Dynamical Monte-Carlo Simulation of Surface Kinetics
V. Guerra∗ and J. Loureiro∗
∗
Centro de Física dos Plasmas, Instituto Superior Técnico, 1049-001 Lisboa, Portugal
Abstract. This work presents a theoretical model for surface reactions of atoms, in which all the elementary steps involved
in surface atomic recombination are included. The system is modeled by a dynamical Monte Carlo scheme, as outlined in [1],
where Monte Carlo time is related with real time. The Monte Carlo results are compared with those of a phenomenological
model for the average fractional coverage of adsorption sites, based on a set of rate balance equations. The results of both
models are shown to agree in the limit of low coverage of adsorption sites.
I. INTRODUCTION
The interaction of atoms with walls in low-temperature plasmas can have a strong influence on the gas-phase
concentrations of different species. For example, the surface kinetics of N and O atoms is important to understand
the plasma reactors used for chemical synthesis, air pollution cleaning, or surface treatments of various materials [2].
Such a knowledge can also be important in studying the performance of heat shields on reusable space vehicles [3].
In this work we develop a dynamical Monte Carlo model to describe heterogeneous chemical reactions occurring in
gas discharges. In the classic approach to study reaction kinetics, analytical models which generally consider spatially
averaged concentrations of adsorbed reactants are utilized. The advantage of our approach arises from the intrinsic
microscopic detail of a Monte Carlo simulation, which easily provides physical insight into the basic phenomena
occurring in the system. Furthermore, the atomistic nature of Monte Carlo models will allow naturally the inclusion
of some more complex effects, not considered here, such as spatial in-homogeneity, diffusion rates dependent upon
surface coverage and surface adsorbate-adsorbate interactions.
Recently, Monte Carlo methods have been utilized in the investigation of many problems related to surface kinetics,
such as dissociative adsorption, [4] surface abstraction [5] or the simple case of adsorption-desorption equilibrium [1].
Although Monte Carlo methods are mostly associated with obtaining the equilibrium properties for model systems,
Fichthorn and Weinberg have presented the theoretical basis for a dynamical Monte Carlo method able to study
dynamical phenomena, provided a set of conditions is satisfied. In what concerns the treatment of time, this method is
similar to the one presented in [6] for the case of spatially homogeneous reactions. Here, we extend the method initially
developed in [1] to a much more complicated system, by considering most of the microscopic processes involved in
surface recombination. We will consider as working system the recombination of ground state nitrogen N( 4 S) atoms
on silica surfaces.
The organization of this paper is as follows. In section II we describe the basis of the dynamical Monte Carlo model
for surface recombination. In section III we briefly present a phenomenological model based on rate balance equations
for the average coverage of adsorption sites, similar to the ones developed in [8, 9]. In section IV we present and
discuss the results obtained using both approaches. Finally, in section V we summarize the main conclusions.
II. DYNAMICAL MONTE CARLO METHOD FOR HETEROGENOUS
RECOMBINATION
a) Foundations of the dynamical Monte Carlo method
Following the discussion in [1], the dynamical Monte Carlo method provides a numerical solution that describes
both static and dynamic properties of the system. Furthermore, a relationship between Monte Carlo time and real time
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
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is clearly established. To do so, the Monte Carlo time step must be derived from the transition probabilities of the
various microscopic events, with these probabilities formulated as rates with physical meaning. The time resolution
is accomplished then in a scale at which no two events may occur simultaneously and the algorithm used must be
consistent with the theory of Poisson processes.
In a Poisson process any particular event that becomes possible at time t can potentially occur at any later
time t + ∆t, with a uniform probability, which is based on its rate and is independent of the events before time
t. Consequently, the task of the Monte Carlo algorithm is to create a chronological sequence of distinct events
separated by certain interevent times. Since the microscopic dynamics yielding the exact times of various events
is not modeled in this approach, the chain of events and corresponding interevent times must be constructed from
probability distributions weighting appropriately all possible outcomes. Essentially, the system may undergo certain
distinctive events E = {e1 , e2 , · · · , ek }, which can be characterized by average transition rates R = {r 1 , r2 , · · · , rk }.
