—and dissimilar—effective-medium models incorporating dependent scattering

We compare the predictions of two recently derived effective-medium models for the effective refractive index of a turbid suspension of particles. The two formulas are notoriously dissimilar; both are based on the quasi-crystalline approximation, but the approximations used beyond this point are entirely different. Nevertheless, for dilute suspensions both reduce to the well-established van de Hulst formula. The dissimilarities between the formulas are evident for dense suspensions, where dependent-scattering effects are important. When they might coincide is, therefore, not clear. The purpose of this work is to explore the range of particle parameters and volume fractions for which both models are applicable. Our results show that, rather surprisingly, the models produce very similar curves of the real and imaginary parts of the effective refractive index for volume fractions up to 0.4 and for particles comparable to, and larger than, the wavelength, as well as for a fairly large range of refractive-index contrasts between the particles and the surrounding medium. These results significantly increase our confidence in the validity of both models.

Experiment and theory of the complex refractive index of dense colloidal media.

The complex refractive index is analyzed by measuring its scattering attenuation μ s, group index n g, and group velocity dispersion (GVD) for 100nm diameter silica nanoparticles dispersed in water. The experiments were performed for wavelengths between 410nm and 930nm. The experimental results were compared with different mixing models for the complex refractive index of colloidal suspensions. The group index linearly scaled with the volume fraction both in experiment and for all tested models. It was found that the GVD has a nonlinear dependence on volume fraction in agreement with the coupled dipole model of Parola etal. [J. Chem. Phys.141, 124902 (2014)JCPSA60021-960610.1063/1.4895961] The scattering attenuation is in good agreement with both the coupled dipole model and the low frequency quasi-crystalline approximation [J. Electromagn. Waves Appl.2, 757 (1988)JEWAE50920-507110.1163/156939388X00053] that take particle correlations into account. With an iterative fitting procedure of all the data based on both the coupled dipole model and the quasi-crystalline approximation, the refractive index, porosity, and size of the nanoparticles were determined. We determined that the coupled dipole model is in best agreement with the data.

Effective nonlocal behavior of peridynamic random structure composites subjected to body forces with compact support and related prospective problems

We consider a static problem for statistically homogeneous matrix linear bond-based peridynamic composite materials (CMs) subjected to body force with compact support. Estimation of the effective displacements is performed by the exploitation of the most popular tools and concepts used in conventional local elasticity of composite materials (CMs) with their adaptation to peridynamics. The method is based on estimation of a perturbator introduced by one inclusion inside the infinite peristatic matrix subjected to the body force. The statistical averages of both the displacements and stresses are estimated by summation of these perturbators for all possible locations of inclusions. It leads to obtaining of a new general integral equation (GIE) where a renormalizing procedure is not required for the involved absolutely convergent integrals. It allows us to estimate not only the effective nonlocal constitutive equation but also to evaluate the statistical field averages (inhomogeneous and nonlocal) inside the phases at the fine scale that is critically important for advanced modeling (e.g. for any nonlinear phenomena potentially considered in the future). In particular, in the generalized effective field method (EFM) proposed, the effective field is evaluated from self-consistent estimations using closing of a corresponding integral equation in the framework of the quasi-crystalline approximation. In so doing, the classical effective field hypothesis is relaxed, and the hypothesis of the ellipsoidal symmetry of the random structure of CMs is not used. The method proposed is a promising ingredient at the construction of provably robust data-driven surrogate equations describing both the effective behavior and effective displacement concentration operator inside inclusions. Some opportunities for generalizations of the method proposed are considered. Numerical results for the estimation of effective deformations are obtained for one-dimensional (1D) statistically homogeneous peridynamic composite bar with the prescribed self-equilibrated body forces.

