Particle Size Distribution Function Research Articles

Magnetic neutron scattering from spherical nanoparticles with Néel surface anisotropy: atomistic simulations.

A dilute ensemble of randomly oriented non-interacting spherical nanomagnets is considered, and its magnetization structure and ensuing neutron scattering response are investigated by numerically solving the Landau-Lifshitz equation. Taking into account the isotropic exchange interaction, an external magnetic field, a uniaxial magnetic anisotropy for the particle core, and in particular the Néel surface anisotropy, the magnetic small-angle neutron scattering cross section and pair-distance distribution function are calculated from the obtained equilibrium spin structures. The numerical results are compared with the well known analytical expressions for uniformly magnetized particles and provide guidance to the experimentalist. In addition, the effect of a particle-size distribution function is modelled.

Phase-field model of grain boundary diffusion in nanocrystalline solids: Anisotropic fluctuations, anomalous diffusion, and precipitation

The anisotropic phase-filed model of grain boundary diffusion and precipitation of solute in nanocrystalline solids has been developed. In this model, the Cahn–Hilliard equation is generalized for the anisotropic phase-field diffusion of solute and anisotropic compositional fluctuations. It is found that dynamics of solute concentration profile demonstrates the anomalous diffusion behavior with scaling parameters depending on the mobility ratio and microstructure of a solid solution. It is noteworthy that the increase in source concentration can slow down the concentration front propagation due to uphill diffusion or formation of a new phase. Parameters of grain boundary diffusion control the precipitation dynamics. In particular, a decrease in transverse diffusion coefficient is responsible for longer incubation time, and lower rates of nucleation and nuclei growth in comparison with the case of isotropic solute transport near grain boundaries. Transport properties of boundary and bulk are responsible for the formation of the bimodal size distribution function of second phase particles and specific kinetics of average radius and number density.

Image histogram decomposition method for particle sizing – A numerical simulation study

Scanning probe microscopy is a useful tool in nanoscience. The effective application of nanotechnologies in various fields requires a knowledge of the characteristic attributes of nanoparticles such as shape, dimensions and statistical distribution, and a wide spectrum of experimental and theoretical methods based on various principles have been developed to determine these characteristics. Image histograms offer a global overview of the characteristics of an image. Their shape can encode specific statistical properties of displayed objects such as the distribution function in the case of similar and scalable objects. The model of height histogram presented here proposes a method which solves the long-term problem of processing images of extremely dense particle distributions. The method is based on the principle of the superposition of histograms of individual particles whose topographic surface is described by a parametric model. The resulting height histogram is defined by a convolution of the model of the particle histogram with the distribution function of particle size, with this construction forming the basis of the regression model. The parameters of the distribution function can be obtained via the optimization of the model. The method has been tested on artificially generated configurations of particles of various shapes and size distributions. Each of these configurations creates a topographic surface which is transformed into an image, and the heights obtained from the image allow a histogram to be calculated. Firstly, various configurations of particles are simulated without the presence of any disruptive influences. Next, several experimental effects are evaluated separately (for example, the background, particle shape irregularity and particle overlap). The decomposition of the histogram by the regression model on artificially generated images shows the robustness of the method with respect to particle density, partial horizontal overlap, randomly generated backgrounds and random fluctuations in particle shape. However, the method is sensitive to uniform changes in particle shape, a factor which limits its use to particles with known parametric models of their shape which allow the means of their parameters to be estimated.

Evolution of a Polydisperse Ensemble of Spherical Particles in a Metastable Medium with Allowance for Heat and Mass Exchange with the Environment

Motivated by a wide range of applications in various fields of physics and materials science, we consider a generalized approach to the evolution of a polydisperse ensemble of spherical particles in metastable media. An integrodifferential system of governing equations, consisting of a kinetic equation for the particle-size distribution function (Fokker–Planck type equation) and a balance equation for the temperature (concentration) of a metastable medium, is formulated. The kinetic equation takes into account fluctuations in the growth/reduction rates of individual particles, the velocity of particles in a spatial direction, the withdrawal of particles of a given size from the metastable medium, and their source/sink term. The heat (mass) balance equation takes into account the growth/reduction of particles in a metastable system as well as heat (mass) exchange with the environment. A generalized system of equations describes various physical and chemical processes of phase transformations, such as the growth and dissolution of crystals, the evaporation of droplets, the boiling of liquids and the combustion of a polydisperse fuel. The ways of analytical solution of the formulated integrodifferential system of equations based on the saddle-point technique and the separation of variables method are considered. The theory can be applied when describing the evolution of an ensemble of particles at the initial and intermediate stages of phase transformation when the distances between the particles are large enough, and interactions between them can be neglected.

