Langmuir 1997, 13, 1001-1009
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Diffusion of Interacting Lattice Gases on Heterogeneous Surfaces† Elmarie´ C. Viljoen and Christian Uebing* Max-Planck-Institut fu¨ r Eisenforschung, D-40074 Du¨ sseldorf, Germany Received September 19, 1995. In Final Form: January 2, 1996X The Monte Carlo method is used to simulate diffusion of interacting lattice gases on heterogeneous surfaces. The fluctuation and Kubo-Green methods are utilized for determining the tracer and chemical surface diffusion coefficients. Simulations are carried out on a square array of 64 × 64 lattice sites applying periodic boundary conditions. Surface heterogeneity is introduced in terms of the bivariate trap model with random topography. Simulations are performed for a trap concentration θtrap ) 0.2 and various values of the trap binding energies 2, 1 and the nearest neighbor interaction energy φNN. For φNN < 0 (repulsion) the system exhibits c(2 × 2) ordering at half coverage and low temperatures. Due to the low compressibility of the well-ordered c(2 × 2) lattice gas phase, the chemical diffusion coefficient is large under these circumstances. The effects of surface heterogeneity, 1 * 2, are largely pronounced at low total coverages θ and low temperatures T where most of the adatoms are trapped by the deep traps. At higher coverages the effects are relatively small albeit significant. The existence of deep traps disturbs c(2 × 2) ordering at θ ) 0.5 and low temperatures. It is shown that the breakdown of the order substantially affects surface diffusion.
1. Introduction During the last years the investigation of surface diffusion has attracted considerable interest because of its central importance for many surface processes such as adsorption, desorption, catalytic reactions, melting, roughening, and crystal and film growth. While in previous work most emphasis has been on ideally flat surfaces, relatively little attention has been paid to imperfect surfaces. Even single crystal surfaces are never perfectly flat but contain defects causing structural and energetical heterogeneity. Experimental investigations of surface diffusion on such surfaces are a challenge for state of the art surface science. Recently some new techniques for measuring surface diffusion coefficients of adsorbates on macroscopic surfaces have been developed1 and it is obviously of interest to increase our general understanding of the influence of energetical heterogeneity on surface diffusion. In the present work a lattice gas model for the study of surface diffusion is introduced. In general, lattice gas models are applicable to many interesting problems of diffusion on homogeneous and heterogeneous surfaces. Sapag, Pereyra, Riccardo, and Zgrablich have studied the tracer diffusion of noninteracting adsorbates on different heterogeneous surfaces by means of Monte Carlo modeling.2 For the simulations these authors applied random trap models (RTM) as well as random barrier models (RBM). Monte Carlo simulations have also been used by Mak, Anderson, and George to study the collective or chemical diffusion of noninteracting adsorbates on heterogeneous surfaces.3 The simulations are performed for * To whom correspondence may be addressed: Max-PlanckInstitut fu¨r Eisenforschung, Postfach 140 444, D-40074 Du¨sseldorf, Germany; phone, 49-211-6792-290; fax, 49-211-6792-268; email,
[email protected]. † Presented at the Second International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland/Slovakia, September 4-10, 1995. X Abstract published in Advance ACS Abstracts, September 15, 1996. (1) Gomer, R. Rep. Prog. Phys. 1990, 53, 917. (2) Sapag, K.; Pereyra, V.; Riccardo, J. L.; Zgrablich, G. Surf. Sci. 1993, 295, 433. (3) Mak, C. H.; Anderson, H. C.; George, S. M. J. Chem. Phys. 1988, 88, 4052.
