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CFD simulation of nozzle characteristics in a gas aggregation cluster source
Vacuum 129 (2016) 105e110
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
CFD simulation of nozzle characteristics in a gas aggregation cluster source Lianhua Zhang a, b, *, Jianxiong Shao a, Ximeng Chen a a b
School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China €t Freiburg, Freiburg 79104, Germany Physikalisches Institut, Universita
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 January 2016 Received in revised form 19 April 2016 Accepted 20 April 2016 Available online 22 April 2016
Computational fluid dynamics (CFD) simulation for the sharp-edged nozzle was made using the OpenFOAM source code. Jet dynamics characteristics for the gas stream from a sharped-edged nozzle are then simulated under two-dimensional and axisymmetric steady state condition. Gas and particle velocities in a jet are obtained under different input and boundary conditions to provide an insight into the jet characteristics. For the range of downstream distances considered, the results indicate that a jet is characterized by an initial rapid decay of the axial velocity at the jet center while the cross-sectional flow evolves toward a top-hat profile downstream. Numerical simulation of the free-jet expansion from six types of nozzles indicated that the flow fields had significant differences in the early stages of the expansion. The effects of nozzle geometry on the nanoparticles were investigated. Simulation results indicated that the geometry of nozzle affected the cluster penetration efficiency and beam diameter. The extent to which to the Brownian diffusion can affect the particle extraction from nozzle is investigated. Simulations have shown that the Brownian motion perturbs the clusters from the trajectories dictated by the carrier gas and increases the rate of cluster deposition on the nozzle. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Axisymmetric flow Beam width Brownian motion CFD Nozzle geometry Particle trajectory Penetration
1. Introduction With the rapid development of nanotechnology, various gas aggregation nanocluster sources were developed [1,2]. The concept of gas aggregation originally presented by Sattler et al. [3] was further developed by Haberland et al. [4,5]. Such sources can produce a large number of charged clusters and nanoparticles are dragged by the flow of the carrier gas and extracted from the source through an orifice to the other system with a substantially lower pressure. These fluids are highly viscous and their properties vary with temperature and pressure. It is important to control their parameters in the studies and industrial applications of these produced clusters. Therefore, the clusters are typically collimated into beams by expansion through a nozzle. Apart from the expansion conditions (stagnation pressure, temperature, and effective nozzle diameter), the formation of clusters depends essentially on the geometry of the nozzle and the carrier gases. Some important investigations have been reported on understanding the nozzle
* Corresponding author. School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China. E-mail address: [email protected] (L. Zhang). http://dx.doi.org/10.1016/j.vacuum.2016.04.020 0042-207X/© 2016 Elsevier Ltd. All rights reserved.
dynamic characteristics through theoretical [6e14] and experimental [11e19] studies. Important parameters of the free-jet expansion: stagnation pressure P0, the smallest aperture diameter D, and initial gas temperature T0 [20]. These parameters determine the transport of the nanoparticles from the source to the other system with a lower pressure. The present work is to gain a fundamental knowledge of jet dynamic characteristics and the effect of nozzle geometry on nanoparticle trajectories. For this purpose, CFD analysis is found to be a viable approach because direct measurement of particle velocities and visualization of particle trajectories are very difficult for the high speed and small dimensions involved.
2. Theory and modeling In the Haberland gas aggregation nanocluster source [5], the material is sputtered from the target to form nanoparticles, which are dragged by the flow of the carrier gas. Since the pressure inside the source usually ranges from several Pa to hundreds of Pa and the velocity of the carrier gas flow inside the source is dozens of cm/s, the carrier gas can be treated as the viscous laminar flow in the cylindrical part of the source. Under such conditions, the
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L. Zhang et al. / Vacuum 129 (2016) 105e110
nanoparticles have the much bigger cross-section than the atoms of the carrier gas and the nanoparticle velocity is equal to the drift velocity of the gas. The source was operated at liquid nitrogen temperature under the pressure of 100 Pa. Fixed pressure boundary condition of 1 Pa was imposed at the exit boundary of computational domain downstream of the orifice. The diameter of the source is 100 mm. The viscosity of the helium was computed based on kinetic theory. The particle-free gas flow (axisymmetric, steady, compressible, laminar, and viscous) was calculated by solving the Navier-Stokes equations. Then particles were introduced into the system to obtain their trajectories and velocities. 2.1. Governing equations The governing equations have been solved for steady state conditions using the open source CFD code OpenFOAM [21]. The trajectories of helium gas and inclusion discrete particle are calculated by solving the force balance equation with the forces acting on the particle:
! g rp r d! mp ! ! ¼ FD ð m m P Þ þ FB þ dt rp
(1)
FD is the drag force for per unit particle mass. FB is the Brownian force per unit particle mass. For sub-micron sized particles, a form of Stokes' drag law is available. In this case, FD is defined as:
FD ¼
18n ; d2p rp Cc
(2)
where Cc is the Cunningham correction to Stokes' drag law [22] and can be calculated as:
Cc ¼ 1 þ
2l 1:257 þ 0:4eð1:1dp =2lÞ ; dp
(3)
m is the fluid phase vewhere l is the molecular mean free path; ! m P is the particle velocity; n is the molecular viscosity of the locity; ! fluid; r is the fluid density; rp is the density of the particle; dp is the particle diameter; Re is the relative Reynolds number and defined as
rdp j! m ! m Pj Re ¼ : n
(4)
The mean free path of molecule is given as
kT
l ¼ pffiffiffi 2 2pdm P
(5)
In Equation (5), dm, k, P, and T, respectively, are the helium molecule diameter, the Boltzmann constant, the gas pressure and the gas temperature.
