Self-preserving Size Distribution Research Articles

Continuous gas-phase synthesis of iron nanoparticles at ambient conditions with controllable size and polydispersity

HypothesisBy altering aerosol growth dynamics with unipolar charges, one can obtain aerosols with narrow particles size distributions, a highly desirable feature in applications of functional nanoparticles. ExperimentsUnlike liquid colloid systems, aerosol particles in the free molecular regime undergo coarsening due to Brownian coagulation and will eventually attain a self-preserving size distribution with a typical geometric standard deviation of 1.46 - 1.48. We developed a novel continuous one-step aerosol synthesis reactor that produces iron nanoparticles from ferrocene at ambient conditions, which confines the site of precursor breakdown and particle formation in the downstream vicinity of a positive corona discharge. FindingsWe demonstrated that the particle size could be controlled within 3 - 10 nm with a suppressed geometric standard deviation (1.15 - 1.35). The as-produced iron nanoparticles were successfully used as catalyst for the growth of single-walled carbon nanotubes with a narrow diameter range. With a transient aerosol dynamics model, we showed that a fraction (as small as 0.1%) of unipolar-charged particles could have a significant impact on the aerosol growth dynamics, which eventually results in a narrower particle size distribution with smaller size and higher number concentrations.

The invariant solution of Smoluchowski coagulation equation with homogeneous kernels based on one parameter group transformation

The governing equation of self-preserving size distribution can be obtained by similarity transformation from the Smoluchowski coagulation equation. To deal with the variable upper limit of convolution, the one parameter group transformation is introduced to get the invariant form of the governing equation. However, whether the equation has an invariant solution depending on asymptotic property of the coagulation kernel at the lower-end boundary. For homogeneous kernel, it can be discussed with three classes, type I has no similarity solution, type II has similarity solution which should be analyzed, and type III has similarity solution. Under the existence of similarity solution, an improved iDNS algorithm is developed with fourth order Runge–Kutta method to get the invariant solution of Smoluchowski coagulation equation more efficiently. Based on the invariant solution, a new definition of entropy in statistical physics is proposed to distinguish the convergence of Smoluchowski coagulation equation mathematically. The new entropy is a simple function of algebraic mean volume and has a maximum. At long time, the entropy production approaches to zero, which means the convergence of solution for Smoluchowski coagulation equation mathematically, and the Cercignani conjecture is almost true for Smoluchowksi coagulatin equation.

Solution of Smoluchowski coagulation equation for Brownian motion with TEMOM

The particle number density in the Smoluchowski coagulation equation usually cannot be solved as a whole, and it can be decomposed into the following two functions by similarity transformation: one is a function of time (the particle k-th moments), and the other is a function of dimensionless volume (self-preserving size distribution). In this paper, a simple iterative direct numerical simulation (iDNS) is proposed to obtain the similarity solution of the Smoluchowski coagulation equation for Brownian motion from the asymptotic solution of the k-th order moment, which has been solved with the Taylor-series expansion method of moment (TEMOM) in our previous work. The convergence and accuracy of the numerical method are first verified by comparison with previous results about Brownian coagulation in the literature, and then the method is extended to the field of Brownian agglomeration over the entire size range. The results show that the difference between the lognormal function and the self-preserving size distribution is significant. Moreover, the thermodynamic constraint of the algebraic mean volume is also investigated. In short, the asymptotic solution of the TEMOM and the self-preserving size distribution form a one-to-one mapping relationship; thus, a complete method to solve the Smoluchowski coagulation equation asymptotically is established.

