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The atomic size effect in surface segregation
Thin Solid Films, 129 (1985) 161-180 GENERAL
161
FILM BEHAVIOUR
THE ATOMIC SIZE EFFECT IN SURFACE SEGREGATION A. D.
VAN
LANGEVELD
Department of Physics, Eindhoven Netherlands) (Received October
University of Technology,
17,1983; accepted
March
P.O. Box 513. 5600 MB Eindhoven
(The
11,1985)
The contribution of lattice strain in the process of surface segregation is treated in the present paper in a more consistent model than previously. Equilibrium conditions for surface segregation in terms of a single- and a two-layer model are demonstrated to depend on the effective regular solution parameter. It is also shown that the equilibrium conditions developed are a more general formulation of those already presented in the literature.
1. INTRODUCTION
The presence of the surface of solids contributes to the Gibbs free energy of the system considered (see for example ref. 1, Chapter 12). With most of the solids used in practice, the contribution of the surface free energy to the Gibbs free energy of the system as a whole can generally be considered to be negligibly small. However, with solids used in modern integrated circuit technology and in heterogeneous catalysis the surface free energy is rather high, since an important part of the atoms in the crystallites will be surface atoms. This is an unavoidable consequence of the extreme diminution in the size of the crystallites of the catalytic active metal and the miniaturization of the components in electronic semiconductor devices. In pure metals dispersed on a support the Gibbs free energy of the system can be lowered by (a) a decrease in the metal surface area by a growth of the mean crystallite size,(b) a change in the surface structure by for example faceting of the surface2s3 and (c) bond relaxation between surface atoms and their neighbours4@j. Single-phase alloys, or metals with traces of impurities, have an additional means of decreasing the surface free energy: surface segregation, as a result of which the composition of the (intercrystalline) surface can differ quite significantly from that of the bulk. Since many properties of alloys (or metals with trace impurities), such as their catalytic activity and selectivity, their mechanical strength and hardness and their electrical conductance, depend strongly on the composition of the (intercrystalline) surface, alloying offers an interesting possibility to modify the properties of pure metals. However, the negative consequences of the added or undesired admixture have to be considered too. One example illustrates that. 0040-6090/85/$3.30
0 Elsevier Sequoia/Printed
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A. D. VAN LANGEVELD
162
Recycled materials can contain other contaminants than the natural raw materials. For good quality control of the recycled products it is often important to know which of these contaminants does not lead to deterioration of the properties of the product. Obviously, for various reasons a fundamental knowledge of the process of surface segregation is of major importance for future applications of new materials. An appropriate way to prepare ultraclean alloys for fundamental surface studies is by the preparation of thin alloy films by evaporating metals onto a suitable substrate such as Pyrex, mica or quartz. An important advantage of this method of preparation is the easy cleaning procedure of the metals prior to the evaporation step, the short anneal period necessary to obtain equilibrium compared with that for massive foils and the relatively low costs since only small amounts of the pure metals have to be used. The theoretical basis of surface segregation in liquid solutions was derived about a century ago by Gibbs’. When solids are studied, then in order to obtain practical results the broken bond approximation is generally used in surface segregation calculations * l1 According to this mode1 bonds are formulated in terms of a bond between two atoms (i.e. a pairwise binding energy), in the nearestneighbour approximation as well as in the next-nearest-neighbour model. In the former, only bonds between nearest neighbours are taken into account, and the binding energy cii of a bond between like atoms can be calculated from the enthalpy AHSubi of sublimation of the pure component and the number Z of nearestneighbour atoms : s..II = -I
2 AHSUb. Z
(1)
It should be noted that, since AHsubi > 0, cii (and analogously ciio) values are all negative by definition and are expressed in energy units per mole of bonds, formulated below as kilocalories per mole. The binding energy of the pairs of unlike atoms can be related to the enthalpy of mixing on the formation of the solution or to the activity coefficients of both components in the solution”. Generally, calculations on surface segregation can be performed in various approximations: (a) the idea1 solution model; (b) the zeroth-order regular solution model, as presented for example by Guggenheimi3; (c) the first-order approximation of the regular solution model as presented for example by Brongersma et ~1.‘~ The most important limitations of the models mentioned are the following assumptions. (i) In approximations (a) and (b) a random distribution of the atoms of both components over the available lattice points is assumed, while in model (c) shortrange order is taken into account. (ii) “The molecules (in our case the atoms of an alloy) are sufficiently alike in size and shape to be interchangeable in a lattice or a quasi-lattice.” (Ref. 13, p. 31.) The latter limitation was apparently considered not to be of major importance for liquid solutions, since Guggenheim acknowledged size differences of up to about 25’7: in the diameter of spherical molecules without, however, making
163
THE ATOMIC SIZE EFFECT IN SURFACE SEGREGATION
corrections for it (ref. 13, p. 24). Probably for this reason the difference in the size of the atoms of both components for solid solutions of metals was generally also neglected. However, especially in solid solutions a difference in the size of the atoms may have important consequences for the thermodynamics of these systems (ref. 1, p. 116; ref. 2) and consequently for the process of surface segregation, as will be demonstrated below. 2.
THE MODELS
AS PRESENTED
IN THE LITERATURE
TO TAKE A DIFFERENCE
IN THE
ATOMIC RADII INTO ACCOUNT
2.1. The correction as proposed by Miedema Recently, Miedema is has proposed to correct the enthalpy of sublimation of the component with the larger atoms in an alloy by multiplying its value by the ratio of the cross-sectional areas of the smaller and the larger atoms. His argument was that larger atoms expose a larger surface area in the alloy, as represented in Fig. 1.
Fig. 1. A schematic representation proposed by Miedema”.
of an oversized
A atom in the surface
of a matrix
of B atoms,
as
However, in the framework of the broken bond approximation the difference in the energy of an atom in the surface and in the bulk of a metal is simply caused by the number of missing neighbours of a surface atom, and the surface area is of no importance to iti as long as both types of atoms are sufficiently alike in size to be interchangeable on their lattice points. In addition, the correction in the form as proposed will also meet some other fundamental problems. First, it influences the energy sAAfor all bonds of the pairs of larger atoms in the surface as well as in the bulk. This is caused by the fact that the sublimation enthalpy in the final equation used in surface segregation calculations is composed of the contributions of the partial molar enthalpies of component A in the bulk and in the surface. Secondly, the question arises why the binding energy sAAof the pairs of larger atoms is corrected and not the energy .sBBof the smaller atoms, since the choice is rather arbitrary. In addition, changes in sAA(caused by the correction) also cause changes in Ebb, since .sABis represented by the regular solution parameter 52in the calculations; this parameter is numerically calculated by assuming that
A more consistent approach (but still not correct as we shall argue later) would correct the bonds of all pairs of atoms in the surface. In doing so, the bond strength sABcould be corrected by the mean cross-sectional area of both types of atoms. In the latter case, however, all bonds in the surface will be expressed in energy units per
164
A. D.
VAN
LANGEVELD
mole of bonds per unit surface area, while all contributions in the bulk are formulated as energy units per mole of bonds. Evidently, fundamental problems are now encountered, since the partial molar enthalpy contributions from the bulk and the surface will have different dimensions. Of course, when the difference in energy of a single atom in the surface and in the bulk of a metal is derived from the surface tension or the surface free energy, the cross-sectional area of an atom has to be considered, since the data obtained experimentally are generally expressed in terms of energy per surface area (see for example ref. 17). In any case, the correction proposed by Miedema for the broken bond model will face fundamental problems, and it can only be used in a semi-empirical model. 2.2. Luttice strain relaxation on segregation Friedel McLeani8 derived the relation E
= _ 24nGKr3(r, cl
-r,,)’
(3K +4G)r,’
uccording to the theory of‘ McLean
and
(2)
for the elastic energy E,, released when an odd-sized atom is transferred from the bulk onto the strain-free surface of the alloy. In this equation K is the bulk modulus of the solute, G is the shear modulus of the solvent and rO and r, are the radii of the solvent and the solute atoms in their corresponding pure elements respectively. An important disadvantage of this relation is the presence of the parameter r, the actual radius of the solute atom in the alloy, which is not known u priori. Friedel” eliminated r by minimizing the energy of the system. His final equation is =
E
el
_24~GKrorl(r,-ro12 3Kr,
+ 4Gr,
It should be noted that according to both equations E,, is always negative; strain causes the solute component to segregate to the surface irrespective of the size. Various researchers* 5~20~22 ha ve used these relations to predict the segregation in dilute binary alloys in which the atomic radii of the two components differed, but the predictions were not always successful. Apart from some practical problems, e.g. that the calculations can only be performed for dilute solutions, there are also some fundamental objections against the formulations as presented above. First, it was assumed in the derivation of eqns. (2) and (3) that the lattice strain caused by the difference in the atomic radii fully relaxes when an odd-sized atom segregates to the grain boundary of the crystallite or the outermost surface of the sample. It is self-evident, however, that for an atom in the surface (as is assumed below and elsewhere in the model description of the surface segregation reaction) only part of the elastic strain energy is released. Secondly, it was assumed that the odd-sized solute atom behaves as a spherical particle with a compressibility corresponding to that of the bulk of the pure element and that the cavity of the solvent has an isotropic character. Of course, on an atomic scale neither the solute nor the solvent will behave isotropically. In addition, it is highly questionable whether bulk elastic constants of the pure elements can be used for calculations on the strain energy of isolated solute atoms’*.
THE ATOMIC
SIZE EFFECT
IN SURFACE
165
SEGREGATION
The last and most important drawback of this correction is that it does not discriminate between oversized and undersized solute atoms. It is generally accepted, however, that a compression of a bond length causes a larger increase in the energy of the system than an expansion by the same amount. Probably for the various reasons mentioned above many of the surface segregation calculations based on eqns. (2) and (3) resulted in incorrect quantitative predictions for the segregating component. 2.3. The interatomic potentialfunction in surface segregation calculations Tsai et a1.23 and Abraham and coworkers 24*25have extensively discussed the use of an interatomic potential function in surface segregation calculations concerning dilute binary alloys. The calculations were based on a change in the effective strength of the bonds between the solute and solvent atoms due to a compression (or expansion) of these bonds between the oversized (or undersized) solute atoms and their surrounding matrix. The effective bond strength of the pairs of atoms was formulated as a product of a Lennard-Jones 6-12 potential function and a bond strength of the pairs of atoms at their corresponding equilibrium, i.e. strain free, distance Rijo. Using the pairwise correction for lattice strain, Tsai et a1.23 and Abraham and coworkers24S25 succeeded in making correct predictions for the segregating component in a large number of dilute alloys with a widely varying ratio of the radii of the solute and the solvent atoms. However, they made their predictions again only for very dilute alloys. 3.
