Conditions for the size spectrum of aerosols to be self-preserving

A one-dimensional population balance governing the particle size distribution in a system where particles change their sizes due to both coagulation and individual growth is studied for the possibility of a similarity solution. It is found that when a dimensionless parameter γ is a constant, then the population balance can be transformed into an ordinary integrodifferential equation. Conditions under which the latter equation may have a self-preserving solution are established. Examples related to aerosol dynamics in the continuum regime, the free-molecule regime and the intermediate regime are discussed in detail.     

Brownian Coagulation of Fractal Agglomerates: Analytical Solution Using the Log-Normal Size Distribution Assumption

An analytical solution to Brownian coagulation of fractal agglomerates in the continuum regime that provides time evolution of the particle size distribution is presented. The theoretical analysis is based on representation of the size distribution of coagulating agglomerates with a time-dependent log-normal size distribution function and employs the method of moments together with suitable simplifications. The results are found in the form that extends the spherical particle solution previously obtained by K. W. Lee (J. Colloid Interface Sci. 92, 315-325 (1983)). The results show that the mass fractal dimension has a significant effect on the size distribution evolution during coagulation. When the obtained solution was compared with numerical results, good agreement was found. The self-preserving size distribution of nonspherical agglomerates is discussed. Copyright 2000 Academic Press.     

Brownian coagulation at high concentration

Particle growth by Brownian coagulation at high concentration in the continuum regime is investigated by solving the Langevin dynamics (LD) equations for each particle trajectory of polydisperse suspensions. By monitoring the LD attainment of the self-preserving size distribution (SPSD), it is shown that the classic Smoluchowski collision frequency function is accurate for dilute particle volume fractions, phis, below 0.1%. At higher phis, coagulation is about 4 and 10 times faster than for the classic theory at phis = 10 and 20%, respectively. For complete particle coalescence upon collision, SPSDs develop even in highly concentrated suspensions (up to phis = 35%), as with dilute ones, but are broadened with increasing phis. At high particle concentration, an overall coagulation rate is proposed that reduces to the classic one at low concentration. Detailed collision frequency functions are also obtained at various phis values. Fractal-like agglomerates undergoing coagulation at constant fractal dimension attain an SPSD only temporarily because their effective volume fraction continuously increases, approaching gelation in the absence of restructuring or fragmentation.     

The self-preserving size distribution theory. II. Comparison with experimental results for Si and Si3N4 aerosols

The gas to particle synthesis route is a relatively clean and efficient manner for the production of high-quality ceramic powders. These powders can be subsequently sintered in any wanted shape. The modeling of these production systems is difficult because several mechanisms occur in parallel. From theoretical considerations it can be determined, however, that coagulation and sintering are dominant mechanisms as far as shape and size of the particles are considered. In part I of this article an extensive theoretical analysis was given on the self-preserving size distribution theory for power law particles. In this second part, cumulative particle size distributions of silicon and silicon nitride agglomerates, produced in a laser reactor, were determined from TEM pictures and compared to the distributions calculated from this self-preserving theory for power law particles. The calculated distributions were in fair agreement with the measured results, especially at the high end of the distributions. Calculated and measured particle growth rates were also in fair agreement. Using the self-preserving theory an analysis was made on the distribution of annealed silicon agglomerates, of interest in applications to nanoparticle technology.     

A parametric study of cohesive particle agglomeration in a shear flow—numerical simulations by the discrete element method

In this study, we investigate numerically the shear-induced agglomeration of cohesive inertial particles in a simple shear flow. We conduct a series of numerical simulations by implementing a soft-sphere collision model. Post-processing tools are developed in-house to analyze the results of the simulations in terms of transient and terminal numbers of primary particles and agglomerates, magnitudes and distributions of agglomerate size, and fractal dimension as functions of the salient physical parameters. The obtained numerical results compared with existing transient models suggest that, as the system transitions from formation of duplets to larger agglomerates, in most cases the number of agglomerates in the system reveals clear maxima. The results also show evidence of agglomerate densification, as weaker agglomerates with lower fractal dimensions are broken up and stronger ones with higher fractal dimensions are formed. Furthermore, we found that a simple theoretical model proposed by, among others, Chimmili et al. is able to predict the temporal development of the agglomerate size properly.     

