Effect of Wall Characteristics on the Behaviors of
Reflected Gas Molecules in a Thermal Problem
Kyoji Yamamoto, Hideki Takeuchi and Toru Hyakutake
Department of Mechanical Engineering, Faculty of Engineering,
Okayama University, Okayama 700- 8530, Japan
Abstract. The gas-wall interaction is investigated for the thermal problem between two walls whose temperatures are
different each other. The method of analysis is based on the molecular dynamics as well as DSMC method. The wall
consists of platinum molecules and the gas is taken to be argon. Three cases of wall surface characteristics are
considered: smooth surface, rough surface in molecular scale and the surface with adsorbates. The energy
accommodation coefficients and the jump coefficients are obtained. The discussions are also made on the velocity
distributions of the reflected molecules.
INTRODUCTION
The gas-wall interaction is one of the important problems in rarefied gas dynamics. The study is necessary to
specify the boundary condition for the reflected molecules at a solid surface. The diffuse reflection or the Maxwelltype boundary condition has been assumed generally. However, the diffuse reflection may not be valid at a very
clean or high temperature wall, in an ultra vacuum, in a very high speed flow etc. The accommodation coefficient
should be specified for the Maxwell-type boundary condition and the coefficient may be derived by the study on the
gas-wall interaction. Many studies have been made using the molecular dynamics method to analyze the behavior of
the reflected molecules. [1]-[4] One of the present authors has been studying the rarefied gas flow between two
parallel walls by molecular dynamics method combined by DSMC method to investigate effects of the gas-wall
interaction on the flow. [5]-[8]
In the present study, we apply the same method as in previous papers [5]-[8] to investigate the effect of wall
characteristics on the behavior of the reflected gas molecules. For this purpose, we take three different surface
structures, that is, a smooth surface, a rough surface and the surface with adsorbates. The wall consists of platinum
molecules and the gas is taken to be argon. The adsorbed molecule is assumed to be xenon. We analyze the thermal
problem between two parallel walls whose temperatures are different each other.
METHOD OF ANALYSIS
We consider a slightly rarefied gas between two walls. Let the distance between two walls be L and the reference
number density of the gas be n0. The temperature of the lower wall (TW) is taken to be 300K and the upper wall (TU)
has 360, 450 or 600K. We assume that the gas molecular is diffusely reflected at the upper wall and hence we
analyze the gas-wall interaction at the lower wall of 300K temperature. The Monte-Carlo method [9] is applied to
the analysis of gas motion between two walls, and the gas molecule is assumed to be a hard sphere for this analysis.
Then the mean free path of the molecule is given by lHS = ( 2π d 2 n0)-1. Here, d is the diameter of a molecule. The
Knudsen number defined by Kn = lHS /L is taken to be 0.2 in the present analysis. We apply Bird’s method [9] for the
interaction between gas molecules in the Monte-Carlo simulation. We take 80 cells between two walls. When a
molecule hits the lower wall surface in the Monte-Carlo simulation, we switch to the analysis based on the molecular
dynamics method at this point to trace the motion of the gas molecule, which is going to interact with molecules of
the wall. We use the standard molecular dynamics method.
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
1008
GAS-WALL INTERACTION
We assume that the wall consists of a thin platinum layer. The surface of the layer is set on the (111) plane. We
take six Pt molecules in the X-direction and another six molecules in the Z-direction on the surface. Four molecular
sheets are taken normal to the surface. This makes one segment of the platinum wall. The periodic condition is
applied to the direction parallel to the surface as is usually assumed in the molecular dynamics. This is a model of a
smooth surface. The rough surface has a protuberance on the smooth surface. The protuberance consists of double
deck. The lower layer next to the surface has six Pt molecules making a triangle and further one Pt molecule is
placed above these six molecules. Argon molecule is taken as the gas molecule in the present study. It is found that
the argon molecule is not physically adsorbed but is reflected from the platinum wall surfaces of 300K temperature
after interacting with wall molecules. On the other hand, if we take xenon molecule as the gas molecule, the
molecule is found to be physically adsorbed on the smooth platinum surface of 300K. Actually, nine xenon
molecules are adsorbed on one section of the smooth surface consisting of 6 × 6 Pt molecules parallel to the wall
surface. The physically adsorbed xenon molecules are moving around on the surface. We shall take this as the wall
surface with adsorbates.
