Cosegregation-Induced Formation of Two-Dimensional Surface

Fe-15% Cr-N(100), VC on Fe-3% V-C(100) and on Fe-3% Si-0.04% V-C(100), and TiC on Fe-6%. Al-0.5% Ti-C(100). These surface compounds are epitaxially ...
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Langmuir 1996, 12, 23-28

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Cosegregation-Induced Formation of Two-Dimensional Surface Compounds on Multicomponent Alloy Surfaces† Christian Uebing‡ Institut fu¨ r Festko¨ rperforschung, Forschungszentrum Ju¨ lich, D-52425 Ju¨ lich, Germany Received September 1, 1994. In Final Form: November 29, 1994X The chemical composition of multicomponent alloy surfaces may exhibit significant deviations from the bulk composition due to thermally activated segregation and cosegregation processes. In many systems cosegregation phenomena result in the formation of two-dimensional surface compounds such as CrN on Fe-15% Cr-N(100), VC on Fe-3% V-C(100) and on Fe-3% Si-0.04% V-C(100), and TiC on Fe-6% Al-0.5% Ti-C(100). These surface compounds are epitaxially stabilized on the bcc(100) alloy surfaces, their thicknesses are about one or two atomic layers. In the present work we report results of a Monte Carlo study on the cosegregation-induced formation of surface compounds on bcc(100) surfaces. The simulations are performed utilizing a three-dimensional lattice gas model with two free (100) surfaces and periodic boundary conditions in x and y directions. It is assumed that the lattice consists of two types of lattice sites M and X. The metal sites M form a body-centered cubic (bcc) lattice, whose quasi-octahedral interstices constitute the nonmetal sublattice. The M sites are accessible to either MA or MB atoms, while the nonmetal sites either are occupied by X atoms or remain empty. Pairwise repulsive nearest and attractive next nearest neighbor interactions between MA-X and MB-X atoms are considered as well as up to fourth nearest neighbor X-X repulsions. The simulations indicate that cosegregation-induced formation of the surface compound MBX is basically due to preferential next nearest MB-X neighbor attractions. With an increase of the strength of the preferential next nearest MB-X attractions, surface compound formation is accompanied by a first-order phase transition. Depending on the relative magnitude of the nearest neighbor M-X repulsions, we observe a strong MB subsurface enrichment, which has been verified by XPD for the CrN surface compound on Fe-15% Cr-N(100). For repulsive fourth nearest X-X neighbor interactions, our lattice gas model shows c(2×2) ordering of the X atoms on the bcc(100) surface.

1. Introduction Understanding the major physical and chemical processes at solid surfaces is an important goal of state-ofthe-art surface science. Many material properties are decisively controlled by structure and composition of the surface being subject of adsorption and segregation processes as well. The latter denote the preferential enrichment of one component of a multicomponent alloy at a surface.1-6 Since bulk diffusion of the segregants is required for surface segregation to occur, this phenomenon especially affects high-temperature material properties such as catalytic activity and oxidation resistance. Many detailed investigations of surface adsorption and segregation phenomena have been performed on iron and related steel substrate surfaces; see, e.g., refs 7-11. In multicomponent alloys M-X1-X2 ... with unlike segregants X1, X2 ..., complex segregation behavior could † Presented at the symposium on Advances in the Measurement and Modeling of Surface Phenomema, San Luis, Argentina, August 24-30, 1994. ‡ Present address: Max-Planck-Institut fu ¨ r Eisenlorschung, D-40074 Du¨sseldorf, Germany. X Abstract published in Advance ACS Abstracts, January 1, 1996.

