Numerical Study of Molecular Scattering on the Thermal
Equilibrium Surface with Adsorbates
Jun Matsui
Department of Systems Design, Division of Systems Research, Faculty of Engineering,
Yokohama National University, 79-5 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, JAPAN
Abstract The process of gas molecules, which stick and escape over a solid surface, is analyzed by numerical
simulation. The history of energy of gas molecule sticking on clean surface is compared with that on the surface
with adsorbed gas molecules. The energy and velocity distribution when gas molecule escape from surface is
almost same as the Boltzmann distribution, but the average of them differs from the wall temperature. Some group
of these escaping molecules are not in equilibrium with the surface. The length of sticking time is important in the
process of escaping from the clean surface. Also, the molecule with which the escaped molecule contact at last is
the important parameter of those differences on the adsorbed surface.
1. INTRODUCTION
The adsorbed molecules have a strong effect on the scattering process of the gas molecule on the surface. Though
many studies have done on clean surface experimentally or numerically[1,2,3,4], this effect of adsorbates is not
studied well [5,6].
The author has studied such scattering process on the surface with adsorbed[7], and analyzed the amount of
transferred energy between gas and surface at the first collision. However, while the gas molecule is sticking or
when it escapes from the surface, the microscopic process is not understood well.
In this study, the motion of gas and surface molecules is simulated numerically under the condition of thermal
equilibrium. At first, the scattering process of gas molecule on the clean surface is analyzed. And next, the process
on the 'dirty' surface on which adsorbed gas molecules exist is discussed. Xenon is used as gas and adsorbed
molecules and Platinum as surface molecules.
2. NUMERICAL METHOD
The molecular motion is simulated with the classical molecular dynamics method. The classical Newton's
equation is integrated numerically in this method. For accurate integration, we use the Bulirsch-Store scheme[8].
The Lennard-Jones inverse 6-12 power potential,
-12
-12
f (r) = 4e ( r / s ) - ( r / s )
(1)
[
]
, is used as the interaction potential between molecules. The parameters of potential between molecules are shown
in Table 1. In this study Xenon is the gas molecule and Platinum is used as the solid wall molecule. The standard
value of the energy, ER, is† 1.265x10-18[J], and that of the time, tR, is 1.284x10-13[s]. The standard length, LR, is
2.85x10-10[m].
In the simulation of 'dirty' surface, there are sticking gas molecules on the surface, we use 2 blocks of solid
molecules. At first, the surface molecules and gas molecules are put as Fig. 1. There are two solid walls that make a
closed space for gas molecules. Each wall exists at the area of z<0 and z>ZW. This distance ZW is 10 LR or 20 LR.
Each wall consists of 6 layers, which have 144 molecules each. These molecules are located to make a (111) surface.
A periodic condition is applied to both x and y direction, so that these molecules form an infinite surface virtually.
Some gas molecules are put between these two walls with some initial velocity.
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
957
Then, the motion of all molecules is simulated under the control keeping the sum of kinetic energy. This control
makes the temperature of the system at constant.
After about 10000 steps of calculation, when this system is in the thermal equilibrium status, the control is
stopped. At this time, some of gas molecules are sticking on the surface, and other gas molecules are flying in the
space between walls. The position, kinetic energy, and potential energy of gas molecule are recorded during the
simulation. The equilibrium temperature of the wall is set 600[K], and the number of gas molecule is 1, 48, or 150.
In the simulation of clean surface, we use one wall of solid molecule. The upper wall is erased, and only one gas
molecule starts to collide to the solid wall. The initial gas speed is same as the gas of temperature is 300[K], and
the initial direction of velocity was chosen randomly.
TABLE 1. Parameters for Intermolecular Potentials
Molecules
s [Angstrom]
e [eV]
Xe - Pt
Pt - Pt
2.85
2.523
0.0275
0.325
Xe - Xe
3.886
0.0242
FIGURE 1. Initial Position of Surface Molecules and Gas Molecules.
3. RESULTS ON THE CLEAN SURFACE
3.1 Scattering Process
Figure 2 shows two examples of scattering process of one Xenon gas molecule on the Platinum surface. In this
case, the surface is completely clean. The abscissa of the graph is time, and the z component of position of gas
molecule is shown in upper part of the figure. The potential energy, kinetic energy, and the sum of these two, are
shown in lower part.
6
2
potential energy
total energy
kinetic energy
'Escaped'
E/ER
0.02
'Escaped'
0.00
0.00
-0.02
'Excited'
-0.02
-0.04
15
tC (contacting time)
4
0
0.04
potential energy
kinetic energy
total energy
E/ER
0.02
z/LR
tC
z/LR
5
4
3
2
1
0
0.04
-0.04
20
25
30
35
20
t/tR
40
60
80
100
120
140
t/tR
(a) Simple Collision on the Wall
(b) Process of Sticking and Escaping
FIGURE 2. Example of Process of Colliding, Sticking, and Escaping of the Gas Molecule on the Clean Surface.
