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desorption model for disordered surfaces
Chemical Physics North-Holland
163 ( 1992) 307-3 12
A novel adsorption/desorption J. Mai, Th. Koslowksi
model for disordered surfaces
and W. von Niessen
Inmtut ftir Physlkalrsche und Theoretlsche Chemre der Tcr, Hans-Sommer-Strasse 10, l4’-3300 Brunswick, German} Received
18 December
199
1
A new model for the adsorption and desorption of molecules on a surface is introduced. This model disorder. It shows the strong influence of disorder for low temperatures whtch has also been observed duce a mean-field ansatz which describes the global system properties m the high temperature limit. structures and for global system properties at low temperatures the neglect of disorder is not vahd different predrction of the system properttes.
1. Introduction Surfaces play an important role in various kinds of physical, chemical and biological processes. Therefore much effort has been undertaken to understand the fundamental processes of the interaction between surfaces and molecules. A very important example is the field of heterogeneous catalysis. Most of the work concentrates on perfect surfaces without any defects. Recent experiments show, however, that the behaviour of these systems depends strongly on surface defects [ 11. It is now realized that electronic properties such as the surface core level shift, the surface states and the band structure may change completely. As a result the sticking coefficient (which describes the interaction between molecules in the gas phase and the surface) is strongly influenced. A well-known case is the adsorption of HZ on W ( 111) where the sticking coefficient varies between zero and unity [ 2 1. Another example is the strong dependence of the reaction probability on the number of steps on a surface. Hopster et al. [ 31 have shown that the reaction probability of CO on Pt ( 111) is reduced on stepped surfaces. This is a very important result for industrial catalysis. For these reasons we introduce a simple model which describes the adsorption and desorption of gaseous molecules on a surface which may be strongly disordered. It is important to note that the disordered Hamiltonian used to describe the surface does 0301-0104/92/$05.00
0 1992 Elsevrer Science Pubhshers
takes into account surface in expenments. We mtroFor the study of adsorbate and leads to a completely
not include structures like steps or kinks. Therefore effects purely arising from topological features like multiple bonding of adsorbed molecules to surface sites cannot be understood in the framework of this model. We study the changing behaviour of the system as a function of a disorder parameter. In section 2 we present the adsorption model and in section 3 the details of the Monte Carlo simulation. Section 4 deals with the interpretation of the results. In section 5 a mean-field ansatz for this system is introduced. Conclusions are derived in section 6.
2. A model for adsorption In this section, a simple model describing the disordered metal surface, the adsorbed molecule and the interaction between surface and molecule is presented. All interactions are of a one-electron tightbinding type. The model for the metal surface is based on a square lattice, each surface atom is interacting with its four nearest neighbours. Disorder is introduced by the application of the Anderson model [ 41, the model most frequently used for studies of disorder effects. The Anderson Hamiltonian reads fi=
c IS,)%(S,l+ I
B.V. All nghts reserved.
1 Is,>K,(s,I.
l.J+l
J. Mar et al. /AdsorptIon on duordered surfaces
308
The transfer matrix elements are constant ( V,,= V= 10 eV ), E,is uniformly distributed within an interval [ - WV/2, + WV/2]. It is the source of disorder in the Hamiltonian. 1s,) is an atomic orbital localized at position i of the surface. All atomic orbitals are assumed to be orthogonal. Each site feeds one electron into the system, so finally half of the surface states are occupied by two electrons. The molecule to be adsorbed on the surface is modelled by a two-level system (TLS). The eigenvalues of the TLS are given by Etp= - AE and E,, = AE. Numerical studies have been performed for AE= 2 V and AE= 4 V. The eigenfunctions corresponding to E,pand are ItP)=(lfl)+ltZ))l@ and It,>= G, ( I tl ) - )tz) )/$. The TLS can be interpreted as a diatomic molecule, with highest occupied,molecular orbital (HOMO) I tP) and lowest unoccupied molecular orbital (LUMO) 1t,) . The atomic orbital I tl ) is localized on the first atom of the diatomic molecule, I t2) is localized on the second atom of the diatomic molecule. The simplest physical realization of the TLS described here corresponds to the hydrogen molecule. A diagram of the energies of the surface and the TLS is presented in fig. 1. The lower level is occupied by two electrons. Whenever a molecule is adsorbed on a surface site i, the TLS and surface energies change. These changes in energy are calculated by second order perturbation theory. The geometry of the adsorption is shown in fig. 2. The only new matrix element resulting from adsorption has been set to V,,= 4.5 eV, connecting one atom of the diatomic molecule and a surface site. In-
E
AET
-----
AE
-
_-_
Fig. 1. Energy diagram For details see text.