Simulations of TiO2 nanoparticles synthesised off-centreline in jet-wall stagnation flames

A theoretical analysis of the formation of titanium dioxide (TiO2) nanoparticles from titanium tetraisopropoxide (TTIP) in premixed, jet-wall stagnation flames was performed to investigate the variation of the particle properties as a function of deposition radius. Two different TTIP loadings (280 and 560 ppm) were studied in two flames: a lean flame (equivalence ratio, ϕ=0.35) and a stoichiometric flame (ϕ=1.0). First, the growth of particles was described using a spherical particle model that was fully coupled to the conservation equations of chemically reacting flow and solved in 2D using the finite volume method. Second, particle trajectories were extracted from the 2D simulations and post-processed using a hybrid particle-number/detailed particle model solved using a stochastic numerical method. In the 2D simulations, the particles were predicted to have mean diameters in the range 3–10 nm, which is consistent with, but slightly less than experimental values observed in the literature. Off-centreline particle trajectories experienced longer residence times at higher temperatures downstream of the flame front. Two particle size distribution (PSD) shapes were observed. In the lean flame, a bimodal PSD was observed due to the high rates of inception and surface growth. In contrast, the stoichiometric flame was dominated by coagulation and the particles quickly attained a self-preserving size distribution. The PSDs were found to be different beyond a deposition radius of approximately one and a half times the nozzle radius due to a small degree of aggregation; this may impact the synthesis of nanoparticles using jet-wall stagnation flames for novel applications. Suggestions are made for future work, not least including the need for the predicted radial behaviour to be tested experimentally.

An Approximate Analytical Solution for the Knudsen Number-Dependent Self-Preserving Size Distributions of Coagulating Fine Particles in the Transition Regime

An approximate analytical solution for the size distribution width of fine particles coagulating in the transition regime is derived. It is assumed that the particle size distribution can be represented by a time-dependent log-normal function, which passes through a series of quasi-self-preserving states. The solution extends a previous solution presented for the near-continuum regime to cover the whole transition regime by adopting Dahneke’s collision kernel based on the flux-matching theory. Good agreement was obtained when the new solution was compared to numerical calculations. The effective degree of homogeneity of collision kernel is defined and used for a comprehensive analysis of the Knudsen number-dependent behavior of the quasi-self-preserving size distributions. Although the new solution is valid for the whole particle size range from the free-molecule regime (Knudsen number larger than ~50) through the transition regime (Knudsen number between ~1 and ~50) to the continuum regime (Knudsen number smaller than ~1), it is practically applicable in the transition regime because the self-preserving behavior in the continuum regime or in the free-molecule regime had previously been investigated and understood well.

Polymerization and Collision in High Concentrations for Brownian Coagulation

Aggregation always occurs in industrial processes with fractal-like particles, especially in dense systems (the volume fraction, ϕ>1%). However, the classic aggregation theory, established by Smoluchowski in 1917, cannot sufficiently simulate the particle dynamics in dense systems, particularly those of generat ed fractal-like particles. In this article, the Langevin dynamic was applied to study the collision rate of aggregations as well as the structure of aggregates affected by different volume fractions. It is shown that the collision rate of highly concentrated particles is progressively higher than that of a dilute concentration, and the SPSD (self-preserving size distribution) is approached (σg,n≥1.5). With the increase in volume fraction, ϕ, the SPSD broadens, and the geometric standard is 1.54, 1.98, and 2.73 at ϕ=0.1, 0.2, and 0.3. When the volume fraction, ϕ, is higher, the radius of gyration is smaller with the same cluster size (number-based), which means the particle agglomerations are in a tighter coagulation. The fractal-like property Df is in the range of 1.60–2.0 in a high-concentration system. Knowing the details of the collision progress in a high-concentration system can be useful for calculating the dynamics of coagulating fractal-like particles in the industrial process.

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.

Surface growth, coagulation and oxidation of soot by a monodisperse population balance model

A monodisperse population balance model (MPBM) is developed here that capitalizes on the rapid attainment of the self-preserving size distribution and asymptotic fractal-like structure of agglomerates by coagulation to simulate their evolution with only three equations. Total agglomerate carbon molar, C, number (N) and area (A) concentrations are tracked. The model accounts for the polydispersity of agglomerates by enhancing their collision frequency by that of their self-preserving size distribution based on the radius of gyration in the free molecular regime. Scaling laws from detailed discrete element modeling (DEM) simulations are used to describe the fractal-like morphology of the agglomerates. The MPBM predicts the evolution of soot fv, N and average mobility and primary particle diameters during surface growth and agglomeration in laminar premixed ethylene flames as well as soot oxidation in a tube reactor within 30% of detailed DEM, sectional population balance simulations and measurements. Thus, when self-preserving size distribution and asymptotic structure of agglomerates are attained, this simple MPBM has unprecedented accuracy and can be readily interfaced with computational fluid dynamic (CFD) to model soot formation in combustion devices or process design and optimization for the synthesis of carbonaceous agglomerate nanoparticles.