Fractal-like Aggregates: Relation between Morphology and Physical Properties

A number of modern technological applications require a detailed calculation of the physical properties of aggregated aerosol particles. For example, in probing soot aerosols by the method called laser-induced incandescence (LII), the soot clusters are suddenly heated by a short, powerful laser pulse and then cool down to the temperature of the carrier gas. LII sizing is based on rigorous calculation of the soot aggregate heat-up and cooling and involves prediction of laser light absorption and energy and mass transfer between aggregated particles and the ambient gas. This paper describes results of numerical simulations of the mass or energy transfer between the gas and fractal-like aggregates of N spherical particles in either the free-molecular or continuum regime, as well as the light scattering properties of random fractal-like aggregates, based on Rayleigh-Debye-Gans (RDG) theory. The aggregate geometries are generated numerically using specially developed algorithms allowing "tuning" of the fractal dimension and prefactor values. Our results are presented in the form of easily applicable scaling laws, with special attention paid to relations between the aggregate gyration radius and the effective radius describing various transport processes between the aggregates and the carrier gas. Copyright 2000 Academic Press.     

Cluster-cluster aggregation kinetics and primary particle growth of soot nanoparticles in flame by light scattering and numerical simulations

The agglomeration kinetics of growing soot generated in a diffusion atmospheric flame are here studied in situ by light scattering technique to infer cluster morphology and size (fractal dimension D(f) and radius of gyration R(g)). SEM analysis is used as a standard reference to obtain primary particle size D(P) at different residence times. The number N(P) of primary particles per aggregate and the number concentration n(A) of clusters are evaluated on the basis of the measured angular patterns of the scattered light intensity. The major finding is that the kinetics of the coagulation process that yields to the formation of chain-like aggregates by soot primary particles (size 10 to 40 nm) can be described with a constant coagulation kernel beta(c,exp)=2.37x10(-9) cm3/s (coagulation constant tau(c) approximately = 0.28 ms). This result is in nice accord with the Smoluchowski coagulation equation in the free molecular regime, and, vice versa, it is in contrast with previous studies conducted by invasive (ex situ) techniques, which claimed the evidence in flames of coagulation rates much larger than the kinetic theory predictions. Thereafter, a number of numerical simulations is implemented to compare with the experimental results on primary particle growth rate and on the process of aggregate reshaping that is observed by light scattering at later residence times. The restructuring process is conjectured to occur, for not well understood reasons, as a direct consequence of the atomic rearrangement in the solid phase carbon due to the prolonged residence time within the flame. Thus, on one side, it is shown that the numerical simulations of primary size history compare well with the values of primary size from SEM experiment with a growth rate constant of primary diameter about 1 nm/s. On the other side, the evolution of aggregate morphology is found to be predictable by the numerical simulations when the onset of a first-order "thermal" restructuring mechanism is assumed to occur in the flame at about 20 ms residence time leading to aggregates with an asymptotic fractal dimension D(f,infinity) approximately = 2.5.     

Hybrid monte carlo method for simulation of two-component aerosol coagulation and phase segregation

