Direct simulation Monte Carlo method for gascluster ion beam technology

Z. Insepov *, I. Yamada

LASTI, Himeji Institute of Technology, 3-1-2 Kouto, Kamigori, Ako, Hyogo 678-1205, Japan

Abstract

A direct simulation Monte Carlo method has been developed and applied for the simulation of a supersonic Ar gas

expansion through a converging–diverging nozzle, with the stagnation pressures of P0 ¼ 0:1–10 atm, at various tem-

peratures. A body-fitted coordinate system has been developed that allows modeling nozzles of arbitrary shape. A wide

selection of nozzle sizes, apex angles, with diffuse and specular atomic reflection laws from the nozzle walls, has been

studied. The results of nozzle simulation were used to obtain a scaling law

P0T19=80 daLb

n ¼ const:

for the constant mean cluster sizes that are formed in conical nozzles. The Hagena�s formula, valid for the conical

nozzles with a constant length, has further been extended to the conical nozzles with variable lengths, based on our

simulation results.

� 2002 Elsevier Science B.V. All rights reserved.

PACS: 02.70.Uu; 47.60.+I; 36.40; 82.30.Nr

Keywords: Direct simulation Monte Carlo; Body-fitted coordinate; Supersonic nozzle; Cluster size

1. Introduction

Cluster formation in a supersonic nozzle flow is

a challenging problem for both theory [1–7] and

experiment [8–16]. The gasdynamics theory basedon Navier–Stokes equations cannot adequately

treat this problem due to a highly non-equilibrium

nature of the cluster formation phenomenon.

The direct simulation Monte Carlo (DSMC)

method [17–23] is a probabilistic method, and the

number of simulating ‘‘particles’’ in a DSMC-

model could be many orders of magnitude less

than the real molecules� number. DSMC couldtreat supersonic expansion of gases with inherent

non-equilibrium processes, e.g. with chemical re-

action [20], and it can take into account the for-

mation of small clusters [21,22].

The aim of this work is to study supersonic gas

expansion in the nozzles of arbitrary shape and

to predict average cluster sizes by comparing our

simulation results with available experimentaldata.

*Corresponding author. Tel.: +81-791-58-0412; fax.: +81-

791-58-0242.

E-mail address: insepov@lasti.himeji-tech.ac.jp (Z. Inse-

pov).

0168-583X/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0168-583X(02)01870-0

Nuclear Instruments and Methods in Physics Research B 202 (2003) 283–288

www.elsevier.com/locate/nimb

2. Body-fitted coordinate system

A new three-dimensional body-fitted coordinate(BFC) system has been developed and used within

the DSMC method, which is applied to study an

axially symmetric supersonic gas flow inside the

conical nozzles. The new method could also pre-

dict various flow parameters like flow density,

temperature, pressure, and in principle, cluster size

distribution function for different radial distances

from the nozzle central axis.The BFC method is a well-known method of

gasdynamics and it was first proposed for studying

aerodynamics external flows. It has also been

shown to successfully characterize a complicated

gasdynamics behavior including shock waves

[18,19]. BFC is also known as the boundary-con-

forming method [23] and has not been used for

supersonic nozzle simulation by DSMC before.In DSMC, the physical space must be divided

into small cells. Since the discretization of a nozzle

volume is a complex problem, the BFC method

transforms the physical space into a rectangular

space, which is easily divided into uniform cells.

The DSMC method consists of two processes: (a)

the collisions between the particles in a cell are

counted (step 1), (b) the particles are moved fromprevious position to a new one, according to the

equations of motion (step 2). For step 1, all the

particles within the cell must be identified and

BFC accelerates this step significantly. For step 2,

the BFC method uses the same equations of mo-

tion as for real space, with the corrections due to

the coordinate system transformations [19].

The cell sizes of dxc � 10�5–10�4 m were usedfor our simulations, depending on the total nozzle

length. The equations of motion were numerically

solved with the time step dt � 1–10 ns that was

obtained by a compromise between Eq. (1) and a

reasonable length of computation on a PC.

dt � 0:01� dxc=ðU þ CÞ: ð1Þ

Here, dxc is the cell size, (U þ C) is the total

particle velocity, C is the sound speed. The total

computing time was chosen in such a way that the

simulated particles were able to pass through the

whole nozzle many times [17].

Diffusive, with full accommodation, and spec-

ular boundary conditions [17] were applied to-

gether with BFC, as it was suggested in [19].

For most simulations at low pressure, the totalcomputing time was of about a few days. The

computing time has trend to increase significantly

with the gas pressure. Therefore, a great deal

of our results was obtained at pressures below

5 atm.

