Small Particle Limit Research Articles

Efficient point-based simulation of four-way coupled particles in turbulence at high number density.

In many natural and industrial applications, turbulent flows encompass some form of dispersed particles. Although this type of multiphase turbulent flow is omnipresent, its numerical modeling has proven to be a remarkably challenging problem. Models that fully resolve the particle phase are computationally very expensive, strongly limiting the number of particles that can be considered in practice. This warrants the need for efficient reduced order modeling of the complex system of particles in turbulence that can handle high number densities of particles. Here we present an efficient method for point-based simulation of particles in turbulence that are four-way coupled. In contrast with traditional one-way coupled simulations, where only the effect of the fluid phase on the particle phase is modeled, this method additionally captures the back-reaction of the particle phase on the fluid phase, as well as the interactions between particles themselves. We focus on the most challenging case of very light particles or bubbles, which show strong clustering in the high-vorticity regions of the fluid. This strong clustering poses numerical difficulties which are systematically treated in our work. Our method is valid in the limit of small particles with respect to the Kolmogorov scales of the flow and is able to handle very large number densities of particles. This methods paves the way for comprehensive studies of the collective effect of small particles in fluid turbulence for a multitude of applications.

Angular momentum transfer from swift electrons to non-spherical nanoparticles within the dipolar approximation

In this work, we study the angular momentum transfer from a single swift electron to non-spherical metallic nanoparticles, specifically investigating spheroidal and polyhedral (Platonic Solids) shapes. While previous research has predominantly focused on spherical nanoparticles, our work expands the knowledge by exploring various geometries. Employing classical electrodynamics and the small particle limit, we calculate the angular momentum transfer by integrating the spectral density, ensuring causality through Fourier-transform analysis. Our findings demonstrate that prolate spheroidal nanoparticles exhibit a single blueshifted plasmonic resonance, compared to spherical nanoparticles of equivalent volume, resulting in lower angular momentum transfer. Conversely, oblate nanoparticles display two resonances — one blueshifted and one redshifted — resulting in a higher angular momentum transfer than their spherical counterparts. Additionally, Platonic Solids with fewer faces exhibit significant redshifts in plasmonic resonances, leading to higher angular momentum transfer due to edge effects. We also observe resonances and angular momentum transfers with similar characteristics in specific pairs of Platonic Solids, known as duals. These results highlight promising applications, particularly in electron tweezers technology.

Vorticity suppression by multiphase effects in shock-driven variable density mixing

Shock-driven variable density mixing has been frequently explored through the single-phase Richtmyer–Meshkov instability. Here, such mixing is considered when driven by a multiphase component, the Shock-Driven Multiphase Instability (SDMI). The simple case of a solid particle seeded gas in a cylindrical region surrounded by clean gas is studied. It has been previously shown that the particle-phase can lag behind the gas, diminishing vorticity deposition. In this letter we present theoretical analysis of the vorticity deposition, and a new model predicting the circulation deposition for an SDMI as a function of particle relaxation distance and hydrodynamic mixing strength. The theory is founded on a simplified vorticity equation, advection and multiphase source terms, using simple drag models to predict the particle dynamics, and scaling the results using existing circulation models for the Richtmyer–Meshkov instability in the small particle limit. The model is compared to new high-fidelity experimental data, and previous experiments and simulations, finding good agreement. This model provides the first theoretical prediction of mixing suppression in the SDMI.

Quantum electrodynamics with a nonmoving dielectric sphere: quantizing Lorenz–Mie scattering

We quantize the electromagnetic field in the presence of a nonmoving dielectric sphere in vacuum. The sphere is assumed to be lossless, dispersionless, isotropic, and homogeneous. The quantization is performed using normalized eigenmodes as well as plane-wave modes. We specify two useful alternative bases of normalized eigenmodes: spherical eigenmodes and scattering eigenmodes. A canonical transformation between plane-wave modes and normalized eigenmodes is derived. This formalism is employed to study the scattering of a single photon, coherent squeezed light, and two-photon states off a dielectric sphere. In the latter case, we calculate the second-order correlation function of the scattered field, thereby unveiling the angular distribution of the Hong–Ou–Mandel interference for a dielectric sphere acting as a three-dimensional beam splitter. Our results are analytically derived for a dielectric sphere of arbitrary refractive index and size with a particular emphasis on the small-particle limit. As shown in Phys. Rev. A 108, 033714 (2023)PLRAAN1050-294710.1103/PhysRevA.108.033714, this work sets the theoretical foundation for describing the quantum interaction between light and the motional, rotational, and vibrational degrees of freedom of a dielectric sphere.