The paper presents the development of a hybrid Monte Carlo (MC) method for the simulation of the simultaneous coagulation and phase segregation of an immiscible two-component binary aerosol. The model is intended to qualitatively model our prior studies of the synthesis of mixed metal oxides for which phase-segregated domains have been observed in molten nanodroplets. In our previous works (J. Aerosol Sci.32, 1479 (2001); Chem. Eng. Sci.56, 5763 (2001); submitted for publication) we developed sectional and monodisperse models where the internal state of the aerosol particles was described. These methods have certain limitations and it is difficult to include additional physical effects into the framework. Our new approach combines both constant volume and constant number Monte Carlo methods. Similar to our previous models, we assume that the phase segregation is kinetically controlled. The MC approach allows us to compute the mean number of enclosures (minor phase) per droplet, average enclosure volume, and the width of the enclosure size distribution. The results show that asymptotic behavior of enclosure distribution exists that is independent of initial conditions, which is very close to the continuum self-preserving distribution. Temperature is a key parameter because it allows for a significant change in the internal transport rate within each droplet. In particular, increasing the temperature significantly enhances the Brownian coagulation rate and lowers the number of enclosures per droplet. As a result, the MC results indicate that the growth of the minor phase can be moderated quite dramatically by small changes in system temperature. These results serve to illustrate the utility of this synthesis approach to the controlled growth of nanoparticles through the use of a majority matrix to slow down the encounter frequency of the minor phase and therefore its particle size.     

Numerical simulation and experimental verification of particle coagulation dynamics for a pulsed input

Mathematical simulation of particle coagulation dynamics was carried out using improved sectional modeling techniques for a system with a pulsed input of primary particles. The methodological improvement included the modification of the size density function based on a realistic assumption of particle size distributions, the application of a new and comprehensive curvilinear collision model, and special adjustment for the mass transfer of a doublet of particles that were very different in size. The simulation results demonstrated that the rectilinear model over-predicted the rate of particle coagulation and that the degree of over-prediction increased as the particles increased in size and the system became more heterogeneous. The coagulation rate increased remarkably as the fractal dimension of the particle aggregates decreased. The curvilinear model and the fractal scaling relationship in place of the rectilinear model and the Euclidean sizing geometry are two important modifications to the conventional Smoluchowski modeling approach. However, both modifications, rather than only one of them, should be applied together to produce more accurate and realistic simulations of coagulation dynamics. As indicated by the simulation, the importance of fluid shear rate to particle coagulation is reduced according to the curvilinear model compared to that previously described with the rectilinear model. As particles increased in size, the role of shear rate in coagulation became even less significant according to the curvilinear view of particle collisions. The results of numerical simulations in terms of the evolution of particle size distributions compared reasonably well with the observations of the jar-test coagulation experiments, which suggested the applicability of the modeling system, including the modified curvilinear-fractal approach, established in the present study.     

High-pressure liquid dispersion and fragmentation of flame-made silica agglomerates

The influence of primary particle diameter and the degree of agglomeration of flame-made silica agglomerate suspensions in aqueous solutions is studied by high-pressure dispersion (up to 1500 bar) through a nozzle with a 125 microm inner diameter. These particles were produced (4-15 g/h) by oxidation of hexamethyldisiloxane (HMDSO) in a coflow diffusion flame reactor. Their average primary particle size (10-50 nm) and degree of agglomeration were controlled by varying the oxygen and precursor flow rates. The particles were characterized by nitrogen adsorption, electron microscopy, and small-angle X-ray scattering. Hydrodynamic stresses break up soft agglomerates and yield hard agglomerate sizes in the range of 100-180 nm, as characterized by dynamic light scattering. Soft agglomerates exhibited decreasing light scattering diameters with increasing dispersion pressure, while hard agglomerates were insensitive.     

Asymptotic Particle Size Distributions Attained during Coagulation Processes

The effects of the collision kernel on the self-preserving size distribution and on the gelation phenomenon of aerosol coagulation were investigated. An analytical solution for the asymptotic width of log-normally preserving size distribution during coagulation was obtained as a function of the degree of homogeneity using arbitrary shape of homogeneous collision kernels. From the solution obtained, it was shown that when the degree of homogeneity is larger than 1, self-preserving size distribution does not exist, and gelation occurs. A very accurate numerical coagulation simulation method, the sectional method, was also used for calculating the self-preserving particle size distribution for some specific classes of coagulation kernels and the results were compared with the analytical solution obtained by the log-normal method. Copyright 2001 Academic Press.     

