- Email: [email protected]
On the sticking coefficient of large molecules on (100) surfaces of cubic metals
308
Surface
ON THE STICKING COEFFICIENT OF CUBIC METALS
OF LARGE MOLECULES
Science 233 (1990) 308-316 North-Holland
ON (100) SURFACES
I. JAGER Imtitut
fiir Metdlphysrk,
Received
6 December
Montctnunwersrttit.
1989; accepted
A-8700
for publication
Leohen,
Austrru
5 March
1990
The sticking coefficient of molecules requiring several adsorption sites is calculated for various sizes and shapes of model molecules. The calculation is based on the assumption of a rigid simple quadratic lattice of adsorption sites such as the four-fold hollow sites of the (100) faces of cubic metals, The surface is considered as containing a fraction (1 - cs) of inactive sites capable of nearest- and next-nearest-neighbor interactions of which the nnn interaction is always repulsive whereas the nnn interaction may vary in sign and magnitude. The model thus describes (a) the dependence of the sticking coefficient of molecules (dissociating on adsorption) on the coverage and (b) the dependence of the initial sticking coefficient of large molecules on a pre-coverage of the surface with poisoning atoms
1. Introduction The adsorption of large molecules on metal surfaces is one of the central problems of heterogeneous catalysis and therefore of major technical importance. It is well known that most molecules require a certain number of adjacent sites for adsorption [l]. This number - usually varying between 1 and 20 ~ is associated with the size of an otherwise non-descript “ensemble” of adsorption sites and is usually identified with the gradient of a plot of the logarithm of the sticking coefficient versus the logarithm of the surface concentration of vacant sites. This method often yields fractional numbers which are hard to accept within this concept. The main reason for this discrepancy is probably that the underlying eq.
(with S: the sticking probability, S,: the initial sticking coefficient, c,: the surface concentration of vacant sites, i.e. the number of unoccupied sites
00.0.
[email protected] l 0.0. 0.000 a.0000
l 0a0. 00.0. 00.0. 00.0. boOoOo
d
a)
d)
b) Fig. 1. Adsorption
0039-6028/90/$03.50
n; 1990
sites (see text). Elsevier Science
Publishers
Fig. 2. Perfect long-range order: Zd-phases at T= 0. (a) c(2 x 2)-phase: R c 0.5; (b) L-phase: R z 0.5; (c) Da,,,-phase: R > 0; (d) Da.,,-phase: R > 0. B.V. (North-Holland)
I. Jtiger / Sticking coefficient of large moIeeules on (100) faces
Fig. 3. The correction
factor
K,
for the ensemble
ofcuh~cmetals
shown in the insert and the value R = J2/J, the value of J, (in thermal units).
divided by the total number of sites, and n: the order of adsorption = the number of sites in the ensemble) rest on the assumption that the atoms forming the precoverage (1 - cS) are independent and the probability of finding n adjacent sites vacant is for obvious combinational reasons the nth power of the probability of finding one site, c,. In reality, however, adsorbed atoms generally interact more or less strongly, either directly or by altering the electronic structure of the host metal [2]. That this is by no means a negligible effect is shown by the rich variety of two-dimensional
given. The parameter
309
of the curves
is
surface phases displayed by most kinds of adsorbed atoms [3]. It can be easily imagined that any interaction between adsorbed atoms invalidates eq. (1); e.g. if there is a repulsive interaction between nearestneighbor adatoms, then the probability of finding two adjacent unoccupied sites is no longer the square of c,. Eq. (I) must therefore by augmented by a correction factor K which in turn depends on the interatomic interactions J, .and the shape of the ensemble E necessary for adsorption as well as the coverage c,.
I. /tiger
310
/ Stlckrng
s= Kh(J,, c,).s,,c:‘.
coefficrentof large
(2)
The aim of this paper is to calculate K,. for various ensembles and a number of different cases of interatomic interactions.