Evolution of Statistical Characteristics of Particle Size Distribution Functions for Micropowders during Their Temporal Treatment in Planetary-Type Ball Mill

Evolution of Statistical Characteristics of Particle Size Distribution Functions for Micropowders during Their Temporal Treatment in Planetary-Type Ball Mill

Modeling soot particles as stable radicals: a chemical kinetic study on formation and oxidation. Part II. Soot oxidation in flow reactors and laminar flames

Recent experimental and theoretical studies have shed light on the persistent radical behavior of large polycyclic aromatic hydrocarbons (PAHs) and soot particles. The first detailed soot model developed based on these findings is here proposed. While its good performances in predicting soot formation and growth when soot oxidation is negligible are presented in the companion paper of this work (Part I), the model is here validated against 45 cases of particle size distribution functions (PSDF) and soot volume fraction under oxidizing conditions. Notably, we show that by considering stable radical particles and aggregates it is possible to reproduce soot oxidation by O2 in plug flow reactors under very diluted conditions, whereas existing models could not predict particle consumption in the absence of particle surface activating species. Therefore, this work paves the way for a different approach to soot modeling that could help to better interpret soot reactivity. Oxidation of soot produced from different fuels (ethylene, n-heptane, and toluene, i.e. leading to different particle graphitization) at different temperatures (950–1073 K) and O2 concentrations (1000–10000 ppm) is successfully modeled adopting reference oxidation rates of gas-phase resonantly stabilized PAH radicals, featuring analogous reactivity compared to soot species, and by also accounting for different particle hydrogenation levels. Good model performances are also obtained in both rich and lean premixed ethylene flames (Tmax∼1480–1510 K) and in two sets of counterflow ethylene soot formation/oxidation flames (SFO, Tmax∼2560–2825 K), where OH radical becomes the dominant oxidizer. Finally, model limitations emerging in some of the cases considered are discussed through a sensitivity analysis and with references to other oxidation mechanisms proposed in the literature in order to highlight possible future improvement of the model predictive capability

Prediction of saturated hydraulic conductivity of sandy soil using Sauter mean diameter of soil particles

AbstractSaturated hydraulic conductivity of soil, Ksat, is a critical parameter for mathematical modelling of groundwater and soil water flow. Current empirical Ksat models, such as Kozeny–Carman model, Hazen models, or empirical approaches based on soil pedotranfer functions (PTFs) rely on the specific surface area, certain characteristic particle size(s), for example, d5, d10, d15, d50, etc., or the fractionation of sand, silt and clay. In this study, we proposed an alternative modelling framework to estimate Ksat from void ratio and arithmetic mean diameter of soil particles. We suggested to describe the particle size distribution (PSD) using a continuous mathematical distribution function, for example, the lognormal distribution function, and to obtain the Sauter mean diameter (dS) by calculating the moments of the PSD function. The proposed empirical equations were tested against the measured Ksat values for 54 sand samples with available measured PSD and void ratio as reported in Toumpanou et al. and validated using 49 sandy soil samples obtained from the Unsaturated Soil Database (UNSODA). We found that the dS was more relevant to Ksat than the number mean diameter (dN) and mass mean diameter (dM) (the RMSE was 0.486 for Ksat ‐ dN; 0.187 for Ksat ‐ dS; and 0.212 for Ksat ‐ dM, respectively). We also found that the proposed equation with dS had superior performance over the Hazen equation and PTF‐based models. Our results indicated that the Ksat calculated using dS, which was either estimated from discrete experimental data or calculated from a continuous lognormal PSD function, best matched with the measured values for coarse‐textured soils. Future studies should focus on developing empirical relationships for soil water retention curves and other soil properties based on continuous PSD functions.Highlights A modelling framework to estimate Ksat from the void ratio and arithmetic mean diameter is proposed. Sauter mean diameter, dS, can be obtained by calculating the moments of the PSD function. Ksat model using dS performs better than that using dN, dM, d5, d10, or d50 or PTF‐based models.

