Surface Science 150 (1985) 209-225

North-Holland, Amsterdam

209

A THEORETICAL MODEL FOR COMBINED SURFACE SEGREGATION AND EVAPORATION

C.D. STINESPRING

Aerodyne Research, Inc., 45 Manning Road, Billerica, Massachusetts 01821, USA

and

W.F. LAWSON

Us Department of Enerw, Morgantown Energy Technology Center, Collins Ferry Road, Morgantown, West Vqinia 26505, USA

Received 15 June 1984; accepted for publication 21 September 1984

A theoretical model describing the combined effects of surface segregation and evaporation in binary alloys is developed. Using the model, parameter regimes dominated by segregation,

evaporation, or competitive segregation and evaporation are described, and general criteria for

performing meaningful segregation measurements are discussed. The predictions of the theory are

compared with and found to be qualitatively similar to the measurements of Webber and

Chadwick for the Ni-Cu system. Studies aimed at rigorously testing the theory are also outlined.

1. Introduction

In recent years there has been much interest, both theoretical and experi- mental, in surface segregation in binary alloys systems [l-7]. In a number of these systems, evaporation of the segregant has been observed [l-3]. As noted by Webber and Chadwick [l], such evaporative losses lead to bulk depletion of the segregant, and, as a result, true equilibrium cannot be obtained at the surface. Thus, since conventional UHV methods of obtaining segregation parameters (e.g. the segregation enthalpy) require the measurement of the equilibrium surface composition as a function of temperature [4], segregant evaporation may impose severe limitations on these methods. Therefore, it is important to have a clear understanding of the effects of evaporation and the conditions under which they are significant in any study of surface segregation.

The most consistent and generally accepted theoretical treatment of the combined surface segregation and evaporation is that of Lea and Seah [5]. In their model, however, the key input parameter, the surface enrichment ratio, must be determined experimentally and is subject to considerable uncertainty if segregant evaporation occurs.

0039-6028/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

In this paper, a theoretical model for combined surface segregation and evaporation is presented. Based on a description of the segregation process developed by Hofman and Erlewein [6], this model utilizes input parameters which may in principle be calculated using theoretical analyses of solids and surfaces [4,8,9] and, in certain cases, which may be established using indepen- dent experiments [lO,ll]. We have employed the theory to establish parameter regimes over which behavior is dominated by segregation, evaporation, or competitive segregation and evaporation, and we critically discuss the results of recent experimental measurements [l,ll] for the Ni-Cu alloy system. A rigorous validation of the model has not been attempted. We do, however. indicate how such a validation may be accomplished and suggest improve- ments in the model which may be incorporated for specific alloy systems.

2. Theoretical development

The segregation model developed by Hofman and Erlewein [6] is based upon a free energy diagram similar to that shown in fig. 1. Here the free energy of the segregant is plotted as a function of depth in the ahoy. The indices 0, 1. 2 ‘..., etc. identify the gas phase or vacuum (0), the surface (1) and the interior atomic planes (2, 3,. . . ,etc.) of solid. AC, is the free energy of diffusion, and AC, is the free energy of segregation. The energy gradient at the surface arises from bond breaking and lattice strain [8,9] and is the driving force for the segregation process. Diffusion, however, is the mechanism by which segrega-

DISTANCE Fig. 1. Free energy diagram used to calculate the combined effects of segregation and evaporation.

C. D. Stinespring, W. F. Lawson / Surface segregation and evaporation 211

tion occurs. The energy zero in this diagram is defined at the minimum of the interior free energy. Thus, AG, is negative while AG, is positive.

The evaporation process may be incorporated into the model by introducing

AC,, the free energy of vaporization. As in conventional thermodynamic treatments, AG, is defined relative to the minimum of the interior free energy and is positive. The free energy barrier to evaporation is, therefore, given by

