Dynamic Molecular Collision Model for N2—He Mixture
Takashi Tokumasu*, Yoichiro Matsumoto^, Kenjiro Kamijo* and Mamoru Oike*
* Tohoku University, 2-1-1, Katahim, Aoba-ku, Sendai, Miyagi 980-8577, JAPAN
^The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, JAPAN
Abstract.
Dynamic Molecular Collision (DMC) model is extended to calculate collisions of N2 and He. An intermolecular potential is obtained as the sum of the two potentials between each atom of a N2 molecule
and He. Lennard-Jones (12-6) potential is used as the interatomic potential and potential parameters are
determined by combination rule. N2~He collisions are simulated in many cases by Molecular Dynamics
(MD) method in order to construct the collision model between N2 and He. A collision cross section is
determined based on a diffusion coefficient and a probability density function of energy after collision is
determined by the MD method. In the present paper, moreover, this model is applied to simulations of the
free jet expansion of N2~He mixture by Direct Simulation Monte Carlo (DSMC) method and the flow field
is analyzed. Especially the number density and energy distributions at the axis of a free jet are analyzed in
detail.
INTRODUCTION
When a jet expands into a region of finite pressure, a complicated flow pattern develops. According to the
progress of techniques to make a molecular beam for thin films produced in semiconductor fabrication, it is
required to analyze the structure of this high-velocity free jet expansion. Seed gases are mixed with a large
amount of helium, He, in order to accelerate the seed gases. In this jet, the lighter molecules (He) collide with
the heavier molecules (seed gas) and properties of each gas (He and seed gas) become different from each other.
Especially in the case that the seed gas consists of diatomic molecules, temperatures of each degree of freedom
(translation, rotation and vibration) may become different. Therefore, it is important to analyze the free jet
of binary gas mixture which consists of He and diatomic molecules.
Owing to the rapid progress of computer, numerical simulations are effective to analyze such flows. However,
it is not valid to use the Navier-Stokes equation in order to simulate such flows because they are generated at low
pressure condition, where Knudsen number is large. Numerical methods which can treat molecular collision
must be used to simulate these rarefied gas flows. The Direct Simulation Monte Carlo (DSMC) method is
widely used to simulate these rarefied gas flows [1]. In this method, collisions of molecules are simulated in
a probabilistic manner. Therefore the accuracy of this method depends on the probability density function,
which is called collision model. Owing to the previous researches, the method can treat monatomic rarefied
gas flows accurately. For diatomic rarefied gas flows phenomenological models, for instance Larsen-Borgnakke
model [2], are mainly used for its simplicity. However, it is doubtful to simulate strong nonequilibrium flows,
for instance shock wave or free jet expansion, by the phenomenological models.
Dynamic Molecular Collision (DMC) model [3] is constructed and it is verified that diatomic rarefied gas
flows can be simulated accurately by this model. This model is based on collision dynamics and it is reported
that this model can simulate even nonequilibrium flows accurately. For this reason, it is possible to extend this
model to collisions of He and diatomic molecules in order to simulate free jet expansions of He and diatomic
gas mixture. In the present paper the DMC model is extended in order to simulate the N2~He collisions. The
VHS model [4] is used to simulate the He-He collisions. From these results, the effect of source and background
pressure on the structure of free jet expansion is analyzed. Especially, the effect of mass separation on each
temperature distribution and local rotational energy distribution on the axis of the jet is investigated. Moreover,
the effect of the ratio of number density of N2 to all molecules on energy distribution is also investigated.
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
645
COLLISION MODEL AND ITS APPLICATION
Extension of the DMC model for N2—He Collisions
In the present paper a large number of collisions of N2~He are simulated by the Molecular Dynamics (MD)
simulation in order to obtain a large body of database about collisions of N2~He. An intermolecular potential
between N2 and He is defined as the sum of the two potentials between He and nitrogen atom, N, which consists
of a nitrogen molecules. Lennard- Jones (12-6) potential mentioned below is assumed as an interatomic one
between N and He.
A
A,
JV^-HeV2
/^-HeA6!
0N-He = 4^N-He j (~^— ) ~ (~^r~ ) J
where r is the distance between N atom and He.
