Dynamical Monte-Carlo Simulation of Surface Kinetics

Transcription

1 Dynamical Monte-Carlo Simulation of Surface Kinetics V. Guerra and J. Loureiro Centro de Física dos Plasmas, Instituto Superior Técnico, Lisboa, Portugal Abstract. This wor 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 inetics 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 nowledge can also be important in studying the performance of heat shields on reusable space vehicles [3]. In this wor we develop a dynamical Monte Carlo model to describe heterogeneous chemical reactions occurring in gas discharges. In the classic approach to study reaction inetics, 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 inetics, 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 woring 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

2 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 tas 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 = {e 1,e 2,,e }, which can be characterized by average transition rates R = {r 1,r 2,,r }. If the system is comprised of N species, they can be partitioned among the various possible transition events as N = {n 1,n 2,,n }, where n i is the number of species capable of undergoing a given event e i with rate r i. 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 = {w 1,w 2,,w }. The transition probabilities should be constructed in terms of R by creating a dynamical hierarchy of transition probabilities, as w i = r i /r Max, with r Max sup{r i }. 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 n i r i, i.e., τ j = 1 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 % 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 bac to the gas phase, N N +. (3) The characteristic time for desorption is τ d = ν 1 d exp(e d /RT w ), (4) where ν d and E d are the frequency factor and the activation energy for surface desorption, respectively, and T w denotes the wall temperature. The reversibly adsorbed atoms can also diffuse on the surface from one site to the other, overcoming a potential barrier E D, which is lower than the desorption activation energy E d, N N (diffusion). (5)

3 The characteristic time for surface diffusion is τ D = ν 1 D exp(e D/RT w ). (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 N 2 +. (7) The two atoms can recombine with a probability P R = exp( E R /RT w ). (8) The second mechanism is the recombination between two adsorbed atoms. This is usually nown as the Langmuir- Hinshelwood mechanism (L-H), whose schematic representation is N + N N 2 +. (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 P R of E-R recombination. c) Rates of elementary processes In order to perform the dynamical Monte Carlo calculations we only need to now 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, φ N = v N [N], (10) 4 in which v N = and of the sticing probability 1 and the density of physisorption sites [F], For thermal desorption from irreversible sites we use (4), whereas for diffusion we may use (6), 8T g πm N, (11) r 1 = 1 φ N [F]. (12) r 2 = τ 1 d, (13) r D = τ 1 D. (14) The rate for adsorption in irreversible sites is calculated in the same manner as adsorption in reversible sites, that is through equation (12), r 3 = 3 φ N, (15) [S] with [S] denoting now the total density of chemisorption sites. According to [8], we tae 1 = 3 = 1, ν d = s 1, E d = 51 KJ mole 1, ν D = s 1 and E D = 0.5E d.

4 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 r 4 = P R r 3, (16) where P R is given by equation (8), with E R = 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, r D, and the recombination probability P R, so that if an atom diffuses to an occupied site it will recombine with probability P R. We assume here that if the diffusing atom does not recombine it will desorb from the surface. We further assume the same probability P R 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 framewor of a phenomenological theory for the average coverage of adsorption sites, as described, e.g., in [8, 9]. Let F v and S v denote vacant physisorbed and chemisorbed sites, N f and N s physisorbed and chemisorbed nitrogen atoms, N and N 2 gas-phase atoms and molecules, and [F] = [F v ] + [N f ] and [S] = [S v ] + [N s ] the total surface densities of reversible and irreversible sites, respectively. According to the discussion above, we have [F] cm 2 and [S]/[F] = , as proposed in [8]. Half of the distance between two irreversible sites is then b a cm. (17) The list of reactions corresponding to the processes described by equations (2), (3), (5), (7) and (9) can hence be written as N + F 1 v Nf, (18) N 2 f N + Fv, (19) N + S 3 v Ns, (20) N + N 4 s N2 + S v, (21) N f + S 5 v Ns + F v, (22) N f + N 6 s N2 + F v + S v. (23) Therefore, the corresponding master equations for the coverage of reversible and irreversible sites, θ f = [N f ]/[F] and θ s = [N s ]/[S] are dθ f = (1 θ f )[N] 1 θ f 2 θ f (1 θ s )[S] 5 θ f θ s [S] 6, (24) dt and dθ s = (1 θ s )[N] 3 θ s [N] 4 + θ f (1 θ s )[F] 5 θ f θ s [F] 6, (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 1 4 related with r 1 r 4 by 1 = r 1 /[N], 2 = r 2, 3 = 3 /[N] and 4 = r 4 /[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). Λ D represents the average distance traveled by a physisorbed atom at the surface due to diffusion, so that Λ D = 4D s τ N, (26) where D s is the diffusion coefficient, D s = a2 τ D, (27)

5 8x10-3 7x10-3 (a) (b) 6x x θ f 4x10-3 θ s 3x10-3 2x x x10-5 2x10-5 3x10-5 4x x10-4 2x10-4 3x10-4 4x10-4 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 E d = 63.8KJ mole 1. and τ N τ d 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 D = 1 ΛD 2 4 b 2 a2 b 2, (28) where a 2 /b 2 is the probability for direct impingement on irreversible sites. Using this procedure, the rates for reactions (22) and (23) are, respectively, 5 = D (29) τ N [S] and 6 = P R 5. (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 T g = 500 K, T w = 350 K and gas phase atomic concentration [N] = 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 θ s, as a function of time, starting from an empty surface at t = 0, for the case of E d = KJ mole 1, E D =0.5 E d 31.9 KJ mole 1, ν D =10 11 s 1 and 3 =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 θ s in the case of a higher desorption energy, E d = = 66.3 KJ mole 1, E D =0.5 E d 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 tae 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,

6 2.0x10-2 (a) 1.5x (b) θ f 1.0x10-2 θ s 5.0x x10-5 4x10-5 6x10-5 8x10-5 1x x10-4 2x10-4 3x10-4 4x10-4 FIGURE 2. Fractional coverage of reversible (a) and irreversible (b) sites as in figure 1, but assuming E d = 66.3KJ mole (a) C 0.8 (b) C B A 0.6 θ f 0.1 (A) Monte Carlo with diffusion 1 site (B) Monte Carlo with diffusion 2 sites (C) Monte Carlo with diffusion 4 sites B A θ s (A) Monte Carlo with diffusion 1 site (B) Monte Carlo with diffusion 2 sites (C) Monte Carlo with diffusion 4 sites 0 2x10-4 4x10-4 6x10-4 8x x10-4 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 sticing 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, θ s 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 θ s for E d = = 76.5 KJ mole 1, E D =0.5 E d 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 eep 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.

7 V. CONCLUSIONS In this wor 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, (1991). 2. Katu, I., Noguchi, K., and Numada, K., J. Appl. Phys., 62, (1989). 3. Capitelli, M., editor, Molecular Physics and Hypersonic Flows, vol. C-482 of NATO ASI Series, Kluwer Academic Publishers, Resnyansii, E. D., Myshlyavtsev, A. V., Elohin, V. I., and Bal zhinimaev, B. S., Chem. Phys. Letters, 264, (1997). 5. Sholl, D. S., J. Chem. Phys., 106, (1997). 6. Gillespie, D. T., J. Comput. Phys., 22, (1976). 7. Ziff, R. M., Gulari, E., and Barshad, Y., Phys. Rev. Lett., 56, (1986). 8. Kim, Y. C., and Boudart, M., Langmuir, 7, (1991). 9. Gordiets, B. F., Ferreira, C. M., Nahorny, J., Pagnon, D., Touzeau, M., and Vialle, M., J. Phys. D: Appl. Phys., 29, (1996).