The influence of the cooling rate on interface segregation

Surface Science 0 North-Holland

14 (1978) 656-666 Publishing Company

THE INFLUENCE

OF THE COOLING RATE ON INTERFACE

SEGREGATION

I. J&GER Erich-Schmid-Institut fiir Festktirperphysik der asterreichischen Akademie der Wissenschaften, c/o Montanuniversitcit Leoben, Jahnstrasse 12, A-8700 Leoben, Austria Received

20 July 1977; manuscript

received

in final form 16 January

1978

The influence of the cooling rate on interface segregation is discussed theoretically. Criteria are obtained for a “freeze-in” of the segregation level assuming a state of equilibrium before quenching. The predictions of the theory are found to be in agreement with the work of Burton, Helms and Polizzotti on the difference in segregation levels between elevated and ambient temperature. Since the assumption of a dilute solid solution is used, the validity of the results is restricted to either very low bulk concentration or to elements which are not too strongly surface-active.

1. Introduction Interface segregation usually occurs only at elevated temperatures whereas there are reasons not to use surface sensitive techniques at such temperatures. Therefore the specimen under investigation is usually quenched to room temperature before application of methods such as AES, SIMS or others, commonly just by switching off the heater current, i.e. at a more or less unknown cooling rate. Values reported range from some ten to several hundred degrees K per second. In many papers dealing with interface (surface or grain boundary) segregation one reads things like: “This (cooling rate) is believed to be fast enough for diffusion to freeze in”. The results of Burton et al. [l] show clearly that this is not necessarily so. These authors investigated surface segregation of Au to Ni(ll1) surface and found large differences between the enrichment rations measured at annealing temperature and that measured at ambient temperature on a quenched sample. The aim of this paper is therefore to investigate this problem theoretically and to establish criteria for the conservation of surface enrichment during quench.

2. Theoretical

considerations

Three major assumptions are made throughout the following: (a) Before quenching bulk and interface are in thermodynamic equilibrium. 656

I. Jiiger / Cooling rate and interface segregation

657

(b) The bulk concentration of the segregant and (or) the segregation enthalpy is low enough that the formulae for dilute solid solutions are valid. (c) The cooling rate is a constant and there are no thermal gradients in the sample. Let the equilibrium temperature before quenching be TA then the assumptions (a) and (b) yield the interface concentration of segregant: COA)

= cb

exd-ahr,,,lRTd ,

ce being the bulk concentration and MSeg the enthalpy the configurational entropy. The assumption (c) means that T(t)=

TA-

of segregation

excluding

i-t

throughout cooling or at least until T is low enough that no more diffusion controlled processes can take place. This assumption is clearly an oversimplification because the time dependence of the temperature during quenching obeys rather a power or exponential law, but since it is difficult to record it with sufficient precision the values of -T reported are mean values and any further discussion is meaningless. The change of the interface segregation level during quenching cannot be easily investigated experimentally mainly because of the many parameters entering the problem. These are: the diffusion data DO, Q; the atomic jump distance a; the segregation enthalpy AHSeg; the equilibrium (annealing) temperature TA; the cooling rate -?‘. Furthermore the first four parameters can be varied only by choosing suitable couples of solvent and segregate; therefore only certain combinations of these are possible. In a theoretical investigation, however, each parameter can vary independently of the other ones. A theoretical investigation of interface segregation is mainly the problem of finding and solving a proper equation of diffusion. In principle, Fick’s second law can be extended to take into account. the presence of a driving force acting on the migrating atoms yielding Smoluchowksi’s well known equation ct = D(c,,

+ c,ux + Ck),

(3)

and the suffixes x, t denoting differentation with respect to with u = AHSeg (x)/RT, space and time respectively. The validity of the differential equation (3) depends on the tacit assumption that changes of either c or u over atomic distances are negligibly small. This is true throughout the whole crystal with one major exception; in

