PHYSICAL REVIEW B VOLUME 61, NUMBER 22 1 JUNE 2000-II Surface-induced disorder in body-centered-cubic alloys F. F. Haas Institut fu¨r Physik, Universita¨t Mainz, D-55099 Mainz, Germany F. Schmid Max-Planck-Institut fu¨r Polymerforschung, Ackermannweg 10, D-55021 Mainz, Germany K. Binder Institut fu¨r Physik, Universita¨t Mainz, D-55099 Mainz, Germany Received 16 November 1999; revised manuscript received 10 February 2000 We present Monte Carlo simulations of surface-induced disordering in a model of a binary alloy on a bcc lattice which undergoes a first-order bulk transition from the ordered DO3 phase to the disordered A2 phase. The data are analyzed in terms of an effective interface Hamiltonian for a system with several order parameters in the framework of the linear renormalization approach due to Bre´zin, Halperin, and Leibler. We show that the model provides a good description of the system in the vicinity of the interface. In particular, we recover the logarithmic divergence of the thickness of the disordered layer as the bulk transition is approached, we calculate the critical behavior of the maxima of the layer susceptibilities, and demonstrate that it is in reason- able agreement with the simulation data. Directly at the 110 surface, the theory predicts that all order parameters vanish continuously at the surface with a nonuniversal, but common critical exponent 1. How- ever, we find different exponents 1 for the order parameter ( 2 , 3) of the DO3 phase and the order parameter 1 of the B2 phase. Using the effective interface model, we derive the finite size scaling function for the surface order parameter and show that the theory accounts well for the finite size behavior of ( 2 , 3), but not for that of 1. The situation is even more complicated in the neighborhood of the 100 surface, due to the presence of an ordering field which couples to 1. I. INTRODUCTION 3, and with short range forces in Ref. 4 . Wetting phenomena are also present in alloys which un- First-order phase transitions in the bulk of systems can dergo a discontinuous order-disorder transition in the bulk.5,6 drive a variety of interesting wetting phenomena at their sur- In many cases, surfaces are neutral with respect to the sym- faces and interfaces. They have attracted much attention over metry of the ordered phase, but reduce the degree of ordering many years,1 and are still very actively investigated.2 Promi- due to the reduced number of interacting neighbors. The sur- nent examples are the wetting of a liquid on a solid substrate faces can thus be wetted by a layer of disordered alloy, i.e., at liquid-vapor coexistence, or the wetting of one component ``surface induced disorder'' SID occurs.7,8 The situation is of a binary fluid below the demixing temperature on the reminiscent of liquid-vapor wetting; however, the underlying walls of a container. These systems are representatives of a symmetry in the system restricts the possible wetting sce- generic situation, which has been studied in particular detail: narios significantly. Three phases coexist, substrate, liquid and vapor. The sub- We shall illustrate this for systems with purely short range strate acts as inert ``spectator'' which basically provides the interactions: We consider a Landau free energy functional of ``boundary conditions'' for the liquid-vapor system. The the form liquid-vapor transition can be described by a single order parameter e.g., the density , which can take two equilibrium bulk values at coexistence the liquid density or the gas den- F m dr dz g 0 2 m 2 f b m r ,z ... sity . Obviously, the liquid phase will only wet the substrate if it is preferentially adsorbed by the latter. As one ap- proaches the liquid-vapor coexistence from the vapor side, dr fs m r ,z 0 .... 1 different scenarios are possible, depending on the substrate interactions and on the temperature: Either the liquid film Here the vector m subsumes the relevant order parameters, covering the substrate remains microscopic at coexistence the z axis is taken to be perpendicular to the surface, and dr ``partial wetting'' , or it grows macroscopically thick integrates over the remaining spatial dimensions. The offset ``complete wetting'' . The transition from partial to com- of the bulk free energy density f b(m) is chosen such that plete wetting can be first order or continuous ``critical wet- f b(mb) 0 in the bulk. The surface contribution fs(m) ac- ting'' . Since critical wetting is only expected on certain sub- counts for the influence of the surface on the order param- strates at a specific temperature, it is rather difficult to eter, i.e., the preferential adsorption of one phase or in the observe experimentally an experimental observation of criti- case of SID the disordering effect. In mean field approxima- cal wetting with long range forces has been reported in Ref. tion, the functional 1 is minimized by the bulk equation 0163-1829/2000/61 22 /15077 15 /$15.00 PRB 61 15 077 ©2000 The American Physical Society 15 078 F. F. HAAS, F. SCHMID, AND K. BINDER PRB 61 FIG. 2. Cahn construction schematic for surface-induced dis- order in a system with a one component order parameter m a at bulk coexistence and b off bulk coexistence. Dashed line shows surface term f s (m) for critical wetting; dotted line for partial wet- ting. ally, one has to deal with a number of order parameters and other coupled fields, which interact in a way that may not always be transparent. If the surface under consideration FIG. 1. Cahn construction schematic for a second-order wet- does not have the symmetry of the bulk lattice with respect ting transition: a Critical wetting, b partial and complete wetting; to the ordered phases, the interplay of order parameters and c off bulk coexistence, approaching critical wetting. Insets show surface segregation creates effective ordering surface the corresponding order parameter profiles. See text for more ex- fields,11­14 which may affect the critical behavior at the planation. surface.14,15 In the case of a one component order parameter, such a field drives the system from critical wetting to partial d2m wetting. When several order parameters are involved, this is g i not necessarily the case.16­18 More subtle effects can lead to dz2 i f b m , 2 surface order even at fully symmetric surfaces.19­21 which describes the motion of a classical particle of mass g Experimentally, surface induced disorder has been inves- in the external potential f tigated at the 100 surface of Cu b(m) , subject to the boundary 3Au.22­25 A number of condition at z 0 studies have provided evidence that the order parameter right at the surface vanishes continuously as the bulk transition is dm dm approached,22­24 and established the relation with the exis- g i tence of a disordered surface layer of growing thickness.25 dz if s m with g dz 2g f b m . 