Nonequilibrium surface segregation

Nonequilibrium Surface Segregation M. J. S P A R N A A Y Philips Research Laboratories, 5600 M D Eindhoven, The Netherlands Received December 7, 1979; accepted February 25, 1980 The process of surface segregation of a mixture AzBI_x, starting from the situation in which the composition parameter x is uniform up to the surface, is discussed. It is shown that for an " i d e a l " behavior (the nonideality energy parameter w is zero) the process can easily be understood in terms of a well-known diffusion law. F o r positive values of w (the mixture AxB~-x shows a tendency toward demixing), the diffusion is slower than for negative values of w. Illustrations are given for w = +2kT and w = - 2 k T and it is seen that here the difference is considerable. INTRODUCTION

It has been amply demonstrated that a number of alloys (with the general formula Axnl-x) show surface segregation (1-6), i.e., the bulk value of the composition parameter x is different from that at the surface. Assume a single crystal of such an alloy is cleaved. At the moment of cleaving, time t = 0, the value of the composition parameter at the newly formed surface is the same as that in the bulk. There is no equilibrium and therefore there is a gradient of the chemical potential of c o m p o n e n t A and of that of c o m p o n e n t B. Atoms A and B of these components begin to move in a direction indicated by the gradients until a final, equilibrium, state is established, in which the chemical potentials have uniform values up to the surface. We shall consider only the one-dimensional case, where the flows of the atoms A and B are in the z direction, perpendicular to the plane of the surface. In this article we describe the diffusion behavior assuming the regular solution model (7) of the AxBx-x alloy. The final equilibrium state is characterized by a surface composition A~BI_~, in which as a general rule y :~ x. F o r simplicity it will be assumed here that only the outermost atomic layer has this composition parameter

y. Effects of a deviating composition of the second, third, etc. layers (8, 9) are left aside. We use expressions of the chemical potentials and of the diffusion equation, which are a direct consequence of the regular solution model. In this way we obtain a consistent treatment of the equilibrium and the nonequilibrium properties o f the alloy. THE MODEL

The model is as follows. There are N lattice points per unit volume, and N s lattice points per unit area at the surface of a given crystal plane. A lattice point is either occupied by an atom A or an atom B. The number of nearest neighbors of an atom is z, i.e., z is the coordination number. At the surface the coordination number is z' and z' < z, because at one side neighbors are missing, a fraction mz say. It is also useful to introduce a coordination n u m b e r / z , which is the n u m b e r of nearest neighbors in a plane parallel to the surface. It is seen that l + 2 m = 1. Concerning the cohesive energy of the crystal only nearest-neighbor interaction energies EAA, EBs, and E~8 will be considered. These energies, between two neighboring A atoms, or B atoms, or an atom A and a neighboring atom B, are counted negative in the case of an attraction. 607

Journal of Colloid and Interface Science, Vol. 75, No. 2, June 1980

0021-9797/80/060607-05502.00/0 Copyright © 1980by Academic Press, Inc. All rights of reproduction in any formreserved.

608

M.J. SPARNAAY

The total number of nearest neighbors per unit volume is 1 / 2 z N . Introducing zX as the number o f A B pairs per unit volume, the total energy E per unit volume (no surface effects) is:

E=~1 Z(NAEAA + N B E B B )

+ Xw

[1]

where 1 w = -z(2EAB

2

-- Eaa -- EBB)

[2]

and NA = Nx;

NB =N(1-x).

[3]

For the parameterX one has in the simplest approximation: X -

NANa

N

[4]

- Nx(1 - x).

With this approximation Guggenheim (7) obtained for the chemical potential ~ab of component A in the bulk of the alloy: IxbA = tZ° + k T ln x + w ( 1 - x )

2.

[5]

For/~Bb, the chemical potential of component B in the bulk of the alloy, one has: I ~ = tz ] + k T ln ( 1 -

x) + w x 2.

