L. T. WLLLE and J. VENNIE: Kinetics of Surface Segregation in Binary Alloys 443

phys. stat. sol. (b) 131, 443 (1985)

Subject classification: 1.6; 21.1

Laborutorium voor KristuElogrufie eTh Studie van de Yuste Stoj, Rijksuniversiteit Gent1)

Kinetics of Surface Segregation in Binary Alloys

BY L. T. WILLE and J. VENNIK

Dedicated to Prof. Dr. Dr. h. c. Dr. E. h. P. GORLICH on the occasion of his 80th birthday

A new theory for the kinetics of surface segregation in binary alloys is proposed, which takes into account the concentration in the subsurface layers. This more realistic segregation profile is derived within the regular solution pair-bonding model. The surface composition as a function of time is determined analytically and it is in qualitative agreement with the experimentally observed behaviour.

Eine neue Theorie fur die Kinetik der Oberflachensegregation in binaren Verbindungen wird vor- geschlagen, die die Konzentration in den Suboberfl6chenschichten berucksichtigt. Dieses reali- stischere Segregationsprofil wird im Rahmen des regularen Losungs-Paarbindungsmodells abge- leitet. Die Oberflachenzusammensetzung als Funktion der Zeit wird analytisch bestimmt und ist in qualitativer Ubereinstimmung mit dem experimentell beobachteten Verhalten.

1. Introduction

The segregation of one of the constituents t o planar defects in binary alloys is a phenomenon with important consequences. In the case of grain boundaries [I] it can cause embrittlement, whereas a t a free surface [ 2 ] it influences the catalytic behaviour. Experimental studies of surface segregation are often hindered by preferential sputtering or because the segregation kinetics are not known. The standard theory of these kinetics is due to McLean [3] in the case of grain boundaries. Obviously surface segregation can be treated completely similarly, the only difference being that the solute diffuses to the surface from one side only. McLean’s original theory has been extended by several authors [4 to 61. In this paper we propose an alternative theory in which the actual composition profile is approximated more closely. It will be our aim to investigate the consequences of this approach, rather than obtain exact agree- ment with experiments.

McLean [3] considered the grain boundary at temperature TI as a reservoir with a concentration Cgb in equilibrium with the crystal interior a t a concentration C,, the ratio being 01, = C,,/C,. After quenching to a temperature T,, at which the equili- brium concentration ratio is a,, the change in grain boundary concentration with time is described by a one-dimensional diffusion equation. [t is assumed that the concen- tration in the grain boundary is equal to 0 1 ~ times the concentration in the adjacent crystal plane, during this process of diffusion. The interface condition expresses that the flux through the interface equals the rate of accumulation of the segregant in the grain boundary. In this way the problem is completely determined and it can he solved by the method of Laplace transforms. It is found that the grain boundary con- centration builds up rather quickly but that i t approaches equilibrium slowly.

l) Krijgslaan 281, B-9000 Gent, Belgium.

444 L. T. WILLE and J. VENNIK

This theory was applied to the case of surface segregation by Lea and Seah [41. These authors also took into account the possibility of evaporation of the segregant at the surface, assuming the rate of evaporation to be proportional to the solute content in the surface layer. The alloy system FeSn was studied with this method and it was found that the assumption of a constant a2 during segregation was completely inadequate. Moreover in reality a2 might vary over many orders of magnitude depend- ing on the adsorption relation. Using an experimentally obtained relation between a2 and the surface concentration, which gives a much larger initial value for aZ, satisfactory experimental agreement was reached.

A further analysis of the segregation kinetics was made by Rowlands and Woodruff [5], focussing on the driving forces behind surface segregation. Only parameters of direct physical interest such as diffusion coefficients and the heat of segregation entered this theory. Again the PeSn-system was investigated and the agreement with the data of Lea and Seah [4] was satisfactory. A similar, somewhat more phenomeno- logical theory was proposed independently by Brailsford [6].