-
1+ Two-level
E,,,ltu>
Et, , ItI >
- system
of the surface and the two-level
system.
Two- level system
Surface Fig. 2. Geometry
of adsorption
and site labels.
dices are written whenever necessary and have the following meaning: s denotes surface, t corresponds to the TLS. i is a surface site, (Ycounts surface eigenvalues Es,. c,, is the eigenvector component at site i corresponding to the eigenvector of E,. Due to an adsorption of the TLS at site i, the energy of surface eigenstate E, is changed by (2) The energy change is weighted by the squared eigenvector coefficient corresponding to site i in eigenstate cy. The lowest level of the TLS exhibits an energy change of (3) The total adsorption energy is given by the sum of the changes of the energies of all occupied orbitals. All contributions of first order perturbation theory equal zero. Higher order perturbation terms have not been included, because the adsorption energy at site i would not be independent of adsorption processes taking place at neighbour sites. From a chemical point of view, the model described above corresponds to the linear combination of molecular orbitals, treated by second order perturbation theory. Similar models have been studied in the literature to calculate the geometry of adsorption on ordered surfaces [ 5 1. Whenever disorder is introduced to a Hamiltonian, localization of electron eigenfunctions may occur [ 4 1. For the system described above, electron localization leads to an adsorption energy which depends on the place of adsorption. Localization is more important in low-dimensional systems like chains or surfaces than it is for three-dimensional
J. MaI et al. /Adsorption on disordered surfaces
materials. There is still an open discussion whether extended states can occur in two dimensions at all. The localization at an arbitrary small amount of disorder is required by the scaling theory of localization [ 6 1. There have been considerable doubts about the validity of this theory arising from numerical studies of models like quantum percolation [ 71 or continuous random networks [ 8 1. For the Anderson model, states in the middle of the surface energy band that look extended on a length scale comparable to the size of the systems studied in this article have been computed by Yoshino and Okazaki [ 91 for a small amount of disorder ( W/ V< 6.5). However, localized states will always arise at the band edges of the surface density of states and thus have an influence on the adsorption energy. A Lanczos algorithm [ 10.1 1 ] has been used to compute the eigenvalues and eigenfunctions of the surface Hamiltonian, which are required for the calculation of the adsorption energy within the frame of perturbation theory. As all of the eigenvectors of a large symmetric matrix had to be computed, a storage efficient version of the algorithm recently developed by two of the present authors [ 121 has been used.
3. The simulation We calculate the adsorption energies by the procedure described above for a surface which is represented by a two-dimensional square lattice with periodic boundary conditions. On this lattice a Monte Carlo simulation for an adsorption/desorption model is performed. The following rules are used: For a randomly chosen site the number of occupied nearest neighbours z is calculated. The transition probabilities that a vacant cell becomes occupied (adsorption) and that an occupied cell becomes vacant (desorption) in the next time step are given by W, =min{l,
exp(E,,,-Z,,)}
(Adsorption),
(4)
and WI =min(l,
exp( -Eads+=Enn)J
(Desorption)
, (5)
respectively. All adsorption energies are given in units of kT. If the central cell is vacant and WI is larger
309
than a random number c ([E [ 0,1 ] ), adsorption takes place. The equivalent rule holds for the desorption event for an occupied central cell. These transitions take the interaction between a molecule and the surface and between the adsorbed molecules into account. This procedure is performed on a 50 x 50 lattice with 10000 Monte Carlo steps where one Monte Carlo step is defined as one update on the average for each cell of the lattice. We perform this simulation for different parameter sets: In all simulations we use the interaction energy between adsorbed molecules E,, = 2kT where k is the Boltzmann constant and T is the temperature. We study only the non-trivial case of repulsive adatom interaction. In the case of attractive adatom interactions the surface will become completely covered by molecules. For the temperature we choose T= 500 K and T= 1000 K.