658

I. Jtiger / Cooling rate and interface segregation

the vicinity of an interface. It can be shown that AHses (x) for interface segregation vanishes everywhere except for a few atomic layers next to the interface [3]. For segregants not too strongly surface active, the force acting on the atoms in the second, third, .. . layer can be neglected (this assumption is made throughout the following), because its contribution is very small compared to the contribution at the interface layer. (“Interface” shall be used as a synonym for surface, grain boundary, all behaving similarly with respect to segregation.) So Smoluchowski’s equation (3) is valid everywhere except for the interface layer (furthermore referred to as “first layer”) and the adjacent one (the “second layer”). The diffusional flux between those two atomic layers has to be calculated by means of the atomic jump frequencies [2] neglecting evaporation: (4) cl and c2 denote the concentrations of the segregant in the first and second atomic layer respectively and a is the atomic jump distance. The change of c2 with time is given by C2t

= -c1t

-

div j (bulk)

(5)

Here j denotes the diffusional flux from the bulk of the material towards the second layer. Due to the absence of a driving force, this flux is governed by Fick’s second law,,but unfortunately the standard expression [4] :

-div j(bulk)

= -(fyi2

-!i

~~~~~2,

(6)

is valid only for the case of constant diffusion coefficient D.For time dependent D a similar expression can be derived according to Crank [4] by rearranging Fick’s second law

NO

ct=y-

a

a2

and introducing a2 CT = o(t)

(7)

cxx *D(t)ct=cxx’

Ct.

the dimensionless

coordinates

x (in units of a) an T by

(8)

This leads to -

CT -

cxx,

(9)

I. Jiiger / Cooling rate and interface segregation

659

with

T =

$j D(i) dt'.

00)

0

With these new coordinates

eqs. (4), (5) and (6) read

Cl7 = caW(7) - @-1(r),

(11)

7 c2Az) _I- (7_z)1/2’

c27=-cl&20

dz

(12)

using the abbreviation (13)

w(r) = exp ]-AkJ~T(r)l.

The problem is now to find or at least to approximate the inverse transformation to eq. (10) in order to evaluate W(r). Let us assume a linear relation between T and t T(t) = TA - i-t,

(14)

then the time dependence

D(t) =Do exp[-Q/R

of D,

(15)

WI,

yields (using eq. (10)) T = (Do/a2) _/ exp [-Q/ZZ(T,

- kt)] dt.

(16)

0

This integral can be evaluated in closed form by means of the exponential E,(z) (see e.g. Abramowitz [5]), yielding

r=$$(Taenp(-&)

-~~exp(-j$-)+~~I(j$)-~I(&)]]

integral

.(17)

TF denotes T(t) = TA - Tt, the temperature at the upper boundary of the integral in eq. (16). Furthermore the exponential integral E,(z) can be approximated [5] by E,(z) = (1 - o/z) e-‘/z,

(18)

with CY = 0.87627.

(19)

660

I. Jiiger / Cooling rate and interface segregation

This leads to

using the abbreviation

121) or

1).

hT2=2h(;+$-($Finally an approximation

for ln(T,ITA)

(22) is found [6] : (23)

A slight rearrangement of the equations shows that the six parameters quoted earlier enter the computation only through the following three dimensionless combinations:

QBTA,

(24)

Pz = -(a2Do)(i’/TJ,

(251

P,= -A&.$RTA.

(26)

Pz =

The range of these parameters can be estimated: P2 from lo-*’ to lo-l4 and P3 from 0 to 5. sidered, because strongly surface active etements dilute solid solution.) With these parameters and using eqs. (22) and Yt> = T*A + 4 + 10 +;T;p:fA(;)with

-P&--T

Pl - 4)l

P, usually ranges from 20 to 40, (Higher values of P3 are not conusually violate the assumption of a (23), T(t) is approximated 112

)

by (27)

a 1.

exp(Pd

With the aid of eqs. (27,28) the set of simultaneous differential equations (I 1) and (12) is solved by a numerical method of the Runge-Kutta type evaluating B’(T) by eq. (13) at each step and computing the integral involved in eq. (12) whenever

I. Jiiger / Cooling rate and interface segregation

661

needed by the trapezoidal approximation. These computations were done on an UNIVAC 494 (RZ Graz) and an IBM 1130 (RZ Leoben) computer varying Pr , Pz and P3 systematically with the only restriction being the limited CPU time.

