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.     

Modeling Orthokinetic Coagulation in Spatially Varying Laminar Flow

An orthokinetic coagulation model including the effects of agglomeration and local stress-induced aggregate breakup was developed. This model was used to simulate coagulation in the flow between two eccentrically located and rotating cylinders. Four methods of modeling coagulation in the flow system were examined. The first technique used a volume-weighted average of the local strain rates, while a second method used an equivalent volume-weighted power (G). A third method treated each volume element as a separate batch reactor and determined a final volume-averaged floc population. The final modeling technique applied mass transfer between each of the elements. Results indicated that substantial differences in average particle diameters and populations were generated with each of the methods, especially where mass transfer between the elements was considered. It was concluded that mass transfer between regions of varying flow strain rate and/or velocity gradient should be included in accurate coagulation modeling. Copyright 2000 Academic Press.     

Bubble particle heterocoagulation under turbulent conditions

An analytical model that enables the calculation of the flotation rate constant of particles as a function of particle size with, as input parameters, measurable particle, bubble, and hydrodynamic quantities has been derived. This model includes the frequency of collisions between particles and bubbles as well as their efficiencies of collision, attachment, and stability. The generalized Sutherland equation collision model and the modified Dobby-Finch attachment model developed previously for potential flow conditions were used to calculate the efficiencies of particle-bubble collision and attachment, respectively. The bubble-particle stability efficiency model includes the various forces acting between the bubble and the attached particle, and we demonstrate that it depends mainly on the relative magnitude of particle contact angle and turbulent dissipation energy. The flotation rate constants calculated with these models produced the characteristic shape of the flotation rate constant versus particle size curve, with a maximum appearing at intermediate particle size. The low flotation rate constants of fine and coarse particles result from their low efficiency of collision and low efficiencies of attachment and stability with gas bubbles, respectively. The flotation rate constants calculated with these models were compared with the experimental flotation rate constants of methylated quartz particles with diameters between 8 and 80 micro m interacting with gas bubbles under turbulent conditions in a Rushton flotation cell. Agreement between theory and experiment is satisfactory.     

A study on shear coagulation and heterocoagulation

Coagulation and heterocoagulation of spherical particles in a simple shear flow have been studied by means of trajectory analysis. Effects of particle size and size ratio have been extensively examined. Some new features of shear coagulation and heterocoagulation have been recognized. Primary and secondary shear coagulations differ in many aspects. The nonequatorial effective capture cross-sections can be attributed to the pure secondary shear coagulation. Shear stability can be low at small and large particle size ranges due to secondary and primary shear coagulation, respectively, with a high stability range at medium particle sizes. A second range of high stability may appear at even larger sizes. On the basis of relative coagulation rate, shear heterocoagulation between particles of the same material but different sizes may or may not be favored over the respective homocoagulations. In the case of primary coagulation, the two homocoagulations are favored over the heterocoagulation. The opposite is true in the case of secondary coagulation. In a suspension composed of particle species of different materials and different sizes, if the larger particles are less stable at an assumed same size with the smaller particles, the homocoagulation of the larger species is still favored over heterocoagulation in the case of primary shear coagulation. In the case of secondary shear coagulation, the heterocoagulation may be favored over the homocoagulation of the larger species, if the above mentioned stability difference is not very large, the particle size difference is not small, and the size of the larger particles is within a certain range.     

沉降对不同粒径聚苯乙烯胶乳球异向聚集的影响

R.Folkersma等报道了在微重力环境下2μm聚苯乙烯(PS)胶乳球的异向聚集速率有明显增大的结果，本文作者之一孙祉伟等的实验结果却与此有很大差异。为此作者在孙祉伟等的实验基础上对1,2,3μmPS以及1μm+2μmPS胶乳球混和体系的相对聚集速率进行了研究。作者对原有实验装置进行了改进，并验证了改进后的实验装置的可行性。用密度匹配法实现模拟微重力条件，用快聚集过程中浊度随时间的变化表示相对聚集速率。结果表明，重力引起的沉降对所研究体系聚集速率的影响是很小的，在实验误差范围内可以忽略。作者认为与Stein等结果显著不同的原因是二者使用的样品表面性质不同以及实验方法的差异。     

Flocculation kinetics of precipitated calcium carbonate

When the percentage of filler in paper is increased, the optical properties are improved and the production cost lowered. However, fillers weaken paper strength by decreasing the fibre–fibre bonded area. Little is known about the optimum filler floc size or filler floc properties to allow developing optimum paper characteristics. Consequently, the kinetics of aggregation of scalenohedral precipitated calcium carbonate (PCC) filler was studied using various polymers (flocculants, coagulants and dry strength agents). The sodium salt of partially hydrolysed polyvinyl formamide copolymerized with acrylic acid (PVFA/NaAA) or C-starch lead to floc sizes, less sensitive to dosage within a certain range. Results from stability ratios correlate with PCC particle size. The change in particle size measured by photometric dispersion analysis (PDA) correlates well with the change in PCC particle size measured by light scattering/diffraction. Kinetic calculations show the orthokinetic aggregation times to be consistent with the experimental PDA results. The main uncertainty in the orthokinetic times is estimating the effective shear rate. It is proposed that the bridging surface area of PCC particles, the area which can form bonds between PCC particles or aggregates, should be used to study the kinetics of PCC aggregation, and not the total or projected surface area. In polymer induced aggregation, the PCC particle size increases to a plateau value with increasing polymer dosage. Two regions are most pronounced for C-PAM, PVFA/NaAA and A-starch. Region I corresponds to bridging flocculation. Region II is where the particle size reaches a plateau, and not the expected maximum predicted by classical polymer bridging theory or charge neutralisation theory, likely because of a competition between particle aggregation and polymer adsorption.     

