VOLUME 42, NUMBER 15 PHYSICAL REVIEW LETTERS 9 APRIL 1979

Prediction of Surface Segregation in Binary Alloys Using Bulk AlloyVariables

John C. HamiltonDepartment of Materials Science and Engineering, Cornett University, Ithaca, ¹w Fork 14853

(Received 3 November 1978)

A simple graphical approach is proposed which correctly predicts the occurrence orabsence of surface segregation for virtua11y a11 binary a1loys that have been studied ex-perimentally. The ruIe is based on elemental variables re1ated to the work functions andelectron densities of the elements. These variables have been proposed by Miedema andused by him to predict binary-alloy heats of solution and elemental surface energies. Thegraphical condition for segregation is derived analytically and is valid for both substitu-tional and interstitial alloys.

The surface enrichment of one component of abinary alloy is known as surface segregation ifthe alloy is in a single-phase region of the phasediagram. Experimentally it must be distinguishedfrom precipitation which occurs if the solubilityof the solute is exceeded. Segregation has re-ceived considerable attention because of its rela-tion to temper embrittlement phenomena andcatalysis by alloys and because it is readily ob-served using ultrahigh-vacuum techniques andsurface-sensitive spectrscopies. The purpose ofthis Letter is to demonstrate that bulk variablesproposed by Miedema allow simple graphical pre-diction of the occurrence of surface segregationin binary alloys. In addition, the graphical condi-tion for segregation is derived analytically fromthe surface energies of the two alloy components.

The heat of segregation, q „g,may be definedas the change in enthalpy that occurs when a sol-ute atom is exchanged between a bulk and sur-face site. Since the number of configurations ofthe system is reduced by confining an impurityatom to the surface, segregation will decreasethe configurational entropy. Segregation willoccur only if it lowers the free energy. Thus ifwe assume that the total entropy of segregationis dominated by the configurational entropy, seg-regation can only occur if q„gis negative. Atsufficiently high temperatures the entropy termwill dominate in the free energy; consequentlythe impurity atoms will be uniformly distributed.As the temperature is lowered segregation willoccur if q,« is negative. Various authors havepredicted q„gfrom surface energies, ' bond-breaking theories, elastic-strain theories, andcombination of these effects. ' Recently a thoroughdiscussion of surface segregation in terms ofsurface energies, heat of solution, and straineffects has been published. ' Two other predic-tive theories do not involve estimation of q„&.Burton and Machlin have proposed a rule basedon the melting curve of the bulk alloy which al-

lows prediction of the occurrence of segregation. 4

Kerker, Moran-Lopez, and Bennemann havepredicted segregation to transition-metal sur-faces using a tight-binding-type electronic theo-ry. ' Their theory is based on electronic alloyparameters including the number of d electronsper atom and the difference in the d-electron en-ergies of the two alloy components.

The graphical prediction of segregation pro-posed in this Letter is based on extensive workby Miedema and co-workers on the heats of solu-tion of binary alloys. ' In their work y*, the em-pirically adjusted electron work function, and

nzs, the electron density of the pure metal at theboundary of the signer-Seitz cell are used as pa-rameters. For equal molar volumes, the heat ofsolutions is found to be

q„)= -P b y*) + Q( bn ws'I') + 8&,

where P, Q, and R are constants, 9= 1 for alloysof simple metals with transition metals and 0= 0otherwise. In this formula Ay*= y~* —y~* andbnws~I'= (n ws I') ~ —(nws~Is) g where subscriPts 8and A represent the solute and solvent, r espec-tively. Miedema interprets Eq. (1) physically byconsidering the analogous problem of joining twomacroscopic pieces of metal. ' The interfacialenergy includes an attractive term due to chargetransfer between metals with differing work func-tions. The energy associated with the resultingdipole layer is represented by P(hy~)'. -Inorder to match electronic wave functions at theinterface, energy is required to alter the pure-metal electron densities. This repulsive energyis represented by Q(~n ws'I')'. Other interpreta-tions have been suggested; Hodges, for example,has proposed a model involving charge exchangebetween the s and d orbitals of the solute andsolvent atoms. ' In spite of difficulties in itsinterpretation, Eq. (1) ha. s been successfullyused to predict heats of solution of a wide varietyof binary alloys. ' The parameters cp* and n~ ' '

1979 The American Physical Society 989

VOLUME 42, +UMBER 15 PHYSICAL RKVIKW LKTTKRS 9 A,PRIL 1979

TABLE I. Solvent-solute pairs for which experimen-tal data is available. In all cases the element in pa-rentheses is the dilute component. For Pd(C) there isno segregation to individual surface sites although asurface phase does form due to interaction betweencarbon atoms on the surface (see Ref. 10).