As for the interaction potential between molecules, we take the Morse potential for Pt-Xe and Pt-Ar interactions
given by [2], [10]
φ Pt − Xe (r ) = ε P − X {exp[− 2σ P − X (r − r0 P − X )] − 2 exp[− σ P − X (r − r0 P − X )]},
(1)
ε P− X / k = 319.1K ,
o

o


σ P − X = 1.05 Α −1 , 
r0 P − X = 3.20 Α ,
φ Pt − Ar (r ) = ε P− A {exp[− 2σ P − A (r − r0 P− A )] − 1}2 ,
(2)
(3)
ε P − A / k = 134.7 K ,

, 

o

= 4.6 Α .
o
σ P − A = 1.6 Α
r0 P − A
−1
(4)
Further, we take the Lennard-Jones potential for Pt-Pt, Xe-Xe, and Ar-Ar interactions:
 σ 12  σ  6 
φ ij (r ) = 4ε   −    .
 r 
 r  
(5)
Here, r is the intermolecular distance and the numerical values involved in the potentials are listed in Table 1. [11] In
this Table, m is the mass of a molecule and k is the Boltzmann constant.
We shall use the Lennard-Jones potential between Xe and Ar whose interaction parameters ε and σ are given by
the following empirical combining laws [11]
σ Xe− Ar =
1
(σ Xe + σ Ar ) , ε Xe− Ar = ε Xeε Ar .
2
(6)
When a gas molecule hits the wall surface in the process of Monte-Carlo simulation, we put the molecule at a
distance of 8 σ Pt above the most upper Pt molecular layer, where σ Pt is the position of minimum Pt potential in Eq.
(5). The initial velocity of this molecule has been calculated by the Monte-Carlo simulation. Newton’s equations of
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TABLE 1. Values included in Lennard-Jones potential.
o
Pt
Xe
Ar
σ [Α]
ε / k [K]
2.523
4.10
3.405
3771.5
221
119.8
m [g]
3.24 × 10-22
2.18 × 10-22
6.63 × 10-23
motion of molecules are integrated with Verlet’s method. The time step of integration is taken to be 4.54 × 10-15 s.
When the gas molecule returns to the position which is more than 8 σ Pt away from the wall surface and is not
influenced by the wall molecules, we consider that the molecules have been completely reflected, and resume the
Monte-Carlo calculation there. The molecular dynamics method was applied to trace the motion of all molecules
impinging on the wall surface.
RESULTS
The calculation was made of the case that the lower wall temperature (TW) is 300K and the upper wall
temperature (TU) is 360K, 450K or 600K.
A. Temperature distributions
Figures 1~3 show the temperature distributions T/TW in case of TU =360K, 450K and 600K, respectively. Here TW
is the lower wall temperature taken to be 300K. The abscissa y is normalized by the mean free path lHS of the hard
sphere molecule, and y =0 is on the lower wall surface. In these figures, the filled circle means the distribution in
case of the smooth wall, the open circle is that of the rough wall and the triangle is for the adsorbed wall. It will be
seen from these figures that the adsorbed wall shows lower temperature distribution in all cases. On the other hand,
the smooth and rough walls show higher temperature distributions and these are almost the same when the wall
temperature difference is large (TU =450 and 600K). However, at small temperature difference (TU =360K), the
distribution of the rough wall is higher than that of smooth wall.
We can draw a line representing the asymptotic temperature distribution which is valid outside the Knudsen layer.
The Knudsen layer is of the order of the mean free path and it diminishes rapidly from the wall surface. We may
consider that the Knudsen layer extends up to y ≈ 0.6 from the wall surface. Then, the asymptotic line is obtained by
the least squares method using the data of 0.6< y <4.5, and is shown as a solid line in Figs.1~3.
1.5
1.2
1.4
T/Tw
T/T w
1.3
1.1
1.2
TU =360K
TU =450K
smooth
rough
Xe adsorbed
1
0
2
smooth
rough
Xe adsorbed
1.1
1
0
4
y
2
4
y
FIGURE 1. Temperature distribution (TU =360K).
FIGURE 2. Temperature distribution (TU =450K)
1010
2
1.8
T/T w
1.6
1.4
TU =600K
smooth
rough
Xe adsorbed
1.2
1
0
4
y
FIGURE 3. Temperature distribution (TU =600K).
