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Spatial atomic distribution in small bimetallic clusters
Science 156 ( 1985) 404-409 North-Holland. Amsterdam
Surface
404
SPATIAL ATOMIC DISTRIBUTION CLUSTERS *
IN SMALL BIMETALLIC
C.A. BALSEIRO Centro Atbmico
Bnriloche. S.C. Bariloche, Argentina
and J.L. MORAN-Li)PEZ
**
Institute ftir Festktirperforschung Germany
der Kernf#~.~chungsanlage
Received
for publication
10 July 1984; accepted
Within the (55 atoms) is equivalent and For a system temperatures.
Jiilich,
D
- 5170 Jiilich, Fed, Rep. of
20 July 1984
regular solution model the spatial distribution of atoms in cube-octahedral clusters studied. All the sites located at the same distance from the center are considered the atoms distributed at random with a characteristic concentration x,( i = 1.2. 3.4). that orders at low temperatures, we calculate the equilibrium values at different
There is a large number of publications reporting that the best catalysts are obtained in the form of small particles. In particular, bimetallic clusters are used to improve the selectively for certain chemical reactions [l-3]; examples of those catalysts are RuCu, OsCu, PtRd, PtIr, etc., small particles. In a small particle a large fraction of its atoms is exposed to chemical reactions. Therefore, knowledge of their spatial distribution is necessary to understand its catalytic activity. Generally, it is assumed that the two kinds of atoms are distributed at random in the whole particle. However, this assumption could be valid only at high temperatures. At low temperatures it is expected that a kind of ordering should occur. Here, we study within the regular solution model [4] the spatial distribution of atoms in bimetallic cube-octahedral clusters with 55 atoms. As shown in fig.
* Work partially supported by Consejo Mexico-Argentina exchange program No. ** On leave from Departamento de Fisica. IPN, Apdo. Postal 14-740, 07000 Mexico,
National de Ciencia 140113 H21-003. Centro de investigation DF, Mexico.
0039-6028/85/$03.30 @ Elsevier Science Publishers (North-~oiland Physics Publishing Division)
y Tecnologia,
B.V.
y Estudios
through
the
Avanzados
de1
1, this number of atoms forms a perfect cube-octahedra in which the surface atoms are afi the second (6), third (24) and fourth (12) nearest neighbours of the central atom. The first nearest neighbours (12) are also shown in the figure and in the particle they lie below the surface. We call the first, second, third and fourth shell all the sites corresponding to the first, second, third and fourth nearest neighbours respectively and we denote the number of nearest neighbours lying in she11 j to an atom in shell i by Zij. Those values are given in table 1. We consider a particle with NA atoms of type A and 55 - NA atoms of type B. distributed in all the different shells. The concentration in shell i is given by xj = e*,i&?
0)
where #A,i and Nj are the number of atoms A and the total number of sites in the i th shell, respectively.
Fig. 1. The first (solid circles), second (2). third (3) and fourth (4) nearest atoms in a 5%atom cube-octahedra.
neighbours
of the central
406
C.A. Balseiro. J. L. Morrin-Lhper
/ Spatial atomic distrihutron
By assuming now that the contributions to the energy are the short-range pairwise interactions UAA, U,, and U,,, the internal energy of the system can be written U= 12W[ X”X, + 2x; + 2X,X, + 4x,x,
+ 2x,x,
+ 2x: +x,x4
+ 4x3+
+ +(A - l)( x0 + 12x, + 4x, + 10x, + 5x,)],
(2)
where w = u,,
+ u,,
- 2u,,
A = (Ca,
- &)/‘I+‘.
.
