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Two stages of binding energy between vacancies and in atoms in an Al matrix
Materials Science and Engineering, 19 (1975) 95--103 © Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands
Two Stages o f Binding Energy between Vacancies and In Atoms in an A1 Matrix
J.E. EPPERS(~N, P. FURNROHR and V. GEROLD
Max-Planck-Institut fi~r Metallforschung, Institut fi~r Werkstoffwissenschaften und Institut fiir Metallkunde, Universita't Stuttgart (Germany) (Received November 8, 1974)
SUMMARY
The complex composition dependence of the isochronal recovery of quenched-in electrical resistivity in dilute binary alloys of In in A1 can be explained b y the existence of t w o stages of bonding between vacancies and the solute In atoms. The present paper reports the application of this model to the experimental determination of the binding energy for the vacancy - In atom pair and of the energy with which a second vacancy is b o u n d to such a mixed pair. It is found that the binding energy associated with the pair is 0.29 + 0.02 eV and that for the second vacancy in the vacancy-rich cluster is 0.22 + 0.03 eV. The b o d y of previously reported pair binding energies for this system is summarized, and it is shown that, if the older data are corrected using the currently accepted best values for the various required parameters, a reasonable degree of consistency results, the consensus value being in satisfactory agreement with the present determination of the binding energy for the vacancy In atom pair.
1. INTRODUCTION
When small concentrations of In are added to A1 or to certain Al-alloys, the characteristic annealing behaviour following quenching is markedly altered. As Silcock [1] has pointed out, these p h e n o m e n a can best be attributed to a tendency for the In atoms to form a bond with the lattice vacancies which effectively prevents a fraction of the vacancies from participating individually in subsequent transport processes at low annealing temperatures. There have been several experimental ef-
forts to determine the strength of this bonding between a vacancy and an In atom in an A1 matrix; however, the reported binding energies, to be discussed presently, range from as low as 0.17 to a high of 0.42 eV. It is true that a number of these efforts might be classified as indirect in that the determinations were based upon the effect of small ternary additions of In to A l - Zn or A l - Cu alloys. But it is distressing to note the lack of agreement among those who have used binary alloys, since one might have expected that the interpretation would, in this case, be inherently more straightforward. On the basis of an analysis of the isochronal recovery of a series of quenched, dilute binary alloys, Epperson, Ffirnrohr and Gerold [2] have recently shown that the use of a dilute binary alloy is n o t necessarily sufficient to insure that the measurements can be interpreted meaningfully in terms of bonding between single vacancies and single In atoms alone. Rather, they found convincing evidence that, in addition to the vacancy - In atom pairs, a less stable vacancy-rich cluster is formed below a certain temperature when the vacancy concentration exceeds the impurity level. The possible occurrence of this vacancy-rich species severely restricts the alloy composition that can be used if, for example, electrical resistivity measurements are to be interpreted quantitatively according to current analytical schemes. With this in mind, the two-stage tight binding model outlined in reference 2 has been used as a guide in selection of alloy composition and experimental technique to permit the binding energies for the two stages of vacancy - In atom association to be measured. Knowledge of these energy parameters is a necessary prerequisite for doing a realistic mathematical analysis in order to test more quantitatively our under-
96 standing of the recovery processes in this alloy system. Because of this need and because of the more general interest in the matter of vacancy - impurity atom association, an itemized justification of the analysis will be presented, along with a discussion of the results and their correlation with the previous determinations for the Al - In system. Since it will be necessary to make occasional reference to the tight binding model, a brief description is in order here. It is assumed that during the quench the vacancies interact with the In atoms and, because of the high binding energies between the vacancies and impurities, a metastable equilibrium distribution forms consisting of single vacancies (v), isolated impurity atoms (i), bound nearest-neighbor vacancy - impurity atom pairs (vi), and triplets consisting of two vacancies and one impurity atom (v2i). The isochronal recovery of quenchedin resistivity is observed to occur in four stages which, in the order of increasing annealing temperature, are attributed to: (1) coagulation of " u n b o u n d " vacancies into dislocation loops, (2) decomposition of the vacancy-rich clusters into bound vi pairs and free vacancies, (3) dissolution of the dislocation loops and (4) precipitation of the In. For the present purpose, it is convenient to consider the limiting distribution of these four defect species in terms of three characteristic composition regions which are determined by the total concentration (all concentrations Cj are given in atom fractions, unless otherwise specified) of impurity atoms (Cio) and of vacancies (Cvo) as represented schematically in Fig. 1. This limiting distribu-
150
2. EXPERIMENTAL The electrical resistivity measurements were made using thin foil-like samples. The starting material was 99.999% A1 and 99.9999% In. The preparation of the alloys and the basic quenching, annealing and measuring techniques have been described elsewhere [ 2]. 2.1. Determination o f the mixed pair binding energy In order to determine the binding energy for the vacancy - In atom pair, the apparent energy of vacancy formation was measured for an alloy containing 474 at. p.p.m. In, an alloy well out in region I as characterized in Fig. 1. All electrical resistivity measurements were made at liquid nitrogen temperature after quenching from temperatures in the range of 408-563°C, controlled to +2 deg C. The reasons for this choice of alloy and range of quenching temperatures will be discussed in more detail in the following section.
CVo= lOOppm
,o
/,..
/
',,% ./ o
tion is defined as that resulting when the maxim u m number of vacancies are bound, each in the stablest configuration consistent with the total vacancy and impurity content. While in this oversimplified case the triplets are confined to regions II and III, it will be found in the real case that the vacancy-rich distribution spills over somewhat into region I as well. The practical consequence of this is that the Lomer assumptions [3] are not adequately fulfilled when the vacancy concentration is nearly equal to or exceeds the total tight binding impurity content but are nearly satisfied for alloys well out in region I.
iV'
"_._.L/_ ................. Io0 Ci, (ppm)
2O0
Fig. 1. Limiting distributions for vacancies, impurity atoms, pairs and triplets as a function of alloy content assuming complete partitioning and a constant total vacancy concentration for the various alloys. The subscripts denote the defect species.
2.2. Determination o f the second stage binding energy It was argued in reference 2 that the rate controlling step for the recovery stage centered at about 30°C is the thermal decomposition of vacancy-rich triplets into vacancy - In atom pairs and free vacancies, the latter then migrating to sinks. Hence, the second stage binding energy was determined by measuring the activation energy for migration of the defect responsible for this recovery stage in an alloy containing 70 at. p.p.m. In after quenching from 530°C. These choices too will be justified in the following section.