Fig. 1 shows the geometry of a conical nozzle

that was modeled in this work. The shape of these

devices was chosen to be similar to that usedat Kyoto University (Japan) and for Karlsruhe

(Germany) groups� experiments, in order to com-

pare simulation results with experiment. In this

figure, P0 and T0 are the pressure and the temper-

ature in the stagnation chamber. The two hori-

zontal arrows show the direction and the relative

magnitude of the flow velocity u. The nozzle couldbe characterized by the following parameters: dth –the throat diameter, D – the exit diameter, and h is

the apex angle for the conical type of a nozzle, and

L is the total nozzle length.

The studied nozzles had similar entrance sec-

tions characterized by the length of the inlet

cone section and the length of the cylindrical

throat. These parameters along with the throat

diameter were similar to ones in Hagena�s work[8,9].

The conical apex angles and the lengths of exit

nozzle section were used as variable parameters. In

total, more than 20 nozzles with different lengths

and exit apex angles were simulated for different

stagnation pressures from 0.1 to 10 atm, at room

temperature.

Fig. 1. A typical conical nozzle is shown that has the following

parameters [8,9]: d ¼ 0:14–0:16 mm, D ¼ 1–3 mm, L ¼ 24:5

mm. The apex angles were varied between 5� and 15�.

284 Z. Insepov, I. Yamada / Nucl. Instr. and Meth. in Phys. Res. B 202 (2003) 283–288

3. Simulation results

Figs. 2–5 show the results for a typical nozzlewith a small apex angle. The independent variable

is the x-axis along the flow of a simulated particle

from the central x-axis. The following flow char-

acteristics are shown here: the flow velocity uðxÞ in

m/s (Fig. 2), the flow temperature T ðxÞ in K (Fig.

3), the flow density qðxÞ in kg/m3 (Fig. 4), and the

flow pressure PðxÞ in Torr (Fig. 5), along the x-axis(that is given in reduced units), after 2� 105 time

steps. In these plots, the solid line (1) shows the

dependence of the appropriate variable along theconical wall, and the circles (2) are the variables

calculated along the central x-axis.

Fig. 2. U–velocity plot for the conical nozzle with the para-

meter from [8]. Line 1 shows the variable for the cells near the

nozzle wall, line 2 – along the centerline of the flow.

Fig. 3. Temperature plot for the same nozzle as in Fig. 2. Lines

1 and 2 have the same meaning as in Fig. 2.

Fig. 4. Density plot for the same nozzle as in Figs. 2 and 3.

Lines 1 and 2 have the same meaning as in Fig. 2.

Fig. 5. Pressure plot for the same nozzle as in Figs. 2–5. Lines 1

and 2 have the same meaning as in Fig. 2.

Z. Insepov, I. Yamada / Nucl. Instr. and Meth. in Phys. Res. B 202 (2003) 283–288 285

3.1. Hagena’s analysis

By analyzing a large number of experimentswith various gases, carried out at Karlsruhe Nu-

clear Center for many years, Hagena and co-

workers [8,9] have found a few useful scaling laws

for the flow parameters. As they have suggested,

the mean cluster size remains almost constant with

increasing the stagnation temperature T0 if the

stagnation pressure P0 and the nozzle throat dia-

meter deq are related to each others along with therelation (2):

P0T19=80 d0:8

eq ¼ const:;

deq ¼ dth= tan h=2ð Þ:ð2Þ

Here, P0 is the stagnation pressure, T0 is the

stagnation temperature, deq is the equivalent sonicthroat diameter [10], and dth is the throat diameter

for a conical nozzle. An important feature of this

relation is that the power exponents entering into

(2) could be obtained from the temperature gra-

dients dT=dx, or the transit times, dt ¼ dx=v0 alongthe x-axis centerline for a nozzle.

The Hagena�s formula (2) is a further develop-

ment of the two fundamental relations for gasexpansion through a nozzle: (a) isentropic relation

P0Tc=ð1�cÞ0 ¼ const.; (b) the line with equal bimo-

lecular processes, P0Tð1:5c�1Þ=ð1�cÞ0 ¼ const. Here, c is

the ratio of specific heats and P0, T0 are the sourcegas pressure and temperature. Formula (2) uses

the fact that both cluster size and cluster mass flux

density increase with increasing the source pres-

sure P0 and decreasing the source temperature T0.Such changes in the stagnation conditions would

lead to the crossing the saturation line at higher

densities, thus favoring the condensation processes

[8,9].