Dry Snow Parameter Retrieval With Ground-Based Single-Pass Synthetic Aperture Radar Interferometry

In this article, we investigate the potential of using single-pass InSAR model-based approaches to retrieve dry snow parameters. Two InSAR scattering models of dry snow are considered: the dense-medium random volume over ground (RVoG) model and the simple variant of the full penetration (FP) model. A quasi-crystalline approximation (QCA)-based extinction analysis confirms the negligible extinction dependence of the InSAR observables at L/C/X-band for fresh dry snow. The FP models the low-frequency (L/C/X-band) InSAR phase as a single constraint of snow depth and density, which can be supplemented by an extra observation (e.g., InSAR coherence or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> depth/density). The single-pass InSAR models and inversion approaches were validated using X-band InSAR data collected from a tower-based three-frequency (X/Ku-low/Ku-high) fully polarimetric TomoSAR system, where a multi-frequency polarimetric InSAR analysis and ground-to-volume ratio-based snow condition analysis were conducted. We also analyzed the sensitivity and error propagation of the single-pass InSAR phase and coherence in measuring dry snow depth/density. It was found that the X-band HH-pol FP-modeled single-pass InSAR phase along with RVoG-modeled coherence or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> depth is capable of measuring snow water equivalent (SWE) with a 23–26 mm uncertainty (13–15%) and a 20–26 mm bias (12–15%) for dry snow SWE of 0.2 m, and with an optimal perpendicular baseline on the order of a tenth of the snow depth (0.8 m) at our test site. This single-pass InSAR approach with the FP model is potentially useful and thus needs further investigation for large-scale dry snow retrieval with a wide range of snow conditions using ground-based/airborne/spaceborne low-frequency (L/C/X-band) InSAR observations.

Propagation of scalar waves in dense disordered media exhibiting short- and long-range correlations.

Correlated disorder is at the heart of numerous challenging problematics in physics. In this work we focus on the propagation of acoustic coherent waves in two-dimensional dense disordered media exhibiting long- and short-range structural correlations. The media are obtained by inserting elastic cylinders randomly in a stealth hyperuniform medium itself made up of cylinders. The properties of the coherent wave is studied using an original numerical software. In order to understand and discuss the complex physical phenomena occurring in the different media, we also make use of effective media models derived from the quasicrystalline approximation and the theory of Fikioris and Waterman that provides an explicit expression of the effective wave numbers. Our study shows a very good agreement between numerical and homogenization models up to very high concentrations of scatterers. This study shows that media with both short- and long-range correlations are of strong interest to design materials with original properties.

Temperature-dependent particle stability behavior and its effect on radiative transfer in water/SiO2 nanofluids

Radiative transfer is one of the methods of energy transport that includes in a wide range of applications and we feel it in our daily lives. Thermal radiation transfer plays an effective role in the utilization of renewable energy. The radiative and optical properties, as well as the nature of the radiative scattering, are the basic principles of the thermal radiation transfer. The unique properties of nanofluids offer the unmatched potential for use in energy utilization, the working temperature has a dominant effect on the stability and radiative properties of such type of suspensions. In this research, the radiative transfer (optical properties, the independent and dependent scattering, and radiative properties) in water/SiO2 nanofluids are investigated; taking into consideration the effect of working temperature on the stability of the particles. The effect of the temperature on the stability ratio and particle agglomeration is determined by estimating the radius of gyration of particle agglomerates using the scaling law based on the stability (DLVO) method. The single-scattering approximation (SSA) is used to calculate the radiative properties in the case of independent scattering, while the quasi-crystalline approximation (QCA) is used for this purpose in the case of dependent scattering. The results show that the temperature has a significant effect on the stability of particles and radiative transfer in nanofluids. It was observed by comparing the results from the two approximation methods in the Rayleigh regime. Particle size affects the physical and scattering cross-sectional areas which give a general understanding of the scattering mechanism from small to large particles.

Light scattering and absorption by two-dimensional arrays of nano and micrometer monodisperse spherical silver particles

The problem of light scattering and absorption is considered for a free-standing two-dimensional array (monolayer) of nano- and micrometer spherical silver particles. To solve this problem for short- and imperfect long-range ordered arrays the statistical theory of multiple scattering of waves is used. The key points of the developed semi-analytical method are described. The method is based on the quasicrystalline approximation and the multipole expansion of electromagnetic fields and the dyadic Green's function in terms of vector spherical wave functions. Results for the angular distribution of scattered light, the coefficients of coherent transmission and reflection, incoherent scattering, and absorption of a monolayer are obtained for a spectral range corresponding to the nanoparticle plasmonic resonance. Some features of light propagation through two-dimensional arrays of particles with different spatial order are investigated and discussed. The results are in close agreement with the existing experimental data for a partially ordered array and the calculations for a regular array obtained under the extended low-energy electron diffraction theory.