97 3. PRINCIPLE O F ANALYSIS AND E X P E R I M E N T A L RESULTS
3.1. T h e binding e n e r g y o f a v a c a n c y - I n a t o m pair
For a dilute f.c.c, alloy with binary impurity concentration Cio, the total equilibrium vacancy concentration at temperature T~ is given approximately by the Lomer expression [3] : Cvo(Ta) = A e x p ( - - E ~ / k T a ) × [1 -- 12 Cio + 12 Cio e x p ( E ~ ) / k T , ) ] ,
(1)
where A is the entropy factor, E~ is the formation energy of a vacancy in the pure solvent, and EB (1) is the nearest-neighbor binding energy between a single vacancy and a single substitutional impurity atom. It is essential to recognize that this expression is based u p o n the assumption that only single vacancies, isolated impurity atoms and b o u n d nearest-neighbor pairs of these elementary defects exist in the alloy. It is also implicitly assumed that the formation entropy is the same for free and b o u n d vacancies. Except for the possibility of creating significant concentrations of divacancies at the higher annealing temperatures, the assumptions with respect to the kinds of defects present at T , are probably adequately fulfilled for the dilute A l - In alloys. Complications which would confuse the measurements can arise, however, during quenching unless caution is exercised in the choice of alloy composition. On the basis of the isochronal recovery measurements reported in reference 2 and a more detailed mathematical analysis to be presented [4], it is concluded that the 474 p.p.m. In alloy being used here is far enough o u t in region I to avoid difficulties with the vacancy-rich defects. To effect the analysis, one begins b y noting that during quenching the system is partitioned, the pairs t e n d i n g to form to the maximum extent possible consistent with the In content and the vacancy concentration at the quenching temperature. Thus, following quenching (and partitioning) the resistivity can be represented formally according to Matthiessen's rule by ~'~q = ~ o + PvCv + PiCi + P v i e v i ,
(2)
where pv, Pi and Pvi are the specific resistivities of the defects indicated by the respective subscripts, and ~2o is the resistivity contribution
from the unperturbed matrix at the measuring temperature. There is no obvious way to heat treat these A l - In alloys so as to break up the vacancy - In atom pairs and, at the same time, leave all the In in solution [2]. Rather, it appears necessary to choose a reference state such that the precipitation of In is virtually complete. For this purpose an anneal of 30 min at 240°C suffices, since it is estimated that less than 10 at. p.p.m. In remains in solution [2]. In this reference condition, the resistivity of the alloy can be represented by ~2re~ = ~2o + ~ p ,
(3)
where ~p is the resistivity due to the precipitate particles. The contribution from the In remaining in solution has been omitted because the concentration is so small and, as will be argued presently, simple substitution of an In atom for one of A1 produces little change in the electrical resistivity. Thus, the excess resistivity introduced by a perfect quench is given approximately by /kp = PvCv + Pi(Cio - - Cvi) + PviCvi - - ~~p.
(4)
This presupposes that the resistivity measurements are not influenced b y changes in the vibrational spectrum, a requirement that is best fulfilled by making measurements at low temperature such as that of liquid nitrogen. It has already been specified that for the present alloy Cio >> Cvo, therefore Cv is nearly zero and the first term on the right side of eqn. (4) essentially vanishes. After the indicated reference anneal, the In precipitates are rather massive [5] and large precipitates present in such a small volume fraction contribute little to the resistivity [6]. Then for this special case, the resistivity can be adequately accounted for b y A p = (Pvi - - pi)Cvi -b PiCio.
(5)
It should be noted that according to this expression, the dependence of Ap on the quenching temperature (T,) is due entirely to the Cvi term. Hence, a plot of the experimental values of Ap versus Tq should give a measure of the constant term, piC~o. Such a plot is shown in Fig. 2 where it is seen that the measured Ap values sensibly approach zero for the lower quenching temperatures. Electron microscopy has revealed no evidence of In precipitation after a 15 min anneal at 400°C, b u t is not clear h o w far one is entitled to extrapolate
98
Cz.= ,~74 ppm i
~
I0'
!' J 0
' rq r'c J
Fig. 2. The quenched-in electrical resistivity Ap for an Al - 474 p.p.m. In alloy for a series of quenching temperatures. The ~ o values represent the difference between the resistivity in the quenched state and following a 30 min anneal at 240°C, all measurements being made at liquid nitrogen temperature.
below this temperature. However, the essence of this conclusion can be demonstrated by the following experiment. A series of alloys with different In content were quenched from 490°C into room temperature water and allowed to age at ambient temperature for 34 h, thus eliminating the u n b o u n d vacancies. The resistivity was measured at liquid nitrogen temperature in this condition and again after a 30 min anneal at 240°C. The resulting resistivity differences are shown in Fig. 3. The steep change in the low concentration range is due to the increasing concentration of bound defects with increasing In con-
(. 700
tent. The nearly constant value of the quenchedin resistivity b e y o n d a b o u t 100 at. p.p.m. In is due primarily to the annealing o u t of vacancyIn atom pairs. Our conclusion from above that the resistivity contribution from an isolated substitutional In atom is small compared with that from the b o u n d defects is consistent with these observations, otherwise one should find a positive slope in the higher concentration range also. The slight negative slope here is probably due to the spilling over of some of the vacancy-rich clusters into region I. That such spilling over occurs will be demonstrated in reference 4. Fortunately, the presence of these few vacancy-rich clusters produces a resistivity error which tends to compensate for a small positive contribution from the Pi Cio term in eqn. (5). In effect, it has been argued that, for this region I alloy, the vacancy population existing in equilibrium at Tq has, during the quench, reacted with the impurity species to form vacancy - In atom pairs and that these mixed pairs are responsible for the observed quenchedin resistivity. Based on the foregoing arguments, the quenched-in resistivity can be expressed as Ap = xA Pvi exp (--E~/kTq) × [1 -- 12 Cio + 12 Cio exp(--EBa)/kTq)],
(6) where × is the fraction of vacancies retained by the quench and is assumed to be independent of Tq. An Arrhenius plot of Ap versus 1/Tq does not yield a strictly straight line for the Lomer expression as Burke [7] has pointed out; however, a few numerical calculations suffice to convince one that the deviation is so small as to be undetectable experimentally and, hence, to a very good approximation the above expression can be written as
Ap = A' exp(--E*/kTq),
200
Ci~ppm ) 300
400
Fig. 3. The quenched-in resistivity retained at room temperature for a series of Al - In alloys quenched from 490°C. A full description is given in the text.
500
(7)
where E* is the effective energy of vacancy formation. A least-squares fit of the experimental data in the given temperature range from 408 ° to 563°C, shown in Fig. 4 in the form of an Arrhenius plot, yields the value of 0.66 eV for E*. The experimentally determined E~ can be related to a binding energy for the vacancy - In atom pairs b y computing the apparent energy of vacancy formation for a range of pair bind-
99 Cv°
C~°
'°I
=A'e'KD(-ET/kT) =4.74x I0 ~
E[(AI)=O.76eV e~(AI) = e 24 Computed foe 397 "C q, T~ 547°C
08
06
E~ (eV) 04
E~f = 0.66eV 02 0.~
i
= T~
IXIOOO)
Fig. 4. Arrhenius p l o t o f the quenched-in resistivity of an Al - 474 p.p.m. In alloy.
o
~
o~
o~o
E(BI) (eV) t2
ing energies, using the experimental temperature range. Such calculations have been carried o u t according to eqn. (1) over the range from 397 ° to 547°C and the effective energy of vacancy formation obtained by fitting the calculated results to the expression given by eqn. (7). The results shown in Fig. 5a were obtained for a 474 p.p.m, alloy using 0.76 eV for the formation energy of a vacancy in the pure solvent, a value that is widely accepted* for A1 (Simmons and Balluffi [8], Federighi [9], Takamura [10], Chen and Meshii [11]). As can be seen the experimental value for E* corresponds to a pair binding energy (EB (1)) amounting to a b o u t 0.29 eV with the error estimated n o t to exceed +0.02 eV. If the value of X in eqn. (6) is known, the experimental value for A' in eqn. (7) can be related to the specific electrical resistivity of a b o u n d mixed pair. So as to do an order of magnitude calculation, let us assume that X equals one. Then
I0
A'64_8 i.......
