Log-normal Size Distribution Function Research Articles

Analytical Model for Determination of Size-Distribution of Colloidal Silver Nanoparticles from Surface Plasmon Resonance Wavelength and Dielectric Functions.

In this work it is shown that the size of silver nanoparticles in a colloidal solution can be determined only from the wavelength of the surface plasmon resonance and material and medium dielectric functions. The size dependence of dielectric functions of silver nanoparticles becomes noticeable in nanoparticles which are smaller than 30 nm in size, which is in accordance with Mie scattering theory applicability. The novelty of this work is in the development of an analytical model for the determination of the size of silver nanoparticles derived from applying shift functions to the UV-Vis spectra, resulting in well-known characteristic diameters of log-normal size distribution function. The purpose of these shift functions is the reconstruction of experimental UV–Vis spectra from simulated ones based on the Beer–Lambert law and log-normal distribution function in order to find the mode diameters of colloidal silver nanoparticles. The introduction of Lagrangian analogue of extinction cross section explains the redshift constant characteristic for given nanoparticle material and the size distribution of nanoparticles. Therefore, the size determination of colloidal silver nanoparticles is possible only through UV–Vis spectroscopy.

Realization of Self-Preserving Size Distribution Theory for the Evolution of Tropospheric Atmospheric Aerosols Through an Inverse Gaussian Distribution

AbstractThe inverse Gaussian distributed method of moments (IGDMOM; J. Atmospheric Sci. 77 (9): 3011-3031, 2020) was developed to analytically solve the kinetic collection equation (KCE) for the first time. Using the IGDMOM, we obtained both new analytical and asymptotic solutions to the KCE. This is shown for both the free molecular and continuum regime collision frequency functions. The new analytical solutions are highly suitable for demonstrating the self-preserving size distribution (SPSD) theory. The SPSD theory is considered one of the most elegant research works in atmospheric science for aerosols or small cloud droplets. It was initially discovered by Friedlander (J. Meteorology 17 (5): 479-483, 1960) and then developed by Lee (J. Colloid Interface Sci. 92 (2): 315-325, 1983) with an assumption of the time-dependent lognormal size distribution function. In this study, we demonstrate that the SPSD theory of coagulating atmospheric aerosols can be presented in a simpler and more rigorous theoretical way, which is realized through the introduction of the IGDMOM for describing aerosol size distributions. Using the IGDMOM, the new formulas for the SPSD, as well as the time required for aerosols to reach the SPSD, are analytically provided and verified. Furthermore, we discover that the SPSD of atmospheric aerosols undergoing coagulation is only determined using a shape factor variable, 𝛺, which is composed of the first three moments at an initial stage. This study has critical implications for developing tropospheric atmospheric aerosol or small cloud droplet dynamics models and further verifies the SPSD theory from the viewpoint of theoretical analysis.

Multi-level micromechanical analysis of elastic properties of ultra-high performance concrete at high temperatures: Effects of imperfect interface and inclusion size

Ultra-high performance concrete (UHPC), as a typical multi-phase composite material, is vulnerable to mechanical degradation and explosive spalling at high temperatures. In this work, the temperature-dependent elastic modulus of UHPC is estimated through the step-by-step homogenization from gel matrix level to ultra-high performance fiber-reinforced concrete level. Totally three forms of C-S-H gels are considered to determine the elastic properties of the initial matrix, and the phase transformation and volume fractions of individual phases at high temperatures are calculated based on the hydration and dehydration kinetics models. Moreover, the effects of imperfect interface and inclusion size are taken into account by introducing the spring-layer interface model and log-normal size distribution function to the homogenization scheme. The predicted elastic moduli are validated by analytical and experimental results, showing that accurate predictions can be obtained at up to 800°C. Interestingly, the imperfect interface matters when high-modulus inclusions are embedded in the low-modulus matrix, and the effect of imperfect interface on effective elastic modulus varies a lot at different microscopic levels. The findings highlight the synergistic effects of imperfect interface and inclusion size on the elastic properties, which is helpful to the mechanical design of UHPC at high temperatures.

