Solid State Communications, Vol. 84, No. 6, pp. 663-667, 1992. Printed in Great Britain.

0038-1098/92 $5.00 + .00 Pergamon Press Ltd

SURFACE SEGREGATION IN SMALL BIMETALLIC PARTICLES

S. Modak* and B.C. Khanra*

Saha Institute of Nuclear Physics, Sector 1, Block - - AF, Bidhannagar, Calcutta -700 064, India

(Received 21 July 1992 hy C.N.R. Rao)

A tight-binding electronic theory has been used to study the surface segregation behavior of small bimetallic particles having the shape of a cubooctahedron. For P t - l b (lb = Cu, Ag, Au) systems it is found that lb atoms segregate to the surface of the cluster. The extent of segregation is found to be less for smaller particles. The results of our model agree very well with the Monte Carlo (MC) simulation results of Strohi and King.

I. INTRODUCTION

IT HAS BEEN well established that studies on surface segregation in binary alloys are important for understanding their catalytic activity and selectivity [I, 2]. With this in mind various theoretical as well as experimental studies have already been performed to find out the surface composition of binary alloys [3-7]• These studies were mostly concerned with semi-infinite systems. But in actual catalytic processes alloy catalysts are normally used in the form of small particles [8-1 I]. Hence, it is very important to study the surface segregation behavior of small bimetallic particles. So 'far only a few theoretical model calculations [12-15] have been done to determine the electronic structure and alloy composition at the surface of small bimetallic particles. In this paper a tight binding electronic theory has been proposed to study the electronic structure and surface com- position of P t - l b (Ib = Cu, Ag, Au) clusters having the shape of a cubooctahedron.

2. PARTICLE M O R P H O L O G Y

The particle shape considered in this calculation is a perfect-f.c.c, cubooctahedron with both (1 0 0) and (11 I) faces being present on the surface. For this type of structure there are five kinds of inequivalent sites present on the surface [I 6]. A schematic diagram is shown in Fig. I. Please note, the corner sites C 6, two types of edge sites, C~, C~, and two types of face

• ~ 4 ' ~ • • • sites C~ and C s are the mequwalent surface s~tcs for

the cubooctahedron particles. For the notations C I we have considered the [I 6]. Besides, t.here are always some atoms in the interior of the particle which have coordination number twelve. Among the sites on the surface inequivalency arises mainly from the count of total number of nearest neighbors. In any com- plete cubooctahedron number of lowest coordinated sites (which has six nearest neighbors) is always twentyfour. The number of other inequivalent sites on the surface varies with the size of cluster and these have coordination number seven, eight and nine etc.

3. T H E O R E T I C A L MODEL

in our theoretical model we have calculated the average alloy composition at the surface of bimetallic particle of composition A,.Bt_.,. The alloy composi- tion at the various inequivalent positions is deter- mined by minimizing the free energy F(x,i ,, xq:, xq,, .X'q~. x,l~, X,h, ) of the system with respect to xu.. X,r_, xu,, x,~,. x,~ and x.,. Here, xq, represents the coverage of A atom of the alloy in qith inequivalent site and is expressed as

N A

"' ( I ) .v,/, = N,-'--~, "

where Nu, represents the total number of qi sites and N A gives the number of A-atoms in those qi sites.

q,

This minimization of free energy F is done with the constraint that the average alloy concentration of the cluster, i.e.

6 * To whom all correspondence should be addressed. Z N'4 * Present address: Department of Chemical Engineer- ,t, N'4

ing. Iowa State University. 231 Sweeny Hall. Ames. x - 't'=l---~ - IA 50011 USA. N N

(2)

663

664 S U R F A C E S E G R E G A T I O N IN SMALL B I M E T A L L I C P A R T I C L E S Vol. 84, No. 6

Fig. 1. Face-centered cubic (f.c.c.) cubooctahedron of 586 atoms.

where (i is thc site energy and ~/ is the interaction energy between thc ith a n d / t h site. Let q represent the incquivalent sites.