0
0
020
O40
~)
(~0
(ev)
Fig. 5. C o m p u t e d E~ and A' according to eqns. (1) and (7) in the t e x t ; (a) the effective f o r m a t i o n energy E~ and (b) the e n t r o p y factor A ' , each as a f u n c t i o n of the pair binding energy. The parameters used in the c o m p u t a t i o n s are given in the figure.
776
= - Z po, * (In quenching t y p e experiments, the values E~ = 0.76 eV and E v = 0.62 eV (the activation energy f r o m vacancy migration) are c o m m o n l y q u o t e d . However, in diffusion m e a s u r e m e n t s (Peterson and Rothman [12]), it is n o w generally accepted that for Al the activation energy for diffusion, Q = E~ + E v , is 1.26 eV. This discrepancy is at present n o t understood, but we may be dealing with either " e f f e c t i v e " energy parameters which are representative o f m o r e c o m p l e x entities than single vacancies or with parameters which are actually t e m p e r a t u r e dependent.
(b)
(8)
where Po is the specific electrical resistivity o f the A1 matrix, and A' can be obtained from Fig. 5b, which was c o m p u t e d as discussed in the previous paragraph. Using the indicated value for A' (3.65) and the experimentally determined 0.21 #~2 cm for po, one finds Pvi at liquid nitrogen temperature to be a b o u t 45 p~2 cm (i.e., 0.45 # ~ cm/at.% b o u n d pairs). Because no a t t e m p t was made to account for vacancy
100
loss during the quench, this should be regarded as a preliminary result which represents the lower limit for Pvi. Even so, it is worth noting that this is of the proper order of magnitude when compared with the value of 110 #~2 cm reported for a vacancy bound to a Ge atom in an A1 matrix ~[13]. 3.2. The binding energy f o r the vacancy-rich defects
The extent to which the excess vacancies are bound in the vacancy-rich clusters cannot be determined by an adaptation of the Lomer expression because the basic assumptions are not fulfilled as was discussed in a previous section. Rather, this second stage binding energy has been determined by the change of slope method (Damask and Dienes [14] ). Given an instantaneous change in temperature during an isothermal anneal, the energy of migration, Era, for a singly activated process can be determined from the limiting slopes, S~, at the point where the temperature was changed from the relationship, ln(Sz/S2) = _
E m (l/T1 -- l/T2).
k
(9)
It is known from reference 2 that a quenching temperature of 530°C and an alloy with 70 p.p.m. In will approximately maximize the recovery stage attributed to the partial decomposition of the vacancy-rich defects. In principle, the measurements can be made anywhere in the range from 0 ° to a b o u t 75°C without encountering interference from other recovery processes. In fact, the temperature range within which useful measurements can be made is much more restricted owing to practical considerations. That is, at the lower temperatures the recovery process is t o o slow and at the higher temperatures too rapid to permit a sufficiently accurate determination of the slope. Figure 6 shows the results of one such change of slope experiment for this alloy. The results of four independent determinations yield the value of 0.84 + 0.03 eV for the effective energy of migration. Since it is thought that the rate controlling step for the process responsible for this recovery stage is the thermal decomposition of the vacancy-rich clusters into mixed pairs and single vacancies, the binding energy is given by the difference between the effective activation energy of migration measured for this
I,\ ~
- s,
Tq = 530 C Ern--086 eV±OO3eV
\ ~
r~.,8.79"c
E "g<1
I i
I To :3a5 2 "C I I I I I
-3
o
,b 2'o ]o ~ to (rnin)
5'o ~b
Fig. 6. Segments of isothermal ageing curves for an alloy consisting of 70 p.p.m. In in A] quenched from 530°C from which the activation energy of migration for the second recovery stage is derived.
alloy and the energy for migration of a single vacancy in A1, which is taken to be 0.62 eV after Federighi [9]. This yields a value of 0.22 + 0.03 eV for the second stage binding energy, EB (2). While this is a rather substantial binding energy, it is, nonetheless, significantly smaller than the value of 0.29 + 0.02 eV found for the vacancy - In atom pair as discussed earlier. It will be shown in reference 4 that these t w o binding energies give a quite satisfactory description of the dependence on alloy composition and on quenching temperature of the low temperature isochronal recovery for an extended range of dilute A l - In alloys. According to the present model, the specific electrical resistivity of the triplet complex is given by the initial slope of the curve in Fig. 3. From this, one finds that Pv2i = 8 1 / ~ 2 cm (i.e., 0.81 p~2 cm/at.% complex), again referred to liquid nitrogen temperature. There are no k n o w n experimental measurements with which Pv2i can be compared; however, it should be pointed out that this value is fully consistent with the earlier argument that substitution of In for A1 has a minimal effect on the electrical resistivity, whereas the b o u n d vacancies make a substantial contribution. One might note also that according to the theoretical calculations by Dexter [15] and by Asdente and Friedel [16] for somewhat larger vacancy clusters, the resistivity increment per vacancy decreases slowly with increasing number of vacancies in the cluster, in agreement with the present observations.