PARAMETERIZATION OF MAGNETIC NANOPARTICLES MATHEMATICAL MODEL USING EVOLUTIONARY ALGORITHMS

In this paper, the evolutionary algorithms approach is applied to the parameterization of a mathematical model describing the Mössbauer spectra of nanogranular (or nanoparticle) magnetic systems. These systems exhibit physical properties very different from bulk specimens being of great interest for material science and its use as biosensors, magneto sensors, data storage, and magnetic fluids. The purpose of this work is to compare the performance between the Differential Evolution and the Evolutionary Strategies algorithms to optimize the model parameters which best fit the experimental Mössbauer spectra of nanoscale magnetic particles. Spectra of two samples (??iron foil and NiFe2O4 nanoparticles) were recorded, at room temperature, by a conventional Mössbauer spectrometer using a scintillation detector in transmission geometry with a 57Co/Rh source. Fits to Mössbauer spectra were done using spin hamiltonians to describe both the electronic and nuclear interactions; a model of superparamagnetic relaxation of two levels (spin ½) and stochastic theory; a lognormal particle size distribution function as well as a dependency of the magnetic transition temperature and the anisotropy constant on particle diameter. The evolutionary algorithms have been implemented using Python programming language. For comparison, the two algorithms obey the termination criterion of 6,000 evaluations of the objective function. The results presented show the efficiency of these algorithms in the optimization of the parameters and on the fits of the spectra.

ON SPIN HAMILTONIAN FITS TO MÖSSBAUER SPECTRA OF NiFe2O4 NANOPARTICLES SYNTHESIZED BY CO-PRECIPITATION

Nanocrystalline NiFe2O4 particles prepared by chemical co-precipitation method were studied using magnetic measurements, 57Fe Mössbauer spectroscopy, X-ray diffraction, and transmission electron microscopy. Fits to Mössbauer spectra, in the range of 4.2 K – 300 K, were done using spin hamiltonians to describe both the electronic and nuclear interactions, a model of superparamagnetic relaxation of two levels (spin ½) and stochastic theory, a log-normal particle size distribution function as well as a dependency of the magnetic transition temperature and the anisotropy constant on particle diameter. We have used evolutionary strategies to fit the more complex Mössbauer spectra line shapes. The nanoparticles have an average size of 7 nm and exhibit superparamagnetism at room temperature. The saturation magnetization (Ms) at 4.2 K was determined from M vs. 1/H plots by extrapolating the value of magnetizations to infinite fields, to 24.21 emu/g and coercivity to 3.15 kOe. A magnetic anisotropy energy constant (K) 1.9´105 J/m3, at 4.2 K, were calculated from magnetization measurements. The synthesis, characterization, and functionalization of magnetic nanoparticles is a highly active area of current research located at the interface between materials science, biotechnology, and medicine. Superparamagnetic iron oxides nanoparticles have unique physical properties and have emerged as a new class of diagnostic probes for multimodal tracking and as contrast agents for magnetic resonance imaging (MRI).

Frequency Dependence of Initial Heat Generation in Granular Magnetite Nanoparticles

The frequency dependence of heat generation in granular magnetite nanoparticles was studied using vibrating sample magnetometry and the nanoTherics Magnetherm™ hyperthermia testing system. First, the particle size and its distribution were obtained by fitting the M-H curve to the classical Langevin function weight-averaged with a modified log-normal size distribution function. Next, the ac power dissipation model for monodispersed nanoparticles was extended to the polydispersed case, and the volumetric heating rate was predicted as a function of the frequency of the applied field under the assumption of the Neel relaxation mechanism. Finally, the temperature increase caused by the application of an ac magnetic field was measured experimentally at frequencies of 50, 112, 261, 335, 474 and 523 kHz, with the root-mean-squared field fixed to 250 Oe, and the results were compared with those from theoretical calculation. The frequency dependence of the initial heating rates was found to be in good agreement with the results predicted by using the polydispersed power dissipation model.