Scattering and Absorption of Light by a Monolayer of Spherical Particles under Oblique Illumination

The problem of light scattering and absorption by a monolayer of homogeneous spherical particles at an arbitrary angle of incidence of a plane wave, which determines the phase shift between the averaged fields in the particles, has been solved. The solution is based on the quasi-crystalline approximation of the theory of multiple scattering of waves, the mean-field approximation, and the multipole expansion of the fields and the tensor Green function in terms of vector spherical wave functions. Formulas for determining the angular distribution of the incoherently scattered field intensity, the incoherent scattering, absorption, coherent transmission, and reflection coefficients have been derived. The dependences of these characteristics on the direction of illumination are illustrated using partially ordered monolayers of dielectric and semiconductor particles as an example. It is shown that: (i) under oblique illumination the angular distribution of the scattered radiation in azimuthal scattering angle for any polarization of the incident wave (except for p and s) is asymmetric relative to the plane of incidence, (ii) the maximum of the angular distribution of the radiation transmitted through the monolayer can approach its normal with increasing angle of incidence and/or concentration of particles, (iii) there exist conditions under which the bulk of the scattered radiation is concentrated in the direction opposite to the direction of the incident radiation, and (iv) the dependence of the monolayer absorption coefficient on the illumination angle has extrema whose positions and magnitudes are determined by the polarization of the incident wave.

Generalized effective fields method in peridynamic micromechanics of random structure composites

We consider a static problem for statistically homogeneous matrix linear bond-based peridynamic composite materials (CMs). For the media subjected to remote homogeneous volumetric boundary loading, one proved that the effective behavior of this media is governed by conventional effective constitutive equation as in the local elasticity theory. It was made by the most exploitation of the popular tools and concepts used in conventional elasticity of composite materials (CMs) with their adaptation to peridynamics. This is extraction from the material properties a constituent of the matrix properties. Effective moduli are expressed through the introduced new notion of the average local polarization tensor. The average is performed over the surface of the extended inclusion phase rather than over an entire space. Any spatial derivatives of displacement fields are not required. The basic hypotheses of locally elastic micromechanics are generalized to their peridynamic counterparts. In particular, in the generalized effective field method (EFM) proposed, the effective field is evaluated from self-consistent estimations by the use of closing of a corresponding integral equation in the framework of the quasi-crystalline approximation. In so doing, the classical effective field hypothesis is relaxed, and the hypothesis of the ellipsoidal symmetry of the random structure of CMs is not used. One demonstrates some similarity and difference with respect to other methods (the dilute approximation and Mori-Tanaka approach) proposed before in micromechanics of peridynamic CMs. Comparative numerical analyses of these methods are performed for 1D case.

Arctic and subarctic snow microstructure analysis for microwave brightness temperature simulations

