Characterization of the defects responsible for the precipitation of silver from dilute PbAg solid solutions

Aeta metall, mater. Vol. 42, No. 12, pp. 4049-4058, 1994

Pergamon

0956-7151(94)00190-1

Copyright © 1994ElsevierScienceLtd Printed in Great Britain.All rights reserved 0956-7151/94 $7.00 + 0.00

CHARACTERIZATION OF THE DEFECTS RESPONSIBLE FOR THE PRECIPITATION OF SILVER FROM DILUTE Pb-Ag SOLID SOLUTIONS A. MENA'P, H. A M E N Z O U 2, G. MOYA m and J. BERNARDINI 2 ~Laboratoire de Physique des Matrriaux, Ecole Normale Suprrieure Rabat, B.P. 5118, Maroc, 2Laboratoire de Metallurgie, URA 443, Facult6 des Sciences St Jrr6me, 13397 Marseille Cedex 20 and 3Laboratoire de Physique des Matrriaux, Facult6 des Sciences St Jrr6me, 13397 Marseille Cedex 20, France (Received 1 October 1993; in revised form 9 February 1994)

Abstract--The diffusion coeti~cientsof silver (D *) have been measured during its precipitation from dilute Pb-Ag solid solutions (92 x 10-6 at/at and 184 x 10-6 at/at) by using ~°Ag radiotracer between 327 and 363 K. The diffusion activation energy Qo, = 49.4 kJ/mol is smaller than the diffusion activation energy (Qo = 60 kJ/mol) of silver in pure lead but is in fair agreement with the activation energy for precipitation of silver previously measured by the authors (Ea = 50 kJ/mol). Evidence that the (Pb-Ag) pair is the defect responsible on the one hand for silver migration during the precipitation in these dilute alloys and on the other hand for silver diffusion in pure lead and lead-silver solid solutions is reviewed.

1. INTRODUCTION Previous investigations of Pb-(noble metal) alloys have indicated that solute atoms in these alloys diffuse several orders of magnitude faster than the host atoms. This behaviour has been reviewed by several authors [1-3] and it is generally agreed that for a solute X the transport occurs through the motion of some interstitial defect Xi by the so-called "dissociative mechanism" where the exchange between the practically non mobile substitutional X s atoms and the Xi atoms requires the participation of vacancies. In metallic systems the dissociative diffusion of X atoms is generally governed by the transport properties of Xi solutes as the dislocation density is large enough to provide a sufficiently fast rate of vacancy generation. The effective diffusion coefficient D is given by the equation:

C~qDi D = ~C~q + C~

(l)

where D i is the diffusion coefficient of X~ atoms, and C~q and C~ the equilibrium concentrations of Xi and Xs atoms. One of the still debated questions is how the interstitial configurations of the X~ atoms look like and how they move. The diffusion mechanisms involved to explain this fast diffusion are deduced either from some diffusion measurements performed at high temperature in pure lead and lead-(noble metal) solid solutions or from studies of the kinetics of noble metals precipitation at low temperatures. In the second case, the solute precipitation is generally analysed by measuring the resistivity change, after quenching from high temperatures, of rich alloys

which are more or less concentrated. These two sets of results, obtained by equilibrium and non equilibrium techniques, are difficult to compare. Moreover the equilibrium measurements in lead and lead solid solutions can be only performed at high ternperature because of the very small solubility of noble metals in lead [3]. Whatever the temperature range of the diffusion measurements, the Arrhenius plots of the solutes diffusivity D(X) in pure lead are always linear indicating that a single diffusion mechanism, characterized by an apparent diffusion activation energy QD, is predominant. The experimental value of QD relative to the diffusion of silver in lead is very well known between 361 and 576 K. As other authors, [4-7], we obtained in a previous study [8] Q D - 60 kJ/mol. This value is unusually higher than those measured for Cu (33.6kJ/mol [9]) and for Au (42 kJ/mol [10]) diffusion in lead; it implies that the diffusion of Ag in lead occurs by a more complex mechanism than a simple interstitial diffusion mechanism. Diffusion measurements in lead-silver solid solutions could theoretically give some information about the diffusion mechanism [l l, 12]. However these experiments suffer from the lack of solute solubility: the slight difference in Ag diffusitivity [7, 8, 13, 14], between pure lead and lead-silver solid solutions, always remains in the order of the experimental errors (about 10%). Indirect measurements of silver diffusivity at lower temperatures, obtained by studying the kinetics of precipitation of silver from Pb(Ag) supersaturated solid solutions, have led to different values of the activation energy of precipitation even in the dilute alloys