If the system is comprised of N species, they can be partitioned among the various possible transition events as
N = {n1 , n2 , · · · , nk }, where ni is the number of species capable of undergoing a given event e i with rate ri . Thus, a
particular configuration of the system at a particular time can be characterized by the distribution of N over R. This
distribution is constructed by a Monte Carlo algorithm which selects randomly a certain event, among the various
possible events available at each time, and that affects the chosen event with an appropriate transition probability
from W = {w1 , w2 , · · · , wk }. The transition probabilities should be constructed in terms of R by creating a dynamical
hierarchy of transition probabilities, as wi = ri /rMax , with rMax ≥ sup{ri }. If a sufficiently large system is utilized to
assure that the independence of various events is achieved, the Monte Carlo algorithm effectively simulates the Poisson
process, and the passage to real time can be achieved in terms of N and R. This can be done provided that at each trial
j at which an event is realized time is updated with an increment τ j selected from an exponential distribution with
parameter λ = ∑i ni ri , i.e.,
1
τ j = − ln(U) ,
(1)
λ
where U is a uniform random number between 0 and 1. This procedure ensures that a direct and unambiguous
relationship between Monte Carlo and real time is established.
b) Application to heterogeneous recombination
Let us now consider the microscopic events involved in surface atomic recombination, as well as the surface
characteristics. In our model the surface is divided into various cells of radius a of the order of one Angstrom, each
of them representing an adsorption site that can hold atoms either reversibly or irreversibly. Every site has 4 nearest
neighbors, so that the surface can be regarded as a lattice of adsorption sites and the surface is totally covered with
adsorption sites. A lattice 750x750 is used in the calculations. The present simulation accounts for the mechanisms of
adsorption, thermal desorption, surface diffusion, and recombination.
Adsorption is the formation of a bond between the atom and the solid surface, and can be schematically represented
by
N + ∗ −→ N ∗ ,
(2)
where ∗ represents an empty adsorption site, N is a gas phase atom, and N ∗ denotes the adsorbed atom. Adsorption in a
reversible site is associated here with physisorption, whereas that in an irreversible one corresponds to chemisorption.
It is assumed that 0.2% of the surface is covered by irreversible sites [8].
The reversibly adsorbed atoms vibrate on the surface. At sufficiently high temperatures the vibration can extract the
adsorbed atom from the potential well, and the atom desorbs back to the gas phase,
The characteristic time for desorption is
N ∗ −→ N + ∗ .
(3)
τd = νd−1 exp(Ed /RTw ) ,
(4)
where νd and Ed are the frequency factor and the activation energy for surface desorption, respectively, and Tw
denotes the wall temperature. The reversibly adsorbed atoms can also diffuse on the surface from one site to the
other, overcoming a potential barrier ED , which is lower than the desorption activation energy Ed ,
N∗
N∗
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(diffusion).
(5)
The characteristic time for surface diffusion is
τD = νD−1 exp(ED /RTw ) .
(6)
It is assumed that a physisorbed atom can diffuse to each of the 4 neighboring sites with the same probability.
The chemisorbed atoms cannot desorb or diffuse so that they can be lost only by atomic recombination. The
recombination of atoms on the surface can occur through different mechanisms. The first one is the recombination
between an atom from the gas phase impinging on an occupied adsorption site. The free gas atom impinging on
the surface may react with an adsorbed atom, resulting in a recombined molecule. This is the so-called Eley-Rideal
mechanism (E-R),
N ∗ + N −→ N2 + ∗ .
(7)
The two atoms can recombine with a probability
PR = exp(−ER /RTw ) .
(8)
The second mechanism is the recombination between two adsorbed atoms. This is usually known as the LangmuirHinshelwood mechanism (L-H), whose schematic representation is
N ∗ + N ∗ −→ N2 + ∗ .
(9)
In practice, L-H recombination occurs when an adsorbed atom diffuses to an occupied neighboring site. In this case,
the two atoms can recombine with the same probability PR of E-R recombination.
c) Rates of elementary processes
In order to perform the dynamical Monte Carlo calculations we only need to know the rates of each elementary
process. We shall use modified rates for some processes, in order to reduce the computation time, but let us start by
describing the rates corresponding to a typical DC or HF nitrogen discharge in a silica surface.
The rate for adsorption in physisorption sites, in (site)−1 s−1 , will depend on the flow of atoms from the gas phase
to the wall,
hvN i[N]
φN =
,
(10)
4
in which
s
8kTg
hvN i =
,
(11)
π MN
and of the sticking probability k10 and the density of physisorption sites [F],
k10 φN
.
