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.     

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.     

凝并和成核机理下颗粒尺度分布的Monte Carlo求解

颗粒的凝并和成核现象影响其尺度分布,现有的MonteCarlo方法描述颗粒尺度分布的时间演变过程存在若干困难.提出了一种新的多重MonteCarlo(MMC)算法,基于时间驱动,利用加权的虚拟颗粒的思想,在模拟过程中保持虚拟颗粒总数不变和计算区域体积不变.利用该算法对“常凝并核,一阶成核”的情况下颗粒尺度分布的时间演变过程进行了数值求解,所得结果与数值解相符,表明MMC算法具有高且稳定的计算精度.另外,MMC算法由于跟踪比实际颗粒数目少得多的虚拟颗粒而具有较低的计算代价.     

The self-preserving size distribution theory. I. Effects of the Knudsen number on aerosol agglomerate growth

Gas-phase synthesis of fine solid particles leads to fractal-like structures whose transport and light scattering properties differ from those of their spherical counterparts. Self-preserving size distribution theory provides a useful methodology for analyzing the asymptotic behavior of such systems. Apparent inconsistencies in previous treatments of the self-preserving size distributions in the free molecule regime are resolved. Integro-differential equations for fractal-like particles in the continuum and near continuum regimes are derived and used to calculate the self-preserving and quasi-self-preserving size distributions for agglomerates formed by Brownian coagulation. The results for the limiting case (the continuum regime) were compared with the results of other authors. For these cases the finite difference method was in good in agreement with previous calculations in the continuum regime. A new analysis of aerosol agglomeration for the entire Knudsen number range was developed and compared with a monodisperse model; Higher agglomeration rates were found for lower fractal dimensions, as expected from previous studies. Effects of fractal dimension, pressure, volume loading and temperature on agglomerate growth were investigated. The agglomeration rate can be reduced by decreasing volumetric loading or by increasing the pressure. In laminar flow, an increase in pressure can be used to control particle growth and polydispersity. For D(f)=2, an increase in pressure from 1 to 4 bar reduces the collision radius by about 30%. Varying the temperature has a much smaller effect on agglomerate coagulation.     

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.     

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.     

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.     

磨盘碾磨聚丙烯粒度分布与接枝率的研究

用分形几何方法研究了磨盘碾磨中聚丙烯 (PP)的粉碎和固相力化学接枝 .用粒度分析仪测定经磨盘碾磨聚丙烯的粒度分布 ,用分形理论处理实验数据 .结果表明 ,PP粒度分布存在无标度区 ,具有线性分形特征 ,磨盘碾磨对PP粒子分形行为有较大影响 ,聚丙烯粒度分布的分维值随碾磨次数的增加而增大 .磨盘碾磨中聚丙烯固相力化学接枝的实验结果表明 ,N 羟甲基丙烯酰胺在聚丙烯表面上的接枝率亦随碾磨次数增加而增加 ,即与聚丙烯粒度分布的分维值相关 .因此 ,可用分数维定量描述聚丙烯粒子在磨盘碾磨中的粉碎规律 ,揭示了N 羟甲基丙烯酰胺在聚丙烯表面固相力化学接枝反应的本质 .     

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.     

Bivariate Extension of the Quadrature Method of Moments for Modeling Simultaneous Coagulation and Sintering of Particle Populations

We extendthe application of moment methods to multivariate suspended particle population problems-those for which size alone is insufficient to specify the state of a particle in the population. Specifically, a bivariate extension of the quadrature method of moments (QMOM) (R. McGraw, Aerosol Sci. Technol. 27, 255 (1997)) is presented for efficiently modeling the dynamics of a population of inorganic nanoparticles undergoing simultaneous coagulation and particle sintering. Continuum regime calculations are presented for the Koch-Friedlander-Tandon-Rosner model, which includes coagulation by Brownian diffusion (evaluated for particle fractal dimensions, D(f), in the range 1.8-3) and simultaneous sintering of the resulting aggregates (P. Tandon and D. E. Rosner, J. Colloid Interface Sci. 213, 273 (1999)). For evaluation purposes, and to demonstrate the computational efficiency of the bivariate QMOM, benchmark calculations are carried out using a high-resolution discrete method to evolve the particle distribution function n(nu, a) for short to intermediate times (where nu and a are particle volume and surface area, respectively). Time evolution of a selected set of 36 low-order mixed moments is obtained by integration of the full bivariate distribution and compared with the corresponding moments obtained directly using two different extensions of the QMOM. With the more extensive treatment, errors of less than 1% are obtained over substantial aerosol evolution, while requiring only a few minutes (rather than days) of CPU time. Longer time QMOM simulations lend support to the earlier finding of a self-preserving limit for the dimensionless joint (nu, a) particle distribution function under simultaneous coagulation and sintering (Tandon and Rosner, 1999; D. E. Rosner and S. Yu, AIChE J., 47 (2001)). We demonstrate that, even in the bivariate case, it is possible to use the QMOM to rapidly model the approach to asymptotic behavior, allowing an immediate assessment of when previously established asymptotic results can be applied to dynamical situations of current/future interest. Copyright 2001 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.     