Beneficiation properties of ash-and-slag dumps

Purpose. Development of a mathematical model for the indices of TPP ash-and-slag waste separation in closed flotation circuit when applying recleaning operations of enriched product. Methodology. The methods for calculating separation indices using the separation characteristics of separation processes and calculating the indices of the release of valuable mineral using statistical relationships between the particle size of the inclusion, ash content and particle size distribution function were used. Findings. Separation characteristics of rough and recleaning flotation operations were preliminarily determined experimentally, relations that allow calculating the indices of the release of coal particles in ash-and-slag dumps were defined, based on which the functions of particle distribution by their content of valuable mineral fractional composition were found. The separation characteristics of the connections of the separation blocks, including several operations of cleaning of the enriched froth product, were determined. Calculations to determine the quantitative and qualitative separation indices were performed. Originality. Analytical relationships to determine the indices of the release of coal inclusions in the process of material from ash-and-slag dumps grinding were obtained. Practical value. The established dependencies make it possible to justify the required grinding coarseness to achieve the specified parameters of the enrichment process based on the values of the beneficiation properties of ash-and-slag dumps.

Settling of copper-rich suspended particles from acid drainage neutralization as a function of chemical composition and particle size distribution

The attenuation of total and dissolved metals from acid drainage (AD) results from the interaction of physical and chemical processes occurring during and after AD neutralization in impacted rivers. Chemical removal occurs when dissolved metals are transferred into fine particles or flocs, while physical removal occurs when metal-rich suspended particles are deposited from the water column onto the riverbed. Most works studying metal attenuation in rivers focus on characterizing chemical and physical removal processes separately, yet these processes often interact and take place concurrently. The fate of copper (Cu) in particle suspensions formed from partial neutralization of AD was studied using a settling column coupled to a device that measured particles size distribution in situ, with minimal floc disruption. Chemical composition, particles size distribution, and total suspended solids (TSS) were measured as particles settled. The physical removal of freshly formed particles was enhanced by aluminum (Al), promoting the formation of larger particles and higher rates of TSS removal. However, Fe promoted higher partition of Cu onto particles, controlling its chemical removal. While the rate of Cu association to particles increased over time, TSS settled out from solution within the initial hour, regardless of the chemical composition within the range of tested conditions. Therefore, different remediation strategies may be applied depending on the goal for removal (e.g., removal of TSS, removal of Cu) and water composition. These strategies must consider chemical conditions (i.e., pH and concentrations of Al and Fe), settling times, and mixing conditions during particles formation for an optimal removal.

Ultra-dispersed particles of water-soluble and water-insoluble substances formed from gas-droplet flows

The work is devoted to the experimental study of the ultra dispersed particles formation process, which is essentially a fundamental problem of heat and mass transfer in gas-droplet flows under phase transformation conditions. Various methods of generating and controlling the gas-droplet flow parameters are considered. The description of the created equipment and diagnostic methods for studying gas-droplet flows is given. The particle size distribution functions are obtained using an aerosol particle spectrometer. The possibility of controlling the parameters of ultrafine particles of water-soluble and water-insoluble substances (medications) formed under evaporation of microdroplets of solutions and suspensions is shown experimentally. An attempt to reconstruct the distribution functions of microdroplets and their average dimension from measurements of the parameters of ultrafine particles formed after complete evaporation of the droplets is made.

Development of optical diagnostic method for colloidal solutions based on elastic light scattering

The work is devoted to colloidal solutions diagnostics by the optical method based on elastic light scattering by particles of the dispersed phase. The scheme of the developed optical electronic complex for recording the intensity of radiation scattered by colloidal particles is presented. The experimental study results of the elastic scattering of laser radiation on spherical nanometer-sized aluminum oxide particles suspended in water are represented. A method for measuring the scattering indicatrix based on the registration of radiation scattered in various directions and subsequent computer processing has been proposed. A method for reconstructing the particle size distribution function of colloidal solution dispersed phase by comparing the measured scattering indicators and their computer models has been developed. The method has been tested, as a result of which the size distribution function of aluminum oxide nanopowder particles has been restored.