THE
REGULAR
COMPONENTS
SOLUTION
HAVE DIFFERENT
MODEL MOLAR
EXTENDED
TO
SYSTEMS
OF
WHICH
THE
VOLUMES
In principle this model is an extension of the calculations of Tsai et al. and Abraham and coworkers to more concentrated alloys. An analysis of the available literature (see earlier) revealed that it should not be particularly difficult to modify the regular solution model in such a way that it allows the molar volumes of both pure components to differ. Using the potential functions as presented by Tsai et a1.23 and Abraham and coworkers24,25 an attempt will be made to combine both approaches. For simplicity an additional assumption with regard to the mean lattice point distance in the three-dimensional point lattice of the alloy will be made. Although the assumption does not always hold exactly for bimetallic alloys, it will be assumed that the interatomic distance R, in the alloy is determined by the bulk composition of the alloy, i.e. by the Vegard law: R, = x,,~R,,’
+ xbBRBBo
(4)
where xbAand xbBare the bulk molar fractions of A and B respectively and RAA” and R BB ’ are the lattice point distances in both pure components. Also, as a consequence of this assumption, the volume effect that occurs on mixing equals zero, and hence the enthalpy H of the system equals the energy E. The next point to be discussed is the choice of the potential function tiij for the interaction between atoms i and j to be used in the calculations. In the literature various types of potential functions have been proposed by Morse, Mie, Lennard-
166
A. D. VAN LANGEVELD
Jones etc. Since the choice is somewhat arbitrary in any case, a Lennard-Jones 6-12 potential function will be used throughout, since it has already been shown to be convenient in surface segregation calculations on dilute binary alloys” 25. The only shape-determining parameters of it are the intrinsic equilibrium distance Rijo and the intrinsic binding energy sijo at the same distance. The intrinsic equilibrium distance Rijo of a pair of atoms will be defined by RijO = riO+rjO
(5)
where rio and rjo are the atomic radii in the pure elements i and j respectively. Subsequently, the effective binding energy qj(Rb) of the pairs of atoms is formulated as follows:
(6) Alternatively, the use of a shorthand function yields
notation
for the Lennard-Jones
6612 potential
Q~(RJ = qjo$ij
(7)
A representation of the effective interatomic binding energy as a function of the interatomic distance is presented in Fig. 2. It should be noted that, as a consequence of the formulation as presented above, the effective binding energy Ebb for all bonds will be a function of the bulk alloy composition.
1 Fig. 2. The effective binding energy qj(R,) of a pair of atoms as a function atomic distance RJR,,‘, Rijo being the intrinsic equilibrium distance.
of the normahzed
inter-
167
THE ATOMIC SIZE EFFECT IN SURFACE SEGREGATION
The basic characteristics of the regular solution model presented below are as follows. (1) The broken bond model will be used throughout, i.e. only bonds between nearest-neighbour atoms are assumed to be of relevance. (2) The solution is assumed to be single phase with a random distribution of all atoms over the lattice points. Consequently, the numbers of the various types of bonds can be calculated by means of simple statistics. Also, the ideal configurational entropy contribution can be calculated simply. It should be noted that the excess configurational entropy contribution and the excess internal entropy contribution are neglected in the present model. (3) The distance between the lattice points is assumed to be determined by the Vtgard law. (4) The binding energies of the pairs of atoms are assumed to be a product of a binding energy sijo at the intrinsic equilibrium distance Rijo and the Lennard-Jones 6-12 potential function tiij. (5) A non-ideal behaviour with respect to the various effective binding energies will be taken into account:
The enthalpy Hb of the bulk of such a solution consisting of 1 mol of lattice points can be formulated as follows: Hb = z{*x,2&,,0 +A* + x,(1 -x&*B”+AB
+3(1-
xb)2sBBo+BB >
(8)
where Z is the number of nearest-neighbour atoms in the bulk and xt, stands for the bulk molar fraction of component A. As already mentioned, sAAo,sBBoand .sABoare all negative by definition. However, in some cases the effective binding energy sijotjij can become positive as a result of a compression of the bond. It can be rationalized that when 1.1 > rAo/rgo > 0.89 (of course for the specific case that a Lennard-Jones 6-12 function is used) all binding energies will be negative and consequently Hb < 0 i2. Of course, H,, equals zero when the numerical values of the effective binding energies cancel. The binding energy .sABocan be derived from experimental data related to the enthalpy of the bulk of the solution, i.e. from the enthalpy AHmiX of mixing or from the activity coefficients Y* and yB. Problems arising in these calculations when the molar volumes of the two components differ will be discussed elsewhere’ 2. 4.
SURFACE SEGREGATION
4.1. Surface segregation in a regular solution as deJined earlier Because of relaxation effects, changes in the interatomic distance, the shape of the interatomic potential function and the intrinsic binding energy in the topmost layers of atoms can be expected to occur. Clearly, these effects are related. In order to perform surface segregation calculations on a system as defined in Section 3, the following assumptions concerning the surface phase of the alloy will be made. (1) The lattice point distance in the surface phase equals that in the bulk of the alloy.
168
A. D. VAN LANGEVELD
(2) The potential function in the surface of the alloy equals that in the bulk. (3) Because of relaxation effects, the binding energy of the atoms in the topmost atomic layer to their neighbouring atoms will increase by a factor 1 + 6. (4) The number of atomic layers that the surface phase consists of depends on the numerical value of the effective regular solution parameter Q(R,). The following remarks relate to these assumptions. (1) It is generally accepted that in pure elements and alloys the interatomic distances in the surface differ from that of the bulk26. However, for reasons of simplicity this difference will be neglected. Recent results of Sachtler et aL2’ suggest our assumption to be reasonable, even for alloys in which the radii of the two types of atoms differ by a maximum of about loo! and in which the surface composition differs from that of the bulk. (2) In fact this assumption is an evident simplification. Bohnen5 has recently demonstrated in qualitative terms that the shape of the interatomic potential function in the surface of a pure metal is different from that in the bulk. Since quantitative data on this effect are not yet available in the literature, the effect will be neglected for the moment. (3) Data in the literature for this correction are also rather scarce. Since the correction is easily introduced, it has nevertheless been introduced below in the various equations on the surface segregation reaction. It should be noted that the relaxation parameter S is assumed to be equal for the various types of bonds between the surface atoms and their neighbouring atoms. which is also a simplification. (4) As demonstrated by Defay and Prigogine2s, only for binary solutions in which the enthalpy of the exchange reaction A-A + B-B +Z 2ApB equals zero can a single-layer model be used to describe in a formally correct way the surface segregation reaction. In such a single-layer model it is assumed that only the composition of the topmost layer of atoms or molecules differs from that of the bulk. For an ideal system the effective regular solution parameter Q(R,) which is formulated as Q(%)
= &AB(Rb)~lil&AA(Rb)f&BB(Rb)}
(9)
equals zero. For non-ideal solutions the single-layer model is in principle not correct and only leads to an approximate value of the surface composition. It should be stressed that the enthalpy of mixing in the formation of an alloy (which can be related to a function containing the effective regular solution parameter) does not necessarily equal zero when Q(R,) = 0 and vice versa. Since the most important property of a solution is the interchange energy a(&), which determines amongst other things the number of atomic layers that the surface phase consists of and whether a zeroth-order or a first-order regular solution model (see ref. 13) should be used to describe the solution correctly, the terminology ideal solution will be reserved exclusively for those systems in which L&R,) = 0. When !2(R,) # 0,and the surface phase is assumed to consist of the two topmost layers of atoms or molecules, it can be demonstrated that the composition of the second layer generally differs only slightly from that of the bulk (indicating that the two-layer model is a
THE ATOMIC
SIZE EFFECT
IN SURFACE
169
SEGREGATION
reasonable approximation). However, the composition of the topmost atomic layer as calculated from this two-layer model can differ quite significantly from that for the single-layer model. In contrast, the surface free energy of the solution seems to be influenced only moderately by the choice of the number of atomic or molecular layers in the surface phase ‘* . Finally it should be remarked that, in those cases when the single-layer model predicts only’a moderate surface segregation for solutions in which Q(R,) # 0, the two-layer model will predict a composition of the topmost atomic or molecular layer which will be very close to that for the single-layer model. From the arguments above it can be concluded that surface segregation in binary solutions can be predicted quantitatively using either a single-layer or a twolayer model, depending on the characteristics of the solution. The attractiveness of the single-layer model is that it is relatively simple to make numerical calculations with it, while the calculations of the various equations for two-layer models, which will result in a better quantitative prediction for some solutions, are generally much more complicated. Pioneering work in the field of multilayer model calculations on alloys, in which the atomic radii of the two components are assumed to be equal, has been presented by Williams and Nason”. 4.2. Surface segregation in a single-layer model The derivation of the equation determining the equilibrium of segregation in a single-layer model is easy to perform. The basis is a consideration of the surface segregation as a chemical reaction : A(bulk) + B(surf) = A(surf) + B(bulk) The equilibrium of segregation is determined by the minimum of the total Gibbs free energy G, which is the sum of the contributions of the bulk (G,,) and the surface (G,) phases: G, = G,+G,
(10) When the minimum of G, is sought it is also necessary to take into consideration the additional conditions which make the variables dependent. The minimum of G, is determined by the condition =, =, ax, ax, and for semi-infinite systems this reduces to
(11)
8% % (12) ax, ax, i.e. at equilibrium the chemical potential of a component in the surface equals that for the bulk. The Gibbs free energy of the bulk can easily be calculated when the enthalpy of the bulk mentioned earlier (eqn. (8)) is used. The Gibbs free energy of the surface can be formulated as G, = (1 +~)Z,{+X,~E~~~ ti*A+Xlu +(I +~)Z”cx,xb&**“~**+{xl(l
-xl)k3°~*B+~(1 -x,)+xt.U
+(1-x,)(l-~,)s,,~$,,]+RT{~~lnx~+(1-x~)ln(l-x~)}
-X1)2G3oh3B~
+
--x,)hOhJ+
(13)
170
A. D. VAV LANGEVELD
where Z, and plane parallel Introduction results in the molar fraction
Z, are respectively the numbers of nearest neighbour atoms in the to the surface and above or below the plane parallel to the surface. of the free energy of the surface and the bulk into eqn. (12) finally following relation which can be used to calculate the equilibrium of component A in the topmost atomic layer:
ICIAA){z-(l+az, +Z,)) 9&BB0$B” - &AA0 x I%-(1
+@(Z,x, +-Z&J-${Z-(1$6)(Z,
-Z{3x,2c*,oIc/,*‘+xb(l
-%Ja,,“~*R’+~(l
+2wq
x +Z,.))]-
-xb)2&riB0~rni’} + (14)
where tiij’ represents
the derivative
of tiij with respect to xb, i.e.