2. Method In general two trends can be observed: (a) Repulsive interactions yielding (at sufficiently low temperatures) long-range ordered “network”
y ++
molecules
on (100)
facesoJcuhtc metals
structures, like c(2 x 2) p(2 x 2), etc. increase the blocking power of a certain coverage cS, even at temperatures where no LRO exists. For example: a perfect c(2 x 2) structure covers the whole surface at c, = 0.5 in such a way that it is quite impossible to find two adjacent unoccupied lattice sites. If the degree of order is not quite so perfect (at higher temperatures) or the coverage is less than 0.5 the inhibiting effect is less but of course the trend remains. (b) Attractive interactions between adatoms decrease the blocking power because with increasing interaction strength the
R=.5
KE
1
o.lcs_ l,;___:_-_l---s-;~ , _.-,$., .7/ / I’ _$ , ’ /’ I ' I./' I' I 2 /'
Fig. 4. The correction
_gicr- _/’
factor K,
’
for the ensemble shown in the insert and the value R = J,/J, given. The parameter the value of J, (in thermal units).
of the curves is
I. Jtiger / Stickingcoefficientof largemoleculeson (100) faces of cubic metals
atoms tend to form islands leaving the remaining lattice more and more cleared from obstacles. In the extreme case of very low temperatures all blocking atoms form a macroscopic compact cluster which occupies the fraction (1 - c,) of all sites leaving the rest, c,, unoccupied. In this case S = S,,c;
(3)
for all sizes of ensembles, n. The complete solution of the problem, on the other hand, is by no means simple. A macroscopic ensemble of interacting adatoms on a surface, subject to a variable
0 'K,
#t
chemical potential, is isomorphic to the two-dimensional Ising model with arbitrary interactions in a magnetic field [4]. This problem has no gcnera1 solution except for some very special cases [.5], therefore some restrictions are necessary: (a) only nearest- and next-nearest-neighbor interactions are allowed; (b) the nearest-neighbor interactions always are repulsive. These conditions are sufficient for the occurrence of only second-order phase transitions [6], except in the rather improbable case of dominant
IO
‘t”++ R=.25
311
IKC #
R=.25
R;25
Fig. 5. The correction
factor
K,
for the ensemble
shown in the insert and the value A = JJJ, the value of J, (in thermal
units).
given. The parameter
of the curves
is
1. J&w
312
/ Sticking coefficient of large molecules on (100) faces
IO KE
#
of cubicmetals
R=O
04 0 1
#
1
R=O
R=O
KE
0.1
Fig. 6. The correction
KE
0.5
factor
K,
for the ensemble
shown in the insert and the value R = .&‘.I, given. The parameter the value of J, (in thermal units).
next-nearest-neighbor attraction where a firstorder phase transition may occur. They therefore permit the use of an approximative method which would break down in the vicinity of a first-order phase transition but works well enough in spite of second-order ones. This was checked by comparison with Monte Carlo calculations [8] and the results of series expansions [IO]. Moreover conditions (a) and (b) cover the majority of adsorption phenomena where c(2 x 2) and p(2 X 2) structures arise [3].
of the curves
is
The model chosen therefore is as follows: We consider a simple quadratic rigid lattice of adsorption sites (no reconstruction is admitted), such as e.g. the lattice of the four-fold hollow sites on (100) faces of cubic crystals. A certain fraction (1 - cS) of these sites is blocked by some sort of atoms interacting among each other with Jr (nearest neighbors) and J2 (next-nearest neighbors); the chemical potential necessary to establish c, is not considered but assumed to exist. J, and J, are in thermal units (= J,/kT, J,/kT).