Determination of the Error in Transferring of Length Unit’s Size when Measuring the Nanoparticles’ Diameter Using an Analyzer of Particles’ Differential Electrical Mobility

The quality of nanomaterials and nanotechnologies is largely determined by the stability of the applied technologies, which, to a large extent, depend on the constancy of particle sizes. In this regard, metrological problems arise that are associated both with measuring the dimensions of the microstructure of aerosols, suspensions and powders, and with ensuring the uniformity of measurements when transferring a unit of a physical quantity from a standard to working measuring instruments. The purpose of this work was to determine and calculate the error in transferring the size of a unit of length when measuring the diameter of nanoparticles.An analyzer of differential electric mobility of particles was determined as a reference measuring instrument for which the calculation was made. It allows the separation of aerosol particles based on the dependence of their electrical mobility on the particle size. In combination with a condensation particle counter, it allows you to scan an aerosol and build a particle size distribution function. This measurement method is the most accurate in the field of measuring the diameters of particles in aerosols, therefore, the error in the transmission of particle size must be set as for a standard.The paper describes the physical principles of measurement by this method and presents an equation for determining the diameter of nanoparticles. Based on this equation, the sources of non-excluded systematic error were identified. Also, an experimental method was used to determine the random component of the measurement error of nanoparticles and to calculate the error in transferring the size of a unit of length when measuring the diameter of nanoparticles.The obtained results will be used for metrological support of standard samples of particle size, ensuring traceability of measurements of aerosol particle counters and for aerosol research.

Physical and chemical characterization of two kinds of coal-derived soot

The chemical composition, oxidation reactivity, nanostructure, and particle size of two coal-derived soot were experimentally investigated. The soot samples were collected from Yinni (YN) and Datong (DT) single coal combustion flames through a nitrogen-quenched sampling tube. The collected soot samples were then tested on an X-ray photoelectron spectroscopy (XPS) and a thermogravimetric analyzer (TGA) to obtain their surface chemistry and oxidation reactivity. XPS analysis showed that extensive C and O, as well as trace other elements, including S, Na, Fe, and Si, etc., appeared on the surface of the two coal-derived soot. The sp2/sp3 content of YN soot was higher than that of DT soot, indicating the better reactivity of YN soot. The results of TGA experiments showed that the oxidation reaction constant of YN soot was indeed larger than DT soot. Interestingly, the two coal-derived soot contained a portion of unburned ash after the TGA oxidation experiments.The nanostructure and particle size distribution function of soot generated in YN and DT single coal combustion flames were also studied by a self-designed thermophoresis sampling system combined with scanning electron microscopy (SEM) and an online dilution probe in conjunction with scanning mobility particle sizer (SMPS). From the SEM images collected from a relatively low height of the single coal flames, a small-sized individual nascent soot and a large-sized and liquid-like drops were observed. With the increase of flame height, the primary particle size of the two kinds of coal-derived soot aggregates decreased due to oxidation, while the primary particle number increased because of agglomeration. The soot aggregates sampled from higher heights of the two single coal flames had a relatively larger particle size but lower concentration.

Bench tests for microscopic theory of Raman scattering in powders of disordered nonpolar crystals: Nanodiamonds and beyond

AbstractRecent Raman data on nanocrystallite arrays are revised within the microscopic theory for Raman peaks positions and broadening (linewidth). The theory combines the elasticity theory‐like approach for optical phonons used in order to evaluate the Raman peaks structure and the Green's function method applied for the phonon lines broadening. These theories are supported by the atomistic calculations within the dynamical matrix method for optical phonons and by the bond polarization model used to calculate the Raman intensities. The experimental data on four various nanopowders are analyzed with the use of this theory. The large width of the Raman peak in nanoparticles as compared with the corresponding peak in bulk materials and the width inverse dependence on the particle size previously observed by other researchers are explained within the framework of the theory. It is shown that the theory is capable to extract confidently from the Raman data four important microscopic characteristics of the nanopowder including the mean particle size, the variance of the particle size distribution function, the strength of intrinsic disorder in the particle, and the effective faceting number that parameterizes the particle shape.