AC, - AG,. In the manner of Hofman and Erlewein [6], we use the free energy diagram

and reaction rate theory to determine the flux, j,,, * ,, of segregant atoms between adjacent planes. These are given by

j,, = N,Xir~i exp[ - (AGv - AG,)/RT], 04

j,, = u;~X,W,,V, exp[ -(AC, - AG,)/RT], (lb)

j,, = a,2X2W2,v2 exp[ -AG./RT], (lc)

j,, = a;2X2W23v2 exp[ -AG,/RT], (14

i,,+l = a,F24%,r~~v, ev-AGdRTl. (14 where N, is the total number of available surface sites per unit area, X, is the concentration of segregant species in the i th plane, vi is the frequency of the normal vibrational mode resulting in a jump from the ith to the (i f 1)th plane, K,;*i is the corresponding jump probability, a, is the lattice constant for the ith plane, R is the universal gas constant, and T is the absolute temperature. Using these equations, the rate of change of each X, may be written

dXi/dt = (_&i -_k -jlO)~?V (2a)

dX,/dt = (j3, -_k +_k -j2i)ai9 (2b)

hX,/dr= (jj+l,, -_L+, +L,,, -ji,i-l)af. (2c) This set of coupled, nonlinear, first-order differential equations must be

solved numerically in order to describe the time dependent behavior of the system. In the numerical approach subsequently described, calculated or measured values for vi and a, could readily be used to improve the accuracy of the calculations. Following Hofman and Erlewein [6], however, we assumed that vi = v and a, = a (i.e., vi and a, are constant for all planes). In this approximation, eqs. (1) and (2) may be combined to yield

d& AG, - AG, + AG, -= dr

W2,X2 - WI2 X, exp - N,a’X, exp Pa)

d& -= w32x3- w23x2 + w12xl exp dr (3b)

212 C. D. Stinespring, W. F. Lawson / Surface segregarion and evaporation

where the bulk diffusivity has been taken as

D = a2v exp( -AG,/RT), (4)

and the dimensionless time characteristic of bulk diffusion has been defined as

r = tap2D. (5)

In their discussion of the jump probability, Hofman and Erlewein [6] considered a number of functional forms and used the one which gave the best fit for experimental data for the Cu-Sn system [12]. This choice will be discussed later in some detail. Without reference to a specific system, however, it is assumed that

WI, = 1 - x,. (6)

Combining eqs. (3) and (6) yields a set of coupled, non-linear, first-order differential equations for the concentration of the segregant in each atomic plane. in numerically solving these equations for the time-dependent behavior, we have employed a self-starting, fourth-order, predictor-corrector time-march- ing technique. The initial value problem requires the specification of the initial segregant concentrations, X,, and two independent coefficients (Y = AGJRT,

the segregation parameter, and 6 = (AC, - AC, + AG,)/RT, the evaporation parameter, If (Y, 6, or Xj( r = 0) are such that the concentration changes rapidly near the surface, then the time step must be decreased commensurately. To simulate an infinite bulk, enough layers are explicitly included so that the innermost layers experience no appreciable concentration changes over the total elapsed time of the integration.

3. Results

In discussing the detailed results of the calculations, our primary objective is to demontrate the general importance and influence of the dimensionless parameters (Y and 8. As seen from eq. (3a), as (Y, the segregation parameter, becomes more negative, back diffusion from plane 1 to plane 2 decreases, and segregation increases. Likewise, as 6, the evaporation parameter, becomes more negative, the evaporation loss from the surface decreases. To demonstrate the interplay between the segregation and evaporation processes, we will show that, for appropriate values of the parameters (Y and S, the behavior may be described as segregation-controlled, evaporation-controlled, or competitive.

C. D. Stinespring, W. F. Lmvson / Slrrface segregation and evaporarion 213

To provide the background for this discussion, it is helpful to review the predictions of the segregation model as developed by Hofman and Erlewein [6] and to emphasize several of their observations. Specifically, in the absence of evaporation (S = - cc), the equilibrium concentration in each layer is obtained from eq. (3) for d Xi/d7 = 0. For our particular choice of the jump probability, the result is

=&exp(--a),fori=l, b

and

X5=X i b, for i>,2. @I

Eq. (7) shows the dependence of the equilibrium surface concentration, Xfq, on the segregation parameter, a, and the bulk segregant concentration, Xb. The kinetics of the approach to equilibrium are given by the numerical solution of eq. (3). Figs. 2 and 3 show typical results for these calculations. In fig. 2, the surface ~ncentration is plotted as a function of 7 for S = - cc (i.e., no evaporation), two values of a, and X, = 0.1. Fig. 3 shows the corresponding three dimensional plots of concentration versus T for the 2nd, lOth, 18th, etc. layers. For the given values of a and Xb = 0.1, it may be see that the surface concentration rapidly approaches the equilibrium values given by eq. (7) by depleting the near surface layers. The near surface layers, in turn, approach equilib~um by out-diffusion of the segregant from deeper layers. It is evident

1.00 a7

0 100 200 380 400 500 6 81

TiME(a2/D) Fig. 2. Segregant surface concentration versus dimensionless time for X, = 0.1, S = - a3 (no evaporation), and (a) a = - 12 and (b) (I = - 4.