Self-preserving size distribution and collision frequency of flame-made nanoparticles in the transition regime

Coagulation of polydisperse primary particles is the dominant agglomerate growth mechanism in flame reactors and in the low-temperature or high-pressure regions of combustion engines. Here, the evolution of agglomerate size distribution (SD) and collision frequency function of coagulating soot and silica primary particles is investigated with simultaneous surface growth or coalescence using discrete element modeling (DEM). The growth of polydisperse soot and silica nanoparticles is captured from the free molecular up to the transition regime in excellent agreement with microscopy and mobility measurements in diffusion flames and flame spray pyrolysis (FSP) reactors. The broad agglomerate size distributions of soot and silica formed in the free molecular regime narrow by coagulation in the transition regime, attaining their mobility-based quasi-self-preserving or signature SD with the same geometric standard deviation of 1.48 ± 0.03, in excellent agreement with soot or silica SDs measured from diffusion flames and the CAST generator or from a FSP reactor, respectively. The coagulation rate of both soot and silica agglomerates is about twice as much higher as that of monodisperse agglomerates in the transition regime, regardless of material or combustion conditions. The DEM-derived collision kernel enhanced by a factor of 1.82 ± 0.35 can be interfaced with method of moments and monodisperse population balance models to improve the design of aerosol flame reactors and combustion engines.

A novel moment method using the log skew normal distribution for particle coagulation

A novel moment method is proposed to predict the detailed size distribution for particle coagulation. The method uses the log skew normal distribution (LSND) to approximate the actual size distribution. The LSND is a four-parameter distribution that generalizes the conventional log-normal distribution to allow for asymmetrical characteristics. The moment closure is achieved by applying the Gauss-Hermite quadrature for the integral terms in the moment equations. With this approach, no further treatments are needed for specific coagulation kernels. Then the method is validated by comparing it with other recognized numerical methods for Brownian coagulation in the continuum regime and the free-molecular regime. The results show that the present method well predicts the detailed size distribution over the entire coagulation time for both regimes. Moreover, the tails of the self-preserving size distribution are well reproduced by the present method. The asymptotic characteristics of Brownian coagulation are also analyzed based on the LSND. The results validate the LSND in analytically representing the self-preserving size distribution and predicting the asymptotic geometric standard deviation.

A new approximation approach for analytically solving the population balance equation due to thermophoretic coagulation

The analytical solution of the population balance equation (PBE) for coagulation by differential settling velocities remains a challenging issue due to the absolute value in the collision kernel. A geometric mean approximation approach is first proposed in this study to handle the integration of the absolute value and convert the moment equation into an integrable form. Using this approach along with the log-normal method of moments, the PBE is first analytically solved for thermophoretic coagulation in the low Knudsen number limit. The present analytical solution is validated by comparing the size distribution parameters with the sectional method. The maximum relative errors in this study are quite acceptable as compared with previous analytical solutions for Brownian coagulation. Moreover, the self-preserving size distribution predicted by the present analytical solution compares well with the sectional method and previous numerical studies on thermophoretic coagulation. Based on the analytical solution, the characteristics of the size distribution evolution during thermophoretic coagulation are investigated theoretically. The results show that the coagulation process is faster with a larger initial geometric standard deviation and the time required to reach the self-preserving distribution increases with the difference between the initial and asymptotic geometric standard deviations.