Multiparticle sintering dynamics: from fractal-like aggregates to compact structures

Multiparticle sintering is encountered in almost all high temperature processes for material synthesis (titania, silica, and nickel) and energy generation (e.g., fly ash formation) resulting in aggregates of primary particles (hard- or sinter-bonded agglomerates). This mechanism of particle growth is investigated quantitatively by mass and energy balances during viscous sintering of amorphous aerosol materials (e.g., SiO(2) and polymers) that typically have a distribution of sizes and complex morphology. This model is validated at limited cases of sintering between two (equally or unequally sized) particles, and chains of particles. The evolution of morphology, surface area and radii of gyration of multiparticle aggregates are elucidated for various sizes and initial fractal dimension. For each of these structures that had been generated by diffusion limited (DLA), cluster-cluster (DLCA), and ballistic particle-cluster agglomeration (BPCA) the surface area evolution is monitored and found to scale differently than that of the radius of gyration (moment of inertia). Expressions are proposed for the evolution of fractal dimension and the surface area of aggregates undergoing viscous sintering. These expressions are important in design of aerosol processes with population balance equations (PBE) and/or fluid dynamic simulations for material synthesis or minimization and even suppression of particle formation.     

Synthesis of monodisperse anionic submicron polystyrene particles by stabilizer-free dispersion polymerization in alcoholic media

In this work, monodisperse polystyrene (PS) particles were synthesized in ethanol/water medium using sodium salt of styrene sulfonic acid and 2,2′-azobis(isobutyronitrile) as ionic comonomer and nonionic initiator, respectively. The polymerization was carried out at low agitation speed, and no stabilizer (or surfactant) was added to the polymerization medium. This polymerization system (stabilizer-free dispersion polymerization) was initiated as a homogeneous solution of monomer, comonomer, medium, and initiator. With the production of free radicals, polymerization developed into a heterogeneous system. The effect of various polymerization conditions on the size and size distribution of the obtained particles was evaluated. The experimental results showed that with an increase in ethanol content, the size of the particles increased while no significant change was observed in particle size distribution. Furthermore, with increasing ionic comonomer content, the size of the particles decreased and their size distribution became broader. Moreover, it was observed that addition of an electrolyte to the polymerization medium also increased the particles’ size and broadened their size distribution. It is noteworthy to point out that the coagulation occurred in higher amounts of electrolyte. Finally, it is concluded that the polar component of Hansen solubility parameter of the polymerization medium affects the particle size and particle size distribution greatly.     

Characterisation of structure and aggregation processes of aquatic humic substances using small-angle scattering and X-ray microscopy

Aquatic humic substances (HS), an important part of the dissolved organic carbon in freshwater systems, are polyfunctional natural compounds with polydisperse structure showing strong aggregation/coagulation behaviour at high HS concentrations and in the presence of metal ions. In this study, small-angle neutron scattering (SANS), small-angle X-ray scattering (SAXS) and X-ray microscopy (XRM) were applied to characterise the structure and aggregation processes of HS in solution. In SAXS and XRM the high brilliant synchrotron radiation was used as X-ray source. Applying small-angle scattering, information about the size distribution and shape of aquatic HS was obtained. Spherical HS units were found which were stable in a wide concentration range in a kind of "monomeric" state almost independent of pH and ionic strength. At higher concentrations they formed chain-like agglomerates or disordered HS structures. In studies on the coagulation behaviour of HS after addition of copper ions, a linear relationship between Cu(2+) concentration and the formation of large disordered HS-Cu(2+) agglomerates was obtained. By using X-ray microscopy, single "huge" particles were found in older solutions and in solutions with high HS concentrations. Over a threshold Cu(2+) concentration of approx. 300 mg/L, the formation of an extensive HS-Cu(2+) network structure was observed within a few minutes. The presented structures show the ability of the methods used to characterise processes between diluted phase and suspended matter, which play an important role particularly in the region of phase interfaces.     