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surfaces with randomly arranged traps or with blocks of traps. In some of these cases where analytical expressions for the diffusion coefficients can be derived, the Monte Carlo results have been compared with analytical predictions.3 In many cases even lattice gas models are too complicated for exact analytical treatments of the diffusion problem especially if ad-ad interactions are taken into consideration, and only in some exceptional cases general equations for the description of surface diffusion can be derived.4,5 However, the Monte Carlo method constitutes a powerful tool to analyze surface diffusion, and it has been shown in previous publications that this method is extremely valuable to improve our understanding of adatom diffusion on plain and stepped surfaces.6-8 The aim of the present work is to investigate the mutual influence of surface heterogeneity and ad-ad interactions on surface diffusion. For that purpose we study the bivariate trap model which is probably the simpliest way to describe energetical surface heterogeneity. Pairwise nearest neighbor repulsions are considered which cause c(2 × 2) ordering of the lattice gas at low temperatures. We obtain the tracer and the chemical surface diffusion coefficients by well-established standard methods (KuboGreen and fluctuation method) using Monte Carlo modeling. In section 2 the lattice gas model and the numerical procedure are described in detail. In section 3 our basic results for the diffusion of noninteracting and interacting lattice gases on homogeneous and heterogeneous surfaces are presented and discussed. 2. Procedure In the present work we examine the effect of particleparticle interactions on the diffusion of lattice gases on heterogeneous surfaces. For this purpose the Monte Carlo method is used as already mentioned. All simulations are carried out on a square lattice with 64 × 64 sites and periodic boundary conditions. The calculations are performed on the supermassive parallel Intel Paragon supercomputer of the Ju¨lich research center. (4) Zhdanov, V. P. Surf. Sci. Lett. 1985, 149, L13. (5) Reed, D. A.; Ehrlich, G. Surf. Sci. 1981, 102, 588. (6) Uebing, C.; Gomer, R. J. Chem. Phys. 1991, 95, 7626. (7) Uebing, C.; Gomer, R. Surf. Sci. 1994, 306, 419. (8) Uebing, C. Phys. Rev. B 1995, 49, 13913.
© 1997 American Chemical Society
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for the lattice gas with and without nearest neighbor interactions, respectively. The occupation of lattice sites by adsorbates is described by local occupation variables ci defined as
ci )
{
1, if site i is occupied 0, if site i is vacant
(2)
Double occupancy of lattice sites is excluded. 2.2. The Monte Carlo Simulation of Surface Diffusion. The basic steps of surface diffusion are jumps of adatoms from filled initial sites i to adjacent vacant sites j. As in ref 6 the activation energy for such jumps is calculated as the energy difference between saddle point S and single site energy of the initial site i
(3)
E ) S - i Figure 1. Schematic drawing of the periodic potentials used for the simulation of surface diffusion on heterogeneous surfaces. The minima of the potential constitute the adsorption sites. Simulations are performed for the so-called bivariate trap model where there are two types of adsorption sites, shallow and deep traps with adsorption energies 1 and 2, respectively. Nearest neighbor interaction energies are assumed to modify the adsorption energies (dashed lines) but not the saddle point energies s.
2.1. Surface Heterogeneity. In general, surface heterogeneity can be represented by probability distributions of site specific adsorption and saddle point energies, PAd(i) and PS(S).9-11 Obviously, the systematic treatment of realistic probability distributions PAd and PS would go far beyond the scope of the present paper. Therefore the considerations will be restricted to the simplest model of a heterogeneous surface, the well-known bivariate trap model. The characteristic features of our lattice gas model are schematically outlined in Figure 1. It is assumed that there are two types of adsorption sites, shallow and deep traps with adsorption energies 1 and 2, respectively. The various trap sites are randomly distributed. It is assumed that the concentration of the deep traps is c2 ) 1 - c1 ) 0.2. Such a situation is believed to be appropriate for adsorbates on, e.g., alloy surfaces, where the various types of trap sites are generated by the different chemical environments of the adsorption sites. It would be quite interesting to study the effect of c2 on the surface diffusion in some detail. However, this task will exceed the scope of the present paper. All adsorption sites are separated by wells of the periodic potential. The saddle point energies are uniformly given by S throughout the whole lattice. The relative minima of the periodic potential are modified by pairwise nearest neighbor interaction energies φNN. However, for the sake of simplicity we assume that the saddle points are not affected by particle-particle interactions. The lattice gas Hamiltonian can be written as
cj ) ∑i cii ) -∑i ci(i° + φNN∑ NN
H)-
(1)
where i and i° denote the site specific adsorption energies (9) The saddle points of the periodic potential constitute the barriers which diffusing particles need to overcome during their random walk between nearest neighbor sites. The barrier heights are frequently referred to as bond energies. Details of the site-bond terminology are described in refs 10 and 11. (10) Mayagoitia, V.; Rojas, F.; Pereyra, V.; Zgrablich, G. Surf. Sci. 1989, 221, 394. (11) Mayagoitia, V.; Rojas, F.; Riccardo, J. L.; Pereyra, V.; Zgrablich, G. Phys. Rev. B 1990, 41, 7150.