958
In the case of Fig. 2(a), the gas molecule collides once on the surface, and it escapes soon. The contacting time
of this process is also shown as tC. This period, tC, is defined by the time between two peaks of kinetic energy. One
peak is the beginning of collision, and the other is escaping. Though the total energy becomes very small after
collision in Fig. 2(a), but it is still positive. So this molecule is judged that it is not sticking on the surface.
In this study, the time when a gas molecule escape is defined by both energy and position. When the total energy
of gas molecule is positive, and when its z position is more than 2.0 LR, the gas molecule is recognized as the
escaped.
In the case of Fig. 2(b), the gas molecule sticks to the surface at t=35 tR, and its total energy is about -0.03 ER
while sticking. When t is about 80 tR, the molecule gets some energy from surface molecules, and its total energy
becomes near -0.02ER. It is not enough to escape from the surface, but the gas molecule is now under somewhat
excited status. When t is about 125 tR, the gas molecule gets more energy again and it can escape from the surface,
after going though the potential well. A lot of molecules that escaped from the clean surface experienced such an
intermediate 'excited' status.
3.2 Energy Distribution after Escape
The end of contacting time is detected for the escaped molecules. The total energy of the gas molecule is sampled
at that time. Figure 3 shows the relation of length of the contacting time and the total energy of gas molecule at
escaping. Clearly we can divide these samples into two groups. The group A is consist of the samples whose
contacting time is very short. And thier energy at escaping is high. After a small gap in the abscissa, there are a lot
of samples whose energy distribution is similar. We call these samples as "group B," and divide this group B into
3 parts as shown in Fig. 3.
group A
50
group B2
group B1
-3
40
E/ER x10
group B3
30
20
10
0
0
50
100
150
tC/tR
FIGURE 3. Relation between the Contacting Time and Energy at Escaping
TABLE 2. Average of Energy at Escaping from Surface
Group A
Group B1
Group B2
Group B3
Range of the Contacting time [tR]
0-5
5-67.5
34.5-115.5
69.63-170
Number of Samples
601
300
300
300
Average of E [K]
471
514
524
558
Table 2 shows the statistical value of each group. The unit of average of the total energy at escaping is converted
into temperature for comparison. The gas molecule whose contacting time is smaller than 5tR is grouped as A. In
this simulation, 601 samples was recognized as this group. The average of energy in this group is 471[K] that is
very different from the temperature of wall, 600[K]. Here, the initial temperature of gas molecule is 300[K], and the
gas molecules in group A is collided only once, as shown in Fig. 2 (a). So the accommodation with the surface is
not enough.
Group B1, B2, and B3 have the same number of samples, 300. The average of energy becomes larger as the
sticking time becomes longer. The long sticking time makes the gas molecules much accommodated.
959
The distributions of energy just after escaping are shown in Figure 4. The histogram in the figure is the result of
present simulation. The curves show the Boltzmann distribution of energy,
Ê E ˆ
1
FT (E) =
E expÁ ˜.
(2)
2
k T
(k B T )
Ë
B
¯
In the figure, the Boltzmann distribution of wall temperature F600(E) is shown also, and distribution using the
average temperature of samples is drawn, too. Though the average of energy of group A is far from the wall
†
temperature,
the distribution in Fig. 4 is almost same as F471(E).
In Fig. 4(b), the distribution of result is almost same as F600(E). The chi-square test indicates that this
distribution in Fig. 4 (b) is the Boltzmann distribution when the level of significant is 5%. So the molecule whose
contacting time is longer than 70 tR, is thought to be in the thermal equilibrium with the surface.
50
80
Result of simulation
f600(E)
f471(E)
40
20
20
0
F558(E)
30
N
N
60
Result of simulation
F600(E)
40
10
0
10
20
-3
30
40
0
50
0
10
E/ER x10
20
-3
30
40
50
E/ER x10
(a) Group A
(b) Group B3
FIGURE 4. Energy Distribution of Gas Molecules at Escaping from the Surface.
3.3 Velocity Distribution after Escape
Figure 5 shows the distribution of velocity of gas molecule at escaping. The normal velocity has been accelerated
by the potential well at the end of tC, so the normal velocity component must be reduced considering its potential
energy at escaping, to get an accurate velocity at escaping.
When the gas molecule is in thermal equilibrium, the distribution function of each component should be
f n (v n ) = 2b 2v n exp(-b 2v n2 )
and
f t (v t ) = 2b 2v t exp(-b 2v t2 )
Ê 2k B T ˆ-1/2
m
.