2. The Equilibrium Segregation Profile

In this paper we will investigate the consequences of using a more realistic omposition profile in the surface region. Most theories of segregation are based on the regular solution model and the pair-bonding treatment as introduced by Swalin [7] in the study of bulk thermodynamics in binary alloys. Assuming nearest neighbour pair interactions wij (i, j = A, B) between the atoms of an AB alloy, the equilibrium concentration in the different layers parallel to the surface is described by a set of transcendental equations [8],

1 - CB -' + log ?I} ; n = I, 2, ... . Here cB denotes the bulk concentration and Acn is the excess concentration in the n-th layer,

Ac, = cn - CB , (2) where cn is the concentration in the n-th layer and the surface is taken to be the zeroth layer. The interaction energies are

VAB = WAA - WBB , (3 )

(4)

which is the heat of mixing of the alloy. These are the only parameters in the model. Often these quantities are obtained from tabulated values of thermodynamic proper- ties of alloys. However, it would be very interesting to derive them from an electronic st,ructure calculation (e.g. tight binding or pseudopotential method). Finally ni, is

i.e. the difference in heats of sublimation of the pure metals A and B, and

w = WAA + WBB - ~ W A B ,

Kinetics of Surface Segregation in Binary Alloys 445

Fig. 1. Composition profile within the regular so- lution pair-bonding model (dashed line) and ex- ponential fit (full line)

-

-

-

- -

layer

the number of nearest neighbours in a layer and nout is the number of nearest neigh- bours in an adjacent layer.

The set of equations (l), which shows that the alloy can be considered as a system of (interacting) ferrnions, has been obtained IS] by using a mean-field approximation to the Concentration functional theory proposed by Gyorffy and Stocks [9]. Similar equations have been obtained by a number of authors [lo to 121. It is evident from ( l a ) that the surface composition is determined by V ~ B , i.e. the component with the lowest heat of sublimation segregates to the surface. We will assume that c, are the concentrations of this component in the different layers, so that co > 0. When w = 0, the alloy forms an ideal solution and (1 b) shows that only the surface layer is enriched. This IS evident since it is the only layer with a deficiency of nearest neighbours. For non-ideal solutions two cases can be considered depending on the sign of w. When w > 0 the alloy has a miscibility gap and the bulk value is approached monotonically. This is the case for a clustering alloy, like CuNi. When w < 0 the alloy is an ordering one (e.g. CuZn) and the composition profile behaves in an oscillating manner. In both cases the bulk value is reached within a few layers [lo to 121. Recently the model has been extended to treat the concentration profile of irradiatied alloys 1131 and to determine the atomic distribution in bimetallic clusters [ 141.

In Fig. 1 we show the calculated composition profile a t the (111) surface of an f.c.c. crystal, using (I), for an alloy with bulk concentration cB = 0.3 and parameters w = 0.0162 eV, V AB = 0.1 eV, T = 770 K. These values are taken from [13]. We note that the bulk concentration is reached monotonically after approximately five layers.

3. Segregation Kinetics

When the alloy forms an ideal solution the McLean model using a variable concen- tration ratio [5] might very well be valid since only the surface layer is enriched. It is our purpose to investigate the consequences of using a composition profile that approximates the above results for non-ideal clustering solutions. To this end we consider the concentration in the different layers, cn( t ) , as a continuous variable c(z, t ) (the crystal is located in z > 0) and we will approximate the solutions of (1) by an exponential profile,

C(Z, 0) = CB + C; exp (-P,z) . ( 5 )

The constants c: and fll, which are temperature dependent, should be obtained from it fit to the numerical solution of (1).

446 L. T. WILLE and J. VENNIK

In Pig. 1 we have also plotted an exponential fit to the segregation profile. We note that the assumption of an exponential profile gives an adequate description in this case. The parameters have the values c: = 0.62 and PI = 1.354546.