The associated jump probability PJ is given in essence (for details in practice see ref 6) by
PJ ∝ exp(-E/kBT)
(4)
The procedure for simulating jumps in the canonical ensemble is the following. First for a given lattice gas configuration an initial site i is randomly picked. If filled, an adjacent final site j is randomly selected. If the destination is vacant, a jump may occur with probability PJ (eq 4), otherwise no jump occurs. Thermodynamic equilibrium is established before starting a diffusion run at the desired fixed coverage θ. Approach to equilibrium is monitored by following the total energy and is assumed to occur when this quantity fluctuates about an average value. Typically 1000 Monte Carlo steps (MCS) are required to establish equilibrium (1 MCS consists of N × N interrogations of lattice sites). However, at very low temperatures when c(2 × 2) ordered lattice gas phases are formed the equilibrium required up to 200 000 MCS. 2.3. Determination of Surface Diffusion Coefficients. The tracer diffusion coefficient D* is determined from measurements of the mean square displacements of N tagged adatoms ∆ri(t) according to1,12
(
1 1
N
)
∑〈[∆ri(t)]2〉 tf∞ 4t N i)1
D* ) lim
(5)
Here t is elapsed time. The ∆ri(t) values are expressed in units of the lattice constant a0. The chemical diffusion coefficient D is determined via two different approaches, the fluctuation method as well as the Kubo-Green method. In essence, the fluctuation method measures the particle number autocorrelation function fn(t)/fn(0) for a small probed region embedded in the whole two-dimensional lattice. In refs 13 and 14 it has been shown in detail how the decay of fn(t)/fn(0) can be used to obtain the chemical surface diffusion coefficient, which we call DF. Thus this method is a computer simulation of the field emission fluctuation method13 used experimentally for determining adsorbate diffusion coefficients. For the autocorrelation function we can write
〈δN(t)δN(0)〉
fn(t) fn(0)
)
〈(δN)2〉
(6)
where N is the number of adatoms in the probed area. (12) Tringides, M.; Gomer, R. Surf. Sci. 1986, 166, 419. (13) Gomer, R. Surf. Sci. 1973, 38, 373. (14) Mazenko, G.; Banavar, J. R.; Gomer, R. Surf. Sci. 1981, 107, 459.
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Details of this method are presented in ref 7 and will not be repeated here. In the present work we used as 16 × 16 square probe for the determination of the particle density autocorrelation function. In order to obtain a meaningful overall chemical diffusion coefficient DF, we used autocorrelation functions based on up to 3 × 107 MCS taking full advantage of the computational power of the supermassive parallel Intel Paragon supercomputer. The second method of determining the chemical diffusion coefficient D is based on the Kubo-Green equation, here written as15
DKG )
(
)
∂µ/kBT ∂ ln θ
T
DJ )
[
]
〈(δN)2〉 〈N〉
-1
DJ
(7)
where µ is chemical potential, θ coverage, and DJ a “jump diffusion coefficient”. 〈(δN)2〉 is the mean square number fluctuation in an area A containing on average 〈N〉 particles. [∂(µ/kBT)/∂ ln θ]T is the well-known thermodynamic factor. Equation 7 is in fact another way of writing the Kubo-Green equation1 for D if DJ is taken as
DJ )
N
1
N
∫dt〈∑vi(0)∑vj(t)〉 4t i)1
(8)
j)1
It can be shown that DJ can also be written as
DJ )
N
1 1
〈(
4t N
∆ri)2〉 ∑ i)1
(9)
which makes the connection with a diffusion coefficient more explicit, particularly by comparison with the expression for the tracer diffusion coefficient D*
D* )
1 1
N
∑
4t Ni)1
(∆ri)2
(10)
As in previous simulations,6 calculations are carried out in terms of D°, the chemical diffusion coefficient for zero interactions between adsorbates on a homogeneous lattice (Langmuir gas). In the present work Kubo-Green and fluctuation methods yield numerically identical results for the chemical diffusion coefficient apart from statistical errors, which are more pronounced for DF. Therefore, we will present Kubo-Green values DKG throughout this paper. 3. Results and Discussion In the following sections we will describe and discuss the mutual influence of surface heterogeneity and of pairwise nearest neighbor repulsions on the surface diffusion. All simulations of surface diffusion on heterogeneous surfaces to be reported here are done for a concentration of the deep traps c2 ) 1 - c1 ) 0.2. We will start with the investigation of surface diffusion on a homogeneous square surface without any particleparticle interactions (Langmuir gas). It is noted that for this special case the diffusion behavior is well-known and largely understood. Thus the only justification to run a simulation for such a parameter set is to test the correctness of the improved Monte Carlo algorithm and the statistical accuracy of the sampling. Figure 2 shows the coverage dependence of the normalized tracer and chemical diffusion coefficients. D*/D° decreases monotonic with increasing coverage. In order (15) Reed, D. A.; Ehrlich, G. Surf. Sci. 1981, 105, 603.