(1) Blakely, J. M. Introduction to the Properties of Crystal Surfaces; Pergamon: Oxford, 1973. (2) Blakely, J. M.; Shelton, J. C. In Surface Physics of Materials; Blakely, J. M., Ed.; Academic Press: New York, 1975; Vol. I. (3) Blakely, J. M. CRC Crit. Rev. Solid State Mater. Sci. 1978, 7, 333. (4) Grabke, H. J.; Viefhaus, H. In Surface Segregation and Related Phenomena; Dowben, P. A., Miller, A., Eds.; CRC Press: Boca Raton, FL, 1990. (5) King, T. S. In Surface Segregation and Related Phenomena; Dowben, P. A., Miller, A., Eds.; CRC Press: Boca Raton, FL, 1990. (6) Segregation may also occur at grain boundaries and other interfaces but this topic will not be treated here. (7) Grabke, H. J. Z. Phys. Chem. (Munich) 1985, 100, 185. (8) Grabke, H. J. Mater. Sci. Eng. 1980, 42, 91. (9) Viefhaus, H.; Tauber, G.; Grabke, H. J. Scr. Metall. 1975, 9, 1181. (10) Grabke, H. J.; Paulitschke, W.; Tauber, G.; Viefhaus, H. Surf. Sci. 1977, 63, 377. (11) Grabke, H. J.; Viefhaus, H.; Tauber, G. Arch. Eisenhu¨ ttenwes. 1978, 49, 391.

0743-7463/96/2412-0023$12.00/0

arise. According to Guttmann’s classification scheme,12 competition for the available surface sites (site competition) or cosegregation of the different segregants must be expected. On the basis of a regular solution approach, Guttmann proposed theoretical models for the description of such segregation phenomena in multicomponent alloys.12-14 Cosegregation has to be expected in systems displaying attractive interactions between different solutes or between solutes and base metal M. However, such interactions between different chemical species may also cause precipitation according to relevant bulk phase diagrams of the alloys.15 Moreover, in the presence of different adspecies at a surface, phase transition behavior is to be expected as a consequence of the various interactions between segregants. Upon surface cosegregation the formation of twodimensional surface compounds is possible. The thicknesses of such compounds are in the range of one or two atomic layers. In principle, surface compounds can also be formed by coadsorption of the constituent components from the residual gas atmosphere or by adsorption-induced segregation, where both components originate from different reservoirs. The outline of this paper is as follows. First we will briefly review experimental studies of surface cosegregation and surface compound formation on iron-based bcc alloy surfaces (section 2). Then we will present and discuss recent results of a detailed Monte Carlo study which is aimed to determine the driving force of surface compound formation (section 3). (12) Guttmann, M.; McLean, D. In Interfacial Segregation; Johnson, W. C., Blakely, J. M., Eds.; American Society for Metals: Metals Park, OH, 1977. (13) Guttmann, M. Metall. Trans. A 1977, 8, 1383. (14) In quaternary or even more complex systems, combinations of these models are needed for a detailed description of segregation phenomena as is discussed in ref 15. (15) Uebing, C.; Viefhaus, H.; Grabke, H. J. In Surface Segregation and Related Phenomena; Dowben, P., Miller, A., Eds.; CRC Press: Boca Raton, FL, 1990; Chapter 9, p 241.

© 1996 American Chemical Society

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Uebing

Table 1. Surface Compounds Formed upon Cosegregation of the Constituent Components on (100) Surfaces of Iron-Based bcc Alloys compound

alloy

orientation

ref

TiC VC VC V(C,N) CrN CrC

Fe-6% Al-0.5% Ti-C Fe-3% V-C Fe-3% Si-0.04% V-C Fe-3% V-C,N Fe-15% Cr-N Fe-15% Cr-C

(100) (100) (100) (100) (100) (100)

