PHYSICAL REVIEW 8 VOLUME 32, NUMBER 6 15 SEPTEMBER 1985

Free energy of surface segregation in binary alloys

Yi-Chen Cheng, C. J. Wu, and R. C. ChiangDepartment of Physics, Xationa/ Taiwan University, Taipei, Taiwan, Republic of China

(Received 18 December 1984; revised manuscript received 21 February 1985}

The pair-bond approximation and the mean-field approach are used to derive an expression for the freeenergy of surface segregation in binary alloys. It is suggested that a correction term, in addition to the sur-face energy term and the heat of solution, should be included in the free energy of segregation in order toobtain a better quantitative agreement between theory and experiment. Thks correction term is aconcentration-dependent quantity and therefore cannot be obtained from pure-metal properties. The theoryis applied to the Cu-Ni alloy, and if the correction term is properly chosen, then a good quantitative agree-ment with experiment is found for all in-depth surface compositions. The free energy of segregation ob-tained by this method is in qualitative agreement with that calculated on surface core-level shifts, This isnot the case if the correction term is not considered.

During the past decade the phenomenon of surface segre-gation in binary alloys has attracted a great deal of attention,and considerable progress has been made both theoreticallyand experimentally. This surface property is important inmany scientific and technological fields, such as catalysis,corrosion, and chemisorption; it is also closely related to thephenomenon of grain-boundary segregation, which is ofgreat importance in metallurgy. Recent advances in experi-mental techniques, such as Auger electron spectroscopy(AES), low-energy ion scattering (LEIS), and ultravioletphotoemission spectroscopy (UPS), make the surface mea-surements more reliable. This in turn stimulates a greatdeal of theoretical studies.

The theory of surface segregation for an A„Bl „binaryalloy usually can be summarized in the following form 9

1 —x),

Xbexp( —AF„/ks T),

1 —Xb

where x& is the concentration of the A atoms in the ~thlayer, xb =x is the bulk concentration corresponding to

(A, =O is the surface layer), ks is the Boltzmannconstant, and T is the absolute temperature. The quantityhF& denotes the free energy of segregation for the ~thlayer. Thus, if 4E& & 0, then x&) xb and the A atoms be-come concentrated in the P th layer. Of course, the most im-portant layer is the surface layer and, therefore, the mostimportant quantity is AJ'o. If AI" o can be expressed in termsof some known pure-metal properties, then the surfacecomposition of an alloy can be predicted from the knownpure-metal properties.

The most common and easiest way to derive Eq. (1) is touse the pair-bond approximation, a quasichemical ap-proach in which the total energy of the system is treated asthe sum of nearest-neighbor pair-interaction energies. AFocan easily be obtained as a sum of two terms. ' The firstterm is the difference of (pure-metal) surface energies ofthe two composing elements. The second term comes fromthe mixing of the two elements and is called the heat ofsolution. Since usually it is the surface energy term thatdominates, this theory predicts that the element havinglower surface energy segregates at the surface. Althoughthe predictions are usually in good agreement with experi-ment qualitatively, quantitative agreements are still notsatisfying.

In this paper we assert that the pair-bond approximationis only a crude approximation and that one needs to add athird correction term 4@ to EFo to take into account theresidual effects not considered in the approximation. Sincethe theoretical calculation of AP may not be easy, we try toestimate 5@ by fitting the experimental data Our . theory isapplied to the Cu-Ni alloy, and 5@ is chosen such that thetheoretical prediction of Eq. (1) agrees with the experimen-tal data of the first-layer concentration. We then find thefollowing two interesting results:

(1) The theoretical calculations of the concentration ofthe second and third layer also agree with experiment.

(2) The free energy of segregation, AFO, agrees qualita-tively with that which is obtained by Mukherjee, Moran-Lopez, Kumar, and Bennemann" by calculating the surfacecore-level shifts. '

When the correction term is not considered (AP = 0), theagreements noted above disappear (along with the agree-ment of the first-layer concentration). This shows that 5@plays a significant role in the theory of surface segregation.We note that 5@ is a concentration-dependent quantity andtherefore cannot be obtained from pure-metal properties.

