A new surface segregation isotherm description and its application to binary alloys

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Surface Science 331-333 (1995) 799-804

ELSEVIER

A new surface segregation isotherm description and its application to binary alloys L.Z. Mezey a, W. Hofer b,, a Institute of Physics, TUBudapest, Budafoki {~t8-10, H-1111, Budapest, Hungary b Institut ffir Allgemeine Physik, TU Wien, Wiedner Hauptstrasse 8-10 / 134, A-1040 Wien, Austria

Received 21 July 1994; accepted for publication 6 December 1994

Abstract In previous years a new theory, modem thermodynamic calculation of interface properties (MTCIP), was developed. Its general equations on the thermodynamic equilibrium were solved in a first approximation (MTCIP-1A) and applied to several dilute binary alloy systems in good agreement with experimental results, even in the case of surface reactions with the environment. Here, first a more developed second approximation (MTCIP-2A) is shortly outlined, allowing, among other new results, for any number of interface sublayers, and being valid in the total compositional range of the bulk. Then, by some simplifications, a new segregational isotherm is developed from it. Using the two dilute limits as described by the MTCIP-2A, then even with a single surface composition (measured at any bulk composition and temperature) this segregational isotherm permits to obtain surface compositions over the total bulk compositional range. Results are shown for the low-index surfaces of PtNi at temperatures of about 1100 K, mostly in better agreement with experimental values than other present theories. Keywords: Alloys; Equilibrium thermodynamics;Low index single crystal surfaces; Nickel; Platinum; Surface segregation

1. Introduction Due to the large practical and theoretical significance of the subject considerable progress has been achieved in the last ten to twenty years in the calculation of the surface properties of solids. In the case of the surface composition, too, several valuable results have been achieved (for a review see e.g. Ref. [1]). However, there are still important problems to

* Corresponding author. Fax: +43 1 586 4203.

be solved. For instance, in the relatively simple case of the free surfaces of binary bimetallic alloys the problem of calculating the surface chemical composition in the thermodynamic equilibrium state (TES) is not solved so far with general validity. (Below we show several discrepancies between calculated and measured values.) There are cases where not even the component segregating to the surface is correctly predicted by some calculations. In this work a new method for the calculation of the TES composition of binary alloys (in principle valid for non-metallic ones, too) is outlined. This is based on a new, more developed form of our earlier

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L.Z. Mezey, W. Hofer / Surface Science 331-333 (1995) 799-804

thermodynamic calculations. It will be shown to work properly for the example of PtNi alloys, which are both important in practice and challenging for theoretical calculations.

2. A short outline of the MTCIP theory The more basic modern thermodynamic calculation of interface properties (MTCIP) theory [2,3], in its part on the TES, gives a system of equations as necessary conditions of this state for a thermodynamic system consisting of the interface and the two bodies separated by it. From these equations those involving the component chemical potential can be written for the special case of binary mixtures (consisting of components i = 1 and 2) as

Xl=X~l)(1-X/]~S-~-~) q/exp( qtA/lx] / = 1...m.

) (2.1)

def

Here X = X 2 is the atomic fraction of component 2 (e.g. Pt for PtNi), /

refers to the atomic sublayer /def

/

/

and (1) to the bulk, q = ~b2/thl is the ratio of the partial molar surface area of this component to the other one and lz~ is the excess chemical potential of component i defined as usually in respect to the bulk of the pure component i. The symbol Ai denotes the difference of this quantity in the surface layer / in respect to the bulk of the mixture. Consequently, it contains the effects of the broken bonds above the surface, too (non-bonding term [4]). In our previous work (MTCIP-1A, first approximation) first a monolayer, then a multilayer approach was used for solving Eq. (2.1) for dilute binary alloys. The published version of the latter (MTCIPML) [5], however, neglected the non-bonding (or surface tension) term for PtNi (where this neglection has some physical basis). In the following the present improvements of MTCIP-2A (second approximation) over MTCIP-ML (the latest published version of the approaches to the MTCIP) are shortly outlined. Then the "segregational isotherm method" will be deduced from the MTCIP-2A and applied to low-index surfaces of PtNi alloys.

