EVOLUTION OF NUMBER CONCENTRATION OF NANO- PARTICLES UNDERGOING BROWNIAN COAGULATION IN THE TRANSITION REGIME*

WANG Yu-ming

Institute of Fluid Measurement and Simulation, China Jiliang University, Hangzhou 310018, China,

E-mail: wangyuming126@126.com

LIN Jian-zhong

China Jiliang University, Hangzhou 310018, China

State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China

EVOLUTION OF NUMBER CONCENTRATION OF NANO- PARTICLES UNDERGOING BROWNIAN COAGULATION IN THE TRANSITION REGIME*

WANG Yu-ming

Institute of Fluid Measurement and Simulation, China Jiliang University, Hangzhou 310018, China,

E-mail: wangyuming126@126.com

LIN Jian-zhong

China Jiliang University, Hangzhou 310018, China

State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China

Evolution of number concentration of nanoparticles undergoing Brownian coagulation in the transition regime is studied theoretically and numerically. The results show that the curves of particle size distribution move toward the area with large particle diameters, the curve peak becomes lower and the range that particle diameters cover becomes wider as time elapses. In the process of coagulation the particles with small diameter disappear gradually and the particle size distribution remains a log-normal distribution. The change rate of the particle size distribution is more appreciable at the initial stage than that at the final stage. The initial Knudsen number has a significant effect on the coagulation rate which increases with decreasing the initial Knudsen number. The larger the initial geometric standard deviation is, the smaller the curve peak is, and the wider the area that curves cover is. The initial geometric standard deviation has a significant effect on the particle size distribution which can remain a self-preserving state when the initial geometric standard deviation is smaller than 2. With the increase of the diversity of initial particle size, the particle size distribution does not obey the log-normal distribution any more as time elapses.

nanoparticles, Brownian coagulation, number concentration, numerical simulation

Introduction

Numerous applications of two-phase flow carrying nanoparticles can be found in the field of engineering, for example, the system of an air cleaner, ejector scrubbers, air-spray systems, vehicle exhaust plumes and at the outflow of combustion products. In many of these processes the distribution of the dispersed particle is a controlling factor in the efficiency and the stability of the process. Brownian coagulation is an important phenomenon for nanoparticles because it results in a decrease in number concentration and an increase in particle size, while many properties ofnatural or man-made nanoparticles depend on their size distribution. Brownian coagulation was described theoretically by Smoluchowski who put forward a scheme to calculate discretized particles. After that, Muller derived an integro-differential equation, i.e., the Particle General Dynamic Equation (PGDE) to describe the evolution of number concentration of particles. It is, however, difficult to solve the PGDE as long as the Brownian coagulation is involved because of the complexity of the PGDE[1]. Fortunately, the solution of the PGDE in a specific particle size regime, i.e., the free molecule regime or the continuum regime, can be obtained by using the moment method, for example, Pratsinis’s Moment Method (PMM)[2], the Quadrature Method Of Moments (QMOM)[3]and the Sectional Method (SM)[4]. Furthermore, Yu et al.[5,6]put forward a new approach, i.e., the Taylor- series Expansion Method Of Moments (TEMOM), to solve the PGDE undergoing Brownian coagulation. Liu and Lin[7], Gan et al.[8]performed numerical simulation ofnanoparticle coagulation in a Poiseuille flow and a planar jet flow using the TEMOM, respectively. Although the TEMOM was proved to be more economical than other methods in computational cost without losing accuracy, the usage of the TEMOM was still restricted in the free molecule regime or the continuum regime.