3 The related case of ``interface induced disorder'' has been studied among other in Cu-Pd, where the width of antiphase If the order parameter has just one component, this equa- boundaries was shown to diverge logarithmically as the tem- tion can be solved graphically by the Cahn construction.9 perature of the transition to the disordered phase was ap- This is illustrated in Fig. 1 for the case of a continuous wet- proached from below.26 ting transition. The corresponding order parameter profiles The first simulation studies of surface induced disorder in are shown as insets. Complete wetting is encountered if different systems have reproduced the continuous decrease f s (m) crosses 2gfb at the outer side of the minimum cor- of the surface order parameter at the bulk first order responding to the adsorbed phase. Partial wetting is found if transition,27,18,16 and the logarithmic growth of a disordered the crossing point is located between the two minima Fig. layer near the surface.16 A detailed study of surface induced 1 b . Critical wetting connects the two regimes, i.e., f s (m) disorder at the 111 surface of CuAu has been published crosses 2g f b right at the adsorbed phase minimum of fb recently by Schweika et al.28 The critical behavior of various Fig. 1 a . Figure 1 c shows a case where the system is off quantities has been analyzed, and critical exponents were bulk coexistence. found which agree well with the theory of critical wetting. Now, let us consider the case of surface induced disorder. Most notably, Schweika et al. observe nonuniversal expo- Here, several equivalent ordered phases exist, and the or- nents, as predicted by renormalization group theories of wet- dered state breaks a symmetry. For neutral surfaces which do ting phenomena.30,31 In contrast, Monte Carlo simulations of not discriminate between the ordered phases, f b and fs have critical wetting in a simple Ising model have given results the same symmetry. This implies that f s is extremal in the which were more consistent with mean field exponents.29 disordered phase (m 0), i.e., f s is zero at m 0 and thus This latter finding has intrigued theorists for some time, and crosses 2g f b at m 0 Fig. 2 . Comparing that with the a number of theories have been put forward to account for scenario sketched above Fig. 1 , we find that surface in- the unexpected lack of fluctuation effects.32­34 The nonuni- duced disordering corresponds to either partial or critical versality of the exponents observed by Schweika et al. seems wetting6-the symmetry of the surface interactions excludes to indicate that the fluctuations are restored in the case of the possibility of complete wetting.10 The off-coexistence SID. Alternatively, it may also stem from a competition of situation Fig. 2 b resembles that in Fig. 1 c . different length scales associated with the local order param- Alloys which exhibit surface induced disorder thus seem eter and the local composition.17,18,35 particularly suited to study critical wetting. Unfortunately, In the present work, we study surfaces of a bcc-based the simplification due to the symmetry of the system often alloy close to the first order transition from the ordered DO3 goes along with severe complications in other respect: Usu- phase to the disordered phase. Our work is thus closely re- PRB 61 SURFACE-INDUCED DISORDER IN BODY-CENTERED- . . . 15 079 lated to that of Schweika et al. It differs in that the order In the linearized theory, the partition function of the parameter structure in the bcc case is much more complex Hamiltonian 4 is approximated by than in the fcc alloy: whereas only one three dimensional order parameter drives the transition considered by Schweika et al., we have to deal with two qualitatively different order Z D l e dr ( l)2/8 1 dr V0 l . 6 parameters, which are entangled with each other in a rather intriguing way. In fact, we shall see that one of them behaves It is convenient to switch from the real space r to the Fourier as expected from the theory of critical wetting, whereas the space q . The integration over short wavelength fluctuations other exhibits different critical exponents, which do not fit with wave vector q 1, where is arbitrary, is then into the current picture. straightforward: One separates l into a short wavelength part A similar system has been investigated some time ago by l (q ) l(q ) (q 1) and a long wavelength part l¯ l l , Helbing et al.16 The systems studied there were rather small, and a detailed analysis of the critical behavior was not pos- and exploits the relation V0(l¯ x) exp xd/dl V0(l¯), to ob- sible. Helbing et al. report evidence for the presence of a tain the unrescaled coarse grained potential38 logarithmically growing disordered layer at the 100 surface as phase coexistence was approached. In retrospect, this re- 1 V¯ dq q l 2 V sult seems surprising, since the 100 surface breaks the sym- l¯ D l exp 4 2 8 l d dl 0 l¯ metry with respect to one of the order parameters, and we 2 know nowadays that this nucleates an ordering surface field. exp , d 2 V In order to elucidate the influence of this ordering field in 2 dl l¯ , 7 more detail, we have thus considered both the 110 surface, which has the full symmetry of the bulk lattice, and the 100 with surface. Our paper is organized as follows: In the next section, we 1/ 2 dq 1 2 ln / , 8 provide some theoretical background on the theory of wet- , q 1/ q2 ting in systems with several order parameters. Section III is devoted to some general remarks on order-disorder transi- where is a microscopic cutoff length. After rescaling r tions in bcc alloys, and to the presentation of the model and r / , V¯ (l) V (l) d 1V¯ (l¯ ), and noting that the the simulation method. Our results are discussed in Sec. IV. roughness exponent is zero for capillary waves in d 3 We summarize and conclude in Sec. V. dimensions, this can be rewritten as II. EFFECTIVE INTERFACE THEORY 2 V dhe h2/2 2 , V OF SURFACE-INDUCED DISORDER l 2 2 0 l h . 9 , A. General considerations Renormalizing the potential V0(l) thus amounts to convolut- We have already sketched one of the popular mean field ing it with a Gaussian of width 2 , ,30 which is the width of approaches to wetting problems in the introduction. Since the a free interface on the length scale parallel to the interface. bulk of the system is not critical, one can expect fluctuations In the case of a bound interface, a natural choice for is , to be negligible for the most part. Only the fluctuations of the the parallel correlation length of the interface. Since the re- local position l(r ) of the interface between the growing sur- maining fluctuations after the renormalization should be face layer and the bulk phase remain important.36 As the small on this length scale, the procedure can be made self interface moves into the bulk, capillary wave excursions of consistent by equating with its mean field value larger and larger wavelengths become possible. These intro- duce long-range correlations parallel to the surface, charac- d2 terized by a correlation length 4 V l / 2 1 at , 10 which diverges at wetting. dl2 In light of these considerations, fluctuation analyses often l l replace the Landau free energy functional 1 by an effective where the average position of the interface l is the position interface Hamiltonian30,36,37 of the minimum of V (l). Note that the renormalized free energy density per area 2 is of order unity. The singular H l /k part F BT dr 1 s of the total interface free energy thus scales like Fs 8 l 2 V0 l . 