(3) (4)
A particle with W > 0 will tend to develop ordered temperatures and positive values of A mean a tendency A to the surface. The configurational entropy is given by
structures [4] at low [5] to segregate atoms
N, !
szkF1n(q - N~,,)!N~,,! Now, in order to get the spatial minimize the free energy F=
U-
configuration
at a given
temperature
TS,
with respect centration
(6) to all concentrations
with the constraint
that
is kept fixed. The results for a particle with parameters W = 4 and A figs. 2 and 3. In fig. 2 we show the equilibrium values A-atoms, hi,,, as a function of the total number of A-atoms for different temperatures. The temperature is measured in
the average
J=o
0
_
1 2 3 4
1 0 0 0
con-
= 0.5 are shown in of the number of in the cluster, NA. units of k.
Table 1 Number of nearest neighbours lying in shell j to an atom in shell I. Z,,,: the last column total number of nearest neighbours i
we
j=l
J=2
J=3
j=4
7
12 4 4 2 1
0 2 0 1 0
0 4 4 2 4
0 1 0 2 0
12 12 8 I 5
(T) is the
C.A. Balseiro, J. L. Morirn- Lirpez / Spatial atomic distribution
407
At low temperatures (2’ = l), as one starts to substitute B-atoms by A-atoms in the cluster, they go to the first and third shell. But when one gets to NA = 11, the equilibrium configuration is obtained by moving the A-atoms to the second and fourth shell. Then, just after the two shells are full, additional substitutions drive the A-atoms to the first and third shell.
_.....- 3StiELL
-
1 SHELL
-I-
ZSHELL
---
3SHELL
I
20-
I
-
1SHELL
-I-.-
ZSHELL
,
I
I
I
I
a
TZL
I
.-* ,
---3SHELL . ..... &SHELL
15-
1
4'
Y
-
/+
+
/+
.J’
Fig. 2. The number of atoms NA., in the different A-atoms NA and for three different temperatures. A = 0.5 in units of k.
shells as a function of the total number of The parameters we used were W= 4 and
408
C.A. Balseiro, J.L. Morirn-Lbpez
/ Spaiial atomic distribution
At higher temperatures (T = 2) the kind of transition observed at NA = 11 takes place at NA = 15 but it is not so sharp. Finally at T = 4 the occupation in all shells is a smooth function of NA and practically no order is observed in the whole range of NA. To illustrate the temperature dependence of the spatial arrangement we show in fig. 3 a particle with NA = 15 and at the different temperatures considered. One clearly sees that at low temperatures (T = 1) the atoms are arranged in an ordered way, NA,z = 6, NA 4 = 9. As one increases the temperature a kind of order-disorder transformation takes place and at T = 4 practically all A-atoms are randomly distributed. A similar model for surface segregation in small particles with a tendency to phase separation at low temperatures (W-c 0) was studied before [6]. However, in that model. all the atoms lying at the surface were assumed equivalent. As N~=l5 T= 1
Fig. 3. Spatial three different
T=
2
T=
4
distribution of A-atoms temperatures.
(black
circles)
in a cube-octahedra
with
NA = 15 and for
C.A. Botseiro, J.L. Morhtipez
/ Spatial atomic distribution
409
we showed here, in a small cluster, as in the one we considered, it is important to distinguish the surface atoms that belong to the second, third and fourth shell since the coordination number is different (column T of table 1). In conclusion, we have shown that the spatial distribution of atoms in bimetallic clusters suffer appreciable changes as a function of temperature and that the assumption generally made about a random distribution of atoms is valid only at high temperatures. It is a pleasure to thank Professor K.H. Bennemann for helpful discussions.
References [l] [Z] [3] [4] [5] (61
J.H. Sinfelt, J. Catalysis 29 (19873) 308. J.H. Sinfelt, Platinum Metals Rev. 20 (1976) 114. J.H. Sinfelt, Rev. Mod. Phys. 51 (1979) 569. R.A. Swalin, Thermodynamics of Solids (Willey, New York, 1972). J.L. Morb-Lbpez and L.M. Falicov, Phys. Rev. B18 (1978) 2542. D. TomBnek, S. Mukherjee and K.H. Bennemann, Phys. Rev. B28 (1983) 665.