3. A short outline of the MTCIP-2A approach

3.1. Inclusion of the non-bonding (surface tension) factor This factor is to account for the interatomic bonds missing over a free surface as compared with the bulk. In the MTCIP-1A it was used already in a first approximation [4,6] and for further calculational details we refer to those works. Here, however, it is described in a more refined form as outlined below (Sections 3.2-3.4, 3.8). The inclusion of this term gives A//z~ as

A~=ff/G'+(,j=~-2 _

a/'/+J/x~'/+J ( X / + i ) ) (3.1)

Here G~ is the molar internal free enthalpy of atomisation (its calculation is given in Ref. [4] in detail). The symbol otI'l+) represents the fraction of the "effective" neighbours (see Section 3.2) of an atom of i in sublayer ee which are found in the sublayer / + j, where the local composition is given by X I+s. The quantity ~ / i s the relaxed (TES) value of these fractions for the neighbours missing above the surface (see Section 3.3). The last term in Eq. (3.1) accounts for the (reference) bulk conditions of A~z~.

3.2. The use of the effective neighbour concept In all our previous papers the nearest-neighbour approach was used. Here we account for the second, third, etc. ones by giving a weight 3 to the second ones, which is to reflect, however, the influence of the further ones too. Then the number of effective neighbours (EN) of an atom is defined as def

z = z 1 + 3z 2,

(3.2)

with z I and z z being the number of first and second nearest neighbours, respectively. A value of 8 = 0.5 (cf. Ref. [7]) will be used here.

3.3. Multilayer description of the interlayer coupling In the current EN description, for low-index planes, missing (first or second) neighbours are found

L.Z. Mezey, W. Hofer/ Surface Science 331-333 (1995) 799-804 in general for the two topmost sublayers f = 1 and 2. For these the fractions of the missing EN are given by a I = 0/1,0 + o/1,-1 and a 2 = 0/2,0 ( o / f = 0 f o r / > 2). Surface relaxation is then described separately and leads with its previous treatment [4,6] to We= 0///(1 + 0//). For consistency, the difference A a / = c ~ / - ~ / (a formal increase in the fraction of remaining EN due to surface relaxation, reflecting increased bonding strengths) is then redistributed among the effective neighbours below the respective layer according to their a / , t + j values.

3.4. Effect of surface reconstruction in the pure components on their surface tension terms Surface reconstructions are reported for low-index Pt surfaces [8], but not in the case of Ni. At our temperatures of interest (T--- 1100 K) these reconstructions are absent on Pt(ll0), too. Since reconstructions, as all changes at the surface, are to lead to a lower value of the surface free energy, they will be reflected in a lower value of ~1 in Eq. (3.1) (see the relation of these quantities in Refs. [4,6]). Calculating this quantity by selecting the most reliable experimental values from those of the surface tensions [9] and using the method published before [4,6] leads to an ~1 value for Pt which is lower by a factor of 0.9249 as compared with the (unreconstructed) Ni value. Therefore, in place of a common value, ~ 1 i = ~1 (with 6 = 0.5) a n d ~ 1 t = 0.9249~ 1 will be used for the pure elements for (100) and (111) planes.

3.5. The real mixture description concentrated alloys

for more

The molar excess free enthalpy of mixing G e may be described by [10] G e =X(1-X)(ao(Z

) +aa(T)X+a2(T)X

2)

def

= X(1 - X ) f r ( X ) .

801

a s, giving a j ( T ) = a ~ - Tas. For finding the constants, limiting values of f r ( X ) and of the analogously defined other two quantities are used for X Xpt "-> 0 and X ---> 1 in the way published earlier [12]. In addition, a third value is obtained in each case by using the tabulated value at X = 0.5. =

3.6. Lattice distortion release (LDR) for more concentrated alloys The description of LDR is done in a straightforward generalisation of the procedure published for binary dilute alloys [12]. This procedure leads to an excess chemical potential function in the surface which differs from the bulk one. The constants aj are treated

def as aj =

ajc + a jd with chemical and distor-

tional parts. For smaller Ni atoms solved in Pt there is no considerable lattice distortion, consequently /.£~qi(X = 1) = a 0 + a 1 + a 2 = a S + a~ + a~ is taken. Then, as previously, the chemical part /z~tc(X = 0) = a S is obtained [12] as 3/z~i(X= 1). Finally, at higher values of X=Xpt, for which a 2 becomes significant in Eq. (3.3) (the case of Ni solved in Pt, see above), a 2 consists in fact as the chemical part only: a 2 = a c2. From these three conditions all a tC can be obtained, permitting the description of LDR for any concentration. In this work for the low-index faces of PtNi, LDR will be used within the topmost sublayer ( f = 1), as well as for the contribution to A~x~ from the sublayer below ( f = 1, j = 1 in Eq. (3.1)), since LDR is connected with missing neighbours at the surface (there is no LDR in the bulk). In the case of the (110) surface with the most open structure and with a relatively high portion of missing EN above the surface even for f = 2, LDR is used also within that sublayer. For the sublayers with LDR the component partial molar surface areas will be calculated as follows.