101 4. C R I T I C A L C O M P A R I S O N W I T H P R E V I O U S L Y REPORTED
EXPERIMENTAL
VACANCY - INDIUM
ATOM BINDING ENERGIES
2 / s u m m a r y of the previously reported binding energies for vacancy - In atom pairs is given in Table 1, and a brief discussion of these earlier data is worthwhile. First, the value determined b y Entwistle, Fell and Koo [17] based on hardness measurements was intended as a rough approximation and, as such, is probably reasonable. The values of a b o u t 0.36 to 0.39 eV reported by the Japanese workers from measurements on ternary systems (Ohta and Hashimoto [18] and Ohta, Hashimoto and Tanimoto [19] ) appear to be erroneously high owing to their having used incorrect values for the binding energies between vacancies and the major solute atoms. Perry and Plumbridge [20] have pointed o u t that the value of 0.39 eV based on the A l - Zn work mentioned above would be reduced to 0.26 eV by using the currently accepted value of 0.06 eV rather than their value of 0.18 eV for the vacancy - Zn atom binding energy. In like manner, Ohta e t a l . [19] used the value of 0.20 eV for the vacancy - Cu atom binding energy; whereas, Perry and Entwistle [21] have recently quoted a value of 0.05 eV. Having made these corrections, it is interesting that all the older results based on the ternary alloys are n o w in substantial agreement with
the more recent ternary determinations by Entwistle and Perry [22], by Plumbridge and Entwistle [23] and b y Plumbridge [24], the range of values being from 0.24 to 0.28 eV. The upper end of this range is slightly lower than the present best estimate of 0.29 eV, y e t is statistically consistent within our estimated error of -+ 0.02 eV. Based u p o n the present viewpoint, the value of 0.42 + 0.04 eV reported by Duckworth and Burke [25] is thought to be anomalously high because their analysis assumed that the only species of b o u n d defects in their alloy was the vacancy - In atom pair. It will be shown [4] that in a 46 p.p.m. In alloy, which they used, a b o u t two-thirds of the In atoms are associated with two vacancies, with essentially all the others existing in the form of mixed pairs. Hence the effective binding energy is given by the weighted sum; EB~ ~ 0.42 eV if one uses our lower limit of 0.19 eV for EB ~2) or E~ ~ - 0.46 eV if the upper limit of 0.25 eV is used. Each of these possibilities is consistent with their reported value. Duckworth, Ramachandran and Burke [26] later estimated the binding energy to be a b o u t 0.27 eV based on loop density measurements. The values derived by Plumbridge [27,28] from his work on binary A l - In alloys are significantly lower than any of the other reported values, including his o w n more recent work [24], and it does not appear possible to
TABLE 1 S u m m a r y o f r e p o r t e d b i n d i n g e n e r g i e s * f o r v a c a n c y - I n a t o m p a i r s in A1 E~In (eV)
Alloy system
0.175 -- 0.2 0.39 0.39 0.36 0 . 4 2 -+ 0 . 0 4
A1 A1 A1 A1 A1
0.27 0.25 0.27
+ 0.02
A1 - Cu - I n A1 - C u - I n A1 - I n
0.18 0.24
- - 0.21 --0.28
-
Cu Zn Zn Cu In
-
In In In In
Method
References
Hardness Time to peak resistivity T i m e t o p e a k a n d initial r a t e Initial r a t e Apparent energy of vacancy formation Initial r a t e Initial rate Loop density (EM)
E n t w i s t l e , Fell a n d K o o [ 17 ] Ohta and Hashimoto [18] Ohta, Hashimoto and Tanimoto [19] O h t a , H a s h i m o t o a n d T a n i m o t o [ 19] D u c k w o r t h a n d B u r k e [ 25 ]
Entwistle and Perry [22 ] Perry and Entwistle [21 ] Duckworth, Ramachandran and Burke [26]
0.176 -- 0.205 0.26
+ 0.01
A1- In ~Al-Cu-In~
~Al ~ Zn- - In Al - In A1-Zn-In
Isothermal resisitivitydecay at 0°C Plumbridge [27 ] Initialrate in post reversion ageing Plumbridge and Entwistle [23 ] Isothermal resisitivity decay at P l u m b r i d g e [ 28 ] --21.3°C I n i t i a l r a t e in p o s t r e v e r s i o n ageing P l u m b r i d g e [ 2 4 ]
* The range of values or error limits are those given by the authors cited.
102
reconcile these data with the present and earlier results. While it is agreed that he has correctly attributed the isothermal decay of quenched-in resistivity at 0°C and --21.3°C to the annihilation of unassociated vacancies at sinks, it appears, on the basis of the twostage tight binding model, that even more vacancies are b o u n d to In atoms than he assumed. This, taken in combination with the inherent difficulties with vacancy losses during quenching, which he discussed at some length [27], suggests that this analysis may be seriously in error. The proximity of his higher value to the currently determined second stage binding energy is purely fortuitous. Hence, the cumulative evidence would indicate that the effective binding energy for the vacancy - In atom pair is no less than about 0.25 eV and no greater than a b o u t 0.30 eV, the correct value probably being nearer the upper end of this range. There is only one other known experimental value with which the second stage binding energy can be compared. Siegel [29] has reported a value for the activation energy of migration for the corresponding recovery stage in a 50 p.p.m. In alloy to be 0.87 eV. This is in satisfactory agreement with the value of 0.84 + 0.03 eV for the 70 p.p.m. In alloy on which the present determination of the second stage binding energy of 0.22 + 0.03 eV is based. The evidence presently available tends to suggest that the correct value for EB ~2) is nearer the higher value of 0.25 eV as will be discussed more fully in reference 4.
5. CONCLUSIONS AND GENERAL DISCUSSION
The essence of the two-stage tight binding model proposed for the dilute A l - In alloys is the existence of two b o u n d defect species, vacancy - In atom pairs and vacancy-rich clusters (probably triplets consisting of two vacancies associated with one In atom). The two required binding energies for vacancies with In atoms in an A1 matrix have been measured, and the pair binding energy is found to be 0.29 + 0.02 eV and that for a second vacancy b o u n d to such a mixed pair is 0.22 + 0.03 eV. It has been shown that most of the previous determinations of pair binding energies for this system are in reasonable agreement when reconsidered from the present state of know-
ledge. These investigations have, however, been based on studies of quenched materials, the energy parameters being inferred from measurements of various non-equilibrium properties. Thus the question of whether the discrepancy between the high binding energies derived from quenching t y p e experiments and the vanishingly small values deduced from high temperature measurements [ 12] is real or only apparent remains unanswered. The general agreement that is now emerging from the quenching type experiments emphasizes the need for a concerted effort to answer this fundamental question. It may well be that the binding energies determined from the quenching type experiments should be designated as interaction energies or binding free energies, because a term due to the vibrational entropy is always included. There are no known experimental efforts to assess the significance of the entropy contribution to the apparent binding energies for this system. However, it is known from the theoretical work by Girifalco and Kuhlmann-Wilsdorf [ 30] and Kuhlmann-Wilsd o f f [31] that the interaction energy associated with a vacancy and a stress field in a metal is a complex function whose magnitude is dependent on temperature. This temperature dependence is associated with an entropy term which arises because of changes of the elastic constants and, hence, of the vibrational spectrum in the immediate vicinity of the lattice defects. All too commonly, the potential consequence of such an entropy effect is ignored when attempting a critical comparison of binding energies determined in widely separated temperature regions.
ACKNOWLEDGEMENTS
We wish to express o u t thanks to Carmen Ruano for valuable comments concerning this manuscript.
REFERENCES 1 J.M. Silcock, Phil. Mag., 4 (i959) 1187. 2 J.E. Epperson, P. Fiirnrohr and V. Gerold, Phil. Mag., 29 (1974) 1189. 3 W.M. Lomer, in Vacancies and Other Point Defects in Metals and Alloys, Inst. Metals, London, 1958, p. 79.
103 4 J.E. Epperson, P. Fiirnrohr and V. Gerold, to be published. 5 J.M. Silcock, J. Inst. Metals, 84 (1955 - 56) 19. 6 P. Fiirnrohr, Diplomarbeit, Universit~it Stuttgart, 1970. 7 J. Burke, J. Less-Common Metals, 28 (1972) 441. 8 R.O. Simmons and R.W. Balluffi, Phys. Rev., 117 (1960) 52. 9 T. Federighi, in Lattice Defects in Quenched Metals, Academic Press, New York, 1965, p. 217. 10 J. Takamura, in Lattice Defects in Quenched Metals, AcademicPress, New York, 1965, p. 521. 11 K.Y. Chen and M. Meshii, Mater. Sci. Eng., 16 (1974) 115. 12 N.L. Peterson and S.J. Rothman, Phys. Rev., B1 (1970) 3264. 13 M. Beller, P. Ffirnrohr and V. Gerold, Phys. Status Solidi, (a) 17 (1973) 435. 14 A.C. Damask and G.J. Dienes, Point Defects in Metals, Gordon and Breach, New York, 1963, p. 145. 15 D.L. Dexter, Phys. Rev., 103 (1956) 107. 16 M. Asdente and J. Friedel, J. Phys. Chem. Solids, 11 (1959) 115. 17 K.M. Entwistle, J.H. Fell and K.I. Koo, J. Inst. Metals, 91 (1962 - 63) 84.
