Three-Dimensional Simulation of Brownian Motion of Nano-Particles In Aerodynamic Lenses

Aerosol Science and Technology ISSN: 0278-6826 (Print) 1521-7388 (Online) Journal homepage: https://www.tandfonline.com/loi/uast20 Three-Dimensional Simulation of Brownian Motion of Nano-Particles In Aerodynamic Lenses Omid Abouali, Abbas Nikbakht, Goodarz Ahmadi & Saieedeh Saadabadi To cite this article: Omid Abouali, Abbas Nikbakht, Goodarz Ahmadi & Saieedeh Saadabadi (2009) Three-Dimensional Simulation of Brownian Motion of Nano-Particles In Aerodynamic Lenses, Aerosol Science and Technology, 43:3, 205-215, DOI: 10.1080/02786820802587888 To link to this article: https://doi.org/10.1080/02786820802587888 Published online: 09 Feb 2009. Submit your article to this journal Article views: 917 Citing articles: 7 View citing articles Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalinformation?journalcode=uast20

Aerosol Science and Technology, 43:205 215, 2009 Copyright c American Association for Aerosol Research ISSN: 0278-6826 print / 1521-7388 online DOI: 10.1080/02786820802587888 Three-Dimensional Simulation of Brownian Motion of Nano-Particles In Aerodynamic Lenses Omid Abouali, 1 Abbas Nikbakht, 1 Goodarz Ahmadi, 2 and Saieedeh Saadabadi 1 1 Department of Mechanical Engineering, Shiraz University, Shiraz, Iran 2 Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, New York, USA A computer code for analyzing nano-particle motions in an aerodynamic particle beam focusing system was developed. The code uses an accurate three-dimensional model for the Brownian diffusion of nano-particles in strongly varying pressure field in the aerodynamic lens system. Lagrangian particle trajectory analysis was performed assuming a one-way coupling model. The particle equation of motion used included drag and Brownian forces. The prediction of the 3-D model for penetration efficiency, beam divergence angle, beam diameter and radial cumulative fraction were evaluated and were compared with those of the axisymmetric models. The simulation results showed that for particle diameters less than 30 nm in helium gas, the Brownian force could significantly affect the beam focusing and particle penetration efficiency. Some potential errors in the naïve usage of axisymmetric model were discussed. It was shown that the earlier axisymmetric models lead to the incorrect mean square radial displacement of Brownian particles. The present 3-D approach, however, leads to the correct value of the radial mean square displacement of 4Dt. The effect of the inlet orifice and relaxation region on the performance of the lens system was also investigated. It was shown that the major losses of the 4 to 10 nm particles occur in the inlet orifice and relaxation region walls. For 15 to 30 nm particles, the main losses occur at the inlet orifice walls. Some alterations of the shape of the inlet orifice were examined and a new design is suggested to reduce the loss of the particles at the inlet flow control orifice. INTRODUCTION Aerodynamic lenses are formed by combinations of properly designed axisymmetric contractions and expansions, and Received 7 May 2008; accepted 29 October 2008. The work of OA and AN was supported by a grant from Iranian Nanotechnology Initiative Council (INI) and the work of GA was supported by a grant from US Department of Energy (NETL) and by NYS- TAR through the Center for Advanced Material Processing (CAMP) of Clarkson University. The authors would like to thank Professor McMurry of University of Minnesota and Dr. Wang for providing the detail design of their inlet orifice. Address correspondence to Omid Abouali, Department of Mechanical Engineering, Shiraz University, Shiraz, Fars 71348-51154, Iran. E-mail: abouali@shirazu.ac.ir are used for generating focused particle beams. Particles in a critical size range passing through a sequence of contractions drift towards the axis and form a narrow beam. Particles larger than the critical size range are removed from the stream by inertial impaction on the lens wall, while smaller particles follow the flow streamlines and are not focused. The diameters of the focused particle beams are controlled with appropriate sizing of sequence of lenses of varying diameters, operating pressure, Brownian motion, and particle shape. Jayne et al. (2000), Zhang et al. (2002), Kane and Johnston (2000), and Kane et al. (2001) described the use of aerodynamic lenses for producing focused aerosol particle beams for online characterization of fine particles. Murphy and Sears (1986 1987) reported producing narrow particle beams by expanding an aerosol from atmospheric pressure to low backpressures through a nozzle. Computational and experimental studies of aerodynamic lenses were first performed by Liu et al. (1995a, b). They showed highly collimated particle beams could be produced without sheath air. They also investigated the effect of Brownian motion and lift