17 18 19 15, 18 15, 20, 21 21

2. Experimental Studies of Surface Cosegregation In ternary or even more complex alloy systems the joint enrichment of two (or more) solutes at surfaces and grain boundaries is possible. During the late 1970s the study of these so-called cosegregation phenomena has attracted considerable interest, which basically has been driven by the need of understanding the influence of cosegregation processes on the temper embrittlement of steels.12 The first systematic investigation of such cosegregation phenomena has been performed by Dumoulin and Guttmann using dilute ternary iron-based alloys Fe-M-X with M as metallic solutes (Ni, Cr, V, Ti, Mo) and X as metalloid impurities (Sb, P, S, C, N).16 Their pioneering work demonstrates clearly that cosegregation can be unequivocally attributed to strong attractive interactions between the metallic and nonmetallic solutes M and X, e.g., between Ni-Sb, Ti-Sb, Mo-P, Ti-P, Cr-N, Mo-N, V-N, MoC, V-C, Ti-C, and Ti-S.16 The possibility of segregation and cosegregation induced formation of two-dimensional compounds has been discussed in detail at first by Guttman.13 Almost one decade later Viefhaus, Peters, and Grabke provided the first experimental evidence of a two-dimensional surface compound formed upon cosegregation of the constituent components. These authors studied the surface segregation under ultrahigh vacuum (UHV) conditions as well as the initial stages of growth and composition of protecting oxide- and oxycarbide layers of Fe-6% Al-0.5% Ti-C(100) alloy single crystals.17 Under UHV conditions a twodimensional surface compound TiC is formed at elevated temperatures as an interlayer between substrate surface and a segregated p(1×1) Al layer. In oxidizing atmospheres a titanium oxycarbide layer Ti(C,O) is formed, which is covered by a very dense and well-adherent epitaxial Al2O3 layers. Since then surface compound formation has been studied in detail for (100) oriented body-centered cubic (bcc) iron-based alloys (Table 1). The surface compounds TiC, VC, V(C,N), CrN, and CrC are epitaxially arranged on the relevant substrate surface;15,17-21 their proposed structures are derived from the (100) plane of the rock salt structure (the structure of the three-dimensional bulk compounds TiC, VC, V(C,N), and CrN) as is shown in Figure 1. In all cases the surface compounds homogeneously cover the bcc(100) substrate surface. The thickness of the surface compounds has been determined by quantitative evaluation of Auger spectra and by sputter depth profiling at about one to two atomic layers.18,21 Very recently X-ray photoelectron diffraction (XPD) has been utilized to clarify the structure of the CrN surface (16) Dumoulin, P.; Guttmann, M. Mater. Sci. Eng. 1980, 42, 249. (17) Viefhaus, H.; Peters, J.; Grabke, H. J. Surf. Interface Anal. 1987, 10, 280. (18) Uebing, C.; Viefhaus, H.; Grabke, H. J. Surf. Sci. 1992, 264, 114. (19) Uebing, C.; Viefhaus, H. Surf. Sci. 1990, 236, 29. (20) Uebing, C.; Viefhaus, H.; Grabke, H. J. Appl. Surf. Sci. 1988, 32, 363. (21) Uebing, C. Surf. Sci. 1990 225, 97.

Figure 1. Schematic drawing of the bcc(100) surface and of the (100) plane of the three-dimensional CrN structure (NaCl type).

Figure 2. Structure model for the segregated CrN surface compound on Fe-15% Cr-C,N(100). This compound consists of two layers of Cr atoms (light gray) and N atoms at the surface. Shown is a vertical cut through the crystal along the [011]bcc azimuthal direction.