Where dij is the Kronecker delta function, and
SO ¼
216vkB T 2 r p2 rd5p rp Cc
T is the absolute temperature of the fluid; v is the kinematic viscosity, Cc is the Cunningham correction and kB is the Boltzmann constant. Amplitudes of the Brownian force components are of
rffiffiffiffiffiffiffiffiffi pSO Fbi ¼ zi Dt
(8)
where zi are zero-mean, unit-variance-independent Gaussian random numbers. The amplitudes of the Brownian force components are evaluated at each time step. The energy equation must be enabled in order for the Brownian force to take effect. Brownian force is intended only for laminar simulations. 2.3. Boundary conditions and solution methodology The geometry of computational domain with boundary conditions is shown in Fig. 1. A pure nozzle is considered as a twodimensional, steady axisymmetric flow that has passed through a nozzle before entering the vacuum as a free jet. Because the jet is assumed to be axisymmetric, symmetry conditions are applied and only the upper half of flow domain was solved. The CFD simulation starts from inlet and enters the vacuum, and ends after the jet has travelled 100 mm downstream. In the CFD model, sections AB and BC are treated as slip walls, whereas sections BD and DE as shown in Fig. 1 are considered as free boundaries for which pressure conditions are used. After a converged solution was obtain, nanoparticles were added into the pure gas. As it is a normal expectation in a gas aggregation cluster source that particles are entrained through the chamber and nozzle exit, the initial particle velocity at the inlet was set to the velocity of its surrounding gas, and the particle velocities were calculated using the discrete phase model. 3. Results and discussion 3.1. Dynamic characteristics of gas flow filed A set of tests with different initial pressures and nozzle diameters has been carried out. The CFD results for the jet flow with inlet pressures of 90, 100, 110 and 120 Pa and nozzle diameters of 4.8, 5.0 and 5.2 mm were obtained and some representative plots are given in Figs. 2 and 3. Fig. 2(a) shows the variation of jet axial velocity at the jet center with the axial distance from the inlet and under different inlet pressures (90e120 Pa). All the curves indicate that there is initial rapid decay in the jet velocity within the axial distance. The velocity
2.2. Brownian force For sub-micron particles, the effects of Brownian motion can be optionally included in the additional force term. The components of the Brownian force are modeled as a Gaussian white noise process with spectral intensitySn,ij
Sn; ij ¼ SO dij
(7)
(6) Fig. 1. Computation domain and boundary conditions of sharp-edged nozzle.
L. Zhang et al. / Vacuum 129 (2016) 105e110 700
600
10
500
8
300 200
500
6
4
d=4.8mm d=5.0mm d=5.2mm
p=100Pa
600
p=90Pa p=100Pa p=110Pa p=120Pa
10
8
400 Axial velcoity (m/s)
d=5.0mm
Axial velcoity (m/s)
700
Axial velocity (m/s)
Axial velocity (m/s)
800
400
107
300
6
4
2
200
2 0 -0.10
-0.08
100
100
0 -0.10
-0.08
-0.06
-0.04
Axial distance x (m)
0 -0.10
-0.06
-0.04
-0.02
Axial distance x (m)
-0.02
-0.05
0.00
0.05
0 -0.10
0.10
-0.05
0.00
Axial distance x (m)
0.05
0.10
Axial distance x (m)
Fig. 2. The axial velocities of gas flow along the axis of the orifice system in dependence on pressure and the diameter of the orifice.
700
700
600
600
400
500
Axial velocity (m/s)
500
Axial velocity (m/s)
x=10mm x=30mm x=50mm x=70mm x=90mm
d=5.0mm p=100Pa
300
200
100
x=10mm p=100Pa
400
d=4.8mm d=5.0mm d=5.2mm
300
200
100
0 0.00
0.01
0.02
0.03
0.04
0 0.00
0.05
Radial distance r (m)
(a)
0.01
0.02
(b)
0.03
0.04
0.05
Radial distance r (m)
Fig. 3. The calculated radial profile of the gas flow velocity.
of a gas flow u after the orifice is comparable with the sound speed pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u0 ¼ kgT=m (g ¼ 5/3 for helium, m is the atom mass, and u0 ¼ 822 m/s at T ¼ 200 K).
Q ¼ pr02 Nu
(9)
where Q is the flow rate, r0 is the orifice radius, N is the number density of atoms inside the magnetron chamber, and u is the
effective beam velocity. We note the number density of atoms decreases after the orifice while the flow velocity u is close to the speed of sound u0. u/u0 z 0.84, which shows little variation with gas flow [23]. Fig. 2(b) shows the effect of nozzle diameters at an inlet pressure of 100 Pa. It is noticed that a reduction in nozzle diameter corresponds to a slight more rapid decay of the jet velocity along the jet axis. The expansion of the gas flow after orifice is calculated in 700
700
500
10
400 Axial velocity (m/s)
200
8
6
4
5
400
300
200
4
Axial velcoity (m/s)
Axial velcoity (m/s)
500
Axial velocity (m/s)
600
gas velcoity Dp=05nm Dp=10nm Dp=20nm Dp=40nm
600
300
p=100Pa, Dp=05nm gas velocity D=5.2mm D=5.0mm D=4.8mm
p=100Pa, D=5.0mm
3
2
1
0 -0.10
2
-0.08
-0.06
-0.04
-0.02
Axial distance x (m)
100
0 -0.10
100 -0.08
-0.06
-0.04
-0.02
Axial distance x (m)
0 -0.10
-0.05
0.00
Axial distance x (m)
0.05
0.10
0 -0.10
-0.05
0.00
Axial distance x (m)
Fig. 4. Axial flow and particle velocities along the axis of the orifice system.