4 2 . From the renormalized Hamiltonian 4 , Here all lengths are given in units of the bulk correlation length 1 / b in the phase adsorbed at the surface, the parameter H l /k dq 1 is the dimensionless inverse of the interfacial tension BT 4 2 0 8 q2 1 l q 2, 11 k 2 BT/4 b , 5 we can now calculate the distribution probability to find the interface at a position h, and the potential V0(l) describes effective interactions be- tween the interface and the surface. The wetting transition is 1 thus identified with a depinning transition of the interface P 2 h h l 0 H 2 e h2/2 , 12 from the surface. 2 15 080 F. F. HAAS, F. SCHMID, AND K. BINDER PRB 61 and the joint probability distribution that the interface is found at h and h at two points separated by r from each zz dr dhdh mbare z h mbare z h other dr P(2) h,h ,r P h P h , 20 P(2) h,h ,r h l 0 h l r H expand the joint probability P(2)(h,h ,r) in powers of 1 (r) g(r)/ 2 K 0(r/ )/ ln( / ), 2 g 0 2 g r 2 hh h h 2 h h 2 P(2) h,h ,r P h P h 1 r 2 exp 4 g , 0 g r ... 4 g 0 g r ... 13 1 h2 h 2 h2h 2 2 1 r 2 *** , 2 4 where 21 / dq and recall drrK g r l 0 l r 0(r) 1 and drrK0(r)2 1/2. If the in- H eiq r / 14 trinsic width of the profile m 0 q2 1 bare(z) is small compared to , the intrinsic profile can be approximated by a simple step is the height-height correlation function of the interface and profile in the interfacial region, mbare(z) mb (z), where mb is the bulk order parameter. One then obtains 2 g 0 2 ln / . 15 2 2 2 2 2 z l 2 An analogous expression has been derived by Bedeaux and zz mb e (z l )2/ 2 *** . 2 2 4 4 Weeks for a free liquid-gas interface in a gravitational 22 field.39 In three dimensions, the height-height correlation function for r and is a Bessel function K, So far, these results are valid for infinite systems. The restriction to finite lateral dimension L affects the interface g r 2 K distribution P(h) 12 in two ways: It introduces a lower 0 r/ . 16 cutoff /L in the integrals over q e.g., Eq. 14 , and the We assume that the average order parameter profile mean position of the interface the zeroth mode is no longer m(z) is given by the average over mean field order param- fixed at the minimum of the renormalized potential, but dis- eter profiles mbare(z l), centered around the local interface tributed according to exp L2V (h) . The width of the dis- positions l, which are distributed according to the distribu- tribution function P(h) is now given by tion function P(l) d2V / dq h 2 L2 m z dlP l m bare z l . 17 /L q2 1 dh2 h l The distribution functions P(h) and P(2)(h,h ,r) can then 2 ln ln 1 2 4 2. 23 be used to calculate various characteristics of the profile. L L For example, the effective width of the order parameter profile, W 1/(2 m / z) l , is broadened by P(h) and di- B. Bare and renormalized effective interface potential verges according to40,41 We shall now apply these general considerations to a spe- cific potential V0(l), designed to describe systems with short W2 W 2 2 range interactions and several order parameters and nonor- 0 2 , 18 dering densities. Effective interface potentials for systems where W with two order parameters have been derived by Hauge35 and 0 denotes the ``intrinsic width'' of the mean field Kroll and Gompper.17 Their approach can readily be gener- profile, W0 1/(2dmbare /dz) z 0. alized to the case of arbitrary many order parameters and Another quantity of interest is the layer-layer susceptibil- nonordering densities. We choose the coordinate system in ity, which describes the order parameter fluctuations at a given distance from the surface, the order parameter and density space m such that m 0 in the phase which wets the surface, and that the coordinate axes mi point in the directions of the principal curvatures of 2 the bulk free energy function f zz dr m 0 m r z m z . 19 b(m). Close to this phase, f b can then be approximated by the quadratic form Since it has the dimension of a square length, one deduces g 1 immediately that 2 2 zz scales like in the interfacial region. f b m mi , 24 For a more detailed analysis, we rewrite 2 i 2 zz as i PRB 61 SURFACE-INDUCED DISORDER IN BODY-CENTERED- . . . 15 081 where is the field which drives the system from coexist- C. Free energy scaling ence, and the i have the dimension of a length. We number Now our task is to determine the coordinate axes i (i 0) such that the self-consistently by use of i are arranged in Eq. 10 , which will yield the scaling behavior of the singu- descending order. The largest of these dominates the corre- lar part of the surface free energy, F 2 . Before general- lations at large distances and is thus the correlation length s izing to several order parameters, we shall briefly discuss the b , i.e., 0 b 1 in our units. The surface contribution has situation in a system with only one length scale the form 0. The formal alikeness of the more general theory with this often 1 discussed special case can thus be highlighted. Moreover, f s m hi,1mi cijmimj . 25 many of the results derived for one order parameter carry i 2 i j over directly to the case of several order parameters. Following Hauge and Kroll/Gompper, we now assume that In a system with one order parameter, the singular free the actual profile from the adsorbed phase to the bulk phase energy has the scaling form is close to the profile of a free interface between these two 2 phases. Close to the surface region, we thus approximate the Fs 8 f 0 , 30 former by the test function where the scaling function f ( 0) depends on the dimension- less parameter mi z vi exp z l / i 26 at z l), where l denotes the position of the effective inter- 0 C0 ( 1)/2a0 with C0 8 /2b. 31 face. Inserting this into Eq. 1 with Eqs. 24 and 25 , we Depending on the value of 0, one can distinguish between obtain the effective interface potential different regimes: 2 V 0 1: f 0 1/2g1 2 0 0 l aie l/ i bije l(1/ i 1/ j) l 27 i i j with g1 x 1 x 2 x2 *** for l 0, with ai hi,1vi and bij 12 (cij g ij / i)vivj . This expression is of course only valid for large l. Notably, it fails complete wetting , 32 at l 0, since the true potential V0(l) must diverge there. We shall suppose that the leading term b 1 2 00 b in the second sum 2 *** is positive and dominates over the more rapidly decaying 0 1: f 0 1 2 0 8 0 terms, and disregard the latter in the following. At 0 or in mean field approximation , the interface critical wetting, field like , 33 is flat, and its position is given by the minimum of V0(l). 2 2 At nonzero , the potential has to be renormalized as 0 1: f 0 0/2 1/(1 )g2 0/2 1/(1 )... described in the previous section. Now the renormalization 3x is straightforward if the fluctuations are sufficiently small 2 7 x2 with g2 x 1 that the interface position l at wetting is well in the 2 1 8 1 2 asymptotic tail of the potential weak fluctuation limit . According to a criterion introduced by Bre´zin, Halperin, *** partial wetting . 34 and Leibler,30 this is true as long as 2 The point 0 dle (l l )2/2 V0(l) 0 0 is the critical wetting point. If one ap- 2 proaches this point from the partial wetting side a0 0 on dle (l l )2/2 V0(l), i.e., the coexistence line 0, the parallel correlation length 2 2 2 diverges with the well-known nonuniversal exponent l 0 and / i l 0 28 for all i . For i 1/2, the first inequality enforces the sec- 2 /b 1/2(1 ) a0 1/(1 ), 35 ond one. In a system with one order parameter, it leads to the well-known inequality 1/2.30,31 As we shall see shortly, and the distance between the average position of the inter- this condition is also sufficient to ensure the validity of the face and the surface diverges asymptotically like weak fluctuation limit in a system with several order param- l 1 2 / 1 ln a eters. Since in our simulations turns out to be much 0 . 