Figure 2. Normalized surface diffusion coefficients vs surface coverage θ for the Langmuir gas with site exclusion. Shown are the tracer diffusion coefficient D*/D° (×) and the chemical diffusion coefficient obtained via the Kubo-Green method DKG/ D° (O). The dashed line gives the vacancy availability factor V ) 1 - θ of the Langmuir gas.
to rationalize this finding we note that the tracer diffusion coefficient can be defined as the product of a tracer correlation factor f,16,17 a vacancy availability factor V, and an average jump probability 〈Pj〉18,19
D* ) fV〈Pj〉
(11)
For the Langmuir case the average jump probability does not depend on the local environment of a diffusing adatom and is fully determined by eq 4. The vacancy availability factor for the Langmuir gas with site exclusion is trivially known as V ) 1 - θ. Thus we have
D*/D° ∝ f(1 - θ)
(12)
It is quite obvious that the results for D*/D° show substantial deviations from 1 - θ especially at high coverages indicating that f markedly differs from unity.19 Figure 2 also shows that D/D° is equal to unity for all surface coverages. Due to the normalization of the Monte Carlo data, this behavior is trivially expected. Thus the only justification to run a simulation for such a parameter set is to test the correctness of the program and the statistical accuracy of the sampling. 3.1. Effect of Surface Heterogeneity. Next we will investigate surface diffusion for a noninteracting lattice gas on a heterogeneous surface. As already mentioned we consider the existence of shallow and deep traps, the concentration of the latter being c2 ) 0.2. In the absence of nearest neighbor interactions the exact solution of the thermodynamics is trivially known. For the calculation of site specific surface coverages we express the chemical potentials of adatoms on the various trap sites µ1, µ2 according to (16) LeClaire, A. D. In Physical ChemistrysAn Advanced Treatise; Eyring, H., Henderson, D., Jost, W., Eds.; Academic Press: New York, 1970; Vol. 10. (17) Murch, G. E. Philos. Mag. 1981, A43, 871. (18) Murch, G. E.; Thorn, R. J. J. Phys. Chem. Solids 1977, 38, 789. (19) Kehr, K.; Binder, K. In Applications of the Monte Carlo Method in Statistical Physics, Topics in Current Physics; Binder, K., Ed.; Springer-Verlag: Berlin, 1987; Vol. 36, p 181.
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Figure 3. Site specific surface coverage θ1, θ2 vs total surface coverage θ for a noninteracting lattice gas on a heterogeneous surface with two types of adsorption sites, deep traps (2, c2 ) 0.2) and shallow traps (1, c1 ) 1 - c2). Results are shown for three characteristic temperatures, expressed in terms of (2 1)/kBT: (×) (2 - 1)/kBT ) 1.20; (+) (2 - 1)/kBT ) 2.41; (*) (2 - 1)/kBT ) 4.82. Symbols denote Monte Carlo results, while solid lines represent calculations according to eqs 13-15.
( ) ( )
µ1 ) µ0 - 1 + kBT ln
θ1 1 - θ1
µ2 ) µ0 - 2 + kBT ln
θ2 1 - θ2
(13)
Here θ1 and θ2 are the coverages of the corresponding shallow and deep trap sites. µ0 is the chemical potential of the noninteracting Langmuir gas. At thermodynamic equilibrium all chemical potentials are equal and we have
ln
[
]
θ2 1 - θ1 (2 - 1) ) 1 - θ2 θ1 kBT
(14)
The total coverage θ is given by
θ ) c2θ2 + (1 - c2)θ1
(15)
The coverage dependence of the site specific surface coverages θ1, θ2 is shown in Figure 3 for three representative temperatures. First it is noted that there is excellent agreement between Monte Carlo results (symbols) and theoretical values calculated according to eqs 13-15. Two general facts stand out: For a given total surface coverage θ2 is always substantial larger than θ1, and θ2 increases with decreasing temperatures while θ1 decreases. At low temperatures (2 - 1)/kBT ≈ 4.82 the deep traps are almost saturated for θ J c2 while the shallow traps remain empty for θ j c2. Figure 4 shows the coverage dependence of the normalized tracer and chemical diffusion coefficients together with the normalized mean square fluctuations 〈(δN)2〉/ 〈N〉. At high but finite temperatures such as (2 - 1)/kBT ) 1.20 the deviations from Langmuir behavior (T f ∞) are small: The chemical diffusion coefficient does not depend much on coverage while the tracer diffusion coefficient decreases monotonic. However, their absolute values are slightly reduced with respect to the Langmuir
Figure 4. Normalized surface diffusion coefficients and mean square fluctuations 〈(δN)2〉/〈N〉 vs surface coverage θ for the bivariate trap model. Shown are results for different temperatures, expressed in terms of (2 - 1)/kBT.