Mixing rule for calculating the effective refractive index beyond the limit of small particles.

Considering light transport in disordered media, the medium is often treated as an effective medium requiring accurate evaluation of an effective refractive index. Because of its simplicity, the Maxwell-Garnett (MG) mixing rule is widely used, although its restriction to particles much smaller than the wavelength is rarely satisfied. Using 3D finite-difference time-domain simulations, we show that the MG theory indeed fails for large particles. Systematic investigation of size effects reveals that the effective refractive index can be instead approximated by a quadratic polynomial whose coefficients are given by an empirical formula. Hence, a simple mixing rule is derived which clearly outperforms established mixing rules for composite media containing large particles, a common condition in natural disordered media.

The ignition of fine iron particles in the Knudsen transition regime

A theoretical model is considered to predict the minimum ambient gas temperature at which fine iron particles can undergo thermal runaway–the ignition temperature. The model accounts for Knudsen transition transport effects, which become significant when the particle size is comparable to, or smaller than, the molecular mean free path of the surrounding gas. Two kinetic models for the high-temperature solid-phase oxidation of iron are analyzed. The first model (parabolic kinetics) considers the inhibiting effect of the iron oxide layers at the particle surface on the kinetic rate of oxidation, and a kinetic rate independent of the gaseous oxidizer concentration. The ignition temperature is solved as a function of particle size and initial oxide layer thickness with an unsteady analysis considering the growth of the oxide layers. In the free-molecular limit (small particles), the thermal insulating effect of transition heat transport can lead to a decrease of ignition temperature with decreasing particle size; however, the presence of the oxide layer slows the reaction kinetics, and its increasing proportion in the small-particle limit can lead to an increase of ignition temperature with decreasing particle size. This effect is observed for sufficiently large initial oxide layer thicknesses. The continuum transport model is shown to predict the ignition temperature of iron particles exceeding an initial diameter of 30 µm to a difference of 3% or less (30 K or less) when compared to the prediction of the transition transport model. The second kinetic model (first-order kinetics) considers a porous, non-hindering oxide layer, and a linear dependence of the kinetic rate of oxidation on the gaseous oxidizer concentration. The ignition temperature is resolved as a function of particle size with the transition and continuum transport models, and the differences between the ignition characteristics predicted by the two kinetic models are identified and discussed.

Synthetic photometry for carbon-rich giants

Context. The properties and the evolution of asymptotic giant branch (AGB) stars are strongly influenced by their mass loss through a stellar wind. This, in turn, is believed to be caused by radiation pressure due to the absorption and scattering of the stellar radiation by the dust grains formed in the atmosphere. The optical properties of dust are often estimated using the small particle limit (SPL) approximation, and it has been used frequently in modelling AGB stellar winds when performing radiation-hydrodynamics (RHD) simulations. Aims. We aim to investigate the effects of replacing the SPL approximation by detailed Mie calculations of the size-dependent opacities for grains of amorphous carbon forming in C-rich AGB star atmospheres. Methods. We performed RHD simulations for a large grid of carbon star atmosphere+wind models with different effective temperatures, luminosities, stellar masses, carbon excesses, and pulsation properties. Also, a posteriori radiative transfer calculations for many radial structures (snapshots) of these models were done, resulting in spectra and filter magnitudes. Results. We find that, when giving up the SPL approximation, the wind models become more strongly variable and more dominated by gusts, although the average mass-loss rates and outflow speeds do not change significantly; the increased radiative pressure on the dust throughout its formation zone does, however, result in smaller grains and lower condensation fractions (and thus higher gas-to-dust ratios). The photometric K magnitudes are generally brighter, but at V the effects of using size-dependent dust opacities are more complex: brighter for low mass-loss rates and dimmer for massive stellar winds. Conclusions. Given the large effects on spectra and photometric properties, it is necessary to use the detailed dust optical data instead of the simple SPL approximation in stellar atmosphere+wind modelling where dust is formed.