Passive microwave (PMW) remote sensing has proven to be a useful approach to characterize the volume of seasonal snowpack in remote northern regions at the synoptic scale. Modeling emitted microwave brightness temperatures (TB) is made possible using a physical radiative transfer model that takes into account microstructural and stratigraphic structure of the snowpack. However, prescribing the microstructure remains a difficult task. This paper aims to find proper microstructure parametrization and the snow emission model formulation that best optimize TB simulations for Arctic and Subarctic snowpacks. Surfaced-based radiometric measurements in conjunction with in-situ snowpack characterization were used for testing different configurations based on the Snow Microwave Radiative Transfer model (SMRT), with two electromagnetic models (Dense Media Radiative Transfer Quasi Crystalline Approximation, DMRT, and Improved Born Approximation, IBA) and two microstructure description theories (Sticky Hard Sphere, SHS, and Exponential, Exp). We compare the performance of three configurations (DMRT-SHS, IBA-SHS and IBA-Exp) with a unique large dataset (119 snowpits with concomitant microwave ground-based radiometer observations) covering a wide range of Arctic and Subarctic snow types in Northern and Eastern Canada. Results show that the input measured microstructure parameters must be scaled up in order to better match simulated and observed TB at 11, 19, 37 and 89 GHz. We show that the IBA-Exp gives the best results, with a Root-Mean-Square Error (RMSE) lower by up to 30% for Subarctic snow and 24% for Arctic snow compare to the other model configurations we used. In addition, we undertake a complementary experiment on isolated homogeneous snow slabs to investigate the sensitivity of the scaling factor to snow microstructure. The retrieved microwave correlation length appears significantly different than the in-situ Debye correlation length. At high frequencies, the observed variability of these scaling factors with frequency and snowpack types means that density, SSA and estimated correlation length seem insufficient to appropriately fully characterize snow microstructure for microwave modeling purposes.

A discussion on the effective permittivity of multi-component medium derived by Maxwell–Garnett, strong fluctuation and quasicrystalline-CP modeling

To evaluate the average dielectric property, the effective permittivity (EP) of multi-component media in low frequency region is the first issue. The most classical formula is the Maxwell–Garnett (MG) model. However, it needs guidelines to restrict the applicability of MG model. Later on, the strong fluctuation theory (SF model) and QCA-CP model (quasi-crystalline approximation with coherent potential) have been studied. Similarities and difference between these models need to be clarified. In this paper, all these models are derived in a consistent way. Numerical examples show the difference between the EPs of these models. It is found that when the dielectric perturbation is not large (dielectric particle is not quite different from background) and especially, the fractional volume of inclusions is not large (e.g. less than 0.15), all these models are good enough to evaluate EP. But, if these pre-conditions are not well met, the MG model becomes unsuitable. Thus the SF and QCA-CP models are better than MG in either cases of strongly or weakly fluctuating cases. If the discrete spherical particles are well defined, e.g. size, complex dielectric constant, the QCA-CP would be a rigorous model for the calculation of EP and extinction coefficient.

Evaluation of Brightness Temperature Sensitivity to Snowpack Physical Properties Using Coupled Snow Physics and Microwave Radiative Transfer Models

There are multiple existing microwave radiative transfer models (RTMs) to simulate the brightness temperature (Tb) of snowpacks. It is still challenging to have consistent Tb responses from RTMs due to individual physical formulations of the snowpack scattering process. This article examines three of the widely-used multi-layer RTMs: 1) the microwave emission model of layered snowpacks (MEMLS); 2) the dense media radiative transfer based on the quasi-crystalline approximation (QCA) of Mie scattering of densely packed sticky spheres (DMRT-QMS); and 3) the Helsinki University of Technology (HUT) model. Interestingly, these models yield slightly different Tb responses when driven by the same physical snowpack properties. Tb variations, dependent on the choice of RTMs, are then evaluated to improve the understanding of model differences in microwave emission from a snowpack. We first perform a sensitivity study of the Tb predictions from the three RTMs as a function of snow grain sizes, densities, and depths. While Tb from all three RTMs decreases with increasing snow grain sizes, it is found that a scaling factor is required to have the same amount of Tb attenuation for small grain sizes within the Rayleigh scattering regime. For larger grain sizes, however, a scaling coefficient is not enough to match the model outputs due to the different scattering assumptions of the RTMs. For a single snow layer with increasing snow depths and densities, all three RTMs exhibit Tb attenuations arising from the increase in path lengths and optical depths. Further evaluations are conducted by feeding the three RTMs with the output of a snow physics model driven by in situ weather forcing in a coupled simulation. Outputs of this coupled model include snowpack physical properties and Tbs. By using snow stratigraphy observations, another set of Tb simulation is also conducted with RTMs driven by in situ snowpit observations. The snow physics outputs from the coupled case are compared against in situ snow stratigraphy observations. And both Tb simulations are compared against ground-based microwave observations from the European Space Agency (ESA) Nordic Snow Radar Experiment (NoSREx) 2009–2012. For three consecutive years, the in situ driven Tbs have 21.0-K root-mean-squared error (RMSE) while the coupled simulations have 24.7-K RMSE. However, after isolating the dry snow period and excluding diurnal melting snow conditions, 12.2- and 6.3-K RMSEs are achieved from in situ and coupled cases, respectively, in the 2011 water year.