(15) Since the physical meaning of the various contributions in eqn. (14) is obscured by the complicated form, the relation will now be simplified and analysed in the simplified form. First, it will be assumed that the surface relaxation parameter 6 = 0, and also that rAo = rgo (as a consequence of the latter, tiij = 1 and tiij = 0 for all bonds). The resulting equation is identical with that presented by for example Wynblatt and Ku’0321 and Seah22:
It should be noted that since all atoms are now always at their intrinsic equilibrium distance, the superscript zero to the binding energies has been omitted. If it is further assumed that the system behaves ideally, i.e. 52 = 0, eqn. (16) reduces to (17) 01
x1( 1 -xb) =
x,(1 -.x1)
exp
-
.a& BB
-
E&4)
2RT
WV
Clearly, in eqns. (14), (16) and (17) the term
represents the ideal entropy of mixing contribution of the surface and the bulk phases. The first term in all the expressions represents the difference in the binding energy (in energy per mole of bonds) of the pairs of like atoms, which in eqns. (16), (17) and (17a) can be related to the difference in the enthalpy of sublimation. The term containing G(R,) obviously represents the influence of the non-ideal behaviour of the solution. The last term in eqn. (14) collects all derivatives which are present since lcIij is a function of xb.
171
THE ATOMIC SIZE EFFECT IN SURFACE SEGREGATION
The results of some calculations on a single-layer model as described above are presented in Fig. 3. The calculations were performed on an f.c.c. structure of a hypothetical alloy system of which the difference in the enthalpies of sublimation equals zero and AHmiX= 0, i.e. in the case that R,,’ = R,,’ the solution behaves ideally. In these calculations it is assumed that the atoms of component A, of which the molecular fraction is indicated along the axes, are the larger ones. In the case that R O = R,,’ the surface composition equals that of the bulk. However, when a digrence in the radii is introduced, the solute is always expelled to the surface (for dilute solutions). It should also be noted that, as a consequence of the asymmetry of the potential function used, the larger atoms are always expelled to the surface over a wider range of concentrations than the smaller atoms. 1.0
0
0.5
1.0 -x
b
alloy system with a varying ratio RAAo/RBBo: ,R,,'= 1.05R,,";...,R,,o = l.lOR aa’. The intrinsic binding energies aijo of all pairs R AA'- RBBo.--1 are assumed to be equal, the enthalpy of sublimation of both pure components being 100 kcal mol- r. The calculations were performed for a temperature of 750 K on a (111) crystallographic plane on an f.c.c. lattice, the surface relaxation parameter 6 being assumed to be zero. Fig. 3. The surface
composition
of a hypothetical
As also argued elsewhere12, strain not only affects the thermodynamics of a binary solid solution in surface segregation but also all parameters related to the enthalpy of the bulk of the solution, e.g. the enthalpy of mixing. This effect has been neglected in the calculations discussed above. In a consistent approach, however, strain effects have to be taken into account in the evaluation of &ABofrom the enthalpy of mixing data. In Fig. 4 the results are presented for the system described
172
A. D. “AN LANGEVELD
0
0.5
1.0 -x
b Fig. 4. The surface composition of a (11 I) plane of a hypothetical alloy system with the effect of lattice strain on AH”‘” taken into account (--) and neglected (. .). The enthalpy of sublimation of both pure elements is 100 kcal mol-’ and the enthalpy of mixing is zero. In the calculations an equilibration temperature of 750 K was assumed.
above with RAAo/RBBo = 1.O and 1.1, in which the strain effects on AH mix are taken into account and are neglected. Clearly the differences in the calculated surface composition using a consistent and a non-consistent approach cannot be neglected. It should be stressed here that the differences between the consistent and the nonconsistent approach are caused only by the stronger intrinsic binding energy of the A-B pairs in the consistent approach. The stronger A-B band also results in a depletion of the smaller solute B at x,, _- 0.90; in the bulk a B atom has more energeticallyfavourable A neighbours than in the surface. It should be noted that the difference between no strain effect and the consistent approach in which the strain is taken into account is rather limited for this specific case. In other cases, however, the difference can be quite large, as will be shown later. In Fig. 5 the results of calculations on the (111) plane of the Cu-Au system, in which the atomic size effect is taken into account and is neglected, are presented. In these calculations the enthalpies of sublimation derived from the Handbook qf Chemistry and Physics 33 have been used, resulting in sAuAuo = - 14.7 kcal mol ’ and cCuCuo = - 13.4 kcal mol ‘. The binding energy of the pairs of Cu-Au atoms was derived from Miedema’ 5, resulting in an estimated value of - 18.1 kcal molt ’ for EcuAuo. Experimental data derived from the literature29P32 are also presented. Despite the relatively large scatter in the experimental data, the model in which lattice strain is taken into account, as presented in this paper, is in better agreement
THE ATOMIC SIZE EFFECT
0
IN SURFACE
173
SEGREGATION
0.5
1.0 -x
b Fig. 5. The calculated surface composition of Au-Cu alloys as a function of the bulk molar fraction with lattice strain taken into account (--) and negelected (...). The calculations were performed for a temperature of 750 K and a (111) crystallographic plane on an f.c.c. structure. Experimental data are shown for comparison: +, Losch and Kirschnerz9; 0, McDavid and Fain3’; X, van Santen et ~1.~~; q , Sparnaay and Thomas3’.
with these data than is the more simple model in which atomic size effects are neglected*. It should also be noted that, according to t-heresults of the simple model, the solute is depleted from the surface at both ends of the diagram. As above, this effect is the result of the extremely strong bond between pairs of unlike atoms. The influence of the numerical values of other parameters, such as the surface relaxation parameter, the difference in the enthalpy of sublimation and the enthalpy of mixing on the surface composition of binary solutions, will not be discussed here, since in those respects the model presented in this paper behaves as a conventional regular solution. For an extensive discussion on these points we refer to Williams and Nason”. 4.3. The surface free energy of an alloy within theframework of the single-layer model In order to calculate the difference in the Gibbs free energy of the various crystallographic planes of an equilibrated alloy, eqn. (13) can be used. However, for a comparison all energies have to be divided by the surface area of 1 mol of lattice points since the Gibbs free energy of the surface phase is calculated for 1 mol of lattice points. * Note added in proof.Also, the results of these calculations those based on an electronic theory39.
are in excellent quantitative
agreement
with
174
A. D.
VAN
LANGEVELD
Of course, a more elegant parameter is the surface free energy 7, the free energy required to create the surface. It allows a direct comparison of the various crystallographic planes of an alloy mutually, and of the alloys with those of pure metals. In Fig. 6 the process of creating a surface from a bulk material has been represented schematically. In order to describe the process in terms of thermodynamics, the solid before the creation of the surface has been divided in three sections, the first of which (1) contains the future surface phase with a corresponding Gibbs free energy content Gb( I), while the two other sections (2 and 3) are and will stay bulk. After the creation of the surface, the energy content of the surface phase will be G,( I ).
G#)
G$)
G&l
Gb(2)
Fig. 6. A representation a bulk phase.
ofthe creation
ofa surface phase with a geometrical
surface area A (2 x +A) out of
By comparison with Fig. 6 it is self-evident that the change in the Gibbs free energy on the formation of a surface of area A (A being the geometric surface area of 1 mol of lattice points in the topmost atomic layer) is given by AG = G,( 1) + G,(2) + G,(3) - Gb( 1) - G,(2) - G,(3)
(18)
or, since 1’ = AC/A, ?A = Gs(1)-G,Cl)
(19)
Ofcourse, all bonds to the bulk that the layer is connected with also have to be taken into account in G,( 1) and Gb( 1). G,( 1) is the Gibbs free energy of a surface layer as given by eqn. (13) while Gb( 1) can simply be derived from eqn. (8) by adding the entropy contribution. The final equation is
YA = (1+~)(Z,{~X,~E,,~~,,+X~(~ -xx,)~,,~~,,+~(l -x~)~E~OIC/,~) +
+z”c~,x,~**“~,*+~-~~(~-x,)+x,(1 + ( 1-
x1
)( 1 - x&Bijo&J)
+:(1--~,)2E,,oll/,,}+RT{~1
-x,lnx,-(l-x,)ln(l-x,)}
-.x,)}hJ0h3+
- z” ,2Xb2h0$.4A
+
x,(1
-
%&4B0~,4R
+
lnx,+(l-xi)ln(l-xi)(20)
THE ATOMIC SIZE EFFECT
IN SURFACE
175
SEGREGATION
As mentioned above, eqn. (20) can be used to calculate the surface free energy once the various parameters are known. However, it can also be of use in making an estimate of the surface relaxation parameter when the surface free energy is accurately known. 4.4. Surface segregation in a two-layer model The conditions for equilibrium of this model are more complex, since the Gibbs free energy of the system has to be at its minimum as a function of two mutually dependent variables, namely x1 and x2, which both depend on xb. As in the singlelayer model above, the Lagrange method has to be used for that reason, leading to the following two conditions determining the equilibrium of segregation: dG,
aG,
-=o
(21)
ax, ax, aG,
dG,
-=o
(22)
ax, ax,
where G, and G, are the Gibbs free energies of the surface and the bulk phase respectively. Introduction of expressions for these two energies into eqns. (21) and (22) finally leads to the following relations containing x1 and x2 which have to be resolved : ~(&88°~BB--*Ao~AA){Z-_(1 x [Zx,-(1+6)(Z,x, --Wx,2&4011/*.4’+xb(l
+@V,
+&)I
+z,x,)-+{z-(l+6)(z, --&*B”$*e’++(l
+2Q2(&) x +Z,)}]
-
-xtJ2s,ri0~,,‘~+ (23)
The similarity of eqn. (23) of the two-layer model and eqn. (14) of the single-layer model should be noted. In fact, eqn. (23) can be reformulated into an equilibrium condition for a single-layer model by introducing x2 = xb. This is to be expected since the basic eqn. (12) for the equilibrium condition of the single-layer model is synonymous with the basic eqn. (21) of the two-layer model. Interesting systems to test the two-layer model presented above are the Pt-Cu and Pd-Cu alloy systems. Both form a continuous series of solid solutions, the atomic radii differ by about 7.5% and the effective regular solution parameter differs from zero as is demonstrated in Table I. The binding energy parameters of the various pairs of atoms are presented in Table II. The experimental results for the Pt-Cu 14*34*35and the Pd-Cu 36 alloy systems are compared with those from theoretical considerations in Fig. 7 and Fig. 8