compound formed on Fe-15% Cr-N(100).22,23 In XPD the angle-resolved intensities of core level photoelectrons at usually high kinetic energies (Ek > 400 eV) are measured along high symmetry directions and analyzed for forward scattering enhancements.24-26 These generally occur along close-packed crystallographic directions and thus reveal the structure of, for example, an adsorbed overlayer.27-30 The proposed structure model of the CrN surface compound (Figure 2) shows a double layer of Cr and a single layer of N atoms located ≈0.5 Å above the first Cr layer.22,23 The Cr-Cr interlayer spacing is expanded by 16% relative to bulk Cr. It is argued that the expanded Cr-Cr interlayer spacing represents a transition between bcc and fcc structure.23 3. The Lattice Gas Model In this section a three-dimensional lattice gas model with two free (100) surfaces and periodic boundary conditions in x and y directions is introduced which allows for the first time the treatment of surface segregation and cosegregation phenomena on ternary alloys MA-MB-X. It is assumed that the lattice consists of two types of lattice sites M and X. The metal sites M form a body-centered cubic lattice, whose quasi-octahedral interstices constitute (22) Uebing, C.; Scheuch, V.; Kiskinova, M.; Bonzel, H. P. Surf. Sci. 1994, 321, 89. (23) Scheuch, V.; Kiskinova, M.; Bonzel, H. P.; Uebing, C. Phys. Rev. B, in press. (24) Fadley, C. S. Surf. Sci. 1984, 16, 275. (25) Egelhoff, W. F., Jr. Crit. Rev. Solid State Mater. Sci. 1990, 16, 213. (26) Chambers, S. A. Adv. Phys. 1991, 40, 357. (27) Wesner, D. A.; Coenen, F. P.; Bonzel, H. P. Phys. Rev. B 1989, 39, 10770. (28) Knauff, O.; Grosche, U.; Bonzel, H. P.; Fritzsche, V. Mol. Phys. 1992, 76, 787. (29) Grosche, U.; Hamadeh, H.; Knauff, O.; David, R.; Bonzel, H. P. Surf. Sci. 1993, 281, L341. (30) Bonzel, H. P. Prog. Surf. Sci. 1993, 42, 219.

Formation on Surface Compounds

Langmuir, Vol. 12, No. 1, 1996 25

bridge and on top positions shown in Figure 3b. However, in all cases discussed below the surface concentration of X atoms on these sites is negligible. It is useful to apply local occupation variables ciM, ciX for the relevant sublattice sites according to

ciM ) 1, for MB atoms on the ith site of the M sublattice 0, for MA atoms on the ith site of the M sublattice (1)

{

and

ciX ) 1, for X atoms on the ith site of the X sublattice 0, for vacant sites of the X sublattice (2)

{

The Hamiltonian of the three-dimensional lattice gas model is given by

ciM - µX ∑ ciX ∑ M X (1) i j (1) i j c c φM φM X ∑ M X X ∑(1 - cM)cX NN NN j (2) i j (2) i c c φ φM X ∑ M X M X ∑ (1 - cM) cX NNN NNN i j (2) i j c c φ φ(1) ∑ XX X X XX ∑ cXcX - ... +H0 NN NNN

H ) -∆µm

B

Figure 3. (a) Schematic drawing of some of the irregularly shaped octahedral interstices within the unit cell of the bcc structure. Black circles represent M atoms. The octahedral interstices are labeled according to their short axis. (b) Schematic drawing of the available coordination sites for X atoms at the bcc(100) surface which are considered in the present work.

the nonmetal sublattice. The M sites are accessible to either MA or MB atoms, while the nonmetal sites are either occupied by interstitials X or remain empty. The threedimensional lattice studied in this paper is constituted by up to 64 × 64 × 64 sites (interstitials plus metal sites). The quasi-octahedral interstices of the bcc structure are irregularly shaped (Figure 3). Each X atom in such an interstice is surrounded by two nearest metal neighbors MA, MB at a distance a/2 with a as cubic lattice constant and four next nearest metal neighbors at a distance a/x2. The short axis is directed along one of the cubic axes. By symmetry there are three types of octahedral interstices which in the following will be denoted Ox, Oy, and Oz (the indices denote the directions of the short axis). The quasi-octahedral interstices of the bcc structure are located at the midpoints of the edges and at the centers of the faces of the unit cube (Figure 3). There are six octahedral interstices in the unit cell, two of each kind. In the bulk of the bcc structure all quasi-octahedral interstices are equivalent because of the cubic symmetry. However, at the surface the bulk symmetry is broken and thus these interstices are no longer equivalent. In Figure 3b the situation is schematically shown for the bcc(100) surface. Here the interstices Oz constitute the 4-fold hollow positions of the surface which are the most stable coordination sites, e.g., for N and C on Fe-15% Cr(100).31-34 The 4-fold hollow positions are surrounded by four next nearest M neighbors located at the corners of the unit square and one nearest M neighbor below. In the present work we also allow the occupation of the 2-fold (31) Fast, J. D. Gases in Metals; Macmillan Press: London, 1976. (32) Fast, J. D. Interactions of Metals and Gases Vol. II, Kinetics and Mechanisms; Macmillan Press: London, 1971. (33) Dijkstra, L. J. Philips Res. Rep. 1947, 2, 357. (34) Williamson, G. K.; Smallman, R. E. Acta Crystallogr. 1953, 6, 361.