These parameters, crN_He and ^N-He are determined by combination rule. Using <JN_N = 3.17 x 10~10
m, eN_N = 6.52 x 1(T22 J [3] and crHe_He = 2.60 x 1CT10 m, eHe-He = L44 x 10~22 J I 5 ]> these Pa~
rameters are determined as ^N-He = ^'^ x 10~10 m' £N-He = 3-06 x 10~22 J. At initial condition, a N2
molecule is placed at (3<7N_jjemjje/ (^He + ?^N2) > ^mHe/ (mHe + m N 2 ) > ^) an(^ a ^e m°lecule is placed at
(— 30N_He m N 2 / (^He + m N 2 ) ' ~^ m N 2 / (mHe + m N 2 ) ' 0) where 6 is the impact parameter. The momentum
of N2 and He is consistent with each other during the simulation if the initial velocity is chosen so that the
initial momenta of the two molecules are the same. Considering this condition, the initial relative velocity,
V
N ' ^He> anc^ initial angular velocity of N2, u-^ , are given by
/m N
kTtr
o;N2?y = 0,
^N 2 ,z = 0,
(4)
where Ttr and Trot are the relative translational and rotational energy, respectively, m is the mass of molecule,
I is the momentum inertia and k is Boltzmann constant. The impact parameter b and the initial Euler angle
of N2 molecules are chosen so that a realistic collision probability is reproduced [3] . The detailed simulation
method about translational and rotational motion of molecules is referred in Ref. [3]. After the simulation
is completed, final relative velocity and angular velocity of molecule is recorded. The simulation condition
about Ttr and Trot are given in the same manner of Ref. [3] and the large body of data about N2-He collision
is obtained. The probability density function of energy after collision is shown in Fig. 1. Initial energy is
Tjje = 400 K, T^2 = 400 K and Trot = 400 K. In the present paper the amount of exchange of energy at
each degree of freedom have a relation, i.e., Ae tr jj e : Ae tr ^ 2 : Aerot^2 = 1 : 7 : —8. As shown in this
figure, the probability density function of energy after collision has a peak near the initial energy and decreases
exponentially.
Using these data, the DMC model for N2~He collision is constructed in a same manner for N2~N2 collision
[3]. A total collision cross section of N2-He is determined so that a diffusion coefficient obtained by this
cross section is consistent with the experimental results [6]. Therefore, the total collision cross section, <JT, is
obtained by
<JT =
3_______ /2fcT re
- |) nrefDref \ m12
where nre/ is a reference number density, mi2 is a reduced mass between N2 and He, Tref is a reference
temperature, Dref is a reference diffusion coefficient, a is a parameter of the VHS model [1] and g is a relative
velocity. These values are chosen to be nre/ = 2.69 x 1025 1/m3, mi2 = 3.50, Dref = 0.607 cm2/s, a = 11.45
and Tref = 273.15 K. [7]
646
0.05
v
•Bo 0.04
c
"rot
0.03
c/3
0.02
3 0.01
•§
! -o
20
o
20
40
60
80
Energy [-]
FIGURE 1. The probability density function of energy after collision.
VHS Model for He-He Collisions
Collision models between monatomic molecules are very simple than those of diatomic molecules and many
collision models are introduced [4], [8]. In the present paper, Variable Hard Sphere (VHS) model [4] is used as
the cross section of He. The parameters about He are also referred in Ref. [4]. In order to verify the validity of
this model energy distributions at an equilibrium condition is simulated using this model and the results are
compared with theoretical distributions. The results are shown in Fig. 2. A calculation condition is T = 300
K, P = 1.013 x 105 Pa and the number of molecules is N = 7337. In this figure the energy is reduced by 2kT.
As shown in this figure, the results are very consistent with the theoretical results and it can be said that this
model can calculate the binary collision well.
~ i i i i i i i r
I
I
I
I
I
I
: Maxwellian
Distribution
(DSMC)
(DSMC)
erot (DSMC)
§
3
1.5
C
£
£
i
</3
I
I
L
If 0 - 5
•i-H
•s
X)
£ o
0
0.5
1
1.5
2.5
Energy [-]
FIGURE 2. The energy distribution at equilibrium condition.
647
u
4
4
r Arithmetic
Domain
r Geometric
Domain
5d
Orifice
z Geometric Domain
z Arithmetic Domain
FIGURE 3. Simulation domain.