αE
TU =360K
TU =450K
TU =600K
Experiment
0.49
0.43
0.41
2
TABLE 2. Accommodation and Jump Coefficients.
Smooth wall
Rough wall
α E //
σ
0.29
0.22
0.19
0.70
0.62
0.60
N
αE
α E //
σ
0.37
0.41
0.39
α E =0.55
0.25
0.29
0.25
0.49
0.50
0.51
β
3.4
3.6
4.0
N
β
αE
5.0
4.4
3.7
0.75
0.71
0.68
Adsorbed wall
α E //
0.66
0.62
0.59
σ
N
0.78
0.77
0.74
α E =0.85
β
1.9
1.9
2.0
B. Accommodation coefficients and jump coefficients
The accommodation coefficient α E of the kinetic energy of molecules is defined by
αE =
Ei − E r
,
Ei − E w
(7)
where Ei and Er are the kinetic energies of the impinging and reflecting molecules, respectively. Ew is the kinetic
energy of the molecule to be reflected diffusely with the wall temperature. The value α E calculated is shown in
Table 2. We define the accommodation coefficient α E // of the kinetic energy of molecules parallel to the wall
surface:
α E // =
Ei // − Er //
,
Ei // − Ew //
(8)
where E// means the parallel component of the kinetic energy to the wall surface. The value of α E // is also shown in
Table 2. It can be seen that the accommodation coefficient α E for the adsorbed wall is the largest among three types
of the wall. The accommodation coefficients for the smooth and rough walls are almost the same, but the
accommodation coefficient of the rough wall is a little bit smaller than that of the smooth wall. Especially, it is so at
lower temperature. Table 2 also shows the accommodation coefficients obtained by experiments [12], [13] on the
bottom of the Table. The left-hand value (0.55) was obtained by first heating the test specimen of a platinum wire
over 1000 oC and then cooling it down to the room temperature. Therefore, the platinum surface may not be so much
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contaminated by other molecules. Another experimental value (0.85) at the bottom right corner of Table 2 was
obtained for the platinum wire without special treatment. Therefore, the platinum surface may be contaminated with
other molecules and is compared with the present calculation for the adsorbed wall although the adsorbed molecules
in the experiment may not be xenon. It will be seen that the experimental values are well compared with the present
values. Table 2 also contains the normal momentum accommodation coefficient σ N defined by
σN =
pi − p r
,
pi − p w
(9)
where p is the normal momentum and the subscripts i, r, w mean the values of incident, reflected and diffusely
reflected molecule, respectively.
We are analyzing the case of Kn =0.2, which is considered to be slightly rarefied gas. In this case, the flow can
be treated as the slip flow. That is, the main region of the flow can be considered as a continuum and the rarefaction
appears near the wall surface whose distance is of the order of the mean free path. This region is called the Knudsen
layer. Outside the Knudsen layer, we can analyze the flow by the so-called slip boundary condition. We shall
calculate the slip (jump) coefficients using the asymptotic temperature distributions obtained in the present
calculation and shown as solid lines in Figs. 1~3. The jump boundary condition at the wall surface is written as
 ∂T 
T (0 ) − Tw = β lt 
 ,
 ∂Y  0
(10)
where β is the jump coefficient and lt is the mean free path defined by the thermal conductivity κ as
lt =
4κ
5 p0
Tw
2R
, p0 = ρ 0 RTw .
(11)
Here R is the gas constant and ρ 0 is the reference density. It is shown that lt is related to lHS by lHS =16 lt /(15 π ).
[9] Using the temperature distributions shown as solid lines in Fig.1~3 and Eq. (10), we can calculate the jump
coefficient β . The results are shown in Table 2.
C. Potential at the rough wall
At first glance, we may consider that the argon molecule is strongly accommodated at a rough surface compared
with a smooth surface. However, we have seen that the argon molecule is not so well affected at the rough wall,
especially, when temperature difference is small. We here explain why the argon molecule has weak accommodation
at the rough wall. For this purpose, we depict the potential distribution between the rough platinum surface and an
argon molecule. Figure 4 shows the present model of the rough surface. There are seven platinum molecules of
double deck on the smooth surface. Figure 5 shows the potential distribution of the surface to an argon molecule at a
height of 1.8 σ Pt from the top platinum molecule consisting the protuberance. The potential is negative at this level
and has a deep negative well near the protuberance. Therefore, an argon molecule gets an attractive force to the
protuberance and approaches to it. Figure 6 shows the potential distribution at the height of 0.9 σ Pt from the
protuberance. The potential takes large positive value, especially, near the protuberance. The argon molecule will get
a strong repulsive force here and may be reflected from the wall surface without having much interaction with the
wall. That is, the argon molecule with small kinetic energy coming to the wall surface first moves towards the
protuberance by the attractive force and then it is repelled by the strong repulsive force caused by the protuberance.