18 M. Ohta and F. Hashimoto, J. Phys. Soc. Japan, 19 (1964) 1987. 19 M. Ohta, F. Hashimoto and T. Tanimoto, J. Phys. Soc. Japan, 21 (1966) 562. 20 A.J. Perry and W.J. Plumbridge, Phil. Mag., 25 (1972) 139. 21 A.J. Perry and K.M. Entwistle, J. Inst. Metals, 96 (1968) 344. 22 K.M. Entwistle and A.J. Perry, J. Inst. Metals, 94 (1966) 24. 23 W.J. Plumbridge and K.M. Entwistle, J. Inst. Metals, 97 (1969) 232. 24 W.J. Plumbridge, Aluminium, 49-8 (1973) 540. 25 F.C. Duckworth and J. Burke, Phil. Mag., 14 (1966) 473. 26 F.C. Duckworth, T.R. Ramachandran and J. Burke, Intern. Conf. on Vacancies and Interstitials in Metals, J~ilich-Conf. 2, Vol. 1, 1968 p. 185. 27 W.J. Plumbridge, Phil. Mag., 20 (1969) 707. 28 W.J. Plumbridge, Scripta Met., 4 (1970) 85. 29 R.W. Siegel, Tagung der Deutschen Physikalischen Gesellschaft, Miinster, 1973. 30. L.A. Girifalco and D. Kuhlmann-Wilsdorf, J. Appl. Phys., 35 (1964) 438. 31 D. Kuhlmann-Wilsdorf, J. Appl. Phys., 36 (1965) 637.
Two Stages o f Binding Energy between Vacancies and In Atoms in an A1 Matrix
J.E. EPPERS(~N, P. FURNROHR and V. GEROLD
Max-Planck-Institut fi~r Metallforschung, Institut fi~r Werkstoffwissenschaften und Institut fiir Metallkunde, Universita't Stuttgart (Germany) (Received November 8, 1974)
SUMMARY
The complex composition dependence of the isochronal recovery of quenched-in electrical resistivity in dilute binary alloys of In in A1 can be explained b y the existence of t w o stages of bonding between vacancies and the solute In atoms. The present paper reports the application of this model to the experimental determination of the binding energy for the vacancy - In atom pair and of the energy with which a second vacancy is b o u n d to such a mixed pair. It is found that the binding energy associated with the pair is 0.29 + 0.02 eV and that for the second vacancy in the vacancy-rich cluster is 0.22 + 0.03 eV. The b o d y of previously reported pair binding energies for this system is summarized, and it is shown that, if the older data are corrected using the currently accepted best values for the various required parameters, a reasonable degree of consistency results, the consensus value being in satisfactory agreement with the present determination of the binding energy for the vacancy In atom pair.
1. INTRODUCTION
When small concentrations of In are added to A1 or to certain Al-alloys, the characteristic annealing behaviour following quenching is markedly altered. As Silcock [1] has pointed out, these p h e n o m e n a can best be attributed to a tendency for the In atoms to form a bond with the lattice vacancies which effectively prevents a fraction of the vacancies from participating individually in subsequent transport processes at low annealing temperatures. There have been several experimental ef-
forts to determine the strength of this bonding between a vacancy and an In atom in an A1 matrix; however, the reported binding energies, to be discussed presently, range from as low as 0.17 to a high of 0.42 eV. It is true that a number of these efforts might be classified as indirect in that the determinations were based upon the effect of small ternary additions of In to A l - Zn or A l - Cu alloys. But it is distressing to note the lack of agreement among those who have used binary alloys, since one might have expected that the interpretation would, in this case, be inherently more straightforward. On the basis of an analysis of the isochronal recovery of a series of quenched, dilute binary alloys, Epperson, Ffirnrohr and Gerold [2] have recently shown that the use of a dilute binary alloy is n o t necessarily sufficient to insure that the measurements can be interpreted meaningfully in terms of bonding between single vacancies and single In atoms alone. Rather, they found convincing evidence that, in addition to the vacancy - In atom pairs, a less stable vacancy-rich cluster is formed below a certain temperature when the vacancy concentration exceeds the impurity level. The possible occurrence of this vacancy-rich species severely restricts the alloy composition that can be used if, for example, electrical resistivity measurements are to be interpreted quantitatively according to current analytical schemes. With this in mind, the two-stage tight binding model outlined in reference 2 has been used as a guide in selection of alloy composition and experimental technique to permit the binding energies for the two stages of vacancy - In atom association to be measured. Knowledge of these energy parameters is a necessary prerequisite for doing a realistic mathematical analysis in order to test more quantitatively our under-
96 standing of the recovery processes in this alloy system. Because of this need and because of the more general interest in the matter of vacancy - impurity atom association, an itemized justification of the analysis will be presented, along with a discussion of the results and their correlation with the previous determinations for the Al - In system. Since it will be necessary to make occasional reference to the tight binding model, a brief description is in order here. It is assumed that during the quench the vacancies interact with the In atoms and, because of the high binding energies between the vacancies and impurities, a metastable equilibrium distribution forms consisting of single vacancies (v), isolated impurity atoms (i), bound nearest-neighbor vacancy - impurity atom pairs (vi), and triplets consisting of two vacancies and one impurity atom (v2i). The isochronal recovery of quenchedin resistivity is observed to occur in four stages which, in the order of increasing annealing temperature, are attributed to: (1) coagulation of " u n b o u n d " vacancies into dislocation loops, (2) decomposition of the vacancy-rich clusters into bound vi pairs and free vacancies, (3) dissolution of the dislocation loops and (4) precipitation of the In. For the present purpose, it is convenient to consider the limiting distribution of these four defect species in terms of three characteristic composition regions which are determined by the total concentration (all concentrations Cj are given in atom fractions, unless otherwise specified) of impurity atoms (Cio) and of vacancies (Cvo) as represented schematically in Fig. 1. This limiting distribu-
150
2. EXPERIMENTAL The electrical resistivity measurements were made using thin foil-like samples. The starting material was 99.999% A1 and 99.9999% In. The preparation of the alloys and the basic quenching, annealing and measuring techniques have been described elsewhere [ 2]. 2.1. Determination o f the mixed pair binding energy In order to determine the binding energy for the vacancy - In atom pair, the apparent energy of vacancy formation was measured for an alloy containing 474 at. p.p.m. In, an alloy well out in region I as characterized in Fig. 1. All electrical resistivity measurements were made at liquid nitrogen temperature after quenching from temperatures in the range of 408-563°C, controlled to +2 deg C. The reasons for this choice of alloy and range of quenching temperatures will be discussed in more detail in the following section.
CVo= lOOppm
,o
/,..
/
',,% ./ o
tion is defined as that resulting when the maxim u m number of vacancies are bound, each in the stablest configuration consistent with the total vacancy and impurity content. While in this oversimplified case the triplets are confined to regions II and III, it will be found in the real case that the vacancy-rich distribution spills over somewhat into region I as well. The practical consequence of this is that the Lomer assumptions [3] are not adequately fulfilled when the vacancy concentration is nearly equal to or exceeds the total tight binding impurity content but are nearly satisfied for alloys well out in region I.
iV'
"_._.L/_ ................. Io0 Ci, (ppm)
2O0
Fig. 1. Limiting distributions for vacancies, impurity atoms, pairs and triplets as a function of alloy content assuming complete partitioning and a constant total vacancy concentration for the various alloys. The subscripts denote the defect species.