4049

4050

MENAI et al.: SILVER PRECIPITATION FROM DILUTE Pb-Ag

(C^g < 400 x 10-6at/at). Cohen and Turnbull [15] found, by resistivity measurements, that precipitation of P b - A g (200 and 300 x l0 -6 at/at) occurs between 314 and 360 K with an activation energy E a = 61.9 kJ/mol. This value being not so far from the diffusion activation energy QD, the authors concluded that the migrating defect is the single Ag interstitial Ag i which is neither associated with a vacancy nor another atom. In a 380 x 10-Sat/at Pb(Ag) alloy, Nozato et al. [16] found two resistivity recovery steps. The first, below

showing the same shape and the same silver concentration as those previously used in the flux calorimetry study [17], which gave the rate constant k and the order n of the precipitation reaction. This set of experiments was completed by some measurements in lead-silver solid solutions in the absence of silver precipitation at high temperature and in supersaturated P b - A g alloys where the precipitation of the excess of silver was achieved at the diffusion temperature before the ll°Ag diffusion measurement.

283 K (E a = 37.7 kJ/mol), is attributed to the AG~ migration. The second (283-423 K, E a = 50.2 kJ/tool) is associated with the migration of (Ag-V) pairs. Later, Mena'/ et al. studied 92 x 10 -6 and 184 x 10-6at/at Pb(Ag) alloys by calorimetric measurements using a flux microcalorimeter [17]. They found that the precipitation occurs, as a first order reaction, in the Cohen temperature range but with the Nozato precipitation energy Ea: an interpretation in terms of (Ag-Pb) interstitially migrating pairs in mixed dumbbell configuration of defect, consisting of a silver and a lead atom sharing one lattice site, is given. This interpretation has been recently verified by using the positron lifetime technique. No defect-positron trapping is in effect observed in dilute quenched alloys (CAs -- 96 X 10 -6 at/at) while a clear increase in the positron lifetime due to a vacancy-type defect is only induced in concentrated quenched (CAs ~ 920 x 10 -6 at/at) P b - A g alloys [18]. Thus from Mena'f's and Nozato's results, it seems that the activation energy E~ deduced from precipitation studies after quenching is lower than that obtained from diffusion measurements at higher temperatures in pure lead and Pb--Ag solid solutions, To summarize, different sets of silver diffusion experiments in lead, performed by using a radiotracer between 361 and 576 K, in the presence of the equilibrium concentration of the defects responsible for silver diffusion, lead to an activation energy QD ~- 60 kJ/mol; on the contrary, the activation energy E~ linked to silver precipitation, measured by other techniques at lower temperatures with a non equilibrium concentration of defects, lies between 50.2 and 61.9 kJ/mol. The purpose of the present work is (i) first to check the existence of a difference between Qo and Ea values, by measuring these quantities in the same samples, with the same techniques over the same range of temperature and then (ii) to characterize the defects responsible for silver atoms transport in dilute lead silver alloys in both equilibrium and non equilibrium conditions. To this end, we decided to measure the diffusion of silver during its precipitation by using H°Ag radiotracer in P b - A g ( 9 2 x 1 0 - 6 a t / a t ) a n d (184x 10-rat/at). To measure the diffusion coefficient of a solute during its precipitation the rate constant k of the precipitation reaction has to be known. Diffusion measurements were therefore performed on samples