214 C. D. Srrnespring. W. F. Lcrwson / Surfuce srgregution und eurrporutwn

from figs. 2 and 3 that, although the value of the equilibrium surface con- centration depends on a, the kinetics of the approach to equilibrium are primarily diffusion-controlled and relatively insensitive to variations in CL

Figs. 4 and 5 show calculations corresponding to figs. 2a and 3a but for X, = 0.01. Because the bulk concentration is lower, segregant depletion of the near surface layers is more extensive, and the kinetics of the approach to

equilibrium are somewhat slower than that shown in figs. 2 and 3 for X,, = 0.1. Thus, for non-evaporative situations, the segregation behavior is very sensitive

238

600 2

600 2

Fig. 3. Segregant concentration, X, versus dimensionless time and depth in atomic layers for (a) a=-l2,X,=O.l,and6=-co;(b)a=-4,X,=O.l,andS=-~0.

C. D. Stinespring, W. F. Lawson / h/ace segregahm and evaporation 215

to the bulk and near surface concentrations of the segregant. It is not surprising that departures from this behavior may be substantial in situations

where evaporation leads to the loss of segregant from the surface, near surface, and possibly even the bulk.

The general effects of evaporation on segregation are illustrated in figs. 6 and 7 which show X, versus T and the corresponding three dimensional plots

0 2000 4000 6000 6000 10000 12

TIME(a*/D)

Fig. 4. Segregant surface concentration versus dimensionless time for a = - 12, X, = 0.01, and s=-co.

12000 2

Fig. 5. Segregant concentration, X, versus dimensionless time and depth for a = - 12, X, = 0.01, andS=-m.

216 C. D. Stmespring, W.F. Lawon / Surface segregation and euaporcrtlon

for X, = 0.1, CY = - 12, and several values of 6. From a comparison of figs. 2a, 3a, 6a and 7a, it is evident that for 6 6 - 12 evaporation is not significant on a time scale commensurate with the approach to equilibrium in the non-evapora- tive case. Behavior such as this is referred to as segregation-controlled. For 8 = -4 evaporation is significant on the same time scale as the approach to equilibrium as evidenced by a comparison of figs. 2a, 3a, 6b and 7b. Behavior such as this is termed competitive. Finally, for 6 = - 1, the surface concentra- tion decreases rapidly from its initial value. This behavior is referred to as evaporation-controlled. The three-dimensional plots shown in fig. 7 indicate that, once evaporation becomes important, the initial near surface depletion propagates into the bulk with longer annealing times.

4. Discussion

In formulating the model, we have made relatively simple approximations for a,, v, and u/;,; As indicated previously, data for ai and Y,, as they become available, may be readily incorporated into the calculations. This could be especially important for the surface and near surface layers where reconstruc- tion may occur. Moreover, these considerations, along with possible modifica- tions in the value of the free energy of the near surface layers, represent a first step in describing multilayer segregation using this model.

0 00;v

0 100 200 300 400 500 6 0

TIME(a*/D)

Fig. 6. Segregant surface concentration versus dimensionless time for a = - 12, X, = 0.1, and (a)

S=-72,(b)6=-4,and(c)6=-1.

C. D. Shespring, W. F. Lawson / Surface segregation and evaporation 217

0 238

600 2

0.1

238 0

600 2 b

238

600 2

Fig. 7. Segregant concentration, X, versus dimensionless time and depth in atomic layers for (a) a=-12,Xb=0.1,andS=-12;(b)n-- 12, Xb = 0.1, and S = - 4; (c) = = - 12, Xb = 0.1, and

6=-l.

