- Email: [email protected]
Temperature programmed desorption by dimers from a surface
L611
Surface Science 187 (1987) L611-L615 North-HoUand, Amsterdam
SURFACE SCIENCE LETTERS TEMPERATURE PROGRAMMED D E S O R P T I O N BY DIMERS FROM A SURFACE
J.J. LUQUE and A. CORDOBA Departamento de Termologia, Facultad de Fisica, Universidad de Sevilla, Sevilla, Spain Received 4 November 1986; accepted for publication 23 April 1987
The desorption by dimers from a surface is studied. Migration on the lattice and interaction betweenneighboring adatoms can exist. Starting from a master equation with transition probabilities in the Arrhenius form, a set of kinetic equations is obtained. For a square lattice, when lateral interaction is repulsive and surface mobility operates, rate desorption curves first exhibit a maximum, next a shoulder, and finally a long tail. For the linear chain and all the other cases of the square lattice these curves show only a maximum.
A technique for studying adsorption phenomena is to perform a desorption process under a linear temperature program and to analyze the desorption rate curves.
Our aim in this paper is to study desorption by pairs of monomers which are jointly desorbed from a surface, granting that interaction exists between neighbor adatoms on the lattice. Particularly, we shall analyze the influence on the desorption rate exerted by surface mobility. We consider dimer desorption from, or readsorption onto, a rectangular lattice. Adsorbed dimers are dissociated and the resulting monomers can migrate by hopping from one site to an adjacent vacant site. We also assume that interaction exists between nearest neighbors. Due to this interaction, an elemental process of adsorption-desorption or migration involves eight sites, as is shown in fig. 1. These elemental mechanisms and their respective probabilities determine the model. Then a master equation is formulated and the transition probabilities are chosen in the Arrhenius form. If the pair where the process takes place is parallel to the x-axis (as in fig. 1), the probabilities are: l'J/dXsorption(l, k ) = A x e - t 2 " + t ' x + l ' ' y ) / k r ,
l - - 0 , 2; k = 0 , 4 Wa~isorpfion(/, k ) = arx e - ( 2 ' ' - t ' x - * ' y ) / k r ,
x ( , Wn~igration 7/, m , n ,
m , ) = A x,, e + [ ( , , , - , , ' ) , ~ + ( , , - , , ' ) , x ] / k T
0039-6028/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(1)
L612
J.J. Luque, A. C6rdoba / TPD by dimers from a surface
C
O--< yT , C Fig. 1. Sites involved in an elemental process of adsorption-desorption or migration.
(the + and - signs correspond to mobility towards right and left hand side, respectively) n = 0, 1; n ' = 0 , 1; m = 0 , 2; m ' = 0 , 2, where l and k denote respectively the number of particles connected to the central pair in the x-axis and y-axis directions; n ( n ' ) and m ( m ' ) denote respectively the number of particles joined in the x-axis and y-axis directions to the left (right) particle of the central pair; e and e' are the respective activation energies due to the substrate for desorption and adsorption; ~ and ey represent the lateral interaction between nearest neighbors which contributes to the activation energies (we assume opposite contributions for adsorption and desorption); A x, A" and A~' are constants, k is the Boltzmann constant and T denotes temperature. When the central dimer is parallel to the y-axis, the transition probabilities may be written in a similar way by a suitable change of the x and y subscripts. Following the guideline of ref. [1], adapted to a two-dimensional case, one can obtain a hierarchy of kinetic equations for densities of groups involving one, two, three, etc., sites of the lattice. Specifically, the time evolution of monomers and pairs involves densities of groups of eight sites disposed as in fig. 1. We assume the closure approximation
PI ] Pl IIPl I1 L(~J L ( ~ ) ! 0 ) L @ J
(2)
P(@)3P(O)3P(e)P(O)P(O)P(@) where P( .) denotes the probability of finding the configuration between brackets on the lattice. Then we obtain a closed set of three kinetic equations governing the time evolution of the densities of monomers, n+ = 0, and pairs, n++ and n~+. These equations are solved numerically.