FREEJET EXPANSION OF N2-HE MIXTURE
Simulation Method
In simulations of free jet expansions a simulation domain is axisymmetric and simulations are carried out
in two-dimensions. Figure 3 shows the simulation domain. The radial direction is denoted by r and the axis
direction is denoted by z. An orifice is set at z = 0 and the diameter of the orifice is chosen to be d = 5
mm. The left and right side of the orifice is upstream and downstream domain, respectively. A pressure
and temperature at the upstream domain are P0 and T0, respectively and those of downstream domain are
POO and TOO, respectively. The size of upstream domain is ru x zu = d x d and that of downstream domain
is rd x rd = Sd x 20d. First, N2 and He molecules are placed at upstream and downstream domain at
respective equilibrium condition. The number of N2 and He molecules in upstream domain are obtained by
Nu N2 = sP0Vu/kT0 and Nu jje = (1 — s)P0Vu/kT0, respectively and those in downstream molecules are
Nd = sPooVd/kToQ and Nd = (1 — •s)P00Vd/^T00, respectively, where Vu and Vd is a volume of respective
domain and s is the ratio of number density of N2 to all molecules. Assuming the central angle of the domain,
#, these values are obtained by Vu = 0.5d3$ and Vd = 640d3#. The molecules are placed by rui = d\fR,
%ui = —dR, rdi = Sd^/R and zji = 20ctR, where subscript i is the value of molecule i and R is the random
number uniformly distributed in the range of (0,1). An initial velocity and angular velocity are given by adding
the thermal velocity and the flow velocity of each domain. Initial flow velocity in r direction is chosen to 0 and
that in z direction is chosen from the inlet Mach number obtained by
J_
(6)
M
where S and 5* is the area of upstream boundary and the orifice, respectively and 7 is the ratio of specific
heat. The motion of molecules in two-dimension is calculated in the way introduced by Bird [1]. In order to
simulate collisions of molecules, the simulation domain must be divided by a large number of small cells. The
size of the cell near the orifice must be very small while the one far from the orifice may be large because
the size of the cell must be less than a local mean free path. Therefore the simulation domain is divided in a
geometric manner near the orifice and it is divided in an arithmetic manner far from the orifice. In the present
paper, the downstream simulation domain in the range of 0 < r < 3d (hereafter called r geometric domain) is
divided in a geometric manner and the downstream simulation domain in the range of 3d < r < Sd (hereafter
called r arithmetic domain) is divided in an arithmetic manner. About z direction, the downstream simulation
domain in the range of 0 < z < 5d (hereafter caller z geometric domain) is divided in a geometric manner
648
and the downstream simulation domain in the range of 5d < z < 2Qd (hereafter caller z arithmetic domain) is
divided in an arithmetic manner. In the upstream domain, the domain is divided in an arithmetic manner in
both r and z direction.
During the simulation, molecules collide at a wall of the orifice. As a surface condition, diffuse reflection
condition at T = T0 is assumed. Free stream condition is used at the upstream and downstream boundaries.
The number of molecules which enter the simulation domain, however, is different at every position because
the flow velocity at the boundary is different as shown in Fig. 3, especially at the downstream z boundary
at low Knudsen number. In this paper the number of molecules which enter the upstream and downstream
domain is calculated at every cell. The flow velocity at the boundary is assumed as the same velocity in the
cell at the boundary and is calculated iteratively. Using the flow velocity at the boundary of ith cell, L^, the
number of molecules of each species which enter the simulation domain from ith boundary, is obtained by
where n is the number density of each species, Si is an area of the ^th boundary, (3 = y/m/2&T, m is a mass
of each molecule, K(x) = [exp(— x2) — ^/TTX {1 + erf(x)}] and erf(x) is an error function. The thermal velocity
and rotational energy of these molecules are obtained by T = T0 and T = T^ at upstream and downstream
boundary, respectively. Simulations are performed until the free jet reaches steady state and properties of the
flow in each cell are sampled and recorded. Simulations are performed with reduced units, i.e., length in A0,
energy in 2kT0 and mass in m^2
RESULTS AND DISCUSSIONS
First two simulations are performed by changing a source and a background pressure so that the ratio of
the source to the background pressure keeps constant. In the present paper the ratio is chosen to be 100.