force on the particle beam diameter downstream of the nozzle. Absence of multidimensional gas velocity field reduces the accuracy of their particle trajectory analysis especially away from the axis. Zhang et al. (2002) characterized particle beam collimation in a single aerodynamic lens and an individual nozzle. They found that the maximum particle displacement and particle loss occurs at a particle Stokes number near unity. The performance characteristics of the lens and the nozzle were found to depend on their geometry, flow Reynolds number, and particle Stokes number. Zhang et al. (2004) studied the gas particle flows through an integrated aerodynamic-lens nozzle. They found that the inlet transmission efficiency (η t ) for particles in the intermediate size range with diameters of 30 500 nm was about unity. (They defined the inlet transmission efficiency as the fraction of the particles reaching the target relative to the number of particles at the inlet upstream. Here the target was a 2 mm diameter plate in a mass spectrometer, which is located 240 mm downstream of the nozzle.) The transmission efficiency gradually reduced to 205

206 O. ABOUALI ET AL. about 40% for large particles and to almost zero for very small particles with dp 15 nm. Approximation in the Brownian force computation was the major limitation of their work. Wang et al. (2005) developed a numerical simulation methodology that was able to accurately characterize the focusing performance of aerodynamic lens systems. They demonstrated the ability of aerodynamic lenses to focus sub 30 nm spherical unit density particles by a new design and arrangements of lenses and using helium as the carrier gas. Wang et al. (2006a) developed an Aerodynamic Lens Calculator Module for design of aerodynamic lens systems. The calculator module could be used to design the key dimensions of a lens system including pressure limiting orifice, relaxation chamber, focusing elements, spacers, and the accelerating nozzle. Wang et al. (2006b) presented an experimental study of nanoparticle focusing using aerodynamic lenses with helium as carrier gas. Measurements were done for three types of particle materials: vacuum pump oil, sodium chloride, and proteins in the 3 30 nm diameter range. The particle transport efficiency through the aerodynamic lens and the beam width downstream of the lens system were measured. Their results showed that their lens system can deliver nanoparticles in the size range of 3 30 nm from atmosphere pressure to high vacuum with high transmission efficiencies and narrow beam diameters. All earlier computational modeling studies of aerodynamic lenses were restricted to the assumption of axisymmetric flows where the radial particle trajectories were evaluated. The aerodynamic lenses typically include a vacuum pump attached to the sidewall of the intermediate chamber that makes the flow three-dimensional (non-axisymmetric) at least in the intermediate chamber. When the flow is not axisymmetric, then the particle trajectories also deviate from being axially symmetric. In addition, the Brownian motion is generated by three-dimensional stochastic excitations. The analysis of trajectories of Brownian particles in axisymmetric flows, however, requires certain care. Abouali and Ahmadi (2007) presented a simulation using a three-dimensional computer model for the airflow field and particle motion in aerodynamic lenses. Their simulation results showed that the assumption of axisymmetric flow downstream of the nozzle was a reasonable approximation when the suction pump of the intermediate chamber was installed far from the nozzle. They, however, used a constant slip correction factor in the expression for the Brownian force, which rendered their simulation results for sub 10 nm particles to a qualitative description. Wang et al. (2005) augmented FLUENT TM by a UDF (User Defined Function) and included the variable slip correction factor in their simulation of nano-particle motions in aerodynamic lenses. Their study, however, were limited to an axisymmetric analysis of radial particle trajectories. In this study a three-dimensional computer model for particle motion in aerodynamic lenses was developed. The axisymmetric form of this computer model was developed before by Nikbakht et al. (2007). For evaluating the axisymmetric flow field in the aerodynamic lens and downstream of the nozzle, FLUENT TM (version 6) software was used. To avoid the limitation of the commercial software, in the present study, a new computer code for analyzing the 3-D particle trajectories in multistage aerodynamic lenses was developed. The new code included the appropriate variable slip correction factor in the expression for the drag force and the Brownian excitation. Particle trajectories inside the inlet orifice, relaxation region and lenses, through the outlet nozzle, and downstream of the nozzle were analyzed using the developed