Analysis of Recrystallization of Fine-Crystalline Corundum in a Supercritical Water Medium Using the Lognormal Particle Size Distribution Function

The mechanism of synthesis of fine-crystalline corundum from aluminum hydroxide under induced nucleation conditions in a supercritical water fluid at 400°C and initial pressure of 26.4 MPa followed by exposure in the synthesis medium was studied. The crystal size distribution was analyzed by electron microscopy. The use of the lognormal function to describe the crystal size distribution of the synthesized corundum particles was substantiated. The possibility of using the lognormal particle size distribution function and the time dependence of its parameters to reveal the routes of product formation was analyzed. The dimensional spectrum of microcrystals has four components, whose appearance is associated with different routes of product formation. The number of distribution components remains unchanged during prolonged exposure, but their average characteristics have different time dependences. The mass redistribution during recrystallization, generally leading to a decrease in the average crystal size, is explained by differences in the crystal structure mobility of different components. It was concluded that the states of corundum in the newly formed crystals and in the overgrown layer on the particles of the inducing additive are different.

Semi-empirical model for indirect measurement of soot size distributions in compression ignition engines

This work proposes a semi-empirical model, which provides soot particle size distribution functions emitted by compression ignition engines. The model is composed of a phenomenological model based on the collision dynamics of particle agglomerates and an empirical model, which provides key input parameters such as primary particle size and a mathematical relationship between the size of the agglomerate and number of primary particles. The phenomenological model considers the relevant fluid-dynamics phenomena influencing the collision frequency function. It is observed that Brownian motion is the predominant phenomenon and in a much lesser degree inertial turbulent motion. The experimental model requires air/fuel ratio, engine speed, soot density and mean instantaneous in-cylinder pressure. A Dirac delta is used as a seed for the agglomerate size function whose magnitude depends on the soot volume concentration and the mean primary particle size at each engine operation condition. In a further step, the obtained modelled agglomerate size functions are fitted to lognormal size distributions defined by the modelled mean size and standard deviation. Modelled lognormal agglomerate size distribution functions are validated with respect to experimental distributions obtained using a Scanning Mobility Particle Sizer (SMPS).

Design Rules for the Taylor‐Couette Disc Contactor

AbstractThe Taylor‐Couette disc contactor (TCDC) is a stirred liquid‐liquid phase contactor which is suitable for applications in bioseparations. For liquid‐liquid reactor design, information about the specific mass transfer area is inevitable. Therefore, the drop size distribution and holdup in the TCDC were investigated under various operating conditions and appropriate correlations for the prediction of these parameters have been determined. Experimental data of drop size distributions were correlated with lognormal, Gaussian, and Weibull drop size distribution functions.

Diffuse Reflectance Spectroscopic Approach for the Characterization of Soil Aggregate Size Distribution

Assessment of soil structure and soil aggregation remains a challenging task. Routine methods such as dry- and wet-sieving approaches are generally time consuming and tedious, which calls for a robust, fast, and nondestructive method of soil aggregate characterization. Over the last two decades, diffuse reflectance spectroscopy (DRS) has emerged as a rapid and noninvasive technique for soil characterization. Combined with chemometric and data-mining algorithms, it provides an effective way of measuring several soil attributes and has the added advantage of being amenable to a remote sensing mode of operation. The objective of this study was to determine if the DRS approach could be used as a rapid, noninvasive technique to estimate soil aggregate characteristics. The DRS approach was examined for the estimation of soil aggregate characteristics such as the geometric mean diameter and two statistical parameters of the lognormal aggregate size distribution (ASD) functions using 910 soil samples from India representing three important soil groups. Results showed that the geometric mean diameter and the median aggregate size parameter provided excellent predictions, with ratio of performance deviation (RPD) values ranging from 1.99 to 2.28. The RPD value for the standard deviation of the ASD ranged from 1.36 to 1.72, suggesting moderate prediction. It was further observed that soil aggregates influence the incident electromagnetic radiation on soils primarily in the visible region and to some extent the shortwave- and near-infrared regions. Electronic transitions of Fe-bearing minerals, clay minerals, and C–H functional groups of organic matter may be responsible for modifying the spectral reflectance from soils in addition to the self-shadowing effects of surface roughness. The results of this study suggest that the chemometric approach may be combined with DRS to estimate soil aggregate size characteristics.