We consider a semi-infinite solid binary alloy composedof A and B atoms with a plane surface. The pair-bond ap-proximation is used to describe the total energy of the sys-tem. The grand Hamiltonian of the system can easily bewritten down after introducing the chemical potentials p,~and p,~. By considering the translational invariance parallelto the surface, we can obtain the following result of themean-field approximation"

ZJ Z„J 4p, 'mq=tanh ' m„+ "

(mq~t+m~ ~)+, X~1,kg T AT AT

Zs Js ZoJ Apsmo= tanh mo+ mr+kg T AT AT

Here J is the heat of mixing of the alloy and Ap,' is the

chemical potential difference. " Z, is the number of'thenearest neighbors within each layer and Z„ is the number ofthe nearest neighbors between adjacent layers. ThusZ =Z, +2Z„ is the total number of the nearest neighbors.

198S The American Physical Society

32 BRIEF REPORTS 4225

By noting that m& = 2x& —1, we easily obtain 1.0

exp( —hF„'/kT),1 —x], 1 —xb

(3) 0.9

0.8

where xb is the bulk concentration of the A atoms, i.e.,xp =x . In Eq. (3) we have the free energy of segregation'5[the lLF 's are primed and for the reason see Eq. (6)],

~FO = (lTAQA rrBQB) + [4(ZJxq —Z„Jx~ —Z, J,xp)

0.7

0.5

and

—Z„J—Z, (J—J,)], (4)

0. 3

b, Fp = KFp + hltI(x),

AF), = hF)', , A. ~ 1(6)

We choose 5@ by fitting the experimental data of the con-centration of the first layer of a given alloy and then com-pare the theoretical predictions of the -concentration of thesecond and third layers with experiment. We also comparethe form of AFO with that which is obtained by anothertheoretical work which uses a different approach.

We apply our theory to the Cu-Ni alloy, as the atomic size

AFAR =4[ZJxb —Z, Jx), —Z„J(x),+t+x), ))], g~ I,where cr~ ~~~ and a~ ~~~ are, respectively, the surface energyand the surface area per atom for a pure A (8) metal. Thesecond term in the expression of AFO is called the heat ofsolution.

As we have mentioned in the preceeding paragraphs thatthe pair-bond approximation is inadequate to describe thewhole surface effects, we propose to add additional term Altlto AFO in order for the theory to better agree with experi-ment. Therefore, the correct free energy of segregation canbe written as

0. 2

0. 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Xb

FIG. 2. The same plot as in Fig. 1, but using Eq. (6) (6@F0}in-stead.

difference between Cu and Ni atoms is small and the strainenergy effect can be neglected. Kumar, 7 using the fact thatCu-Ni alloy is a nonregular solid solution and that the heatof mixing J is a concentration-dependent quantity, studiesthe surface-segregation phenomenon of the Cu-Ni alloy atT = 937 K. He uses the usual pair-bond approximation withshort-range order effect included. Compared with the ex-perimental result of Watanabe, Hashiba, and Yamashina, 'his calculated result, in the surface layer, shows a reason-able agreement with experiment in the low-Ni-concentrationregion. However, in the high-Ni-concentration region thereis a large discrepancy. Moreover, in the second layer his

1.0

0.9

0.8

- 0.1

0 2&I

0.7 -0.3-

0.6

0.5 -05-

0.4 -0.6-

0.3 -0.7

0.2 -0.8

0. 1 - 0.9

I I I I I I

0 0.1 0.2 0.3 0.4 0:5 0.6 0.7 0.8 0,9 1.0Xb

-1.0

I I I I I I I

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0FIG. 1. Calculated surface composition for the Cu-Ni alloy by us-

ing Eqs. (3)-(5), (5/=0}, and T =973 K. xb is the bulk Ni con-centration. ( ), ( ), and (- —-), respectively, denotethe Ni concentration in the A. =O, 1, and 2 layers. The surface istaken to be a (111) surface. 0, , and 5 are experimental resultsof Watanabe et aI. (Ref. 16} in the X = 0, 1, and 2 layers, respective-ly.

FIG. 3. Plot of 4@/k~T vs the bulk Ni concentration xb atT =937 K. The definition of 5@ is given in Eq. (6). The values of5@ are obtained by fitting the experimental data of the Ni concen-tration in the P = 0 layer (Fig. 2}.