case indicating that diffusion is slowed down as adatoms are adsorbed at deep traps. Upon decreasing the temperature the effect of the deep traps becomes more pronounced. At (2 - 1)/kBT ) 4.82 the measured values of D* and DKG are very low up to θ ≈ 0.2 where the deep traps are nearly saturated. At high total coverages DKG increases monotonic but seems to level off at a saturation value as θ approaches unity. In contrast, D* goes through a relative maximum around θ ≈ 0.4. In order to explain the maximum in the coverage dependence of D*, we note that the occupation of shallow trap sites (after saturation of the deep ones) causes an increase of the average jump probability 〈Pj〉 (eq 11) which accounts for the increase of D* at θ J 0.2. For larger values of θ, however, the (1 - θ) behavior of the vacancy availability factor V dominates the tracer diffusion coefficient. The coverage dependence of the normalized mean square fluctuations 〈(δN)2〉/〈N〉, which represent the inverse of the thermodynamic factor (eq 7), shows the expected 1 - θ behavior at infinite temperatures, i.e. (2 - 1)/kBT ) 0. At lower temperatures there are deviations from ideal behavior, especially at low surface coverages, where most of the adatoms are trapped by the deep traps. However, these deviations are relatively small indicating that the chemical diffusion coefficient is basically determined by the jump diffusion coefficient (eq 7). The effect of surface heterogeneity on the temperature dependence of surface diffusion is shown in Figure 5. It is quite obvious that tracer and chemical diffusion coefficients exhibit well-pronounced decreases as temperature decreases. This is true with one exception, namely, the behavior of D* at high coverages where diffusion out of deep traps probably does not play an important role. At the lowest coverage studied, θ ) 0.1, diffusion depends strongly on temperature indicating that diffusion is largely dominated by the diffusion out of the deep traps. In order to rationalize these findings, we present activation energies of tracer diffusion EA* obtained upon
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a
b Figure 6. Normalized surface diffusion coefficients and mean square fluctuations 〈(δN)2〉/〈N〉 vs surface coverage θ for a lattice gas with nearest neighbor repulsive interactions, φNN, on a homogeneous square lattice. Shown are results for different temperatures, expressed in terms of |φNN|/kBT: (b) |φNN|/kBT ) 0; (×) |φNN|/kBT ) 1.20; (+) |φNN|/kBT ) 2.41; (*) |φNN|/kBT ) 4.82.
Figure 5. (a) Arrhenius plot of the normalized surface diffusion coefficients D*, DKG for various surface coverages as indicated. Simulations are performed for the bivariate trap model with c2 ) 0.2. Temperature is expressed in terms of (2 - 1)/kBT. (b) Activation energies EA* and EA for the tracer and chemical surface diffusion obtained from (a).
differentiation of the Arrhenius plots of Figure 5a. Note that EA* is expressed in units of (2 - 1). Positive EA* values correspond to the intuitively expected behavior: Diffusion is slowed down as the temperature decreases. Except for θ ) 0.1 all EA* values are relatively small, EA* < 0.3(2 - 1). As already mentioned this behavior can be attributed to the fact that the deep traps are nearly saturated at such total coverages. The effect of the deep traps becomes more and more negligible as coverage increases since then the fraction of adatoms on shallow trap sites increases. The mobility of such adatoms is substantially larger and, therefore, it can be concluded that mobile adatoms on shallow trap sites dominate surface diffusion. However, at θ ) 0.1 almost all adatoms are trapped at low temperatures and diffusion necessarily requires detachment from deep trap sites. At higher temperatures a substantial fraction of the adatoms are
located at shallow traps which causes a net decrease of the activation energy for tracer diffusion. 3.2. Effect of Repulsive Nearest Neighbor Interactions on Surface Diffusion on Homogeneous Surfaces. We have investigated the effect of repulsive nearest neighbor interactions, φNN, on lattice gas diffusion by using a homogeneous square lattice. Thus the lattice gas Hamiltonian is basically given by eq 1 but with i° being fixed for all adsorption sites. This system, which has been studied in detail in the course of earlier work,6 has been reexamined for two reasons: First, we wanted to verify the numerical results of the earlier work by using the improved Monte Carlo algorithm on the massive parallel Intel Paragon computer, and second, it appeared to be necessary to obtain a substantially larger number of data points especially at low temperatures for the comparison between homogeneous and heterogeneous lattices. The phase diagram of the lattice gas with nearest neighbor repulsion (which is shown in ref 20) shows c(2 × 2) ordering around half coverage below a critical temperature21
|φNN|/kBTc ) 1.76
(16)
The phase boundary separating c(2 × 2) ordered and disordered lattice gas structures is well-known,20 and therefore, diffusion simulations can be carried out in the existence range of ordered and disordered structures. Figure 6 shows the coverage dependence of surface diffusion coefficients and the normalized mean square fluctuations 〈(δN)2〉/〈N〉 for several representative temperatures. The first thing to note is that surface diffusion is accelerated in the presence of nearest neighbor repulsive (20) Binder, K.; Landau, D. P. Phys. Rev. 1980, B21, 1941. (21) Onsager, L. Phys. Rev. 1944, 65, 117.