I. Jdger / Sticking coefficient of
313
large moleculeson (100) faces of cubic metals
Then we calculate the probability of finding the following ensembles of adjacent unoccupied sites (cf. figs. la-ld): (a) two sites, (b) a square of sites, (c) two adjacent squares, and (d) a square of squares, using Kikuchi’s method [7] with a square as the basis figure. The probabilities of larger compounds (c, d) were calculated as products of the probabilities of the components divided by the probabilities of connecting components counted twice. For instance the probability of two adjacent squares (c) is approximated as the square of the probability of one square divided by the probability of one pair of sites. Obviously this procedure neglects longer-range correlations but since only nearest- and next-nearest-neighbor interactions are taken into account the error thus introduced may be neglected [S].
3. Display of results Figs. 3-8 show the correcting factor K, calculated for the ensembles (a)-(d) for various values of R = .&/.I1 (note that Jr 2 0 means repulsion) as a function of the concentration of vacant sites c,. The trends mentioned above can be easily verified: For R = 1 where according to the ground state phase diagram [9] a layered phase (L, sometimes called superantiferromagnetic) according to fig. 2b centered around c, = 0.5, and two degenerate phases (figs. 2c and 2d) centered at c, = 0.25 and 0.75, respectively are to be expected the pair correction does not deviate from 1 down to c, = 0.35 and then starts to decrease more or less steeply due to the more or less perfect state of D,,,,-phase. Note that here as in all cases of second-order phase transitions the actual phase transition does not manifest itself by any irregularity of shortrange-order properties (such as probabilities of ensembles) and is therefore not shown in the diagrams. The correction factors for squares and combinations of squares, on the other hand, start deviating from 1 as early as c, = 0.85 due to the
fact that on a perfect D,,,,-phase no square of vacant sites can be found. For R = 0.5 the diagrams look qualitatively similar to those of R = 1 except for a slight indentation of the pair correction factor around c, = 0.5. This comes from the fact that R = 0.5 is a limit where neither the L-phase not the c(2 X 2)phase are stable and capable of displaying longrange order, but in short-range-order features a certain trend towards the c(2 x 2)-phase (fig. 2a) can be seen (see below). At R = 0.25 the influence of the c(2 X 2)-phase (which, if perfect, blocks pair adsorption entirely) can be seen in the pair correction factor. Again the larger-ensemble correction factors are qualitatively similar to those above. R = 0 is the limiting case of vanishing nextnearest-neighbor interaction. This case has been treated in earlier papers [lo] and is only included for the sake of completeness. This value of R is also the lower limit of the range of existence of the L-phase, therefore the diagrams show no features around c, = 0.75 or c, = 0.25. At R = -0.5 and - 1 the correction factors for squares and combinations thereof show the most remarkable feature of values greater than 1. This comes, as mentioned earlier, from the attractive interaction between next-nearest neighbors which tends to stabilize the c(2 x 2)-phase around c, = 0.5 but on the other hand can cause the surface coverage to become two-phase as described above, preferably around c, = 0.75 and c, = 0.25 [4]. From eqs (2) and (3) it can be seen that the limit of the correction factor for any ensemble should be lim K, J, + m
= cl-”
(for c, = 0.75 or c, = 0.25).
(4) This is displayed in the diagrams only for c, = 0.75, due to a but not for c, = 0.25. This is probably breakdown of Kikuchi’s method near a tricritical point, but this not a very interesting point is view of practical applications.
4. Application of the results
in
The results displayed two different ways:
in figs. 3-8
may be used
I.Jiiger / Stding
314
T
+-+
coefficrent of large molecules on (100) faces of cubic metals
IO
R=-.S
4
R=-.5
KE
(b)
Fig. 7. The correction
factor
Kf for the ensemble
shown in the insert and the value R = J,/J, the value of J, (in thermal units).