176
A. D.
I THEEFFECTIVE
VAN
LANGEVELD
TABLE
FUNCTION
REGULAR
OF THE BULK
SOLUTION
PARAMETER
MOLAR FRACTION
-‘(h
-R(R,)
0. I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
2.2 2.4 2.6 2.8 3.0 3.3 3.6 3.Y 4.2
TABLE
OF THE
Pt-CU
ANU
THE
PddCu
ALLOY
SYSTEM
AS A
OF COPPER
(kcal (mol bonds))
‘)
1.8 I.9 1.9 2.0 2.1 2.2 2.3 2.4 2.5
II
THE BINDING
ENERGIES
E,,’
OF THE VARIOUS
PAIRS OF ATOMS AT THElR
INTRINSIC
EQUILIBRIUM
DISTANCE
RiiO Atom pair -E,,” (kcal (mot bonds)
‘)
PttPt 22.5
pttcu 19.9
CuPd 15.7
cu-cu 13.4
Pd-Pd 15.0
1.0 -
‘b
Fig. 7. The surface composition of PttCu alloys as a functton of the bulk composition at a temperature of 1000 K with lattice strain taken into account (upper full curve) and with neglect of the size effect (m m).All calculations were performed for a (111) crystallographic plane on an f.c.c. lattice. Experimental data are shown for comparison: 8, Brongersma et al.“‘; x, Kelley et a[.35; & van Langeveld and Pane?.
177
THE ATOMIC SIZE EFFECT IN SURFACE SEGREGATION
respectively. Clearly, for both alloy systems the experimental data are in good agreement with those from theoretical considerations in which the surface relaxation parameter 6 is assumed to be zero. For the Pt-Cu system the best fit between experimental data and theoretical considerations at 1000 K could be obtained by use of a surface relaxation parameter 6 = 0.05. Also, the introduction of next-nearest-neighbour interactions, as proposed by Gijzeman3’, leads to only minor corrections in the surface composition. The reason for this is that the creation of a consistent model in which next-nearest-neighbour interactions are taken into account will for the most part cause the effect to cancel itself out3s.
10
xs
I
05
0
10
05 -x
b
Fig. 8. The surface composition of Pd-Cu alloys as a function of the bulk composition at a temperature of 670 K with the difference in the atomic radii taken into account (-) and with neglect of the atomic size effect (---). Experimental data are shown for comparison: I, derived from Auger analysis; $, obtained from photoelectron work function measurements.
4.5. The surface free energy of an alloy within the framework of the two-layer model In a similar way to that in the single-layer model, the surface free energy is given by the relation YA = G,(l)-G,(l)
(251
The only difference is that the surface phase as well as the reference bulk phase consist of two atomic layers. Again all bonds in the surface layers and of the surface phase to the bulk have to be taken into account; the same holds for the calculation of the bulk reference state.
178
A. D.
VAN
LANGEVELD
The Gibbs free energy of the surface is given by G,(l) = (1+6)Z,{3x,2~,,oll/,,+x1(1 ++(I -x,)2%H3011/,,~+(l +1x1(1 -x,)+x2(1 + z,
{~~22E**ohA
-x,)E,,~$,,+ +6)z,[x,x,&,,01C/AA+
-.~,)h3°~AB+(1 +x2(1
+Z”Cx2.~,&**“~AA+{.~2(l
-x,)(1
-.u2)“BBo$Bs]+
- X2)E*Ro$,4B + )( 1 - .Y2)2C”BoIC/“”; +
-.YlJ+-)cb(l
+(l-x,)(l--~,)&,,O~,,I+RT(xl
-x,)}t.,,O$,,+
Inx,+(l-.x,)ln(l-x,)+
+x,lnx,+(l-x,)ln(l-.x2))
(26)
If use is made of the enthalpy of the bulk of an alloy, as given by eqn. (8) the surface free energy can be formulated as
?lA = (1 +~)(z,{~x,2~,,o~,,+x1(1 +z,[x,x2E,,0 +(I -x,)(i
-x&‘4B”$*B+*(1
$AA++,(1-X2)+x2(1
-_x,)5BBOf/JBBi+
-x,)}EA80$.4B+
-~~~~~~~“I(/~~I~+~1{3X~2~**oIC/**+.~2(1 -X2)E,4Bo$AB +
+~(l-~2)2~,,oll/,,}+ZyCX2X~E~Ao~*A+{X2(l-.Y~)+X~(l-X2)j x “.4B”~.4L3+(l
-x2)(1
-.%)E”““$mI
-2Z{~x,~e.4,0$,,
x s4B01ClAB + 3( 1 - %)2%3RotiBB)+RT{x, +x~ln.~-,+(1-x,)ln(l-.u2)-2x,lnx,+(1-x,)In(l-.u,))
x
+.x,(1 -Yb) x
Inx,+(l--x,)ln(l-.u,)+ (27)
The calculation of the surface free energy of various crystallographic planes can be of use to check their mutual stability. In pure metals the densely packed planes are more stable than the open planes. Since in alloys the surface segregation of the component with the lower surface energy is stronger the more open (rough) the surface is, it is necessary to investigate whether for alloys the rough surfaces are not more stable than the more densely packed surfaces. Equation (27) has been successfully used to demonstrate that of the (111) and (100) surfaces of Pt-Cu alloys the (111) faces are the more stabIej4. The last point is of particular importance for a comparison of model calculations and experimental data on polycrystalline surfaces which mainly consist of the most stable crystallographic planes. 5.
CONCLUSIONS
Lattice strain in binary alloys, caused by the difference in the atomic size of the two components, can seriously influence the equilibrium of surface segregation. The introduction of an interatomic potential function in the pairwise binding energy results in effective binding energies qj(Rb) of the pairs of atoms considered. A subsequent use of these effective binding energy parameters leads to equilibrium conditions for surface segregation in terms of a single- and a two-layer model. The equations determining the equilibrium of surface segregation are similar to those
THE ATOMIC SIZE EFFECT
IN SURFACE
SEGREGATION
179
already presented in the literature, the only difference being that in the derivation of the equations given in this paper the difference in the molar volumes of the two components is taken into account. Using the single-layer model, predictions were made for the Cu-Au system, while for the Cu-Pd and the Cu-Pt systems it was shown that the two-layer model has to be applied for model calculations. For all systems, experimentally obtained data are in reasonable agreement with the theoretical predictions presented in this paper. ACKNOWLEDGMENTS
The investigations were supported by the Netherlands Foundation for Chemical Research with financial aid from the Netherlands Organization for the Advancement of Pure Research. The author is grateful to Dr. B. E. Nieuwenhuys for stimulating discussions. REFERENCES
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
R. A. Swalin, Thermodynamics of Solids, Wiley, New York, 1962. C. Herring, Phys. Reo., 82 (1951) 87. J. van der Veen, Thesis, Delft University, 1981. A. Crucq, L. Degols, G. Lienard and A. Frennet, Surf Sci., 80 (1979) 78. K. P. Bohnen, Surf. Sci., 115 (1982) L96. R. P. Gupta, Phys. Rev. B, 23 (1981) 6265. J. W. Gibbs, Collected Works, Yale University Press, New Haven, CT, 1957. M. J. Kelley and V. Ponec, Prog. Surf Sci., II (198 I) 139. Y. W. Lee and H. I. Aaronson, Surf. Sci., 95 (1980) 227. V. S. Sundaram and P. Wynblatt, SurJ Sci., 52 (1975) 569. F. L. Williams and D. Nason, Surf. Sci., 45 (1974) 377. A. D. van Langeveld and A. J. Staverman, Z. Metallkd., 76 (1985) 54. A. E. Guggenheim, Mixtures, Clarendon, Oxford, 1952. H. H. Brongersma, M. J. Sparnaay and T. M. Buck, Surf Sci., 71(1978) 657. A. R. Miedema, Z. Metailkd., 69 (1978) 455. D. P. Woodruff, The Solid Interface, Cambridge University Press, 1973, p. 5. A. S. Skapski, J. Chem. Phys., 16 (1948) 389. D. McLean, Grain Boundaries in Metals, Clarendon, Oxford, 1957. J. Friedel, Ado. Phys., 3 (1954) 446. P. Wynblatt and R. C. Ku, Surf. Sci., 65 (1977) 511. P. Wynblatt and R. C. Ku, in W. C. Johnson and J. M. Blakely (eds.), Interfacial Segregation, American Society for Metals, Metals Park, OH, 1977. M. P. Seah, J. C&d., 57 (1979) 450. N. H. Tsai, G. M. Pound and F. F. Abraham, J. Caral., 50 (1977) 200. F. F. Abraham, N. H. Tsai and G. M. Pound, Surf Sci., 83 (I 979) 406. F. F. Abraham and C. R. Brundle, J. Vat. Sci. Technol., I8 (1981) 506. D. P. Woodruff, in D. A. King and D. P. Woodruff (eds.), The Chemical Physics of Solid Surfaces and Heterogeneous Caralysis, Vol. 1, Clean Solid Surfaces, Elsevier, Amsterdam, 198 1. J. W. A. Sachtler, M. A. van Hove, J. B. Biberian and G. A. Somorjai, Surf. Sci., 210 (1981) 19. R. Defay and I. Prigogine, Trans. Faraday Sot., 46 (1950) 199. W. Losch and J. Kirschner, J. VU. Sci. Technol., I5 (1978) 1541. J. M. McDavid and S. C. Fain, Jr., Surf. Sci., 52 (1975) 161. R. A. van Santen, L. H. Toneman and R. Bouwman, Surl Sci., 47 (1975) 64.
180
32
33 34 35 36 37 38 39
A. D. VAN LANGEVELD
M. J. Sparnaay and G. E. Thomas. Surf. Sci., 135 (1983) 184. Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL, 58th edn.. 1977-1978. A. D. van Langeveld and V. Ponec, Appl. SIG-[ Sri., 16 (1983) 405. M. J. Kelley, D. G. Swarzfager and V. S. Sundaram, J. Vat. Sci. Technol.. 16 (1979) 664. A. D. van Langeveld, H. A. C. M. Hendrickx and B. E. Nieuwenhuys, Thin Solid Films, 109 (198 13) 179. 0. L. J. Gijzeman, SW/. Sci., 150 (1985) I A. D. van Langeveld, Thesis, University of Leiden, 1983. C. A. Balseiro and J. L. MoranL6pez. Phj~. Rev. B. 2/ (1980) 349.