Ultrafine Particulate Matter in Methane-Air Premixed Flames With Oxygen Enrichment

A complementary computational and experimental study is carried out on the formation of ultrafine particulate matter in premixed laminar methane air flames. Specifically, soot formation is examined in premixed stretch-stabilized flames to observe soot inception and growth at relatively high flame temperatures common to oxygen enriched applications. Particle size distribution functions (PSDF) measured by mobility sizing show clear trends as the equivalence ratio increases from Φ = 2.2 to Φ = 2.4. For a given equivalence ratio, the measured distribution decreases in median mobility particle size as the maximum flame temperature increases from approximately 1,950–2,050 K. The median mobility particle size is 20 nm or less for all flame conditions studied. The volume fraction decreases with increasing flame temperature for all equivalence ratio conditions. The Φ = 2.2 condition is close to the soot inception limit and both number density and volume fraction decrease monotonically with increasing flame temperature. The higher equivalence ratio conditions show a peak in number density at 2,000 K which may indicate competing soot inception processes are optimized at this temperature. Flame structure computations are carried out using detailed gas-phase combustion chemistry of the Appel, Bockhorn, Frenklach (ABF) model to examine the connection of the observed PSDF to soot precursor chemistry. Agreement between measured and computed flame standoff distances indicates that the ABF model could provide a reasonable prediction of the flame temperature and soot precursor formation for the flames currently studied. To the first order, the trends observed in the measured PSDF could be understood in terms of computed trends for the formation of benzene, naphthalene and other soot precursors. Results of the current study inform particulate matter behavior for methane and natural gas combustion applications at elevated temperature and oxygen enriched conditions.

The influence of non-stationarity and interphase curvature on the growth dynamics of spherical crystals in a metastable liquid

This manuscript is concerned with the theory of nucleation and evolution of a polydisperse ensemble of crystals in metastable liquids during the intermediate stage of a phase transformation process. A generalized growth rate of individual crystals is obtained with allowance for the effects of their non-stationary evolution in unsteady temperature (solute concentration) field and the phase transition temperature shift appearing due to the particle curvature (the Gibbs-Thomson effect) and atomic kinetics. A complete system of balance and kinetic equations determining the transient behaviour of the metastability degree and the particle-radius distribution function is analytically solved in a parametric form. The coefficient of mutual Brownian diffusion in the Fokker-Planck equation is considered in a generalized form defined by an Einstein relation. It is shown that the effects under consideration substantially change the desupercooling/desupersaturation dynamics and the transient behaviour of the particle-size distribution function. The asymptotic state of the distribution function (its 'tail'), which determines the relaxation dynamics of the concluding (Ostwald ripening) stage of a phase transformation process, is derived. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

Water Uptake by Evaporating pMDI Aerosol Prior to Inhalation Affects Both Regional and Total Deposition in the Respiratory System.

As pulmonary drug deposition is a function of aerosol particle size distribution, it is critical that the dynamics of particle formation and maturation in pMDI sprays in the interim between generation and inhalation are fully understood. This paper presents an approach to measure the evaporative and condensational fluxes of volatile components and water from and to solution pMDI droplets following generation using a novel technique referred to as the Single Particle Electrodynamic Lung (SPEL). In doing so, evaporating aerosol droplets are shown capable of acting as condensation nuclei for water. Indeed, we show that the rapid vaporisation of volatile components from a volatile droplet is directly correlated to the volume of water taken up by condensation. Furthermore, a significant volume of water is shown to condense on droplets of a model pMDI formulation (hydrofluoroalkane (HFA), ethanol and glycerol) during evaporative droplet ageing, displaying a dramatic shift from a core composition of a volatile species to that of predominantly water (non-volatile glycerol remained in this case). This yields a droplet with a water activity of 0.98 at the instance of inhalation. The implications of these results on regional and total pulmonary drug deposition are explored using the International Commission of Radiological Protection (ICRP) deposition model, with an integrated semi-analytical treatment of hygroscopic growth. Through this, droplets with water activity of 0.98 upon inhalation are shown to produce markedly different dose deposition profiles to those with lower water activities at the point of inspiration.