0.05
0.10
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Refs. [11e13]. In Fig. 3(a), the jet axial velocity profiles in five cross-section at axial distance x ¼ 10, 30, 50,70 and 90 mm are given to illustrate the variation in jet axial velocity across and along the jet flow. The velocity profile at close to the jet exit (x ¼ 10 mm) again shows a rapid drop towards the outlet. It appears that at x ¼ 90, the velocity has approached a balanced value so that the velocity variation across the jet is minimal. The jet velocity profiles within the jet domain flatten out as the distance from the nozzle exit increases. Fig. 3(b) shows the influence of nozzle diameter on jet axial velocity profile at axial location of x ¼ 10 mm it is noticed that the crosssectional flow profile has same tendency that is essentially irrespective of the nozzle diameter. The particle shape was assumed to be spherical and the CFD study used gold nanoparticles of four different diameters, 5, 10, 20 and 40 nm. Fig. 4(a) shows the particle velocity at the jet center along the jet axis for each of the four particle sizes together with the gas velocity for a comparison purpose. The particles were released at the inlet and the inlet is at a distance of about 10 cm away from the orifice, it is reasonable to assume that at the beginning the clusters drift velocity coincides with the atom drift velocity [9]. The figure indicates that the decay of particle velocities with the axial distance is in a similar way to the gas velocity except the 20 nm and larger particles. The gas velocity decays more rapidly than the particle velocity, and the velocities of small particles decelerate more quickly than the velocities of large particles. It is evident that the probability of collision is higher for larger clusters: the collision cross section increases with the cluster radius as rc2 [9] while the cluster mass Mfrc3 [24]. Thus, even with a higher probability of collision, acceleration of larger clusters is less effective than the smaller ones. The velocities of the nanoparticles are strongly dependent on their mass. As it was observed in Refs. [11e13]. However, when the cluster velocities are in the range of about 80e180 m s1 [17], there is only a weak cluster mass dependence, which is not in agreement with theoretical predications [9]. The discrepancies are explained by the interaction of clusters with buffer gas atoms outside the aggregation chamber during the passage from orifice towards the detector [17]. This interaction is not included in the theoretical model [9]. Fig. 4(b) shows the effect of nozzle diameter on the particle velocities. The plots for the three diameters can almost be represented by a single curve, although a smaller diameter is associated with a slightly higher particles velocity. From this phenomenon, we can get that
the gas flow velocity weakly depends on the orifice diameter. The experiment evaluation of argon flow velocity was reported by Ref. [13] and weakly depends on the orifice diameter. The simulation data for the other inlet pressures (90e120 Pa) yield identical trends as those shown in Fig. 5(b). It can also be noticed from Fig. 4 that the particle velocity approaches the corresponding jet velocity as the jet flows away. The particle velocity profiles at five downstream locations x ¼ 10, 30, 50, 70 and 90 mm are plotted in Fig. 5 for the case of p ¼ 100 Pa, D ¼ 5 mm and Dp ¼ 5 nm. The figure shows that the
No. 1, Sharp-edged orifice
No. 2, Orifice with a conical converging throat
No. 3, Capillary orifice
No. 4, Laval orifice 300
250
Axial velcoity (m/s)
p=100Pa, D=5.0mm, Dp=05nm 200 x=10mm x=30mm x=50mm x=70mm x=90mm
150
No. 5, Orifice with converging-diverging throat
100
50
0 0.00
0.01
0.02
0.03
0.04
0.05
Radial distance r (m) Fig. 5. The calculated particle velocity profiles at five downstream locations.
No. 6, Orifice with a conical diverging section after the throat Fig. 6. Six different shapes of the orifices.
L. Zhang et al. / Vacuum 129 (2016) 105e110 1.0
109
1.0
0.9
0.8
short convergent nozzle sharp-edged nozzle capillary nozzle laval nozzle convergent-divergent nozzle divergent nozzle
0.8
Penetration
Penetration
0.9
short convergent nozzle sharp-edged nozzle capillary nozzle laval nozzle convergent-divergent nozzle divergent nozzle
0.7
0.6
0.5
(b) With diffusion
0.4
(a) No diffusion 0.7 1
10
1
dp(nm)
10
dp(nm)
Fig. 7. Comparison of particle penetration through six types of the nozzles: (a) No diffusion; (b) with diffusion.