B

A

A

(3)

(1) (1) (2) (2) , φM , φM , and φM are nearest (index 1) where φM AX BX AX BX and next nearest neighbor interactions (index 2) between adjacent MA-X and MB-X pairs. Correspondingly, (i) φXX denote X-X couplings. In the present work we consider up to fourth nearest neighbor X-X interactions. The various sums of eq 3 run over all pairs of nearest and more distant neighbors once. ∆µM ) µMB - µMA and µX are chemical potentials. H0 contains terms independent of the ciM, ciX, which are irrelevant to the subject of the present paper. All the pairwise interaction energies of eq 3 are assumed to be independent of temperature and concentration. In the present work it is assumed that the M-X interaction energies are repulsive at short distances, (1) (1) (2) φM , φM < 0, and attractive at long distances, φM , AX BX AX (2) φMBX > 0. All the M-X and X-X interaction energies are additive, i.e., the strength of a atom-atom bond does not depend on the actual number of interacting atoms in the vicinity. This latter assumption is certainly not justified since the occupation of octahedral interstices in the bulk requires significant distortion of the surrounding M lattice as indicated by X-ray diffraction.35 However, the inclusion of nonadditive interaction energies would definitely exceed the scope of the present paper. Moreover it is assumed that the bond lengths rMX between interstitials and metal atoms are conserved during the segregation process (rigid lattice). Thus we do not consider the possibility of segregation induced surface relaxation. The Monte Carlo simulations have been carried out in the grand canonical ensemble, i.e., with the chemical potentials ∆µM, µX and temperature T as independent variables. In essence, the Monte Carlo procedure consists of the following steps: (1) Initially a fixed number of MA, MB, and X atoms are thrown at random onto the lattice. (2) The sample is then allowed to establish thermal equlibrium for a given set of parameters (∆µM, µX,

(35) Jack, K. H. Proc. R. Soc. London, A 1951, 208, 200.

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Uebing

(i) (i) (i) φM , φM , φXX , and T) applying the Metropolis imporAX BX tance sampling algorithm.36,37 Step 2 is repeated until the total energy of the system starts to fluctuate around mean values. Typically 1000 Monte Carlo steps suffice to allow equilibration (one Monte Carlo step (MCS) is defined as 64 × 64 × 64 interrogations of lattice sites at random), whereas a substantially increased number of MCS (>105) were necessary in the vicinity of phase transitions at low temperatures. (3) Then up to 500 additional MCS are carried out to determine reasonable averages of the measured bulk (xMB, xX), surface (θMB, θX), sub and subsurface (θsub MB , θX ) concentrations of the segregants. As usual these quantities are given by the average values of the relevant local occupation variables, e.g.

xMB ) 〈ciM〉, for all M bulk sites θMB ) 〈ciM〉, for all M surface sites

(4)