The average LDOS in any incquivalent site is

oq( E ) " ' f = .~qp,~ (E) + (1 - .vq)pq (E). (9)

Now. the total internal energy of the bimetallic particle is

E = Z NqEq. (10) q

where Eq is the internal energy of an atom in q th inequivalent position and Nq is the number of such sites in the particle. Eq is estimated from the formula

I;' I

f Eq = Epq( E ) d E. ( 1 I )

where E r denotes the Fermi energy.

remains fixed. Here N is the total number of atoms . constituting the said bimetallic particle of which the

number of A-atoms is N '~ and number of B-atoms is N t~ so that

N = N 4 +Nt~ (3)

The free energy F of the bimetallic particle at tem- perature T has been estimated from the expression

F = E - TS. (4)

where E is the internal energy and S is the con- figurational entropy of the particle. S is cvaluated as

S = k~ Z InN, i4, (N~'~ Nq:,), . (5) q¢

To calculate the internal energy of the system we have first calculated the local density of states (LDOS) at each inequivalent site of the cluster from the diagonal elcment of the one particlc Green's function Gii(E) a s

pi l (E)=__l Im G J(E) . i = A . B . (6) re

The diagonal Grecn's function Gii (E) is evaluatcd by Clus te r -Bethc-Lat t ice mcthod [6, 17] from Dyson's equation

1 1 I G - E----S- ~ - E + 2 HG (7)

with thc Hamiltonian

H = Z e i ] i ) ( i l + Z Z v , . , l i ) ( . j l . (8) t i t

i ¢ j

4. RESULTS A N D DISCUSSIONS

The calculations are performed with the d-band parameters only in view of the fact that the d electrons play the most important role in the physico-chemical behavior of transition metals and their alloys. The input parameters needed for the calculation are the d-bandwidth, the d-band center and the d-band filling of each component of the alloy. These values have been taken mainly from [18]. For Pt we consider the d-band filling with 9.7 electrons [19], while for Ib atoms it is 10.0 electrons [18]. The band widths for Pt is taken to be 7.3eV while for Cu, Ag and Au it is 2.8 eV. 3.6eV and 5.3 eV respectively. The mean energy difference b for P t -Cu , P t - A g and Pt Au are taken to be 1.41eV, 0.15eV and 0.11 eV respectively. The calculations have been carried out for two different sizes of cubooctahedral particles - - one having 201 atoms and the other having 586 atoms.

In Fig. 2 we plot the typical density of states curves for atoms located at different inequivalent sites of a cubooctahedron (586 atoms cluster) - - the average composition of the cluster being Pt05Ag05. Here, one important aspect may be worth noticing - - the surface sites give rise to narrower band because of the smaller number of nearest neighbors. It may further be noticed that. the corner site having six nearest neighbors, gives the highest peak and a typical interior atom with twelve nearest neighbors gives rise to a very flat band. Due to topographical change but still having the same total number of atoms in the particle our density of states curves for different inequivalent sites are not well resolved near

Vol. 84, No. 6 SURFACE SEGREGATION IN SMALL BIMETALLIC PARTICLES 665

6.00

o

0

o~ o 4.00

u')

o

2.00 £:3

0.00 -5.00

.. . . . . . Face sites ,g', Edge ,, ^/a ", . . . . . . Corner , , /,'Y!\ ' '

I

-3:oo -,.'oo ,.o0 Energy (eV)

!

3.00 E F

Fig, 2. Comparison of thc LDOS for thc different inequivalent sites of the cubooctahcdron particlc of 586 atoms of Pt()..~Ag05.

the half maximum. But they are well resolved at thc top and bottom of the band. On the whole, the density of states curves may thus be considered to reflect very significantly the nonequivalence of thc sites of the particle.

In Fig. 3 we have shown the avcrage concen- tration of Pt atoms on the surface of cubooctahedral particlcs at temperature T = 550 K. Wc have con- sidered particles having 201 and 586 atoms rcspcct- ively. We have compared our results with those obtained by MC simulation technique by Strohl and King [12].

The data presented in Fig. 3 reveal a good amount of physical properties which may bc dcscribcd as follows:

(i) Group Ib metals always scgregatc to thc surface irrespective of the particle composition.

(ii) For larger particles the scgrcgation is marc (compare for example, a, a' with h, h' respectively). This is because for smaller particles there always remains some constraints which restrict thc cxtent of segregation. The constraints arisc mainly from two reasons: (1) The number of available Ib atoms in thc particle may not be sufficient to cover the surface sites though they have higher probability to cover the surface sites. (2) The number of surfacc sites available for exchanging an interior lb atom with a Pt surfacc atom may not be sufficient for smaller particles.

(iii) For Pt0.91b0.1 composition the segregation is insensitive to the details of the group Ib metals, i.e. the segregation is of the same ordcr of magnitude for

B

o ~

1.0 ~ 0.8

. 0.6

0.4 o

0.2 O1

m cu m A g E ~ E ,.