[6]

ponents. These assumptions have as a practical consequence that the mechanism of the diffusion consists only of an exchange process of atoms A and B, the total number Nt of available sites remaining constant. Thus we exclude effects arising from interstitial sites and boundaries between crystallites, such as were considered in Refs. (3) and (4). These effects may lead to local accumulations of the atoms A and B. With respect to assumption (b), impurity atoms should be absent. Assumptions (a) and (b) imply that the diffusion process can be described by a single equation, Eq. [7]. It is not necessary to use an additional diffusion equation with NB and /xB. In order to solve the nonequitibrium problem, the expression for /x] should be used and introduced in Eq. [7]. The omission of the superscript b will be justified in the next section. When the energy parameter w is taken as zero and when diffusion in only the z direction is considered, Fick's second law is obtained. This equation is frequently used for diffusion problems, also for the kinetics of segregation (11). However, in the segregation case, there often is a rather strong interaction between neighboring atoms. These are even at the basis of the explanation of the segregation phenomenon. Therefore, an ideal behavior, as implied by w = 0, is unlikely. When EAB is more negative than 1/2(EAA + EBB), i.e., when w < 0, there is a tendency toward alloying, when w > 0, there is atendency toward demixing.

Equations [5] and [6], in which k is the Boltzmann constant and T the absolute temperature, contain the terms /~o and ~o. They are the chemical potentials of the pure metals A and B, respectively. In the regular solution model they are equal to 1/2ZEAa and 1/2zEBB. According tothe principles of irreversible thermodynamics (10) and taking the set of Nt = N V + N S A lattice points (V is the T H E S E G R E G A T I O N PROCESS volume, A the surface area) as our reference The relation between the equilibrium frame, one has the following diffusion value at the surface of the composition equation: parameter, y, and the bulk composition ONa/Ot = D / k T div [NA grad /£a] [7] parameter, x, was found by Guggenheim as follows: From Eqs. [5] and [6] one obtains: in which D is the diffusion constant. This equation is valid, when the following ~ _ g ~ __ ~,o _ ~ o assumptions are made: (a) There is no depleX tion or accumulation of particles in any part + k T In ~ + w(1 - 2x). [8] 1--x of the crystal; (b) There are only two c o r n Journal of Colloid and Interface Science, Vol. 75, No. 2, June 1980

NONEQUILIBRIUM

SURFACE

Composition ~aramet:er

F o r t h e o u t e r m o s t s u r f a c e a t o m i c l a y e r it w a s f o u n d t h a t (6, 7): tz~ -

txsB = k T In

609

SEGREGATION

1.0

7.0-

t=O

-~= ~

.Y=&9

Y 1 -

y

0.s

+ w[l(1 - 2y) + m(1 - 2x)].

0.5

[9] x_l

T h e v a l u e o f y is d e t e r m i n e d b y t h e e q u i librium condition: /-Zb - -

/ £ Bb =

JL('~I - -

]'£~ =

t"/'A - -

/£B"

x=(2f

o

i t l J l l l l

5h

[10]

i

0

---~z

i

i

2h

I

b)

[

5h

i

i

i

----z

Chemical ~ofentfal

F o r u s e in E q . [7] a s e p a r a t e e x p r e s s i o n o f /z~ is n e e d e d . This is f o u n d to b e : Iz~ = IX° + k T l n

y T

+ w[l(1 - y ) 2

+m(1

-y)(!

- 2x)].

[11]

F r o m t h e e q u i l i b r i u m c o n d i t i o n /zab = /z~ = /Za t h e c o n s t a n t / z ° b e c o m e s :

,

,

,

,

I

,

,

5h

h 2h X

o)

,

,

,

,

,

h 2h

I

,

,

5h

---~ z

d)

--~z

tz ° = tz ° + k T l n -

Y -

w[l{(1

+m{(1

-

y)Z _

-y)(1

(1

-

x) z}

- 2x)-

2(1 - x ) 2 } ] .

[12]

We already noted that just after cleavage o f a c r y s t a l t h e c o m p o s i t i o n is u n i f o r m u p to t h e s u r f a c e a n d t h a t in t h a t c a s e t h e r e must be a gradient of the chemical potentials n e a r t h e s u r f a c e . In o r d e r to c a l c u l a t e this g r a d i e n t w e a s s u m e t h a t at t h e s u r f a c e (z = 0), w h e r e t h e a t o m i c s u r f a c e l a y e r is located, the value of/x~ can be found by r e p l a c i n g y b y x in Eq. [11]: tZ~A(t = O) = tx ° + k T l n x

+ w[(1 - m)

× (1 - x ) z - m ( l

-x)].