The asymptotic stability of the Taylor-series expansion method of moment model for Brownian coagulation

In the present study, the linear stability of population balance equation due to Brownian motion is analyzed with the Taylor-series expansion method of moment. Under certain conditions, the stability of the Taylor-series expansion method of moment model is reduced to a well-studied problem involving eigenvalues of matrices. Based on the principle of dimensional analysis, the perturbation equation is solved asymptotically. The results show that the Taylor-series expansion method of moment model is asymptotic stable, which implies that the asymptotic solution is uniqueness, and supports the self-preserving size distribution hypothesis theoretically.

Flame synthesis of functional nanostructured materials and devices: Surface growth and aggregation

Combustion is essential to the manufacture of carbon black, fumed oxides, optical fibers and, recently, new high-value products like carbon nanotubes, nanosilver and biomagnetic nanofluids that are driven to market predominantly by start-ups. This technology is attractive for material synthesis for its proven scalability as it does not involve the tedious steps of wet chemistry and can readily form stably metastable compositions and high purity products.Recent advances in aerosol and combustion sciences reveal that coagulation and sintering and/or surface growth control product particle size and morphology through the high temperature particle residence time, self-preserving size distribution and power laws for fractal-like particles. This motivates synthesis of an array of unique particle compositions and morphologies primarily by spray combustion leading to new catalysts, gas sensors, bio-materials and, most recently, to hand-held devices such as breath analysis sensors for monitoring chronic illnesses. In particular, multi-scale process design integrating mesoscale and molecular dynamics facilitates understanding of combustion product development.The latter also contributes to understanding of aggregation and surface growth of nascent soot, a bona fide nanostructured material! So here nascent soot dynamics, after nucleation or inception, are investigated through accounting of soot agglomeration and surface growth by acetylene pyrolysis. Neglecting the fractal-like nature of soot underestimates its mobility diameter and polydispersity up to 40%. The evolution of nascent soot structure from spheres to aggregates is quantified by the mass fractal dimension and mass–mobility exponent, in excellent agreement with microscopic and mass–mobility measurements in a standard burner-stabilized stagnation ethylene flame. Surface growth chemically bonds the constituent primary particles of these aggregates, while the effect of soot volume fraction on soot morphology is elucidated. Based on aggregate projected area, a scaling law is derived for determining the primary particle size of nascent soot aggregates from mass–mobility measurements rather than tedious image counting.

Coagulation of Agglomerates Consisting of Polydisperse Primary Particles.

The ballistic agglomeration of polydisperse particles is investigated by an event-driven (ED) method and compared to the coagulation of spherical particles and agglomerates consisting of monodisperse primary particles (PPs). It is shown for the first time to our knowledge that increasing the width or polydispersity of the PP size distribution initially accelerates the coagulation rate of their agglomerates but delays the attainment of their asymptotic fractal-like structure and self-preserving size distribution (SPSD) without altering them, provided that sufficiently large numbers of PPs are employed. For example, the standard asymptotic mass fractal dimension, Df, of 1.91 is attained when clusters are formed containing, on average, about 15 monodisperse PPs, consistent with fractal theory and the literature. In contrast, when polydisperse PPs with a geometric standard deviation of 3 are employed, about 500 PPs are needed to attain that Df. Even though the same asymptotic Df and mass-mobility exponent, Dfm, are attained regardless of PP polydispersity, the asymptotic prefactors or lacunarities of Df and Dfm increase with PP polydispersity. For monodisperse PPs, the average agglomerate radius of gyration, rg, becomes larger than the mobility radius, rm, when agglomerates consist of more than 15 PPs. Increasing PP polydispersity increases that number of PPs similarly to the above for the attainment of the asymptotic Df or Dfm. The agglomeration kinetics are quantified by the overall collision frequency function. When the SPSD is attained, the collision frequency is independent of PP polydispersity. Accounting for the SPSD polydispersity in the overall agglomerate collision frequency is in good agreement with that frequency from detailed ED simulations once the SPSD is reached. Most importantly, the coagulation of agglomerates is described well by a monodisperse model for agglomerate and PP sizes, whereas the detailed agglomerate size distribution can be obtained by scaling the average agglomerate size to the SPSD.