4226 BRIEF REPORTS 32

0.5 5.0

0 4)go 03-

&II

0.2—

4.0—

0.1—

l I I I I I I I I

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

FIG. 4. The surface core-level shifts 4,' (Ni) for the Cu-Ni alloyat T=673 K calculated by Mukherjee etal. (Ref. 11). x& is thebulk Ni concentration.

2.0—

result seems to agree well with experiment for all Ni con-centrations, but it is misleading to interpret that his theoryis good for predicting the concentration of the second layer.The reason is that the concentration of the second layerstrongly depends on that of the first layer. Since there is alarge discrepancy in the first layer in the high-Ni-concentration region, the agreement in the second layerdoes not mean much. In Fig. 1 we plot the concentration ofthe first three layers of the Cu-Ni alloy by using Eqs.(3)-(5). The values of J's used are the same as Ref. 7,and J,/J is taken to be 1.2. The result is almost the same asthat of Ref. 7, except that the second layer deviates morefrom the experimental values. In Fig. 2 we use Eq. (6) andfit the experimental values of the surface layer by making

an adjustable parameter. We see that all first threelayers fit very well with the experimental results. Theparameter 5@ we obtained is plotted in Fig. 3. We see that5@ varies smoothly as the concentration of Ni increases. Inorder to make our theory more sensible, we compare ourresult of AFO with that obtained by another theoretical workwhich uses a different approach. Recently, Mukherjeeet al. " used an electronic theory to calculate the surfacecore-level binding energy shifts (SCLS) for the Cu-Ni alloy.This SCLS is shown to be equal to the heat of surface segre-gation' for the Cu-Ni alloy. Since the entropy of surfacesegregation is negligible, the SCLS is approximately equalto the free energy of surface segregation for Cu-Ni alloy.The result of Mukherjee et al. " at T =673 K is shown inFig. 4. The magnitude of the SCLS increases slowly as thebulk Ni concentration increases from zero concentration. Itreaches a maximum at around 90% of Ni atoms and thendecreases again. Our results of AEO, using Eq. (4) (4/= 0)and Eq. (6) (5@&0), are plotted in Figs. 5(a) and 5(b),respectively. It is interesting to note that when 5/=0, thevalue of AFO increases monotonically as the bulk Ni concen-tration increases. This is clearly quite different from that ofFig. 4. However, when h$ is taken into account [Fig.5(b)], AFO does have a maximum as the bulk Ni concentra-tion varies, and the general shape agrees, at least qualita-

1.00

I l I t I I

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Xb

FIG. 5. The free energy of surface segregation AFO vs the bulkNi concentration at T =937 K. (a) 5/=0 and (b) 6/~0.

This work is supported by the National Science Council ofthe Republic of China under Grant No. NSC73-0201-M002a-01.

tively, with that of Fig. 4.For the Cu-Ni alloy the origin of 5@ may be interpreted

as follows. The crystal structure is a fcc lattice with 10 or 11electrons per atom (the number of electrons varies with theconcentration). The cohesive energy contains a term thatvaries with the number of electrons per atom. Iri the sur-face layer there is also such a contribution, but it is differentfrom that of the bulk value because the surface band struc-ture is different from that of the bulk. This difference has acontribution to 6J'0, which cannot properly be taken into ac-count by the broken bond picture at the surface. The orderof magnitude of AP shown in Fig. 3 is quite reasonable forthis effect.

In conclusion, we have shown that if the values of AJ'0are chosen such that the theory agrees with the experimentin the first-layer concentration, then the theoretical predic-tions of the second- and third-layer concentration also agreewell with experiment. The variation of UFO (chosen by themethod noted above) with respect to the bulk concentrationalso qualitatively agrees with the theoretical work of Mu-kherjee et al. " on surface core-level shifts, whereas thevalues of AFO derived from the pair-bond approximation donot have such properties.

32 BRIEF REPORTS 4227

'Present address: Yuin-Lin Institute of Technology, Hu-Wei,Yuin-Lin, Taiwan, Republic of China.

Present address: Department of Physics, Tung-Hai University,Taichung, Taiwan, Republic of China.

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