2.2. Determination o f the second stage binding energy It was argued in reference 2 that the rate controlling step for the recovery stage centered at about 30°C is the thermal decomposition of vacancy-rich triplets into vacancy - In atom pairs and free vacancies, the latter then migrating to sinks. Hence, the second stage binding energy was determined by measuring the activation energy for migration of the defect responsible for this recovery stage in an alloy containing 70 at. p.p.m. In after quenching from 530°C. These choices too will be justified in the following section.
97 3. PRINCIPLE O F ANALYSIS AND E X P E R I M E N T A L RESULTS
3.1. T h e binding e n e r g y o f a v a c a n c y - I n a t o m pair
For a dilute f.c.c, alloy with binary impurity concentration Cio, the total equilibrium vacancy concentration at temperature T~ is given approximately by the Lomer expression [3] : Cvo(Ta) = A e x p ( - - E ~ / k T a ) × [1 -- 12 Cio + 12 Cio e x p ( E ~ ) / k T , ) ] ,
(1)
where A is the entropy factor, E~ is the formation energy of a vacancy in the pure solvent, and EB (1) is the nearest-neighbor binding energy between a single vacancy and a single substitutional impurity atom. It is essential to recognize that this expression is based u p o n the assumption that only single vacancies, isolated impurity atoms and b o u n d nearest-neighbor pairs of these elementary defects exist in the alloy. It is also implicitly assumed that the formation entropy is the same for free and b o u n d vacancies. Except for the possibility of creating significant concentrations of divacancies at the higher annealing temperatures, the assumptions with respect to the kinds of defects present at T , are probably adequately fulfilled for the dilute A l - In alloys. Complications which would confuse the measurements can arise, however, during quenching unless caution is exercised in the choice of alloy composition. On the basis of the isochronal recovery measurements reported in reference 2 and a more detailed mathematical analysis to be presented [4], it is concluded that the 474 p.p.m. In alloy being used here is far enough o u t in region I to avoid difficulties with the vacancy-rich defects. To effect the analysis, one begins b y noting that during quenching the system is partitioned, the pairs t e n d i n g to form to the maximum extent possible consistent with the In content and the vacancy concentration at the quenching temperature. Thus, following quenching (and partitioning) the resistivity can be represented formally according to Matthiessen's rule by ~'~q = ~ o + PvCv + PiCi + P v i e v i ,
(2)
where pv, Pi and Pvi are the specific resistivities of the defects indicated by the respective subscripts, and ~2o is the resistivity contribution
from the unperturbed matrix at the measuring temperature. There is no obvious way to heat treat these A l - In alloys so as to break up the vacancy - In atom pairs and, at the same time, leave all the In in solution [2]. Rather, it appears necessary to choose a reference state such that the precipitation of In is virtually complete. For this purpose an anneal of 30 min at 240°C suffices, since it is estimated that less than 10 at. p.p.m. In remains in solution [2]. In this reference condition, the resistivity of the alloy can be represented by ~2re~ = ~2o + ~ p ,
(3)
where ~p is the resistivity due to the precipitate particles. The contribution from the In remaining in solution has been omitted because the concentration is so small and, as will be argued presently, simple substitution of an In atom for one of A1 produces little change in the electrical resistivity. Thus, the excess resistivity introduced by a perfect quench is given approximately by /kp = PvCv + Pi(Cio - - Cvi) + PviCvi - - ~~p.
(4)
This presupposes that the resistivity measurements are not influenced b y changes in the vibrational spectrum, a requirement that is best fulfilled by making measurements at low temperature such as that of liquid nitrogen. It has already been specified that for the present alloy Cio >> Cvo, therefore Cv is nearly zero and the first term on the right side of eqn. (4) essentially vanishes. After the indicated reference anneal, the In precipitates are rather massive [5] and large precipitates present in such a small volume fraction contribute little to the resistivity [6]. Then for this special case, the resistivity can be adequately accounted for b y A p = (Pvi - - pi)Cvi -b PiCio.
(5)
It should be noted that according to this expression, the dependence of Ap on the quenching temperature (T,) is due entirely to the Cvi term. Hence, a plot of the experimental values of Ap versus Tq should give a measure of the constant term, piC~o. Such a plot is shown in Fig. 2 where it is seen that the measured Ap values sensibly approach zero for the lower quenching temperatures. Electron microscopy has revealed no evidence of In precipitation after a 15 min anneal at 400°C, b u t is not clear h o w far one is entitled to extrapolate
98
Cz.= ,~74 ppm i
~
I0'
!' J 0
' rq r'c J
Fig. 2. The quenched-in electrical resistivity Ap for an Al - 474 p.p.m. In alloy for a series of quenching temperatures. The ~ o values represent the difference between the resistivity in the quenched state and following a 30 min anneal at 240°C, all measurements being made at liquid nitrogen temperature.
below this temperature. However, the essence of this conclusion can be demonstrated by the following experiment. A series of alloys with different In content were quenched from 490°C into room temperature water and allowed to age at ambient temperature for 34 h, thus eliminating the u n b o u n d vacancies. The resistivity was measured at liquid nitrogen temperature in this condition and again after a 30 min anneal at 240°C. The resulting resistivity differences are shown in Fig. 3. The steep change in the low concentration range is due to the increasing concentration of bound defects with increasing In con-
(. 700
tent. The nearly constant value of the quenchedin resistivity b e y o n d a b o u t 100 at. p.p.m. In is due primarily to the annealing o u t of vacancyIn atom pairs. Our conclusion from above that the resistivity contribution from an isolated substitutional In atom is small compared with that from the b o u n d defects is consistent with these observations, otherwise one should find a positive slope in the higher concentration range also. The slight negative slope here is probably due to the spilling over of some of the vacancy-rich clusters into region I. That such spilling over occurs will be demonstrated in reference 4. Fortunately, the presence of these few vacancy-rich clusters produces a resistivity error which tends to compensate for a small positive contribution from the Pi Cio term in eqn. (5). In effect, it has been argued that, for this region I alloy, the vacancy population existing in equilibrium at Tq has, during the quench, reacted with the impurity species to form vacancy - In atom pairs and that these mixed pairs are responsible for the observed quenchedin resistivity. Based on the foregoing arguments, the quenched-in resistivity can be expressed as Ap = xA Pvi exp (--E~/kTq) × [1 -- 12 Cio + 12 Cio exp(--EBa)/kTq)],
(6) where × is the fraction of vacancies retained by the quench and is assumed to be independent of Tq. An Arrhenius plot of Ap versus 1/Tq does not yield a strictly straight line for the Lomer expression as Burke [7] has pointed out; however, a few numerical calculations suffice to convince one that the deviation is so small as to be undetectable experimentally and, hence, to a very good approximation the above expression can be written as
Ap = A' exp(--E*/kTq),
200
Ci~ppm ) 300
400
Fig. 3. The quenched-in resistivity retained at room temperature for a series of Al - In alloys quenched from 490°C. A full description is given in the text.