2. MATERIALS AND EXPERIMENTAL TECHNIQUES The experimental materials and techniques employed were described in an earlier publication concerning the bulk diffusion of silver in lead [8]. Briefly: • Present silver diffusion experiments have been performed in Pb(Ag) single crystals obtained by diffusion of inactive silver, from both sides of the specimens, in pure lead single crystals at different temperatures Ts. This process allows saturated solid solutions to be obtained which can be used either at temperature T higher than TS (to measure silver diffusion in equilibrium solid solutions) or at temperature T lower than Ts (to measure silver diffusion in supersaturated solid solutions). Samples were held at each of the temperatures Ts for sufficient time to achieve solid solubility level greater than 90% in the middle of the specimen about 2 mm thick. After the heat treatment, excess solute at the surface was removed by dissolution in aqua regia and the actual alloy composition checked by dissolving and analysing one or two slices ( g 30/~m thick) from both front surfaces of the sample by Inductively Coupled Plasma Spectrometry. Results of these chemical analyses which agreed closely with the expected alloy concentrations were utilized in further analysis of the results. • After polishing the surface and before the diffusion annealing, solid solutions were always annealed for 6 h at Tq = 573 K and water quenched (except when ll°kg diffusion had to be performed after the precipitation of the excess of silver: in that case, a long heat treatment at the intended diffusion ternperature was carried out before ll°Ag diffusion). This treatment is very important. As we will see later, it fixes the concentration and the nature of defects present at the beginning of silver precipitation. • All the heat treatments were performed under the flow of ultra pure 6N hydrogen in a vertical furnace to enable rapid quenching ( > 10,000°C/s) of the samples by gravity. • The radiotracer ll°Ag (specific activity around 2mCi/mg) was chemically deposited from an acidified (0.1 N) nitrate solution. The subsequent procedures sectioned the samples, weighting and counting as is usually done in radiotracer diffusion techniques; sectionment was

MENAI et al.: SILVER PRECIPITATION FROM DILUTE Pb-Ag performed by chemical dissolution. After each section, radiotracer activity was measured by first counting the residual activity of the sample (Gruzin's method [19]) and then the activity of each dissolved slice, with a Ge semi-conductor counter, The experimental errors on the diffusion coefficients have been estimated to _ 10% on D values [8] and + 20% on D* ones to take into account the additional error linked to the use of the rate constant k in the calculation of this parameter,

4051

been completed at the intended diffusion temperature T. In that case, B* atoms diffuse in a saturated solid solution in the presence of solute precipitates when no compound (or intermediate phase) exists in the binary bulk phase diagram A-B. In the present case, the solution (3) will remain valid owing to the small solubilities of silver in lead.

3.3. Diffusion of B* during the precipitation of B atoms from the supersaturated solid solutions A(B) In this case, the diffusing substance is partly immobilized by the irreversible chemical reaction of precipitation. To deduce the diffusion coefficients D*

3. ANALYSIS OF THE DIFFUSION PROFILES Three types of experiments can be performed in A(B) solid solutions obtained by diffusion of the inactive B solute at a temperature T = T~. In the first type of measurements the radiotracer B* diffuses in saturated (or under-saturated) solid solutions at temperatures T >/Ts. In the two other cases the radiotracer B* diffuses at temperatures T lower than T~, either after the precipitation of the excess of inactive solute or during this precipitation,

3.1. Diffusion of B* in (under)-saturated A(B) solid solutions

at temperature T from the penetration profiles, the rate of removal of diffusing substance, which is characterized by its rate constant k corresponding to T, should be known. The equation for diffusion in one dimension is [20]

OC O2C ~ = D* ~ - kC"

provided that the diffusion coefficient D* is assumed to be constant ([20], p. 124) If the irreversible reaction is a first order reaction (n = 1) with initial and boundary conditions

As in a pure metal, the radioactive solute concentration of B* at the depth x in a saturated (or under-saturated) solid solution yields the Fick's second law

(4)

t = 0 > 0

j',

C = 0

for x > 0

C = C o for x = 0

the solution of the equation (4) is ([20], p. 125)

&C

82C

8t - D 8x 2 .

(2)

C =k

C~ e x p ( - k t ' ) dt' + C~ e x p ( - k t ) .