code. The results of 3-D model were compared with those of axisymmetric model and discussed. It was shown that the earlier axisymmetric models of particles motion in the aerodynamic lenses predict higher penetration efficiency, lower particle beam radius, and lower beam divergence angle compared with the present 3-D simulation results. (Note that the penetration efficiency is defined as the ratio of the particles passing through the nozzle to the number of particles entering the aerodynamic lens system.) The effect of the pressure reducing orifice and relaxation region on the performance of the lens system was also evaluated. MODEL DESCRIPTION The aerodynamic lens system designed by Wang et al. (2006b) composed of a pressure reducing orifice, a relaxation chamber, multistage focusing elements and an outlet nozzle. The aerosol flow passes through the inlet orifice, which reduces the pressure from atmospheric pressure to the lens operating pressure of about 520 Pa. Particles are focused into a tight beam as they pass through the aerodynamic lens system and are delivered into a vacuum chamber. The cross sectional and perspective views of the schematic of the aerodynamic lens system is shown in Figure 1. Wang et al. (2005) described a computational model of the aerodynamic lens system without the inlet orifice and relaxation chamber. Here, both cases including and excluding the orifice and relaxation chamber in the lens system are studied. The model of the lens system without the pressure-reducing orifice and relaxation region is studied first and the results of the 3-D and the axisymmetric models are compared. The complete model of the lens system including the inlet orifice and relaxation chamber is also studied. The diameter of the lens tube is 10 mm and adjacent focusing elements are 15 mm apart. The contraction diameters of the three orifices that are used in the analysis are, respectively, 1.26, 1.64, and 2.33 mm. At the end of the three focusing elements, the gas passes through a nozzle with a diameter of 2.76 mm. The outlet boundary is also shown in Figure 1 and a chamber pressure of 1Pa at the nozzle downstream is imposed. A pressure-reducing orifice (O keefe, 7 sapphire orifice type J, aperture diameter 71 µm) limits the aerosol flow rate with helium as the carrier gas into the lens system at 88 cm 3 /min. The pressure drops from atmosphere pressure to about 520 Pa after the pressure-reducing orifice was predicted by the simulation, which is in agreement with the measurement reported by Wang et al. (2006b). It should be emphasized that a mass flow rate boundary condition was used at the inlet of the pressure-reducing orifice in the numerical simulation.

3-D SIMULATION OF BROWNIAN MOTION IN AERODYNAMIC LENSES 207 FIG. 1. (a) Cross section of the aerodynamic lens system including the inlet orifice and relaxation chamber proposed by Wang et al. (2006b) and used in present study. (b) Wire-frame view of the aerodynamic lens system without the inlet orifice and the relaxation chamber used in the present study for comparisons of3-d and axisymmetric models. (Figure provide in color online.) The mean free path of the gas downstream of the nozzle is of the order of the nozzle size. While using the continuum flow assumption is questionable, nevertheless, it was used as a rough approximation. Earlier studies by Abouali and Ahmadi (2007) showed that the flow in the lens and the nozzle is axisymmetric, and the flow downstream of the nozzle can also be treated approximately as being axisymmetric. GOVERNING EQUATIONS For adilute gas-particle flow in an aerodynamic lens, a oneway interaction model is used. That is, it is assumed while gas will carry the particles, the concentration and size of the particles are too small to affect the gas flow. Under this condition, the gas flow field can first be evaluated and then be used for evaluation of the particle trajectories. The axisymmetric compressible viscous laminar flow field was evaluated using the upwinding method. Details of the governing equations for an axisymmetric flow condition are given in FLUENT TM Users Guide (2003), and hence need not be repeated here. The Lagrangian equation of particle motion in Cartesian coordinate is given as dx p dt = V p, [2] where V p is the particle velocity vector, F D is the drag force per unit mass, F B is the Brownian force per unit mass, g is the acceleration of gravity and x p is the particle position vector. Note that the flow is in laminar regime and the particles are assumed spherical. In the 3-D model, the particle equations of motion were solved in Cartesian coordinates and the motion of the particles in cylindrical coordinate were evaluated using the standard transformation, r 2 = x 2 + y 2,θ = tan 1 (y/x), z = z. Drag Force The expression for the modified Stokes drag force per unit mass including the Cunningham correction is given by: F D = 3µC D Re 4ρ p d 2 p C (V V p ), [3] c dv p dt = F D + F B + g, [1] where V is the fluid velocity vector, d p is particle diameter, ρ p is the particle density, µ is the coefficient of viscosity, C c is the