500
(7)
where E* is the effective energy of vacancy formation. A least-squares fit of the experimental data in the given temperature range from 408 ° to 563°C, shown in Fig. 4 in the form of an Arrhenius plot, yields the value of 0.66 eV for E*. The experimentally determined E~ can be related to a binding energy for the vacancy - In atom pairs b y computing the apparent energy of vacancy formation for a range of pair bind-
99 Cv°
C~°
'°I
=A'e'KD(-ET/kT) =4.74x I0 ~
E[(AI)=O.76eV e~(AI) = e 24 Computed foe 397 "C q, T~ 547°C
08
06
E~ (eV) 04
E~f = 0.66eV 02 0.~
i
= T~
IXIOOO)
Fig. 4. Arrhenius p l o t o f the quenched-in resistivity of an Al - 474 p.p.m. In alloy.
o
~
o~
o~o
E(BI) (eV) t2
ing energies, using the experimental temperature range. Such calculations have been carried o u t according to eqn. (1) over the range from 397 ° to 547°C and the effective energy of vacancy formation obtained by fitting the calculated results to the expression given by eqn. (7). The results shown in Fig. 5a were obtained for a 474 p.p.m, alloy using 0.76 eV for the formation energy of a vacancy in the pure solvent, a value that is widely accepted* for A1 (Simmons and Balluffi [8], Federighi [9], Takamura [10], Chen and Meshii [11]). As can be seen the experimental value for E* corresponds to a pair binding energy (EB (1)) amounting to a b o u t 0.29 eV with the error estimated n o t to exceed +0.02 eV. If the value of X in eqn. (6) is known, the experimental value for A' in eqn. (7) can be related to the specific electrical resistivity of a b o u n d mixed pair. So as to do an order of magnitude calculation, let us assume that X equals one. Then
I0
A'64_8 i.......
0
0
020
O40
~)
(~0
(ev)
Fig. 5. C o m p u t e d E~ and A' according to eqns. (1) and (7) in the t e x t ; (a) the effective f o r m a t i o n energy E~ and (b) the e n t r o p y factor A ' , each as a f u n c t i o n of the pair binding energy. The parameters used in the c o m p u t a t i o n s are given in the figure.
776
= - Z po, * (In quenching t y p e experiments, the values E~ = 0.76 eV and E v = 0.62 eV (the activation energy f r o m vacancy migration) are c o m m o n l y q u o t e d . However, in diffusion m e a s u r e m e n t s (Peterson and Rothman [12]), it is n o w generally accepted that for Al the activation energy for diffusion, Q = E~ + E v , is 1.26 eV. This discrepancy is at present n o t understood, but we may be dealing with either " e f f e c t i v e " energy parameters which are representative o f m o r e c o m p l e x entities than single vacancies or with parameters which are actually t e m p e r a t u r e dependent.
(b)
(8)
where Po is the specific electrical resistivity o f the A1 matrix, and A' can be obtained from Fig. 5b, which was c o m p u t e d as discussed in the previous paragraph. Using the indicated value for A' (3.65) and the experimentally determined 0.21 #~2 cm for po, one finds Pvi at liquid nitrogen temperature to be a b o u t 45 p~2 cm (i.e., 0.45 # ~ cm/at.% b o u n d pairs). Because no a t t e m p t was made to account for vacancy
100
loss during the quench, this should be regarded as a preliminary result which represents the lower limit for Pvi. Even so, it is worth noting that this is of the proper order of magnitude when compared with the value of 110 #~2 cm reported for a vacancy bound to a Ge atom in an A1 matrix ~[13]. 3.2. The binding energy f o r the vacancy-rich defects
The extent to which the excess vacancies are bound in the vacancy-rich clusters cannot be determined by an adaptation of the Lomer expression because the basic assumptions are not fulfilled as was discussed in a previous section. Rather, this second stage binding energy has been determined by the change of slope method (Damask and Dienes [14] ). Given an instantaneous change in temperature during an isothermal anneal, the energy of migration, Era, for a singly activated process can be determined from the limiting slopes, S~, at the point where the temperature was changed from the relationship, ln(Sz/S2) = _
E m (l/T1 -- l/T2).
k
(9)
It is known from reference 2 that a quenching temperature of 530°C and an alloy with 70 p.p.m. In will approximately maximize the recovery stage attributed to the partial decomposition of the vacancy-rich defects. In principle, the measurements can be made anywhere in the range from 0 ° to a b o u t 75°C without encountering interference from other recovery processes. In fact, the temperature range within which useful measurements can be made is much more restricted owing to practical considerations. That is, at the lower temperatures the recovery process is t o o slow and at the higher temperatures too rapid to permit a sufficiently accurate determination of the slope. Figure 6 shows the results of one such change of slope experiment for this alloy. The results of four independent determinations yield the value of 0.84 + 0.03 eV for the effective energy of migration. Since it is thought that the rate controlling step for the process responsible for this recovery stage is the thermal decomposition of the vacancy-rich clusters into mixed pairs and single vacancies, the binding energy is given by the difference between the effective activation energy of migration measured for this
I,\ ~
- s,
Tq = 530 C Ern--086 eV±OO3eV
\ ~
r~.,8.79"c
E "g<1
I i
I To :3a5 2 "C I I I I I
-3
o
,b 2'o ]o ~ to (rnin)
5'o ~b
Fig. 6. Segments of isothermal ageing curves for an alloy consisting of 70 p.p.m. In in A] quenched from 530°C from which the activation energy of migration for the second recovery stage is derived.
alloy and the energy for migration of a single vacancy in A1, which is taken to be 0.62 eV after Federighi [9]. This yields a value of 0.22 + 0.03 eV for the second stage binding energy, EB (2). While this is a rather substantial binding energy, it is, nonetheless, significantly smaller than the value of 0.29 + 0.02 eV found for the vacancy - In atom pair as discussed earlier. It will be shown in reference 4 that these t w o binding energies give a quite satisfactory description of the dependence on alloy composition and on quenching temperature of the low temperature isochronal recovery for an extended range of dilute A l - In alloys. According to the present model, the specific electrical resistivity of the triplet complex is given by the initial slope of the curve in Fig. 3. From this, one finds that Pv2i = 8 1 / ~ 2 cm (i.e., 0.81 p~2 cm/at.% complex), again referred to liquid nitrogen temperature. There are no k n o w n experimental measurements with which Pv2i can be compared; however, it should be pointed out that this value is fully consistent with the earlier argument that substitution of In for A1 has a minimal effect on the electrical resistivity, whereas the b o u n d vacancies make a substantial contribution. One might note also that according to the theoretical calculations by Dexter [15] and by Asdente and Friedel [16] for somewhat larger vacancy clusters, the resistivity increment per vacancy decreases slowly with increasing number of vacancies in the cluster, in agreement with the present observations.