There exists a transition regime, where the effects of gas slip on coagulation should be taken into account, between the free molecule regime and the continuum regime, which makes the theoretical analysis rather complicated. The dynamic behavior of nanoparticles in the transition regime has been addressed previously by Pratsinis[9]. Nychyporuk et al.[10]reported controlled formation of a monolayer of separated Si nanoparticles at the surface of a Si wafer using extremely short monopulsed anodization in the transition regime and controlled formation of a monolayer or doublelayer of laterally separated Si nanoparticles at the surface of Si wafers with the use of short pulse anodization in the transition regime, and compared with two other anodization regimes: nanopore formation and electropolishing[11]. Cho et al.[12]experimentally determined the mobility diameter of transition regime agglomerates with 3, 4, and 5 primary particles. Yu and Lin[13]extended a newly proposed TEMOM to solve agglomerate coagulation due to Brownian motion in the entire size regime. The mass fractal dimension is found to play an important role in determining the decay of agglomerate number and the spectrum of agglomerate size distribution, but the effect decreases with decreasing agglomerate Knudsen number. Shin et al.[14]investigated the effect of particle morphology on unipolar diffusion charging of nanoparticle agglomerates consisting of multiple primary spheres. The results showed that the electrical capacitance of chain-like agglomerates becomes significantly larger compared to that of spheres with the same mobility diameter as particles become larger. However, the effects of initial particle parameters on the size distribution of coagulating particles have not been addressed in the previous studies. The nanoparticles with different initial Knudsen numbers and initial geometric standard deviations based on particle size will have different relationships between the number concentration and the particle size when particle Brownian coagulation takes place. Therefore, the aim of present work is to study theoretically and numerically the Brownian coagulation of nanoparticles in the transition regime and give the effects of initial Knudsen number and initial geometric standard deviation based on particle size on the size distribution of coagulating particles.

1. Theory foundation

The transport of the nanoscale particles dispersed throughout the fluid is governed by the PGDE. The PGDE describes particle dynamics under the influence of various physics and chemical phenomena: convection, diffusion, coagulation, surface growth, nucleation and other internal/external forces. The time evolution of spherical nanoparticle size distribution can be described based on the PGDE as follows[13]

where K is the collision coefficient in the continuum regime, C(v) is the Cunningham slip correction factor by taking into account the effects of the gas slip for small particles which can be shown in the following form

where A=1.591, Kn is the Knudsen number (Kn=λ/r, where λ is the mean free path of air molecules and r is the particle radius).

In order to obtain n(v,t), the unknown parameters N, vgand σ in Eq.(4) should be calculated in advance. There have been several attempts to derive an analytical expression for the time evolution of a nanoparticle size distribution during coagulation. In the transition regime, the dimensionless time, geometric mean volume and geometric standard deviation can be expressed as functions of time:

Fig.1 Relationship between dimensionless total number of particles and dimensionless time

2. Numerical method

Substituting Eq.(12) into Eqs.(6)-(11), we can obtain the relationships between vgand σ with dimensionless time KN0t as shown in Figs.2 and 3.

From Fig.1 it can be seen that the total number of particles decreases with time because particles collide and coagulate to form the larger ones. The Brownian coagulation also can lead to the growth of the particle volume as shown in Fig.2 where the curve becomessteeper afterKN0t=7, indicating that the particle coagulation takes place more rapidly afterKN0t=7. Figure 3 shows that the geometric standard deviation based on particle size increases with time because multi-size of particles is formed in the process of coagulation.

Fig.2 Relationship between dimensionless geometric mean particle volume and dimensionless time

Fig.3 Relationship between the geometric standard deviation based on particle size and dimensionless time

SubstitutingN,vgandσinto Eq.(4), we can obtain the distribution of particle number concentration as a function of elapsed time.

Fig.4 Particle size distribution as a function of dimensionless time

3. Results and discussions

In the calculation the mean free path of air moleculesλis 65 nm, the initial Knudsen number and initial geometric standard deviation based on particle size change from 0.93 to 2.17 and from 1 to 3, respectively.