208 O. ABOUALI ET AL. Cunningham correction factor given as (Hinds 1999) C c = 1 + 2λ d p [1.257 + 0.4exp( 1.1(d p /2λ))], [4] and Re, the particle Reynolds number, is defined as Re = V V p d p, [5] ν where ν is the kinematic viscosity of the gas. In Equation (3), the drag coefficient C D, which accounts for the Reynolds number correction to the Stokes drag, is given as (Hinds 1999) C D = 24 Re (1 + 0.15Re0.687 ). [6] Note that the nonlinear corrections are expected to be significant only for larger particles with appreciable slip velocity. For nano-particles with small slip velocity, Re is very small and the second term in Equation (6) is negligible. The exception is for the region at the orifices where the gas-solid slip velocity may be large. In most parts of the lenses system, however, the slip velocity is very small. In Equation (4) the Cunningham correction factor depends on gas mean free path. There is a significant change in the gas mean free path in the aerodynamic lens due to the large variations in the gas pressure. Therefore, the Cunningham correction factor must be calculated using the correct gas mean free path along particle trajectories for an accurate simulation of the drag and Brownian forces. The mean free path of helium is given as (Hirschfelder et al. 1954) λ(µm) = kt 2πd 2 m P = 45.3T P. [7] In Equation (7), d m,k,p,and T, respectively, are the helium molecule diameter (0.258 nm), the Boltzmann constant, the gas pressure (Pascal), and the gas temperature (Kelvin). Brownian Force When a small particle is suspended in a fluid, it is subjected to the imbalanced random impacts of the gas molecules that cause the nano-particles to move on an erratic path, which is known as the Brownian motion. A Gaussian white noise stochastic process can model the random impacts of the molecules. White noise is a zero mean Gaussian random process with a constant power spectrum given by (Li and Ahmadi 1993) S nn = 2kTβ πm p = 2β2 D π kt, with D = βm = ktc c [8] 3πµd p where k = 1.38 10 23 J/K is the Boltzmann constant, β = 3πµd p /C c m p is the inverse of the particle relaxation time, m p is the mass of the particle, subscript n denotes the white noise excitation, and D is the diffusion coefficient. The procedure suggested by Ounis and Ahmadi (1993) and Li and Ahmadi (1993) for simulating the Brownian motion was used in the computer code developed in this study. To start the process, a time step tisselected. (The time step should be much smaller than the particle relaxation time.) Then pairs of uniform random numbers U 1 and U 2 (between 0 and 1) are generated and are transformed to a pair of unit variance zero mean Gaussian random numbers. This is done using the Box- Muller transformations (Box and Muller 1958; Papoulis 1965): G 1 = 2ln U 1 cos (2πU 2 ), [9] G 2 = 2ln U 1 sin (2πU 2 ). [10] The amplitude of the Brownian force per unit mass in x, y, and z directions at each time step is then evaluated by π Snn F bi = G, [11] t where S nn is given by Equation (8), and G is a unit variance, zero mean Gaussian random number given in Equations (9) and (10). The entire generated sample of the Brownian force is then shifted by U t, where U is a uniform random number between zero and one. RESULTS Using a computational modeling approach, the flow fields in a class of aerodynamic lenses for the configuration designed by Wang et al. (2005, 2006b) are simulated. The grid sensitivity analysis was performed and the size of the grid was increased until the solution was independent of further refinement of the grid. Consequently, a computational grid of 70,000 elements was used for simulating gas flow in the axisymmetric model of the aerodynamic lens without the inlet orifice, and 120,000 elements were used for complete model of lens system including the pressure reducing inlet orifice. The CPU time for the above cases was 3 and 10 h, respectively, on a2ghz P 4 CPU. A mass flow rate boundary condition (2.98 10 7 kg/s) for helium at the inlet of the aerodynamic lens system was used. The pressure drops from atmosphere pressure to about 520 Pa after the pressure-reducing orifice in the complete model of the lens system was predicted by the simulation, which is in agreement with the measurement reported by Wang et al. 2006b. Fixed pressure boundary condition of 1 Pa was imposed at the exit boundary of computational domain downstream of the nozzle. The viscosity of the helium was computed based on kinetic theory. In the following sections, the differences for particle motion in the 3-D and axisymmetric models for the lens system without the inlet orifice and the relaxation chamber are first discussed. This is followed by the analysis of the inlet orifice and relaxation region, using the complete model of the lens system.