(3.3)

From this the component excess chemical potentials /x~ are obtained by standard thermodynamical methods [10]. Similar formulae with constants a T and ajS are used, then, for the molar excess heat H e and entropy S e of mixing. The data available for PtNi [11] allow only for temperature independent a~ and

3.7. The partial component molar surface areas In previous calculations the component partial molar surface areas ~bi of an alloy were approximated by the respective pure component values ~b°, the calculation of which was published in detail [4].

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L.Z. Mezey, W. Hofer/ Surface Science 331-333 (1995) 799-804

In a better description a change with the composition is included as ~)i(Xi ) = ~o + a,(1 - X / ) 2 ( i = 1,2).

(3.4)

This is based on the generalised Gibbs-Duhem equation of the theory of mixtures [10], with the second term being an excess quantity. With Xpt--+ 0 LDR arises (e.g., by elevating of the oversize Pt atoms over the Ni atoms of the topmost sublayer; "squeezing out"). Then the area occupied by a surface Pt atom diminishes to that of a Ni atom, and we will obtain {#pt(Xpt ~- 0) = (/70t + a , = ~b°i, permitting the calculation of a,. 3.8. Surface reconstruction on the mixture surfaces On PtNi, at T ~ 1100 K, surface reconstructions similar to those of pure Pt (but absent on pure Ni) were found [13] for the (100) and (111) surfaces when, as usually, they are rich in Pt. To account for this dependence on the Pt surface content, the quantities ~ connected with the broken bonds are to be taken composition dependent. In a first approach this dependence is described in the form of that of ~bi. Then for Pt we have ~p1t = ~a (unreconstructed) for Xpt --->0 (on Ni) and alt = 0.9249~ 1 at Xpt = 1. Due to the analogy with Eq. (3.4) this leads to an increase for Ni on pure Pt of A ~ i =0.0751~ 1. Only the limiting values will be used in this work (see Section 4), the Pt-rich ones for (100) and (111) with known reconstructions at higher temperatures and the unreconstructed ones for (110).

4. Deduction of the basic formulae of the segregational isotherm method

The MTCIP-2A method outlined so far permits a description of the quantities in the basic Eq. (2.1) for any T and X °), using a system of coupled, non-linear equations (3.1). This system will be simplified now to give a set of independent, non-linear equations, permitting to describe each X ! in its dependence on X (~) separately. Thus we establish a "segregational isotherm" for each sublayer / :

XI(X(1);T) Let the exponent in Eq. (2.1) be denoted as E/. This quantity obviously depends on T, on the bulk

composition X (1) and in general on all the surface sublayer compositions given by X i (see Eq. (3.1)). The latter, however, are - in the TES - actually determined again by T and X (1), so that in fact we have E l = EI(T,X(i)). The dependence of E I on X (1) might be described as follows. Let first, in place of X (1) the volume fraction y(1) (often used in contemporary real mixture descriptions [14] for physical reasons) be introduced. With the pure component atomic volumes V/°(T)= V/ (described, e.g., in Ref. [4]) we have Y(~)=X(21)V2//(X[I)v1 + X2(1)V2). Then the dependence of E i on Y{~) will be described by means of Legendre polynomials: E I ( Y (1)) = Y'~c~Pk(Y(1)), / = 1...m.