Analytical approximation to the complex refractive index of nanofluids with extended applicability.

Currently, there are available a few simple analytical approximations to the complex effective refractive index that may be used for nanofluids. Namely, the Maxwell Garnett mixing formula with scattering corrections, the Maxell Garnett Mie approximation, the Foldy-Lax approximation and the small particle limit of the quasi-crystalline approximation. These approximations are valid either for very small nanoparticles (below a few nanometers in radius) or for very dilute nanofluids (below about 1% in particles' volume fractions) and therefore, do not cover the whole domain of particle suspensions referred to as nanofluids. Here we propose a new simple analytical approximation based on local field corrections to the Foldy-Lax approximation. The new mixing formula coincides with the mentioned approximations when they are expected to be valid and provides physically sound predictions when the mentioned approximations are no longer valid, within the realm of nanofluids. We compare predictions of the analytical approximations considered in this work with experimental data published earlier for nanofluids of polystyrene in water.

Spatial Order and Absorption of light by Monolayer of Silicon Nano- and Submicrometer-Sized Particles

The effect of the spatial order of a monolayer of monodisperse spherical crystalline silicon nano- and submicrometer-sized particles upon its absorption coefficient is theoretically investigated. The calculation method is based on the quasicrystalline approximation of the theory of multiple scattering of waves and multipole expansion of electromagnetic fields and tensor Green function in terms of vector spherical wave functions. The results can be used for an enhancement of light harvesting in a solar cells design.

Absorption of light by a particulate monolayer: Effect of ordering, concentration, and size of particles

Absorption of light by a free standing monolayer of monodisperse crystalline silicon (c-Si) spherical particles is investigated using the quasicrystalline approximation (QCA) of the theory of multiple scattering of waves. The simulation results for the partially ordered monolayer and the highly ordered monolayer with imperfect triangular lattice are presented in the ranges of particle diameters from 0.05 µm to 1 µm, monolayer filling factors from 0.1 to 0.7, and wavelengths of incident light from 0.3 µm to 1.12 µm. The spatial pattern of a monolayer affects the absorption coefficient most strongly when the particle sizes are comparable with the wavelength. The spatial-order-induced peaks in the absorption spectra of a highly ordered monolayer are studied. The spectral absorption by such a monolayer can be more than 30 times greater than that of the partially ordered monolayer in the narrow wavelength intervals. The integrated over the terrestrial solar spectral irradiance “Global tilt” ASTM G173-03 absorption coefficient is analyzed. In the wavelength range of 0.3 µm ≤ λ ≤ 1.12 µm the integral absorption can be almost twice as large as that for the partially ordered monolayer. In the wavelength range of small absorption index of c-Si (0.8 µm ≤ λ ≤ 1.12 µm) the integral absorption coefficient of highly ordered monolayer can be more than 10 times greater than for the partially ordered monolayer.

Scattering of Polarized and Natural Light by a Monolayer of Spherical Homogeneous Spatially Ordered Particles under Normal Illumination

Expressions for determining the optical characteristics of a monolayer of homogeneous monodisperse spherical particles under normal illumination by a plane electromagnetic wave with arbitrary polarization and nonpolarized radiation have been obtained. They are based on the quasi-crystalline approximation (QCA) of the theory of multiple scattering of waves (TMSW) and the multipole decomposition of fields and the tensor Green function in terms of the vector spherical wave functions. The influence of the polarization state of the incident wave on the angular structure of the radiation that is scattered by a partially ordered monolayer and a monolayer with a imperfect lattice has been considered.