˜ =
Ë m ¯
2k B T
, while b = Á
Here fn is the distribution of normal velocity component, vn. And ft is that of tangential component, vt. The average
of vn under Boltzmann distribution is,
†
vn =
Ú
•
0
v n f n dv n =
p
pk†B T
=
.
2b
2m
†
Using this relation, the average of velocity can be converted into the temperature,T. Same formula can be used
between average of vt and T.
Table 3 shows these converted value of average of velocity in normal and tangential direction. In the samples of
†
group A, the average is far from that of wall temperature. The normal velocity component, vn, is very high, while
average of vt is very low. The averages of normal component in the groups B1,B2, B3 are almost same as 600[K]
that shows the quick accommodation to the surface. On the other hand, the tangential component is slowly
accommodated to the wall temperature. This tendency of tangential component is same as that of total energy.
The result of simulation and the theoretical functions are compared in Fig. 5. The distribution function has very
similar shape as that of Boltzmann distribution.
TABLE 3. Average of Normal and Tangential Component of Kinetic Energy
Group A
Group B1 Group B2
Group B3
Average of Normal Velocity[K]
Average of Tangential Velocity[K]
703
299
633
435
960
581
600
505
569
50
50
Result of simulation
f600(v)
40
Result of simulation
f600(v)
40
f633(v)
f435(v)
N
30
N
30
20
20
10
10
0
0.0
0.1
0.2
0.3
0
0.0
0.4
0.1
0.2
vn/vR
(a) Normal Velocity
0.3
0.4
vt/vR
(b) Tangential Velocity
FIGURE 5. Velocity Distribution of Group B1
4. RESULTS ON THE DIRTY SURFACE
4.1 Scattering Process
Figure 6 shows two examples of scattering process on the dirty surface. In these cases, 48 gas molecules are put
between two walls, whose distance is 20 LR. About 34 molecules are sticking in average. When more gas molecules
are put, the number of sticking molecules is saturated at 78. So in the case of figure 6, the surface is partially
covered by the adsorbed gas molecules. The clean area can be found somewhere on the surface.
In Fig. 6(a), the gas molecule is once sticking on the solid surface, then it makes a slight collision with other
sticking gas molecules at t=310 tR, and then it escapes from the surface. During this process, the total energy of this
gas molecule becomes positive at t=230 tR once, but this molecule remains sticking. Only the tangential kinetic
energy is large in this case, but the normal velocity is too small to escape.
The gas molecule escapes directly from the surface in the case of Fig. 6(b). In these figures, the potential energy
while the gas molecule sticks is relatively higher than that was shown in Fig. 2(b) of clean surface. Also, the
kinetic energy is larger, and the height of jump in z direction is much higher. On the dirty surface, the stuck
molecule seems to be always in some 'excited' status. Even when the sticking molecules are not affected by other
gas molecules directly, the existence of other gas gives such effect indirectly.
tC (contacting time)
4
z/LR
z/LR
6
2
0.06
0
0.04
potential energy
total energy
kinetic energy
potential energy
total energy
0.04
kinetic energy
0.02
E/ER
E/ER
0.02
0.00
0.00
-0.02
-0.02
-0.04
180
5
4
3
2
1
0
-0.04
200
220
240
260
280
300
320
160
t/tR
180
200
220
240
260
t/tR
(a) Escape Colliding with Adsorbed Molecule
(b) Escape from Clean Area
FIGURE 6. Example of Process of Colliding, Sticking, and Escaping of the Gas Molecule on the Dirty Surface.
4.2 Energy Distribution after Escape
Figure 7 shows the relation between the length of contacting time and total energy at escaping. The escaped gas
molecules are divided into two groups. Molecules in the group C escapes from clean surface area, like shown in
Fig. 6(b). These molecules are not affected by other gas molecules when they escape from the surface. When a gas
molecule collides with other gas molecule in escaping, it is grouped into D. Fig. 7(a) shows the distribution of
961
group C. Some samples whose contacting time tC is short shows different distribution to others. So, as done in
Fig. 3, these samples are divided into two small groups, C1 and C2.
In Fig. 7(b), though there is a small gap where no sample is found at t=22tR, the difference between the
distribution of energy before this time and after is not clear. When the gas molecule is affected by other gas, the
dependency of short contacting time on the energy is relatively small.
The distribution of energy at escaping is shown in Fig. 8. The result of group C is compared with the
Boltzmann distribution in Fig. 8(a). The total distribution of group C is almost similar to the Boltzmann
distribution. There are a few samples whose energy is negative in this distribution. This was caused by small
fluctuation of potential energy of the gas molecule.