3-D SIMULATION OF BROWNIAN MOTION IN AERODYNAMIC LENSES A) Comparisons of 3-D and Earlier Axisymmetric Models In this section, the computational results for the particle motions in the gas stream of the lens system without the inlet orifice are presented. Figure 2 shows sample particle trajectories in the FIG. 2. online.) 209 aerodynamic lens system for particle diameters of 2, 6, and 10 nm with and without inclusion of the Brownian force. (In these figures for illustrative purposes only 25 particles are injected at the inlet.) It is seen that most of the 2 nm particles are deposited Sample trajectories through the aerodynamic lens system with and without Brownian motion for 2, 6, and 10 nm particles. (Figure provided in color

210 O. ABOUALI ET AL. on the walls due to Brownian diffusion. In the absence of Brownian diffusion all 2 nm particles pass through the nozzle. Diffusion appears to be significant for 6 and 10 nm particles, but these particles stay closer to the lens axis with no particles being lost to the lens walls. For 30 nm particles and larger (not shown here) no significant differences can be seen between the cases with and without Brownian motion. So 30 nm particles are highly focused near the axis and a small skimmer can be used between the aerodynamic lens system and the detection chamber. This figure shows that the Brownian effects reduces the effectiveness of the aerodynamic beam focusing for particles of the order of 10 nm or smaller for the range of pressure condition considered in the present study. To check the accuracy of the Brownian simulation procedure, a series of simulations were performed and dispersion of nanoparticles injected from the centerline of a constant velocity pipe and channel flows was analyzed. The simulation results for 3-D and axisymmetric flow in a pipe and channel flow models are compared with the exact solution of the diffusion equation for these simple cases. Figure 3 show the dispersion of 100 nm particles with a particle-to-fluid density ratio of 2000 and flow velocity of 1 m/s. For evaluating the mean-square displacement of the particles shown in Figure 3, ensembles of 5000 and 1000 samples, respectively, for the 3-D and axisymmetric models were used. The variances of particle displacement due to Brownian diffusion in Cartesian coordinate direction x i as obtained from the solution of the diffusion equation is given as σ 2 x i = 2Dt, [12] where D is the particle diffusivity given by Equation (8). The variance of the radial displacement of the particles that are emitted from a source at r = 0 can be found form the solution of the diffusion equation in cylindrical coordinates given as σ 2 r = 4Dt. [13] As noted before, the mean square radial displacement of the Brownian particles is twice that of one-dimensional Cartesian direction This result also follows from the fact that r 2 = x 2 + y 2, the mean-square displacement in the radial direction is given as r 2 = x 2 + y 2 = 2x 2 = 4Dt, where the isotropy of the Brownian movement is assumed. Abouali and Ahmadi (2007) provided additional discussion of these results. Figure 3 shows that the time evolution of the mean-square radial displacements of the Brownian particles predicted by the present 3-D Lagrangian model is in good agreement with the exact solution given by Equation (13). Similarly the mean-square displacement in the y-direction obtained by the two-dimensional Lagrangian model for two-dimensional Cartesian coordinates is in close agreement with the expression given by Equation (12). These results show that the present procedure for simulating Brownian motions in y and R directions are quite accurate. To provide further validation of the developed 3-D computer model and comparison with the earlier axisymmetric models, a series of simulation for particle transport in a pipe under laminar flow conditions were performed. A 10 mm diameter tube with a length of 20 mm at a pressure of 500 Pa with average velocity of 5 m/s was analyzed. The simulation results of earlier axisymmetric models, as well as the 3-D model for penetration efficiencies are compared with those predicted by the empirical correlation of Gormley Kennedy (Gormley and Kennedy 1949) in Figure 4. This figure shows that the 3-D model predictions are in good agreement with the experimental data. The earlier axisymmetric model, however, overestimates the penetration for FIG. 3. Comparison of the simulated mean-square particles radial and vertical displacements as predicted by axisymmetric and 3-D models with the exact solutions for 100 nm particles injected at the axis of a pipe with a uniform flow. (Figure provided in color online.) FIG. 4. Comparison of experimental data with axisymmetric and 3-D models for penetration of particles. (Figure provided in color online.)