Interface integral technique for the thermoelasticity of random structure matrix composites

We consider linear thermoelastic composite media, which consist of a homogeneous matrix containing a statistically homogeneous random set of aligned homogeneous heterogeneities of non-canonical (i.e. non-ellipsoidal) shape. The representations of the effective properties (effective moduli, thermal expansion, and stored energy) are expressed through the statistical averages of the interface polarization tensors introduced apparently for the first time. The properties of the interface polarization tensors are described. The new general integral equations connecting the stress and strain fields in the point being considered with the stress and strain fields in the surrounding points are obtained for the random fields of heterogeneities. The method is based on a recently developed centering procedure where the notion of a perturbator is introduced in terms of boundary interface integrals estimated by the method of fundamental solution for a single inclusion inside the infinite matrix. This enables us to reconsider basic concepts of micromechanics such as effective field hypothesis, quasi-crystalline approximation, and the hypothesis of “ellipsoidal symmetry.” Effective properties (such as effective moduli, thermal expansion, and stored energy) as well as the first statistical moments of stresses in the phases are estimated for statistically homogeneous composites with the general case of the inclusion shape. The results of this reconsideration are quantitatively estimated for some modeled statistically homogeneous composites reinforced by aligned homogeneous heterogeneities of non-canonical shape. The explicit new representations of the effective thermoelastic properties and stress concentration factor are expressed through some building blocks ( perturbators) described by numerical solutions for one heterogeneity inside the infinite medium subjected to the homogeneous remote loading. Some new effects are detected that are impossible in the framework of a classical background of micromechanics.

Collective modes of two-dimensional classical Coulomb fluids.

Molecular dynamics simulations have been performed to investigate in detail collective modes spectra of two-dimensional Coulomb fluids in a wide range of coupling. The obtained dispersion relations are compared with theoretical approaches based on quasi-crystalline approximation, also known as the quasi-localized charge approximation, in the plasma-related context. An overall satisfactory agreement between theory and simulations is documented for the longitudinal mode at moderate coupling and in the long-wavelength domain at strong coupling. For the transverse mode, satisfactory agreement in the long-wavelength domain is only reached at very strong coupling, when the cutoff wave-number below which shear waves cannot propagate becomes small. The dependence of the cutoff wave-number for shear waves on the coupling parameter is obtained.

Effect of dependent scattering on light absorption in highly scattering random media

In the last a few decades, the approximate nature of radiative transfer equation (RTE) leads to a bunch of considerations on the effect of dependent scattering in random media, especially in particulate media composed of discrete scatterers. This effect usually indicates those deviations of RTE from experimental and exact numerical results due to electromagnetic wave interference. Here we theoretically and numerically demonstrate the effect of dependent scattering on absorption in disordered media consisting of highly scattering scatterers. By making comparisons between the independent scattering approximation-radiative transfer equation (ISA-RTE) approach and the full-wave coupled dipole method (CDM), we find that deviations between the two approaches increase as the scatterer density increases. The discrepancy also grows with the optical thickness of the whole random media. To quantitatively take dependent scattering effect into account, we develop a theoretical model of the dependent-scattering corrected radiative properties, based on the path-integral diagrammatic technique and the quasi-crystalline approximation (QCA) in the multiple scattering theory. The model results in a more reasonable agreement with numerical simulations. The present work is of practical importance in correctly modeling light absorptance in random media and interpreting the experimental observations in various applications for random media, such as solar energy concentration, micro/nanofluids, structural coloration, etc. It also has profound implications for the coherent scattering physics in random media with absorption.

On the extinction coefficient of light in non-absorbing nanoparticle suspensions.

We assess the validity and possible use of a simple formula for the complex effective refractive index of a colloidal suspension of very small particles obtained from the quasi-crystalline approximation. This approximation takes into account the so-called dependent scattering effects, which are strongest, in relative terms, in suspensions of non-absorbing nanometer-sized particles. We test experimentally the predictions of the model for the extinction of light in dispersions of sodium dodecyl sulfate micelles in tri-distilled water. We find that the theoretical model can be fitted to the experimental data and provide estimates of the micelles' size and the minimum distance between each other in suspension.