4. Results The simulation described above is performed on a disordered surface according to the scheme described above. The mean value and the standard deviation of the adsorption energies for the different parameters are listed in table 1. With increasing disorder the mean values of the adsorption energies decrease for all temperatures and all AE. As the energies are given in kT the adsorption energy at T= 500 K is twice as high as the adsorption energy at T= 1000 K. The adsorption energy decreases with increasing AE. A high value of W favours strong spatial fluctuations in the adsorption energy which vary from 10% to 50%. In fig. 3 the coverage 8 of the molecules (average number of molecules per lattice site) is shown as a function of the disorder parameter W. With increasing l+’ the coverages decrease because the adsorption energy of the surface decreases and the desorption event becomes more important (see table 1). This effect is clearly much more important for lower temperatures (T= 500 K) and for smaller A.E (AE=2.0) because the difference in the adsorption energy is larger than for AE=4.0. Therefore the coverage is more influenced by the disorder parameter W at low temperatures and low A/Z. Next we would like to study the influence of spatial fluctuations of the adsorption energy. To this end all
J. MaI et al. /Adsorptron
310 Table I Mean values and standard W
deviations
of the adsorption
on duordered surfaces
energies of the surface for different T=lOOOK
T= 500 K
4.0 8.0 12.0 16.0 20.0
T and AE
AE=2v
AE=4v
AE=zv
AE=4v
-11.4kO.9 _ 10.0t2.0 -8.8f2.7 -7.8k3.1 -7.113.3
-7.1 kO.3 -6.5k0.9 -5.9+ 1.2 -5.4k 1.5 -4.9f1.6
-5.7f0.4 -5.OfO.l -4.4i 1.3 -3.9f 1.5 -3.5+ 1.6
-3.5kO.2 - 3.2 f 0.4 - 3.0 f 0.6 -2.7k0.7 -2.5f0.8
10
jI_:::::
08
06 8
V
v’
04
O.L--
02
02.-
00 I-L
8
12 W
16
20
Oat 1
8
12 W
16
20
Ftg. 3. Mean value of the steady state coverage 8 as a function of the disorder parameter W for the unaveraged adsorption energoes of the surface for different T and AE. The symbols denote: (o) T=500K,E=2V; (*) T=500K,E=4V; (A) T=lOOOK, E=2V; (V) T= 1000 K, E=4V.
Fig. 4. Mean value of the steady state coverage 8 as a function of the disorder parameter W for the averaged adsorption energies of the surface for different T and AE. The symbols denote: ( o ) T=500 K, E=2V; (*) T=500 K, E=4V; (A) T=lOOO K, E=2V; (V) T=lOOOK,E=4C:
simulations were performed with the surface calculated by the procedure described above with an averaged surface (with respect to the adsorption energy) on which all cells have the same adsorption energy which is the mean value of the original surface. The results for this ordered (averaged) surface are shown in fig.‘4. Comparing the results of fig. 3 and 4 a strong influence of disorder on the coverage can be seen for T= 500 K and AL?= 2.0. This can be easily understood from the fact that higher temperatures smooth out these disorder effects and higher AE leads to smaller fluctuations in the adsorption energy. Partial coverages S, are shown as a function of the
disorder parameter W in fig. 5 (disordered surface) and fig. 6 (ordered surface). Here 19,is the coverage by molecules with i occupied nearest neighbours. We introduce 8, as a simple measure for the developed adsorbate structure. From fig. 5 one can see that with increasing W the number of isolated molecules (So) and the number of molecules which have only one neighbour increases and that the number of molecules with more than one neighbour decreases. Therefore at high values of the disorder parameter W a non-compact adsorbate structure is obtained. This behaviour can be understood from the fact that the adsorption energy decreases with increasing W. At
J. Ma1 et al. /Adsorptron on disordered surfaces
I L
8
12
16
20
311
cules with no or a single neighbour can remain on the surface. For the averaged surface (fig. 6) the same tendency holds as for the surface with local fluctuations. But without these fluctuations nearly all adsorbed molecules are isolated at high values of IV. Due to spatial fluctuations regions can exist with a high adsorption energy in which pairs and triples of molecules appear. The largest difference between the averaged and the un-averaged surface appears at high values of W because the effects of local fluctuations are most important here. Such local effects do not appear on the averaged surface. This is an example for the important influence of local effects which leads to a totally different structure whereby global measures (like the coverage) are nearly unaffected.
W Fig. 5. Mean value of the steady state partial coverages e, as a function of the disorder parameter Wfor the unaveraged adsorption energies of the surface for T= 500 K and AE= 4V. The symbolsdenote: (0) i=O: (A) r=l; (V) 1=2; (*) 1=3; (X) 1=4.
5. A mean-field approximation In this section we introduce a simple mean-field ansatz which is able to describe the steady state coverage to a certain degree. For this purpose we use the following differential equation which takes into account only the mean number of the occupied neighbours and the mean value of the adsorption energy : &(l-@)exp(E,,,-4E,,@) -@exp(4E,,,@-E,,,) .