(a) Consider a molecule requiring one of the ensembles (a-d) for adsorption and dissociation into a number of like atoms. Then eq. (3) together with the displayed correction factors K, describes the dependence of the sticking probability S of this molecule on the fraction of the surface (1 - cS) already covered by the atoms it consists of. For example: certain hydrocarbons C,H,. require a comparatively large ensemble of sites; on adsorption they dissociate into x carbon atoms whereas (at least in a certain range of tempera-
given. The parameter
of the curves
is
tures} the y hydrogens desorb [ll]. Then this theory is able to explain why usually the adsorption saturates (i.e. S -+ 0) at rather low coverages. (b) Consider a surface pre-adsorbed with a certain amount (1 - c,) of atoms blocking the adsorption sites. Then eq. (3) together with the diagrams yields the initial sticking coefficient for molecules requiring certain ensembles for adsorption. This is part of an explanation why frequently rather small amounts of blocking atoms can “poison” a catalyst completely [12]. It is, however, not the whole
I. JCger / Sricking coesficient of large molecules on (JOO) faces
ofcubicmetals
315
R=-1.
(b) 10
335
R=-I.
’
KE
C&l
I
Gi
, .
, 0.
.I -l@'
Id) Fig. 8. The correction
factor
K,
for the ensemble
shown in the insert and the value R = J2/J1 given. The parameter the value of J, (in thermal units).
explanation, because the activity of a catalyst usually consists of several steps: adsorption of the reactant(s) - possibly more than one! - diffusive formation of the reaction product(s) - desorption of the latter. Poisoning atoms may in principle influence any of those steps, presumably the adsorption and diffusion steps, and it should be a very interesting but extremely complex task to calculate the influence of interacting poison atoms on the adsorption of more than one kind of molecule and the necessary diffusional motion of the parts thereof afterwards.
of the curves is
5. Conclusion The probabilities of certain ensembles of adsorption sites are calculated as a function of the fraction (1 - cS) of sites blocked by interacting atoms. It is shown that and why the simple combinatorial power-law-dependence of the sticking coefficient on the fraction of unoccupied sites is no longer valid. The model calculation is for the following two reasons no complete description of the poisoning of a catalyst by preadsorption of blocking atoms.
316
I. Jiiger / Sttcking coefficient of large molecules on (100) faces of cubic metals
(a) It neglects that for catalytical purposes usually a second reactant is to be adsorbed. (b) It neglects that the initial sticking coefficient is no unique measure for the number of large molecules that can be packed on a surface. This number depends strongly on the diffusional rearrangement of molecules adsorbed at first at random (cf. e.g. the packing of hand rods on a simple quadrate lattice [I 31). The complete solution of the problem of the poisoning of a catalyst, including the blocking of diffusion paths by far exceeds the computational resources available.
References [l] W.H.M. Sachtler, Catal. Rev.-Sci. Eng. 14 (1976) 193. [2] T.L. Einstein, Crit. Rev. Solid State Mat. Sci. 7 (1978) 261.
(31 G.A. Somarjai and F.J. Szalkowski, J. Chem. Phys. 54 (1971) 389; K.H. Lau and W. Kohn, Surf. Sci. 75 (1978) 69; J. Lopez, J.C. Le Bosse and J. Rosseau-Violet. J. Phys. C 13 (1980) 1139. [4] K. Binder and D.P. Landau, Surf. Sci. 61 (1976) 577. [5] B.M. McCoy and T.T. Wu, The Two-Dimensional Ising Model (Harvard University, Cambridge, MA, 1973). [6] S. Katsura and S. Fujimori, J. Phys. C 7 (1974) 2506. [7] R. Kikuchi, Phys. Rev. 81 (1951) 988. [8] I. Jlger, to be published. [9] P.A. Slotte, J. Phys. C 16 (1983) 2935. [IO] 1. Jlger, Z. Metallkd. 76 (1985) 147. [ll] M.A. Chesters, B.J. Hopkins, P.A. Taylor and R.I. Winton, Surf. Sci. 83 (1979) 181. [12] T.E. Fischer and S.R. Kelemen, J. Catal. 53 (1978) 24. V. Ponec, Catal. Rev.-Sci. Eng. 18 (1978) 151. [13] R.J. Baxter, J. Math. Phys. 9 (1968) 650.