Change in particle size distribution of fractal agglomerates during Brownian coagulation in the free-molecule regime

Time evolution of particle size distribution of fractal agglomerates undergoing Brownian coagulation in the free-molecule regime was investigated. A simple analytical solution for the size distribution change was obtained by using the assumption that the size distribution during the coagulation process can be represented by a time-dependent log-normal function. The derived solution consists of three parameters of the log-normal distribution function. This study is believed to provide the first analytical solution for all the parameters of the log-normal distribution of fractal agglomerates undergoing coagulation in the free-molecule regime. To validate the derived solution, numerical computations were performed. The results were compared with the analytical solution and good agreement was obtained.     

Growth of Fractal Aggregates in the Intermediate Regime of Particle Accommodation

The growth dynamics of fractal aggregates was studied within the framework of continuum model in the self-consistent mean field approximation. The regime that is intermediate between the diffusion-limited aggregation and reaction-limited aggregation was considered. The dependence of aggregate fractal dimension on the attachment probability of particles during their collisions with an aggregate was obtained. In the limiting cases, the values of fractal dimension coincide with those determined earlier. The domain of the values of attachment probability was revealed where several regions characterized by their own values of fractal dimension were specified in the structure of growing cluster. Physical nature of the emergence of various regions in the aggregate structure was discussed.     

Evaporation-induced formation of fractal-like structures from nanofluids

After the nanofluids are fully dried, the self-assembled nanoparticles can form various structures on the substrate. The fractal-like patterns are among them. The two-dimensional Kinetic Monte Carlo model is developed to predict the drying patterns of the nanofluids in an open domain, where the dewetting front shrinks from the edge toward the center. The simulation reveals that the initially dispersed particles can be deposited into an isotropic branched structure which remains frozen after full evaporation of the base fluid. The well-developed fractal-like particle aggregates are different from the fractal cavities obtained in the previous closed domain simulation. The present prediction of the fractal particle aggregation is verified by the experiments with the water-based nanofluids. The images taken using a scanning electron microscope prove that the evaporation-induced branched microstructures are formed by the nanoparticles as the water is totally dried.     

Particle Size Distribution Measurement and Assessment of Agglomeration of Commercial Nanosized Ceramic Particles

ABSTRACT

Commercially available powders with primary particle sizes of 10–100?nm (transition aluminas, Boehmites, and a commercially available silica dispersion) have been studied using three different instruments, a photocentrifuge (Horiba CAPA 700), photon correlation spectroscopy (PCS, Malvern Zetasizer 4), and an x-ray disc centrifuge (XDC, Brookhaven X-ray Disc Centrifuge BI-XDC). Particle size distributions for five powders were collected with each instrument and in conjunction with nitrogen adsorption measurements an agglomeration factor calculated. An example is given to show the importance of using a light scattering correction when measuring particle size distributions with a photocentrifuge for a gamma alumina powder where the uncorrected data can overestimate the D _V50 by up to a factor of 10 depending on the powder. The importance of the assessment of agglomeration and how different treatments such as milling modifies the agglomeration factor F _AG is illustrated for an “as received” and attrition milled gamma alumina. Results are discussed with respect to the assumptions and limitations of the different instruments. Results are presented after consideration of the hydrodynamic density in the sedimentation methods, and light scattering for the optical based methods. For narrow size distributions in the 15–25?nm range all three instruments show very a good correlation. When the size range approaches the 40–100?nm regime PCS is very sensitivity to small populations of agglomerates. The instrument giving the best resolution in the 10–100?nm range was found to be the XDC. The speed of measurement should also be born in mind and varies enormously from several minutes for the PCS to several hours for the sedimentation techniques. To assess the accuracy of the measured sizes a model spherical silica powder was analyzed on all the instruments as well as by image analysis. The results with the silica powder showed how the accuracy of the sedimentation methods depends strongly on a knowledge of the suspended particles hydrodynamic density. This can be effected greatly by particle or agglomerate porosity and the thickness of the electrical double layer in the aqueous dispersions investigated. The results with the silica suggest accuracy's on the size better than ±20% is difficult without an accurate hydrodynamic density whereas consistency between methods for narrow size distributions can be better than 5% for median volume diameters.     