One simulation condition is PQ = 100 Pa and P^ = 1 Pa (hereafter called case A) and the other simulation
condition is PQ = 1000 Pa and POO = 10 Pa (hereafter called case B), respectively. The source and background
temperature are chosen to be 400 K and the ratio of number density of N2 to all molecules, s, is chosen to be
0.1. The central angle, #, is chosen to be 2.0 x 10~n rad and 2.0 x 10~12 rad in case A and B, respectively. In
this condition the initial number of molecules are both 312,351. In case A the r geometric domain is divided
into 75 cells and the r arithmetic domain is divided into 25 cells. The z geometric domain is divided into 100
cells and the z arithmetic domain is divided into 50 cells. The upstream simulation domain is divided into
75 x 75 cells. In case B the r geometric domain is divided into 200 cells and the r arithmetic domain is divided
into 100 cells. The z geometric domain is divided into 300 cells and the z arithmetic domain is divided into
200 cells. The upstream simulation domain is divided into 300 x 300 cells. It is confirmed that all cell size is
less than the respective local mean free path. The time step, At is chosen to 1.0 and 0.1 in reduced units in
case A and B, respectively.
The ratio of number density of He to N2 at the axis of the jet is shown in Fig. 4. This figure shows that
the ratio of number density in the zone of silence is less than 9, which is the value at equilibrium condition. In
case B, the value increase beyond 9 and get the maximum near z = 30 mm. In a free jet, it is reported that
the mass separation effect leads to an enrichment of the heavier molecule (N2) on the axis of the free jet and
an enrichment of the lighter molecule (He) on the shock wave [9] . As shown in this figure a mach disk shock
exist near z = 30 mm and He molecules accumulate at the mach disk shock in case B while it doesn't exist in
case A. Moreover, the ratio of number density of He to N2 in case A is less than that in case B although the
ratio of the source to the background pressure is same. A diffusion velocity of binary mixture is proportional
to the diffusion coefficient obtained by
m He m N2
(p/n)2 16nroi2
where p = mjj e nfj e + ^N 2 n N 2 > n = nHe + n N 2 > mi2 = m He m N 2 /( m He + m N 2 )> °"12 is the cross section
between molecule He and N2 and Q^2' is the collision integral. As shown in Eq. (8), the diffusion coefficient
increases as the pressure decreases because the number density decreases. Therefore the mass separation effect
is greater in case A and the ratio of number density of He to N2 in case A is less than in case B.
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14
12
10
0
_L
_L
_L
_L
20
40
60
80
Z - direction [mm]
100
FIGURE 4. The ratio of number density of He to N2 at the axis of the free jet.
The z direction translational temperatures of N2 and He and rotational temperatures of N2 are shown in
the left side of Fig. 5 and the velocities of N2 and He are shown in the right side of Fig. 5. The upper figures
show the results in case A and the lower figures show the results in case B.
First, as shown in this figure, the temperatures of each degree of freedom in case A are more different from
each other than those in case B. This occurs because there aren't enough collision in case A. The detailed
explain is as follows. The translational motion of molecules can be distinguished into two parts, the one which
contribute to the translational temperature and the other which contribute to the the flow velocity. In a free jet
the molecules at higher pressure expand to lower pressure domain and the energy of molecules which contribute
to the translational temperature is transferred to the energy which contribute to the flow velocity. However,
these molecules collide with the molecules at lower pressure domain and the energy which contribute to the
flow velocity is transferred to the energy which contribute to the translational temperature. At the axis of
free jet the energy transfer between z direction flow velocity and z direction translational temperature mainly
occurs and therefore nonequilibrium condition is formed. If there are enough collision the nonequilibrium
condition relaxes and the temperature of respective degree of freedom becomes the same values. However, the
downstream mean free path in case A is about 6.3 mm and every molecule collide only several times in the
simulation domain while the molecules in case B collide 10 times more than that in case A. For this reason the
temperatures of each degree of freedom in case A is more different from each other than those in case B.
The rotational temperature in case B becomes much smaller than that in case A. The rotational energy of
N2 molecules can decrease only by collision at which translational energy is small. As shown in the right side
of Fig. 5, the gas expands in 0 < z < 5 ~ 10 mm and in this region translational temperature is small. As
mentioned above, the number of collision of each molecules in case A is too small to decrease the rotational
temperature. However, the number of collision of each molecules in case B is enough to make the rotational
temperature almost the same value of translational temperature. For this reason, the rotational temperature
decreases as the source to the background pressure is increase at the same ratio of source and background
pressure. The rotational energy distribution at the minimum rotational temperature point is shown in Fig. 6.