161
FILM BEHAVIOUR
THE ATOMIC SIZE EFFECT IN SURFACE SEGREGATION A. D.
VAN
LANGEVELD
Department of Physics, Eindhoven Netherlands) (Received October
University of Technology,
17,1983; accepted
March
P.O. Box 513. 5600 MB Eindhoven
(The
11,1985)
The contribution of lattice strain in the process of surface segregation is treated in the present paper in a more consistent model than previously. Equilibrium conditions for surface segregation in terms of a single- and a two-layer model are demonstrated to depend on the effective regular solution parameter. It is also shown that the equilibrium conditions developed are a more general formulation of those already presented in the literature.
1. INTRODUCTION
The presence of the surface of solids contributes to the Gibbs free energy of the system considered (see for example ref. 1, Chapter 12). With most of the solids used in practice, the contribution of the surface free energy to the Gibbs free energy of the system as a whole can generally be considered to be negligibly small. However, with solids used in modern integrated circuit technology and in heterogeneous catalysis the surface free energy is rather high, since an important part of the atoms in the crystallites will be surface atoms. This is an unavoidable consequence of the extreme diminution in the size of the crystallites of the catalytic active metal and the miniaturization of the components in electronic semiconductor devices. In pure metals dispersed on a support the Gibbs free energy of the system can be lowered by (a) a decrease in the metal surface area by a growth of the mean crystallite size,(b) a change in the surface structure by for example faceting of the surface2s3 and (c) bond relaxation between surface atoms and their neighbours4@j. Single-phase alloys, or metals with traces of impurities, have an additional means of decreasing the surface free energy: surface segregation, as a result of which the composition of the (intercrystalline) surface can differ quite significantly from that of the bulk. Since many properties of alloys (or metals with trace impurities), such as their catalytic activity and selectivity, their mechanical strength and hardness and their electrical conductance, depend strongly on the composition of the (intercrystalline) surface, alloying offers an interesting possibility to modify the properties of pure metals. However, the negative consequences of the added or undesired admixture have to be considered too. One example illustrates that. 0040-6090/85/$3.30
0 Elsevier Sequoia/Printed
in The Netherlands
A. D. VAN LANGEVELD
162
Recycled materials can contain other contaminants than the natural raw materials. For good quality control of the recycled products it is often important to know which of these contaminants does not lead to deterioration of the properties of the product. Obviously, for various reasons a fundamental knowledge of the process of surface segregation is of major importance for future applications of new materials. An appropriate way to prepare ultraclean alloys for fundamental surface studies is by the preparation of thin alloy films by evaporating metals onto a suitable substrate such as Pyrex, mica or quartz. An important advantage of this method of preparation is the easy cleaning procedure of the metals prior to the evaporation step, the short anneal period necessary to obtain equilibrium compared with that for massive foils and the relatively low costs since only small amounts of the pure metals have to be used. The theoretical basis of surface segregation in liquid solutions was derived about a century ago by Gibbs’. When solids are studied, then in order to obtain practical results the broken bond approximation is generally used in surface segregation calculations * l1 According to this mode1 bonds are formulated in terms of a bond between two atoms (i.e. a pairwise binding energy), in the nearestneighbour approximation as well as in the next-nearest-neighbour model. In the former, only bonds between nearest neighbours are taken into account, and the binding energy cii of a bond between like atoms can be calculated from the enthalpy AHSubi of sublimation of the pure component and the number Z of nearestneighbour atoms : s..II = -I
2 AHSUb. Z
(1)
It should be noted that, since AHsubi > 0, cii (and analogously ciio) values are all negative by definition and are expressed in energy units per mole of bonds, formulated below as kilocalories per mole. The binding energy of the pairs of unlike atoms can be related to the enthalpy of mixing on the formation of the solution or to the activity coefficients of both components in the solution”. Generally, calculations on surface segregation can be performed in various approximations: (a) the idea1 solution model; (b) the zeroth-order regular solution model, as presented for example by Guggenheimi3; (c) the first-order approximation of the regular solution model as presented for example by Brongersma et ~1.‘~ The most important limitations of the models mentioned are the following assumptions. (i) In approximations (a) and (b) a random distribution of the atoms of both components over the available lattice points is assumed, while in model (c) shortrange order is taken into account. (ii) “The molecules (in our case the atoms of an alloy) are sufficiently alike in size and shape to be interchangeable in a lattice or a quasi-lattice.” (Ref. 13, p. 31.) The latter limitation was apparently considered not to be of major importance for liquid solutions, since Guggenheim acknowledged size differences of up to about 25’7: in the diameter of spherical molecules without, however, making
163
THE ATOMIC SIZE EFFECT IN SURFACE SEGREGATION
corrections for it (ref. 13, p. 24). Probably for this reason the difference in the size of the atoms of both components for solid solutions of metals was generally also neglected. However, especially in solid solutions a difference in the size of the atoms may have important consequences for the thermodynamics of these systems (ref. 1, p. 116; ref. 2) and consequently for the process of surface segregation, as will be demonstrated below. 2.
THE MODELS
AS PRESENTED
IN THE LITERATURE
TO TAKE A DIFFERENCE
IN THE
ATOMIC RADII INTO ACCOUNT
2.1. The correction as proposed by Miedema Recently, Miedema is has proposed to correct the enthalpy of sublimation of the component with the larger atoms in an alloy by multiplying its value by the ratio of the cross-sectional areas of the smaller and the larger atoms. His argument was that larger atoms expose a larger surface area in the alloy, as represented in Fig. 1.
Fig. 1. A schematic representation proposed by Miedema”.
of an oversized
A atom in the surface
of a matrix
of B atoms,
as
However, in the framework of the broken bond approximation the difference in the energy of an atom in the surface and in the bulk of a metal is simply caused by the number of missing neighbours of a surface atom, and the surface area is of no importance to iti as long as both types of atoms are sufficiently alike in size to be interchangeable on their lattice points. In addition, the correction in the form as proposed will also meet some other fundamental problems. First, it influences the energy sAAfor all bonds of the pairs of larger atoms in the surface as well as in the bulk. This is caused by the fact that the sublimation enthalpy in the final equation used in surface segregation calculations is composed of the contributions of the partial molar enthalpies of component A in the bulk and in the surface. Secondly, the question arises why the binding energy sAAof the pairs of larger atoms is corrected and not the energy .sBBof the smaller atoms, since the choice is rather arbitrary. In addition, changes in sAA(caused by the correction) also cause changes in Ebb, since .sABis represented by the regular solution parameter 52in the calculations; this parameter is numerically calculated by assuming that
A more consistent approach (but still not correct as we shall argue later) would correct the bonds of all pairs of atoms in the surface. In doing so, the bond strength sABcould be corrected by the mean cross-sectional area of both types of atoms. In the latter case, however, all bonds in the surface will be expressed in energy units per
164
A. D.
VAN
LANGEVELD
mole of bonds per unit surface area, while all contributions in the bulk are formulated as energy units per mole of bonds. Evidently, fundamental problems are now encountered, since the partial molar enthalpy contributions from the bulk and the surface will have different dimensions. Of course, when the difference in energy of a single atom in the surface and in the bulk of a metal is derived from the surface tension or the surface free energy, the cross-sectional area of an atom has to be considered, since the data obtained experimentally are generally expressed in terms of energy per surface area (see for example ref. 17). In any case, the correction proposed by Miedema for the broken bond model will face fundamental problems, and it can only be used in a semi-empirical model. 2.2. Luttice strain relaxation on segregation Friedel McLeani8 derived the relation E
= _ 24nGKr3(r, cl
-r,,)’
(3K +4G)r,’
uccording to the theory of‘ McLean
and
(2)
for the elastic energy E,, released when an odd-sized atom is transferred from the bulk onto the strain-free surface of the alloy. In this equation K is the bulk modulus of the solute, G is the shear modulus of the solvent and rO and r, are the radii of the solvent and the solute atoms in their corresponding pure elements respectively. An important disadvantage of this relation is the presence of the parameter r, the actual radius of the solute atom in the alloy, which is not known u priori. Friedel” eliminated r by minimizing the energy of the system. His final equation is =
E
el
_24~GKrorl(r,-ro12 3Kr,
+ 4Gr,
It should be noted that according to both equations E,, is always negative; strain causes the solute component to segregate to the surface irrespective of the size. Various researchers* 5~20~22 ha ve used these relations to predict the segregation in dilute binary alloys in which the atomic radii of the two components differed, but the predictions were not always successful. Apart from some practical problems, e.g. that the calculations can only be performed for dilute solutions, there are also some fundamental objections against the formulations as presented above. First, it was assumed in the derivation of eqns. (2) and (3) that the lattice strain caused by the difference in the atomic radii fully relaxes when an odd-sized atom segregates to the grain boundary of the crystallite or the outermost surface of the sample. It is self-evident, however, that for an atom in the surface (as is assumed below and elsewhere in the model description of the surface segregation reaction) only part of the elastic strain energy is released. Secondly, it was assumed that the odd-sized solute atom behaves as a spherical particle with a compressibility corresponding to that of the bulk of the pure element and that the cavity of the solvent has an isotropic character. Of course, on an atomic scale neither the solute nor the solvent will behave isotropically. In addition, it is highly questionable whether bulk elastic constants of the pure elements can be used for calculations on the strain energy of isolated solute atoms’*.
THE ATOMIC
SIZE EFFECT
IN SURFACE
165
SEGREGATION
The last and most important drawback of this correction is that it does not discriminate between oversized and undersized solute atoms. It is generally accepted, however, that a compression of a bond length causes a larger increase in the energy of the system than an expansion by the same amount. Probably for the various reasons mentioned above many of the surface segregation calculations based on eqns. (2) and (3) resulted in incorrect quantitative predictions for the segregating component. 2.3. The interatomic potentialfunction in surface segregation calculations Tsai et a1.23 and Abraham and coworkers 24*25have extensively discussed the use of an interatomic potential function in surface segregation calculations concerning dilute binary alloys. The calculations were based on a change in the effective strength of the bonds between the solute and solvent atoms due to a compression (or expansion) of these bonds between the oversized (or undersized) solute atoms and their surrounding matrix. The effective bond strength of the pairs of atoms was formulated as a product of a Lennard-Jones 6-12 potential function and a bond strength of the pairs of atoms at their corresponding equilibrium, i.e. strain free, distance Rijo. Using the pairwise correction for lattice strain, Tsai et a1.23 and Abraham and coworkers24S25 succeeded in making correct predictions for the segregating component in a large number of dilute alloys with a widely varying ratio of the radii of the solute and the solvent atoms. However, they made their predictions again only for very dilute alloys. 3.