Solvent(solute)

Segregatingcomponent Reference

Ag(Au)Ag(cu)Ag(Pb)

Au(Ag)Au(Cu)AU(Pd)Au(S )

co(c)

Cu(Ag)CU(AQ)

cu(Ni)

Fe(C)Fe(cr)Fe(Cu)'Fe(Mn)Fe(Ni)Fe(Sn)Fe(Zr)

Ni(Au)Ni(c)Ni(cu)Ni(Fe)Ni(Pd)

os(Pt)

Pd(Ag)Pd(Au)I d{c)Pd(V)

AgAuAuSn

AuNone

cCrCQ

MnNiSnZr

AucCuNiPd

AgAuCNone

111213

13,1415, 161718

10

12.1315, 1619

20211322232425

26-2829, 3019,312332

34171035

(ev3

—1.0 — ~

—2.0—0.4 -0.2

~ SURFACE ENRICHMENT OF SOLUTE

& NO SURFACE ENRICHMENT

+ SURFACE ENRICHMENT OF SOLVENT

I I I I I

0 0.2 0.4ly

(DENSITY UNITS) ~

of A, as follows':

9 seg ~ sol+ 9 vap Qads

FIG. 1. Plot of Ap*=ys~ —p„~and Dnws~~3=(nws~ )s—(n~s' )z for the binary alloys listed in Table I. Thedistinction between systems showing no segregationand those showing surface enrichment of the solvent isexperimentally difficult for dilute alloys and is not like-ly to be reliable. The separation line is plotted fromEq. (6) of the text.

Pt(Au)Pt(c)Pt(Cu)Pt(cr)Pt(Fe)Pt(Ni)Pt(ah)Pt(sn)

Rh(AI)Rh(Pt)

Zr(Fe)

AuNoneCuNoneNoneNonePtSn

Fe

3637383939394041

4240

have also been used to predict the type of siteoccupied by ion-implanted impurity atoms inberyllium. '

In this Letter it is assumed that the heat of seg-regation can be represented as some function ofAcp* and ~&s' '. The heat of segregation is re-lated to q „p,the heat of vaporization of B, andto q,d„the heat of adsorption of 8 on the surface

~so&+Hg on surf g Hg in g ~ (2)

H~,„,„z~- H~ i„~is the heat required to trans-fer an impurity atom 8 from its pure elementalstate to the surface of A. It is expected that thisheat will have a functional dependence on Ay*and ~»' ' since the formation of the surfaceA Bbond will inv-olve charge transfer (describedby Ay*) and matching of electron densities at theA Binterface (des-cribed by ~ws'~'). In orderto test this assumption all available data on seg-regation (see Table I) were plotted as shown in

Fig. 1. For each solvent-solute pair a point wasplotted having coordinates Ay* and M»' '."For the lower right portion of this plot, segrega-tion does not occur or surface enrichment of thesolvent occurs. (For dilute alloys these twocases are difficult to differentiate experimental-ly. ) For the upper left portion of this plot, thesolute is enriched at the surface. Both substitu-

990

VOLUME 42, NUMBER 15 PHYSICAr. REVIEW I.ETTERS 9 APRIL, 1979

tional and interstitial alloys exhibit similar de-pendence of segregation behavior on the values ofAy+ and ~ws

The equation of the line which separates seg-regation from nonsegregation in Fig. 1 is derivedas follows. For a monolayer ideal-solution mod-el, the surface concentration of the solute X~'is given by

XB XB (WB YA)a1-X ' 1-X RT

where X~' is the bulk solute concentration, y~and y~ are the surface energies of the solute and

!

solvent, respectively, and a is the average molar

surface area. '44 For this simple model, seg-regation is predicted if y~ —y„&0.

