The aerosol particle collision kernel considering the fractal model of particle motion

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Abstract

A method is presented in which the fractal model of particle motion, distinct from the Langevin equation of motion, has been used to determine the collision kernel of particles. The collision kernels, valid in the continuum and free molecular regions, are reformulated to be dependent on one sixth of the mean square displacement of a particle divided by time instead of diffusion coefficient and the root mean square displacement divided by time instead of the mean velocity. One generalized formula for collision kernels in the transition regime is obtained by equating the two expressions for kernels. The approaching time is calculated using equation describing the particle trajectory. For condensation and monodisperse aggregation the calculated collision kernel is the harmonic average of limiting kernels valid in continuum and free molecular regimes. It is in agreement with the experimentally verified formula of Fuchs–Sutugin. The results for polydisperse aggregation are close to those obtained by the widely accepted method of Fuchs. The obtained formula can also be used to approximate the drag force on particles in the transition regime.

Graphical abstract

The Sherman formula is the model dependence for monodisperse aggregation and condensation kernels. It also approximates the points calculated for polydisperse aggregation.

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Highlights

► Fractal model of the ballistic-diffusive transition is used to calculate aggregation kernel. ► The aggregation kernel is based on mean square displacements of approaching particles. ► The time required to approach two particles is computed by particle trajectory equation. ► The model has experimental confirmation by other experimentally verified formulae.

Introduction

The growth of aerosol particles by collisions either between particles (aggregation) or particles and vapor molecules (condensation) changes the particle size distribution and plays an important role in a number of aerosol phenomena and processes, including new particle formation in the atmosphere (Leppä et al., 2011) and nanomaterials synthesis (Buesser & Pratsinis, 2012). The behavior of these systems is governed by the collision rate which for low volume fractions of particles and vapor molecules is given (Friedlander, 2000) as the product of the collision kernel k and the number concentrations of colliding particles ni and njR=kninj

The form of the aggregation kernel depends on the region where the process occurs. In the continuum region the kernel is proportional to the product of the sum of diffusion coefficients of aggregating particles and the sum of their radii, according to Smoluchowski theory (Smoluchowski, 1916)kdiff=4π(ai+aj)(Di+Dj)

Veshchunov, 2010, Veshchunov, 2012 and Veshchunov & Azarov (2012) consider the applicability of the formula describing particle–particle collisions (Eq. (2)). They show it is relevant for coalescence of small particles with large ones, but for equal sized particles applies only in the limit in which the particle mixing time is substantially faster than the collision time, such that concentration gradients do not develop between particles. This takes place, however, for sufficiently dilute systems, in which the particle radius is much smaller than the inter-particle distance.

The free molecule aggregation kernel can be expressed as (Seinfeld, 1986)kball=π(ai+aj)2ui2+uj2

Their mutual relation can be expressed by the formulakballkdiff=ajui4Di(1+ai/aj)1+uj2/ui21+Dj/Di

The mean thermal velocity of the particle, which arises when deriving the ballistic collision kernel considering the full Maxwell–Boltzmann distributions of the colliding particles (Vincenti and Kruger, 1975, Allen, 1992) is described by the equationui=(8/π)kBT/miwhich corresponds, via a set of equations (Di=kBT/fi; fi=mi/τi; τi=λi/ui) to the diffusive mean free pathλi=(8/π)Di/ui

Using Eqs. (2), (3), the ratio of collision kernels can be expressed as dependent on the diffusive Knudsen numberkdiffkball=π2KnDwhereKnD=1+Dj/Di(1+ai/aj)1+uj2/ui2λiaj

The diffusive Knudsen number takes different form dependent on the ratio of particle radii. Forai=ajKnD=12λawhereas forajaiKnD=λiaj

At a constant temperature, according to Eqs. (5), (6), the diffusive mean free path of a particle is proportional to its diffusion coefficient and to square root of its mass (Dahneke, 1983). The non-fractal particle growth causes the increase of the square root of its mass (miai3/2) and the reduction of the diffusion coefficient in the slip region (Diai2), so the reduction of the diffusive mean free path (λia3/2−2) and the diffusive Knudsen number (KnDai3/23). For larger particles Diai1 and KnDai3/22. For high diffusive Knudsen numbers the aggregation can be approximated by the ballistic model. As the aggregation proceeds KnD diminishes. The continuum approximation is adequate for KnD≅0 which corresponds to negligible diffusive mean free path.

In the transition region neither ballistic nor diffusive model can describe the collision kernel since the diffusive Knudsen number is of the order of unity for atmospheric submicrometer particles. Therefore some trials were undertaken to solve the problem.

Fuchs, 1934, Fuchs, 1959 first proposed a model, in which a vapor molecule moves diffusively if is far from the particle but starts to move ballistically when the distance becomes less than a critical value. This approach results in an analytical expression due to flux-matching procedure, in which the critical distance is of the order of vapor molecule mean free path. As an alternative, Fuchs & Sutugin (1970) interpolated Sahni's (1966) solution to the Boltzmann equation for neutron absorption by a black body.

Moreover, Fuchs (1964) created a model of aerosol coagulation by analyzing the diffusive flux in the nearness of the absorbing sphere. Collisions between particles were also investigated by several other researchers (Sitarski and Seinfeld, 1977, Dahneke, 1983, Sahni, 1983), whereas particle collisions with small entities (condensation) were analyzed by Loyalka, 1973, Loyalka, 1982, Sitarski & Nowakowski (1979) and Seaver (1984).

A number of simulations used the Langevin equation to solve the problem of transition regime collisions (Nowakowski and Sitarski, 1981, Gutsch et al., 1995, Trzeciak et al., 2004, Narsimhan and Ruckenstein, 1985, Heine and Pratsinis, 2007, Isella and Drossinos, 2010). Veshchunov (2010) calculated the aggregation kernel according to the swept zone model as dependent on number of particle jumps between collisions for different diffusive Knudsen numbers. Gopalakrishnan & Hogan (2011) performed the first passage time calculations to determine the collision kernel with the results close to the commonly used expressions for both the aggregation and condensation kernels.

Section snippets

Model

Brownian diffusion is present in many phenomena and processes. Observing the motion of a Brownian particle one concludes that at very short times the motion is ballistic whereas for long times the particle starts to behave according to Einstein's theory. However this movement cannot be described by the known equationr2=6Dtwhere 〈r2〉 is the mean square displacement of the particle position in a given time t, since the formula cannot be applied for any arbitrarily short time (Einstein, 1956).

Verification of the model

The collision kernel derived is first compared to some popular collision kernels using the analysis introduced by Gopalakrishnan & Hogan (2011). The dimensionless kernel H is defined asH=μij2fij(ai+aj)3kwhere μij=mimj/(mi+mj); fij=fifj/(fi+fj), and the diffusive Knudsen number is defined asKnD=kBTμijfij(ai+aj)from whichH=4πKnD2k/kdiff

Using Eqs. (32), (36) and the relation between Knudsen numbersKnD=8πKnDone gets the dimensionless form of kernel derived in this study for monodisperse

Discussion and conclusions

The fractal model of the transition of from ballistic to diffusive motion of a Brownian particle has been used in this paper to calculate the aerosol collision kernel in the transition regime. The simple form of the obtained formula for monodisperse aggregation kernel and condensation makes it possible to compare it directly with the existing kernels via graphical representations. The Cunningham correction factor was chosen to calculate the diffusion coefficients for aggregating particles of

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