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a
b
Figure 8. Site specific surface coverages θ1, θ2 vs total surface coverage θ for a lattice gas with nearest neighbor repulsive interactions, φNN, on a heterogeneous surface with two types of adsorption sites, deep traps (2, c2 ) 0.2) and shallow traps (1, c1 ) 1 - c2). Simulations are performed for (2 - 1) ) |φNN|. Results are shown for several characteristic temperatures, expressed in terms of |φNN|/kBT: (×) |φNN|/kBT ) 1.20; (+) |φNN|/ kBT ) 2.41; (*) |φNN|/kBT ) 4.82.
square fluctuations 〈(δN)2〉/〈N〉 (eq 7). This can be understood by recalling that 〈(δN)2〉/〈N〉 is proportional to the two-dimensional compressibility K of the adlayer
〈(δN)2〉 〈N〉 ) kBTK A 〈N〉
Figure 7. (a) Arrhenius plot of the normalized surface diffusion coefficients D*, DKG for various surface coverages as indicated. Simulations are performed for a lattice gas with nearest neighbor repulsive interactions, φNN, on a homogeneous square lattice. Temperature is expressed in terms of |φNN|/kBT. (b) Activation energies EA* and EA for the tracer and chemical surface diffusion obtained from (a).
interactions. However, this effect is very small at low coverages where the adatoms are far apart (at least on average). At temperatures well above Tc (i.e., for φNN/kBT ) 1.20) the diffusion coefficients vary smoothly upon increasing the total coverage θ. If the temperature is lowered below Tc, then the tracer diffusion coefficient D* exhibits a well-pronounced minimum sharply at half coverage. This behavior is clearly attributed to the c(2 × 2) ordering. In ref 6, where D* was analyzed according to eq 11, it was demonstrated that the minimum of D* basically reflects minima of the average jump probability 〈Pj〉 and the correlation factor f. The most striking finding probably is that the minimum of D* corresponds to a sharp maximum of DKG (i.e., a maximum of the thermodynamic factor (∂µ/kBT)/∂ ln θ) which in turn is associated with a minimum of the mean
(17)
where A is the area. K is smallest for the well-ordered c(2 × 2) lattice gas phase. Figure 7a shows normalized D* and DKG values for various total coverages θ vs inverse temperature, |φNN|/ kBT. At high coverages, θ g 0.7, there is Arrhenius behavior with high activation energies EA, EA* throughout the whole temperature range under investigation (Figure 7b). It should be noted that these are negative activation energies, since the nearest neighbor interactions are repulsive (i.e., φNN < 0).22 For low coverages, θ e 0.3, D* and DKG are weakly activated at high temperatures, while activation energies are close to zero at low temperatures. At half coverage, θ ) 0.5, both D* and DKG are activated above Tc. Below that temperature D* and DKG decreases upon decreasing the temperature indicating that the corresponding activation energies EA, EA* are positive. However, upon further decreasing the temperature DKG increases again and approaches very high values well below Tc while D* decreases. In order to rationalize this striking difference between tracer and chemical diffusion coefficient, we recall that both diffusion represent completely different views of surface diffusion. The tracer diffusion coefficient describes the motion of tagged particles on the surface, and it is quite obvious that the motion of adatoms in the more or less perfectly ordered c(2 × 2) structure requires breaking of the order which costs energy, so that a positive activation energy occurs. The chemical diffusion coefficient, however, describes the decay of natural density fluctuations and is at least partly (22) Negative activation energies EA correspond to the situation that the total activation energy for diffusion is less than it would be for the noninteracting case, i.e., the Langmuir gas, D°.