218 C. D. Stinespring, W. F Lawson / Surface segregution and evaporation

The approximation used for w, in this work is somewhat simpler than that used by Hofman and Erlewein [6] for the Cu-Sn system. They found that

W,, = (1 + X,/X;“) x;“, (9a)

and

wk=l-Xk, k>2, (9b)

for X;l= 0.33 gave the best fit for the Cu-Sn data. Physically, this jump probability limits the surface concentration to a maximum value of X;n and is proportional to the number of available surface sites at a given surface concentration. In this respect, the jump probability used here (eq. (6)) may be considered as a special case of eq. (9) for X;l = 1, and, for a general discussion of the implications of the model as opposed to a specific application, this approximation is quite adequate. Moreover, as indicated by Hofman and

Erlewein [6], the choice of a particular functional form for yk must be based on either a detailed theoretical analysis or the experimental data for a given system.

In comparing experimental and calculated results, it is important to note that surface analytical techniques based on electron spectroscopy sample a number of layers rather than just the surface. Thus, the observed surface concentration should be compared not with Xi but rather with a calculated value of X, which includes contributions from deeper planes [3]. That is,

XY = X,+X,exp[-d/X(E)] +X,exp[-2d/X(E)] + . . . . (IO)

where X(E) is the mean free path for an Auger or photoelectron of energy E

used in the experimental analysis and d is the planar spacing. Using known or estimated values of X(E) and d, XFlc may be determined within the frame- work of the model discussed here.

Experimental studies of surface segregation are generally performed by annealing the sample at elevated temperatures in order to determine the equilibrium surface composition as a function of temperature. Based on these measurements, the free energy of segregation or more precisely the segregation enthalpy is determined. Alternatively, the sample may be heated rapidly to the desired temperature and the approach to the equilibrium surface concentration monitored. From these data, the diffusivity of the segregant may in principle be determined.

In practice, the temperature range for experimental measurements of the segregation enthalpy must be chosen on the basis of three criteria. First, the temperature should be sufficiently high that annealing times are of a reasona- bly short duration. This, for the most part, is determined by the scale factor a2/D in the dimensionless time. Second, as seen from eq. (7), the temperature range must be sufficiently wide that variations in Xrq are observable. This is determined by the variable (Y. Third, the temperature range should be chosen in

C. D. Shespring. W. F. Lawson / Surface segregation and evaporation 219

such a manner as to avoid evaporation. This will be determined by the magnitudes of both (Y and S.

The theory developed here suggests that the behavior of a given system may be classified as segregation-controlled, evaporation-controlled, or competitive. Clearly, the best experimental results will be obtained if the temperature range can be selected such that the behavior falls in the segregation-controlled regime. In certain situations, however, this may not be possible. If evaporation can not be eliminated and the behavior falls in the competitive regime, AC,

may be determined by fitting the measured and calculated time dependence of X, or XFIC versus T if reasonably accurate data are available for AC,, AC,, Do and a. If the behavior falls in the evaporation-controlled regime, it appears that little useful information concerning the segregation process can be ob- tained.

As an example of these considerations, it is instructive to consider the Ni-Cu system. Recently Webber and Chadwick [l] have reported studies of this binary alloy which show the effects of evaporation on the segregation process. For 5% Cu in Ni alloys, their results indicate that Cu evaporation occurs as low as 870 K, and at 1000 K irreversible Cu depletion is observed. In a subsequent report [14], this latter temperature was reduced to 920 K. These data allow a qualitative comparison between experimental results and the predictions of the theory developed here. To make this comparison, however, it

is first necessary to obtain the appropriate values of AC,, AC,, AGv, D,,, and a for the Ni-Cu system. For this purpose, it shall be assumed that the entropy changes associated with the segregation and diffusion processes are negligible compared with that of vaporization. Thus, AC, - AH,, and AC, - AH,,

where AH, and AH,, are the enthalpies of segregation and diffusion, respec- tively.

Taking into account bond breaking and lattice strain, Wynblatt and Ku [8] report calculated values of AH, for a number of binary alloys. For Ni-Cu, they obtained a value of - 5.7 kcal/mole while calculations by Miedema [13] led to a value of - 8.5 kcal/mole. Very recently, Egelhoff [15] has reported experimental measurements of AH, using surface core-level binding-energy shifts which indicate that AH, is approximately - 10.6 kcal/mol. This, how- ever, represents the limiting value of AH, as the concentration of Cu goes to zero, and a value of - 5 kcal/mol is suggested for 10% Cu in Ni alloys. Thus, for 5% Cu in Ni alloys, the intermediate value of -8.5 kcal/mol is not unreasonable and will be used here.