Predictions on dynamic evolution of compositional mixing degree in two-component aggregation

The compositional distribution in two-component aggregative mixing of initially bidisperse particle populations can be described by a Guassian-type function, which is determined by the mixing degree χ (assessed quantitatively by the mass-normalized power density of excess component A), and the overall mass fraction ϕ (a known value from the initial feeding condition) of component A. It is known that χ will reach a steady-state value χ∞ over time (factually, after attaining the self-preserving size distribution), and χ∞ is only relevant to ϕ, namely the feeding condition. However, the dynamic evolution of χ before the attainment of a steady-state value is not exactly known. In this paper, the fast differentially-weighted Monte Carlo method for population balance modeling was used to predict the dependence of time-varied χ on initial feeding conditions through hundreds of systematically varied simulations. It is found that χ is subject to an exponential decay, largely depending on the ratio of steady-state mixing degree and its initial value (χ∞/χ0). With the explored exponential formulas for the dynamic mixing degree, it is possible to attain an optimum control on the compositional distributions during two-component aggregation processes through selecting the initial feeding parameters, and the time needed for reaching a steady-state is investigated.

The self-preserving size distribution of fractal aggregates coagulating by Brownian motion and simultaneous fluid shear at low Peclet numbers: Numerical solutions

The asymptotic size distribution of fractal particles undergoing Brownian and simultaneous shear-induced coagulation has not been investigated. We have addressed this issue by establishing and solving ordinary integro-differential coagulation equations. The self-preserving distribution (SPSD) requires a shear rate decreasing with time according to a formula, which we report, for various fractal dimensions. Under this condition, it is the Peclet number of primary particles that controls the self-preservative characteristics of coagulating aggregates. The size distribution of fractal aggregates at low Peclet numbers of primary particles was determined to be self-preserving. The SPSDs were calculated for aggregates of various fractal dimensions. An upper limit of the Peclet number exists where the SPSD could be obtained. This upper limit decreases with decreasing fractal dimension: from 1.1 for the fractal dimension of 3 to 0.14 for the fractal dimension of 1.8. The Peclet number of particles with the mean volume hydrodynamic radius is a constant during the coagulation process, and is proportional to the Peclet number of primary particles. As the Peclet number increases, the SPSDs will broaden, and the peak value will also increases and drifts to the left. The SPSD is close to a lognormal distribution. This represents a theoretical foundation for the size distribution evolution of coagulating fractal aggregates in flow fields and for the lognormal size distribution assumption of atmospheric aerosols.

Coagulation-agglomeration of fractal-like particles: structure and self-preserving size distribution.

Agglomeration occurs in environmental and industrial processes, especially at low temperatures where particle sintering or coalescence is rather slow. Here, the growth and structure of particles undergoing agglomeration (coagulation in the absence of coalescence, condensation, or surface growth) are investigated from the free molecular to the continuum regime by discrete element modeling (DEM). Particles coagulating in the free molecular regime follow ballistic trajectories described by an event-driven method, whereas in the near-continuum (gas-slip) and continuum regimes, Langevin dynamics describe their diffusive motion. Agglomerates containing about 10-30 primary particles, on the average, attain their asymptotic fractal dimension, D(f), of 1.91 or 1.78 by ballistic or diffusion-limited cluster-cluster agglomeration, corresponding to coagulation in the free molecular or continuum regimes, respectively. A correlation is proposed for the asymptotic evolution of agglomerate D(f) as a function of the average number of constituent primary particles, n̅(p). Agglomerates exhibit considerably broader self-preserving size distribution (SPSD) by coagulation than spherical particles: the number-based geometric standard deviations of the SPSD agglomerate radius of gyration in the free molecular and continuum regimes are 2.27 and 1.95, respectively, compared to ∼1.45 for spheres. In the transition regime, agglomerates exhibit a quasi-SPSD whose geometric standard deviation passes through a minimum at Knudsen number Kn ≈ 0.2. In contrast, the asymptotic D(f) shifts linearly from 1.91 in the free molecular regime to 1.78 in the continuum regime. Population balance models using the radius of gyration as collision radius underestimate (up to about 80%) the small tail of the SPSD and slightly overpredict the overall agglomerate coagulation rate, as they do not account for cluster interpenetration during coagulation. In the continuum regime, when a recently developed agglomeration rate is used in population balance equations, the resulting SPSD is in excellent agreement with that obtained by DEM.