Surface
ON THE STICKING COEFFICIENT OF CUBIC METALS
OF LARGE MOLECULES
Science 233 (1990) 308-316 North-Holland
ON (100) SURFACES
I. JAGER Imtitut
fiir Metdlphysrk,
Received
6 December
Montctnunwersrttit.
1989; accepted
A-8700
for publication
Leohen,
Austrru
5 March
1990
The sticking coefficient of molecules requiring several adsorption sites is calculated for various sizes and shapes of model molecules. The calculation is based on the assumption of a rigid simple quadratic lattice of adsorption sites such as the four-fold hollow sites of the (100) faces of cubic metals, The surface is considered as containing a fraction (1 - cs) of inactive sites capable of nearest- and next-nearest-neighbor interactions of which the nnn interaction is always repulsive whereas the nnn interaction may vary in sign and magnitude. The model thus describes (a) the dependence of the sticking coefficient of molecules (dissociating on adsorption) on the coverage and (b) the dependence of the initial sticking coefficient of large molecules on a pre-coverage of the surface with poisoning atoms
1. Introduction The adsorption of large molecules on metal surfaces is one of the central problems of heterogeneous catalysis and therefore of major technical importance. It is well known that most molecules require a certain number of adjacent sites for adsorption [l]. This number - usually varying between 1 and 20 ~ is associated with the size of an otherwise non-descript “ensemble” of adsorption sites and is usually identified with the gradient of a plot of the logarithm of the sticking coefficient versus the logarithm of the surface concentration of vacant sites. This method often yields fractional numbers which are hard to accept within this concept. The main reason for this discrepancy is probably that the underlying eq.
(with S: the sticking probability, S,: the initial sticking coefficient, c,: the surface concentration of vacant sites, i.e. the number of unoccupied sites
00.0.
[email protected] l 0.0. 0.000 a.0000
l 0a0. 00.0. 00.0. 00.0. boOoOo
d
a)
d)
b) Fig. 1. Adsorption
0039-6028/90/$03.50
n; 1990
sites (see text). Elsevier Science
Publishers
Fig. 2. Perfect long-range order: Zd-phases at T= 0. (a) c(2 x 2)-phase: R c 0.5; (b) L-phase: R z 0.5; (c) Da,,,-phase: R > 0; (d) Da.,,-phase: R > 0. B.V. (North-Holland)
I. Jtiger / Sticking coefficient of large moIeeules on (100) faces
Fig. 3. The correction
factor
K,
for the ensemble
ofcuh~cmetals
shown in the insert and the value R = J2/J, the value of J, (in thermal units).
divided by the total number of sites, and n: the order of adsorption = the number of sites in the ensemble) rest on the assumption that the atoms forming the precoverage (1 - cS) are independent and the probability of finding n adjacent sites vacant is for obvious combinational reasons the nth power of the probability of finding one site, c,. In reality, however, adsorbed atoms generally interact more or less strongly, either directly or by altering the electronic structure of the host metal [2]. That this is by no means a negligible effect is shown by the rich variety of two-dimensional
given. The parameter
309
of the curves
is
surface phases displayed by most kinds of adsorbed atoms [3]. It can be easily imagined that any interaction between adsorbed atoms invalidates eq. (1); e.g. if there is a repulsive interaction between nearestneighbor adatoms, then the probability of finding two adjacent unoccupied sites is no longer the square of c,. Eq. (I) must therefore by augmented by a correction factor K which in turn depends on the interatomic interactions J, .and the shape of the ensemble E necessary for adsorption as well as the coverage c,.
I. /tiger
310
/ Stlckrng
s= Kh(J,, c,).s,,c:‘.
coefficrentof large
(2)
The aim of this paper is to calculate K,. for various ensembles and a number of different cases of interatomic interactions.