36 smaller than 1/2, we shall not discuss the other regimes in The relevant regime for most cases of surface induced the present paper. disorder is however the critical wetting regime, where the In the weak fluctuation limit, the renormalized potential critical wetting point is approached under a finite angle to the takes the form coexistence line in (a0 , ) space. Here the parallel correla- tion length scales like V 2 l a / i b 4 e 2l l. 2 ie l/ i 1 i: i 1/2 with 29 8 1/2 37 The cutoff parameter is of the order of the correlation as approaches zero, the width of the interface diverges length, b 1, and will be dropped hereafter. with 15 082 F. F. HAAS, F. SCHMID, AND K. BINDER PRB 61 where the scaling variables are W2 2 2 2 ln , 38 i Ci (1 2 i)(1 / i)/2 iai 48 and its average position with with C 2*(2 i 1) 2b 1/2 i. l 1/2 ln . 39 i 8 /2 i As in the one-order parameter case, we have to distin- These results can be used to derive the layer-bulk suscep- guish between different regimes depending on the values of tibility of the order parameter in the interfacial region the scaling variables. m m l 1 0 0 0, l D. Symmetry preserving and symmetry breaking surfaces z . 40 l l ln Let us first assume that the effect of nonordering densities In the step approximation m0,bare(z) m0,b (z), the layer- can be disregarded e.g., because the associated length scales bulk susceptibility in the interfacial region can be calculated are small, i 1/2), and consider the case of a symmetry in more detail: preserving surface. No ordering surface fields are then present, i.e., ai hi 0 for all contributions i. The system is m 1 z l thus in a ``multicritical wetting regime,'' where 0,b 2 i 1 for 0,z e (z l )2/2 . 41 2 2 2 ln all i, and the scaling function can be expanded as It has a slightly asymmetric peak of width at z l , the 2 i 1 height of which scales like 1/ . f i 1 i ***. 49 2 The layer-layer susceptibility could already be derived in i 2 i the previous section. It also has a peak at the interface, which The effective interface position l , and the correlation is however a factor of 2 narrower. Its height scales like length are given by Eqs. 39 and 37 as in the case of normal critical wetting. Hence all the results related to inter- 2 2 l l / 1/ ln .... 42 facial properties, such as the interfacial width, the interfacial Next we determine the critical behavior of the order pa- layer susceptibilities, etc., remain unchanged. In particular, rameter at the surface, m the criterion for the validity of the weak fluctuation limit is 0,1 , still 1/2 from Eqs. 28 , 38 and 39 . The surface F 2 1 order parameters obey the power law m s 0,1 h 0,1, 0,1 0,1 a0 2 . 43 2 1 It will prove useful to rederive the exponent mi,1 i,1, i,1 2 i 1 . 0,1 in an alter- ai 2 i 2 2 native way: The surface order parameter in mean field theory i 50 is given by mbare(0) mb exp( l/ 0). Averaging the profile according to Eq. 17 yields Following the lines of Eq. 44 , one also obtains the finite size scaling function m 2 /2 2 0,1 mb e l/ 0 P(l) mbe l / 0 0. 44 2 2 After inserting /2 i 0 1 and using Eqs. 39 and 38 , one re- M i x x x 1 e2 /x i . 51 covers the power law of Eq. 43 with the same exponent 0,1 . The approach has the advantage that it allows for a A whole sequence of surface exponents is thus predicted, straightforward calculation of finite size effects on surface one for each order parameter. In practice, however, one critical behavior: We simply replace the expression 38 for hardly ever measures only one ``pure'' order parameter m i . in the infinite system by Eq. 23 to obtain Instead, one expects to observe some combination of contri- butions with different exponents m i,1 , which will be domi- 0,1 mb 0,1M 0 8 L1/ , 45 nated by the leading exponent 0,1 ( 1)/2 in the with the scaling function asymptotic limit 0. The situation changes when at least one of the ai becomes nonzero at coexistence. This is the case, e.g., at a symmetry M /2 0 x x x 1 e2 /x. 46 breaking surface, where one or several surface fields become nonzero, or even at a symmetry preserving surface if the We are now ready to generalize these results to the case length scale associated with a nonordering density exceeds of several order parameters and nonordering densities. For- half the bulk correlation length, i 1/2. mally, the theory turns out to remain very similar. The self- Let aJe l/ J be the leading nonvanishing term in the po- consistent determination of tential 27 . As one approaches coexistence, 0, the scal- leads to a generalized version of the scaling form for the singular part of the surface free ing variable J increases and one eventually enters a regime energy 30 , J 1. For negative aJ , ( J 1), the wetting becomes partial, i.e., no surface induced disordering takes place. For F 2 s 8 f i , 47 positive aJ , ( J 1), different scenarios are possible, de- PRB 61 SURFACE-INDUCED DISORDER IN BODY-CENTERED- . . . 15 083 pending on the sign and the amplitude of the higher order terms ai , (i J) in Eq. 29 . If they are positive or suffi- ciently small, such that a J / i iaJ 1, 52 the disordered phase wets the surface. The effective interface position l diverges asymptotically like l 2 FIG. 3. bcc lattice with a disordered A2 structure, b B2 struc- J 1 /2 J ln , 53 ture, and c DO3 structure. Also shown is the assignment of sub- the parallel correlation length scales like lattices a,b,c,d. J/4 , 54 here.18 However, l in this work is taken from Eq. 36 rather than determined self consistently, hence the resulting and the scaling function in Eq. 47 takes the form critical exponents differ somewhat from those calculated 1 here. As in the case of the symmetry preserving surface, a f J / i whole set of exponents is predicted by Eq. 56 . In the i 2 1 i J KJ i J i asymptotic limit 0, however, the surface behavior is ex- 1 pected to be governed by the leading exponent 2 J 1 2 J KJ 2 , 55 b b 0,1 1 . 57 with J 2 J We have reinserted the bulk correlation length b 1 here. J / i Finally, we discuss the critical behavior of the surface K J J 2*(1 J i i / J) 1 2 J /2 i . susceptibilities. The corresponding critical exponents can be i i shown to obey simple scaling laws. In the case of the According to Eq.. 28 , the weak fluctuation regime here is surface-bulk susceptibility, the relation follows trivially: bounded by 2 2J , thus encompassing the regime 1/2. mi,1 The criterion 52 is motivated as follows: If one of the i,1 i,1, i,1 1 i,1. 58 higher order ai is negative and large, the interface potential V In the case of the surface-surface susceptibility, it depends on (l) may exhibit a second minimum closer to the surface, the regime under consideration. In the ``critical wetting re- which competes with the minimum at large l and may pre- gimes'' discussed here, the free energy scaling function f can vent the formation of an asymptotically diverging wetting be expressed as a Taylor series in powers of the scaling layer. The inspection of the free energy scaling function 55 reveals that the transition to such a partial wetting regime is variables i or i,J , respectively, and one obtains appropriately described in terms of the combined scaling variable m 2f i,1 i,11 h i,11, i,11 1 2 i,1 . 