Particle design of tolbutamide by the spherical crystallization technique. III. Micromeritic properties and dissolution rate of tolbutamide spherical agglomerates prepared by the quasi-emulsion solvent diffusion method and the solvent change method

With the objective of modifying the micromeritic properties of tolbutamide (i.e., to manufacture a highly functional powder form), particle design was attempted using a quasi-emulsion solvent diffusion (QESD) method, and the micromeritic properties and dissolution rate of the obtained spherical agglomerates were evaluated by comparison with agglomerates prepared by the solvent change (SC) method. For the production of tolbutamide agglomerates by the QESD method, a necessary condition was the addition of a sucrose fatty acid ester to the system as an emulsifying agent. The particle diameter of the agglomerates obtained by the QESD method depended on the size of the initially formed quasi-emulsion droplets, which in turn depended on the viscosity of the solution. In addition, the agglomerates were nearly perfectly spherical in shape. In the QESD method, the quasi-emulsion droplets crystallized instantaneously from the droplet surface inward. The resultant agglomerates were dense, had great mechanical strength and showed excellent flowability due to their perfect spherical shape. On the other hand, the agglomerates produced by the SC method were conglomerates of primary crystals, and fine, needle-like crystals formed on their surface. As a result, these agglomerates had a large specific surface area, and they therefore showed greater solubility than the agglomerates prepared by the QESD method.     

The effect of particle coalescence on the surface area of a coagulating aerosol

The initial stage of particle formation in high temperature processes is characterized by a high density of very small particles undergoing rapid coagulation. When these particles are solid this leads to agglomerates with a high specific surface area. However, at high gas temperatures particle coalescence which is very sensitive to the temperature may reduce the surface area and increase the size of the primary particles. In this paper we generalize the Smoluchowski equation to incorporate the coalescence rate into the aerosol dynamics. Individual agglomerates are characterized by their volume, v, and surface area, a. A Liouville term is added to the coagulation equation determining the movement of the distribution function through a-space due to coalescence. For the rate of coalescence a simple two sphere model has been used. Results for the surface area and the average diameter of the individual primary particles are presented for the case of a collision kernel which is independent of the particle structure. As an example, the theory is applied to fine particle formation in combustion processes under nonisothermal conditions.     

How does surface modification aid in the dispersion of carbon nanofibers?

Small-angle light scattering is used to assess the dispersion behavior of vapor-grown carbon nanofibers suspended in water. These data provide the first insights into the mechanism by which surface treatment promotes dispersion. Both acid-treated and untreated nanofibers exhibit hierarchical morphology consisting of small-scale aggregates (small bundles) that agglomerate to form fractal clusters that eventually precipitate. Although the morphology of the aggregates and agglomerates is nearly independent of surface treatment, their time evolution is quite different. The time evolution of the small-scale bundles is studied by extracting the size distribution from the angle-dependence of the scattered intensity, using the maximum entropy method in conjunction with a simplified tube form factor. The bundles consist of multiple tubes possibly aggregated side-by-side. Acid oxidation has little effect on this bundle morphology. Rather acid treatment inhibits agglomeration of the bundles. The time evolution of agglomeration is followed by fitting the scattering data to a generalized fractal model. Agglomerates appear immediately after cessation of sonication for untreated fibers but only after hours for treated fibers. Eventually, however, both systems precipitate.     

Titanium coated with high-performance nanocrystalline ruthenium oxide synthesized by the microwave-assisted sol–gel procedure

Ruthenium oxide coating on titanium was prepared by the sol–gel procedure from well-defined colloidal oxide dispersions synthesized by the microwave (MW)-assisted hydrothermal route under defined temperature and pressure heating conditions. The dispersions were characterized by dynamic light scattering (DLS) measurements and scanning electron microscopy (SEM). The electrochemical properties were analyzed as capacitive performances gained by cyclic voltammetry and electrochemical impedance spectroscopy and as the electrocatalytic activity for oxygen evolution from acid solution. The obtained dispersions were polydisperse and contained regular particles and agglomerates of increasing surface energy and decreasing particle size as the MW-assisted heating conditions were intensified. Owing to these features of the precursor dispersions, the obtained coatings had considerably improved capacitive performances and good electrocatalytic activity for oxygen evolution at high overpotentials.     

Synthesis of large-scale,narrowly dispersed,highly cross-linked,and spherical latex particles via one-step emulsion polymerization through particle coagulation

Particle coagulation technology is a facile approach to prepare large-scale and narrowly dispersed polymer particles. However, diverse shapes such as ellipsolid, snowman, dumbbell, and trimer among others were obtained if the cross-linker was directly added into the initial reaction mixtures due to the restriction of the highly cross-linking particle fusion process. In this study, we prepared sub-200?nm, narrowly dispersed, highly cross-linked, and spherical latex particles using particle coagulation technology by controlling the relation between the cross-linking net formation and particle coagulation. Depending on the addition time or feeding rate of the cross-linker (divinylbenzene, DVB), the particles with different sizes or shapes were obtained. The later the addition start time of DVB, the narrower the particle size distribution of the latex particles. Alternatively, the increase of the continuing feeding time could also be used to decrease the width of particle size distribution of the ultimate latex. In addition, narrowly dispersed and spherical latex particles also could be directly obtained by advancing the particle coagulation time using 2, 2′-Azobis (2-methylpropionamidine) dihydrochloride as a cationic initiator. Our study presents a new method that will further widen the fields of application of particle coagulation technology.