J.J. Luque, A. Cbrdoba / TPD by dirnersfrom a surface
L613
We have formulated the problem in a general way, so that the interactions in the x-axis and in the y-axis directions may be considered different from each other. Obviously, if both are equal (square lattice), the problem is simplified as n~+ = nY++= , / and there are only two kinetic equations. On the other hand, if Ay, Ay,p Ayi t and ( s vanish, the problem breaks down to that of a set of independent linear chains. We would like to point out that for the latter case the results obtained from the approximation (2) coincide with those obtained from a straightforward formulation of the problem in a linear chain, as in ref. [1]. Thus, eq. (2) may be considered as a reasonable extension of the approximation given by eqs. (12) and (13) of ref. [1]. We have performed numerical calculations only for t h e square lattice and the linear chain. We have dealt with the case where the adsorption process does not operate; only desorption and mobility are effective, and a linear temperature program is applied. We have analyzed the evolution of 8, ,/ and the desorption rate versus temperature for different interaction energies, with and without mobility. When interaction is attractive or does not exist, differences between cases with and without mobility are not significant, except for the final remains when mobility is inhibited, since ,/tends to zero when 0 is different from zero. However, if lateral interaction is repulsive and mobility operates, the desorption rate curves exhibit a lower maximum; next, at a medium stage of the
0.025
SURFACE [
SURFACE(
0.020
c
0.015
c
WITH 'IOBILITY
WITHOUT ~t0BILITY
0.010 w
0.005
LINEARCHAIN
LINEARCHAIN 0
0.015
N
a~
0o010
WI THOUT
WITH
BILI~
0.005
300
dO0
500
GO0
300
400
500
GO0
TEMPERATURE(K) Fig. 2. Desorption rate versus temperature for a square lattice. Ax/temperature rate = 106 K-~; A'~'/Ax =10 -5 (when mobility exists; initial temperature = 250 K; c = 60 kJ m o l - l ; (a) ¢,. = 0; (b) ¢ ~ = - 2 k J m o l - t ; ( c ) ( x = 2 k J m o l -s.
L614
J.J. Luque, A. Cbrdoba / TPD by dimers from a surface
desorption process, these curves have a shoulder, and, finally, a longer tail than those resulting from the other cases. F o r the linear chain the curve does not exhibit the m e d i u m stage for typical values of diffusion energy ( - 1 / 1 0 times the desorption energy). A set of representative results is shown in fig. 2. By analyzing fig. 3 one can see in m o r e detail how the desorption process takes place. Fig. 3(i) shows that the evolution of the n u m b e r of pairs of a d a t o m s (which allow the removal process) does practically not d e p e n d on surface mobility. While for attractive lateral interaction energies the change from n++ = 1 to n++ = 0 is very sharp and takes place in a very n a r r o w t e m p e r a t u r e range, for repulsive energies it takes place in a m o r e gradual form. Fig. 3(ii) shows the evolution of the a d a t o m - v a c a n c y pairs, which allow surface diffusion. This effect is m u c h greater for repulsive than for attractive energies. F o r repulsive energies and for the values of the p a r a m e t e r used in our calculations, if one c o m p a r e s the cases with and without mobility, it can be seen that the effect of diffusion is decisive for temperatures higher than 400 K ( 0 = 2 5 % ) : n + + = 0 and r/isolated adatoms=0 (see fig. 3(iii)), and removal of dimers takes place essentially from isolated a d a t o m s which b e c o m e non-isolated due to diffusion. Thus in the final stage of desorption the process is governed by the surface mobility. On the contrary, for attractive energies this effect is very small and decreases m o r e and m o r e as the interaction energy increases.
0°8 ~so°6 (i)
~0°2
~0o2
(£i)
~ao., 0°2
d
f
8U0ol
(tii)
h
i
300
400
500
TEMPERATURE(K)
Fig. 3. Adatom pairs (i), vacancy-adatom pairs (ii) and isolated adatoms (iii) fractions versus temperature for a square lattice. Ax/temperature rate = 106 K-I; ( = 60 kJ/mol-t; initial temperature = 250 K; (a) %,=2 kJ mo1-1, A'~'/A~ =10-5; (b) (:, = - 2 kJ mol -], A'~'/A x = 10-5; (c) fl, = 2 kJ mo1-1, A~' = 0; (d) (~, = - 2 kJ mo1-1, A~' = 0.
J.J. Luque, A. C6rdoba / TPD by dimers from a surface
L615
In the desorption of dimers, spectra with a peak-shoulder-tail shape have been justified by considering heterogeneous surfaces [2]. Here it is made clear that also these spectra can arise from homogeneous substrates if the adsorbed particles interact with one another repulsively and surface mobility is operative. This result may be interesting for analyzing experimental spectra with a peak-shoulder-tail shape. This work was partially supported by grant PB85-0363 of the CAICyT of the Spanish Government.
References
[1] J.J. Luqu¢ and A. C6rdoba, J. Chem. Phys. 76 (1982) 6393. [2] A. C6rdoba and J.J. Luque, Phys. Rev. B31 (1985) 8111.