59 i,1 a 2i J / i J / i i,J i ( J / i 1) 1 /(2 i J)... J aiaJ . The dominating exponents in the asymptotic limit are 0,1 This quantity has to be large at the point and 0 where the one 0,11 . minimum of V (l) splits up in two. The condition 52 en- III. MODELING ORDER-DISORDER TRANSITION sures that i,J is small for all . IN BCC ALLOYS The wetting is critical with respect to all order parameters mi with length scales i larger than J . As coexistence is Figure 3 shows some typical structures of binary AB approached, they vanish at the surface according to the bcc alloys e.g., FeAl Ref. 42 . It is useful to divide the bcc power law lattice into four fcc sublattices a ­d as indicated in the figure. The phase transitions are then conveniently described in 2 terms of a set of order parameters43 m J i i,1 a i,1, i,1 1 . 56 i i 2 2 i J 1 ca cb cc cd , The finite size scaling function M i(x) is again given by Eq. 51 , with the scaling variable x 4 L2/ 2 ca cb cc cd , 60 J . Note that the exponents i,1 are nonuniversal even in the mean field limit ( 0). This remarkable effect has first been discov- 3 ca cb cc cd , ered by Hauge35 and later studied by Kroll/Gompper in an where c denotes the composition on the sublattice , i.e., fcc Ising antiferromagnet using a mean field the average concentration of one component A there. In the approximation,17 Monte Carlo simulations, and a linear disordered phase, all sublattice compositions are equal and renormalization group study similar to the one presented these order parameters vanish. The B2 phase is characterized 15 084 F. F. HAAS, F. SCHMID, AND K. BINDER PRB 61 by nonzero 1, and the DO3 phase in addition by nonzero 2 3. By symmetry, physical quantities have to be in- variant under sublattice exchanges (a b), (c d), and (a,b) (c,d). The leading terms in a Landau expansion of the free energy F thus read F F 2 2 2 4 0 A1 1 A2 2 3 B 1 2 3 C1 1 C 4 4 2 2 2 2 2 2 2 3 C3 2 3 C4 1 2 3 . 61 We point out in particular the cubic term B 1 2 3. It can be read in two ways. On the one hand, it describes how the FIG. 4. Phase diagram of our model in the T-H plane. Solid B2-order influences the DO3 order: The order parameter 1 lines mark first-order phase transitions; dashed lines second-order breaks the symmetry with respect to individual sign reversal phase transitions. Arrows indicate the positions of a critical end of 2 or 3 and orients ( 2 , 3) such that 2 point cep and a tricritical point tcp . sgn(B 1) 3. Conversely, one can interpret the product 2 3 as an effective ordering field acting on 1. We shall The surface simulations were performed in a L L D come back to this point later. geometry with periodic boundary conditions in the L direc- At the presence of surfaces, the situation is even more tion and free boundary conditions in the D direction, varying complicated. First, we can always expect that one component D from 100 to 200 and L from 20 to 100. In order to handle enriches at the surface, since there are no symmetry argu- systems of that size efficiently, we have developed44 a mul- ments to prevent that. Even if no explicit surface field cou- tispin code,45 which allowed to store the configurations bit- pling to the total concentration c is applied, the component wise instead of bytewise.46 Our Monte Carlo runs had total which is in excess with respect to the ideal stoichiometry of lengths of up to 2 106 Monte Carlo sweeps. the bulk phase 3:1 in the DO3 phase will segregate to the surface. Second, we have already mentioned that the Landau IV. SIMULATION RESULTS expansion of the surface free energy f s depends on the ori- entation of the surface.11,14 The 110 surface has the same We have studied 110 and 100 oriented surfaces at T symmetry with respect to sublattice exchanges as the bulk, 1V/kB close to the first order bulk transition between the hence the Landau expansion of the surface free energy must ordered DO3 phase and the disordered A2 phase. The exact have the form 61 . In case the order is sufficiently sup- bulk transition point was determined previously from bulk pressed at the surface, one can thus hope to find classical simulations by thermodynamic integration,47 H0 /V surface induced disordering here. In the case of the 100 10.00771(1).44 In the presence of such a high bulk field, surface, the symmetry with respect to the exchange the very top layer of a free 110 or 100 surface is com- (a,b) (c,d) is broken. The surface enrichment of one pletely filled with A particles, i.e., Ising spins S 1. Conse- component then induces an effective ordering surface field, quently, the order parameters and the layer susceptibili- which couples to the order parameter 1.12 Other ordering ties vanish there. In the following, we shall generally fields coupling to 2 and 3 are still forbidden by symmetry. disregard this top layer and analyze the profiles starting from The full spectrum of possible ordering surface fields is al- the second layer. lowed in the case of the 111 surface. In order to model these phase transitions, we consider an A. 110... surfaces: DO3 order Ising model of spins Si 1 on the bcc lattice with antifer- romagnetic interactions between up to next nearest neigh- We begin with a detailed discussion of surface induced bors, disordering at 110 surfaces, i.e., surfaces with the full sym- metry of the bulk. Figure 5 shows profiles of the order pa- rameter of DO3 ordering per site H V SiSj V SiSj H Si , 62 2 2 ij ij i 23 2 3 /2. 64 where the sum i j runs over nearest neighbor and i j One clearly sees how a disordered layer forms and grows in over next nearest neighbor pairs. Spins S 1 represent A thickness as the bulk transition is approached. In order to atoms and S 1 B atoms, hence the concentration c of A is extract an interface position l and an effective interfacial related to the average spin S via width W, we have fitted the profiles to a shifted tanh function bulk 1 exp 2 z l /W ... 1. 65 c S 1 /2, 63 23 n 23 The results are shown in Figs. 6 and 7. Sufficiently close to and the field H represents a chemical potential. The param- the bulk transition, at (H0 H)/V 0.005, the data are con- eter 0.457 is chosen such that the highest temperature sistent with the logarithmic divergence predicted by Eqs. which can still support a B2 phase is about twice as high as 39 and 18 . Intuitively, one would expect that an effective the highest temperature of the DO3 phase, like in the experi- interface theory is only applicable if l W, i.e., the width of mental phase diagram of FeAl. The phase diagram of our the interface is smaller than the distance of the interface from model is shown in Fig. 4. the surface. Indeed, Fig. 7 shows that the logarithmic behav- PRB 61 SURFACE-INDUCED DISORDER IN BODY-CENTERED- . . . 