3.1Change of particle size distribution with time

Figure 4 shows the particle size distribution as a function of dimensionless time. In the figuredpis the particle diameter andnv/Nis the dimensionless number concentration. The initial mean particle diameter is 100nm and the initial geometric standard deviation based on particle size is 1. We can see that the initial shape of the curve (KN0t=0) is narrow, and the particles with diameter of 100 nm account for 68%. As time elapses, the curves move toward the area with large particle diameters, the curve peak becomes lower and the range that particle diameters cover becomes wider. The reason is that the particles collide with one another to form larger particle, which results in the diversity of particle diameter. In the process of particle coagulation the particles with small diameter disappear gradually. It also can be seen that the particle size distribution remains a log-normal distribution as time elapses. The change rate of the particle size distribution is larger at the initial stage (KN0t＜7) than that at the final stage (KN0t＞7). AtKN0t=7.7 the particle diameters concentrate between 400 nm to 800 nm, and the particles with diameter under 200 nm disappear.

Fig.5 Effect of initial Knudsen number on the particle size distribution

3.2Effect of initial Knudsen number on the particlesize distribution

The different particle size distributions with different initial Knudsen numbers fromKn0=0.93 70 nm to 30 nm) atKN0t=7 are shown in Fig.5, from where we can see that the initial Knudsen number effects are indeed obvious for the curves not only shift to the area with large particle diameters, but also become wider when the initial Knudsen number decreases. It is reasonable because the smaller the initial Knudsen number is, the larger the initial particle size, so that the particle size distribution covers the area with larger particle diameters. From the figure we to 2.17 (corresponding initial particle radii fromcan see that the initial Knudsen number has a significant effect on the coagulation rate due to the geometric standard deviation changes obviously with the alteration of initial Knudsen number. The initial Knudsen number considered in the present study corresponds to the transition regime in which the gas slip around the particles will appear, so the coagulation is seen to increase with decreasing Knudsen number. That the curve of particle size distribution becomes wider with decreasing Knudsen number supports this view point.

Fig.6 Effect of initial geometric standard deviation on the particle size distribution

3.3Effect of initial geometric standard deviation on the particle size distribution

Figure 6 shows the particle size distribution on the particle diameter as a function of the initial geometric standard deviation,σ0, atKN0t=7. It can be seen that the larger the initial geometric standard deviation is, the smaller the curve peak is, and the wider the area that curves cover is. The difference between curves for different0σis great, which shows that the initial geometric standard deviation has a significant effect on the particle size distribution. In addition, it is noticed that the particle size distributions forσ0=1 andσ0=1.5 remain a lognormal distribution, while the particle size distributions forσ0＞2 deviate from a log-normal distribution and approach a certain value regardless of the initial value of0σ. In other words, the particle size distributions can remain a self-preserving state when the initial geometric standard deviation is smaller than 2. As we know, the initial geometric standard deviation based on particle size represents the diversity of particle size. It is concluded that with the increase of the diversity of initial particle size, the particle size distribution does not obey the log-normal distribution any more as time elapses, which makes the problem of coagulation become complex.

4. Conclusion

The distributions of polydispersed nanoparticles undergoing Brownian coagulation in the transition regime as a function of elapsed time have been studied. The total number of particles, the geometric mean particle volume and the geometric standard deviation based on particle size have been calculated. Based on these the change of particle size distribution with time and the effects of initial Knudsen number and initial geometric standard deviation on the particle size distribution are obtained. The results show that the total number of particles decreases, while the geometric mean particle volume and the geometric standard deviation based on particle size increase with time. The curves of particle size distribution move toward the area with larger particle diameters, the curve peak becomes lower and the range that particle diameters cover becomes wider as time elapses. In the process of particle coagulation the particles with small diameter disappear gradually and the particle size distribution remains a log-normal distribution. The change rate of the particle size distribution is larger at the initial stage than that at the final stage. The initial Knudsen number has a significant effect on the coagulation rate due to the geometric standard deviation changes obviously with the alteration of initial Knudsen number. The coagulation rate increases with decreasing Knudsen number. The larger the initial geometric standard deviation is, the smaller the curve peak is, and the wider the area that curves cover is. The initial geometric standard deviation has a significant effect on the particle size distribution. The particle size distributions can remain a self-preserving state when the initial geometric standard deviation is smaller than 2. With the increase of the diversity of initial particle size, the particle size distribution does not obey the log-normal distribution any more as time elapses.