(6)
Using a Taylor expansion for the exponential function including only the linear term and setting the left side of eq. (6) equal to zero the analytic result for the steady state coverage reads @= I L
El
12 W
16
20
Ftg. 6. Mean value of the steady state parttal coverage @, as a functton of the disorder parameter W for the averaged adsorption energies of the surface for T= 500 K and AE= 4V. The symbolsdenote: (0) r=O; (A) r=l; (V) i=2; (*) 1=3; (x) r=4.
large Wthe adsorption energy is comparable with the repulsive interaction energy of three to four occupied neighbours. In this region the desorption plays the most important role. During the process only mole-
1+ Eads 2(1+2E,,).
The results are shown in table 2 for AE= 2.0 and for AE= 4.0. The first row shows the adsorption energy. The second, third and fourth row show the coverage obtained by the simulation, the results of the numerically solved mean-field ansatz ( 6 ) (using the RungeKutta algorithm) and the analytically obtained values (7). In the fifth row the differences between the simulation and the numerically solved ansatz are listed. The analytic results differ more from the simulation than the numerical results because of the simplification in the representation of the exponential
J. Mar et al /AdsorptIon on drsordered surfaces
312 Table 2 Mean value of the steady state coverage
8 obtained
by the stmulatton
and the mean-field
ansatz for different
I+: T and AE
AE=4J’
AE=2J E ads
8 Llrn
8 MF
8 ana,yt
A8
E ads
8 *un
8 MF
8 anaM
A8
-5.1 - 5.0 -4.4 -3.9 -3.5
0.56 0.52 0.51 0.50 0.49
0.64 0.58 0.53 0.49 0.46
0.67 0.60 0.54 0.49 0.45
0.08 0.06 0.02 0.01 0.03
-3 5 -3.2 -3.0 -2.7 -2.5
0.49 0.48 0.47 0.46 0.45
0.46 0.43 0.42 0.39 0.38
0.45 0.42 0.40 0.37 0.35
0.03 0.05 0.05 0.07 0.07
function by a Taylor expansion with only one term. The results obtained by the mean-field ansatz are in qualitatively good agreement with the results obtained by the simulation. Quantitatively correct results cannot be arrived at by this simple ansatz which does not account for local properties of the surface and the adsorbate structure. We compare the results of the mean-field ansatz only with the results of the simulation at T= 1000 K because the neglect of any local effects is less valid for lower temperatures.
A+A-tO type. It is also possible to introduce the adsorption of different molecules and their reaction. This will be done in the future.
6. Conclusions
References
The important influence of surface defects on system properties which has been observed in experiments can also be found in the present simple theoretical adsorption/desorption model. It shows the fundamental role disorder plays in surface processes. These effects are most important at low temperatures and low AE. For the high temperature limit the neglect of the effect of disorder is justified. For this region a mean-field ansatz describes global measures like the coverage qualitatively correctly but structural properties cannot be determined from this ansatz. For low temperature the behaviour of the system is dominated by disorder effects and correct predictions about the cluster structure of adsorbates can only be obtained by models which take this additional effect of disorder into account. The next straight forward step in a description of a heterogeneous catalysis system is the introduction of a chemical reaction. The model described in this article can be easily extended to study a reaction of the
[ I] K. Wandelt. pnvate commumcatron. [2] Z. Knor. Surface Set. 169 (1986) 317. [3] H. Hopster, H. Ibach and G. Comsa, J. Catal. 46 ( 1977) 37. [4] P.W. Anderson, Phys. Rev. 109 (1958) 1492. [ 51 R. Hoffmann, Sohds and surfaces: a chemrst’s view of bonding in extended structures (VCH Verlag, Wemheim, 1988). and references therem. (61 E. Abrahams, P.W. Anderson, D.C. Licciardello and T.V. Ramakrishnan, Phys. Rev. Letters 42 (1979) 673. [ 71 Y. Meir, A. Aharony and A.B. Hams, Europhys. Letters 10 (1989) 275, Th. Koslowski and W. von Nressen, Phys. Rev. B 42 ( 1990) 10342. [8] Th. Koslowskt and W. von Nressen, J. Phys. Cond. Matter 4 (1992) 1093. [ 9 ] S.Yoshino and M. Okazaki. J. Phys. Sot. Japan 43 ( 1977) 415. [lo] C. Lanczos, J. Res. Natl. Bur. Std. B 45 (1950) 225. [ 11 ] J.K. Cullum and R. Willoughby, Lanczos algonthms for large symmetric eigenvalue problems, Vols. 1, 2 (Birkhluser. Basel, 1985). [ 121 Th. Koslowski and W. von Niessen, submitted for publication.
Acknowledgement This research was supported by the Deutsche Forschungsgemeinschaft and in part by the Fonds der Chemischen Industrie.