Estimates of AC, may be made using data for pure Cu, AC?, and free energy of solution data, AGso,, for Ni-Cu alloys. Following Blakely and Shelton [4], AC, is given by

where AH and AS represent the appropriate enthalpy and entropy terms

220 C. D. Stinespring, W. F. Lawson / Surface segregation and euaporatmn

respectively. For 10% Cu in Ni at 973 K, Hultgren et al. [16] give values of 1.9 kcal/mol for AH,,,and 3.9 cal/mol . K for AS,,,,. Values for AH:” and As:‘” may be obtained from the JANAF data base [17]. For temperatures in the range of 800 to 1200 K, AH:” is 80.0 kcal/mol and As:‘” is 30.6 cal/mol . K. Thus, AG, is given by

AC, = 78.1 kcal/mol - T(26.7 cal/mol . K).

Finally, the lattice constant for pure Ni, 3.52 A, may be used to estimate a, and from radioactive tracer measurements for 64Cu in Ni [19], D,, is 0.57 cm2/s and AH,, is 61.7 kcal/mol for the temperature range of 1323 K to 1633 K. Although the temperature and concentration ranges for the physical data differ somewhat, these values should provide at least some insight into the behavior of the Ni-Cu system.

Fig. 8 shows a plot of XFq versus T obtained using eq. 7 for X, = 0.05 and AH, = - 8.5 kcal/mol. For the estimated value of AH,, it is clear that measurements for T a 700 K are required in order to observe the temperature dependence of X,. The time required to attain equilibrium may be determined using plots of X, versus 7. One such plot for T = 804 K ((Y = - 5.3, S = - cc, X, = 0.05) is shown in fig. 9. From this plot, the dimensionless time required to approach equilibrium is req - 600, and since a2/D is 100 s at 804 K. the

00’,, , , 1 I I I I I I I 1 I I !

500 I000 I500 2000 2: 00

TEMPERATURE (K)

Fig. 8. Calculated equilibrium segregant surface concentration X, versus temperature for X, = 0.05 and AH, = - 8.5 kcal/mol.

C. D. Srinespring W. F. Lawson / Surface segregation and eoaporarion 221

corresponding physical time is t, = q.,(a'/D) - 6 X lo4 s. For other tempera- tures, t,, may be determined in a similar manner. It is convenient to recall from the discussion of fig. 2 that r”4 is not particularly sensitive to changes in T (i.e. a) while a2/D depends exponentially on T-'. Thus the kinetics of the approach to equilibrium at a given temperature are essentially determined by the value of a2/D. These values along with those for t,, at selected tempera- tures assuming res - 600 are summarized in table 1. Based on these values, it is evident that for T > 855 K equilibration times will not be too long and, thus, for this temperature range, the first and second criteria discussed previously are readily satisfied for the Ni-Cu system.

1 00.

0 754

: /---I l-l -z 7T

400

TIME(a*/D)

Fig. 9. Calculated Cu surface concentration versus dimensionless time for 5% Cu in Ni at 804 K.

This calculation was performed using a = - 5.3 (AH, = - 8.5 kcal/mol) and 6 = - cc (no evapora-

tion).

Table 1

Calculated values of a2/D and teq for 5% Cu in Ni alloys

T(K) a 2/D (s)

718 104 759 103 804 102 855 10 914 1

Unfortunately the third criterion is not easily satisfied. This is illustrated in fig. 10 which shows plots of X, versus r that take evaporation into account. These calculations indicate that even for temperatures in the lower end of the desired range the segregation and evaporation processes are competitive, and for temperatures higher than 914 K the behavior may be classified as evapora- tion-controlled. Thus, using the estimated input data, it appears that the third criterion cannot be satisfied for the desired temperature range and, conse- quently, that a determination of AH, for Ni-Cu may not be possible using conventional measurement techniques.