sub θM ) 〈ciM〉, for all M subsurface sites B

4. Results and Discussion In the following sections the results of detailed Monte Carlo simulations for the lattice gas model are presented and discussed. These studies are performed for characteristic sets of interaction energies. Clearly these very simple models do not exhaust, or perhaps even approximate all real situations. Although no Monte Carlo model on a rigid lattice can do full justice to surface segregation and cosegregation phenomena, where lattice distortions probably play an important role, it is possible to analyze the effects of nearest and more distant M-X and X-X couplings on the surface composition, and this is done in the present paper. 4.1. Surface Segregation in the Absence of Preferential M-X Interactions. First we investigate the (1) (1) (2) (2) ) φM < 0 and φM ) φM > 0 with all X-X case φM AX BX AX BX (i) interactions XX ) 0. In that case M and X atoms constitute an ideal solution on their corresponding sublattice. Figure 4 shows the temperature dependence of surface and subsurface concentrations of MB and X atoms for bulk concentrations xMB ) 0.15 and xX ) 10-6. In the absence of preferential M-X interactions MB does not segregate at the bcc(100) surface; i.e., the surface concentration of MB atoms correspond closely to the bulk concentration, θMB ) xMB. The nonmetal atoms X preferentially occupy the 4-fold hollow sites of the surface, which are the most stable coordination sites of the bcc(100) surface. The surface concentration of X atoms on 2-fold bridge and on (2) j top positions is negligible. At temperatures kBT/φM AX 0.055 all 4-fold hollow sites are occupied. Upon increase of the temperature, θX decreases and approaches the bulk (2) concentration xX at kBT/φM ≈ 0.12. Figure 4 also AX demonstrates that the subsurface concentrations θsub MB , correspond closely to the bulk concentrations x , θsub MB xX. X 4.2. Surface Segregation with Preferential M-X Interactions. Next we analyze surface segregation in the case of preferential next nearest neighbor M-X interactions. Without loss of generality, it is assumed (2) are that the next nearest neighbor interactions φM BX (36) Metropolis, N.; et al. J. Chem. Phys 1953, 21, 1087. (37) Binder, K.; Stauffer, D. In Applications of the Monte Carlo Method in Statistical Physics; Binder, K., Ed.; Springer-Verlag: Berlin, 1984.

Figure 4. Temperature dependence of surface (upper panel) and subsurface concentrations (lower panel) for the lattice gas (1) (1) (2) (2) (1) ) φM , φM ) φM ) -φM , xMB ) 0.15, and model with φM AX BX AX BX AX sub sub -6 xX ) 10 : (+) θX, θX ; (×) θMB, θMB . Nearest and more distant neighbor interactions between adjacent X atoms are not considered. The calculations are performed for lattices with 64 × 64 × 64 sites (interstitials plus metal sites).

(1) (1) (2) Figure 5. Same as Figure 4, but for φM ) φM ) -φM and AX BX AX (2) (2) φMBX/φMAX ) 1.06.

stronger than the corresponding interactions between MA (2) (2) and X; i.e., φM > φM > 0. This situation has been BX AX (2) (2) studied for two characteristic ratios φM /φM as is BX AX discussed in detail below. 4.2.1. Surface Cosegregation. In Figure 5 the temperature dependence of surface and subsurface con(2) (2) /φM ) 1.06. The centrations is shown for the case of φM BX AX most striking finding is that at temperatures below (2) kBT/φM ≈ 0.1 both surface concentrations θX and θMB AX are simultaneously enhanced with respect to the bulk concentrations xX and xMB. Cosegregation causes saturation of both surface sublattices at temperatures (2) kBT/φM j 0.05 which is interpreted as the formation of AX a complete layer of the binary surface compound MBX. sub However, the subsurface concentrations θsub MB and θX correspond closely to the relevant bulk concentrations xMB and xX. Comparing Figures 4 and 5 clearly indicates that

Formation on Surface Compounds

(1) (1) (2) Figure 6. Same as Figure 4, but for φM ) φM ) -φM and AX BX AX (2) (2) φMBX/φMAX ) 1.19.