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Figure 9. Snapshots of representative lattice gas configurations for a lattice gas with nearest neighbor repulsive interactions, φNN, on a heterogeneous surface: (a) θ ) 0.1; (b) θ ) 0.35; (c) θ ) 0.5; (d) θ ) 0.7. Simulations are performed for (2 - 1) ) |φNN| at |φNN|/kBT ) 2.41. (O) denotes filled shallow traps; (b) denotes filled deep traps.
governed by the thermodynamic factor of eq 7. In the presence of the well-ordered c(2 × 2) lattice gas phase the mean square fluctuations 〈(δN)2〉/〈N〉 are extremely small.6 Thus the thermodynamic factor and the chemical diffusion coefficient are large, indicating that DKG is dominated by the thermodynamic factor in the stability range of the c(2 × 2) lattice gas phase. 3.3. Effect of Repulsive Nearest Neighbor Interactions and Surface Heterogeneity. Finally we will study surface diffusion for a lattice gas with nearest neighbor repulsive interactions on a heterogeneous surface. As already mentioned surface heterogeneity is introduced through the bivariate trap model with random arrangement of the deep traps. Here we assume 2 - 1 ) |φNN|; i.e., the site energy difference of both types of traps is numerically equal to the absolute value of the nearest neighbor repulsion. Clearly this very simple model does not exhaust, or perhaps even approximate, all real situations. Although no Monte Carlo simulation of a bivariate trap model can do full justice to surface diffusion of interacting species on a heterogeneous surface, it is possible to analyze the general effects of ad-ad interactions and surface heterogeneity on surface diffusion, and this is done in the following section. Figure 8 shows Monte Carlo results for the site specific coverages θ1 and θ2 of shallow and deep traps, respectively. At temperatures well above Tc, i.e., at |φNN|/kBT ) 1.20, the curves are very similar with respect to the corresponding curves of the noninteracting case (Figure 3). The coverage of the deep trap sites, θ2, is always larger than θ1 indicating the preferential occupation of these sites even if nearest neighbor repulsive interactions are present. At temperatures slightly below Tc, i.e., at |φNN|/kBT ) 2.41, the site specific coverages show a striking behavior. For
low total coverages θ2 is substantially larger than θ1 as is intuitively expected. However, if θ approaches half coverage, θ2 goes through a relative minimum which obviously is induced by the nearest neighbor repulsive interactions. Characteristic snapshots of the lattice gas indicates the existence of large c(2 × 2) ordered patches superimposed on the heterogeneous surface (Figure 9c). The ordering is disturbed especially in the vicinity of clusters of deep traps. c(2 × 2) ordered patches are not observed at θ ) 0.1 and 0.7 while there are small clusters of local c(2 × 2) ordering at θ ) 0.35. Upon further decreasing the temperature, i.e., at |φNN|/kBT ) 4.82, θ1 and θ2 behave very similar at low total coverages indicating that the nearest neighbor repulsion prevents the preferential occupation of the deep trap sites. Sharply above half coverage θ2 approaches saturation very rapidly while θ1 shows a plateau at about θ1 ≈ 0.5. Above θ ≈ 0.6, θ1 increases again and approaches saturation. The detailed interpretation of the coverage and temperature dependence of the site specific surface coverages guides us to the conclusion that these quantities are largely influenced by the surface heterogeneity at T > Tc, while the effect of the c(2 × 2) ordering induced by the nearest neighbor repulsive interactions is dominating below Tc. The coverage dependence of the normalized surface diffusion coefficients and of the normalized mean square fluctuations 〈(δN)2〉/〈N〉 is shown in Figure 10a. The general behavior of these quantities is remarkably similar to the corresponding data of the lattice gas with nearest neighbor repulsions on the homogeneous surface. The effects of surface heterogeneity on surface diffusion are relatively small albeit significant, especially in the case of c(2 × 2) ordering. In order to compare the coverage dependence of surface diffusion for the heterogeneous and
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a
b
Figure 11. (a) Arrhenius plot of the normalized surface diffusion coefficients D*, DKG for various surface coverages as indicated. Simulations are performed for a lattice gas with nearest neighbor repulsive interactions, φNN, on a heterogeneous square lattice. Temperature is expressed in terms of |φNN|/ kBT. (b) Activation energies EA* and EA for the tracer and chemical surface diffusion obtained from (a). Figure 10. (a) Normalized surface diffusion coefficients and mean square fluctuations 〈(δN)2〉/〈N〉 vs surface coverage θ for a lattice gas with nearest neighbor repulsive interactions, φNN, on a heterogeneous square lattice. Shown are results for different temperatures, expressed in terms of |φNN|/kBT: (b) |φNN|/kBT ) 0; (×) |φNN|/kBT ) 1.20; (+) |φNN|/kBT ) 2.41; (*) |φNN|/kBT ) 4.82. (b) Comparison of diffusion coefficients and mean square fluctuations for the heterogeneous and homogeneous surfaces according to Figures 6 and 10a.
