Surface Science 79 (1979) 109-116 0 North-Holland Publishing Company

SEGREGATION AND SHORT-RANGE ORDER PROPERTIES AT THE

BOUNDARIES OF TWO-DIMENSIONAL BIMETALLIC CLUSTERS

J.L. MORAN-LbPEZ and L.M. FALICOV Department of Physics *, University o.f California, Berkeley, California 94 720, USA

Received 31 July 1978

A previous theory of concentration and short-range order properties of segregating alloys at the surface is extended to study bimetallic two-dimensional clusters and their boundaries. It is found that the reduced coordination number in the clusters leads to (a) a lowering of the phase separation temperature by large factors (-2 to 4) and (b) a drastic enrichment of the cluster boundary in one of the species. Applications to Au-Ni and Cu-Ni bimetallic clusters are pre- sented.

1. Introduction

The composition and degree of order in an alloy near the spatial boundaries are important in many physical, chemical and metallurgical phenomena (magnetic pro-

perties [l-3], catalytic activity [4-61, corrosive behavior 171, etc.) as shown by a large number of experiments performed in the last years.

In particular, it has been shown that supported two-dimensional bimetallic alloys exhibit catalytic behavior with interesting technical applications [6,8]. It is ob- served that bimetallic clusters in two-dimensional alloys can exist at temperatures such that in the corresponding bulk samples the two components show a very low miscibility. Furthermore, it is observed that only one of the elements (Cu in CuOs

or CuRu) is present at the boundary of the cluster. Recently, theoretical studies of segregation in binary alloys [9-121 and on the

existence of surface miscibility gaps [I l] within the regular solution theory have been presented. This theory assumes the system to be completely random under any circumstances and therefore it is valid only at high temperatures. At low tem- peratures it gives only qualitatively correct results. In particular, it predicts surface segregation for alloys different from Ao. s O.s B even in cases where the differences in bond energies between pure elements, VA, and UBB, is zero [I 11.

Recently, the authors have developed a theory for the surface effects in binary alloys ([ 13,141, hereafter referred to as I and II respectively). We showed that it is

* Work supported in part by the National Science Foundation through grant DMR78-03408.

109

110 J.L. Morcira-Ldpez, L.&f. Falicov /Segregation and short-rarzge order properties

necessary to include both short- and long-range order if we want to describe the sys- tem properly over the whole range of temperatures. The theory is based on pairwise atomic interactions between nearest neighbors only U,,, UAn and Unn and the entropy of the system is calculated in the pair approximation following Kikuchi’s Cluster Variation method 1151. For the sake of consistency, and in the absence of better data, it is assumed that the parameters UAA, lJ,, and lJnn are independent of concentration, coordination number and explicit location of the atoms.

In this paper we apply the theory developed in II to two-d~rnellsion~ clustering alloys. In section 2 we outline the main features of the theory and present the results as applied to Cu,Nir _-x and Au,Ni, _-x for two different lattice structures, square and hexagonal. Section 3 is a summary and discussion.

2. Calculation and results

2. I, Bulk properties

The bulk properties of a segrating alloy are described in detail in II. Here, we

summarize only some of its features. Segregating or clustering alloys A,Br _-x separate at temperatures below a critical

temperature 7’,(x) into two phases cu and 0 with concentrations X, and xp of ele- ment A. The concentrations have the limits X, = 1 and xp = 0 at T = 0.

With the introduction of the molar-fraction parameter <, the long-range order Q, and the short-range order parameters u,, it is possible to calculate selfconsistently the bulk properties. These parameters are defined by

i=A’JN, +p:-p$,, ~,-l-p&&$p$, (u=c~fl). (1)

Here NQ is the number of atoms forming the a-phase, N is the total number of atoms in the system and the symbols & and pf;~ (1,J = A, B) denote the probability of finding an I-atom and an I - J nearest neighbor bond in the v-phase. With the de~nitions (I) the pair and site probabilities can be written as functions of the four

parameters E, n, o,,~o~ and the nominal concentration X. The equilibriunl values for the four parameters for a given x are obtained by

minimizing the free energy F = U-- KS. The internal energy U and the entropy S

can be written [14] as

U($, a,, up) = U(0, 0,O) + $ZNx(l - x)[&, + (1 - [) up] w, (2)

where 2 is the number of nearest neighbors, and W= UAA + UBS - 2UAn. Eq. (3) corresponds to Kikuchi’s expression [15] for the entropy in the pair approxima-

J.L. h&win-Ldpez, L.M. Falicov /Segregation and short-range order properties 111

tion. Minimization of F yields four coupled equations which determine .$, v, u, and

9 The phase separation transition temperature takes its maximum value for x = 0.5

and is given by

kT,(x = 0.5) = &‘I(2 ln[Z/(Z - 2)]}-’ . (4)

The short-range order at this temperature and concentration takes the value

0, = up = a@,> = Z(Z - 1)-l - 1 . (9

2.2. Properties at the boundary

In order to calculate boundary properties we label the sites in each phase with an index n corresponding to the row to which they belong. Row n = 0 is the boundary

(outer row) of the two-dimensional cluster, and all other rows are parallel to it. When there is phase separation, the boundary sites can be part of either phase. We define the boundary molar function accordingly

tB = ~,@YN(B) > (6)

where N,(B) is the number of bounda~ sites which belong to the a-phase and N(B)

is the total number of sites at the boundary. We also define (i) An nth row concentration of element A on phase v (V = cr, fl):

XV&) = PvA@)

(ii) Three different intrarow pair probabilities:

Pf;J(% n) 9 I, J = A, B , [WV = (WI .