It does not have much time to interact with the wall surface. If there is no protuberance, the potential is still small
and negative. The argon molecule comes close to the wall surface and will have much chance to interact with the
surface molecules and to be well accommodated with the wall.
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FIGURE 4. Model of the rough surface.
FIGURE 5.
Interaction potential distribution (1.8 σ Pt).
FIGURE 6. Interaction potential distribution (0.9 σ Pt).
D. Molecular velocity distribution function at the wall surface
We next discuss on the molecular velocity distribution function at the wall surface. We show only the result
when the upper wall temperature TU is 600K, but the similar results have been found in other cases. Figure 7 shows
the velocity distribution function of molecules for the smooth wall; Figs.7 (a) and 7 (b) are the distributions parallel
and normal to the surface, respectively. The open circle means the distribution of the reflected molecule and the
filled circle is that of the incident molecule obtained in the present calculation. Here and below, the molecular
velocity is normalized by the most probable speed Cm = 2 RTw . The curves in Fig.7 (a) are drawn from the
following Maxwell-type boundary condition for the reflected molecule parallel to the wall:
f X+ = α f w + (1 − α ) f X−
(VY
> 0) ,
(
)
f w = exp − V X2 / π
,
(12)
where α is the accommodation coefficient. The value of α is taken to be α E // (=0.19) for the solid curve and α E
(=0.41) for the dashed curve. The distribution f X− for the incident molecule is taken from the present calculation
(filled circle). It will be seen that the solid curve agrees quite well with the distribution of the reflected molecule
shown by open circles. The agreement between the dashed curve and open circles is also not so bad. The curves in
Fig.7 (b) are drawn from the following Maxwell-type boundary condition for the reflected molecule normal to the
wall:
f Y+ = σ f w + (1 − σ ) fY−
(VY
> 0) ,
(
f w = 2VY exp − VY2
)
,
(13)
where σ is accommodation coefficient and f Y− is the distribution function for the incident molecule obtained in the
present calculation. If we take σ = σ N (=0.60), we can draw the solid curve. We obtain the dashed curve by taking
σ = α E (=0.41). We see that the solid curve has a better agreement with open circles than the dashed curve. We may
say that the Maxwell-type boundary condition produces the good velocity distribution of the reflected molecule if
we take a proper accommodation coefficient. One preferable choice of the accommodation coefficient in the thermal
problem may be: (1) we use α E // for the parallel direction of velocity of the reflected molecule and (2) the
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0.6
1
T U =600K
T U =600K
smooth wall
in
out
–––– Maxwell–type
σ N =0.60
– – – Maxwell–type
α E =0.41
smooth wall
in
out
–––– Maxwell–type
α E // =0.19
– – – Maxwell–type
α E =0.41
fY
0.4
fX
0.5
0.2
0
0
–2
0
2
VX
FIGURE 7. (a) Molecular velocity distribution along the wall.
(smooth wall)
0.6
–2
0
1
T U =600K
T U =600K
rough wall
in
out
–––– Maxwell–type
σ N =0.51
– – – Maxwell–type
α E =0.39
rough wall
in
out
–––– Maxwell–type
α E // =0.25
– – – Maxwell–type
α E =0.39
fY
0.4
2
VY
(b) Molecular velocity distribution normal to the wall.
(smooth wall)
fX
0.5
0.2
0
0
–2
0
2
VX
FIGURE 8. (a) Molecular velocity distribution along the wall.
(rough wall)
–2
0
2
VY
(b) Molecular velocity distribution normal to the wall.
(rough wall)
accommodation coefficient of the normal momentum ( σ N ) is used for the normal direction of velocity of the
reflected molecule. The use of the energy accommodation coefficient ( α E ) is also not so bad for the reflected
molecule as is usually taken in the Maxwell-type boundary condition.