As shown in this figure, the distribution in case B apart from Maxwell distribution farther than that in case A
although the nonequilibrium between each temperature is stronger in case A than in case B.
As shown in Fig. 5, the z direction translational temperature overshoots beyond the downstream temperature
although it doesn't in case B. This is caused by the structure of free jet, especially slip line and mach disk.
At low Knudsen number the slip line is formed and therefore the flow velocity at the axis of the free jet is
faster at the end of simulation domain as shown in the right side of Fig. 5. This means that in case B almost
all the energy which contribute to the flow velocity is not transferred to the energy which contribute to the
translational temperature while in case A almost all the energy which contribute to the flow velocity is almost
transferred to the translational temperature. For this reason the translational temperature in case A is higher
than that in case B. Moreover, there are less He molecules at the axis of free jet at higher Knudsen number
owing to the mass separation effect mentioned above and therefore N2 molecules can obtain larger energy than
650
1200
WlOOO
<D
g 800
3
8 600
<U 400
H
Po=WO [Pa]:
200
0
20
_L
40
60
80
100
20
Z - Distance [mm]
40
60
80
100
Z - Distance [mm]
500
1500
20
40
60
80
20
Z - Distance [mm]
40
60
80
100
Z - Distance [mm]
FIGURE 5. The z direction temperatures, rotational temperatures and z direction velocities at the axis of the free
jet.
in case B and the translational temperature of N2 overshoots beyond the downstream temperature.
Next, the ratio of number density of N2 to all molecules is changed and the rotational temperature and
the flow velocity at the axis of free jet is investigated. The other simulation condition is the same as that of
P0 = 1000 Pa mentioned above. The rotational temperature and the flow velocity of N2 is shown in Fig. 7.
As shown in this figure, the rotatioal temperature distribution is almost consistent with each other although
the flow velocity is very different from each other.
CONCLUDING REMARKS
DMC model is extended to calculate collisions of N2-He. An intermolecular potential is obtained as the sum
of the two potentials between each atom of a N2 molecule and He. Lennard-Jones (12-6) potential is used as
the interatomic potential and the potential parameters are determined by combination rule. A total collision
cross section is determined based on a diffusion coefficient and the probability density function of energy after
collision is determined in the same manner of Ref. [3]. This model is applied to the simulation of the free jet
expansion of N2-He mixture. Simulations are performed by changing the source and background pressure under
the constraint that the ratio between them is constant. The results show that the temperature of each degree
of freedom is different from each other at higher Knudsen number (at lower pressure). However, the rotational
energy distribution at the minimum rotational temperature point is closer to the Maxwellian distribution at
higher Knudsen number. Moreover, rotational temperature can be smaller for the lower Knudsen number. At
the higher Knudsen number, the z direction translational temperature of N2 overshoots 2 times more than the
downstream temperature owing to the structure of free jet, especially slip line, mach disk and mass separation
effect. Moreover, it is found that the effect of the ratio of number density of N2 to all molecules on the rotational
651
:A B
:B
: Maxwellian
Distribution
i i i i i i i i i I i i i i i i i i i i i i i i ^i i i i
0
100
200
300
400
500
600
Rotational Energy [K]
FIGURE 6. The rotational energy distribution at the minimum rotational temperature point,
temperature distribution is small.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
Bird G.A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford: Clarendon Press, (1994).
Borgnakke C. and Larsen P.S., J. Comp. Phys. 18, 405-420 (1975)
Tokumasu T., Matsumoto Y., Phys. Fluids 11, 1907-1920 (1999).
Bird, G.A. Prog. Astro. Aero., 74, 239-255 (1981)
Reid, R.C., Prausnitz, J.M. and Sherwood, T.K., The Properties of Gases and Liquids, 3 ed., McGraw-Hill, New
York, (1977).
Nanbu K., J. Phys. Soc. Jpn. 59, 4331-4333 (1990).
Chapman, S. and Cowling, T.G., The Mathematical Theory of Non-Uniform Gases, Cambridge Univ. press, (1970)
Koura, K. and Matsumoto, H., Phys. Fluids A 3, 2459-2465 (1991)
Sebacher, D.I., J. Chem. Phys. 42, 1368-1372 (1965).
20
40
60
80
20
Z - Distance [mm]
40
60
80
100
Z - Distance [mm]
FIGURE 7. The rotational temperature and flow velocity at the respective ratio of number density of N2 to all
molecules at the axis of the free jet.
652