Surface Science 187 (1987) L611-L615 North-HoUand, Amsterdam
SURFACE SCIENCE LETTERS TEMPERATURE PROGRAMMED D E S O R P T I O N BY DIMERS FROM A SURFACE
J.J. LUQUE and A. CORDOBA Departamento de Termologia, Facultad de Fisica, Universidad de Sevilla, Sevilla, Spain Received 4 November 1986; accepted for publication 23 April 1987
The desorption by dimers from a surface is studied. Migration on the lattice and interaction betweenneighboring adatoms can exist. Starting from a master equation with transition probabilities in the Arrhenius form, a set of kinetic equations is obtained. For a square lattice, when lateral interaction is repulsive and surface mobility operates, rate desorption curves first exhibit a maximum, next a shoulder, and finally a long tail. For the linear chain and all the other cases of the square lattice these curves show only a maximum.
A technique for studying adsorption phenomena is to perform a desorption process under a linear temperature program and to analyze the desorption rate curves.
Our aim in this paper is to study desorption by pairs of monomers which are jointly desorbed from a surface, granting that interaction exists between neighbor adatoms on the lattice. Particularly, we shall analyze the influence on the desorption rate exerted by surface mobility. We consider dimer desorption from, or readsorption onto, a rectangular lattice. Adsorbed dimers are dissociated and the resulting monomers can migrate by hopping from one site to an adjacent vacant site. We also assume that interaction exists between nearest neighbors. Due to this interaction, an elemental process of adsorption-desorption or migration involves eight sites, as is shown in fig. 1. These elemental mechanisms and their respective probabilities determine the model. Then a master equation is formulated and the transition probabilities are chosen in the Arrhenius form. If the pair where the process takes place is parallel to the x-axis (as in fig. 1), the probabilities are: l'J/dXsorption(l, k ) = A x e - t 2 " + t ' x + l ' ' y ) / k r ,
l - - 0 , 2; k = 0 , 4 Wa~isorpfion(/, k ) = arx e - ( 2 ' ' - t ' x - * ' y ) / k r ,
x ( , Wn~igration 7/, m , n ,
m , ) = A x,, e + [ ( , , , - , , ' ) , ~ + ( , , - , , ' ) , x ] / k T
0039-6028/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(1)
L612
J.J. Luque, A. C6rdoba / TPD by dimers from a surface
C
O--< yT , C Fig. 1. Sites involved in an elemental process of adsorption-desorption or migration.
(the + and - signs correspond to mobility towards right and left hand side, respectively) n = 0, 1; n ' = 0 , 1; m = 0 , 2; m ' = 0 , 2, where l and k denote respectively the number of particles connected to the central pair in the x-axis and y-axis directions; n ( n ' ) and m ( m ' ) denote respectively the number of particles joined in the x-axis and y-axis directions to the left (right) particle of the central pair; e and e' are the respective activation energies due to the substrate for desorption and adsorption; ~ and ey represent the lateral interaction between nearest neighbors which contributes to the activation energies (we assume opposite contributions for adsorption and desorption); A x, A" and A~' are constants, k is the Boltzmann constant and T denotes temperature. When the central dimer is parallel to the y-axis, the transition probabilities may be written in a similar way by a suitable change of the x and y subscripts. Following the guideline of ref. [1], adapted to a two-dimensional case, one can obtain a hierarchy of kinetic equations for densities of groups involving one, two, three, etc., sites of the lattice. Specifically, the time evolution of monomers and pairs involves densities of groups of eight sites disposed as in fig. 1. We assume the closure approximation
PI ] Pl IIPl I1 L(~J L ( ~ ) ! 0 ) L @ J
(2)
P(@)3P(O)3P(e)P(O)P(O)P(@) where P( .) denotes the probability of finding the configuration between brackets on the lattice. Then we obtain a closed set of three kinetic equations governing the time evolution of the densities of monomers, n+ = 0, and pairs, n++ and n~+. These equations are solved numerically.