surface compound formation is unequivocally attributed to the preferential next nearest neighbor interactions (2) φM . BX 4.2.2. First-Order Phase Transitions. In Figure 6 we visualize the effects of increasing next nearest neighbor (2) on the surface cosegregation. Again interactions φM BX the simulations are performed for xX ) 10-6 and xMB ) 0.15. Two general facts stand out. First the saturation of the MB and X surface sublattices persists up to relatively (2) high temperatures kBT/φM ≈ 0.1; and second the shape AX of the graphs is significantly altered. Abrupt changes of surface concentrations occur at a transition temperature indicating the breakdown of the mutual stabilization of MB and X atoms at the surface. This behavior is attributed to temperature-driven first-order phase transition, i.e., the formation/decomposition of the binary surface compound MBX. Again the subsurface concentrations θsub MB and θsub X correspond closely to the relevant bulk concentrations xMB and xX. 4.3. Effect of the Nearest M-X Interactions. In this section we analyze the effects of the nearest M-X neighbor interactions on the behavior of the lattice gas model. As already mentioned it is assumed that these (1) (1) , φM < 0. The interactions are repulsive; i.e., φM AX BX (2) (2) /φM ) simulations are performed for the situation φM BX AX -6 1.06, xMB ) 0.15, and xX ) 10 which corresponds to the situation studied above (Figure 5). 4.3.1. Subsurface Segregation and Desegregation. First the case of preferential MB-X interactions is inves(1) (1) < φM < 0 (Figure 7). The surface tigated, φM AX BX coverages θMB and θX are very similar to the case studied (1) (1) before, φM ) φM (Figure 5). At low temperatures the BX AX binary surface compound MBX is formed, which is sta(2) bilized by the mutual interactions φM between the BX constituent components. Upon increase of the temperature, the surface coverages decrease toward the corresponding bulk values. The most striking finding of Figure 7 is the fact that the subsurface coverage φsub MB shows significant deviations from the corresponding bulk con(2) centration xMB; i.e., θsub MB > xMB for kBT/φMAX j 0.09. In sub contrast, θX corresponds closely to the corresponding bulk concentration. It is concluded that the subsurface enrichment of MB atoms is due to the preferential nearest (1) neighbor MB-X interactions φM . BX

Langmuir, Vol. 12, No. 1, 1996 27

(1) (2) Figure 7. Same as Figure 4, but for φM ) -φM , AX AX (2) (2) (1) (1) -φMBX/φMAX ) 1.06, and φMBX/φMAX ) 1.06.

(1) (1) Figure 8. Same as Figure 7, but for φM /φM ) 0.94. BX AX

These remarkable findings are very interesting since they reproduce some of the structural characteristics of the CrN surface compound on Fe-15% Cr-N(100). The Monte Carlo simulations for the case of preferential nearest and next nearest neighbor interactions between MB and X demonstrate the formation of the binary surface compound MBX which consists of a double layer of MB and a surface layer of X. These findings are confirmed for CrN on Fe-15% Cr-N(100) by XPD as already mentioned.22,23 We now turn to the case of preferential MA-X interac(1) (1) tions, φM < φM < 0 (Figure 8). Again the surface BX AX concentrations θMB and θX are very similar to the corresponding graphs of Figures 5 and 7. However, at low temperatures the MB concentration in the subsurface layer θsub MB is substantially lowered with respect to the bulk concentration xMB. This subsurface depletion is unequivocally attributed to the preferential MA-X interactions (1) φM , which favor the replacement of MB atoms by MA in AX the subsurface layer. 4.4. Surface Segregation in the Presence of Repulsive X-X Interactions. Finally we address the question how surface segregation is affected by the presence of repulsive X-X interactions. Therefore we

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positions of the bcc(100) surface. In order to rationalize this finding, we note that the 4-fold hollow sites constitute a square lattice with lattice constant a. Thus X atoms on adjacent 4-fold hollow sites are affected by the fourth (4) nearest neighbor interaction φXX . Thus c(2×2) ordering (4) /(2kB ln(x2 - 1)) and is to be expected for T < Tc ) φXX θX ≈ 0.5.38 It is possible to define an order parameter m for the c(2×2) ordered lattice gas on the 4-fold hollow positions as usual by subdividing these positions into two square sublattices R and β and measuring relevant sublattice concentrations according to Figure 9. Temperature dependence of surface concentrations (1) (1) (2) ) φM ) -φM , for the lattice gas model with φM AX BX AX (2) (2) -6 φMBX/φMAX ) 1.06, xMB ) 0.15, and xX ) 10 : (+) θX; (×) θMB. Nearest, next nearest, and third nearest X-X neighbor (2) (3) (1) interactions are considered, φ(1) XX ) φXX ) φXX ) -φMAX/8, (4) φXX ) 0. The calculations are performed for lattices with 64 × 64 × 64 sites (interstitials plus metal sites). The order parameter m for c(2×2) ordering of X atoms on 4-fold hollow positions of the bcc(100) surface (eq 5) is included (dashed line).