After quenching at t = 0 equilibrium is disturbed and there is an atomic rearrange- ment leading to an asymptotic profile

C(X, +a) = CB + C; exp (-2j3,~) . (6)

The factor 2 is introduced for convenience later on. Inspection of (1) shows that

For the description of the kinetics we need a diffusion equation that has (6) as a quasi-stationary solution. This is provided by the equation describing diffusion in the presence of an external field [15],

c: < c;.

where D is the diffusion coefficient (at temperature T,) of the segregating species, which has been tabulated for a variety of elements [15]. The mechanism by which diffusion occurs (vacancy, interstitial, or interstitialcy) is left unspecified in our treatment. Note that i t might prove necessary to define an “effective” diffusion coefficient, since not only the segregant is moving towards the surface, but also the other species is diffusing into the bulk. In fact one might even argue that it would be necessary to use a position-dependent diffusion coefficient, since the concentration in different layers is different. These extra complications will be ignored here.

In order to completely determine the problem we need an additional boundary condition. Evidently we would like to have some kind of interface condition, e.g. no net flux outside the surface. However, the matter is complicated because we can only conserve the total number of segregating particles, whereas we should conserve the total number of solute particles. This extra complication arises because in (1) the bulk is considered as an infinite reservoir, held at a fixed concentration cB. Never- theless we will assume conservation of the total number of segregating particles and we give further some evidence why we expect this to give meaningful results.

Conservation of the total number of segregating particles is expressed as +a

0 I (c(x, t ) - c ~ ) dx = N ; t > 0 , (8)

which upon substitution of (5) and (6) gives

It is now evident that c; might be obtained from a numerical solution of (l), but then ,8, is fixed by (9). The full solution of (1) will be different from (6) in that i t approaches the bulk concentration more slowly. However, in view of the fact that we ultimately want to determine the surface composition as a function of time, i.e. c(0, t ) , we believe that the use of (6) should give accurate results for this observable. The surface layer will be almost exclusively fed by the very first layers underneath i t and this is expressed in (6) and (8). Although (6) is only a quasi-stationarystate, in our opinion i t should describe correctly the segregation kinetics in the surface layer. Nevertheless this is an approximation that has to be kept in mind when performing calculations for real systems. We find some support for this assumption in the ex- perimental results of Ng et al. [16]. These authors found for the CuNi system a slight

Kinetics of Surface Segregation in Binary Alloys 447

depletion in Cu in the subsurface layers. This nonmonotonic behaviour is not expected theoretically. We interpret it as an indication that the surface layer has reached its equilibrium concentration, drawing Cu atoms from the deeper layers, whereas these deeper layers are not yet in thermodynamic equilibrium. Our model of particle conser- vation should give a satisfactory description of this kind of behaviour, even though we cannot find depleted layers, since i t is the excess number of particles that is con- served.

The problem is now completely determined by the diffusion equation (7), which automatically ensures the asymptotic solution (6), in combination with the initial condition ( 5 ) and the particle conservation condition (8). Referring all concentrations relative to cB, we can solve (9) by Laplace transformation [17], putting

+m

0

- c(x, p ) = J e-Pt c(x, t ) dt . (10)

The Laplace transform of (7) is

where we have used (5) and q2 = p / D . The solution of (11) which is bounded a t +co is

Next, M can be determined by substitution of (12) into the Laplace transform of (8). This yields

Inverse Laplace transformation [ 151 yields

+ /31 (/31 - 2/32) Dt) + 4 exp ( - a x 3- /31(/31 - ~ B , ) Dt) * (14) Using units of time 1/j3!D, the surface concentration as a function of time is

c = c(0, t ) = - 1 ci {edc (-@ 4 1 1- t ) - erfc ((1 - &) /i) x 2

448 L. T. WILLE and J. VENNIK

Y Y U \ Y c U

U L

1.L I I I I I

Fig. 2. Surface composition (scaled in order to vary from 0 to 1) as a function of time (in 1//3?D). The equilibrium concentration ratios are (a) cT/cl = 0.1, (b) 0.4, (c) 0.7

In Fig, 2 we show the quantity (c - c!/ci)/(l - cs/cg) as a function of time, for different values of the ratio cf/ci. We note that the equilibrium surface composition is ap- proached faster than in the &Lean model with a constant a1 [4]. This is a consequence of the fact that the concentration in the first layer below the surface is higher than the bulk concentration. It is possible to include evaporation from the surface layer in our framework, but since the algebra becomes quite involved we will not elaborate it here.