163 ( 1992) 307-3 12
A novel adsorption/desorption J. Mai, Th. Koslowksi
model for disordered surfaces
and W. von Niessen
Inmtut ftir Physlkalrsche und Theoretlsche Chemre der Tcr, Hans-Sommer-Strasse 10, l4’-3300 Brunswick, German} Received
18 December
199
1
A new model for the adsorption and desorption of molecules on a surface is introduced. This model disorder. It shows the strong influence of disorder for low temperatures whtch has also been observed duce a mean-field ansatz which describes the global system properties m the high temperature limit. structures and for global system properties at low temperatures the neglect of disorder is not vahd different predrction of the system properttes.
1. Introduction Surfaces play an important role in various kinds of physical, chemical and biological processes. Therefore much effort has been undertaken to understand the fundamental processes of the interaction between surfaces and molecules. A very important example is the field of heterogeneous catalysis. Most of the work concentrates on perfect surfaces without any defects. Recent experiments show, however, that the behaviour of these systems depends strongly on surface defects [ 11. It is now realized that electronic properties such as the surface core level shift, the surface states and the band structure may change completely. As a result the sticking coefficient (which describes the interaction between molecules in the gas phase and the surface) is strongly influenced. A well-known case is the adsorption of HZ on W ( 111) where the sticking coefficient varies between zero and unity [ 2 1. Another example is the strong dependence of the reaction probability on the number of steps on a surface. Hopster et al. [ 31 have shown that the reaction probability of CO on Pt ( 111) is reduced on stepped surfaces. This is a very important result for industrial catalysis. For these reasons we introduce a simple model which describes the adsorption and desorption of gaseous molecules on a surface which may be strongly disordered. It is important to note that the disordered Hamiltonian used to describe the surface does 0301-0104/92/$05.00
0 1992 Elsevrer Science Pubhshers
takes into account surface in expenments. We mtroFor the study of adsorbate and leads to a completely
not include structures like steps or kinks. Therefore effects purely arising from topological features like multiple bonding of adsorbed molecules to surface sites cannot be understood in the framework of this model. We study the changing behaviour of the system as a function of a disorder parameter. In section 2 we present the adsorption model and in section 3 the details of the Monte Carlo simulation. Section 4 deals with the interpretation of the results. In section 5 a mean-field ansatz for this system is introduced. Conclusions are derived in section 6.
2. A model for adsorption In this section, a simple model describing the disordered metal surface, the adsorbed molecule and the interaction between surface and molecule is presented. All interactions are of a one-electron tightbinding type. The model for the metal surface is based on a square lattice, each surface atom is interacting with its four nearest neighbours. Disorder is introduced by the application of the Anderson model [ 41, the model most frequently used for studies of disorder effects. The Anderson Hamiltonian reads fi=
c IS,)%(S,l+ I
B.V. All nghts reserved.
1 Is,>K,(s,I.
l.J+l
J. Mar et al. /AdsorptIon on duordered surfaces
308
The transfer matrix elements are constant ( V,,= V= 10 eV ), E,is uniformly distributed within an interval [ - WV/2, + WV/2]. It is the source of disorder in the Hamiltonian. 1s,) is an atomic orbital localized at position i of the surface. All atomic orbitals are assumed to be orthogonal. Each site feeds one electron into the system, so finally half of the surface states are occupied by two electrons. The molecule to be adsorbed on the surface is modelled by a two-level system (TLS). The eigenvalues of the TLS are given by Etp= - AE and E,, = AE. Numerical studies have been performed for AE= 2 V and AE= 4 V. The eigenfunctions corresponding to E,pand are ItP)=(lfl)+ltZ))l@ and It,>= G, ( I tl ) - )tz) )/$. The TLS can be interpreted as a diatomic molecule, with highest occupied,molecular orbital (HOMO) I tP) and lowest unoccupied molecular orbital (LUMO) 1t,) . The atomic orbital I tl ) is localized on the first atom of the diatomic molecule, I t2) is localized on the second atom of the diatomic molecule. The simplest physical realization of the TLS described here corresponds to the hydrogen molecule. A diagram of the energies of the surface and the TLS is presented in fig. 1. The lower level is occupied by two electrons. Whenever a molecule is adsorbed on a surface site i, the TLS and surface energies change. These changes in energy are calculated by second order perturbation theory. The geometry of the adsorption is shown in fig. 2. The only new matrix element resulting from adsorption has been set to V,,= 4.5 eV, connecting one atom of the diatomic molecule and a surface site. In-
E
AET
-----
AE
-
_-_
Fig. 1. Energy diagram For details see text.