homogeneous surfaces, it is quite useful to draw relative quantities, e.g., D*(het)/D*(homo) (Figure 10b). We first note that for nearly all surface coverages and temperatures these relative diffusion coefficients vary between 0.4 and 0.8, indicating that the existence of deep traps slows down surface diffusion. The probably most striking behavior is clearly visible at low temperatures (|φNN|/kBT ) 4.82) and total coverages around 0.5, where the diffusion data show
a relative minimum of the tracer and a relative maximum of the chemical diffusion coefficient. Figure 10b shows that D*(het)/D*(homo) > 1 under these circumstances, indicating that the minimum of D* is less pronounced. Moreover, we find DKG(het)/DKG(homo) < 1 at θ ) 0.5 indicates that the maximum of DKG is also less pronounced. In addition, the maximum of DKG is substantially broadened for θ j 0.5 indicating that the ordering phenomenon is smeared out over a certain range of surface coverages. The effect of surface heterogeneity on the chemical diffusion coefficient is largely due to substantial changes of the relative mean square fluctuations (Figure 10b, lower panel). We attribute all these findings to the substantial disturbance of the perfect c(2 × 2) ordering in the presence of surface heterogeneity (see Figure 9c). The temperature dependence of surface diffusion for our bivariate trap model in the presence of ad-ad
Surface Diffusion
Langmuir, Vol. 13, No. 5, 1997 1009
interactions does not exhibit qualitative surprises (Figure 11a). The measured diffusion coefficients are slightly reduced with respect to the homogeneous case (Figure 7a) indicating the slowing down of surface diffusion in the presence of deep traps. As already mentioned, the effect of the surface heterogeneity is mostly present at half coverage where the relative minimum of D* and the maximum of DKG are less pronounced. As a consequence the activation energies of surface diffusion (Figure 11b) are very similar to the homogeneous case (Figure 7b) except for θ ) 0.5. We have finally investigated the influence of surface heterogeneity in the vicinity of half coverage by measuring surface diffusion coefficients for various values of (2 1)/|φNN| (Figure 12). It is quite obvious that an increase of the energetical surface heterogeneity strongly influences the chemical diffusion coefficient at θ ) 0.5, while the effect on the tracer is much less pronounced. This finding can be attributed to local disturbances of the c(2 × 2) order at half coverage which as already mentioned is induced by the surface heterogeneity. 4. Summary We have studied surface diffusion phenomena on homogeneous and heterogeneous surfaces of square symmetry. Surface heterogeneity is introduced by means of the so-called bivariate trap model. It is assumed that the deep traps are randomly distributed. Monte Carlo modeling has been utilized for the determination of the tracer and chemical surface diffusion coefficients. In the present work Kubo-Green and fluctuation method are utilized to determine the chemical diffusion coefficient. The work presented here has clearly shown that surface heterogeneity does affect surface diffusion of both noninteracting and interacting adsorbates. The effects are largely pronounced at low total coverages θ and low temperatures T where most of the adatoms are trapped by the deep traps. At higher coverages the effects are
Figure 12. Normalized surface diffusion coefficients vs (2 1)/|φNN| for a lattice gas with nearest neighbor repulsive interactions, ψNN, on a heterogeneous square lattice. Shown are results for different total coverages θ. Simulations are performed for |φNN|/kBT ) 4.82.
relatively small albeit significant. The existence of deep traps disturbs c(2 × 2) ordering at θ ) 0.5 and low temperatures. It is shown that the breakdown of the order substantially affects surface diffusion. Acknowledgment. It is a pleasure to acknowledge many helpful and stimulating discussions with R. Gomer, W. Kehr, V. Pereyra, and G. Zgrablich. This work was supported by the Institut fu¨r Festko¨rperforschung of the Forschungszentrum Ju¨lich. LA950782N