[F]
(12)
r2 = τd−1 ,
(13)
rD = τD−1 .
(14)
r1 =
For thermal desorption from irreversible sites we use (4),
whereas for diffusion we may use (6),
The rate for adsorption in irreversible sites is calculated in the same manner as adsorption in reversible sites, that is
through equation (12),
k 0 φN
r3 = 3
,
(15)
[S]
with [S] denoting now the total density of chemisorption sites. According to [8], we take k 10 = k30 = 1, νd = 1015 s−1 ,
Ed = 51 KJ mole−1 , νD = 1013 s−1 and ED = 0.5Ed .
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The rates for recombination can be easily calculated. For the case of E-R recombination, the rate per second and
occupied irreversible site is simply written as
r4 = PR r3 ,
(16)
where PR is given by equation (8), with ER = 14 KJ mole−1 [8]. L-H recombination does not need a specific rate. In
fact, we already have the rate for diffusion on the surface, rD , and the recombination probability PR , so that if an atom
diffuses to an occupied site it will recombine with probability PR . We assume here that if the diffusing atom does not
recombine it will desorb from the surface. We further assume the same probability PR for L-H recombination occuring
in reversible and in irreversible sites. This corresponds in practice to an underestimation of the effects of recombination
between two physisorbed atoms, since the recombination probability should be higher in the case of two physisorbed
atoms.
III. PHENOMENOLOGICAL MODEL FOR SURFACE RECOMBINATION
The processes of surface recombination can be also modeled in the framework of a phenomenological theory for
the average coverage of adsorption sites, as described, e.g., in [8, 9]. Let Fv and Sv denote vacant physisorbed and
chemisorbed sites, N f and Ns physisorbed and chemisorbed nitrogen atoms, N and N2 gas-phase atoms and molecules,
and [F] = [Fv ] + [N f ] and [S] = [Sv ] + [Ns ] the total surface densities of reversible and irreversible sites, respectively.
According to the discussion above, we have [F] ' 1016 cm−2 and [S]/[F] = 2 × 10−3, as proposed in [8]. Half of the
distance between two irreversible sites is then
a
b∼ √
∼ 2 × 10−7 cm.
(17)
2 × 10−3
The list of reactions corresponding to the processes described by equations (2), (3), (5), (7) and (9) can hence be
written as
k1
N + Fv −→
Nf ,
(18)
k
2
N f −→
N + Fv ,
k
3
N + Sv −→
Ns ,
k4
N + Ns −→ N2 + Sv ,
k
5
N f + Sv −→
Ns + Fv ,
k
6
N f + Ns −→
N2 + Fv + Sv .
(19)
(20)
(21)
(22)
(23)
Therefore, the corresponding master equations for the coverage of reversible and irreversible sites, θ f = [N f ]/[F] and
θs = [Ns ]/[S] are
dθ f
= (1 − θ f )[N]k1 − θ f k2 − θ f (1 − θs )[S]k5 − θ f θs [S]k6 ,
(24)
dt
and
d θs
= (1 − θs)[N]k3 − θs [N]k4 + θ f (1 − θs)[F]k5 − θ f θs [F]k6 ,
(25)
dt
Reactions (18) to (21) are modeled in the same way as in the dynamical Monte Carlo, i.e., with the corresponding rate
constants k1 –k4 related with r1 –r4 by k1 = r1 /[N], k2 = r2 , k3 = k3 /[N] and k4 = r4 /[N].
The differences between the two models arrive from the treatment of diffusion and L-H recombination. Here, the
basic idea is that each irreversible adsorption site is surrounded by a collection zone of radius Λ D < b, with b given
by equation (17). ΛD represents the average distance traveled by a physisorbed atom at the surface due to diffusion, so
that
p
ΛD = 4Ds τN ,
(26)
where Ds is the diffusion coefficient,
Ds =
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a2
,
τD
(27)
8x10
-3
7x10
-3
0.8
(a)
(b)
6x10
-3
5x10
-3
4x10
-3
3x10
-3
2x10
-3
1x10
-3
0.6
0.5
θs
θf
0.7
0.4
0.3
Dynamical Monte Carlo
Phenomenological model
Dynamical Monte Carlo
Phenomenological model
0.2
0.1
0
0.0
0
1x10
-5
2x10
-5
3x10
-5
4x10
0
-5
1x10
-4
t (s)
2x10
-4
3x10
-4
4x10
-4
t (s)
FIGURE 1. Fractional coverage of reversible (a) and irreversible (b) sites as a function of time, using a dynamical Monte Carlo
simulation (—) and a phenomenological model (– –), assuming Ed = 63.8KJ mole−1 .