THE
REGULAR
COMPONENTS
SOLUTION
HAVE DIFFERENT
MODEL MOLAR
EXTENDED
TO
SYSTEMS
OF
WHICH
THE
VOLUMES
In principle this model is an extension of the calculations of Tsai et al. and Abraham and coworkers to more concentrated alloys. An analysis of the available literature (see earlier) revealed that it should not be particularly difficult to modify the regular solution model in such a way that it allows the molar volumes of both pure components to differ. Using the potential functions as presented by Tsai et a1.23 and Abraham and coworkers24,25 an attempt will be made to combine both approaches. For simplicity an additional assumption with regard to the mean lattice point distance in the three-dimensional point lattice of the alloy will be made. Although the assumption does not always hold exactly for bimetallic alloys, it will be assumed that the interatomic distance R, in the alloy is determined by the bulk composition of the alloy, i.e. by the Vegard law: R, = x,,~R,,’
+ xbBRBBo
(4)
where xbAand xbBare the bulk molar fractions of A and B respectively and RAA” and R BB ’ are the lattice point distances in both pure components. Also, as a consequence of this assumption, the volume effect that occurs on mixing equals zero, and hence the enthalpy H of the system equals the energy E. The next point to be discussed is the choice of the potential function tiij for the interaction between atoms i and j to be used in the calculations. In the literature various types of potential functions have been proposed by Morse, Mie, Lennard-
166
A. D. VAN LANGEVELD
Jones etc. Since the choice is somewhat arbitrary in any case, a Lennard-Jones 6-12 potential function will be used throughout, since it has already been shown to be convenient in surface segregation calculations on dilute binary alloys” 25. The only shape-determining parameters of it are the intrinsic equilibrium distance Rijo and the intrinsic binding energy sijo at the same distance. The intrinsic equilibrium distance Rijo of a pair of atoms will be defined by RijO = riO+rjO
(5)
where rio and rjo are the atomic radii in the pure elements i and j respectively. Subsequently, the effective binding energy qj(Rb) of the pairs of atoms is formulated as follows:
(6) Alternatively, the use of a shorthand function yields
notation
for the Lennard-Jones
6612 potential
Q~(RJ = qjo$ij
(7)
A representation of the effective interatomic binding energy as a function of the interatomic distance is presented in Fig. 2. It should be noted that, as a consequence of the formulation as presented above, the effective binding energy Ebb for all bonds will be a function of the bulk alloy composition.
1 Fig. 2. The effective binding energy qj(R,) of a pair of atoms as a function atomic distance RJR,,‘, Rijo being the intrinsic equilibrium distance.
of the normahzed
inter-
167
THE ATOMIC SIZE EFFECT IN SURFACE SEGREGATION
The basic characteristics of the regular solution model presented below are as follows. (1) The broken bond model will be used throughout, i.e. only bonds between nearest-neighbour atoms are assumed to be of relevance. (2) The solution is assumed to be single phase with a random distribution of all atoms over the lattice points. Consequently, the numbers of the various types of bonds can be calculated by means of simple statistics. Also, the ideal configurational entropy contribution can be calculated simply. It should be noted that the excess configurational entropy contribution and the excess internal entropy contribution are neglected in the present model. (3) The distance between the lattice points is assumed to be determined by the Vtgard law. (4) The binding energies of the pairs of atoms are assumed to be a product of a binding energy sijo at the intrinsic equilibrium distance Rijo and the Lennard-Jones 6-12 potential function tiij. (5) A non-ideal behaviour with respect to the various effective binding energies will be taken into account:
The enthalpy Hb of the bulk of such a solution consisting of 1 mol of lattice points can be formulated as follows: Hb = z{*x,2&,,0 +A* + x,(1 -x&*B”+AB
+3(1-
xb)2sBBo+BB >
(8)
where Z is the number of nearest-neighbour atoms in the bulk and xt, stands for the bulk molar fraction of component A. As already mentioned, sAAo,sBBoand .sABoare all negative by definition. However, in some cases the effective binding energy sijotjij can become positive as a result of a compression of the bond. It can be rationalized that when 1.1 > rAo/rgo > 0.89 (of course for the specific case that a Lennard-Jones 6-12 function is used) all binding energies will be negative and consequently Hb < 0 i2. Of course, H,, equals zero when the numerical values of the effective binding energies cancel. The binding energy .sABocan be derived from experimental data related to the enthalpy of the bulk of the solution, i.e. from the enthalpy AHmiX of mixing or from the activity coefficients Y* and yB. Problems arising in these calculations when the molar volumes of the two components differ will be discussed elsewhere’ 2. 4.
SURFACE SEGREGATION
4.1. Surface segregation in a regular solution as deJined earlier Because of relaxation effects, changes in the interatomic distance, the shape of the interatomic potential function and the intrinsic binding energy in the topmost layers of atoms can be expected to occur. Clearly, these effects are related. In order to perform surface segregation calculations on a system as defined in Section 3, the following assumptions concerning the surface phase of the alloy will be made. (1) The lattice point distance in the surface phase equals that in the bulk of the alloy.
168
A. D. VAN LANGEVELD
(2) The potential function in the surface of the alloy equals that in the bulk. (3) Because of relaxation effects, the binding energy of the atoms in the topmost atomic layer to their neighbouring atoms will increase by a factor 1 + 6. (4) The number of atomic layers that the surface phase consists of depends on the numerical value of the effective regular solution parameter Q(R,). The following remarks relate to these assumptions. (1) It is generally accepted that in pure elements and alloys the interatomic distances in the surface differ from that of the bulk26. However, for reasons of simplicity this difference will be neglected. Recent results of Sachtler et aL2’ suggest our assumption to be reasonable, even for alloys in which the radii of the two types of atoms differ by a maximum of about loo! and in which the surface composition differs from that of the bulk. (2) In fact this assumption is an evident simplification. Bohnen5 has recently demonstrated in qualitative terms that the shape of the interatomic potential function in the surface of a pure metal is different from that in the bulk. Since quantitative data on this effect are not yet available in the literature, the effect will be neglected for the moment. (3) Data in the literature for this correction are also rather scarce. Since the correction is easily introduced, it has nevertheless been introduced below in the various equations on the surface segregation reaction. It should be noted that the relaxation parameter S is assumed to be equal for the various types of bonds between the surface atoms and their neighbouring atoms. which is also a simplification. (4) As demonstrated by Defay and Prigogine2s, only for binary solutions in which the enthalpy of the exchange reaction A-A + B-B +Z 2ApB equals zero can a single-layer model be used to describe in a formally correct way the surface segregation reaction. In such a single-layer model it is assumed that only the composition of the topmost layer of atoms or molecules differs from that of the bulk. For an ideal system the effective regular solution parameter Q(R,) which is formulated as Q(%)
= &AB(Rb)~lil&AA(Rb)f&BB(Rb)}
(9)
equals zero. For non-ideal solutions the single-layer model is in principle not correct and only leads to an approximate value of the surface composition. It should be stressed that the enthalpy of mixing in the formation of an alloy (which can be related to a function containing the effective regular solution parameter) does not necessarily equal zero when Q(R,) = 0 and vice versa. Since the most important property of a solution is the interchange energy a(&), which determines amongst other things the number of atomic layers that the surface phase consists of and whether a zeroth-order or a first-order regular solution model (see ref. 13) should be used to describe the solution correctly, the terminology ideal solution will be reserved exclusively for those systems in which L&R,) = 0. When !2(R,) # 0,and the surface phase is assumed to consist of the two topmost layers of atoms or molecules, it can be demonstrated that the composition of the second layer generally differs only slightly from that of the bulk (indicating that the two-layer model is a
THE ATOMIC
SIZE EFFECT
IN SURFACE
169
SEGREGATION
reasonable approximation). However, the composition of the topmost atomic layer as calculated from this two-layer model can differ quite significantly from that for the single-layer model. In contrast, the surface free energy of the solution seems to be influenced only moderately by the choice of the number of atomic or molecular layers in the surface phase ‘* . Finally it should be remarked that, in those cases when the single-layer model predicts only’a moderate surface segregation for solutions in which Q(R,) # 0, the two-layer model will predict a composition of the topmost atomic or molecular layer which will be very close to that for the single-layer model. From the arguments above it can be concluded that surface segregation in binary solutions can be predicted quantitatively using either a single-layer or a twolayer model, depending on the characteristics of the solution. The attractiveness of the single-layer model is that it is relatively simple to make numerical calculations with it, while the calculations of the various equations for two-layer models, which will result in a better quantitative prediction for some solutions, are generally much more complicated. Pioneering work in the field of multilayer model calculations on alloys, in which the atomic radii of the two components are assumed to be equal, has been presented by Williams and Nason”. 4.2. Surface segregation in a single-layer model The derivation of the equation determining the equilibrium of segregation in a single-layer model is easy to perform. The basis is a consideration of the surface segregation as a chemical reaction : A(bulk) + B(surf) = A(surf) + B(bulk) The equilibrium of segregation is determined by the minimum of the total Gibbs free energy G, which is the sum of the contributions of the bulk (G,,) and the surface (G,) phases: G, = G,+G,
(10) When the minimum of G, is sought it is also necessary to take into consideration the additional conditions which make the variables dependent. The minimum of G, is determined by the condition =, =, ax, ax, and for semi-infinite systems this reduces to
(11)
8% % (12) ax, ax, i.e. at equilibrium the chemical potential of a component in the surface equals that for the bulk. The Gibbs free energy of the bulk can easily be calculated when the enthalpy of the bulk mentioned earlier (eqn. (8)) is used. The Gibbs free energy of the surface can be formulated as G, = (1 +~)Z,{+X,~E~~~ ti*A+Xlu +(I +~)Z”cx,xb&**“~**+{xl(l
-xl)k3°~*B+~(1 -x,)+xt.U
+(1-x,)(l-~,)s,,~$,,]+RT{~~lnx~+(1-x~)ln(l-x~)}
-X1)2G3oh3B~
+
--x,)hOhJ+
(13)
170
A. D. VAV LANGEVELD
where Z, and plane parallel Introduction results in the molar fraction
Z, are respectively the numbers of nearest neighbour atoms in the to the surface and above or below the plane parallel to the surface. of the free energy of the surface and the bulk into eqn. (12) finally following relation which can be used to calculate the equilibrium of component A in the topmost atomic layer:
ICIAA){z-(l+az, +Z,)) 9&BB0$B” - &AA0 x I%-(1
+@(Z,x, +-Z&J-${Z-(1$6)(Z,
-Z{3x,2c*,oIc/,*‘+xb(l
-%Ja,,“~*R’+~(l
+2wq
x +Z,.))]-
-xb)2&riB0~rni’} + (14)
where tiij’ represents
the derivative
of tiij with respect to xb, i.e.