particle velocity at a given cross-sections increases and then decreases as the radial distance from the jet center increases. It also shows that the particle velocity decrease with the radial distance becomes smaller at locations further downstream, where the particle velocity is expected to approach a more top-hat profile inside the jet. 3.2. Comparison of performance of different orifices Several different designs of the orifice as shown in Fig. 6 were studied. For the sake of comparison, the minimum size of the throat diameter for all six orifices are kept fixed at 5 mm the corresponding computed penetrations were compared with the sharpedged orifice in Fig. 7. Fig. 7 shows the particle penetration through six types of nozzles with/without Brownian motion. The diffusional losses become significant when the particles are smaller than 10 nm. The penetrations of the particles of 1e10 nm are almost the same for these nozzles except the short convergent nozzle and the sharp-edged nozzle when Brownian diffusion is absent. Three sharp drops occur during the penetration in sharp-edged nozzle, capillary nozzle, and divergent nozzle for the particles larger than 20 nm. This is because a large amount of particles are lost on the wall. When the Brownian motion is present, the penetrations are the same for the particles of 8e60 nm in short convergent nozzle, Laval
nozzle, and convergent-divergent nozzle due to the existence of the convergent section. The convergent part plays an important in the penetration of nanoparticles. For the particles larger than 20 nm, no significant difference is observed in the penetration when the diffusion is absent or present. Fig. 8 shows the radii of particle beam for 1e60 nm particles in six types of the nozzles (a) without diffusion and (b) with diffusion. When the diffusion is absent, the particle beam widths increase for the particles in the sharp-edged nozzle and divergent nozzle. However, for the capillary nozzle, the particle beam width firstly decreases and then increases. For other nozzles, the particle beam width shows nearly the same tendency. However, there is a sharp drop in particle width for the particles larger than 50 nm. This is because a large amount of particles are lost on the wall. When the diffusion is present, the particle beam widths firstly decrease and then increase for the particles smaller than 40 nm in the sharpedged nozzle, capillary nozzle, and divergent nozzle because the diffusion is the dominant factor of the particle width. The particle beam widths firstly decrease rapidly and then increase slowly for the particles in the short convergent nozzle, Laval nozzle, and the convergent-divergent nozzle. Therefore, the effects of aerodynamic focusing are stronger in these nozzles with the convergent section [13]. In recent work [10] about the aerodynamic focusing of the nozzle, the results revealed that the presence of the Brownian diffusion could not erase the aerodynamic focalization effect of the
50
50 short convergent nozzle sharp-edged nozzle capillary nozzle laval nozzle convergent-divergent nozzle divergent nozzle
40
30
20
(a) No diffusion
10
0
Particle beam radius(mm)
Particle beam radius (mm)
40
short convergent nozzle sharp-edged nozzle capillary nozzle laval nozzle convergent-divergent nozzle divergent nozzle
30
20
(b) With diffusion 10
0
1
10
dp(nm)
1
10
dp(nm)
Fig. 8. Particle beam widths in six types of the nozzle. (a) without diffusion; (b) with diffusion.
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focuser; however it lowered the efficiency of the particle focusing and remarkably increased the wall deposition. This is in a good agreement with our simulation results. 4. Conclusions We described a numerical tool to characterize the orifice systems. In the tool, the commercial CFD software OpenFOAM was adopted. The gas flow field was obtained by solving the viscous laminar compressible Navier-Stokes equation. Particles were tracked with the Lagrangian approach, in which it was assumed that the presence of particles did not affect the flow field and that there was no particle-particle interaction. This numerical tool was applied to evaluate sharp-edged nozzle. The study has provided an in-depth understanding of the dynamic characteristics of the gas and particles inside a jet. We calculated the streamlines of the flow inside the nozzles and showed the impact of the Brownian diffusion on perturbing the penetration of the particles. Particle trajectories, penetration, velocity and beam width were studied. The particle beam widths decrease fast and then increase slowly for the particles in the short convergent nozzle, Laval nozzle, and convergentdivergent nozzle. Therefore, the effects of aerodynamic focusing are stronger in these nozzles with the convergent section. The convergent section plays an important role in the penetration of particles and affects the particle beam width. This study provides the essential knowledge to optimize the jet characteristics through optimizing the nozzle design and process parameters, and to mathematically model the jet (and particle) characteristics for eventually modeling and improving the orifice performance. References [1] C. Binns, Surf. Sci. Rep. 44 (2001) 1e49.