Small and large particle limits of the asymmetry parameter for homogeneous, spherical particles

The aerosol single scattering asymmetry parameter g is an important intensive particle parameter in determining aerosol direct radiative forcing and radiation pressure. For homogeneous spherical particles and a complex refractive index that is independent of wavelength, g is solely dependent on the size parameter x, the ratio of the particle circumference to the wavelength, and the complex refractive index m of the particle. Here, we explore this dependency for the small and large particle limits with size parameters much smaller and much larger than one. We show that in the small particle (Rayleigh) regime, g is proportional to x2, and give the proportionality factor as a function of the complex m. In addition, we separate regions in the complex refractive index space where the asymmetry parameter is negative (dominantly backscattering) and where it is positive (dominantly forward scattering). In the large particle limit, the geometric optics regime, g converges to a constant value that is a function of m.

Scaling limits and fluctuations for random growth under capacity rescaling

We evaluate a strongly regularised version of the Hastings-Levitov model HL$(\alpha)$ for $0\leq \alpha<2$. Previous results have concentrated on the small-particle limit where the size of the attaching particle approaches zero in the limit. However, we consider the case where we rescale the whole cluster by its capacity before taking limits, whilst keeping the particle size fixed. We first consider the case where $\alpha=0$ and show that under capacity rescaling, the limiting structure of the cluster is not a disk, unlike in the small-particle limit. Then we consider the case where $0<\alpha<2$ and show that under the same rescaling the cluster approaches a disk. We also evaluate the fluctuations and show that, when represented as a holomorphic function, they behave like a Gaussian field dependent on $\alpha$. Furthermore, this field becomes degenerate as $\alpha$ approaches 0 and 2, suggesting the existence of phase transitions at these values.

Angular dynamics of small nanoparticles induced by non-vortex electron beams

In this work, we present a theoretical study of the angular dynamics of small nanoparticles induced by fast non-vortex electron beams. General expressions for the torque and the angular momentum transferred from an electron to an arbitrary—but small—nanoparticle are obtained using a full-retarded classical electrodynamics approach, within the small particle limit. We applied this methodology to study a particular case of interest: the angular dynamics of spherical nanoparticles with homogeneous and isotropic electromagnetic responses. We analytically calculate the total angular momentum transferred from a swift electron to such nanoparticles, finding that it is electric in nature and it is always in a direction determined by the electron trajectory relative to the center of the nanoparticle. We realize that it is possible to represent the angular momentum transferred as the product of two functions: the extinction cross-section of the nanoparticle and a function that only contains information about the swift electron. We present numerical results for the total angular momentum transferred from a swift electron to an aluminum and a gold nanoparticle. We also present an analysis of the temporal behavior of the torque and the electric dipole moment induced within the nanoparticle by the swift electron. We compare the angular momentum transfer calculated in this work with a previously reported case of vortex beams, finding that, for both aluminum and gold nanoparticles, our results are two orders of magnitude smaller. Finally, we consider a particular case of a frictionless gold spherical nanoparticle of radius a=5nm, obtaining that it can spin with an angular frequency up to 29.3Hz.

The influence of Einstein's effective viscosity on sedimentation at very small particle volume fraction

We investigate the sedimentation of identical inertialess spherical particles in a Stokes fluid in the limit of many small particles. It is known that the presence of the particles leads to an increase of the effective viscosity of the suspension. By Einstein's formula this effect is of the order of the particle volume fraction ϕ. The disturbance of the fluid flow responsible for this increase of viscosity is very singular (like |x|−2). Nevertheless, for well-prepared initial configurations and ϕ→0, we show that the microscopic dynamics is approximated to order ϕ2|log⁡ϕ| by a macroscopic coupled transport-Stokes system with an effective viscosity according to Einstein's formula. We provide quantitative estimates both for convergence of the densities in the p-Wasserstein distance for all p and for the fluid velocity in Lebesgue spaces in terms of the p-Wasserstein distance of the initial data. Our proof is based on approximations through the method of reflections and on a generalization of a classical result on convergence to mean-field limits in the infinite Wasserstein metric by Hauray.