(4.1)

k Our current calculations will be done with k = 0...2. The dilute limits El(O) and E l ( l ) are relatively easy to calculate with the MTCIP-2A outlined before; the value of E I at X (1) = 0.5 will be obtained here by fitting with a single experimental result. (Altematively, a calculated result, obtained e.g. by the MTCIP-2A, might be used.) Then we are in the position to obtain the coefficients c ; in Eq. (4.1). The use of the Legendre polynomials permits to easily improve E I ( Y (1)) by including further experimental (or theoretically calculated) points for which just each new value c k/ has to be determined by fitting to the respective result. Now we can describe E l for any X (1) with 0 < X (1) < 1. Thus, each X / is given by a segregational isotherm obtained from

i --X / )qi(Xl) X/=X

<1) i-.~-~]" ~

E / ( / < I ) ) , / = 1...m. (4.2)

5. Results and discussion

In this work, for each sublayer t , in addition to the values of E i obtained by MTCIP-2A calculations for X---> 0 and X---> 1, only one experimental result for E / at a bulk concentration of 50 at% Pt has been used to determine the coefficients c~ in Eq. (4.1). This experimental value has been taken from

L.Z. Mezey, W. Hofer/ Surface Science 331-333 (1995) 799-804 (a) PtNI(IIO) = I I 0 0 K

(b)

PtNi(100)

(e) PtNi(lll)

- 1100 K

lOO

~0

IO

80

80

IO

C

~60

60

84o

40

N 20

2(3

80 20 4O 60 bulkcomposition[at%Pt]

100

803

i o

- 1100 K

~0 lO ~.0

0

1

I

20

40

0 60

80

100

bulkcomposition[at%Pt]

,

0

I

20 40 60 80 bulkcomposition[at%Pt]

lO0

Fig. 1. Surface versus bulk composition for the topmost atomic layer of the three low-index faces of PtNi alloys at ~ 1100 K. The solid line represents our calculated segregational isotherm at 1120 K ((a) and (b)) and 1170 K (c), which has been fitted for 50 at% Pt in the bulk to the LEED data available for the respective temperature. The filled symbols represent different experimental results at similar temperatures: ( 0 ) : LEED [13,15-19], ( • ) : LEIS [20], ( • ) : LEIS [21-24], ( • ) : IDEAS [25]. The open symbols denote the results of other calculations: (O): TBIM [28,29], (O): EAM [26], ([]): EAM [27], (A): MC [30], ([]): LDA [31]. The dashed line indicates surface concentrations equal to the bulk values.

LEED (low energy electron diffraction) experiments [13,15,16] which are available for all the low-index planes for a temperature of about 1100 K. We note that for the (110) surface the LEED result is given as X 1 = 0 + 6 at% [16]. Since, for thermodynamic reasons, that value cannot be exactly zero, a value of 3 at% (in the middle of the given positive range) was used instead. The results are shown in Fig. 1 in the form of segregational isotherms for the topmost layer ( f = 1) and compared with further experimental data and results of other calculations for a similar temperature. (Since the surface compositions X / are known to change with T according to In X / ~ 1/T [1],

these quantities are not much different for temperatures between 1000 and 1200 K, inside which range the temperatures of the used data are found.) In Table 1 the results for sublayer/--- 2 are presented. The experimental data have been measured by LEED [13,15-19], LEIS (low energy ion scattering) [20-24] and IDEAS (incidence dependent excitation for Auger spectroscopy) [25], and the presented theoretical values have been obtained by the embedded atom method (EAM) [26,27], the tight-binding Ising model (TBIM) [28,29], a Monte-Carlo approach (MC) [30] as well as by recent first-principles calculations using a local density approach (LDA) [31]). Differences found between the individual experi-

Table 1 Surface compositions (in at% Pt) of the second atomic sublayer ( f = 2) for the three low-index faces of PtNi alloys; our calculated segregational isotherm results (which have been fitted for 50 at% Pt in the bulk to the LEED data) are compared with further experimental and theoretical values; theoretical results are denoted by * T (K)

1120 1120 1070 1070 1120

1200 1200

25 at% Pt

78 at% Pt

(110)

10 at% Pt in the bulk (100)

(111)

(110)

(111)

57 52+2 b 45 + 10 51 5:10 48 0 14

5 6+6 c

3 a 5+3

88

51 a 30+5a

8

7

8

5 9

62 + 10

a At 1170 K; b at 1070 K; c at 1050 K.

30

74 72 72

Method

* This work LEED [15,17-19] IDEAS [25] LEIS [23,24] * TBIM [28,29] * EAM [26] * EAM [27]

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L.Z. Mezey, W. Hofer / Surface Science 331-333 (1995) 799-804

mental results are usually within typical experimental errors or may be caused by different environmental conditions (e.g., the presence of even a small carbon contamination leads to less Pt on the surface [13,21]). Considering this, good agreement between our thermodynamic calculations and the experimental data is found. In general the other model calculations mentioned here are giving a realistic picture, too. However, as we can infer from Fig. 1 and Table 1 there are still some quantitative and even qualitative problems in other descriptions. For a further discussion of earlier results we refer to Ref. [5] where a short survey of experimental and theoretical results for PtNi alloys is given.