[13]

A t t = 0 t h e c h e m i c a l p o t e n t i a l in t h e second atomic layer, a distance h away from t h e first a t o m i c l a y e r , still h a s its e q u i l i b r i u m v a l u e . T h i s g i v e s f o r g r a d ]£A at t i m e t = 0 the expression: 1

g r a d I~a(t = 0; Z = 0 ~ h) = ~ - (/~0 _ /Z0A -- w m ( 1

-- X ) ( 2

-- x)).

[14]

F o r the third, f o u r t h , etc. l a y e r , l o c a t e d at

FIG. 1, Profiles of the chemical potential /ZAand of the fraction of atoms A at time t = 0 and at t = (thermodynamic equilibrium) for x = 0.1 and y = 0.9 (w = 0). The dotted line in (a) indicates a surface concentration at t = 0 of (x/y)x, which gives the same gradient of the chemical potential as in the segregation case. z = 2h, z = 3h, etc. t h e c h e m i c a l p o t e n t i a l s h a v e , at t = 0, t h e i r e q u i l i b r i u m v a l u e a n d therefore their gradients are zero. T h e s i t u a t i o n at t = 0 a n d at t = w is s k e t c h e d in Fig. 1. H e r e x = 0.1; y = 0.9 and w = 0 are chosen. T h e a d a p t e d m o d e l o n l y p e r m i t s us to calc u l a t e c h e m i c a l p o t e n t i a l s at v a l u e s o f z, w h i c h are m u l t i p l e s o f h, a l a t t i c e p a r a m e t e r . W e w r i t e z = b h , in w h i c h b is an i n t e g e r . E q u a t i o n [7] c a n n o w b e r e w r i t t e n as: Ox Ot

bh

D - h 2 k T [{XhlZA}bh,t

-- { X A ~ A }(b+l)h,t]

b

~

0

• • " [15]

t=0-"

H e r e x a n d /zA a r e v a r i a b l e s a n d t h e subs c r i p t s i n d i c a t e the v a l u e s o f t i m e a n d p o s i t i o n , at w h i c h t h e s e v a r i a b l e s a r e t a k e n ; Journal of Colloid and Interface Science, VoL 75, No. 2, June 1980

610

M.J. SPARNAAY

I

xbj s Composition

0.3~--~ •

..... W=-2kT - - W - - O ....... W = +2kT

i, ~.

0.2

y = 0.9 x=X4o=0.1

I

I

I

h

2h

3h

FIG. 2. Profiles

of the

,

fraction

I

I

#h 5h 7. =bh di.stonce of atoms

a constant initial concentration in a semiinfinite region bordering a narrow surface region, where the concentration at time t = 0 is different from that in the bulk, the difference decaying according to a 1/(TrDt )ii2 × e x p ( - z 2 / 4 D t ) law. This then is just the case under consideration here, the only difference being that, in order to obtain the surface concentration x0,t of the segregation case, we have to multiply the surface concentration obtained for pure diffusion, by a factor ( y / x ) .

A at time RESULTS

t = 5h2/D f o r w = - 2 k T ; w = 0 ; w = + 2 k T .

A~A at time t and z = bh is the difference tXA(t; Z = bh) - I~A(t; Z = (b + 1)h) and A/zA at time t and z = (b + 1)h is the difference tZA(t; Z = ( b + 1 ) h ) - tza(t; z = (b + 2)h). The equation can be solved numerically by assuming time intervals At, which must be shorter than 0.1h2/D. In order to visualize the solution of the diffusion problem, let us return to Eqs. [13] and [5]. F o r simplicity the contributions containing w are omitted. At time t = 0 one has: tX~A = i~° + k T In x

(t

=

0; z

=

0)

Results of the numerical solution of Eq. [15] are given in Figs. 2 and 3. Figure 2 gives the profile of the fraction of atoms A at t = 5h2/D for x0,0 = 0.1; y = 0.9 with w = + 2 k T (this corresponds to the case of C u - N i alloys at T = 300°K); w = 0, and w =-2kT. F u r t h e r m o r e l = 0.5 and m = 0.25 are taken. In all cases there is a minimum at z = h and such a minimum is characteristic for this type of diffusion. When not only the first, but also the second, third, and fourth (8) layers are involved in the segregation process, this minimum will of course be w e a k e n e d and it can be displaced to a position deeper inside the crystal. The effect of a nonzero value o f w is given in Figs. 2 and 3. We have selected the values

[16]

xaf Surface composition

and t ~ = t z° + k T ln x

(t = 0; z >/h).