Comparison of Measured Particle Lung-Deposited Surface Area Concentrations by an Aerotrak 9000 Using Size Distribution Measurements for a Range of Combustion Aerosols

Surface area in addition to mass concentration is increasingly being emphasized as an important metric representing potential adverse health effects from exposure to inhaled particles. Lung-deposited surface area (SA) concentrations for a variety of aerosols: coal, biomass, cigarette, incense, candle, and TiO2 were measured using an AeroTrak 9000 (TSI Incorporated) and compared with those calculated from number size distributions from a scanning mobility particle sizer (SMPS). Three methodologies to compute the SA concentrations using the International Commission on Radiological Protection's (ICRP) Lung Deposition model and an SMPS were compared. The first method calculated the SA from SMPS size distributions, while the second method used lognormal size distribution functions. A third method generated a closed-form equation using the method of moments. All calculated SMPS SA data against which the measured SA data were compared were generated using the first method only; however, the SA concentrations calculated from each of the three methods demonstrated strong correlations with each other. Overall, results between measured and calculated lung-deposited SA indicated strong positive linear associations (R 2 0.78 - >0.99), moderately dependent on the type of aerosol. In all cases, the measured SA concentrations slightly underestimated those calculated from the SMPS data, with the exception of coal combustion particles. Although some dependency on aerosol material exists, the instrument measuring lung-deposited SA demonstrated consistent reliability across a range of concentrations for a range of materials. For optimal results however, applying a correction factor (CF) before taking the instrument to the field is recommended. Copyright 2013 American Association for Aerosol Research

Spin-glass freezing of maghemite nanoparticles prepared by microwave plasma synthesis

Magnetic properties of 6 nm maghemite nanoparticles (prepared by microwave plasma synthesis) have been studied by ac and dc magnetic measurements. Structural characterization includes x-ray diffraction and transmission electron microscopy. The temperature scans of zero field cooled/field cooled (ZFC/FC) magnetization measurements show a maximum at 75 K. The ZFC/FC data are fitted to the Brown-Néel relaxation model using uniaxial anisotropy and a log-normal size-distribution function to figure out the effective anisotropy constant Keff. Keff turns out to be larger than the anisotropy constant of bulk maghemite. Fitting of the ac susceptibility to an activated relaxation process according to the Arrhenius law provides unphysical values of the spin-flip time and activation energy. A power-law scaling shows a satisfactory fit to the ac susceptibility data and the dynamic critical exponent (zv ≈ 10) takes value between 4 and 12 which is typical for the spin-glass systems. The temperature dependence of coercivity and exchange bias shows a sharp increase toward low temperatures which is due to enhanced surface anisotropy. The source of this enhanced magnetic anisotropy comes from the disordered surface spins which get frozen at low temperatures. Memory effects and thermoremanent magnetization experiments also support the existence of spin-glass behaviour. All these magnetic measurements signify either magnetic blocking or surface spin-glass freezing at high and low temperatures, respectively.