3-D SIMULATION OF BROWNIAN MOTION IN AERODYNAMIC LENSES 211 FIG. 5. Comparison of beam divergence angles as predicted by axisymmetric, Wang et al. (2005) and 3-D models. (Figure provided in color online.) the ranges of diffusion dominant nano-particles. This is as expected because the earlier axisymmetric models underestimate the diffusion of particles in radial direction, because their predicted mean-square radial displacements converge to 2Dt and underestimate the correct value by a factor of two. The main problem of these earlier axisymmetric models for particle diffusion analysis in aerodynamic lenses and pipe flows is that they use a 2-D Cartesian equation for particle motion in radial direction. Earlier, Hagwood et al. (1999) used Ito s calculus for deriving an equation for stochastic particle motion in radial direction. Their analysis leads to the addition of a D/r term to the right-hand side of the governing equation. However, since D/r is singular at the axis, the computational modeling of their equation encounters serious difficulty for application to the configurations that dispersion near the axis is of interest. Particle beam divergence angles as defined by Wang et al. (2005) were evaluated. Accordingly, the particle beam diameter is twice the radius that contains 90% of particles. For evaluating the beam divergence angle, the beam radius at two planes are evaluated and used. Here the plane were selected the same as Wang et al. (2005) at axial distances of 75.3 mm and 85.3 mm from the inlet, which correspond to 1.8 and 5.4 nozzle diameters downstream of the nozzle exit. Then the difference of these two beam radii was divided by the distance between the two planes (here 10 mm) for evaluating the beam divergence angle. Figure 5 compares the simulation results for the divergence angle of the 3-D model with the predictions of earlier axisymmetric models and those of Wang et al. (2005). The theoretical Brownian limit derived by Liu et al. (1995a) is also shown in this figure. The earlier axisymmetric model that underestimates radial direction forces leads to smaller values of the beam divergence angles. The prediction of Wang et al. (2005), is even smaller than to the earlier axisymmetric model, but it is close to the Brownian limit suggested by Liu et al. (1995a). FIG. 6. Comparison of predicted particle beam diameters by axisymmetric and 3-D models with the experimental data at 71 mm downstream of the nozzle. (Figure provided in color online.) Figure 6 compares the present 3-D and earlier axisymmetric simulation results with the experimental data of Wang et al. (2006b) for particle beam diameter at 71 mm axial location downstream of the nozzle. The simulation result of Wang et al. (2005) is also shown in this figure for comparison. All simulations seem to capture the trend of the experimental data but somewhat underestimate the magnitude of the experimental beam diameters. The present model seems to show slightly better agreement with the experimental data when compared with the prediction of Wang et al. (2005). Figure 7 compares the radial distribution of the cumulative fraction of the particles at injection and at the nozzle exit planes as predicted by the present 3-D with the results of Wang et al. (2005). The cumulative particle fraction was normalized to the total number of injected particles (20000 for 3-D model). In addition, as suggested by Wang et al. (2005), the particle radial location was normalized using the inner radius of spacers (5 mm) for the inlet curve and the nozzle radius (1.38 mm) for the outlet curves. It is seen that the cumulative fraction of the particles injected at the inlet for all models is proportional to [2(r/R) 2 -(r/r) 4 ], which is required for a uniform distribution of the particles in a fully developed laminar flow. As noted before the particle beam width is defined as the beam diameter that contains 90% of total particles passing through a section so particle beam diameter can be evaluated from Figure 7. From this figure, it is seen that Wang et al. (2005) approach predicts a higher cumulative fraction for 10 nm particles when compared with the present 3-D model. B) Complete Model of Lens System Including the Inlet Orifice In this section the results of a series of simulations for the complete model of the lens system is presented. Here the inlet orifice and the relaxation chamber are included in the computer model. The inlet orifice reduces the inlet atmospheric pressure to