3. Results A typical result of the computations is plotted in fig. 1. It shows the variation of Q/Q, the interface enrichment factor, with the temperature, The lower curve (A) gives the quenching behaviour, the upper curve (E) the equilibrium. The quenched curve consists of three major parts: At high temperatures (part I) the curve is nearly identical to the equilibrium curve (this part is more or less suppressed in fig. 1 due to the computational limits), delimited by a point of inflection, denoted by an arrow at 2-i. At medium temperatures (part II) the curve deviates more and more from equilibrium until at sufficiently low temperatures the interface enrichment keeps constant because the temperature is too low to permit any more significant diffusion. An unambiguous boundary between parts II and III can of course not be CS/cb

30-m

25.-

20+ A (quenched) m

m

Fig. 1. Variation A= quenched.

4

of the interface

5

6 enrichment

7

8 factor

9

10 1 (orb. units)

c&b

with temperature

,: E= equilibrium;

662

I. JSger / Cooling rate and interface segregation

given, but it turns out that 0.5 to 0.7 Ti is a good estimate. The enrichment factor at ambient temperature depends very sensitively on the cooling rate -?at temperatures around Ti, so a mean value of -fshould be evaluated mainly from a record of T versus t around Ti down to approximately 0.5 Ti. An estimate for Ti (rather ki = Ti/TA) is given by [2] : ki is the solution of the transcendental equation 11.7P*( 1.7 1P3 + P,)/kz = exp [-( 1.71 P3 + P,)/ki] .

(29)

This estimate turns out to be in good agreement with the computations of this paper. In certain cases the solution of eq. (29) yields values of ki greater than unity. This means that the temperature at which the inflection point of the cooling curve occurs would be higher than the annealing temperature and this is impossible because the quenching curve starts at TA. Nevertheless such values are of significance. Fig. 2 shows several cooling curves under various cooling conditions. Curve A is the same as in fig. 1 with a real point of inflection (Tt). Curves B and C correspond to the same physical parameters, equal cooling rate, but lower annealing temperatures (Ti = 0.818T& Z$*= 0.75OTi). Note that for these curves ki is greater

25

n

L

Fig. 2. Variation of the interface conditions (see text).

5

6

enrichment

7

8

factor

9

with

10 T(arh

wits)

temperature

foI various

quenching

I. Jiiger / Cooling rate and interface segregation

663

than unity, therefore Try’ is greater than T%‘. In these cases part I of the quenching curve is fully and part II (fig. 1) is partly suppressed. The same behaviour is shown by the curves A, D, F, and G. They correspond to equal annealing temperature T$D9FyG but different cooling rates (A is cooled slower than D, D slower than F and so on). Notice again the growing suppression of part II with increasing ki; curve G consists nearly completely of part III, the part of the curve, where the surface concentration does not change any more with temperature. This consideration together with a comparison between the values of c,(qu)/c,(TA) and ki in fig. 2 leads to a first criterion for “freeze-in” of the interface enrichment factor: The interface enrichment is greater than 1.2.

factor is preserved through quenching,

if ki from eq. (29)

The results of the computations for varying systematically the three parameters P, and P, are presented in form of contour lines in fig. 3 They show the expected sharp increase in the ratio of c,(qu)/c,(TA) with falling P2 (i.e. slower cooling). They also show the computational limits causing the white fields in figs. 3a and 3b. One feature of the figures is not so easy to understand. An increase in P3 (the segregation enthalpy in units of R7’,) is expected to yield an increase in c,(qu)/ cs(TA) because P3 acts as a driving force on the migrating atoms thus an increase in P3 should result in a stronger diffusional flux towards the interface and therefore in an increase in c,(qu). This is true in general, but the figures show this behaviour only for values of P3 less than 1.75, i.e. in the lower part of figs. 3a-3e. In the upper part of the figures a reverse relationship is found. The reason for this “abnormal” behaviour is as follows: According to eq. (1) the interface enrichment factor varies exponentially with P3 = - AHs,,/RT. Differentation of eq. (1) with respect to temperature (eq. (30)) shows, that the number of diffusing atoms necessary to obtain equilibrium during a change of the temperature by AT, which is proportional to AC,

P,,

AC, = -Q

%

exp(-Ah!,JRT)

AT = $-Ps

ev(+Ps)

AT

(30)

