Extension of the Smoluchowski Theory to Transitions from Dilute to Dense Regime of Brownian Coagulation: Triple Collisions

Veshchunov, M.S.; Тарасов, В. И.

doi:10.1080/02786826.2014.931567

Cited by 14 publications

(4 citation statements)

References 23 publications

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“…The discrepancy between the analytical and numerical results was evaluated by the root mean square error (RMSE), which is shown as a function of φ and R in Supplementary Figure S3. In general, the results show higher values of the RMSE for smaller NPs and larger φ, which is in agreement with previous results: it was shown that the classic aggregation theory works well for extremely dilute solutions, namely φ < 0.1%, which are controlled by binary collisions, whereas are dominating in systems with larger φ, thus leading to faster aggregation is higher than the one predicted by the original model [65]. Clearly, when small NPs are considered, the theoretical model overestimates the mobility of large aggregates.…”

Section: Discussionsupporting

confidence: 92%

“…The setups with smaller NPs exhibit greater deviations from theoretical predictions (see Figure 7A,B for R = 0.78 nm and Figure 7C,D for R = 1.5 nm): while aggregation proceeds faster at low simulation times, after about 2 µs it will proceed more slowly than the Smolochowski model. This result can be interpreted considering that aggregation in highly concentrated solutions could initially proceed faster due to multi-particle collisions, whereas the Smoluchowski theory only considers binary collisions [65], and slow down at higher times due to the reduced mobility of larger aggregates. On the other hand, the largest particles tested presented faster aggregation kinetics with respect to the theoretical model (see Figure 7E,F).…”

Section: Discussionmentioning

confidence: 99%

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Multi-Scale Modelling of Aggregation of TiO2 Nanoparticle Suspensions in Water

et al. 2022

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Titanium dioxide nanoparticles have risen concerns about their possible toxicity and the European Food Safety Authority recently banned the use of TiO2 nano-additive in food products. Following the intent of relating nanomaterials atomic structure with their toxicity without having to conduct large-scale experiments on living organisms, we investigate the aggregation of titanium dioxide nanoparticles using a multi-scale technique: starting from ab initio Density Functional Theory to get an accurate determination of the energetics and electronic structure, we switch to classical Molecular Dynamics simulations to calculate the Potential of Mean Force for the connection of two identical nanoparticles in water; the fitting of the latter by a set of mathematical equations is the key for the upscale. Lastly, we perform Brownian Dynamics simulations where each nanoparticle is a spherical bead. This coarsening strategy allows studying the aggregation of a few thousand nanoparticles. Applying this novel procedure, we find three new molecular descriptors, namely, the aggregation free energy and two numerical parameters used to correct the observed deviation from the aggregation kinetics described by the Smoluchowski theory. Ultimately, molecular descriptors can be fed into QSAR models to predict the toxicity of a material knowing its physicochemical properties, enabling safe design strategies.

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Section: Discussionsupporting

confidence: 92%

Section: Discussionmentioning

confidence: 99%

Multi-Scale Modelling of Aggregation of TiO2 Nanoparticle Suspensions in Water

et al. 2022

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“…Also, agglomerate morphology and structure affect handling and processing and eventually agglomerate performance, 13 while mobility determines agglomerates' transport and dispersal properties. 5 Even though Brownian coagulation is reasonably well understood for fully coalescing particles at both dilute 14 and high concentrations, 15,16 little is known about the dynamics of agglomerates during coagulation, especially in the transition regime where the gas mean free path is comparable to the agglomerate radius. Vemury and Pratsinis 17 showed the attainment of self-preserving size distributions (SPSDs) by coagulation of agglomerates with constant D f in the free molecular and continuum regimes.…”

Section: Introductionmentioning

confidence: 99%

Coagulation–Agglomeration of Fractal-like Particles: Structure and Self-Preserving Size Distribution

2015

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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.

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“…These cases represent the potential directions for expanding the theory. Generally speaking, the evolution of non-spherical particles changes the collision-frequency function and requires developing numerical calculations (see, among others, [49][50][51]). Taking this into account we expect that the theory under consideration is suitable when the shape of the particles or aggregates is close to spherical.…”

Section: The Model Of Coalescence Combined With Coagulationmentioning

confidence: 99%

The influence of Brownian coagulation on the particle-size distribution function in supercooled melts and supersaturated solutions

Alexandrov¹,

Ivanov²,

Аlexandrova³

2018

J. Phys. A: Math. Theor.

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A theoretical description for particle coagulation due to Brownian motion combined with coalescence is presented and experimentally verified. The present theory confirms that the particle-size distribution function possesses a universal state, which is described by the self-similar kinetic equation. A complete analytical solution of this integro-differential equation is found in the case of Brownian coagulation mechanism. An explicit analytical particle-size distribution function, , which approximately satisfies the coagulation equation is derived too. It is shown that a universal particle-size distribution is broader, more flat and bell-shaped than the classical Lifshitz and Slyozov distribution function.

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Extension of the Smoluchowski Theory to Transitions from Dilute to Dense Regime of Brownian Coagulation: Triple Collisions

Cited by 14 publications

References 23 publications

Multi-Scale Modelling of Aggregation of TiO2 Nanoparticle Suspensions in Water

Multi-Scale Modelling of Aggregation of TiO2 Nanoparticle Suspensions in Water

Coagulation–Agglomeration of Fractal-like Particles: Structure and Self-Preserving Size Distribution

The influence of Brownian coagulation on the particle-size distribution function in supercooled melts and supersaturated solutions

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