The average of energy of the group D is 604[K]. Though it is almost same as the wall temperature, 600[K], the
distribution of energy is a little different from the Boltzmann distribution. This difference is caused by samples
with high energy value, whose contacting time is from 100 to 300. The gas molecule whose energy is high can
easily migrate on the surface widely, and it may collide with other gas molecule. Such samples are grouped into
D2. On the other hand, when energy of gas molecule is relatively small, it cannot migrate largely, and it cannot
collide with other gas molecule. Such samples are grouped into C2.
group C1
group C2
group D1
group D2
-3
60
40
E/ER x10
E/ER x10
-3
60
20
40
20
0
0
0
100
200
300
400
0
100
200
tS/tR
300
400
tS/tR
(a) Group C: Escape from Clean Area
(b) Group D: Escape Coliding with Other Gas Molecule
FIGURE 7. Relation between Energy at Escaping and Sticking Time
30
40
Result of simulation
F600(E)
25
15
N
N
30
F572(E)
20
Result of simulation
F600(E)
20
10
10
5
0
0
10
20
30
E/ER x10
-3
40
50
0
60
0
10
20
30
-3
40
50
60
E/ER x10
(a) Group C: Escape from Clean Area
(b) Group D: Escape Coliding with Other Gas Molecule
FIGURE 8. Distribution of Energy at Escaping from Dirty Surface
TABLE 4. Statistical Result of Scattering on Dirty Surface
Escape from Clean Area
Range of tC [ tR]
Number of Samples
Average of Energy [K]
Average of Normal Velocity [K]
Average of Tangential Velocity [K]
Escape Coliding with Other Gas
Molecules
Group C1
Group C2
C (C1+C2) Group D1
Group D2
D(D1+D2)
0-5
30
745
1205
617
5-500
203
546
710
560
0-500
233
572
764
573
22-500
213
638
698
623
0-500
280
604
649
610
962
0-22
67
499
500
571
The average of whole samples, including group C and D, is 590[K]. This shows that the gas molecules, which
escape from surface, are in equilibrium with the surface temperature as a whole.
4.3 Velocity Distribution after Escape
Table 4 shows the statistical results of group C and D. The average of normal velocity component of group C1 is
very high. Some of this reason may be its small number of samples. The gas molecule of C1 makes a collision
once, and then it escapes from the surface. This situation is same as that in group A. So the tendency that the
average of normal velocity is much higher than that of wall temperature is same as that of group A. Also group C2
shows high average value of normal velocity.
Gas molecules in group D1 have short contacting time, so they collide with other gas molecule at first, and soon
they escape. The author showed that when a molecule collides with the adsorbed molecule, the amount of energy,
which is transferred from the colliding gas molecule to others, is larger than the that on solid molecules[7]. So
samples in group D1 lose larger energy than other group, and this causes the low average value of energy, normal
and tangential velocity.
Samples in group D2 shows high average values, especially for normal velocity component. It is similar as
group C2. As shown in Fig. 6, the sticking molecules on the dirty surface are always 'excited' by the existence of
other adsorbed molecules. The vibration along the z axis is much larger than that on the clean surface constantly. It
seems that this excited movement along z axis causes the high average of normal velocity component.
Figure 9 shows the distribution of normal velocity component. The distribution function is not completely same
as that of Boltzmann distribution, it is similar to that in thermal equilibrium. Also, the distribution of tangential
velocity component is similar, as shown in Fig. 10.
40
40
Result of simulation
f600(v)
30
f764(v)
20
N
N
30
10
0
0.0
Result of simulation
f600(v)
f649(v)
20
10
0.1
0.2
0.3
0
0.0
0.4
0.1
vn/vR
(a) Group C
0.4
40
Result of simulation
f600(v)
30
Result of simulation
f600(v)
30
f573(v)
20
N
N
0.3
(b) Group D
FIGURE 9. Distribution of Normal Velocity at Escaping from Dirty Surface
40
10
0
0.0
0.2
vn/vR
20
10
0.1
0.2
0.3
0.4
0
0.0
vt/vR
(a) Group C
0.1
0.2
vt/vR
(b) Group D
FIGURE 10. Distribution of Energy at Escaping from Dirty Surface
963
0.3
0.4
5. CONCLUDING REMARKS
Using the molecular dynamics method, the sticking and scattering process of Xenon molecule on the surface of
Platinum is analyzed.
On the clean surface, when the gas temperature is different from the surface temperature, the velocity component
of colliding gas molecule in normal direction of surface is quickly accommodated with the surface temperature.
However, the velocity in tangential direction goes slowly into thermal equilibrium. When the contacting time is
short, the average of energy when the gas molecule escapes from the surface differs from the surface temperature.
When a lot of gas molecules are adsorbed on the surface, both the kinetic and potential energy of gas molecule
while it sticks is much higher than that on the clean surface.
Though the distribution function of energy and velocity at escaping is almost same as the Boltzmann
distribution, the average value is higher or lower than the surface temperature, according to the pattern of the process
of scattering.
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