J.J. Luque, A. Cbrdoba / TPD by dirnersfrom a surface
L613
We have formulated the problem in a general way, so that the interactions in the x-axis and in the y-axis directions may be considered different from each other. Obviously, if both are equal (square lattice), the problem is simplified as n~+ = nY++= , / and there are only two kinetic equations. On the other hand, if Ay, Ay,p Ayi t and ( s vanish, the problem breaks down to that of a set of independent linear chains. We would like to point out that for the latter case the results obtained from the approximation (2) coincide with those obtained from a straightforward formulation of the problem in a linear chain, as in ref. [1]. Thus, eq. (2) may be considered as a reasonable extension of the approximation given by eqs. (12) and (13) of ref. [1]. We have performed numerical calculations only for t h e square lattice and the linear chain. We have dealt with the case where the adsorption process does not operate; only desorption and mobility are effective, and a linear temperature program is applied. We have analyzed the evolution of 8, ,/ and the desorption rate versus temperature for different interaction energies, with and without mobility. When interaction is attractive or does not exist, differences between cases with and without mobility are not significant, except for the final remains when mobility is inhibited, since ,/tends to zero when 0 is different from zero. However, if lateral interaction is repulsive and mobility operates, the desorption rate curves exhibit a lower maximum; next, at a medium stage of the
0.025
SURFACE [
SURFACE(
0.020
c
0.015
c
WITH 'IOBILITY
WITHOUT ~t0BILITY
0.010 w
0.005
LINEARCHAIN
LINEARCHAIN 0
0.015
N
a~
0o010
WI THOUT
WITH
BILI~
0.005
300
dO0
500
GO0
300
400
500
GO0
TEMPERATURE(K) Fig. 2. Desorption rate versus temperature for a square lattice. Ax/temperature rate = 106 K-~; A'~'/Ax =10 -5 (when mobility exists; initial temperature = 250 K; c = 60 kJ m o l - l ; (a) ¢,. = 0; (b) ¢ ~ = - 2 k J m o l - t ; ( c ) ( x = 2 k J m o l -s.
L614
J.J. Luque, A. Cbrdoba / TPD by dimers from a surface
desorption process, these curves have a shoulder, and, finally, a longer tail than those resulting from the other cases. F o r the linear chain the curve does not exhibit the m e d i u m stage for typical values of diffusion energy ( - 1 / 1 0 times the desorption energy). A set of representative results is shown in fig. 2. By analyzing fig. 3 one can see in m o r e detail how the desorption process takes place. Fig. 3(i) shows that the evolution of the n u m b e r of pairs of a d a t o m s (which allow the removal process) does practically not d e p e n d on surface mobility. While for attractive lateral interaction energies the change from n++ = 1 to n++ = 0 is very sharp and takes place in a very n a r r o w t e m p e r a t u r e range, for repulsive energies it takes place in a m o r e gradual form. Fig. 3(ii) shows the evolution of the a d a t o m - v a c a n c y pairs, which allow surface diffusion. This effect is m u c h greater for repulsive than for attractive energies. F o r repulsive energies and for the values of the p a r a m e t e r used in our calculations, if one c o m p a r e s the cases with and without mobility, it can be seen that the effect of diffusion is decisive for temperatures higher than 400 K ( 0 = 2 5 % ) : n + + = 0 and r/isolated adatoms=0 (see fig. 3(iii)), and removal of dimers takes place essentially from isolated a d a t o m s which b e c o m e non-isolated due to diffusion. Thus in the final stage of desorption the process is governed by the surface mobility. On the contrary, for attractive energies this effect is very small and decreases m o r e and m o r e as the interaction energy increases.
0°8 ~so°6 (i)
~0°2
~0o2
(£i)
~ao., 0°2
d
f
8U0ol
(tii)
h
i
300
400
500
TEMPERATURE(K)
Fig. 3. Adatom pairs (i), vacancy-adatom pairs (ii) and isolated adatoms (iii) fractions versus temperature for a square lattice. Ax/temperature rate = 106 K-I; ( = 60 kJ/mol-t; initial temperature = 250 K; (a) %,=2 kJ mo1-1, A'~'/A~ =10-5; (b) (:, = - 2 kJ mol -], A'~'/A x = 10-5; (c) fl, = 2 kJ mo1-1, A~' = 0; (d) (~, = - 2 kJ mo1-1, A~' = 0.
J.J. Luque, A. C6rdoba / TPD by dimers from a surface
L615
In the desorption of dimers, spectra with a peak-shoulder-tail shape have been justified by considering heterogeneous surfaces [2]. Here it is made clear that also these spectra can arise from homogeneous substrates if the adsorbed particles interact with one another repulsively and surface mobility is operative. This result may be interesting for analyzing experimental spectra with a peak-shoulder-tail shape. This work was partially supported by grant PB85-0363 of the CAICyT of the Spanish Government.
References
[1] J.J. Luqu¢ and A. C6rdoba, J. Chem. Phys. 76 (1982) 6393. [2] A. C6rdoba and J.J. Luque, Phys. Rev. B31 (1985) 8111.