15 085 FIG. 7. Squared interfacial width as estimated from the fit 65 FIG. 5. Profiles of 23 near a 110 surface at temperature T in units of 110 layers vs (H H0)/V. Long dashed line shows 1kBT/V for different fields H in units of V as indicated. The bulk squared interface position l 2 for comparison. transition is at H0 /V 10.00771(1). Zeroth top layer is not shown 23(0) 0; see text . seems rather large for a system which is not critical in the bulk. On the other hand, Fig. 5 shows that the bulk order ior sets in approximately at the value of H where l begins to parameter 23 decreases considerably as one approaches the exceed W. The prefactors of the logarithms in Figs. 6 and 7 phase transition point. This observations suggests that a criti- are predicted to be (r/2 /r) 2 cal point is at least nearby, although preempted by the first b /a0 in the case of l Fig. 6 , and 2 2 order transition from the DO b/a0 in the case of W2 Fig. 7 , where b is 3 phase to the disordered phase. the bulk correlation length, a Next we consider the profiles of the layer susceptibilities 0 the lattice constant, a factor 2 or 2 accounts for the distance of 110 layers from each of the order parameter 23 . They can be determined from the other in units of a simulation data by use of the fluctuation relations28 0, and the parameter r max(1,2 J / b) depends on the length scale J of composition fluctuations see the discussion in Sec. II D . We shall see below that the N total surface data suggest z 1 r/2 (1/r 1/2) 0.618. Insert- kBT z total z total ..., 66 ing this result, one derives 4.5 7 b /a0 5.4 8 from Fig. 6, and b /a0 7.8 8 from Fig. 7. These values do not agree N layer with each other within the statistical error; the interfacial zz k width seems to decrease too fast as one moves away from BT z 2 z 2..., 67 coexistence. Yet the difference seems still acceptable, espe- where is the order parameter under consideration, Nlayer cially considering how small the region of apparent logarith- denotes the number of sites in a layer, and Ntotal the total mic behavior is. It has been observed in other systems,48 that number of sites. Figure 8 shows that both the layer-bulk the vicinity of surfaces also affects the intrinsic width W0 of susceptibility z and the layer-layer susceptibility zz exhibit an interface. Moreover, many nondiverging terms have been the expected peak in the vicinity of the interface Eqs. 41 neglected in Eqs. 39 and 18 which lead to systematic and 22 . The centers of the peaks can be fitted nicely by errors if one is not close enough to H0. We note that b Gaussians of width and / 2, respectively, where is calculated from the width W of the order parameter profile using 2/ W. The wings of the peaks are not Gaussian any more, but asymmetric-the layer susceptibilities are en- hanced at the bulk side of the interface, and suppressed at the surface side. Such an asymmetry has been predicted qualita- tively for z in Eq. 41 , but not for zz cf. Eq. 22 . Even in the case of z , the observed asymmetry is so strong that it cannot be brought into quantitative agreement with the theory. We recall that the linear theory approximates the cap- illary waves of the interface by those of a free interface with some suitable long-wavelength cutoff, i.e., they are taken to be distributed symmetrically about the mean interface posi- tion. The failure of the theory to describe the details of the profiles of z and zz presumably reflects the fact that the capillary waves are in fact asymmetric. Nevertheless, the main features of the profiles are captured by the theory. The centers of the peaks are slightly more distant from the FIG. 6. Position of the interface as estimated from the fit 65 in surface than l in Fig. 6, but the difference is not significant units of 110 layers vs (H H0)/V. up to three layers at (H0 H)/V 0.0007]. According to 15 086 F. F. HAAS, F. SCHMID, AND K. BINDER PRB 61 FIG. 10. Maxima of the layer-layer susceptibility zz per site of the order parameter 23 in units of kBT vs (H0 H)/V, for different system sizes as indicated. Inset shows bare data, with a fit to a power law behavior with unknown exponent dotted line . In the main plot, the bulk contribution to zz has been subtracted. Solid line indicates the slope of (H0 H) 1, and dashed line the whole theoretical prediction including the logarithmic correction. long as the interfacial width is dominated by the intrinsic width W0. In the regime (H0 H)/V 0.005, where the cap- illary wave broadening of the interface becomes significant, the data are also consistent with the logarithmically corrected version max z 1/(H0 H) ln(H0 H) see Fig. 9 . FIG. 8. Profiles of the layer-bulk susceptibility z a and the The analysis of the layer-layer susceptibility is more layer-layer susceptibility zz b per site of the order parameter 23 subtle. From a double logarithmic plot of the raw data, one is in units of kBT, for different fields H in units of V as indicated. tempted to conclude that the predicted 1/(H0 H) behavior Solid line shows the fit of a Gaussian of width a (2/ )1/2W is not valid; the data rather suggest a divergence with a criti- and b /21/2 to the profile corresponding to H 10.004. Zeroth cal exponent 0.63 Fig. 10, inset . However, since we are not top layer is not shown (0) 0; see text . aware of any theoretical explanation which could motivate the theoretical prediction 40 and 42 , the heights of the such an exponent, we believe that the apparent power law peaks should diverge with 1/(H behavior over roughly two decades of (H0 H) is most 0 H) with different loga- rithmic corrections. Our data are shown in Figs. 9 and 10. likely accidental. Looking at the values of zz close to the The maxima of the layer-bulk susceptibility are best fitted by center of the slab Fig. 8 b , one recognizes that the contri- the simple 1/(H H bution of bulk fluctuations to zz is significant even close to 0) behavior, which the theory predicts as H0. The situation is complicated by the fact that the bulk fluctuations increase considerably in the vicinity of H0, al- though their amplitude does not diverge. Within the crude approximation that the capillary waves of the interface and the bulk fluctuations are uncorrelated, one can subtract the latter as ``background.'' The thereby corrected data agree reasonably well with the theory, especially when taking into account the logarithmic correction max zz 1/(H0 H) ln(H0 H) Fig. 10 . We proceed to study the properties of the system directly at the surface. Figure 11 shows the order parameter 23,1 in the first layer recalling that the top zeroth layer is dis- carded as a function of (H0 H) for various system sizes. One notices finite size effects if the dimension L parallel to the interface is small. As long as L is large enough, the data exhibit a power law behavior with the exponent 1 FIG. 9. Maximum of the layer-bulk susceptibility 0.618 4 . We emphasize that 1 clearly differs from 1/2 z per site of the order parameter here. It is close to the value 23 in units of kBT vs (H0 H)/V for different 1 0.64 found by Schweika system sizes as indicated. Solid line shows a fit to a (H0 H) 1 et al. in their simulations of surface induced disorder in fcc behavior, and dashed line the same with logarithmic correction see alloys.28 As discussed in Sec. II D, several factors may lead text . to such a nonuniversal exponent-capillary wave fluctua- PRB 61 SURFACE-INDUCED DISORDER IN BODY-CENTERED- . . . 