(15) Since the physical meaning of the various contributions in eqn. (14) is obscured by the complicated form, the relation will now be simplified and analysed in the simplified form. First, it will be assumed that the surface relaxation parameter 6 = 0, and also that rAo = rgo (as a consequence of the latter, tiij = 1 and tiij = 0 for all bonds). The resulting equation is identical with that presented by for example Wynblatt and Ku’0321 and Seah22:
It should be noted that since all atoms are now always at their intrinsic equilibrium distance, the superscript zero to the binding energies has been omitted. If it is further assumed that the system behaves ideally, i.e. 52 = 0, eqn. (16) reduces to (17) 01
x1( 1 -xb) =
x,(1 -.x1)
exp
-
.a& BB
-
E&4)
2RT
WV
Clearly, in eqns. (14), (16) and (17) the term
represents the ideal entropy of mixing contribution of the surface and the bulk phases. The first term in all the expressions represents the difference in the binding energy (in energy per mole of bonds) of the pairs of like atoms, which in eqns. (16), (17) and (17a) can be related to the difference in the enthalpy of sublimation. The term containing G(R,) obviously represents the influence of the non-ideal behaviour of the solution. The last term in eqn. (14) collects all derivatives which are present since lcIij is a function of xb.
171
THE ATOMIC SIZE EFFECT IN SURFACE SEGREGATION
The results of some calculations on a single-layer model as described above are presented in Fig. 3. The calculations were performed on an f.c.c. structure of a hypothetical alloy system of which the difference in the enthalpies of sublimation equals zero and AHmiX= 0, i.e. in the case that R,,’ = R,,’ the solution behaves ideally. In these calculations it is assumed that the atoms of component A, of which the molecular fraction is indicated along the axes, are the larger ones. In the case that R O = R,,’ the surface composition equals that of the bulk. However, when a digrence in the radii is introduced, the solute is always expelled to the surface (for dilute solutions). It should also be noted that, as a consequence of the asymmetry of the potential function used, the larger atoms are always expelled to the surface over a wider range of concentrations than the smaller atoms. 1.0
0
0.5
1.0 -x
b
alloy system with a varying ratio RAAo/RBBo: ,R,,'= 1.05R,,";...,R,,o = l.lOR aa’. The intrinsic binding energies aijo of all pairs R AA'- RBBo.--1 are assumed to be equal, the enthalpy of sublimation of both pure components being 100 kcal mol- r. The calculations were performed for a temperature of 750 K on a (111) crystallographic plane on an f.c.c. lattice, the surface relaxation parameter 6 being assumed to be zero. Fig. 3. The surface
composition
of a hypothetical
As also argued elsewhere12, strain not only affects the thermodynamics of a binary solid solution in surface segregation but also all parameters related to the enthalpy of the bulk of the solution, e.g. the enthalpy of mixing. This effect has been neglected in the calculations discussed above. In a consistent approach, however, strain effects have to be taken into account in the evaluation of &ABofrom the enthalpy of mixing data. In Fig. 4 the results are presented for the system described
172
A. D. “AN LANGEVELD
0
0.5
1.0 -x
b Fig. 4. The surface composition of a (11 I) plane of a hypothetical alloy system with the effect of lattice strain on AH”‘” taken into account (--) and neglected (. .). The enthalpy of sublimation of both pure elements is 100 kcal mol-’ and the enthalpy of mixing is zero. In the calculations an equilibration temperature of 750 K was assumed.
above with RAAo/RBBo = 1.O and 1.1, in which the strain effects on AH mix are taken into account and are neglected. Clearly the differences in the calculated surface composition using a consistent and a non-consistent approach cannot be neglected. It should be stressed here that the differences between the consistent and the nonconsistent approach are caused only by the stronger intrinsic binding energy of the A-B pairs in the consistent approach. The stronger A-B band also results in a depletion of the smaller solute B at x,, _- 0.90; in the bulk a B atom has more energeticallyfavourable A neighbours than in the surface. It should be noted that the difference between no strain effect and the consistent approach in which the strain is taken into account is rather limited for this specific case. In other cases, however, the difference can be quite large, as will be shown later. In Fig. 5 the results of calculations on the (111) plane of the Cu-Au system, in which the atomic size effect is taken into account and is neglected, are presented. In these calculations the enthalpies of sublimation derived from the Handbook qf Chemistry and Physics 33 have been used, resulting in sAuAuo = - 14.7 kcal mol ’ and cCuCuo = - 13.4 kcal mol ‘. The binding energy of the pairs of Cu-Au atoms was derived from Miedema’ 5, resulting in an estimated value of - 18.1 kcal molt ’ for EcuAuo. Experimental data derived from the literature29P32 are also presented. Despite the relatively large scatter in the experimental data, the model in which lattice strain is taken into account, as presented in this paper, is in better agreement
THE ATOMIC SIZE EFFECT
0
IN SURFACE
173
SEGREGATION
0.5
1.0 -x
b Fig. 5. The calculated surface composition of Au-Cu alloys as a function of the bulk molar fraction with lattice strain taken into account (--) and negelected (...). The calculations were performed for a temperature of 750 K and a (111) crystallographic plane on an f.c.c. structure. Experimental data are shown for comparison: +, Losch and Kirschnerz9; 0, McDavid and Fain3’; X, van Santen et ~1.~~; q , Sparnaay and Thomas3’.
with these data than is the more simple model in which atomic size effects are neglected*. It should also be noted that, according to t-heresults of the simple model, the solute is depleted from the surface at both ends of the diagram. As above, this effect is the result of the extremely strong bond between pairs of unlike atoms. The influence of the numerical values of other parameters, such as the surface relaxation parameter, the difference in the enthalpy of sublimation and the enthalpy of mixing on the surface composition of binary solutions, will not be discussed here, since in those respects the model presented in this paper behaves as a conventional regular solution. For an extensive discussion on these points we refer to Williams and Nason”. 4.3. The surface free energy of an alloy within theframework of the single-layer model In order to calculate the difference in the Gibbs free energy of the various crystallographic planes of an equilibrated alloy, eqn. (13) can be used. However, for a comparison all energies have to be divided by the surface area of 1 mol of lattice points since the Gibbs free energy of the surface phase is calculated for 1 mol of lattice points. * Note added in proof.Also, the results of these calculations those based on an electronic theory39.
are in excellent quantitative
agreement
with
174
A. D.
VAN
LANGEVELD
Of course, a more elegant parameter is the surface free energy 7, the free energy required to create the surface. It allows a direct comparison of the various crystallographic planes of an alloy mutually, and of the alloys with those of pure metals. In Fig. 6 the process of creating a surface from a bulk material has been represented schematically. In order to describe the process in terms of thermodynamics, the solid before the creation of the surface has been divided in three sections, the first of which (1) contains the future surface phase with a corresponding Gibbs free energy content Gb( I), while the two other sections (2 and 3) are and will stay bulk. After the creation of the surface, the energy content of the surface phase will be G,( I ).
G#)
G$)
G&l
Gb(2)
Fig. 6. A representation a bulk phase.
ofthe creation
ofa surface phase with a geometrical
surface area A (2 x +A) out of
By comparison with Fig. 6 it is self-evident that the change in the Gibbs free energy on the formation of a surface of area A (A being the geometric surface area of 1 mol of lattice points in the topmost atomic layer) is given by AG = G,( 1) + G,(2) + G,(3) - Gb( 1) - G,(2) - G,(3)
(18)
or, since 1’ = AC/A, ?A = Gs(1)-G,Cl)
(19)
Ofcourse, all bonds to the bulk that the layer is connected with also have to be taken into account in G,( 1) and Gb( 1). G,( 1) is the Gibbs free energy of a surface layer as given by eqn. (13) while Gb( 1) can simply be derived from eqn. (8) by adding the entropy contribution. The final equation is
YA = (1+~)(Z,{~X,~E,,~~,,+X~(~ -xx,)~,,~~,,+~(l -x~)~E~OIC/,~) +
+z”c~,x,~**“~,*+~-~~(~-x,)+x,(1 + ( 1-
x1
)( 1 - x&Bijo&J)
+:(1--~,)2E,,oll/,,}+RT{~1
-x,lnx,-(l-x,)ln(l-x,)}
-.x,)}hJ0h3+
- z” ,2Xb2h0$.4A
+
x,(1
-
%&4B0~,4R
+
lnx,+(l-xi)ln(l-xi)(20)
THE ATOMIC SIZE EFFECT
IN SURFACE
175
SEGREGATION
As mentioned above, eqn. (20) can be used to calculate the surface free energy once the various parameters are known. However, it can also be of use in making an estimate of the surface relaxation parameter when the surface free energy is accurately known. 4.4. Surface segregation in a two-layer model The conditions for equilibrium of this model are more complex, since the Gibbs free energy of the system has to be at its minimum as a function of two mutually dependent variables, namely x1 and x2, which both depend on xb. As in the singlelayer model above, the Lagrange method has to be used for that reason, leading to the following two conditions determining the equilibrium of segregation: dG,
aG,
-=o
(21)
ax, ax, aG,
dG,
-=o
(22)
ax, ax,
where G, and G, are the Gibbs free energies of the surface and the bulk phase respectively. Introduction of expressions for these two energies into eqns. (21) and (22) finally leads to the following relations containing x1 and x2 which have to be resolved : ~(&88°~BB--*Ao~AA){Z-_(1 x [Zx,-(1+6)(Z,x, --Wx,2&4011/*.4’+xb(l
+@V,
+&)I
+z,x,)-+{z-(l+6)(z, --&*B”$*e’++(l
+2Q2(&) x +Z,)}]
-
-xtJ2s,ri0~,,‘~+ (23)
The similarity of eqn. (23) of the two-layer model and eqn. (14) of the single-layer model should be noted. In fact, eqn. (23) can be reformulated into an equilibrium condition for a single-layer model by introducing x2 = xb. This is to be expected since the basic eqn. (12) for the equilibrium condition of the single-layer model is synonymous with the basic eqn. (21) of the two-layer model. Interesting systems to test the two-layer model presented above are the Pt-Cu and Pd-Cu alloy systems. Both form a continuous series of solid solutions, the atomic radii differ by about 7.5% and the effective regular solution parameter differs from zero as is demonstrated in Table I. The binding energy parameters of the various pairs of atoms are presented in Table II. The experimental results for the Pt-Cu 14*34*35and the Pd-Cu 36 alloy systems are compared with those from theoretical considerations in Fig. 7 and Fig. 8
176
A. D.
I THEEFFECTIVE
VAN
LANGEVELD
TABLE
FUNCTION
REGULAR
OF THE BULK
SOLUTION
PARAMETER
MOLAR FRACTION
-‘(h
-R(R,)
0. I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
2.2 2.4 2.6 2.8 3.0 3.3 3.6 3.Y 4.2
TABLE
OF THE
Pt-CU
ANU
THE
PddCu
ALLOY
SYSTEM
AS A
OF COPPER
(kcal (mol bonds))
‘)
1.8 I.9 1.9 2.0 2.1 2.2 2.3 2.4 2.5
II
THE BINDING
ENERGIES
E,,’
OF THE VARIOUS
PAIRS OF ATOMS AT THElR
INTRINSIC
EQUILIBRIUM
DISTANCE
RiiO Atom pair -E,,” (kcal (mot bonds)
‘)
PttPt 22.5
pttcu 19.9
CuPd 15.7
cu-cu 13.4
Pd-Pd 15.0
1.0 -
‘b
Fig. 7. The surface composition of PttCu alloys as a functton of the bulk composition at a temperature of 1000 K with lattice strain taken into account (upper full curve) and with neglect of the size effect (m m).All calculations were performed for a (111) crystallographic plane on an f.c.c. lattice. Experimental data are shown for comparison: 8, Brongersma et al.“‘; x, Kelley et a[.35; & van Langeveld and Pane?.