(2) (3) Figure 10. Same as Figure 9, but for φ(1) XX ) φXX ) φXX ) (4) (1) φXX ) -φMAX/8.

introduce up to fourth nearest neighbor interactions between adjacent X atoms. For the sake of simplicity we assume that all these interactions are of equal strength; (i) (2) ) -φM /8. Since all these calculations to be i.e., φXX AX presented below have been performed for the case (2) (2) φM /φM ) 1.06, xMB ) 0.15, and xX ) 10-6 it is possible BX AX (i) ) to compare the results with Figure 5; i.e., the case φXX 0. Figure 9 shows the temperature dependence of θMB and (1) (2) (3) (2) (4) θX for the situation φXX ) φXX ) φXX ) -φM /8, φXX ) 0. AX The resulting graph is almost identical to Figure 5. Therefore, in the present case the surface concentrations θMB and θX are not affected by the nearest, next nearest, and third nearest neighbor X-X interactions. However, if additional fourth nearest neighbor X-X interactions (1) (4) (2) ... φXX ) -φM /8) the resulting graph are introduced (φXX AX is substantially different (Figure 10). Again at temper(2) j 0.04, a complete layer of the MBX atures kBT/φM AX surface compound is formed. With increasing temperatures the surface concentrations decrease toward the corresponding bulk concentrations. The most intriguing feature is the existence of a plateau for xX ≈ 0.5 which is associated with c(2×2) ordering of X atoms on 4-fold hollow

m ) |θRX - θβX|

(5)

This quantity is shown in Figures 9 and 10 as dashed line. Obviously c(2×2) ordering occurs in a very narrow range around θX ≈ 0.5 in case of repulsive fourth nearest neighbor (4) . X-X interactions φXX 5. Summary In this paper we have introduced a three-dimensional lattice gas model with two free (100) surfaces and periodic boundary conditions in x and y directions for the description of surface cosegregation phenomena on (100) oriented surfaces of bcc alloys. It is assumed that the lattice consists of two types of lattice sites M and X. The metal sites M form a body-centered cubic lattice, whose quasi-octahedral interstices constitute the nonmetal sublattice. The M sites are accessible to either MA or MB atoms, while the nonmetal sites either are occupied by interstitials X or remain empty. The three-dimensional lattice studied in this paper is constituted by up to 64 × 64 × 64 sites (interstitials plus metal sites). It is shown that in the absence of preferential M-X interactions MB atoms do not segregate at the bcc(100) surface while X atoms do segregate and occupy the 4-fold hollow positions of the surface. The cosegregation-induced formation of the MBX surface compound is basically due to preferential next nearest neighbor MB-X attractions, (2) (2) φM > φM > 0. At low temperatures the surface BX AX compound consists of a complete MBX compound layer. Preferential nearest neighbor MB-X repulsions, 0 > (1) (1) φM > φM , cause a substantial MB subsurface enrichBX AX ment as is observed experimentally for the CrN surface compound on Fe-15% Cr-N(100).22,23 c(2×2) ordering of X atoms on 4-fold hollow sites of the bcc(100) surface is found for repulsive fourth nearest neighbor interactions (4) φXX . Acknowledgment. It is a pleasure to acknowledge many helpful and stimulating discussions with H. J. Grabke, H. Viefhaus, and J. E. Whitten. I wish to thank U. Detert for his helpful assistance on parallelizing and optimizing the Monte Carlo algorithms. LA940699K (38) For a detailed discussion of the exactly solvable two-dimensional Ising model with nearest neighbor coupling see Onsager, L. Phys. Rev. 1944, 65, 117. Yang, C. N. Phys. Rev. 1942, 85, 808. McCoy, B. M.; Wu, T. T. The two-dimensional Ising model; Harvard University Press: Cambridge, MA, 1977.