212 O. ABOUALI ET AL. FIG. 7. Cumulative fraction of 2, 6, 10, and 30 nm particles at the inlet of the aerodynamic lens system and at the nozzle exit. (Figure provided in color online.) a low pressure (520 Pa) in the relaxation region. Figure 8 shows the velocity distribution along the axis of the lens system. It is seen that the helium gas flow reaches to a hypersonic velocity of 1700 m/s downstream of the pressure-reducing orifice. This velocity is even greater than that at downstream of the nozzle in the lens system. Figure 9 compares the predicted penetration efficiencies for the lens system with and without the inlet orifice and the relaxation chamber with the experimental data of Wang et al. (2006b). It is seen that when the inlet orifice and the relaxation chamber are not included in the computational model the penetration efficiency is overestimated. The complete model that includes the pressure-reducing inlet orifice and the relaxation chamber, however, predicts much lower penetration efficiency that is in excellent agreement with the experimental data. These results confirm that the pressure-reducing orifice and the relaxation region play an important role in penetration efficiency of nano-particles in lens system. Figure 10 shows the percent particle losses in the orifice and relaxation regions. Summation of the particle losses in the inlet orifice and relaxation regions is about 60% for 4 nm particles and drops to about 20% for 10 nm particles. This shows that most of the particle losses occur in the inlet orifice and the relaxation regions. Comparison of the summation of the losses in the inlet orifice and relaxation regions with the total loss in Figure 10 shows that most particles smaller than 10 nm are captured in the inlet orifice and relaxation regions. Particle larger than 10 nm, however, are mainly captured in the pressure reducing inlet region. Figure 11 compares the predicted variations of the beam diameters with the experimental data of Wang et al. (2006). This figure shows that the inclusion of the inlet orifice and the relaxation chamber does not affect the model predictions for the particle beam diameter. C) Comparison of Performance of Different Pressure-Reducing Orifices The presented results showed that most of the particle losses occur in the inlet orifice and the relaxation chamber. Therefore, a series of simulation for different inlet orifices and the relaxation

3-D SIMULATION OF BROWNIAN MOTION IN AERODYNAMIC LENSES FIG. 8. Velocity distribution on the axis of the complete lens system including the pressure-reducing orifice. (Figure provided in color online.) 213 FIG. 11. Comparison of predicted particle beam diameters for the lens system with and without the inlet orifice and the relaxation chamber at 71 mm downstream of the inlet. (Figure provided in color online.) chambers was performed. Several different designs of the inlet orifice as shown in Figure 12 were studied. For sake of comparison, the minimum size of the throat diameter for all five orifices are kept fixed at 71 µm. The corresponding computed FIG. 9. Comparison of the penetration efficiency for the lens system with and without inlet orifice and relaxation chamber. (Figure provided in color online.) FIG. 10. Comparison of the percent losses in the inlet orifice and the relaxation region. (Figure provided in color online.) FIG. 12. Five different shapes of the studied pressure-reducing orifices (Figure provided in color online.)

214 O. ABOUALI ET AL. CONCLUSIONS In the present study, three-dimensional particle motions in a multistage aerodynamic lens with the end nozzle and nozzle downstream were studied. Simulation results for beam divergence angle and particle beam diameter were presented and the results were compared with those of the earlier axisymmetric models. The Brownian simulation procedure used in the analysis for three dimensional model was verified by comparison with the exact solutions for uniform flows. Comparison of the present simulation results with available models showed certain inadequacies of the earlier axisymmetric models. In particular, some earlier axisymmetric models led to the incorrect mean-square radial displacement of 2Dt. The present 3-D model, however, predicted the correct particle radial displacement variance of 4Dt. The effect of the inlet pressure-reducing orifice and the upstream relaxation region on the performance of the lens system was also investigated. It was shown that an extremely high velocity region occurs downstream of the inlet orifice. In addition, most of the 4 10 nm particle losses occur in the pressurereducing orifice and the relaxation regions. For the larger particles (15 30 nm), most losses occur in the inlet region of the pressure-reducing orifice. Alternate designs for inlet orifice were also considered and their transmission efficiencies were studied. It was found that the inlet orifice with a conical diverging section led to a better transmission efficiency for particles smaller than 10. For particles in a