In order to explain the separation shown inFig. 1, it is necessary to represent y~ —y~ interms of Ay* and M»' '. The surface energy ofan element may be represented as

3176n ws erg eV5/3 2

(y* —0.6)' cm' (density units)'~' (4)

using an empirical relationship proposed byMiedema. ' The mean values of q* and nws'~'

are defined as q*=—,'(y„*+9 s*) and n ws'~'

= —2[(n ws'~') ~+ (n'~') s]. Neglecting terms higherthan second order in Ay* and ~&s' ', the differ-ence in surface energies is

+ws ) ~ws(q* —Q. 6)' n '~' P —Q. 6 cm' (density units)' ' '

The condition for surface segregation may bewritten

&V* &-' I(P - 0 6)/~ ws" l &~ ws" (6)

Reference 43 gives values for y* and nws~' for57 elements. With the exception of nitrogen, thevalue of (q* —0.6)/nws'~' varies only slightIyfrom element to element. For the 56 remainingelements (cp* —0.6)/n ws'~' has a mean value of2.53 and a standard deviation of 0.31. Conse-quently the value of (y* —0.6)/nws'~' will be closeto 2.53 for any pair of elements not includingnitrogen. In Fig. 1, the separation line definedby Eq. (6) is plotted using this mean value. Ingeneral the agreement with experimental data isexcellent indicating that the surface energy dif-ference is the dominant driving force for seg-regation. 4' This is the first time that the depen-dence of surface energies on cp* and n&s'~' givenby Eq. (4) has been used in a description of sur-face segregation. The success of this approachindicates the validity of Eq. (4) in predicting sur-face energies and demonstrates that strain ener-gies and the heat of solution play a minor role indetermining segregation behavior for most binaryalloys. Figures similar to Fig, 1 have been usedto predict bulk phase properties'; however, theseparation lines generally have not been calcu-lated analytically.

The simple graphical approach proposed in thisLetter allows prediction of surface segregationfor both substitutional and interstitial alloys. Inparticular, the presence or absence of surfacesegregation in carbon-metal alloys is accuratelypredicted for the first time. Since values for y*

and n ws'~' are tabulated for a wide variety of ele-ments, this approach allows a simple predictionof surface segregation for a wide variety of bi-nary alloys.

'S. H. Overbury, P. A. Bertrand, and G. A. Somorjai,Chem. Rev. 75, 547 (1975).

P. %ynblatt and R. C. Ku, to be published in Pro-ceedings of the 19V7 American Society for Metals Mate-rials Science Seminar on Interfacial Segregation.

3A. R. Miedema, Z. Metallkd. 69, 455 (1978).4J. J. Burton and E. S. Machlin, Phys. Rev. Lett. 37,

1433 (1976).'G. Kerker, J. L. Moran-Lopez, and K. H. Benne-

mann, Phys. Rev. B 15, 638 {1977).A. R. Mledema, F. R. deBoer, and P. F. de Chatel,

J. Phys. F 3, 1558 (1973); A. R. Miedema, J. Less-Common Met. 32, 117 {1973);A. R. Miedema, R. Boom,and F. R. deBoer, J. Less-Common Met. 41, 283(1975); R. Boom, F. R. deBoer, and A. R. Miedema,J. Less-Common Met. 46, 271 (1976); A. H, . Miedema,Philips Tech. Rev. 36, 217 (1976).

~C. H. Hodges, to be published.E. N. Kaufmann, R. Vianden, J. B. Chelikowsky,

and J. C. Phinips, Phys. Rev. Lett. 39, 1671 {1977).sJ. C. Shelton and J. M. Blakely, in Su+ace Physics

of Materials (Academic, New York, 1975).~ J. C. Hamilton and J. M. Blakely, to be published."S.H. Overbury and G. A. Somorjai, Surf. Sci. 55,

209 (1976).~ P. Braun and W. Parber, Surf. Sci. 47, 57 {1975).' B. E. Sundquist, Acta Metall. 12, 585 (1964).' G. C. Nelson, J. Vac. Sci. Technol. 13, 512 (1976).'5H. C. Potter and J. M. Blakely, J. Vac. Sci. Technol.

12, 635 (1975).J. M. McDavid and S. C. Fain, Surf. Sci. 52, 161

991

VOLUME 42, NUMBER 15 PHYSICAL REVIEW LETTERS 9 APRIL I979

(1975).7A. Jablonski, S. H. Overbury, and G. A. Somorjai,

Surf. Sci. 65, 578 (1977).' S. H. Overbury and G. A. Somorjai, J. Chem. Phys.

66, 3181 (1977).K. Watanabe, M. Hashiba, and T. Yamashina, Surf.

Sci. 61, 483 (1976).2 H. J. Grabke, W. Paulitschke, G. Tauber, and

H. Viehus, Surf. Sci. 63, 377 (1977).'C. Leygraf, G. Hultquist, S. Ekelund, and J. C.

Erickson, Surf. Sci. 46, 157 (1974).A. Joshi, P. W. Palmberg, and D. F. Stein, Metall.