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.     

Particle interactions in flowing suspensions

For a fully destabilized suspension of non-Brownian praticles in laminar tube flow, the extent of orthokinetic flocculation can be calculated by classical Smoluchowski theory, using the average shear rate $\text{G}$ and the average residence time $\text{t}$. It can be shown very simply that the dimensionless quantity $\text{Gt}$ (and hence the degree of flocculation) depends only on the tube dimension and not on the flow rate. However, calculations based on this approach predict far more flocculation than is observed experimentally. There are two major reasons for the discrepancy: 1) the Smoluchowski treatment of orthokinetic flocculation neglects hydrodynamic interaction between particles, which can be introduced by a semi-empirical method due to van de Ven and Mason and this step leads collision efficiencies which are considerably less than unity and depend both on shear rate and on interparticle forces; 2) the shear rate is not uniform in the tube but varies from zero at the tube axis to a maximum value at the wall. Since the major contribution to the flow comes from regions close to the tube axis, where the shear rate is low, the simple averaging procedure considerably overestimates the degree of flocculation.From experimental measurements on the degree of flocculation of dispersions achieved by laminar flow through narrow tubes at different flow rates it is possible to draw semi-quantitative conclusions concerning particle interaction and the strength of flocs.The effect of helical winding of the tube is briefly considered and shown to give more flocculation than in a straight tube. Some experimental results for latex particles destabilized by cationic polymers flowing through straight and coiled tubes are mentioned.     

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.     

Effect of brownian diffusion on chemical kinetics

A new method of theoretical prediction of the kinetic rate constants of fast chemical reactions in solutions is presented. It takes into account the effect of finite diffusive displacements of the reacting molecules. The approach is based on the solution of the steady-state Fokker–Planck equation by the moments method of Grad developed in the theory of coagulation of aerosol particles. A comparison of the predicted rate constants with the experimental data provided by Schuh and Fischer for the self-reaction of tert-butyl radicals in n-alkanes shows a good correspondence.     

Study on improving the turbidity measurement of the absolute coagulation rate constant

The existing theories dealing with the evaluation of the absolute coagulation rate constant by turbidity measurement were experimentally tested for different particle-sized (radius = a) suspensions at incident wavelengths (lambda) ranging from near-infrared to ultraviolet light. When the size parameter alpha = 2pi a/lambda > 3, the rate constant data from previous theories for fixed-sized particles show significant inconsistencies at different light wavelengths. We attribute this problem to the imperfection of these theories in describing the light scattering from doublets through their evaluation of the extinction cross section. The evaluations of the rate constants by all previous theories become untenable as the size parameter increases and therefore hampers the applicable range of the turbidity measurement. By using the T-matrix method, we present a robust solution for evaluating the extinction cross section of doublets formed in the aggregation. Our experiments show that this new approach is effective in extending the applicability range of the turbidity methodology and increasing measurement accuracy.     

Absolute rate of turbulent coagulation from turbidity measurement

Turbidity method has been applied to assess colloid stability. While the method is simple and easy, it is not straightforward in evaluating absolute coagulation rates of colloidal particles. That is, the method requires the evaluation of the extinction cross section of doublets. Recently, the turbidity measurement has been successfully utilized to evaluate the absolute Brownian coagulation rate constant with the T-matrix method, which calculates the extinction cross section of doublets at arbitral size range. The present work was performed to extend the applicability of the method to turbulent coagulation. To this end, we measured the turbidity change of coagulating latex suspensions in a turbulent flow. The measurement was performed as a function of particle size. From the turbidity vs. time relationship, we evaluated turbulent coagulation rate constant using the T-matrix method. Obtained values of the constants agreed well with theoretical ones, demonstrating the usefulness of turbidity method for turbulent coagulation.     

Effect of Particle Size on Rate of Coalescence of Silica Nanoparticles

In high temperature processes, at high concentrations, aerosol particles grow by collisions and coalescence. The rate of coalescence is an important parameter for predicting the final primary particle size. For some materials, particularly silica, predictions of final primary particle size using coalescence rates based upon bulk material properties are not in agreement with experimental results. One explanation may be that the high internal pressure in very small particles (less than 10 nm in diameter) affects the mobility of material within the particles and hence the coalescence rate. In this Note, a new approach for estimating rates of coalescence of particles in the initial stages of growth is presented. Coalescence of liquid particles is assumed to be rate-limited by atomic mobility, and the effect of internal pressure on diffusivity is considered. Copyright 1999 Academic Press.     