The anticipated difficulties in measuring AH, for Ni-Cu do not preclude the possibility of testing the predictions of the theory against the available experimental data for Ni-Cu. Because of limitations in the data base, however, comparisons between experimental results and theoretical predictions are qualitative at best. Based on the data of Webber and Chadwick [l]. such a comparison indicates that the calculations shown in figs. 9 and 10 overestimate both the equilibrium time and the effect of evaporation. Specifically, Webber and Chadwick suggest, as a qualitative guide, that at 800 K equilibrium segregation should be obtained in approximately 5000 s with evaporation losses becoming important well after this. The predictions given here suggest t eq - 60,000 s at 804 K in the absence of evaporation, and when evaporation

0 50 100 150 200 250 300

TIME(a2/D) Fig. 10. Calculated Cu surface concentration versus dimensionless time for 5% Cu in Ni at (a) 718 K, (b) 759 K, (c) 804 K, (d) 855 K, and (e) 914 K. These calculations include the effects of

evaporation and were performed using AH, = - 8.5 kcal/mol, AC, = 78.1 kcal/mol- T(26.7

cal/mol. K) and AH, = 61.7 kcal/mol.

C. D. Stinespring, W.F. Lmwon / Surface segregation and evaporation 223

effects are included, significant evaporative loss may be observed after only 600 s.

Agreement between calculated and experimental results may be improved if the value of AH, is actually somewhat lower than 61.7 kcal/mol. The quoted experimental uncertainty in this value is f2 kcal/mol [19]. If variations between different measurement techniques are taken into account, however, this uncertainty is more realistically +5 kcal/mol, and since diffusivities in thin films and surface layers are known to be faster than bulk diffusivities, a value of AH, on the order of 56.7 kcal/mol is not unreasonable. Using this value for AH,, t, becomes 2680 s at 804 K in the absence of evaporation. Thus, within the uncertainty in AH,, the predicted equilibration time may be brought into reasonable agreement with the experimental results. Moreover, if evaporation is taken into account using the revised value of AH,, the plot of X, versus r shown in fig. 11 is obtained. In this case, X, approaches an “apparent” ~uilibrium value of 39% Cu from which only minor evaporative loss is observed for t > 200(a2/D) - 900 s. Such a result, if experimentally observed, could be interpreted as an indication of little, if any, evaporative loss.

It is interesting to note that the calculated “apparent” equilibrium con- centration is very close to the observed eq~librium concentration of 32% for temperatures less than 1000 K, and, although the calculations predict an early onset of evaporation loss, this may not be unrealistic. The measurements for

0 25-

0.00 ,,,,,J,,,f ,,,,,,‘,’ j,,,,,,,,,

0 50 100 150 200 250 c

TIME(a*/D)

10

Fig. 11. Calculated Cu surface concentration versus dimensionless time for 5% Cu in Ni. Input

parameters are the same as fig. lob except that now AH, = 56.7 kcat/moi.

224 C. D. Stinespring, W. F. Lawson / Surface segregation and evaporation

Ni-Cu were performed on samples which were slowly cooled [20] from the annealing temperature. Consequently, evaporative losses may be less apparent

than predicted due to “freezing in” effects during the cooling process [21]. Thus, taking into account the anticipated uncertainty in the input parameters and experimental results, the predictions of Ni-Cu behavior are qualitatively similar to the observations of Webber and Chadwick [l].

5. Conclusions

It is evident that the currently available data on Ni-Cu are not sufficient to quantitatively test the theoretical model. Specifically, such a test would require independent measurements of the surface concentration as a function of time as well as AH,, AH,, Do, and AC,. The Ni-Cu system provides definite advantages in this respect. The temperature dependence of AC, is reasonably well established [17] and AH, may be determined as a function of bulk concentration using surface core-level binding-energy shift measurements [15]. Moreover, measurements of AH,, and D, for the diffusion of Cu in Ni thin films rather than bulk Ni could be performed. Finally, the actual measure- ments of the Cu surface concentration versus time could be performed with the sample at the annealing temperature to avoid “freezing in” effects.

Acknowledgments

The authors wish to thank Fiong Kong Mak for assiting with the numerical calculations. In addition, the authors wish to thank G.W. Stewart and K.H. Casleton for many helpful discussions during the course of this work.

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C. D. S!inespring, W. F. Lawson / Surface segregation and evaporatiun 225

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