Trans. 6A, 2160 (1975).3K. Wandelt and G. Ertl, J. Phys. F 6, 1607 (1976).M. P. Seah and C. Lea, Philos. Mag. 3l., 627 (1975).R. S. Polizotti and J. J. Burton, J. Vac. Sci. Tech-

nol. 14, 347 (1977).J. J. Burton, C. R. Helms, and H, . S. Polizotti,

J. Vac. Sci. Technol. 13, 204 (1976).VJ. J. Burton, C. R. Helms, and R. S. Polizotti,

J. Chem. Phys. 65, 1089 (1976).P. Wynblatt and R. C. Ku, Surf. Sci. 65, 511 (1977).

"J.C. Shelton, H. R. Patil, and J. M. Blakely, Surf.Sci. 43, 493 (1974).

L. C. Isett and J. M. Blakely, Surf. Sci. 58, 397(1976).

3'C. R. Helms and K. Y. Yu, J. Vac. Sci. Technol. 12,276 (1975).

3~D. Mervyn, R. J. Baird, and P. Wynblatt, to be pub-lished.

33J. C. Riviere, J. Less-Common Met. 38, 193 (1974).3 B.J. Wood and H. Wise, Surf. Sci. 52, 151 (1975).J. J. Burton, Surf. Sci. 69, 712 (1977).J. A. Schwartz, R. S. Polizotti, and J. J. Burton,

J. Vac. Sci. Technol. 14, 457 (1977).37J. C. Hamilton and J. M. Blakely, J. Vac. Sci. Tech-

nol. 15, 559 (1978).3 J. H. Anderson, P. J. Conn, and S. G. Brandenber-

ger, J. Catal. 16, 326 (1970).3~J. J. Burton and R. S. Polizotti, Surf. Sci. 66, 1

(1977).G. C. Williams and F. Williams, in Proceedings of

American Vacuum Society Meeting, San Francisco, 28November-1 December 1978 (to be published).

4'R. Bouwmann, L. H. Toneman, and A. A. Holscher,Surf. Sci. 35, 8 (1973).

J. H. Anderson, P. J. Conn, and S. G. Brandenber-ger, J. Catal. 18, 404 (1970).

43Values for q * and n &s' have been taken from A. R.Miedema, in P/utonium 1975 and Othe~ Actinides, edit-ed by H. Blank and R. Linder (North-Holland, Amster-dam, 1976), p. 6.

44R. Defay, I. Prigogine, A. Bellemans, and D. H.Everett, Surface Tension and AdsorPtion (Wiley, New

York, 1966).45A. R. Miedema, Z. Metallkd. 69, 287 (1978).

For Pd(C) and Fe(Zr) alloys this model predicts no

segregation, yet segregation is observed experimental-ly. In the case of Pd(C), a surface phase condensationfrom very low carbon coverage to a monolayer of car-bon was observed as the temperature was lowered (Ref.10). This condensation was attributed to carbon-carbonadatom interactions. No segregation to isolated sur-face sites was observed. Since Eq. (5) is the conditionfor segregation to isolated surface sites, the observedsegregation is not inconsistent with the prediction. Inthe case of Zr(Fe) it appears that strain energies mustbe included to correctly predict segregation.

New Theory of Repulsion and Structural Stability in Ionic Crystals

Ramesh Narayan and S. Ramaseshan 'Raman Research Institute, BangaEoxe -560006, India

(Received 27 November 1978)

An ionic crystal is viewed as a collection of compressible ions in polyhedral, space-filling cells. Repulsion arises solely from the increased compression energy at the cellfaces. Eighteeg parameters (two per ion) are determined from the lattice spacings and

compressibilities of the twenty alkali halides. These explain for the first time the ob-served structures of all these crystals as well as their thermal and pressure transitions—a significant advance over previous semiempirical theories.

Mobt theories of cohesion in ionic crystalsconcentrate on calculating the repulsive forces,since the attractive interactions are well under-stood. Born's semiempirical theory' ' modelsthe repulsion energy by simple functions of r,the interionic spacing [e.g. , A. exp(- r/p)], usingbvo parameters per crystal fitted from experi-mental data. Among many later extensions, '

Tosi and Fumi' have carried out detailed calcu-lations on the alkali halides using 27 parametersfor seventeen crystals. 4 Although they obtain agood fit with experimental interionic spacings(r,) and compressibilities (E',), their theory, incommon with all Born-type approaches, requiressome of the parameters to be determined onlyfrom the experimental data on the crystals of in-

992 1979 The American Physical Society