[2] K. Wegner, P. Piseri, H. Vahedi Tafreshi, P. Milani, J. Phys. D. Appl. Phys. 39 (2006) R439eR459. [3] K. Sattler, J. Mühlback, E. Recknagel, Phys. Rev. Lett. 45 (1980) 821. [4] H. Haberland, M. Karrais, M. Mall, Y. Thurner, J. Vac. Sci. Technol. A 10 (1992) 3266. [5] H. Haberland, M. Mall, M. Mossler, Y. Qiang, T. Reiners, Y. Thurner, J. Vac. Sci. Technol. A 12 (1994) 2925. [6] H.R. Murphy, D.R. Miller, J. Phys. Chem. 88 (1984) 4474. [7] D.R. Miller, M. Fineman, H. Murphy, The Free-Jet Expansion From a Capillary Source, in: Belotserkovskii, Kogan, Kutateladze, Rebrov (Eds.), 13th International Symposium on Rarefied Gas Dynamics, 1985, p. 923. [8] D. Miller, Free Jet Sources, in: G. Scoles (Ed.), Atomic and Molecular Beam Methods, 1, Oxford University Press, New York, 1988, pp. 14e53. [9] M. Smirnov Boris, Ibrahimkutty Shyjumon, Rainer Hippler, Phys. Rev. E 75 (Jun 2007) 066402. [10] H. Vahedi Tafreshi, P. Piseri, E. Barborini, G. Benedek, P. Milani, J. Nanoparticle Res. 4 (2002) 511e524. [11] M. Ganeva, P.V. Kashtanov, B.M. Smirnov, R. Hippler, Vacuum 110 (2014) 140e145. [12] Petr A. Skovorodko, Simon A. Brown, Domagoj Belic, AIP Conf. Proc. 1333 (2011) 203. [13] Kousal Jaroslav, Polonskyi Oleksandr, Kylian Ondrej, Choukourov Andrei, Artemenko Anna, Pesicka Josef, et al., Vacuum 96 (2013) 32e38. [14] S. Pratontep, S.J. Carroll, C. Xirouchaki, M. Streun, R.E. Palmer, Rev. Sci. Instrum. 76 (4) (2005) 045103. [15] O. Polonskyi, P. Solar, O. Kylian, M. Drabik, A. Artemenko, J. Kousal, et al., Thin Solid Films 520 (12) (2012) 4155e4162. [16] M. Ganeva, T. Peter, S. Bornholdt, H. Kersten, T. Strunskus, V. Zaporojtchenko, et al., Contrib. Plasma Phys. 52 (10) (2012) 881e889. [17] Ganeva Marina, V. Pipa Andrei, M. Smirnov Boris, V. Kashtanov Pavel, Rainer Hippler, Plasma Sources Sci. Technol. 22 (4) (2013) 045011. [18] O.F. Hagena, W. Obert, J. Chem. Phys. 56 (1972) 1793. [19] J.T. MacDaniels, R.E. Continetti, D.R. Miller, The Effect of Nozzle Geometry on cluster Formation in Molecular Beam Sources, in: A.D. Ketsdever, Muntz (Eds.), 23th International Symposium on Rare Gas Dynamics, 2003, p. 670. [20] S.B. Ryali, J.B. Fenn, B. B. Phy. Chem. 3 (1984) 245. [21] OpenFOAM User’s Guide, OpenFOAM Foundation Ltd. [22] W.C. Hinds, Aerosal Technology, second ed., John Wiley&Sons Inc., 1999. [23] I. Shyjumon, M. Gopinadhan, C.A. Helm, B.M. Smirnov, R. Hippler, Thin Solid Films 500 (2006) 41. [24] M. Seidl, M. Brack, Ann. Phys. 245 (1996) 275e310.
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
CFD simulation of nozzle characteristics in a gas aggregation cluster source Lianhua Zhang a, b, *, Jianxiong Shao a, Ximeng Chen a a b
School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China €t Freiburg, Freiburg 79104, Germany Physikalisches Institut, Universita
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 January 2016 Received in revised form 19 April 2016 Accepted 20 April 2016 Available online 22 April 2016
Computational fluid dynamics (CFD) simulation for the sharp-edged nozzle was made using the OpenFOAM source code. Jet dynamics characteristics for the gas stream from a sharped-edged nozzle are then simulated under two-dimensional and axisymmetric steady state condition. Gas and particle velocities in a jet are obtained under different input and boundary conditions to provide an insight into the jet characteristics. For the range of downstream distances considered, the results indicate that a jet is characterized by an initial rapid decay of the axial velocity at the jet center while the cross-sectional flow evolves toward a top-hat profile downstream. Numerical simulation of the free-jet expansion from six types of nozzles indicated that the flow fields had significant differences in the early stages of the expansion. The effects of nozzle geometry on the nanoparticles were investigated. Simulation results indicated that the geometry of nozzle affected the cluster penetration efficiency and beam diameter. The extent to which to the Brownian diffusion can affect the particle extraction from nozzle is investigated. Simulations have shown that the Brownian motion perturbs the clusters from the trajectories dictated by the carrier gas and increases the rate of cluster deposition on the nozzle. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Axisymmetric flow Beam width Brownian motion CFD Nozzle geometry Particle trajectory Penetration
1. Introduction With the rapid development of nanotechnology, various gas aggregation nanocluster sources were developed [1,2]. The concept of gas aggregation originally presented by Sattler et al. [3] was further developed by Haberland et al. [4,5]. Such sources can produce a large number of charged clusters and nanoparticles are dragged by the flow of the carrier gas and extracted from the source through an orifice to the other system with a substantially lower pressure. These fluids are highly viscous and their properties vary with temperature and pressure. It is important to control their parameters in the studies and industrial applications of these produced clusters. Therefore, the clusters are typically collimated into beams by expansion through a nozzle. Apart from the expansion conditions (stagnation pressure, temperature, and effective nozzle diameter), the formation of clusters depends essentially on the geometry of the nozzle and the carrier gases. Some important investigations have been reported on understanding the nozzle
* Corresponding author. School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China. E-mail address: [email protected] (L. Zhang). http://dx.doi.org/10.1016/j.vacuum.2016.04.020 0042-207X/© 2016 Elsevier Ltd. All rights reserved.
dynamic characteristics through theoretical [6e14] and experimental [11e19] studies. Important parameters of the free-jet expansion: stagnation pressure P0, the smallest aperture diameter D, and initial gas temperature T0 [20]. These parameters determine the transport of the nanoparticles from the source to the other system with a lower pressure. The present work is to gain a fundamental knowledge of jet dynamic characteristics and the effect of nozzle geometry on nanoparticle trajectories. For this purpose, CFD analysis is found to be a viable approach because direct measurement of particle velocities and visualization of particle trajectories are very difficult for the high speed and small dimensions involved.