15 087 FIG. 13. Surface layer-bulk susceptibility per site of the order FIG. 11. Order parameter 23,1 at the surface first layer vs parameter (H 23 vs (H0 H)/V for different system sizes as indi- 0 H)/V for different system sizes L L D as indicated. Solid cated. Solid line marks a power law with the exponent line indicates power law with the exponent 1 0.37. 1 0.618. tions, and/or the presence of a length scale Figure 13 shows the layer-bulk susceptibility at the sur- J b/2, which- competes with the correlation length face for the order parameter 23 . According to Eq. 58 , it b and would have to beassociated with the nonordering composition fluctuations should diverge with the exponent 1 1 1 0.382. In- in the case of the symmetry preserving 110 surface. Using deed, the fit to our data in the region (H0 H)/V 0.02 Eq. 57 , we can derive upper bounds for the capillary pa- yields 1 0.37 5 . In the case of the layer-layer suscepti- rameter, 0.236, and for bility, the theory 59 predicts J , J / b 0.618. 11 1 2 1 0.236, i.e., After applying finite size scaling with the exponents 1 11 does not diverge at the phase transition. In fact, it first and increases as H 1/2 cf. Eq. 45 , the data collapse onto a single 0 is approached, but then decreases for (H0 master curve. The form of the latter can be calculated from H)/V 0.02 not shown . The layer-layer susceptibility at Eq. 45 , the surface here behaves in a similar way as observed by Schweika et al. in their studies of surface induced disorder at the 111 surface of an fcc-based alloy.28 xr/2 /r 23,1L 1 / e2 /x, 68 x 1 /2 B. 110... surfaces: B2 order From the results discussed so far, we conclude that the with x (H0 H)L1/ and r max(1,2 J / b), where the behavior of the order parameter 23 can be understood nicely two unknown proportionality constants are fit parameters and within the effective interface theory of critical wetting. How- 0.236 was used the result is only very barely sensitive to ever, we shall see that this holds only in part for the second the choice of ). Figure 12 shows that the data agree nicely order parameter, 1. with the theoretical prediction. Figure 14 shows profiles of 1 for different fields H. They resemble those of 23 , in particular the inflection point of the profiles is located approximately at the same distance from the surface. The upper part of Fig. 14 displays profiles of the total concentration c of A particles Eq. 63 . They exhibit some characteristic, H-independent oscillations in the first four layers, and the A concentration is slightly enhanced in the disordered region. However, the overall variation is rather small. The layer susceptibility profiles of the order parameter 1 are qualitatively similar to those of 23 and not shown here. Figure 15 demonstrates that the maximum of the layer-bulk susceptibility evolves with the field H as theoretically pre- dicted, max z 1/(H0 H) ln(H0 H) . In the case of the layer-layer susceptibility, the agreement with the theoreti- cally expected behavior ( max bulk zz zz ) 1/(H0 H) ln(H0 H) is not quite as convincing, but the data are still consis- tent with the theory for (H H0)/V 0.01 Fig. 16 . Note FIG. 12. Finite-size scaled plot of the surface order parameter that the bare values of max zz would again rather suggest a 23,1 vs (H0 H)/V for system sizes L L D as indicated. Data were scaled with exponents power law, max 1/2 and 1 0.618. Dashed line zz (H0 H) 0.53 Fig. 16, inset , which is shows the finite size scaling function predicted by Eq. 51 . however most likely accidental. 15 088 F. F. HAAS, F. SCHMID, AND K. BINDER PRB 61 FIG. 16. Maxima of the layer-layer susceptibility zz per site of the order parameter 1 minus bulk contribution in units of kBT vs (H0 H)/V, for different system sizes as indicated. Solid line indi- cates the slope of (H0 H) 1, and dashed line the whole theoreti- cal prediction including the logarithmic correction. Inset shows bare data, with a fit to a power law behavior dotted line . different system sizes do not collapse if one performs finite FIG. 14. Profiles of the total composition c ( S 1)/2 top and of the order parameter size scaling with the exponent 1/2 Fig. 18 a . The col- 1 bottom for different fields H in units of V as indicated. Top zeroth layer is not shown c(0) lapse is significantly better if one assumes that the parallel 1, correlation length diverges with the exponent 1(0) (0) . Thin dashed lines with squares show for com- 0.7 0.05 parison the profiles of 23 from Fig. 5. Fig. 18 b . We have no explanation for these unexpected findings. Hence the behavior of the order parameter 1 in the vi- The discussion in Sec. II has shown that several surface ex- cinity of the interface is similar to that of the order parameter ponents i,1 may be present in a system with several order 23 and consistent with the theory of critical wetting. The parameters. Even though we have argued that only the small- agreement however does not persist when looking right at est exponent should survive in the asymptotic limit 0, the surface. Figures 17 and 18 show how the value of 1 in the other power law contributions may conceivably still the first surface layer depends on (H0 H)/V. A power law dominate the behavior of certain quantities over a wide range behavior is found over one and a half decades of (H0 of . However, the critical exponent should in all cases H)/V, yet the exponent 1( 1) 0.801 differs from that remain invariably 1/2. Our results seem to indicate that of 23,1 , 1( 23) 0.618 Fig. 17 . Moreover, the data for the behavior of the order parameter 1 at the surface is gov- erned by a length scale, which differs from that given of the FIG. 15. Maximum of the layer-bulk susceptibility z per site of the order parameter 1 in units of kBT vs (H0 H)/V for different system sizes as indicated. Solid line shows a fit to a (H0 H) 1 FIG. 17. Order parameter 1 at the surface vs (H0 H)/V for behavior, and dashed line the same with the appropriate logarithmic different system sizes L L D as indicated. Solid line shows correction. power law with the exponent 1 0.801. PRB 61 SURFACE-INDUCED DISORDER IN BODY-CENTERED- . . . 15 089 FIG. 19. Surface layer-bulk susceptibility per site of the order parameter 1 vs (H0 H)/V for different system sizes as indicated. This is demonstrated in Fig. 20. The order parameters and the composition c are defined based on the sublattice occu- pancies on two subsequent layers of distance a0/2, starting from the first layer underneath the surface. The top layer is again disregarded, since it is entirely filled with A or S 1. The profiles of 1 clearly display the signature of an addi- tional ordering tendency at the surface, which in fact reverses the sign of 1 in the top layers. However, the effect is rather weak and does not influence the system significantly deeper in the bulk. The profiles can be analyzed like those at the FIG. 18. Finite-size scaled plots of the order parameter 110 surface, and mean interface positions and mean inter- 1 at the surface vs (H0 H)/V for system sizes L L D as indicated. Ex- ponents are 1 0.801, 0.5 in a , and 0.7 in b . interfacial fluctuations, but which nonetheless diverges as H0 is approached. Note that 0.7 is close to the exponent 0.63 with which the bulk correlation length diverges at an Ising