A new application of a multifrequency submillimeter radiometer in determining the microphysical and macrophysical properties of volcanic plumes: A sensitivity study

In this paper, the potential of a multifrequency submillimeter radiometer to characterize ash plumes in the near‐field of a volcanic eruption is evaluated. The radiometer's sensitivity to mass concentration and particle effective dimension is shown to depend most critically on aerosol altitude and ejected water vapor concentration. There is also some dependence on temperature, aerosol shape and complex refractive index. For this study, the volcanic aerosols are assumed to be randomly oriented solid hexagonal silicates of aspect ratio unity. The T‐matrix method is used to calculate the single‐scattering properties of the aerosols at 36 frequencies between 90 GHz and 880 GHz, and the aerosol bulk scattering properties are derived assuming lognormal size distribution functions. A midlatitude standard summer atmosphere and a perturbed midlatitude summer atmosphere are used to quantify the sensitivity, using the delta‐Eddington two‐stream approximation, of the radiometer to the presence of aerosol. It is shown that at 34 frequencies, between 113 GHz and 880 GHz, the sensitivity to aerosol is a maximum if the following four conditions are satisfied: (i) The altitude of the aerosol layer should be ≫ 3 km, (ii) 0.1 g m−3 < mass concentration < 30 g m−3 (iii) the aerosol effective dimension, De, 20 μm < De < 1000 μm and (iv) water vapor ejected by a volcano into the atmosphere should be < 1000 times greater than the background water vapor concentration. The paper demonstrates the potential usefulness of using spectrally resolved submillimeter measurements in the near‐field of volcanic eruptions to characterize plume properties.

Characterization of the microstructure in random and textured polycrystals and single crystals by diffraction line profile analysis

X-ray or neutron diffraction patterns are simulated by convoluting defect specific profile functions based on continuum theory of elasticity. The defect related profile functions are controlled by the physically mandatory minimum number of parameters: the dislocation density, ρ, the dislocation arrangement parameter, M, one or more parameters describing strain anisotropy where their number depends on the crystal symmetry, the median, m and the logarithmic variance, σ of the log-normal size distribution function, and finally the density of stacking faults, α or the frequency of twin boundaries, β. These parameters are, at the same time, among the most relevant physical parameters describing the microstructure of crystalline materials. The theoretical diffraction patterns are produced by the convolution of the defect related, physically based profile functions in the “extended Convolutional Multiple Whole Profile” (eCMWP) software package. The usage of the software package is demonstrated by the microstructure determination in randomly-textured and textured polycrystalline and single crystals specimens of different materials.

Numerical study on the wavelength-dependent polarization of light scattered by aqueous droplets for RGB spectrums

Based on Lorenz–Mie scattering theory, the transformation matrix of an aqueous droplet is derived analytically. By employing single scattering approximation and log-normal size distribution function, the DOP (degree of polarization) of the light scattered by a unit volume of aqueous droplets for RGB (red, green and blue) spectrums is studied. In our study, the wavelength-dependent complex refractive index for water with fine resolution is novelly obtained by using Piecewise Interpolation method based on Segelstein's refractive index data. The analytical results reveal that extrema of the DOP which is dependent on the relative intensities of the orthogonal polarizations are associated with scattering angles, and, for RGB spectrums, increasing wavelength can alleviate the influence of the resonance on the orthogonal polarizations and the DOP.

Characterization of components of nano-energetics by small-angle scattering techniques

Small-angle scattering (SAS) and ultra small-angle scattering techniques, employing x-rays and neutrons, were used to characterize six different aluminum nanopowders and nanopowders composed of molybdenum trioxide and tungsten trioxide nanoparticles. Each material has different primary particle morphology and aggregate and agglomerate geometry, and each is important to the development of nano-energetic materials. The combination of small-angle and ultra small-angle techniques allowed a wide range of length scales to be probed, providing a more complete characterization of the materials. For the aluminum-based materials, differences in the scattering of x-rays and neutrons from aluminum and aluminum oxide provided sensitivity to the metal core and metal oxide shell structure of the primary nanoparticles. Small-angle scattering was able to discriminate between particle size and shape and agglomerate and aggregate geometry, allowing analysis of both aspects of the structure. Using the results of these analyses and guided by scanning electron microscopy (SEM) images, physical models were developed, allowing for a quantitative determination of particle morphology, mean nanoparticle size, nanoparticle size distribution, surface layer thickness, and aggregate and agglomerate fractal dimension. Particle size distributions calculated using a maximum entropy algorithm or by assuming a log-normal particle size distribution function were comparable. Surface area and density determinations from the small-angle scattering measurements were comparable to those obtained from other, more commonly used analytical techniques: gas sorption using Brunauer–Emmett–Teller analysis, thermogravimetric analysis, and helium pycnometry. Particle size distribution functions derived from the SAS measurements agreed well with those obtained from SEM.