2. Theory and modeling In the Haberland gas aggregation nanocluster source [5], the material is sputtered from the target to form nanoparticles, which are dragged by the flow of the carrier gas. Since the pressure inside the source usually ranges from several Pa to hundreds of Pa and the velocity of the carrier gas flow inside the source is dozens of cm/s, the carrier gas can be treated as the viscous laminar flow in the cylindrical part of the source. Under such conditions, the
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L. Zhang et al. / Vacuum 129 (2016) 105e110
nanoparticles have the much bigger cross-section than the atoms of the carrier gas and the nanoparticle velocity is equal to the drift velocity of the gas. The source was operated at liquid nitrogen temperature under the pressure of 100 Pa. Fixed pressure boundary condition of 1 Pa was imposed at the exit boundary of computational domain downstream of the orifice. The diameter of the source is 100 mm. The viscosity of the helium was computed based on kinetic theory. The particle-free gas flow (axisymmetric, steady, compressible, laminar, and viscous) was calculated by solving the Navier-Stokes equations. Then particles were introduced into the system to obtain their trajectories and velocities. 2.1. Governing equations The governing equations have been solved for steady state conditions using the open source CFD code OpenFOAM [21]. The trajectories of helium gas and inclusion discrete particle are calculated by solving the force balance equation with the forces acting on the particle:
! g rp r d! mp ! ! ¼ FD ð m m P Þ þ FB þ dt rp
(1)
FD is the drag force for per unit particle mass. FB is the Brownian force per unit particle mass. For sub-micron sized particles, a form of Stokes' drag law is available. In this case, FD is defined as:
FD ¼
18n ; d2p rp Cc
(2)
where Cc is the Cunningham correction to Stokes' drag law [22] and can be calculated as:
Cc ¼ 1 þ
2l 1:257 þ 0:4eð1:1dp =2lÞ ; dp
(3)
m is the fluid phase vewhere l is the molecular mean free path; ! m P is the particle velocity; n is the molecular viscosity of the locity; ! fluid; r is the fluid density; rp is the density of the particle; dp is the particle diameter; Re is the relative Reynolds number and defined as
rdp j! m ! m Pj Re ¼ : n
(4)
The mean free path of molecule is given as
kT
l ¼ pffiffiffi 2 2pdm P
(5)
In Equation (5), dm, k, P, and T, respectively, are the helium molecule diameter, the Boltzmann constant, the gas pressure and the gas temperature.
Where dij is the Kronecker delta function, and
SO ¼
216vkB T 2 r p2 rd5p rp Cc
T is the absolute temperature of the fluid; v is the kinematic viscosity, Cc is the Cunningham correction and kB is the Boltzmann constant. Amplitudes of the Brownian force components are of
rffiffiffiffiffiffiffiffiffi pSO Fbi ¼ zi Dt
(8)
where zi are zero-mean, unit-variance-independent Gaussian random numbers. The amplitudes of the Brownian force components are evaluated at each time step. The energy equation must be enabled in order for the Brownian force to take effect. Brownian force is intended only for laminar simulations. 2.3. Boundary conditions and solution methodology The geometry of computational domain with boundary conditions is shown in Fig. 1. A pure nozzle is considered as a twodimensional, steady axisymmetric flow that has passed through a nozzle before entering the vacuum as a free jet. Because the jet is assumed to be axisymmetric, symmetry conditions are applied and only the upper half of flow domain was solved. The CFD simulation starts from inlet and enters the vacuum, and ends after the jet has travelled 100 mm downstream. In the CFD model, sections AB and BC are treated as slip walls, whereas sections BD and DE as shown in Fig. 1 are considered as free boundaries for which pressure conditions are used. After a converged solution was obtain, nanoparticles were added into the pure gas. As it is a normal expectation in a gas aggregation cluster source that particles are entrained through the chamber and nozzle exit, the initial particle velocity at the inlet was set to the velocity of its surrounding gas, and the particle velocities were calculated using the discrete phase model. 3. Results and discussion 3.1. Dynamic characteristics of gas flow filed A set of tests with different initial pressures and nozzle diameters has been carried out. The CFD results for the jet flow with inlet pressures of 90, 100, 110 and 120 Pa and nozzle diameters of 4.8, 5.0 and 5.2 mm were obtained and some representative plots are given in Figs. 2 and 3. Fig. 2(a) shows the variation of jet axial velocity at the jet center with the axial distance from the inlet and under different inlet pressures (90e120 Pa). All the curves indicate that there is initial rapid decay in the jet velocity within the axial distance. The velocity
2.2. Brownian force For sub-micron particles, the effects of Brownian motion can be optionally included in the additional force term. The components of the Brownian force are modeled as a Gaussian white noise process with spectral intensitySn,ij
Sn; ij ¼ SO dij
(7)
(6) Fig. 1. Computation domain and boundary conditions of sharp-edged nozzle.
L. Zhang et al. / Vacuum 129 (2016) 105e110 700
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Fig. 2. The axial velocities of gas flow along the axis of the orifice system in dependence on pressure and the diameter of the orifice.
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Fig. 3. The calculated radial profile of the gas flow velocity.
of a gas flow u after the orifice is comparable with the sound speed pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u0 ¼ kgT=m (g ¼ 5/3 for helium, m is the atom mass, and u0 ¼ 822 m/s at T ¼ 200 K).
Q ¼ pr02 Nu
(9)
where Q is the flow rate, r0 is the orifice radius, N is the number density of atoms inside the magnetron chamber, and u is the
effective beam velocity. We note the number density of atoms decreases after the orifice while the flow velocity u is close to the speed of sound u0. u/u0 z 0.84, which shows little variation with gas flow [23]. Fig. 2(b) shows the effect of nozzle diameters at an inlet pressure of 100 Pa. It is noticed that a reduction in nozzle diameter corresponds to a slight more rapid decay of the jet velocity along the jet axis. The expansion of the gas flow after orifice is calculated in 700
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Fig. 4. Axial flow and particle velocities along the axis of the orifice system.