Acknowledgements This work has been supported by the Austrian Fonds zur F6rderung der wissenschaftlichen Forschung, Projekt Nr. P8147-TEC.

References [1] P.A. Dowben and A. Miller, Eds., Surface Segregation Phenomena, 1st ed. (CRC Press, Boca Raton, FL, 1990). [2] L.Z. Mezey, Surf. Sci. 162 (1985) 510. [3] L.Z. Mezey and J. Giber, Acta Phys. Hung. 66 (1989) 309. [4] L.Z. Mezey and J. Giber, Surf. Sci. 234 (1990) 210. [5] W. Hofer, Fresenius J. Anal. Chem. 346 (1993) 246. [6] L.Z. Mezey and W. Hofer, Surf. Sci. 269/270 (1992) 1135, and references therein. [7] J.F. Nicholas, Austr. J. Phys. 21 (1968) 21. [8] G. Chiarotti, Ed., Physics of Solid Surfaces, Vol. 24A of Landolt-B~Smstein, New Series, Group III (Springer, Berlin, 1993). [9] V.K. Kumikov and Kh.B. Khokonov, J. Appl. Phys. 54 (1983) 1346.

[10] See any textbook on real mixtures, e.g., K. Stephan and F. Mayinger, Mebrstoffsysteme und chemische Reaktionen, Band 2 of Thermodynamik, 12th ed. (Springer, Berlin, 1988). [11] R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser and K.K. Kelley, Selected Values of Thermodynamic Properties of Binary Alloys (American Society for Metals, Metals Park, OH, 1973). [12] L.Z. Mezey and W. Hofer, Surf. Interface Anal. 19 (1992) 618. [13] Y. Gauthier and R. Baudoing, in: Surface Segregation Phenomena, Eds. P.A. Dowben and A. Miller (CRC Press, Boca Raton, FL, 1990) p. 169. [14] See, e.g., V.M. Glazov and L.M. Pavlova, Chemical Thermodynamics and Phase Equilibria, 2nd ed. (Metallurgiya, Moscow, 1988) [in Russian]. [15] Y. Gauthier, Y. Joly, R. Baudoing and J. Rundgren, Phys. Rev. B 31 (1985) 6216. [16] Y. Gauthier, R. Baudoing, M. Lundberg and J. Rundgren, Phys. Rev. B 35 (1987) 7867. [17] R. Baudoing, Y. Gauthier, M. Lundberg and J. Rundgren, J. Phys. C 19 (1986) 2825. [18] Y. Gauthier, R. Baudoing and J. Jupille, Phys. Rev. B 40 (1989) 1500. [19] Y. Gauthier, W. Hoffmann and M. Wuttig, Surf. Sci. 233 (1990) 239. [20] L. de Temmerman, C. Creemers, H. van Hove, A. Neyens, J.C. Bertolini and J. Massardier, Surf. Sci. 178 (1986) 888. [21] P. Weigand, P. Novacek, G. van Husen, T. Neidhart and P. Varga, Surf. Sci. 269/270 (1992) 1129. [22] P. Weigand, W. Hofer and P. Varga, Surf. Sci. 287/288 (1993) 350. [23] P. Weigand, B. Jelinek, W. Hofer and P. Varga, Surf. Sci. 295 (1993) 57. [24] P. Weigand, B. Jelinek, W. Hofer and P. Varga, Surf. Sci. 301 (1994) 306. [25] D. Dufayard, R. Baudoing and Y. Gauthier, Surf. Sci. 233 (1990) 223. [26] M. Lundberg, Phys. Rev. B 36 (1987) 4692. [27] H. Stadler, W. Hofer, M. Schmid and P. Varga, Surf. Sci. 287/288 (1993) 366. [28] G. Tr6glia and B. Legrand, Phys. Rev. B 35 (1987) 4338. [29] B. Legrand, G. Tr6glia and F. Ducastelle, Phys. Rev. B 41 (1990) 4422. [30] J. Eymery and J.C. Joud, Surf. Sci. 231 (1990) 419. [31] I.A. Abrikosov, A.V. Ruban, H.L. Skriver and B. Johansson, Phys. Rev. B 50 (1994) 2039.