Surface
404
SPATIAL ATOMIC DISTRIBUTION CLUSTERS *
IN SMALL BIMETALLIC
C.A. BALSEIRO Centro Atbmico
Bnriloche. S.C. Bariloche, Argentina
and J.L. MORAN-Li)PEZ
**
Institute ftir Festktirperforschung Germany
der Kernf#~.~chungsanlage
Received
for publication
10 July 1984; accepted
Within the (55 atoms) is equivalent and For a system temperatures.
Jiilich,
D
- 5170 Jiilich, Fed, Rep. of
20 July 1984
regular solution model the spatial distribution of atoms in cube-octahedral clusters studied. All the sites located at the same distance from the center are considered the atoms distributed at random with a characteristic concentration x,( i = 1.2. 3.4). that orders at low temperatures, we calculate the equilibrium values at different
There is a large number of publications reporting that the best catalysts are obtained in the form of small particles. In particular, bimetallic clusters are used to improve the selectively for certain chemical reactions [l-3]; examples of those catalysts are RuCu, OsCu, PtRd, PtIr, etc., small particles. In a small particle a large fraction of its atoms is exposed to chemical reactions. Therefore, knowledge of their spatial distribution is necessary to understand its catalytic activity. Generally, it is assumed that the two kinds of atoms are distributed at random in the whole particle. However, this assumption could be valid only at high temperatures. At low temperatures it is expected that a kind of ordering should occur. Here, we study within the regular solution model [4] the spatial distribution of atoms in bimetallic cube-octahedral clusters with 55 atoms. As shown in fig.
* Work partially supported by Consejo Mexico-Argentina exchange program No. ** On leave from Departamento de Fisica. IPN, Apdo. Postal 14-740, 07000 Mexico,
National de Ciencia 140113 H21-003. Centro de investigation DF, Mexico.
0039-6028/85/$03.30 @ Elsevier Science Publishers (North-~oiland Physics Publishing Division)
y Tecnologia,
B.V.
y Estudios
through
the
Avanzados
de1
1, this number of atoms forms a perfect cube-octahedra in which the surface atoms are afi the second (6), third (24) and fourth (12) nearest neighbours of the central atom. The first nearest neighbours (12) are also shown in the figure and in the particle they lie below the surface. We call the first, second, third and fourth shell all the sites corresponding to the first, second, third and fourth nearest neighbours respectively and we denote the number of nearest neighbours lying in she11 j to an atom in shell i by Zij. Those values are given in table 1. We consider a particle with NA atoms of type A and 55 - NA atoms of type B. distributed in all the different shells. The concentration in shell i is given by xj = e*,i&?
0)
where #A,i and Nj are the number of atoms A and the total number of sites in the i th shell, respectively.
Fig. 1. The first (solid circles), second (2). third (3) and fourth (4) nearest atoms in a 5%atom cube-octahedra.
neighbours
of the central
406
C.A. Balseiro. J. L. Morrin-Lhper
/ Spatial atomic distrihutron
By assuming now that the contributions to the energy are the short-range pairwise interactions UAA, U,, and U,,, the internal energy of the system can be written U= 12W[ X”X, + 2x; + 2X,X, + 4x,x,
+ 2x,x,
+ 2x: +x,x4
+ 4x3+
+ +(A - l)( x0 + 12x, + 4x, + 10x, + 5x,)],
(2)
where w = u,,
+ u,,
- 2u,,
A = (Ca,
- &)/‘I+‘.
.