depends stronger than exponentially on P3. Therefore at high values of P3 the diffusional flux, which is roughly proportional to p3, is the less able to bring about the necessary number of atoms to maintain equilibrium, the more P3 increases, because this number increases stronger than exponentially with P3. At low values of PJ, on the other hand, the exponential term in eq. (30) tends towards unity, this invalidates the argument above and causes the expected “normal” behaviour. A more precise criterion for the conservation of the interface enrichment factor during quenching can be established using the results given in fig. 3. In fig. 4 the contour lines for c,(qu)/c,(TA) = 1.05 are plotted as functions of P2 and P3 (axes)

I. Jiiger / Cooling rate and interface segregation

664

s

(b)

L--

$1

3..

2.

l-

0

10.‘B

Ix)

10,

105

10-J’

10.‘6

&5

IU“

fj

Fig. 3. Contour lines of Cs(qu)/cs(TA) functions of PI, P2 and P3.

as

I. Jtiger / Cooling rate and interface segregation

Fig. 4. Contour

lines of cs(qu)/c,(TA)

= 1.05 as functions

ofPI,

665

Pz and PJ.

and Pr (parameter). These were chosen because in the author’s opinion a change of interface enrichment during quenching by five percent of its value at the annealing temperature is tolerable especially in view of the limited precision of surface sensitive techniques such as SIMS, AES etc. Comparing figs. 3a-31 with fig. 4 shows that the range of parameters PI to P3 at the right hand side of the appropriate curves corresponds to a deviation of the interface enrichment after quench from that at equilibrium by less than 5%. Therefore: The interface enrichment factor is conserved throughout quenching, if the corresponding point in jig. 4 lies on the right hand side of the corresponding contour line.

4. Conclusion The experimental work of Burton et al. [l] provides a good test for the validity of the theoretical results. Unfortunately, however, a direct comparison of the results of [l] with the computations reported here is not possible because the surface concentrations in the experiments are much too high for the assumption of a dilute solid solution to be valid *. So some additional computational runs were made using crs = c2 (1 - cr) W(7) - cr W(r),

(31)

which describes the kinetics of Langmuir-McLean segregation for small cb, instead of eq. (11). The results of these runs using the data: De = 0.92 cm2/sec, Q = 55 kcal/mol (from ref. [7]), a = 2.48 A, AHseg = -12 kcal/mol, -T= 50 K/set (from * To the author’s the literature.

knowledge

there

are no other

investigations

of this problem

to be found

in

666

I. Jiiger / Cooling rate and interface segregation

l:

15= 9 .$

*I

m.

AtT . QUENCHED o COMPUTED

00. *a l .9. . * l*.

.

l*..

l*.

l.

Q5

Fig. 5. Auger peak-height ratio Au/Ni(61) as a function of annealing temperature for a Ni crystal containing 1 at% Au. Measurements were made both at the annealing temperature and after quenching to room temperature. (From Burton et al. [ 11, used by permission.) Computed values are shown as open circles. ref. [l]), are shown in fig. 5 (open circles) to be in good agreement with the experimental results of Burton et al. *. (A systematic investigation of the influence of the cooling rate on the interface enrichment factor for the case of high interface concentrations is to be published elsewhere.) In spite of the good agreement, however, it is in general not a good method to measure the enrichment factor after quench and evaluate the enrichment at amrealing temperature by means of fig. 3a-3e, because a slight error in Q or -?can introduce a large error in cs(qu)/cs(TA). As stated earlier the aim of this paper is merely to quantify the statement: If the cooling rate is suffcientZy high, the interface enrichment is “frozen in”.

Acknowledgements The author is indebted to Prof. Dr. H.P. Sttiwe for his continuous work and many valuable discussions.

interest in this

References [ 11 J.J. Burton, C.R. Helms and R.S. Polizzotti, J. Vacuum Sci. Technol. 13 (1976) 204. [2] [3] [4] [S]

H.P. Sttiwe and I. Jager, Acta Met. 24 (1976) 605. I. Jager, Phys. Status Solidi (a) 33 (1976) 167. J. Crank, The Mathematics of Diffusion (Clarendon, Oxford, 1975). M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965). [6] I. Bronstein and K. Semendjajew, Taschenbuch der Mathematik (Leipzig, 1963). [7] Diffusion Data, Vol. 2 (1968). * Owing to the lack of precise information about the effective penetration distance of Auger electrons with energies of 61 and 69 eV supplied by Burton et al., this value had to be estimated to be approximately 1.2 atomic layers in accordance with the author’s remark on p. 205 of ref. [l].