In the present work, the DSMC method has

been applied to calculate temperature gradients for

various conical nozzles. The obtained gradients

were used to generalize the Hagena�s formula tothe nozzles with variable exit lengths by adding a

power-low dependence of the mean cluster size on

the nozzle lengths, Ln The new formula (3) relates

flow parameters and fixes the mean cluster size for

different nozzle exit section lengths:

P0T19=80 daLb

n ¼ const: ð3ÞFig. 6 shows the flow temperature along the

x-axis centerline for four conical nozzles with

various apex angles, obtained at the same initialstagnation parameters for Ar. The throat exit

point is located at the initial point of the x-axis(zero point). As it is known from experiments [8–

16], condensation of supersaturated gases occurs

in the close vicinity and downstream the nozzle

throat. Therefore, the fitting procedure has been

applied within the interval from 1 to 3 throat dia-

meters along the x-axis.The power exponent a in (3) can easily be ob-

tained from the calculated temperature gradients

at various equivalent throat diameters (or apex

angles) drawn in a log–log scale. Comparison be-

tween the theory and two experimental data points

measured by Kyoto and Karlsruhe groups is given

as the first line in Table 1.

The same fitting procedure has been carried outfor the nozzles with variable nozzle lengths and

Fig. 6. Fitting the temperature along the centreline for the

nozzle with variable apex angle.

Table 1

Comparison for parameters fit in formula (3): Kyoto University

experiment [11], Hagena�s work [8,9], and this work

Groups

Fitting

parameters

Kyoto

University [11]

Hagena [8,9] DSMC

[this work]

a 0.8–0.97 1.43 1.5

b 0.2–0.4 – 0.3

286 Z. Insepov, I. Yamada / Nucl. Instr. and Meth. in Phys. Res. B 202 (2003) 283–288

apex angle for the narrow exit cone. Fig. 7(a)–(d)

show the fitting for the temperature gradient in a

nozzle and the results of all fittings are given on the

second line of Table 1.

3.2. Average cluster size and ion current

As formula (3) fixes the ratio for the gasdy-namics variables, it can predict the average cluster

sizes and the ion currents only if one data point is

known from experiment. The above statement

could easily be proved by a gasdynamics analysis

that relates the average cluster size hNi to the right

site of (3).

Formula (3) has been used in this work to

predict the average cluster sizes and ion currentsbased on one experimental data point at 4000 Torr

for the nozzle that was supplied by Kyoto Uni-

versity group [11].

Fig. 8 shows an excellent comparison of our

prediction for the average cluster size given by

formula (3) (solid lines) and the Kyoto University

Fig. 8. Prediction for average cluster sizes by formula (3) are

given as solid lines, along with the level numbers. Circles cor-

respond to the experimental data, from left to right [11]:

hNi ¼ 2� 104, 2:5� 104, 2:7� 104 and 3:3� 104. Squares,

from bottom to top: 3:3� 104, 2:2� 104, 1:5� 104 [11]. Apex

angles and nozzle lengths are given in reduced units.

Fig. 7. Fitting the temperature along the centerline for the variable lengths of the nozzle exit sections.

Z. Insepov, I. Yamada / Nucl. Instr. and Meth. in Phys. Res. B 202 (2003) 283–288 287

experimental data points [11] (circles). Therefore,

we would suggest using it for predicting average

cluster sizes.

A significant aspect of the Hagena�s analysis isthe experimental finding [10] that any conical

nozzle can be characterized by a single parameter,

the throat diameter, of an equivalent sonic nozzle.

Unfortunately, there is no such simple represen-

tation for the nozzles other than the conical ones.

Therefore, the results of this work could not be

easily transferable to Laval nozzles.

4. Conclusions

In this study, the DSMC method has been

employed for simulation of a supersonic gas ex-

pansion and a new three-dimensional BFC system

has been developed for an Ar gas, with pressures at

stagnation chamber up to 10 atm, at room tem-perature. The developed method has been applied

to study an axially symmetric supersonic gas flow

inside the conical nozzles of different lengths and

apex angles, at low and intermediate Ar gas pres-

sures.

Flow density, pressure, flow velocity, tempera-

ture and flow atomic flux were calculated by

DSMC along the central x-axis for nozzles withvarious lengths. The DSMC data for the temper-

ature gradient (transit times) were used to find

power exponents in the scaling low for the flows

and the results of this analysis were in good

agreement with Kyoto group�s and Hagena�s ex-

perimental results.

The Hagena�s analysis for the constant mean

cluster size has been generalized for the conicalnozzles with variable lengths of the exit sections.

Acknowledgements

One of the authors (Z.I.) acknowledges valu-

able discussions on the fundamentals of the

DSMC method by Prof. Donald Baganoff of

Stanford University. This work was supported by

the NEDO project (Japan).

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