Figure 8 shows the molecular velocity distributions at the rough wall surface; Fig.8 (a) is the parallel distribution
to and Fig.8 (b) is the normal distribution to the wall. The solid and dashed curves in Fig.8 (a) are drawn from the
Maxwell-type boundary condition (Eq. (12)) with α = α E // (=0.25) and α = α E (=0.39), respectively. The solid and
dashed curves in Fig.8 (b) and drawn form the Maxwell-type boundary condition (Eq. (13)) with σ = σ N (=0.51) and
σ = α E (=0.39), respectively. In this case, it will be seen that the solid and dashed curves agree well with open
circles.
Figure 9 shows the molecular velocity distributions at the adsorbed wall surface; Fig.9 (a) shows the distribution
parallel to the wall, while Fig.9 (b) is the distribution normal to the wall. The solid and dashed curves in Fig.9 (a) are
drawn from Eq. (12) with α = α E // (=0.59) and α = α E (=0.68), respectively. The solid and dashed curves in Fig.9
(b) are drawn form Eq. (13) with σ = σ N (=0.74) and σ = α E (=0.68), respectively. It is seen that the solid and
dashed curves are almost the same and agree well with open circles.
1014
0.6
1
T U =600K
T U =600K
Xe adsorbed wall
in
out
–––– Maxwell–type
σ N =0.74
– – – Maxwell–type
α E =0.68
Xe adsorbed wall
in
out
–––– Maxwell–type
α E // =0.59
– – – Maxwell–type
α E =0.68
fY
0.4
fX
0.5
0.2
0
0
–2
0
2
VX
FIGURE 9. (a) Molecular velocity distribution along the wall.
(adsorbed wall)
–2
0
2
VY
(b) Molecular velocity distribution normal to the wall.
(adsorbed wall)
CONCLUSIONS
We have analyzed the thermal problem between two walls by DSMC method combined with the molecular
dynamics method for the gas-wall interaction. Three cases of the wall surface are considered, i.e., the smooth surface,
the rough surface in molecular scale and the surface with adsorbates. We took the argon gas molecule and the
platinum wall whose wall temperature is 300K. The following results are obtained:
(1)
(2)
(3)
The accommodation coefficients are 0.49, 0.37 and 0.75 for the smooth, rough and adsorbed walls,
respectively, when the temperature difference between two walls is 60K.
The rough wall has poor accommodation compared with the smooth wall at small temperature difference.
The velocity distribution of the reflected molecules is very close to that of the Maxwell-type boundary
condition if we take proper accommodation coefficients.
REFERENCES
1. Wachman, H.Y., Greber, I. and Kass, G., in Rarefied Gas Dynamics, Progress in Astronautics and Aeronautics, ed. Shizgal,
B.D. and Weaver, D.P., Vol, 158, 479-493 (1994).
2. Matsui, J. and Matsumoto, Y., ibid, 515-524 (1994).
3. Yamanish, N., Matsumoto, Y., in Rarefied Gas Dynamics, ed. Brun, R. et al, 421-428 (1999).
4. Lahaye, R.J.W.E., Stolte, S., Holloway, S. and Kleyn, A.W., J. Chem. Phys., 104, 8301-8311 (1996).
5. Yamamoto, K. and Yamashita, K., in Rarefied Gas Dynamics, ed. Shen, C., 375-380 (1997).
6. Yamamoto, K., Memoirs Fac.Engng. Okayama Univ., Okayama, Japan, 33, 9-17 (1999).
7. Yamamoto, K. and Shiota, T., in Rarefied Gas Dynamics, ed. Brun, R. et al, Vol.1, 397-404 (1999).
8. Yamamoto, K., in Rarefied Gas Dynamics, ed. Bartel, J. and Gallis, A., 339-346 (2001).
9. Bird, G.A., Molecular Gas Dynamics, Oxford Clarendon Press, 73, 118 (1976).
10. Head-Gordon, M., Tully, J.C. et al, J. Chem. Phys., 94, 1516-1527 (1991).
11. Hirschfelder, J.O., Curtiss, C.F. and Bird, R.B., Molecular Theory of Gases and Liquids., Wiley, 165,168 (1964).
12. Mann, W.B, Proc. R. Soc. London, A 146, 776-791 (1934).
13. Admur, I.M. et al, J. Chem. Phys. 12, 159-166 (1944).
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