-
1+ Two-level
E,,,ltu>
Et, , ItI >
- system
of the surface and the two-level
system.
Two- level system
Surface Fig. 2. Geometry
of adsorption
and site labels.
dices are written whenever necessary and have the following meaning: s denotes surface, t corresponds to the TLS. i is a surface site, (Ycounts surface eigenvalues Es,. c,, is the eigenvector component at site i corresponding to the eigenvector of E,. Due to an adsorption of the TLS at site i, the energy of surface eigenstate E, is changed by (2) The energy change is weighted by the squared eigenvector coefficient corresponding to site i in eigenstate cy. The lowest level of the TLS exhibits an energy change of (3) The total adsorption energy is given by the sum of the changes of the energies of all occupied orbitals. All contributions of first order perturbation theory equal zero. Higher order perturbation terms have not been included, because the adsorption energy at site i would not be independent of adsorption processes taking place at neighbour sites. From a chemical point of view, the model described above corresponds to the linear combination of molecular orbitals, treated by second order perturbation theory. Similar models have been studied in the literature to calculate the geometry of adsorption on ordered surfaces [ 5 1. Whenever disorder is introduced to a Hamiltonian, localization of electron eigenfunctions may occur [ 4 1. For the system described above, electron localization leads to an adsorption energy which depends on the place of adsorption. Localization is more important in low-dimensional systems like chains or surfaces than it is for three-dimensional
J. MaI et al. /Adsorption on disordered surfaces
materials. There is still an open discussion whether extended states can occur in two dimensions at all. The localization at an arbitrary small amount of disorder is required by the scaling theory of localization [ 6 1. There have been considerable doubts about the validity of this theory arising from numerical studies of models like quantum percolation [ 71 or continuous random networks [ 8 1. For the Anderson model, states in the middle of the surface energy band that look extended on a length scale comparable to the size of the systems studied in this article have been computed by Yoshino and Okazaki [ 91 for a small amount of disorder ( W/ V< 6.5). However, localized states will always arise at the band edges of the surface density of states and thus have an influence on the adsorption energy. A Lanczos algorithm [ 10.1 1 ] has been used to compute the eigenvalues and eigenfunctions of the surface Hamiltonian, which are required for the calculation of the adsorption energy within the frame of perturbation theory. As all of the eigenvectors of a large symmetric matrix had to be computed, a storage efficient version of the algorithm recently developed by two of the present authors [ 121 has been used.
3. The simulation We calculate the adsorption energies by the procedure described above for a surface which is represented by a two-dimensional square lattice with periodic boundary conditions. On this lattice a Monte Carlo simulation for an adsorption/desorption model is performed. The following rules are used: For a randomly chosen site the number of occupied nearest neighbours z is calculated. The transition probabilities that a vacant cell becomes occupied (adsorption) and that an occupied cell becomes vacant (desorption) in the next time step are given by W, =min{l,
exp(E,,,-Z,,)}
(Adsorption),
(4)
and WI =min(l,
exp( -Eads+=Enn)J
(Desorption)
, (5)
respectively. All adsorption energies are given in units of kT. If the central cell is vacant and WI is larger
309
than a random number c ([E [ 0,1 ] ), adsorption takes place. The equivalent rule holds for the desorption event for an occupied central cell. These transitions take the interaction between a molecule and the surface and between the adsorbed molecules into account. This procedure is performed on a 50 x 50 lattice with 10000 Monte Carlo steps where one Monte Carlo step is defined as one update on the average for each cell of the lattice. We perform this simulation for different parameter sets: In all simulations we use the interaction energy between adsorbed molecules E,, = 2kT where k is the Boltzmann constant and T is the temperature. We study only the non-trivial case of repulsive adatom interaction. In the case of attractive adatom interactions the surface will become completely covered by molecules. For the temperature we choose T= 500 K and T= 1000 K.