4. Conclusions

In summary, we have shown that we can reproduce qualitatively the behaviour of the kinetics of surface segregation in alloys, using a realistic composition profile. Our work has the character of a model calculation, since we have made several ap- proximations in order to keep the calculations analytically tractable. As an improve- ment over other theories, we have assumed an exponential profile perpendicular to the surface. We expect this to be an adequate description of the actual situation in ordering alloys and it certainly is a satisfactory approximation to the solutions of the regular solution pair-bonding model, as described by (1). Furthermore, we have described the kinetics by a one-dimensional diffusion equation, as is customary. One might expect ordering phenomena, both within and perpendicular to the layers, to complicate the matter. Also, in order to obtain the correct quasi-stationary solution we have added an external field term in the diffusion equation, whereas the real profile is due to internal interactions. Our main approximation consists of conservation of the number of segregating particles. This certainly does not give the correct asymp- totic behaviour, but we have argued that i t should give a plausible description of the time evolution of the concentration in the surface layer.

This simple model illustrates that the segregation profile should be taken into account in any realistic theory of the segregation kinetics. We suggest that the dis- crepancy between the McLean model and the experiment is partly due to neglecting the concentration variation in the subsurface layers, and partly to the modified ad- sorption relation during the approach to equilibrium.

References

[l] M. P. SEAH, J. Phys. F10, 1043 (1980). [2] W. M. H. SACHTLER and R. A. VAN SANTEN, Appl. Surface Sci. 3, 121 (1979). [3] D. MCLEAN, Grain Boundaries in Metals, Clarendon Press, Oxford 1957. [4] C. LEA and M. P. SEAH, Phil. Mag. 36,213 (1977). [5] G. ROWLANDS and D. P. WOODRUFF, Phil. Mag. A40,459 (1979).

Kinetics of Surface Segregation in Binary Alloys 449

[6] A. D. BRAILSFORD, Surface Sci. 94, 387 (1980). 171 R. A. SWALIN, Thermodynamics of Solids, Wiley, 1962. [8 ] L. T. WILLE, Ph. D. Thesis, Rijksuniversiteit Gent, 1983, unpublished. [9] B. L. GYORFFY and G. M. STOCKS, Phys. Rev. Letters 60,374 (1983).

[lo] F. L. WILLIAMS and D. NASON, Surface Sci. 45, 377 (1974). [ill J. J. BURTON, E. HYMAN, and D. G. FEDAK, 3. Catalysis 37, 106 (1975). [ l2] D. KUMAR, A. MOOKERJEE, and V. KUMAR, J. Phys. F 6,725 (1976). [13] F. MEJIA-LIRA and 3. L. MORAN-LOPEZ, Surface Sci. 143, L427 (1984). [I41 C. A. BALSEIRO and J. L. MORAN-LOPEZ, Proc. 3rd Internat. Symp. Small Part,icles and

Inorganic Clusters, Berlin (West) 1984, t o be published in Surface Sci. [15] W. JOST, Diffusion, Academic Press, 1960. [IS] Y. S. NG, T. T. TSONG, and S. B. MCLANE, Phys. Rev. Letters 4Z, 588 (1979). [I71 P. 31. MORSE and H. FESXBACH, Methods of Theoretical Physics, McGraw-Hill Publ. Co.,

(Received July 17, 1985)

New York 1953.

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