Heat and mass transfer to/from active particles in a fluidized bed—An analysis of the Baskakov-Palchonok correlation

Heat and mass transfer to or from single active particles surrounded by inert (passive) particles in a fluidized bed has been investigated based on published correlations. Special emphasis is on the application of a proposal by Baskakov, further developed by Palchonok. This representation describes heat and mass transfer as a function of the size ratio of inert to active particles. Two limits have been chosen: the limit of small active particles, where the active and the inert particles are equal, and the limit of large active particles, where the influence of the size of the active particle has vanished. The presentation aims at finding a suitable relationship, describing the size ratio of inert to active particles on heat and mass transfer to/from particles in fluidized beds and to critically evaluate its usefulness. It seems that the agreement between available correlations is qualitative and only approximate estimations can be made. A generalized scheme for calculations is presented. The formulation is made for bubbling fluidization. A discussion is presented on its use in circulating fluidized bed applications for fuel conversion as well.

Extinction, absorption, and scattering of light by plasmonic spheres embedded in an absorbing host medium.

Although the general Lorenz-Mie formalism for spheres in an absorbing host has been developed, no correct analytical expressions in the small-particle limit have been published so far. Here, we derive two sets of analytical expressions for the extinction, absorption, and far- and near-field scattering cross sections of small particles embedded in an absorbing host. One set is a modification of the electrostatic approximation (EA) for an absorbing host, whereas the other represents an improved electrostatic approximation (IEA) based on the generalized Lorenz-Mie theory and a new form of Mie coefficients for the internal field expansion. To illustrate the accuracy of the derived approximations, we consider Au and Ag nanospheres embedded in model hosts (real part of the refractive index, 1.33; imaginary part, 0-0.3), in a lossless poly(methyl methacrylate) (PMMA), and a lossy poly(3-hexylthiophene) (P3HT) matrix. In general, the IEA cross sections agree with those calculated using Lorenz-Mie theory if the particle diameter is not greater than 50 nm. Two small-particle limits are found for the near-field scattering cross sections. When host absorption is negligible, the scattering efficiency scales as the fourth power of the size parameter. In contrast, for nonzero absorption, the scattering efficiency scales as the first power of the size parameter. For a spectrally independent host, an increase in host absorption broadens and suppresses plasmonic peaks. We found an exception to this general tendency for near-field scattering by small (10-50 nm) particles; for these, an increase in host absorption increases the scattering peak. This surprising behavior is explained analytically. For 10-30 nm Au particles in the PMMA and P3HT matrixes, the EA and IEA data perfectly agree with the exact Lorenz-Mie simulations, in contrast to the previously reported conclusions. In particular, replacing PMMA with P3HT shifts the plasmonic peaks of the 10 nm particles from 540 nm to 650 nm and strongly enhances near- and far-field scattering. However, far-field scattering does not contribute to the extinction derived from the generalized optical theorem.

Three-component modelling of C-rich AGB star winds – V. Effects of frequency-dependent radiative transfer including drift★

ABSTRACT Stellar winds of cool carbon stars enrich the interstellar medium with significant amounts of carbon and dust. We present a study of the influence of two-fluid flow on winds where we add descriptions of frequency-dependent radiative transfer (RT). Our radiation hydrodynamic models in addition include stellar pulsations, grain growth and ablation, gas-to-dust drift using one mean grain size, dust extinction based on both the small particle limit (SPL) and Mie scattering, and an accurate numerical scheme. We calculate models at high spatial resolution using 1024 gridpoints and solar metallicities at 319 frequencies, and we discern effects of drift by comparing drift models to non-drift models. Our results show differences of up to 1000 per cent in comparison to extant results. Mass-loss rates and wind velocities of drift models are typically, but not always, lower than in non-drift models. Differences are larger when Mie scattering is used instead of the SPL. Amongst other properties, the mass-loss rates of the gas and dust, dust-to-gas density ratio, and wind velocity show an exponential dependence on the dust-to-gas speed ratio. Yields of dust in the least massive winds increase by a factor 4 when drift is used. We find drift velocities in the range $10\!-\!67\, \mbox{km}\, \mbox{s}^{-1}$, which is drastically higher than in our earlier works that use grey RT. It is necessary to include an estimate of drift velocities to reproduce high yields of dust and low wind velocities.