range of 10 30 nm, the orifice with a conical converging throat exhibited a higher transmission efficiency. FIG. 13. Comparison of the transmission efficiency for the lens system with different orifice configurations. (Figure provided in color online.) transmission efficiencies were compared with that of the O keefe sapphire orifice used by Wang et al. (2006) in Figure 13. Among five simulated orifice configurations, the one with a conical diverging section (4 in Figure 12) leads to the minimum loss of fine particles (smaller than 10 nm). This orifice (4 in Figure 12) has a better transmission efficiency compared with that of O keefe sapphire orifice for sub-10 nm particles. For particles in a range of 10 30 nm, the orifice with a conical converging throat (3 in Figure 12) has better transmission efficiency. The orifices with a simple sudden expansion (2 in Figure 12) and with a converging-diverging throat (5 in Figure 12) have lower transmission performance compared with the original O keefe sapphire orifice. REFERENCES Abouali, O., and Ahmadi, G. (2007). 3-D Simulation of Airflow and Nanoparticle Beam Focusing in Aerodynamic Lenses. Int. J. Engineer. 20:45 54. Abouali, O., and Ahmadi, G. (2007). An Axisymmetric Model For Diffusion of Nano-Particle. IEEE-NEMS Conference, Paper No.317, Bangkok, Thailand. Box, G. E. P., and Muller, M. E. (1958). A Note on the Generation of Random Normal Deviates. Ann. Math. Stat. 29:610-611. FLUENT Users Manual. (2003). Discrete Phase Modeling. Gormley, P. G., and Kennedy, M. (1949). Diffusion from a Stream Flowing through a Cylindrical Tube, Proc. Royal Irish Acad. 52(A):163 169. Hagwood, C., Sivathanu, Y., and Mulholland, G. (1999). The DMA Transfer Function with Brownian Motion a Trajectory/Monte-Carlo Approach, Aerosol Sci. Technol. 30:40-61. Hinds, W. C. (1999). Aerosol Technology. Second Edition, John Wiley & Sons, Inc. Hirschfelder, J. D., Curtiss, C. F., and Bird, R. B. (1954). Molecular Theory of Gases and Liquids, Wiley, New York. Jayne, J. T., Leard, D. L., Zhang, X., Davidovits, P., Smith, K. A., Kolb, C. E., and Worsnop, D. R. (2000). Development of an Aerosol Mass Spectrometer for Size and Composition Analysis of Submicron Particles. Aerosol Sci. Technol. 33:49 70. Kane, D. B., and Johnston, M. V. (2000). Size and Composition Biases on the Detection of Individual Ultrafine Particles by Aerosol Mass Spectrometry. Environ. Sci. Technol. 34:4887 4893. Kane, D. B., Oktem, B., and Johnston, M.V. (2001). Nanoparticle Detection by Aerosol Mass Spectrometry. Aerosol Sci. Technol. 34:520 527. Li, A., and Ahmadi, G. (1993). Deposition of Aerosols on Surfaces in a Turbulent Channel Flow. Int. J. Engng Sci. 31:435 445. Liu, P., Ziemann, P. L., Kittelson, D. B., and McMurry, P. H. (1995a). Generating Particle Beams of Controlled Dimensions and Divergence: I. Theory of Particle Motion in Aerodynamic Lenses and Nozzle Expansions. Aerosol Sci. Technol. 22:293 313. Liu, P., Ziemann, P. L., Kittelson, D. B., and McMurry, P.H. (1995b). Generating Particle Beams of Controlled Dimensions and Divergence: II. Experimental Evaluation of Particle Motion in Aerodynamic Lenses and Nozzle Expansions. Aerosol Sci. Technol. 22:314 324. Murphy, W. K., and Sears, G. W. (1986 1987). Production of Particles Beams. J. Appl. Phys. 85. Nikbakht, A., Abouali, O., and Ahmadi, G. (2007). Nano-Particle Beam Focusing in Aerodynamic Lenses An Axisymmetric Model. Scientia Iranica. 14(3):263 272. Ounis, H., Ahmadi, G., and McLaughlin, J. B. (1993). Brownian Particles Deposition in a Directly Simulated Turbulent Channel Flow. Physics of Fluids A. 5:1427 1432. Papoulis, A. (1965). Probability, Random Variables and Stochastic Processes, McGraw-Hill, New York.

3-D SIMULATION OF BROWNIAN MOTION IN AERODYNAMIC LENSES 215 Wang, X., Gidwani, A., Girshick, S. L., and McMurry, P. H. (2005). Aerodynamic Focusing of Nanoparticles: II. Numerical Simulation of Particle Motion through Aerodynamic Lenses. Aerosol Sci. Technol. 39:624 636. Wang, X., and McMurry, P. H. (2006a). A Design Tool for Aerodynamic Lens Systems. Aerosol Sci. Technol. 40:320 334. Wang, X., and McMurry, P. H. (2006b). An Experimental Study of Nanoparticle Focusing with Aerodynamic Lenses. Int. J. Mass Spectrom. 258:30 36. Zhang, X., Smith, K. A., Worsnop, D. R., Jimenez, J., Jayne, J. T., and Kolb, C. E. (2002). A Numerical Characterization of Particle Beam Collimation by an Aerodynamic Lens-Nozzle System: Part I. An Individual Lens or Nozzle. Aerosol Sci. Technol. 36:617 631. Zhang, X., Smith, K. A., Worsnop, D. R., Jimenez, J., Jayne, J. T., and Kolb, C. E., Morris, J. and Davidovits, P. (2004). Numerical Characterization of Particle Beam Collimation: Part II Integrated Aerodynamic Lens-Nozzle System. Aerosol Sci. Technol. 38:619 638.