B A u ~ a "~

^ A,

n nU (b) (c/) (g)

~0.3 o

.~ 0.2

* 0 . ~

o

o'l

[] Cu lIAg

E~ mAu

- , ^

A

(o) t lb~ )

I (a') Fig. 3. Surface segregation effect of P t - Ib bimetallic particle at T = 550K. (a) and (a') corresponds to our calculations. (h) and (h') corresponds to MC simulation results. (A) For Pt091b0j. (B) For Pt05lbos.

Cu, Ag and Au. But for Pto.51b05 composition the segregation generally varies from one Ib metal to the other. This may be explained in terms of the mixing energies for P t - l b atom pairs (please see the work of Strohl and King [12]).

(iv) in most cases the present tight binding calculation gives results in excellent agreement with thc MC simulation results of Strohl and King. However, for P t -Cu alloy the present calculation shows more copper segregation than the MC predictions [Figure 3(B), parts (a) and (b)].

In Fig. 4 the preferential filling of inequivalent surface sites of 201 atom cluster at temperature T = 550 K is shown. From the figure it is seen that for all the alloys and for any composition of the bimetallic cluster the lb atoms prefer to occupy the lowest coordinated sites first. This is evident from the fact that the Pt atom fraction is lowest for six coordinated sites. The system thus minimizes its free

666 SURFACE SEGREGATION IN SMALL BIMETALLIC PARTICLES Vol. 84, No. 6

,.0 ~ _Z~ Coo;,nated

\ ~X,X-" 'o-8 a 9 . = o . s \ \ \

!o, ~ O.l.

1 . 0 ~

\ \ \ 0.;

0.0 0.2 0.4 0.6 0-8

b Cluster concentration of Ag

1.0 ~ ,~ -o- 6 Coordinated • \\\ - . - 7 , ,

0 . 8

o

K 0.s

_.=

0.4

It,

0.2

0.0 0-2 0.4 0.6 0.8

C Clutter concentration of Au

Fig. 4. Fraction of platinum atoms occupying inequivalent sites for different cluster composition of a 201-atom cubooctahedron particle surface at the temperature T = 550K. (A) Pt-Cu. (B) Pt-Ag. (C) Pt-Au.

1.0

0.9

0.6 Xs

0.6.

0.2

a 0.0

I°t-Cu (586 atom cluster) , , ~ , - . O K / /

o '2"¢P;',27,'a;; '°ulation/ / ° / /

/ / :.#

l 3 " ~ I I

0-2 O-& 0.6 0.8 1.0 x (Pt)

1 Pt-Au (B86 atom cluster) / ~ T=SS0K / /

0.8 ~ The present calculation J

0 o

0.& ,;,° F

0.0 0.2 0.4 0.6 0.8 1.0 b X(Pt)

1.0 Pt Ag(S86otom cluster) / ~

/ /

0.8 ~ The present calculation// /

0.6 ° MC Simuiati°n / /

x / /

0.;

i 0.0 0.2 0.4 0.6 0.8 k0

C X(Pt)

Fig. 5. Scgregation behavior of P t - ib bimetallic cubooctahedron particle of 586 atoms. X, represents the average surface concentration of Pt atoms and X represents the alloy composition of the cluster. (A) Pt-Cu. (B) Pt-Au. (C) Pt-Ag.

energy by pushing the atoms of lower heat of vaporization from the interior to the less bound sites in the surface. Physically, thus one may visualize that the six coordinated sites have the lowest

Vol. 84, No. 6 SURFACE SEGREGATION IN SMALL BIMETALLIC PARTICLES 667

coverage of Pt atoms than the sites coordinated to seven, eight and then nine atoms. These findings also agree with those of MC simulation results [12].

Finally. in Fig. 5 we have plotted the average surface concentration of Pt atoms as a function of alloy composition of the cluster for a particular size of bimetallic particle (586 atom cluster). The MC simulation results has also been indicated. It is seen that the agreement between the two approaches is excellent.

5. CONCLUSIONS

Tight binding methods which are very much in use in predicting successfully various physical proper- ties of transition metals and their alloys, have been used here to predict the segregation behavior in small particles. The main advantage of this scheme is that here are starts with some electronic parameters like, d-bandwidth, number of d-electrons in a metal atom and the mean energies of d-electrons, etc. to find out electronically the most favorable equilibrium states. Nowhere in the calculation have we to use any arbitrary semi-empirical approach to achieve the results. Besides, the calculations are very simple and require less computation compared to the MC simulation technique. In view of the fact that such a simple formalism can give results in excellent agreement with the results of the MC calculations, we feel the present scheme can be used to predict as well very successfully other electronic properties of the small clusters like chemisorption on small clusters, etc.

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