Aggregate size distribution evolution for Brownian coagulation-sensitivity to an improved rate constant

Brownian motion causes small aggregates to encounter one another and grow in gaseous environments, often under conditions in which the coalescence rate (say, spheroidization by "sintering") cannot compete. The polydisperse nature of the aerosol population formed by this mechanism is typically accounted for by formulating an evolution equation for the joint PDF of the state variables needed for describing individual particles. In the simple case of fractal-like aggregates (prescribed morphology and state, characterized just by the number of aggregated spherules, or total aggregate volume), we use the quadrature method of moments and Monte Carlo simulations to show that recent improvements in the laws governing free molecule regime coagulation frequency (rate "constant") of these aggregates cause systematic changes in the shape of the asymptotic aggregate size distribution, with significant implications for the light-scattering power and inertial impaction behavior of such aggregate populations.     

Experimental evaluation of the pressure and temperature dependence of ion-induced nucleation

An experimental system for the study of ion-induced nucleation in a SO(2)/H(2)O/N(2) gas mixture was developed, employing a soft x-ray at different pressure and temperature levels. The difficulties associated with these experiments included the changes in physical properties of the gas mixture when temperature and pressure were varied. Changes in the relative humidity (RH) as a function of pressure and temperature also had a significant effect on the different behaviors of the mobility distributions of particles. In order to accomplish reliable measurement and minimize uncertainties, an integrated on-line control system was utilized. As the pressure decreased in a range of 500-980 hPa, the peak concentration of both ions and nanometer-sized particles decreased, which suggests that higher pressure tended to enhance the growth of particles nucleated by ion-induced nucleation. Moreover, the modal diameters of the measured particle size distributions showed a systematic shift to larger sizes with increasing pressure. However, in the temperature range of 5-20 °C, temperature increases had no significant effects on the mobility distribution of particles. The effects of residence time, RH (7%-70%), and SO(2) concentration (0.08-6.7 ppm) on ion-induced nucleation were also systematically investigated. The results show that the nucleation and growth were significantly dependent on the residence time, RH, and SO(2) concentration, which is in agreement with both a previous model and previous observations. This research will be inevitable for a better understanding of the role of ions in an atmospheric nucleation mechanism.     

Numerical Modeling in Radio Frequency Suspension Plasma Spray of Zirconia Powders

A comprehensive model was developed to investigate the suspension spraying for a radio frequency (RF) inductively coupled plasma torch. Firstly, the electromagnetic field is solved with the Maxwell equations and validated by the analytical solutions. Secondly, the plasma field with different power inputs is simulated by solving the governing equations of the fluid flow coupled with the RF heating. Then, the suspension droplets embedded with nano particles are modeled in a Lagrangian manner, considering feeding, collision, heating and evaporation of the suspension droplets, as well as tracking, acceleration, melting and evaporation of the nano or agglomerate particles. The non-continuum effects and the influence of the evaporation on the heat transfer are considered. This particle model predicts the trajectory, velocity, temperature and size of the in-flight nano- or agglomerate particles. The effects of operating conditions and intial inputs on the particle characteristics are investigated. The statistical distributions of multiple particles’ size, velocity, temperature are also discussed for the cases with and without consideration of suspension droplets collision.     

Direct numerical simulation of aerosol coagulation with van der Waals forces

The van der Waals interaction energy has been computed for a large number of silver and water particle pairs in the range of 1 to 15 nm radius by first calculating the exact, retarded Hamaker factor for semi-infinite bodies and then applying a geometric correction for sphericity. The enhancement of particle collision rates has been deduced from these calculations. These values for the collisional enhancement were used in a direct numerical simulation of the coagulation of silver and water aerosols. Despite the size dependence of the van der Waals enhancement factors, the aerosol size distributions quickly attained nearly self-preserving forms.