0.7 _1

[17]

I

Journal of Colloid and Interface Science, Vol. 75, No. 2, June 1980

~

-

I

"

'

~

7 .'/"

0.6 0.5 -

Equations [16] and [17] have the same standard chemical potential/x °, but they have different concentration terms, k T In ( x / y ) x and k T In x, respectively. The gradient of the chemical potential can n o w be expressed in terms of a concentration gradient (as indicated by the dotted line in Fig. la). The problem is thereby reduced to the problem of seeking a solution of F i c k ' s second law. This solution is well k n o w n for the case of

.

....

/

~

.....

--

<~L/>/ i t + " - . - . -

w=-2kr

-

W =°

VZ.-'" ....

01

0

, , , ,I

50

~ , ,,

FIG. 3. Fractions tion

of

time

w = +2kT.

in

I , , , , l,

100

150

of atoms units

, , ,I

, ~ , ,I

200 -~2 lime

A at the surface

h2/D f o r

w =

,,

250

as a func-

-2kT;

w = 0;

NONEQUILIBRIUM SURFACE SEGREGATION w = +2kT, w = 0, and w = - 2 k T , each taken at the same value for the segregation: x -- 0.1 with y = 0.9. This was done, because the p a r a m e t e r s x and y are experimentally accessible and are therefore the first that should enter the equations. H o w ever, with fixed x and y values and variable w the gradient of/ZA, and therefore the average force acting on the diffusing particles, varies, as is shown for t = 0 by Eq. [14]. With positive w it b e c o m e s smaller and with negative w it b e c o m e s larger. Whereas for w = - 2 k T the segregation process is almost terminated at 200h2/D, this process is, at that m o m e n t , only halfway when w = + 2 k T and one reaches the conclusion that the segregation p r o c e s s in alloys, which show a demixing tendency (positive w), tends to be m u c h slower than in mixtures, which show good alloying properties. E x p e r i m e n t s leading to curves of the surface composition versus time (3) and to curves of the composition versus depth at a given time (11) are available in the literature. E x p e r i m e n t s with single crystal twoc o m p o n e n t alloys, carried out with the technique of low-energy noble-gas ion scattering (or ion scattering s p e c t r o s c o p y , ISS) (6) and aiming at an investigation of the role, possibly played by the interaction energy p a r a m e t e r w are n o w carried out. Results

611

with surfaces of C u - A u alloys, (100), (1 i0), and (111) planes, and with surfaces of polycrystalline C u - N i alloys, at t e m p e r a t u r e s ranging from 100 to 400°K, obtained by using H e + and Ne + b e a m s for scattering and sputtering, will be published shortly. ACKNOWLEDGMENT The author is indebted to Dr. Gary Thomas for many valuable discussions. REFERENCES 1. Blakeley, J. M., and Shelton, J. C., in "Surface Physics of Materials" (J. M. Blakeley, ed.), p. 189. Academic Press, New York, 1975. 2. JabIofiski, A., Adv. Colloid Interface Sci. 8, 213 (1977). 3. Wynblatt, P., and Ku, R. C., Surface Sci. 65, 511 (1977). 4. Johnson, W. C., Chavka, N. G., Ku, R., Bomback, J. L., and Wynblatt, P. P., J. Vac. Sci. Technol. 15, 467 (1978). 5. van Santen, R. A., and Boersma, M. M., J. CataL 34, 13 (1974). 6. Brongersma, H. H., Sparnaay, M. J., and Buck, T. M., Surface Sci. 71,657 (1978). 7. Guggenheim, E. A., "Mixtures." Clarendon Press, Oxford, 1952. 8. Williams, F. L., and Nason, D., Surface Sci. 45, 377 (1974). 9. Meijering, J. L., Acta Met. 14, 251 (1966). 10. e.g., Haase, R., "Thermodynamik der irreversiblen Prozesse." D. Steinkopff, Darmstadt, 1963. 11. Hofmann, S., and Erlewein, J., Surface Sci. 77, 591 (1978).

Journal of Colloid and Interface Science, Vol. 75, No. 2, June 1980