Change in particle size distribution of aerosol undergoing condensational growth: alternative analytical solution for the low Knudsen number regime

Condensational growth of a polydisperse aerosol was studied employing the moment method of log-normal size distribution function. The previous analytical solution for the size distribution of a condensationally growing aerosol (Park et al., J. Aerosol. Sci. 32 (2001) 187) was extended to cover the low Knudsen number transition regime. In order to cover the transition regime, the harmonic mean method was used for obtaining the transition correction factors for mass and heat fluxes. Comparisons were made with the existing solution and good agreement was obtained.

Numerical modeling of silicon oxide particle formation and transport in a one-dimensional low-pressure chemical vapor deposition reactor

A numerical model is presented for particle formation and transport during low-pressure chemical vapor deposition of silicon dioxide films from silane and oxygen. A detailed chemical kinetics approach was used to model silicon oxide clustering that leads to homogeneous nucleation. A moment-type aerosol dynamics model was developed, which includes particle growth by surface reactions and coagulation, and particle transport by convection, diffusion and thermophoresis, assuming a lognormal particle size distribution function. A chemical clustering mechanism was coupled to the aerosol dynamics model in an axisymmetric stagnation–flow reactor. Simulations were conducted to predict steady-state spatial distributions of major particle characteristics such as particle concentration, diameter and volume fraction. The effects of various system parameters were assessed for conditions around 1.5 Torr (200 Pa) , 800°C, 200 sccm and an inlet oxygen-to-silane ratio of 20. Model predictions are shown to be in good agreement with experimental data and indicate that, unlike the case of particle formation in silane pyrolysis, the results are relatively insensitive to temperature. On the other hand, we observe a large sensitivity to pressure change, which is corroborated by experiment.

Analysis of Electrical Coagulation of Unipolar Charged Particles in an Alternating Electric Using Moment Method

A numerical study has been carried out on the evolution of the particle size distribution for unipolar charged particles that experience coagulation in an alternating electric field. The collision frequency function of charged particles was analytically derived. The log-normal size distribution function is utilized for representing a poly-disperse size distribution and the moments of the particle size distribution are used to solve the general dynamic equation considering only AC electric force effect. The results are compared with the effects of brownian coagulation.

Size Distribution Function of Droplet in Liquid-Liquid Mixing and Bubble in Gas-Liquid Mixing, Crystal in Crystallization and Crushed Product in Crushing.

The size distribution function of particles that present as droplets in liquid-liquid mixing, bubbles in gas-liquid mixing, crystals in crystallization, and crushed materials in crushing, is examined based on consideration of the balance of the external and the internal force/energy of the particle, and from the viewpoint of information entropy. Then, a new size distribution function (PSF), which is the product of the original size distribution function and the realizable probability function of the size of particles, is presented. The same formula style as that of newly presented PSF can be derived also by balancing the external energy calculated from the energy spectrum density distribution function (ESF) for the turbulent flow field and the internal force/energy of particles. Additionally, it is clarified that the size distribution of eddies in the turbulent flow can be deduced from the same viewpoint by considering that a large eddy is segmented by external force/energy and the style of the distribution is isomorphic with that of ESF. In addition, the usefulness of newly presented PSF is clarified by fitting it to the data generated by the traditional three common unimodal PSF; Rosin-Rammler, Normal and Log-Normal size distribution functions. The main advantages of the newly presented PSF are that it is defined based on a clear physical background and that, for all practical purposes, it can replace the three common PSF. Therefore, this newly presented PSF provides a means of presenting and comparing the different size distribution patterns in terms of a single mathematical expression.