4. Results The simulation described above is performed on a disordered surface according to the scheme described above. The mean value and the standard deviation of the adsorption energies for the different parameters are listed in table 1. With increasing disorder the mean values of the adsorption energies decrease for all temperatures and all AE. As the energies are given in kT the adsorption energy at T= 500 K is twice as high as the adsorption energy at T= 1000 K. The adsorption energy decreases with increasing AE. A high value of W favours strong spatial fluctuations in the adsorption energy which vary from 10% to 50%. In fig. 3 the coverage 8 of the molecules (average number of molecules per lattice site) is shown as a function of the disorder parameter W. With increasing l+’ the coverages decrease because the adsorption energy of the surface decreases and the desorption event becomes more important (see table 1). This effect is clearly much more important for lower temperatures (T= 500 K) and for smaller A.E (AE=2.0) because the difference in the adsorption energy is larger than for AE=4.0. Therefore the coverage is more influenced by the disorder parameter W at low temperatures and low A/Z. Next we would like to study the influence of spatial fluctuations of the adsorption energy. To this end all
J. MaI et al. /Adsorptron
310 Table I Mean values and standard W
deviations
of the adsorption
on duordered surfaces
energies of the surface for different T=lOOOK
T= 500 K
4.0 8.0 12.0 16.0 20.0
T and AE
AE=2v
AE=4v
AE=zv
AE=4v
-11.4kO.9 _ 10.0t2.0 -8.8f2.7 -7.8k3.1 -7.113.3
-7.1 kO.3 -6.5k0.9 -5.9+ 1.2 -5.4k 1.5 -4.9f1.6
-5.7f0.4 -5.OfO.l -4.4i 1.3 -3.9f 1.5 -3.5+ 1.6
-3.5kO.2 - 3.2 f 0.4 - 3.0 f 0.6 -2.7k0.7 -2.5f0.8
10
jI_:::::
08
06 8
V
v’
04
O.L--
02
02.-
00 I-L
8
12 W
16
20
Oat 1
8
12 W
16
20
Ftg. 3. Mean value of the steady state coverage 8 as a function of the disorder parameter W for the unaveraged adsorption energoes of the surface for different T and AE. The symbols denote: (o) T=500K,E=2V; (*) T=500K,E=4V; (A) T=lOOOK, E=2V; (V) T= 1000 K, E=4V.
Fig. 4. Mean value of the steady state coverage 8 as a function of the disorder parameter W for the averaged adsorption energies of the surface for different T and AE. The symbols denote: ( o ) T=500 K, E=2V; (*) T=500 K, E=4V; (A) T=lOOO K, E=2V; (V) T=lOOOK,E=4C:
simulations were performed with the surface calculated by the procedure described above with an averaged surface (with respect to the adsorption energy) on which all cells have the same adsorption energy which is the mean value of the original surface. The results for this ordered (averaged) surface are shown in fig.‘4. Comparing the results of fig. 3 and 4 a strong influence of disorder on the coverage can be seen for T= 500 K and AL?= 2.0. This can be easily understood from the fact that higher temperatures smooth out these disorder effects and higher AE leads to smaller fluctuations in the adsorption energy. Partial coverages S, are shown as a function of the
disorder parameter W in fig. 5 (disordered surface) and fig. 6 (ordered surface). Here 19,is the coverage by molecules with i occupied nearest neighbours. We introduce 8, as a simple measure for the developed adsorbate structure. From fig. 5 one can see that with increasing W the number of isolated molecules (So) and the number of molecules which have only one neighbour increases and that the number of molecules with more than one neighbour decreases. Therefore at high values of the disorder parameter W a non-compact adsorbate structure is obtained. This behaviour can be understood from the fact that the adsorption energy decreases with increasing W. At
J. Ma1 et al. /Adsorptron on disordered surfaces
I L
8
12
16
20
311
cules with no or a single neighbour can remain on the surface. For the averaged surface (fig. 6) the same tendency holds as for the surface with local fluctuations. But without these fluctuations nearly all adsorbed molecules are isolated at high values of IV. Due to spatial fluctuations regions can exist with a high adsorption energy in which pairs and triples of molecules appear. The largest difference between the averaged and the un-averaged surface appears at high values of W because the effects of local fluctuations are most important here. Such local effects do not appear on the averaged surface. This is an example for the important influence of local effects which leads to a totally different structure whereby global measures (like the coverage) are nearly unaffected.
W Fig. 5. Mean value of the steady state partial coverages e, as a function of the disorder parameter Wfor the unaveraged adsorption energies of the surface for T= 500 K and AE= 4V. The symbolsdenote: (0) i=O: (A) r=l; (V) 1=2; (*) 1=3; (X) 1=4.
5. A mean-field approximation In this section we introduce a simple mean-field ansatz which is able to describe the steady state coverage to a certain degree. For this purpose we use the following differential equation which takes into account only the mean number of the occupied neighbours and the mean value of the adsorption energy : &(l-@)exp(E,,,-4E,,@) -@exp(4E,,,@-E,,,) .