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Refs. [11e13]. In Fig. 3(a), the jet axial velocity profiles in five cross-section at axial distance x ¼ 10, 30, 50,70 and 90 mm are given to illustrate the variation in jet axial velocity across and along the jet flow. The velocity profile at close to the jet exit (x ¼ 10 mm) again shows a rapid drop towards the outlet. It appears that at x ¼ 90, the velocity has approached a balanced value so that the velocity variation across the jet is minimal. The jet velocity profiles within the jet domain flatten out as the distance from the nozzle exit increases. Fig. 3(b) shows the influence of nozzle diameter on jet axial velocity profile at axial location of x ¼ 10 mm it is noticed that the crosssectional flow profile has same tendency that is essentially irrespective of the nozzle diameter. The particle shape was assumed to be spherical and the CFD study used gold nanoparticles of four different diameters, 5, 10, 20 and 40 nm. Fig. 4(a) shows the particle velocity at the jet center along the jet axis for each of the four particle sizes together with the gas velocity for a comparison purpose. The particles were released at the inlet and the inlet is at a distance of about 10 cm away from the orifice, it is reasonable to assume that at the beginning the clusters drift velocity coincides with the atom drift velocity [9]. The figure indicates that the decay of particle velocities with the axial distance is in a similar way to the gas velocity except the 20 nm and larger particles. The gas velocity decays more rapidly than the particle velocity, and the velocities of small particles decelerate more quickly than the velocities of large particles. It is evident that the probability of collision is higher for larger clusters: the collision cross section increases with the cluster radius as rc2 [9] while the cluster mass Mfrc3 [24]. Thus, even with a higher probability of collision, acceleration of larger clusters is less effective than the smaller ones. The velocities of the nanoparticles are strongly dependent on their mass. As it was observed in Refs. [11e13]. However, when the cluster velocities are in the range of about 80e180 m s1 [17], there is only a weak cluster mass dependence, which is not in agreement with theoretical predications [9]. The discrepancies are explained by the interaction of clusters with buffer gas atoms outside the aggregation chamber during the passage from orifice towards the detector [17]. This interaction is not included in the theoretical model [9]. Fig. 4(b) shows the effect of nozzle diameter on the particle velocities. The plots for the three diameters can almost be represented by a single curve, although a smaller diameter is associated with a slightly higher particles velocity. From this phenomenon, we can get that
the gas flow velocity weakly depends on the orifice diameter. The experiment evaluation of argon flow velocity was reported by Ref. [13] and weakly depends on the orifice diameter. The simulation data for the other inlet pressures (90e120 Pa) yield identical trends as those shown in Fig. 5(b). It can also be noticed from Fig. 4 that the particle velocity approaches the corresponding jet velocity as the jet flows away. The particle velocity profiles at five downstream locations x ¼ 10, 30, 50, 70 and 90 mm are plotted in Fig. 5 for the case of p ¼ 100 Pa, D ¼ 5 mm and Dp ¼ 5 nm. The figure shows that the
No. 1, Sharp-edged orifice
No. 2, Orifice with a conical converging throat
No. 3, Capillary orifice
No. 4, Laval orifice 300
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Axial velcoity (m/s)
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No. 5, Orifice with converging-diverging throat
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No. 6, Orifice with a conical diverging section after the throat Fig. 6. Six different shapes of the orifices.
L. Zhang et al. / Vacuum 129 (2016) 105e110 1.0
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short convergent nozzle sharp-edged nozzle capillary nozzle laval nozzle convergent-divergent nozzle divergent nozzle
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Fig. 7. Comparison of particle penetration through six types of the nozzles: (a) No diffusion; (b) with diffusion.
particle velocity at a given cross-sections increases and then decreases as the radial distance from the jet center increases. It also shows that the particle velocity decrease with the radial distance becomes smaller at locations further downstream, where the particle velocity is expected to approach a more top-hat profile inside the jet. 3.2. Comparison of performance of different orifices Several different designs of the orifice as shown in Fig. 6 were studied. For the sake of comparison, the minimum size of the throat diameter for all six orifices are kept fixed at 5 mm the corresponding computed penetrations were compared with the sharpedged orifice in Fig. 7. Fig. 7 shows the particle penetration through six types of nozzles with/without Brownian motion. The diffusional losses become significant when the particles are smaller than 10 nm. The penetrations of the particles of 1e10 nm are almost the same for these nozzles except the short convergent nozzle and the sharp-edged nozzle when Brownian diffusion is absent. Three sharp drops occur during the penetration in sharp-edged nozzle, capillary nozzle, and divergent nozzle for the particles larger than 20 nm. This is because a large amount of particles are lost on the wall. When the Brownian motion is present, the penetrations are the same for the particles of 8e60 nm in short convergent nozzle, Laval
nozzle, and convergent-divergent nozzle due to the existence of the convergent section. The convergent part plays an important in the penetration of nanoparticles. For the particles larger than 20 nm, no significant difference is observed in the penetration when the diffusion is absent or present. Fig. 8 shows the radii of particle beam for 1e60 nm particles in six types of the nozzles (a) without diffusion and (b) with diffusion. When the diffusion is absent, the particle beam widths increase for the particles in the sharp-edged nozzle and divergent nozzle. However, for the capillary nozzle, the particle beam width firstly decreases and then increases. For other nozzles, the particle beam width shows nearly the same tendency. However, there is a sharp drop in particle width for the particles larger than 50 nm. This is because a large amount of particles are lost on the wall. When the diffusion is present, the particle beam widths firstly decrease and then increase for the particles smaller than 40 nm in the sharpedged nozzle, capillary nozzle, and divergent nozzle because the diffusion is the dominant factor of the particle width. The particle beam widths firstly decrease rapidly and then increase slowly for the particles in the short convergent nozzle, Laval nozzle, and the convergent-divergent nozzle. Therefore, the effects of aerodynamic focusing are stronger in these nozzles with the convergent section [13]. In recent work [10] about the aerodynamic focusing of the nozzle, the results revealed that the presence of the Brownian diffusion could not erase the aerodynamic focalization effect of the
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Fig. 8. Particle beam widths in six types of the nozzle. (a) without diffusion; (b) with diffusion.