Microwave absorption by small dielectric and semi-conductor coated metal particles

The presence of the oxide coatings on conducting metal particles is often ignored in the calculations of microwave absorption. However, we find that the optical properties of the coatings play a significant role in enhancing or suppressing the absorption of electromagnetic energy. Here, we solve the Mie scattering equations numerically to separate and quantify the role of electric and magnetic field absorption from microwaves at 2.45 GHz by small metal spheres coated with dielectric or semi-conducting materials. The range of size and conductivities of the metal particles and the optical properties of the coatings are chosen for their practical importance. We also provide simple approximate expressions for the absorption per unit volume by coated spheres in the small particle limit which agrees very well with the exact Mie solution. We have demonstrated that for highly conducting particles coated with a material of low conductivity, the electric field absorption depends only on the optical properties, and the volume fraction of the coating. In contrast, the magnetic field absorption depends only on the properties of the core which is the same as the bare metal. We find that the power absorbed by coated particles via the electric field is maximized when loss tangent, tanδ~1. A key result of this study is that a thin layer of lossy coating (~15% of the particle size) on highly conductive particles will significantly enhance (up to a factor of 105) both the power absorbed from the E-field and the total power absorbed.

Analytically decomposing optical force on a spherical particle in Bessel beams into conservative and non-conservative parts

Based on the recently developed Cartesian multipole expansion theory, we analytically analyze the conservative and non-conservative nature of the optical force acting on a spherical particle of arbitrary size and isotropic composition immersed in the optical Bessel beams of arbitrary orders and polarizations. It is rigorously proved that the conservative force on the particle in Bessel beams aligns in the radial direction transverse to beam propagation, while the non-conservative force is completely non-radial, lying in the azimuthal and longitudinal directions. To the best of our knowledge, our work provides the first analytical partition between the conservative and non-conservative components of the optical force on a particle of arbitrary size and composition placed in a class of extensively employed optical beams in practical optical manipulation, beyond the small particle limit.

Design of acoustic metaclusters for the sound control using optimization and deep learning

We demonstrate a novel method to model finite metaclusters that can steer the energy of an incident wave preferentially toward a given direction. This is accomplished by solving an inverse multiple scattering problem for selecting a desired energy distribution of scattered waves. One can steer the incident energy toward a desired direction using a 2-D configuration of metacluster with finite number of fluid cylinders embedded in a homogeneous fluid medium. For a faster implementation of the method, we approximate the problem to small cylindrical particle limit which corresponds to low frequency scattering. The required mechanical properties of fluid scatterers are defined by T-matrix components obtained by solving linear system of equations. A major challenge in implementing our computational model and applying to design of metaclusters devices is ensuring that the scatterers remain manufacturable using available conventional materials with positive values of mechanical properties. Metaclusters are designed by minimizing the relative error between a given and computed scattering patterns and using advanced optimization algorithms and deep learning.

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.

A DARWIN C-star model grid with new dust opacities

AbstractWe have improved the treatment of dust opacity from the small-particle limit approximation to size-dependent which leads to models with smaller grains, lower dust-to-gas ratios, but about the same mass-loss rates and outflow velocities. The K-magnitudes get brighter, whereas the V-magnitudes can be either brighter or dimmer depending on the wind properties.

From acoustic radiation pressure to three-dimensional acoustic radiation forces.

This paper presents three-dimensional acoustic radiation forces exerted by an arbitrary field on an obstacle as a sum of two parts due to momentum transfers by ingoing and outgoing spherical waves, respectively. This view of momentum conservation establishes a connection of the force with asymmetry of far fields and acoustic radiation pressure exerted by a plane wave on a planar reflector. These geometrical interpretations and physical aspects are useful for finding and interpreting special force conditions by examining far-field patterns and scattering functions of sound fields. Multipole expansion of the force as a function of beam shape coefficients and scattering functions is simplified for connection with complex phase shifts of scattering functions. A relation of the scattering function dependence in the small-particle limit is identified.