type transition in three dimensions. Likewise, the ex- ponent 1 0.801 found here resembles the surface critical exponent of the ordinary transition, 1 0.8.49,50 One might thus suspect that 1 in the disordered surface layer becomes critical at H0. However, such a coincidence would seem rather surprising. Furthermore, we have noted earlier that the combination 2 3 acts as an ordering field on 1, hence 1 cannot become critical as long as 23 is not strictly zero. Figure 19 shows the layer-bulk susceptibility at the sur- face as a function of (H0 H)/V. It decreases as H0 is ap- proached, hence the scaling relation 1 1 1 is obviously not met for the order parameter 1,1 . C. 100... surfaces Finally, we turn to the discussion of 100 surfaces. As already mentioned earlier, 100 surfaces break the symmetry with respect to the order parameter 1, an ordering surface field coupling to this order parameter is allowed and thus usually present.11,14 This field is often closely related to sur- face segregation.11,40 In our case, the excess component A of FIG. 20. Profiles of the total concentration of A top, diamonds , the DO3 segregates in the surface layer and induces a stag- of the order parameters 1 bottom, circles and 23 bottom, gered concentration field in the layers underneath, which is squares at H/V 10.003 filled symbols and at H/V 10.007 equivalent to 1 ordering. open symbols . Zeroth top layer is not shown. 15 090 F. F. HAAS, F. SCHMID, AND K. BINDER PRB 61 Due to the complicated order parameter structure in our system, however, our data could not fully be explained within a theory which traces everything back to the proper- ties of a single interface between a disordered and an ordered phase. The theory provides a satisfactory picture for the be- havior of the order parameter describing the DO3 ordering, 23 , and in general for the structure in the interfacial region. However, it fails to predict the behavior of the order param- eter of B2 ordering, 1, directly at the surface. Our data thus indicate that the fluctuations of 1 at the surface require special treatment. Parry and co-workers34,51 have recently suggested an approach to a theory of wetting based on an effective interface Hamiltonian with two ``interfaces,'' the usual one separating the phase adsorbed at the surface and the bulk phase, and a second one which accounts in an ef- fective way for the fluctuations directly at the surface. Our FIG. 21. Order parameter 23 at the surface vs (H0 H)/V for problem seems to call for such an approach. Unfortunately, different system sizes L L D as indicated. Solid line shows we are far from understanding even the constituting ele- power law with the exponent 1 0.61. ments, the fluctuations of 1 at the wall. We seem to observe facial widths can be extracted to yield figures very similar to a coupling between critical wetting and some kind of surface Figs. 6 and 7. The amplitudes of the logarithmic divergences critical behavior of 1, the origin of which is unclear. can again be used to estimate the bulk correlation length Hence already our simple, highly idealized model exhibits b . From the mean interface position, one calculates 4.9 a complex and rather intriguing wetting behavior. In real 7 alloys, numerous additional complications are present which b /a0 5.8 8 , and from the interfacial width, b /a0 will lead to an even richer and more interesting phenomenol- 7.5 9 , in agreement with the values obtained for the ogy. For example, long range interactions are known to in- 110 surface. Likewise, the study of the layer susceptibili- fluence wetting transitions significantly. The effect of van ties at the interface does not offer new surprises. The der Waals forces on wetting has been investigated in detail.1 maxima of the layer-bulk susceptibilities for both 1 and 23 van der Waals forces are important in liquid-vapor systems grow according to a power law z (H0 H) 1. The layer- or binary fluids, but presumably irrelevant in alloys. Instead, layer susceptibility in the interfacial region seems to grow elastic interactions caused by lattice distortions presumably with a different exponent 0.6 like in the case of the 110 play an important role. surface , yet after subtracting the ``background'' the data are Furthermore, real surfaces are never ideally smooth, but also consistent with the theoretically expected behavior. have steps and islands. We have seen that the orientation of Last, we study how the surface value of the order parameter the surface affects the surface ordering. In our study, we did 23 evolves as the transition H0 is approached. Figure 21 not observe dramatic differences between the 110 surface shows that it vanishes according to a power law with the and the 100 surface. Nevertheless, we expect that the influ- exponent 1 0.61 2 , which is within the error the same ence of the surface orientation on the wetting behavior can exponent as in the case of the 110 surface. As far as the be quite substantial, e.g., in situations with strong surface surface behavior of 23 is concerned, the 100 and the 110 segregation, or if surface orientations are involved which surface are thus basically equivalent. The weak ordering ten- also break the symmetry with respect to the DO dency of 3 order e.g., 1 has an at most slightly perturbing effect on the the 111 surface . Likewise, we can expect that steps and profiles of 23 . islands will affect the ordering and the wetting properties of the alloy. It is well known in general that the wetting behav- V. SUMMARY AND OUTLOOK ior on corrugated or rough surfaces differs from that on We have presented an extensive Monte Carlo study of smooth surfaces.52­54 In addition, even a few steps or islands surface induced disorder in a simple spin lattice model for on an otherwise smooth, but symmetry breaking surface of bcc-based binary alloys. Our work complements earlier an alloy can have a dramatic effect on the ordering behavior, Monte Carlo simulations of Schweika et al.,28 who have since every step changes the sign of the ordering surface studied surface induced disorder in fcc-based alloys within a field. similar model. Like these authors, we observe critical wet- ACKNOWLEDGMENTS ting behavior with nonuniversal exponents. We have dis- cussed our results in terms of an effective interface model We wish to thank M. Mu¨ller and A. Werner for helpful designed to describe a system with several order parameters. discussions. F.F.H. acknowledges financial support by the In such a complex material, nonuniversal exponents may re- Graduiertenfo¨rderung of the Land Rheinland Pfalz, and F.S. sult both from fluctuation effects and from a competition of has been supported from the Deutsche Forschungsgemein- length scales. schaft through the Heisenberg program. PRB 61 SURFACE-INDUCED DISORDER IN BODY-CENTERED- . . . 15 091 1 For reviews on wetting see, e.g., P. G. de Gennes, Rev. Mod. 26 Ch. Ricolleau, A. Loiseau, F. Ducastelle, and R. Caudron, Phys. Phys. 57, 827 1985 ; S. Dietrich, in Phase Transitions and Rev. Lett. 68, 3591 1992 . Critical Phenomena, edited by C. Domb and J.L. Lebowitz 27 V. S. Sundaram, R. S. Alben, and W. D. Robertson, Surf. Sci. 46, Academic Press, New York, 1988 , Vol. 12; M. 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