(3) (4)
A particle with W > 0 will tend to develop ordered temperatures and positive values of A mean a tendency A to the surface. The configurational entropy is given by
structures [4] at low [5] to segregate atoms
N, !
szkF1n(q - N~,,)!N~,,! Now, in order to get the spatial minimize the free energy F=
U-
configuration
at a given
temperature
TS,
with respect centration
(6) to all concentrations
with the constraint
that
is kept fixed. The results for a particle with parameters W = 4 and A figs. 2 and 3. In fig. 2 we show the equilibrium values A-atoms, hi,,, as a function of the total number of A-atoms for different temperatures. The temperature is measured in
the average
J=o
0
_
1 2 3 4
1 0 0 0
con-
= 0.5 are shown in of the number of in the cluster, NA. units of k.
Table 1 Number of nearest neighbours lying in shell j to an atom in shell I. Z,,,: the last column total number of nearest neighbours i
we
j=l
J=2
J=3
j=4
7
12 4 4 2 1
0 2 0 1 0
0 4 4 2 4
0 1 0 2 0
12 12 8 I 5
(T) is the
C.A. Balseiro, J. L. Morirn- Lirpez / Spatial atomic distribution
407
At low temperatures (2’ = l), as one starts to substitute B-atoms by A-atoms in the cluster, they go to the first and third shell. But when one gets to NA = 11, the equilibrium configuration is obtained by moving the A-atoms to the second and fourth shell. Then, just after the two shells are full, additional substitutions drive the A-atoms to the first and third shell.
_.....- 3StiELL
-
1 SHELL
-I-
ZSHELL
---
3SHELL
I
20-
I
-
1SHELL
-I-.-
ZSHELL
,
I
I
I
I
a
TZL
I
.-* ,
---3SHELL . ..... &SHELL
15-
1
4'
Y
-
/+
+
/+
.J’
Fig. 2. The number of atoms NA., in the different A-atoms NA and for three different temperatures. A = 0.5 in units of k.
shells as a function of the total number of The parameters we used were W= 4 and
408
C.A. Balseiro, J.L. Morirn-Lbpez
/ Spaiial atomic distribution
At higher temperatures (T = 2) the kind of transition observed at NA = 11 takes place at NA = 15 but it is not so sharp. Finally at T = 4 the occupation in all shells is a smooth function of NA and practically no order is observed in the whole range of NA. To illustrate the temperature dependence of the spatial arrangement we show in fig. 3 a particle with NA = 15 and at the different temperatures considered. One clearly sees that at low temperatures (T = 1) the atoms are arranged in an ordered way, NA,z = 6, NA 4 = 9. As one increases the temperature a kind of order-disorder transformation takes place and at T = 4 practically all A-atoms are randomly distributed. A similar model for surface segregation in small particles with a tendency to phase separation at low temperatures (W-c 0) was studied before [6]. However, in that model. all the atoms lying at the surface were assumed equivalent. As N~=l5 T= 1
Fig. 3. Spatial three different
T=
2
T=
4
distribution of A-atoms temperatures.
(black
circles)
in a cube-octahedra
with
NA = 15 and for
C.A. Botseiro, J.L. Morhtipez
/ Spatial atomic distribution
409
we showed here, in a small cluster, as in the one we considered, it is important to distinguish the surface atoms that belong to the second, third and fourth shell since the coordination number is different (column T of table 1). In conclusion, we have shown that the spatial distribution of atoms in bimetallic clusters suffer appreciable changes as a function of temperature and that the assumption generally made about a random distribution of atoms is valid only at high temperatures. It is a pleasure to thank Professor K.H. Bennemann for helpful discussions.
References [l] [Z] [3] [4] [5] (61
J.H. Sinfelt, J. Catalysis 29 (19873) 308. J.H. Sinfelt, Platinum Metals Rev. 20 (1976) 114. J.H. Sinfelt, Rev. Mod. Phys. 51 (1979) 569. R.A. Swalin, Thermodynamics of Solids (Willey, New York, 1972). J.L. Morb-Lbpez and L.M. Falicov, Phys. Rev. B18 (1978) 2542. D. TomBnek, S. Mukherjee and K.H. Bennemann, Phys. Rev. B28 (1983) 665.