2. Method In general two trends can be observed: (a) Repulsive interactions yielding (at sufficiently low temperatures) long-range ordered “network”
y ++
molecules
on (100)
facesoJcuhtc metals
structures, like c(2 x 2) p(2 x 2), etc. increase the blocking power of a certain coverage cS, even at temperatures where no LRO exists. For example: a perfect c(2 x 2) structure covers the whole surface at c, = 0.5 in such a way that it is quite impossible to find two adjacent unoccupied lattice sites. If the degree of order is not quite so perfect (at higher temperatures) or the coverage is less than 0.5 the inhibiting effect is less but of course the trend remains. (b) Attractive interactions between adatoms decrease the blocking power because with increasing interaction strength the
R=.5
KE
1
o.lcs_ l,;___:_-_l---s-;~ , _.-,$., .7/ / I’ _$ , ’ /’ I ' I./' I' I 2 /'
Fig. 4. The correction
_gicr- _/’
factor K,
’
for the ensemble shown in the insert and the value R = J,/J, given. The parameter the value of J, (in thermal units).
of the curves is
I. Jtiger / Stickingcoefficientof largemoleculeson (100) faces of cubic metals
atoms tend to form islands leaving the remaining lattice more and more cleared from obstacles. In the extreme case of very low temperatures all blocking atoms form a macroscopic compact cluster which occupies the fraction (1 - c,) of all sites leaving the rest, c,, unoccupied. In this case S = S,,c;
(3)
for all sizes of ensembles, n. The complete solution of the problem, on the other hand, is by no means simple. A macroscopic ensemble of interacting adatoms on a surface, subject to a variable
0 'K,
#t
chemical potential, is isomorphic to the two-dimensional Ising model with arbitrary interactions in a magnetic field [4]. This problem has no gcnera1 solution except for some very special cases [.5], therefore some restrictions are necessary: (a) only nearest- and next-nearest-neighbor interactions are allowed; (b) the nearest-neighbor interactions always are repulsive. These conditions are sufficient for the occurrence of only second-order phase transitions [6], except in the rather improbable case of dominant
IO
‘t”++ R=.25
311
IKC #
R=.25
R;25
Fig. 5. The correction
factor
K,
for the ensemble
shown in the insert and the value A = JJJ, the value of J, (in thermal
units).
given. The parameter
of the curves
is
1. J&w
312
/ Sticking coefficient of large molecules on (100) faces
IO KE
#
of cubicmetals
R=O
04 0 1
#
1
R=O
R=O
KE
0.1
Fig. 6. The correction
KE
0.5
factor
K,
for the ensemble
shown in the insert and the value R = .&‘.I, given. The parameter the value of J, (in thermal units).
next-nearest-neighbor attraction where a firstorder phase transition may occur. They therefore permit the use of an approximative method which would break down in the vicinity of a first-order phase transition but works well enough in spite of second-order ones. This was checked by comparison with Monte Carlo calculations [8] and the results of series expansions [IO]. Moreover conditions (a) and (b) cover the majority of adsorption phenomena where c(2 x 2) and p(2 X 2) structures arise [3].
of the curves
is
The model chosen therefore is as follows: We consider a simple quadratic rigid lattice of adsorption sites (no reconstruction is admitted), such as e.g. the lattice of the four-fold hollow sites on (100) faces of cubic crystals. A certain fraction (1 - cS) of these sites is blocked by some sort of atoms interacting among each other with Jr (nearest neighbors) and J2 (next-nearest neighbors); the chemical potential necessary to establish c, is not considered but assumed to exist. J, and J, are in thermal units (= J,/kT, J,/kT).