and τN ∼ τd is the mean residence time of a physisorbed atom on the surface. Since only 1/4 of the atoms impinging
the surface within a collection zone will reach the irreversible site before desorption (the other 3/4 of the atoms migrate
towards farther distances), we may write the probability that a physisorbed atom reach an irreversible site as
kD0 =
1 ΛD2 a2
− 2 ,
4 b2
b
(28)
where a2 /b2 is the probability for direct impingement on irreversible sites. Using this procedure, the rates for reactions
(22) and (23) are, respectively,
k0
k5 = D
(29)
τN [S]
and
k6 = PR k5 .
(30)
We note that this approach models L-H recombination only between a chemisorbed and a physisorbed atom, while in
dynamical Monte Carlo it can also occur between two phyisorbed atoms. This difference should be negligible in the
limit of low coverage of reversible sites, where the collisions between two physisorbed atoms are scarce.
IV. RESULTS AND DISCUSSION
The calculations were performed for the values of the input parameters already described, with Tg = 500 K, Tw = 350
K and gas phase atomic concentration [N] = 1 × 1015 cm−3 . Unfortunately this set of parameters leads to a very
low occupation of physisorption sites, (∼ 10−5). Therefore, it requires the use of a big lattice in order to reduce the
fluctuations of the statistical description. For this reason, we decided to test the validity of our Monte Carlo scheme
for a set of parameters that allows to reduce both the dimension of the system and the computation time. Thus, figures
1a-b show the fractional coverage of reversible and irreversible sites, θ f and θs , as a function of time, starting from an
empty surface at t = 0, for the case of Ed = 51 × 1.25 ' 63.8 KJ mole−1 , ED =0.5×Ed ' 31.9 KJ mole−1 , νD =1011s−1
and k30 =10−2. The full curves correspond to the dynamical Monte Carlo results, while the dashed ones were obtained
using the phenomenological model. We can see that for the relatively low values of θ f corresponding to this case, there
is no significant difference between the results from both models.
Figure 2a-b shows the coverage θ f and θs in the case of a higher desorption energy, Ed = 51×1.3 = 66.3 KJ mole−1,
ED =0.5×Ed ' 33.2 KJ mole−1. The occupation of physisorption sites is now higher than in the conditions of figure
1. The collisions between two physisorbed atoms start to take place, reducing the average time spent on the surface
by a reversibly adsorbed atom. We remember here that in the phenomenological model two physisorbed atoms do not
interact with each other, while in Monte Carlo when a physisorbed atom arrives to reversible site already occupied,
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2.0x10
-2
0.8
(a)
(b)
0.7
1.5x10
-2
1.0x10
-2
5.0x10
-3
0.6
θs
θf
0.5
0.4
0.3
Dynamical Monte Carlo
Phenomenological model
Dynamical Monte Carlo
Phenomenological model
0.2
0.1
0.0
0.0
0
2x10
-5
4x10
-5
6x10
-5
8x10
-5
1x10
0
-4
1x10
-4
t (s)
2x10
-4
3x10
-4
4x10
-4
t (s)
FIGURE 2. Fractional coverage of reversible (a) and irreversible (b) sites as in figure 1, but assuming Ed = 66.3KJ mole−1 .
0.4
(b)
C
B
(a)
0.8
0.3
C
0.2
B
A
θs
θf
0.6
(A) Monte Carlo with diffusion 1 site
(B) Monte Carlo with diffusion 2 sites
(C) Monte Carlo with diffusion 4 sites
Phenomenological model
0.4
(A) Monte Carlo with diffusion 1 site
(B) Monte Carlo with diffusion 2 sites
(C) Monte Carlo with diffusion 4 sites
Phenomenological model
0.1
A
0.2
0.0
0.0
0
2x10
-4
4x10
-4
6x10
-4
8x10
0
-4
t (s)
2x10
-4
4x10
-4
6x10
-4
8x10
-4
t(s)
FIGURE 3. Fractional coverage of reversible (a) and irreversible (b) sites as a function of time assuming E d = 76.5KJ mole−1 ,
using a phenomenological model (– –) and a dynamical Monte Carlo scheme where recombination is attempted one (A), two (B)
and four (C) times (see text).
either the two physisorbed atoms may recombine or one is desorbed remaining the other sticking on the surface. The
interactions between physisorbed atoms have the effect of reducing the average time spent by these particles at the
surface. That is why the phenomenological model overestimates, in principle, the coverage θ f and, consequently, θs
as well.