[1] KIM D. S., PARK S. H. and SONG Y. M. et al. Brownian coagulation of polydisperse aerosols in the transition regime[J].Journal of Aerosol Science,2003, 34(7): 859-868.

[2] PRATSINIS S. E., KIM K. S. Particle coagulation, diffusion and thermophoresis in laminar tube flows[J].Journal of Aerosol Science,1989, 20(1): 101-111.

[3] LIN Jian-zhong, LIU Yan-hua. Nanoparticle nucleation and coagulation in a mixing layer[J].Acta Mechanica Sinica,2010, 26(4): 521-529.

[4] TALUKDAR S. S., SWIHART M. T. Aerosol dynamics modeling of silicon nanoparticle formation during silane pyrolysis: A comparison of three solution methods[J].Journal of Aerosol Science,2004, 35(7): 889-908.

[5] YU M. Z., LIN J. Z. Binary homogeneous nucleation and growth of water-sulfuric acid nanoparticles using a TEMOM model[J].International Journal of Heat and Mass Transfer,2010, 53(4): 635-644.

[6] YU M. Z., LIN J. Z. Solution of the agglomerate Brownian coagulation using Taylor-expansion moment method[J].Journal of Colloid and Interface Science,2009, 336(1): 142-149.

[7] LIU Song, LIN Jian-zhong. Numerical simulation of nanoparticle coagulation in a Poiseuille flow via a moment method[J].Journal of Hydrodynamics,2008, 20(1): 1-9.

[8] GAN Fu-jun, LIN Jian-zhong and YU Ming-zhou. Particle size distribution in a planar jet flow undergoing shear-induced coagulation and breakage[J].Journal of Hydrodynamics,2010, 22(4): 445-455.

[9] PRATSINIS S. E. Simultaneous nucleation, condensation, and coagulation in aerosol reactor[J].Journal of Colloid and Interface Science,1988, 124(2): 416-427.

[10] NYCHYPORUK T., LYSENKO V. and GAUTIER B. et al. Initial stages of silicon anodization in the transition regime: Nanoparticle formation[J].Applied Physics Letters,2005, 86(21): 213107.

[11] NYCHYPORUK T., LYSENKO V. and GAUTIER B. et al. Silicon nanoparticle formation by short pulse electrochemical etching in the transition

regime[J].Journal of Applied Physics,2006, 100(10): 104307.

[12] CHO K., HOGAN C. J. and BISWAS P. Study of the mobility, surface area, and sintering behavior of agglomerates in the transition regime by tandem differential mobility analysis[J].Journal of Nanoparticle Research,2007, 9(6): 1003-1012.

[13] YU M. Z., LIN J. Z. Taylor-expansion moment method for agglomerate coagulation due to Brownian motion in the entire size regime[J].Journal of Aerosol Science,2009, 40(6): 549-562.

[14] SHIN W. G., WANG J. and MERTLER M. et al. The effect of particle morphology on unipolar diffusion charging of nanoparticle agglomerates in the transition regime[J].Journal of Aerosol Science,2010, 41(11): 975-986.

[15] PARK S. H., LEE K. W. Moment method of log-normal size distribution for coagulation problem: Constant collision kernel mode[J].Particle Science and Technology,2000, 18(4): 293-307.

December 27, 2010, Revised April 1, 2011)

* Project supported by the Major Program of the National Natural Science Foundation of China (Grant No. 10632070).

Biography: WANG Yu-ming (1987-), Male, Master Candidate

LIN Jian-zhong,

E-mail: mecjzlin@public.zju.edu.cn

2011,23(4):416-421

10.1016/S1001-6058(10)60131-X