(iii) Four interrow pair probabilities between neighboring rows:

&(n, m) > [CAB) f (WI .

When the dimensionless parameter

A = (cr,, - &$,)/~~UAB - u,, - &,I (7)

is introduced, we find always that the free energy is minimized when eB= 0 for A < 0 and when & = 1 for A > 0. In other words the boundary corresponds invari- ably to only one phase. We choose the elements such that A > 0, and consequently report only the case in which the boundary belongs to the (Y phase.

By means of equations similar to ey. (1) we define (as in II) intrarow o,(n, n) and interrow u&z, m) short-range order parameters. In terms of these, the internal energy of this layered structure is given by

U=constant + nWaGO [$ZO[(x,(n>}2

112 J.L. Morcin-L&m, L.&f. &licov / 5kgregatian and short-range order properties

+x,(n)(A- l)-;A+x,(n){I -a-&)} o,(n,n)]

t- Z1 [X&I) x,(n + 1) + 4 {x,(n) +.X&I + l);t(A - 1)

-4A +x,(n){ - 1 x,(12 + l))a,(n, II + I)]] ) (8)

where % is the number of sites per SQW, .Za is the number of nearest neighbors to each site located in the same row, and 2, is the number of nearest neighbors to one site which are in one of the adjacent rows. (Z = 2, + ZZ,).

The entropy ~II the layered structure is

0 0.2 0.4 0.6 0.8 1.0

Concentration, x

Fig. 1. Concentration-temperature phase diagrams for segregating bimetallic clusters with parameters corresponding to Au,Nil _-x. Two different two-dimensional structures, square and hexagonal, are shown. The phase separation line is shown in each case together with the bulk fee alloy (dashed line) for the sake of comparison. The concentration on the boundafy of the bimetallic cluster is shown fur various nomitral cluster concentrations.

J.L. Mora’n-Lhpez, L.M. Falicov /Segregation and short-range order properties 113

0 I I I I

0 0.2 0.4 0.6 0.8 1.0

x = 0.1 I I I

,----. 0.3\

/’ ‘\,O 5 - / \

,I’

\ \ \ \ \

Hexagonal

I I I I 0.2 0.4 0.6 0.8

where

Concentration, x

Fig. 2. Diagram as in fig. 1 corresponding to Cu,Nil __x.

The equilibrium values of x,(n), O&I, n) and o,(m, n) are obtained tion of the free energy.

c Au,Ni,_, L 0.4

Temperature, kT/IWI

by minimiza-

Fig. 3. Temperature dependence of the short-range order in the two-dimensional cluster,

ocluster, and its boundary, 000, in a square two-dimensional Au,Nil_, segregating alloy. Results for two different nominal concentrations are shown.

114 J.L. Mor&-Lopez, L.M. Falicoti /Segregation and short-range order properties

We have studied in detail the cases of Au,Ni, _-x and Cu,Nir _-x for two two- dimensional clusters: a square arrangement (2 = 4, Z0 = 2, Zr = 1) and an hexago- nal lattice (Z = 6, Ze = 2, Zr = 2) with a (1, 0} type boundary.

In fig. 1 we show the concentration-temperature phase diagrams for Au,Nir _-x alloys and the concentration at the boundary for different nominal cluster concen- trations. For comparison we also show the fee bulk phase diagram. In fig. 2 we

show equivalent results for Cu,Ni, _-x alloys. The results for the short-range order at the boundary as a function of temperature and for different cluster nominal con-

centrations are shown in fig. 3. The bulk results are shown also for comparison. In all cases the heat of mixing W was estimated from the bulk phase separation transi- tion temperature, 595 K for CuNi [ 161 and 1083 K for AuNi [ 171, and the differ- ence UAA - Unn from the heat of vaporization [9].

3. Discussion

We have applied the theory of surface effects in binary alloys developed previ- ously 113,141 to study segregation and short-range order in two-dimensional bime- tallic clusters. Two examples were considered, Cu,Nir _-x and Au,Nir _-x in square and hexagonal lattices.

The introduction of the molar fraction and long- and short-range order param-

eters make the theory valid over the whole temperature range. A consequence of

the short-range order is the lowering in the transition temperature as compared to that predicted by the regular solution theory [ 181.