I. Jdger / Sticking coefficient of
313
large moleculeson (100) faces of cubic metals
Then we calculate the probability of finding the following ensembles of adjacent unoccupied sites (cf. figs. la-ld): (a) two sites, (b) a square of sites, (c) two adjacent squares, and (d) a square of squares, using Kikuchi’s method [7] with a square as the basis figure. The probabilities of larger compounds (c, d) were calculated as products of the probabilities of the components divided by the probabilities of connecting components counted twice. For instance the probability of two adjacent squares (c) is approximated as the square of the probability of one square divided by the probability of one pair of sites. Obviously this procedure neglects longer-range correlations but since only nearest- and next-nearest-neighbor interactions are taken into account the error thus introduced may be neglected [S].
3. Display of results Figs. 3-8 show the correcting factor K, calculated for the ensembles (a)-(d) for various values of R = .&/.I1 (note that Jr 2 0 means repulsion) as a function of the concentration of vacant sites c,. The trends mentioned above can be easily verified: For R = 1 where according to the ground state phase diagram [9] a layered phase (L, sometimes called superantiferromagnetic) according to fig. 2b centered around c, = 0.5, and two degenerate phases (figs. 2c and 2d) centered at c, = 0.25 and 0.75, respectively are to be expected the pair correction does not deviate from 1 down to c, = 0.35 and then starts to decrease more or less steeply due to the more or less perfect state of D,,,,-phase. Note that here as in all cases of second-order phase transitions the actual phase transition does not manifest itself by any irregularity of shortrange-order properties (such as probabilities of ensembles) and is therefore not shown in the diagrams. The correction factors for squares and combinations of squares, on the other hand, start deviating from 1 as early as c, = 0.85 due to the
fact that on a perfect D,,,,-phase no square of vacant sites can be found. For R = 0.5 the diagrams look qualitatively similar to those of R = 1 except for a slight indentation of the pair correction factor around c, = 0.5. This comes from the fact that R = 0.5 is a limit where neither the L-phase not the c(2 X 2)phase are stable and capable of displaying longrange order, but in short-range-order features a certain trend towards the c(2 x 2)-phase (fig. 2a) can be seen (see below). At R = 0.25 the influence of the c(2 X 2)-phase (which, if perfect, blocks pair adsorption entirely) can be seen in the pair correction factor. Again the larger-ensemble correction factors are qualitatively similar to those above. R = 0 is the limiting case of vanishing nextnearest-neighbor interaction. This case has been treated in earlier papers [lo] and is only included for the sake of completeness. This value of R is also the lower limit of the range of existence of the L-phase, therefore the diagrams show no features around c, = 0.75 or c, = 0.25. At R = -0.5 and - 1 the correction factors for squares and combinations thereof show the most remarkable feature of values greater than 1. This comes, as mentioned earlier, from the attractive interaction between next-nearest neighbors which tends to stabilize the c(2 x 2)-phase around c, = 0.5 but on the other hand can cause the surface coverage to become two-phase as described above, preferably around c, = 0.75 and c, = 0.25 [4]. From eqs (2) and (3) it can be seen that the limit of the correction factor for any ensemble should be lim K, J, + m
= cl-”
(for c, = 0.75 or c, = 0.25).
(4) This is displayed in the diagrams only for c, = 0.75, due to a but not for c, = 0.25. This is probably breakdown of Kikuchi’s method near a tricritical point, but this not a very interesting point is view of practical applications.
4. Application of the results
in
The results displayed two different ways:
in figs. 3-8
may be used
I.Jiiger / Stding
314
T
+-+
coefficrent of large molecules on (100) faces of cubic metals
IO
R=-.S
4
R=-.5
KE
(b)
Fig. 7. The correction
factor
Kf for the ensemble
shown in the insert and the value R = J,/J, the value of J, (in thermal units).