(6)
Using a Taylor expansion for the exponential function including only the linear term and setting the left side of eq. (6) equal to zero the analytic result for the steady state coverage reads @= I L
El
12 W
16
20
Ftg. 6. Mean value of the steady state parttal coverage @, as a functton of the disorder parameter W for the averaged adsorption energies of the surface for T= 500 K and AE= 4V. The symbolsdenote: (0) r=O; (A) r=l; (V) i=2; (*) 1=3; (x) r=4.
large Wthe adsorption energy is comparable with the repulsive interaction energy of three to four occupied neighbours. In this region the desorption plays the most important role. During the process only mole-
1+ Eads 2(1+2E,,).
The results are shown in table 2 for AE= 2.0 and for AE= 4.0. The first row shows the adsorption energy. The second, third and fourth row show the coverage obtained by the simulation, the results of the numerically solved mean-field ansatz ( 6 ) (using the RungeKutta algorithm) and the analytically obtained values (7). In the fifth row the differences between the simulation and the numerically solved ansatz are listed. The analytic results differ more from the simulation than the numerical results because of the simplification in the representation of the exponential
J. Mar et al /AdsorptIon on drsordered surfaces
312 Table 2 Mean value of the steady state coverage
8 obtained
by the stmulatton
and the mean-field
ansatz for different
I+: T and AE
AE=4J’
AE=2J E ads
8 Llrn
8 MF
8 ana,yt
A8
E ads
8 *un
8 MF
8 anaM
A8
-5.1 - 5.0 -4.4 -3.9 -3.5
0.56 0.52 0.51 0.50 0.49
0.64 0.58 0.53 0.49 0.46
0.67 0.60 0.54 0.49 0.45
0.08 0.06 0.02 0.01 0.03
-3 5 -3.2 -3.0 -2.7 -2.5
0.49 0.48 0.47 0.46 0.45
0.46 0.43 0.42 0.39 0.38
0.45 0.42 0.40 0.37 0.35
0.03 0.05 0.05 0.07 0.07
function by a Taylor expansion with only one term. The results obtained by the mean-field ansatz are in qualitatively good agreement with the results obtained by the simulation. Quantitatively correct results cannot be arrived at by this simple ansatz which does not account for local properties of the surface and the adsorbate structure. We compare the results of the mean-field ansatz only with the results of the simulation at T= 1000 K because the neglect of any local effects is less valid for lower temperatures.
A+A-tO type. It is also possible to introduce the adsorption of different molecules and their reaction. This will be done in the future.
6. Conclusions
References
The important influence of surface defects on system properties which has been observed in experiments can also be found in the present simple theoretical adsorption/desorption model. It shows the fundamental role disorder plays in surface processes. These effects are most important at low temperatures and low AE. For the high temperature limit the neglect of the effect of disorder is justified. For this region a mean-field ansatz describes global measures like the coverage qualitatively correctly but structural properties cannot be determined from this ansatz. For low temperature the behaviour of the system is dominated by disorder effects and correct predictions about the cluster structure of adsorbates can only be obtained by models which take this additional effect of disorder into account. The next straight forward step in a description of a heterogeneous catalysis system is the introduction of a chemical reaction. The model described in this article can be easily extended to study a reaction of the
[ I] K. Wandelt. pnvate commumcatron. [2] Z. Knor. Surface Set. 169 (1986) 317. [3] H. Hopster, H. Ibach and G. Comsa, J. Catal. 46 ( 1977) 37. [4] P.W. Anderson, Phys. Rev. 109 (1958) 1492. [ 51 R. Hoffmann, Sohds and surfaces: a chemrst’s view of bonding in extended structures (VCH Verlag, Wemheim, 1988). and references therem. (61 E. Abrahams, P.W. Anderson, D.C. Licciardello and T.V. Ramakrishnan, Phys. Rev. Letters 42 (1979) 673. [ 71 Y. Meir, A. Aharony and A.B. Hams, Europhys. Letters 10 (1989) 275, Th. Koslowski and W. von Nressen, Phys. Rev. B 42 ( 1990) 10342. [8] Th. Koslowskt and W. von Nressen, J. Phys. Cond. Matter 4 (1992) 1093. [ 9 ] S.Yoshino and M. Okazaki. J. Phys. Sot. Japan 43 ( 1977) 415. [lo] C. Lanczos, J. Res. Natl. Bur. Std. B 45 (1950) 225. [ 11 ] J.K. Cullum and R. Willoughby, Lanczos algonthms for large symmetric eigenvalue problems, Vols. 1, 2 (Birkhluser. Basel, 1985). [ 121 Th. Koslowski and W. von Niessen, submitted for publication.
Acknowledgement This research was supported by the Deutsche Forschungsgemeinschaft and in part by the Fonds der Chemischen Industrie.