101 4. C R I T I C A L C O M P A R I S O N W I T H P R E V I O U S L Y REPORTED
EXPERIMENTAL
VACANCY - INDIUM
ATOM BINDING ENERGIES
2 / s u m m a r y of the previously reported binding energies for vacancy - In atom pairs is given in Table 1, and a brief discussion of these earlier data is worthwhile. First, the value determined b y Entwistle, Fell and Koo [17] based on hardness measurements was intended as a rough approximation and, as such, is probably reasonable. The values of a b o u t 0.36 to 0.39 eV reported by the Japanese workers from measurements on ternary systems (Ohta and Hashimoto [18] and Ohta, Hashimoto and Tanimoto [19] ) appear to be erroneously high owing to their having used incorrect values for the binding energies between vacancies and the major solute atoms. Perry and Plumbridge [20] have pointed o u t that the value of 0.39 eV based on the A l - Zn work mentioned above would be reduced to 0.26 eV by using the currently accepted value of 0.06 eV rather than their value of 0.18 eV for the vacancy - Zn atom binding energy. In like manner, Ohta e t a l . [19] used the value of 0.20 eV for the vacancy - Cu atom binding energy; whereas, Perry and Entwistle [21] have recently quoted a value of 0.05 eV. Having made these corrections, it is interesting that all the older results based on the ternary alloys are n o w in substantial agreement with
the more recent ternary determinations by Entwistle and Perry [22], by Plumbridge and Entwistle [23] and b y Plumbridge [24], the range of values being from 0.24 to 0.28 eV. The upper end of this range is slightly lower than the present best estimate of 0.29 eV, y e t is statistically consistent within our estimated error of -+ 0.02 eV. Based u p o n the present viewpoint, the value of 0.42 + 0.04 eV reported by Duckworth and Burke [25] is thought to be anomalously high because their analysis assumed that the only species of b o u n d defects in their alloy was the vacancy - In atom pair. It will be shown [4] that in a 46 p.p.m. In alloy, which they used, a b o u t two-thirds of the In atoms are associated with two vacancies, with essentially all the others existing in the form of mixed pairs. Hence the effective binding energy is given by the weighted sum; EB~ ~ 0.42 eV if one uses our lower limit of 0.19 eV for EB ~2) or E~ ~ - 0.46 eV if the upper limit of 0.25 eV is used. Each of these possibilities is consistent with their reported value. Duckworth, Ramachandran and Burke [26] later estimated the binding energy to be a b o u t 0.27 eV based on loop density measurements. The values derived by Plumbridge [27,28] from his work on binary A l - In alloys are significantly lower than any of the other reported values, including his o w n more recent work [24], and it does not appear possible to
TABLE 1 S u m m a r y o f r e p o r t e d b i n d i n g e n e r g i e s * f o r v a c a n c y - I n a t o m p a i r s in A1 E~In (eV)
Alloy system
0.175 -- 0.2 0.39 0.39 0.36 0 . 4 2 -+ 0 . 0 4
A1 A1 A1 A1 A1
0.27 0.25 0.27
+ 0.02
A1 - Cu - I n A1 - C u - I n A1 - I n
0.18 0.24
- - 0.21 --0.28
-
Cu Zn Zn Cu In
-
In In In In
Method
References
Hardness Time to peak resistivity T i m e t o p e a k a n d initial r a t e Initial r a t e Apparent energy of vacancy formation Initial r a t e Initial rate Loop density (EM)
E n t w i s t l e , Fell a n d K o o [ 17 ] Ohta and Hashimoto [18] Ohta, Hashimoto and Tanimoto [19] O h t a , H a s h i m o t o a n d T a n i m o t o [ 19] D u c k w o r t h a n d B u r k e [ 25 ]
Entwistle and Perry [22 ] Perry and Entwistle [21 ] Duckworth, Ramachandran and Burke [26]
0.176 -- 0.205 0.26
+ 0.01
A1- In ~Al-Cu-In~
~Al ~ Zn- - In Al - In A1-Zn-In
Isothermal resisitivitydecay at 0°C Plumbridge [27 ] Initialrate in post reversion ageing Plumbridge and Entwistle [23 ] Isothermal resisitivity decay at P l u m b r i d g e [ 28 ] --21.3°C I n i t i a l r a t e in p o s t r e v e r s i o n ageing P l u m b r i d g e [ 2 4 ]
* The range of values or error limits are those given by the authors cited.
102
reconcile these data with the present and earlier results. While it is agreed that he has correctly attributed the isothermal decay of quenched-in resistivity at 0°C and --21.3°C to the annihilation of unassociated vacancies at sinks, it appears, on the basis of the twostage tight binding model, that even more vacancies are b o u n d to In atoms than he assumed. This, taken in combination with the inherent difficulties with vacancy losses during quenching, which he discussed at some length [27], suggests that this analysis may be seriously in error. The proximity of his higher value to the currently determined second stage binding energy is purely fortuitous. Hence, the cumulative evidence would indicate that the effective binding energy for the vacancy - In atom pair is no less than about 0.25 eV and no greater than a b o u t 0.30 eV, the correct value probably being nearer the upper end of this range. There is only one other known experimental value with which the second stage binding energy can be compared. Siegel [29] has reported a value for the activation energy of migration for the corresponding recovery stage in a 50 p.p.m. In alloy to be 0.87 eV. This is in satisfactory agreement with the value of 0.84 + 0.03 eV for the 70 p.p.m. In alloy on which the present determination of the second stage binding energy of 0.22 + 0.03 eV is based. The evidence presently available tends to suggest that the correct value for EB ~2) is nearer the higher value of 0.25 eV as will be discussed more fully in reference 4.
5. CONCLUSIONS AND GENERAL DISCUSSION
The essence of the two-stage tight binding model proposed for the dilute A l - In alloys is the existence of two b o u n d defect species, vacancy - In atom pairs and vacancy-rich clusters (probably triplets consisting of two vacancies associated with one In atom). The two required binding energies for vacancies with In atoms in an A1 matrix have been measured, and the pair binding energy is found to be 0.29 + 0.02 eV and that for a second vacancy b o u n d to such a mixed pair is 0.22 + 0.03 eV. It has been shown that most of the previous determinations of pair binding energies for this system are in reasonable agreement when reconsidered from the present state of know-
ledge. These investigations have, however, been based on studies of quenched materials, the energy parameters being inferred from measurements of various non-equilibrium properties. Thus the question of whether the discrepancy between the high binding energies derived from quenching t y p e experiments and the vanishingly small values deduced from high temperature measurements [ 12] is real or only apparent remains unanswered. The general agreement that is now emerging from the quenching type experiments emphasizes the need for a concerted effort to answer this fundamental question. It may well be that the binding energies determined from the quenching type experiments should be designated as interaction energies or binding free energies, because a term due to the vibrational entropy is always included. There are no known experimental efforts to assess the significance of the entropy contribution to the apparent binding energies for this system. However, it is known from the theoretical work by Girifalco and Kuhlmann-Wilsdorf [ 30] and Kuhlmann-Wilsd o f f [31] that the interaction energy associated with a vacancy and a stress field in a metal is a complex function whose magnitude is dependent on temperature. This temperature dependence is associated with an entropy term which arises because of changes of the elastic constants and, hence, of the vibrational spectrum in the immediate vicinity of the lattice defects. All too commonly, the potential consequence of such an entropy effect is ignored when attempting a critical comparison of binding energies determined in widely separated temperature regions.
ACKNOWLEDGEMENTS
We wish to express o u t thanks to Carmen Ruano for valuable comments concerning this manuscript.
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