(a) Consider a molecule requiring one of the ensembles (a-d) for adsorption and dissociation into a number of like atoms. Then eq. (3) together with the displayed correction factors K, describes the dependence of the sticking probability S of this molecule on the fraction of the surface (1 - cS) already covered by the atoms it consists of. For example: certain hydrocarbons C,H,. require a comparatively large ensemble of sites; on adsorption they dissociate into x carbon atoms whereas (at least in a certain range of tempera-
given. The parameter
of the curves
is
tures} the y hydrogens desorb [ll]. Then this theory is able to explain why usually the adsorption saturates (i.e. S -+ 0) at rather low coverages. (b) Consider a surface pre-adsorbed with a certain amount (1 - c,) of atoms blocking the adsorption sites. Then eq. (3) together with the diagrams yields the initial sticking coefficient for molecules requiring certain ensembles for adsorption. This is part of an explanation why frequently rather small amounts of blocking atoms can “poison” a catalyst completely [12]. It is, however, not the whole
I. JCger / Sricking coesficient of large molecules on (JOO) faces
ofcubicmetals
315
R=-1.
(b) 10
335
R=-I.
’
KE
C&l
I
Gi
, .
, 0.
.I -l@'
Id) Fig. 8. The correction
factor
K,
for the ensemble
shown in the insert and the value R = J2/J1 given. The parameter the value of J, (in thermal units).
explanation, because the activity of a catalyst usually consists of several steps: adsorption of the reactant(s) - possibly more than one! - diffusive formation of the reaction product(s) - desorption of the latter. Poisoning atoms may in principle influence any of those steps, presumably the adsorption and diffusion steps, and it should be a very interesting but extremely complex task to calculate the influence of interacting poison atoms on the adsorption of more than one kind of molecule and the necessary diffusional motion of the parts thereof afterwards.
of the curves is
5. Conclusion The probabilities of certain ensembles of adsorption sites are calculated as a function of the fraction (1 - cS) of sites blocked by interacting atoms. It is shown that and why the simple combinatorial power-law-dependence of the sticking coefficient on the fraction of unoccupied sites is no longer valid. The model calculation is for the following two reasons no complete description of the poisoning of a catalyst by preadsorption of blocking atoms.
316
I. Jiiger / Sttcking coefficient of large molecules on (100) faces of cubic metals
(a) It neglects that for catalytical purposes usually a second reactant is to be adsorbed. (b) It neglects that the initial sticking coefficient is no unique measure for the number of large molecules that can be packed on a surface. This number depends strongly on the diffusional rearrangement of molecules adsorbed at first at random (cf. e.g. the packing of hand rods on a simple quadrate lattice [I 31). The complete solution of the problem of the poisoning of a catalyst, including the blocking of diffusion paths by far exceeds the computational resources available.
References [l] W.H.M. Sachtler, Catal. Rev.-Sci. Eng. 14 (1976) 193. [2] T.L. Einstein, Crit. Rev. Solid State Mat. Sci. 7 (1978) 261.
(31 G.A. Somarjai and F.J. Szalkowski, J. Chem. Phys. 54 (1971) 389; K.H. Lau and W. Kohn, Surf. Sci. 75 (1978) 69; J. Lopez, J.C. Le Bosse and J. Rosseau-Violet. J. Phys. C 13 (1980) 1139. [4] K. Binder and D.P. Landau, Surf. Sci. 61 (1976) 577. [5] B.M. McCoy and T.T. Wu, The Two-Dimensional Ising Model (Harvard University, Cambridge, MA, 1973). [6] S. Katsura and S. Fujimori, J. Phys. C 7 (1974) 2506. [7] R. Kikuchi, Phys. Rev. 81 (1951) 988. [8] I. Jlger, to be published. [9] P.A. Slotte, J. Phys. C 16 (1983) 2935. [IO] 1. Jlger, Z. Metallkd. 76 (1985) 147. [ll] M.A. Chesters, B.J. Hopkins, P.A. Taylor and R.I. Winton, Surf. Sci. 83 (1979) 181. [12] T.E. Fischer and S.R. Kelemen, J. Catal. 53 (1978) 24. V. Ponec, Catal. Rev.-Sci. Eng. 18 (1978) 151. [13] R.J. Baxter, J. Math. Phys. 9 (1968) 650.