Particle Production in Reflection and Transmission Mode Laser Ablation: Implications for Laserspray Ionization

Particles were ablated from laser desorption and inlet ionization matrix thin films with a UV laser in reflection and transmission geometries. Particle size distributions were measured with a combined scanning mobility particle sizer (SMPS) and aerodynamic particle sizer (APS) system that measured particles in the size range from 10 nm to 20 μm. The matrixes investigated were 2,5-dihydroxybenzoic acid (DHB), α-cyano-4-hydroxycinnamic acid (CHCA), sinapic acid (SA), 2,5-dihydroxy-acetophenone (DHAP), and 2-nitrophloroglucinol (NPG). Nanoparticles with average diameters between 20 and 120 nm were observed in both transmission and reflection geometry. The particle mass distribution was significantly different in reflection and transmission geometry. In reflection geometry, approximately equal mass was distributed between particles in the 20 to 450 nm range of diameters and particles in the 450 nm to 1.5 μm diameter range. In transmission mode, the particle mass distribution was dominated by large particles in the 2 to 20 μm diameter range. Ablation of inlet ionization matrices DHAP and NPG produced particles that were 3 to 4 times smaller compared with the other matrices. The results are consistent with ion formation by nanoparticle melting and breakup or melting and breakup of the large particles through contact with heated inlet surfaces.

On the Mechanism of Aggregate Formation in Metal-Filled Polymer Composites

Mechanisms of the aggregation of nickel particles in a curing polymer matrix are studied. It is shown that the Brownian diffusion of particles and their orthokinetic coagulation cannot explain the experimental data on the aggregate formation obtained earlier.     

Numerical and similarity solutions for reversible population balance equations with size-dependent rates

Population balance equations (PBEs) for reversible aggregation-fragmentation processes are important to particle agglomeration and dissolution, polymerization and degradation, liquid droplet coalescence and breakup, and floc coagulation and disintegration. Moment solutions provide convenient solutions to the PBEs, including steady state and similarity solutions, but may not be feasible for complex forms of size-dependent rate coefficients and stoichiometric kernels. Numeric solutions are thus necessary not only for applications, but also for the study of the mathematics of PBEs. Here we propose a numerical method to solve PBEs and compare the results to moment solutions. The numeric results are consistent with known steady state and asymptotic long-time similarity solutions and show how processes can be approximated by self-similar formulations.     

Simulation of secondary particle formation in seeded emulsion polymerization: The effect of surface charge density

Mathematical modeling and simulation were carried out to investigate the effects of the surface charge density of seed particles on secondary particle formation and the rate of polymerization in the early stage of emulsifier-free seeded emulsion polymerization of methyl methacrylate. Limited coagulation theory was applied to simulate new particle nucleation. The main factor influencing the capture rate of oligomeric radicals in a growing seed particle was assumed to be the electrostatic repulsion of seed particles. DLVO (Deryagiun-Landau-Verwey-Overbeek) theory was also introduced to estimate the electrical repulsion between seed particles and oligomeric radicals in the aqueous phase. In the case of highly charged seed particles, the adsorption rate of oligomeric radicals in the aqueous phase showed a strong effect on the polymerization rate. The low adsorption of oligomeric radicals results in a low value of the average number of radicals per particle. The surface charge density of seed particles was found to play an important role in limiting the polymerization rate at the beginning of the reaction and even in affecting the formation of secondary particles.     

Role of adsorption in formation of inorganic nanoparticles: Kinetic model

A simple kinetic model is proposed for the formation of inorganic nanoparticles in the presence of additives of readily adsorbing organic compounds. Additives and monomers may occupy the same sites on the surface of a growing particle. The maximum sizes and size distribution of formed particles are estimated under the assumption that the surface curvature of a growing particle has equivalent effects on the rate constants of the growth and adsorption. Equations are derived that relate the polydispersity indices for particle mass and radius distributions to the variances of particle radius distribution. The conditions are determined for the formation of virtually monodisperse nanoparticles.     

Specific features of luminescence of Q-CdS colloids with different sizes

The specific features of luminescence of colloidal solutions of Q-CdS with particles of different size and the regularities of luminescence quenching by quenchers of various nature were studied. The luminescence spectra of Q-CdS consist of several bands, which are shifted to the long-ware region as the particle size increases. The dependence of the integral quantum yield of luminescence on the particle size has a sharp maximum at a particle diameter of ?23Å. A Stem—Volmer-type equation including the adsorption isotherm of the quencher molecules on the surface of the Q-CdS colloidal particles was used to describe the regularities of luminescence quenching of Q-CdS colloidal solutions. The CdS particle size was found to affect the efficiency of luminescence quenching. The regularities of luminescence quenching depend both on the rate constant of electron transfer to the quencher molecules and on the ability of the quencher molecules to be adsorbed on the surface of the CdS colloidal particle.