The higher miscibility in bimetallic clusters observed experimentally, can be interpreted noting that if the number of nearest neighbors is decreased, Tc gets sub- stantially lowered. This is illustrated in Figs. 1 and 2, where, for the same IV, the phase diagrams for the fee crystal (Z = 12) and the square (2 = 4) and the hexago- nal (Z = 6) two-dimensional structures are shown. The reduction factors in the transition temperatures with respect to the fee crystal for the square and hexagonal lattices are 3.8 and 2.22 respectively. (In the regular-solution theory the reduction factors are 3 and 2 respectively.) The importance of these facts in the design of catalysts is obvious.

Figs. 1 and 2 show also the results for segregation as a function of temperature for different nominal concentrations x in Au,Nir _-x and Cu,Nir --x alloys. In both cases, we find a strong segregation and the boundary of the cluster consists almost exclusively of Au- or Cu-atoms for temperatures higher than T, and nominal con- centrations x > 0.1. This effect is stronger in Cu,Nir _-x than in Au,Nir _-x since A(CuNi) > A(AuNi). At temperatures T < T, the a-phase stays near the surface and the P-phase moves inwards. The boundary of the a-phase consists of almost pure A atoms (Au or Cu in our case). This is the two-dimensional version of the models proposed by Sachtler and Dorgelo (161, Poneg [ 191 and Cahn [ZO] for three- Dimensions clustering alloys.

J.L. Mor&-Ldpez, L.M. Falicov /Segregation and short-range order properties 115

Results for the temperature dependence of the short-range order for different nominal concentrations in Au,Nit_, alloys are presented in Fig. 3. These results are for the square lattice. The short-range order in the interior of the cluster ucruster at the transition temperature and for x = 0.5 is ucruster (x = 0.5, T= T,) = 0.333. This value is relatively high and should be taken into account in calculations involv- ing probabilities of finding given atomic configurations around an atom. For exam- ple, in calculating the catalytic activity if Cu-Ni alloys in reactions where C-C bonds are broken, it is assumed that the active sites for these reactions are clusters of atoms with a given geometry (two nearest-neighbor Ni-atoms [21]; four Ni-atoms; three Ni-atoms and one Cu-atom [22] ; etc).

The short-range order at the boundary is very small at low temperatures due to

the strong segregation. It increases with temperature up to a maximum as the enrichment of A decreases and then it goes to zero as T + 00.

In conclusion we would like to point out that our theory is in excellent qualita- tive agreement with experimental observations [8], and that the reduction in coor- dination number 2 of these bimetallic clusters is the predominant effect which leads to their rather unusual segregation properties.

Acknowledgments

On of us (J.L.M.L.) wishes to acknowledge the financial support of CoNaCyT- CIEA de1 IPN (Mexico) in the form of a Postdoctoral Fellowship.

References

[l] J.W. Cable, Phys. Rev. B15 (1977) 3477.

[2] J. Vrijen and S. Radelaar, Phys. Rev. B17 (1978) 409.

[3] R.A. Medina and J.W. Cable, Phys. Rev. B15 (1977) 1539.

[4] J.H. Sinfelt, J.L. Carter and D.J.C. Yates, J. Catalysis 24 (1972) 283.

[S] V. Poneq, Catalysis Rev. Sci. Eng. 11 (1975) 41.

[6] J.H. Sinfelt, Science 195 (1977) 641.

[7] D.A. Vermilyea, Phys. Today (Sept. 1976) 23.

[S] J.H. Sinfelt, Progr. Solid State Chem. 10 (1975) 55.

[9] F.L. Williams and D. Nason, Surface Sci. 45 (1974) 377.

[lo] D. Kumar, A. Mookerjee and V. Kumar, J. Phys. F6 (1976) 725.

[ 1 l] C.R. Helms, Surface Sci. 69 (1977) 689.

[ 121 A. Jabtonski, Advan. Colloid Interface Sci. 8 (1977) 213.

(131 J.L. Moran-Lopez and L.M. Falicov, Phys. Rev. B18 (1978) Sept. 15.

[ 141 J.L. Moran-Lopez and L.M. Falicov, Phys. Rev. B18 (1978) Sept. 15.

[15] R. Kikuchi, Phys. Rev. 81 (1951) 988.

[ 161 W.M.H. Sachtler and G.J.H. Dorgelo, J. Catalysis 4 (1965) 654. [ 171 R. Hultgren, R. Orr, P. Anderson and K. Kelley, Selected Values of Thermodynamic Prop-

erties of Metals and Alloys (Wiley, New York, 1963). [ 181 R.A. Swalin, Thermodynamics of Solids (Wiley, New York, 1972).

116 J.L. Monin-Lbpez, L.M. Falicov /Segregation and short-range order properties

[ 191 V. PoneF, in: Electronic Structure and Reactivity of Metal Surfaces, Eds. F.G. Derouane

and A.A. Lucas (Plenum, New York, 1976).

[20] J.W. Cahn, J. Chem. Phys. 66 (1977) 3667.

[21] J.J. Burton and E. Hyman, J. Catalysis 37 (1975) 114.

[22] J.L. Morin-L6pez and K.H. Bennemann, Surface Sci. 75 (1978) 167.