177
THE ATOMIC SIZE EFFECT IN SURFACE SEGREGATION
respectively. Clearly, for both alloy systems the experimental data are in good agreement with those from theoretical considerations in which the surface relaxation parameter 6 is assumed to be zero. For the Pt-Cu system the best fit between experimental data and theoretical considerations at 1000 K could be obtained by use of a surface relaxation parameter 6 = 0.05. Also, the introduction of next-nearest-neighbour interactions, as proposed by Gijzeman3’, leads to only minor corrections in the surface composition. The reason for this is that the creation of a consistent model in which next-nearest-neighbour interactions are taken into account will for the most part cause the effect to cancel itself out3s.
10
xs
I
05
0
10
05 -x
b
Fig. 8. The surface composition of Pd-Cu alloys as a function of the bulk composition at a temperature of 670 K with the difference in the atomic radii taken into account (-) and with neglect of the atomic size effect (---). Experimental data are shown for comparison: I, derived from Auger analysis; $, obtained from photoelectron work function measurements.
4.5. The surface free energy of an alloy within the framework of the two-layer model In a similar way to that in the single-layer model, the surface free energy is given by the relation YA = G,(l)-G,(l)
(251
The only difference is that the surface phase as well as the reference bulk phase consist of two atomic layers. Again all bonds in the surface layers and of the surface phase to the bulk have to be taken into account; the same holds for the calculation of the bulk reference state.
178
A. D.
VAN
LANGEVELD
The Gibbs free energy of the surface is given by G,(l) = (1+6)Z,{3x,2~,,oll/,,+x1(1 ++(I -x,)2%H3011/,,~+(l +1x1(1 -x,)+x2(1 + z,
{~~22E**ohA
-x,)E,,~$,,+ +6)z,[x,x,&,,01C/AA+
-.~,)h3°~AB+(1 +x2(1
+Z”Cx2.~,&**“~AA+{.~2(l
-x,)(1
-.u2)“BBo$Bs]+
- X2)E*Ro$,4B + )( 1 - .Y2)2C”BoIC/“”; +
-.YlJ+-)cb(l
+(l-x,)(l--~,)&,,O~,,I+RT(xl
-x,)}t.,,O$,,+
Inx,+(l-.x,)ln(l-x,)+
+x,lnx,+(l-x,)ln(l-.x2))
(26)
If use is made of the enthalpy of the bulk of an alloy, as given by eqn. (8) the surface free energy can be formulated as
?lA = (1 +~)(z,{~x,2~,,o~,,+x1(1 +z,[x,x2E,,0 +(I -x,)(i
-x&‘4B”$*B+*(1
$AA++,(1-X2)+x2(1
-_x,)5BBOf/JBBi+
-x,)}EA80$.4B+
-~~~~~~~“I(/~~I~+~1{3X~2~**oIC/**+.~2(1 -X2)E,4Bo$AB +
+~(l-~2)2~,,oll/,,}+ZyCX2X~E~Ao~*A+{X2(l-.Y~)+X~(l-X2)j x “.4B”~.4L3+(l
-x2)(1
-.%)E”““$mI
-2Z{~x,~e.4,0$,,
x s4B01ClAB + 3( 1 - %)2%3RotiBB)+RT{x, +x~ln.~-,+(1-x,)ln(l-.u2)-2x,lnx,+(1-x,)In(l-.u,))
x
+.x,(1 -Yb) x
Inx,+(l--x,)ln(l-.u,)+ (27)
The calculation of the surface free energy of various crystallographic planes can be of use to check their mutual stability. In pure metals the densely packed planes are more stable than the open planes. Since in alloys the surface segregation of the component with the lower surface energy is stronger the more open (rough) the surface is, it is necessary to investigate whether for alloys the rough surfaces are not more stable than the more densely packed surfaces. Equation (27) has been successfully used to demonstrate that of the (111) and (100) surfaces of Pt-Cu alloys the (111) faces are the more stabIej4. The last point is of particular importance for a comparison of model calculations and experimental data on polycrystalline surfaces which mainly consist of the most stable crystallographic planes. 5.
CONCLUSIONS
Lattice strain in binary alloys, caused by the difference in the atomic size of the two components, can seriously influence the equilibrium of surface segregation. The introduction of an interatomic potential function in the pairwise binding energy results in effective binding energies qj(Rb) of the pairs of atoms considered. A subsequent use of these effective binding energy parameters leads to equilibrium conditions for surface segregation in terms of a single- and a two-layer model. The equations determining the equilibrium of surface segregation are similar to those
THE ATOMIC SIZE EFFECT
IN SURFACE
SEGREGATION
179
already presented in the literature, the only difference being that in the derivation of the equations given in this paper the difference in the molar volumes of the two components is taken into account. Using the single-layer model, predictions were made for the Cu-Au system, while for the Cu-Pd and the Cu-Pt systems it was shown that the two-layer model has to be applied for model calculations. For all systems, experimentally obtained data are in reasonable agreement with the theoretical predictions presented in this paper. ACKNOWLEDGMENTS
The investigations were supported by the Netherlands Foundation for Chemical Research with financial aid from the Netherlands Organization for the Advancement of Pure Research. The author is grateful to Dr. B. E. Nieuwenhuys for stimulating discussions. REFERENCES
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
R. A. Swalin, Thermodynamics of Solids, Wiley, New York, 1962. C. Herring, Phys. Reo., 82 (1951) 87. J. van der Veen, Thesis, Delft University, 1981. A. Crucq, L. Degols, G. Lienard and A. Frennet, Surf Sci., 80 (1979) 78. K. P. Bohnen, Surf. Sci., 115 (1982) L96. R. P. Gupta, Phys. Rev. B, 23 (1981) 6265. J. W. Gibbs, Collected Works, Yale University Press, New Haven, CT, 1957. M. J. Kelley and V. Ponec, Prog. Surf Sci., II (198 I) 139. Y. W. Lee and H. I. Aaronson, Surf. Sci., 95 (1980) 227. V. S. Sundaram and P. Wynblatt, SurJ Sci., 52 (1975) 569. F. L. Williams and D. Nason, Surf. Sci., 45 (1974) 377. A. D. van Langeveld and A. J. Staverman, Z. Metallkd., 76 (1985) 54. A. E. Guggenheim, Mixtures, Clarendon, Oxford, 1952. H. H. Brongersma, M. J. Sparnaay and T. M. Buck, Surf Sci., 71(1978) 657. A. R. Miedema, Z. Metailkd., 69 (1978) 455. D. P. Woodruff, The Solid Interface, Cambridge University Press, 1973, p. 5. A. S. Skapski, J. Chem. Phys., 16 (1948) 389. D. McLean, Grain Boundaries in Metals, Clarendon, Oxford, 1957. J. Friedel, Ado. Phys., 3 (1954) 446. P. Wynblatt and R. C. Ku, Surf. Sci., 65 (1977) 511. P. Wynblatt and R. C. Ku, in W. C. Johnson and J. M. Blakely (eds.), Interfacial Segregation, American Society for Metals, Metals Park, OH, 1977. M. P. Seah, J. C&d., 57 (1979) 450. N. H. Tsai, G. M. Pound and F. F. Abraham, J. Caral., 50 (1977) 200. F. F. Abraham, N. H. Tsai and G. M. Pound, Surf Sci., 83 (I 979) 406. F. F. Abraham and C. R. Brundle, J. Vat. Sci. Technol., I8 (1981) 506. D. P. Woodruff, in D. A. King and D. P. Woodruff (eds.), The Chemical Physics of Solid Surfaces and Heterogeneous Caralysis, Vol. 1, Clean Solid Surfaces, Elsevier, Amsterdam, 198 1. J. W. A. Sachtler, M. A. van Hove, J. B. Biberian and G. A. Somorjai, Surf. Sci., 210 (1981) 19. R. Defay and I. Prigogine, Trans. Faraday Sot., 46 (1950) 199. W. Losch and J. Kirschner, J. VU. Sci. Technol., I5 (1978) 1541. J. M. McDavid and S. C. Fain, Jr., Surf. Sci., 52 (1975) 161. R. A. van Santen, L. H. Toneman and R. Bouwman, Surl Sci., 47 (1975) 64.
180
32
33 34 35 36 37 38 39
A. D. VAN LANGEVELD
M. J. Sparnaay and G. E. Thomas. Surf. Sci., 135 (1983) 184. Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL, 58th edn.. 1977-1978. A. D. van Langeveld and V. Ponec, Appl. SIG-[ Sri., 16 (1983) 405. M. J. Kelley, D. G. Swarzfager and V. S. Sundaram, J. Vat. Sci. Technol.. 16 (1979) 664. A. D. van Langeveld, H. A. C. M. Hendrickx and B. E. Nieuwenhuys, Thin Solid Films, 109 (198 13) 179. 0. L. J. Gijzeman, SW/. Sci., 150 (1985) I A. D. van Langeveld, Thesis, University of Leiden, 1983. C. A. Balseiro and J. L. MoranL6pez. Phj~. Rev. B. 2/ (1980) 349.