Eulerian formulation for the triboelectric charging of polydisperse powder flows

We propose an Eulerian formulation for the triboelectric charging of powder flows. Recently, continuum descriptions of the charging of monodisperse and bidisperse powder appeared. Now, we propose the first Eulerian formulation for this type of flows that can fully account for polydisperse particle size distributions. To this end, the joint particle size, velocity, and charge distribution function are solved through the direct quadrature method of moments. The newly established approach includes the uptake of electrostatic charge by the particles when contacting a solid wall and the exchange of charge in-between particles during collisions. The electrostatic field induced by the charge of the particles and the drag forces of the surrounding gas affects the particle dynamics. Test cases of two-dimensional steady channel flows demonstrate the charging of the flow from the walls toward the center and in the downstream direction. Further, the method predicts the effect of variations of the particle size and initial velocity distribution and the charge diffusion coefficient. The ability to handle polydispersity is a step toward the simulation of the electric charge buildup of real powder flows in full-scale technical applications.

Polarimetric sensitivity of light-absorbing carbonaceous aerosols over ocean: A theoretical assessment

Visible-light-absorbing carbonaceous aerosols within the boundary layer affect the radiance and polarization states of the radiation at the top of the atmosphere. Remote sensing from suborbital and satellite-based platforms utilizes these radiance and polarization signals to retrieve the key properties of these aerosols. Recent retrieval algorithms have shown a progressive trend toward including multi-angular and multi-spectral polarimetric measurements to produce better retrieval accuracy in comparison to those using measurements based on a single viewing angle. Here, we perform a theoretical investigation of the top of atmosphere (TOA) radiance-related reflectance factor (bidirectional reflectance factor (BRF)) and the two types of polarimetry-related factors (polarized bidirectional reflectance factor (pBRF) and the degree of linear polarization (DoLP)) for different types of atmospheric light-absorbing carbonaceous aerosols as a function of particle size distribution. We selected three polarimetric bands corresponding to those utilized by NASA's Airborne Multiangle SpectroPolarimetric Imager (AirMSPI)—near-UV (470 nm), visible (660 nm), and near-infrared (865 nm)—for our simulations which were performed over ocean surface using the successive order of scattering (SOS) algorithm coupled to a Lorenz-Mie aerosol optics model. The analysis of particle phase matrix elements indicates a close relationship between the angular dependencies of DoLP and associated phase matrix components at the shortest polarimetric band (470 nm). Using Jacobian analysis, we find that the radiance- and polarimetry-related reflectance factors of weakly light-absorbing aerosols, such as brown carbon, are more sensitive to changes in particle size and imaginary refractive index in comparison with those of black carbon, which is strongly light-absorbing. Our results suggest that the DoLP data could be used by future retrieval algorithms for reliably estimating microphysical properties of absorbing carbonaceous aerosols with imaginary refractive index less than 0.4.

Informational Measures of the Shape of the Amoroso Distributions for the Research of Dispersed Materials

For the particle size distribution function various forms of exponential models are used to construct models of the properties of dispersed substance. The most difficult stage of applied research is to determine the shape of the particle distribution model. For the particle size distribution function various forms of exponential models are used to construct models of the properties of dispersed substance. The most difficult stage of applied research is to determine the shape of the particle distribution model. The article proposes a uniform model for setting the interval of information uncertainty of non-symmetric particle size distributions. Based on the analysis of statistical and information uncertainty intervals, new shape coefficients of distribution models are constructed, these are the entropy coefficients for shifted and non shifted distributions of the Amoroso family. Graphics of dependence of entropy coefficients of non-symmetrical distributions show that distributions well-known are distinguish at small of the shapes parameters. Also it is illustrated for parameters of the form more than 2 that it is preferable to use the entropy coefficients for the unshifted distributions.The material contains also information measures for the well-known logarithmic normal distribution which is a limiting case of distribution Amorozo.