The interaction between physisorbed atoms becomes more significant for even higher energies of desorption, as
it can be seen in figure 3, where we plot the time-evolutions of θ f and θs for Ed = 51 × 1.5 = 76.5 KJ mole−1,
ED =0.5×Ed ' 38.3 KJ mole−1, using our dynamical Monte Carlo as before (curves A) and the phenomenological
model (dashed curves). In order to verify the sensitivity of the results to the hypothesis made for the encounters of
two reversibly adsorbed atoms, we have made the dynamical Monte Carlo calculations assuming as well that when a
diffusing atom reaching an occupied site does not recombine it will diffuse to a neighbor site. If this new site is empty
it will stay on the site, whereas if the new site is occupied it will try a second time to recombine, desorbing to the
surface if recombination fails (curves B). We have still tested the case where a diffusing atom that does not recombine
can keep diffusing and attempting recombination until a maximum of four times (curves C). By increasing the chances
for a diffusing physisorbed atom to stay on the surface we decrease the importance of the collisions between two
physisorbed atoms, and hence the results obtained from the dynamical Monte Carlo and the phenomenological models
are made closer.
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V. CONCLUSIONS
In this work we have extended the dynamical Monte Carlo method presented in [1] for adsorption-desorption equilibrium to the much more complex treatment of atomic surface recombination. We have shown that the transient solution
is well described in the present formulation, since a direct relationship between real time and Monte Carlo time is
achieved. The stationary solution can also be obtained as long as the simulation runs for sufficientely long times to
attain equilibrium.
The Monte Carlo results were compared to the ones obtained from a phenomenological model for the average
coverage of adsorption sites. The effect of varying the mean residence time of physisorbed atoms has been investigated
by changing the activation energy for desorption, from a relatively low value in which the residence time is mainly
determined by thermal desorption, until the opposite situation of a high activation energy where the permanence of
the atoms on the wall is determined by the hypotheses made for diffusion. The differences between both models arrive
from the treatment of surface diffusion and, in particular, from the absence of collisions between physisorbed atoms
in the phenomenological model. We have shown that the results of both models agree except in the limit of very high
occupation of adsorption sites.
Since the purpose of this paper was to develop and validate a dynamical Monte Carlo algorithm for surface
recombination, the values of the coefficients of some processes were changed in order to reduce computation time.
The next step will be to use the actual rates in order to derive realistic recombination probabilities from the model.
Nevertheless, in our opinion the description of various elementary processes involved in heterogeneous recombination
and the validity of the method were already achieved in the present paper.
REFERENCES
1. Fichthorn, K. A., and Weinberg, W. H., J. Chem. Phys., 95, 1090–1096 (1991).
2. Katu, I., Noguchi, K., and Numada, K., J. Appl. Phys., 62, 492–497 (1989).
3. Capitelli, M., editor, Molecular Physics and Hypersonic Flows, vol. C-482 of NATO ASI Series, Kluwer Academic Publishers,
1996.
4. Resnyanskii, E. D., Myshlyavtsev, A. V., Elokhin, V. I., and Bal’zhinimaev, B. S., Chem. Phys. Letters, 264, 174–179 (1997).
5. Sholl, D. S., J. Chem. Phys., 106, 289–300 (1997).
6. Gillespie, D. T., J. Comput. Phys., 22, 403–434 (1976).
7. Ziff, R. M., Gulari, E., and Barshad, Y., Phys. Rev. Lett., 56, 2553–2556 (1986).
8. Kim, Y. C., and Boudart, M., Langmuir, 7, 2999–3005 (1991).
9. Gordiets, B. F., Ferreira, C. M., Nahorny, J., Pagnon, D., Touzeau, M., and Vialle, M., J. Phys. D: Appl. Phys., 29, 1021–1031
(1996).
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