Dependence of Steady-State Compositional Mixing Degree on Feeding Conditions in Two-Component Aggregation

Two-component aggregative mixing of initially bidisperse particle populations results in a Gaussian-type compositional distribution function, which can be fully described by the overall mass fraction of component A (ϕ) and the mass-normalized power density of excess component A (χ, indicating the mixing degree). χ will reach a steady-state value once the self-preserving size distribution is attained. The relation between the steady-state value of χ (χ∞) and its initial value (χ0) has not been investigated before. This paper applies population balance modeling to gain insight into the dependence of χ∞ on initial feeding conditions. By model fitting from hundreds of systematically varied simulations, it is found that χ∞/χ0, which depends on ϕ, can be formalized by the Gaussian-type function for Brownian aggregation in the free-molecular regime as well as in the continuum regime, however with different geometric standard deviations. The present work can help to optimize mixing by properly selecting the initial mass and number concentrations of components A and B in the feed.

Agglomerates and aggregates of nanoparticles made in the gas phase

Gas-phase (aerosol) technology is used widely in manufacture of various nanostructured commodities at tons/hour today. So it is quite promising for synthesis of sophisticated nanoparticles motivating basic and applied research. Frequently such nanoparticles are made as clusters of primary particles (PPs) by chemical reaction, aerosol coagulation, sintering, surface growth and even fragmentation. When PPs are bonded by strong chemical forces, they are termed aggregates. As such they are sought in catalysis, lightguide preform manufacture and, most importantly, as components in electronic devices (sensors, batteries). When PPs and aggregates are held together by rather weak, physical forces, they form agglomerates. These are attractive in nanocomposites and fluid suspensions (paints, nanofluids, bioimaging). Such clusters may have also distinct health effects, beyond those of equivalent spherical particles.Agglomerates and aggregates are characterized by microscopy, electromagnetic scattering and mass mobility measurements in terms of their volume-equivalent radius, radius of gyration and/or mobility radii in the free molecular and continuum regimes along with the corresponding power laws (fractal dimension, Df). Coagulation and sintering largely determine nanoparticle structure. Coagulation of PPs leads to agglomerates of Df=1.78 and 1.91 in the continuum and free-molecular regimes, respectively. The coagulation rate of agglomerates is higher than that of volume-equivalent spheres in the free molecular regime. Agglomerates attain also a self-preserving size distribution by coagulation facilitating process design for their manufacture. Mesoscale simulations elucidate the sintering (or coalescence) of agglomerates to aggregates and narrowing of their PP size distribution. Once agglomerates start to sinter, they follow a power law to aggregates and eventually to compact (spherical) particles, regardless of composition and initial PP size distribution. Aggregate properties are in-between those of the initial agglomerate and the fully coalesced sphere. Finally the stability of agglomerates under ultrasonication, stretching, fluid dispersion, impaction and capillary condensation is highlighted.

Asymptotic behavior of the Taylor-expansion method of moments for solving a coagulation equation for Brownian particles

The evolution equations of moments for the Brownian coagulation of nanoparticles in both continuum and free molecule regimes are analytically studied. These equations are derived using a Taylor-expansion technique. The self-preserving size distribution is investigated using a newly defined dimensionless parameter, and the asymptotic values for this parameter are theoretically determined. The dimensionless time required for an initial size distribution to achieve self-preservation is also derived in both regimes. Once the size distribution becomes self-preserving, the time evolution of the zeroth and second moments can be theoretically obtained, and it is found that the second moment varies linearly with time in the continuum regime. Equivalent equations, rather than the original ones from which they are derived, can be employed to improve the accuracy of the results and reduce the computational cost for Brownian coagulation in the continuum regime as well as the free molecule regime.