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focuser; however it lowered the efficiency of the particle focusing and remarkably increased the wall deposition. This is in a good agreement with our simulation results. 4. Conclusions We described a numerical tool to characterize the orifice systems. In the tool, the commercial CFD software OpenFOAM was adopted. The gas flow field was obtained by solving the viscous laminar compressible Navier-Stokes equation. Particles were tracked with the Lagrangian approach, in which it was assumed that the presence of particles did not affect the flow field and that there was no particle-particle interaction. This numerical tool was applied to evaluate sharp-edged nozzle. The study has provided an in-depth understanding of the dynamic characteristics of the gas and particles inside a jet. We calculated the streamlines of the flow inside the nozzles and showed the impact of the Brownian diffusion on perturbing the penetration of the particles. Particle trajectories, penetration, velocity and beam width were studied. The particle beam widths decrease fast and then increase slowly for the particles in the short convergent nozzle, Laval nozzle, and convergentdivergent nozzle. Therefore, the effects of aerodynamic focusing are stronger in these nozzles with the convergent section. The convergent section plays an important role in the penetration of particles and affects the particle beam width. This study provides the essential knowledge to optimize the jet characteristics through optimizing the nozzle design and process parameters, and to mathematically model the jet (and particle) characteristics for eventually modeling and improving the orifice performance. References [1] C. Binns, Surf. Sci. Rep. 44 (2001) 1e49.
[2] K. Wegner, P. Piseri, H. Vahedi Tafreshi, P. Milani, J. Phys. D. Appl. Phys. 39 (2006) R439eR459. [3] K. Sattler, J. Mühlback, E. Recknagel, Phys. Rev. Lett. 45 (1980) 821. [4] H. Haberland, M. Karrais, M. Mall, Y. Thurner, J. Vac. Sci. Technol. A 10 (1992) 3266. [5] H. Haberland, M. Mall, M. Mossler, Y. Qiang, T. Reiners, Y. Thurner, J. Vac. Sci. Technol. A 12 (1994) 2925. [6] H.R. Murphy, D.R. Miller, J. Phys. Chem. 88 (1984) 4474. [7] D.R. Miller, M. Fineman, H. Murphy, The Free-Jet Expansion From a Capillary Source, in: Belotserkovskii, Kogan, Kutateladze, Rebrov (Eds.), 13th International Symposium on Rarefied Gas Dynamics, 1985, p. 923. [8] D. Miller, Free Jet Sources, in: G. Scoles (Ed.), Atomic and Molecular Beam Methods, 1, Oxford University Press, New York, 1988, pp. 14e53. [9] M. Smirnov Boris, Ibrahimkutty Shyjumon, Rainer Hippler, Phys. Rev. E 75 (Jun 2007) 066402. [10] H. Vahedi Tafreshi, P. Piseri, E. Barborini, G. Benedek, P. Milani, J. Nanoparticle Res. 4 (2002) 511e524. [11] M. Ganeva, P.V. Kashtanov, B.M. Smirnov, R. Hippler, Vacuum 110 (2014) 140e145. [12] Petr A. Skovorodko, Simon A. Brown, Domagoj Belic, AIP Conf. Proc. 1333 (2011) 203. [13] Kousal Jaroslav, Polonskyi Oleksandr, Kylian Ondrej, Choukourov Andrei, Artemenko Anna, Pesicka Josef, et al., Vacuum 96 (2013) 32e38. [14] S. Pratontep, S.J. Carroll, C. Xirouchaki, M. Streun, R.E. Palmer, Rev. Sci. Instrum. 76 (4) (2005) 045103. [15] O. Polonskyi, P. Solar, O. Kylian, M. Drabik, A. Artemenko, J. Kousal, et al., Thin Solid Films 520 (12) (2012) 4155e4162. [16] M. Ganeva, T. Peter, S. Bornholdt, H. Kersten, T. Strunskus, V. Zaporojtchenko, et al., Contrib. Plasma Phys. 52 (10) (2012) 881e889. [17] Ganeva Marina, V. Pipa Andrei, M. Smirnov Boris, V. Kashtanov Pavel, Rainer Hippler, Plasma Sources Sci. Technol. 22 (4) (2013) 045011. [18] O.F. Hagena, W. Obert, J. Chem. Phys. 56 (1972) 1793. [19] J.T. MacDaniels, R.E. Continetti, D.R. Miller, The Effect of Nozzle Geometry on cluster Formation in Molecular Beam Sources, in: A.D. Ketsdever, Muntz (Eds.), 23th International Symposium on Rare Gas Dynamics, 2003, p. 670. [20] S.B. Ryali, J.B. Fenn, B. B. Phy. Chem. 3 (1984) 245. [21] OpenFOAM User’s Guide, OpenFOAM Foundation Ltd. [22] W.C. Hinds, Aerosal Technology, second ed., John Wiley&Sons Inc., 1999. [23] I. Shyjumon, M. Gopinadhan, C.A. Helm, B.M. Smirnov, R. Hippler, Thin Solid Films 500 (2006) 41. [24] M. Seidl, M. Brack, Ann. Phys. 245 (1996) 275e310.