(5)

0 Due to the low solubility of silver in lead [8], the boundary condition appropriate to solve equation (2) is that of a constant source of tracer at the surface, The activity A of ~°Ag, as a function of time t and diffusion depth x is

A(x't)-C(x't)=erfc(~) x Ao Co \x/4Dt ]

(3)

where A 0 i s t h e c o n s t a n t a c t i v i t y a t t h e s u r f a c e ( x = 0 ) and Co the silver solubility at the diffusion temperature T, as no compound exits in the bulk phase diagram Pb~Ag. Some experiments had been performed under these conditions in a previous work concerning silver diffusion in lead [8]. They will be briefly mentioned in the following section. It should be pointed out that before the diffusion of radioactive H°Ag at T >/T s a pre-diffusion anneal at the intended diffusion temperature completed the preparation of each sample,

3.2. Diffusion of B* after the precipitation of the excess of B atoms from the supersaturated solid solutions A(B) The diffusion of radioactive B* atoms can be performed at temperatures T lower than Ts when the precipitation of the excess of inactive B atoms has

The term C~ is the solution of Fick's equation in the absence of precipitation. In the event of semi-infinite medium with constant surface concentration Co, C1 corresponds to the relation (2) and the solution (5) results in ([20], p. 130) -C- = s el x p ( - x

CO

\

k n~k,']erfc( -

~/ D*J

-x

j ( . n~k,'] ( x + 5 exp x erfc ~

\ "V t)~/

x/~

\ 2 ~

)

x~) +

.

(6)

\2x/D*t

When x / ~ is greater than 1, the expression erfc x//~ approaches unity and the solution (6) is reduced to ([20], p. 131)

--=C Co

x/kiD *.

(7)

exp - x

The diffusion coefficients D* can be calculated from the slopes of the activity penetration curves log A/A o - f ( x ) when the rate constants k are known for a first order reaction. It should be noticed that the relation (5) takes into account the radioactive atoms which can precipitate but do not consider any possible isotopic exchange between precipitated atoms and free radioactive atoms.

MENAI et al.: SILVER PRECIPITATION FROM DILUTE Pb--Ag

4052

.-

104

1'

ff

3.1o'

>,

..,=

-

×

/

".¢

~ -

5.103

.,

2.102

/

o.,.

X Ipml o

m

0

' 0

J 200

'

~ " 400

0 0

X cpmJ--.

Fig. 1. Penetration profile of ~l°Ag in a Pb-Ag (92 x 10 -6 at/at) solid solution: T = 500 K; t = 5460 s.

100

X

200

Fig. 2. Penetration profile of II°Ag in Pb-Ag (1520 x l0 -6 at/at) after the precipitation of the excess of silver at 361 K for 120 h: T = 361 K; t = 4.32 105 s.

4. EXPERIMENTAL RESULTS

4.1. Silver diffusion in the undersaturated solid solutions [8]

4.3. Silver diffusion during precipitation of the excess of solute

drawn in Gausso arithmetic coordinates [Fig. l(b)], Annealing conditions and the diffusion coefficients o f '~°Ag are listed in Table I.

Two types of solid solutions respectively obtained by an inactive silver diffusion in pure lead at T s = 419 K and T~ = 446 K were used to carry out the ll°Ag diffusion experiments over the range of temperature 327-381 K. They contained 92 x 10 -6 at/at and 1 8 4 x 10-6at/at. The diffusion of silver was measured during the precipitation of the silver phase which is a first order reaction [17]. Figure 3 shows, as an illustration, the experimental

4.2. Silver diffusion after precipitation of the excess of silver

depth profile observed at 349 K for t = 18.5 x 103 s in P b - A g (92 x 10 -6 at/at). As expected owing to the

A n example of the activity depth profile obtained at 500 K for a diffusion time t = 5460 s in P b - A g (92 x 10-6at/at) is shown in Fig. 1. The solid line in Fig. l(a) is a least squares fit of the data to equation

(3) leading to the linearity of the C/Co vs x plot when

A monocrystalline P b - A g (1520 x 10 -6 at/at) solid solution was first prepared by heating at T = 553 K for 96 h a lead sample covered by silver; it was then annealed at 361 K for 120 h to precipitate most of the silver prior to the diffusion o f the radiotracer ll°Ag performed at the same temperature for 4.32 105 s. The diffusion coefficient of ll°Ag [D=(1.7___0.2)10-14m2-s-~], calculated from the penetration curve which fits the theoretical solution (3), (cf. Fig. 2), is slightly smaller than that measured at the same temperature in pure lead [DpAbg = (2.4 + 0.3)10-14 m 2 . S-1].

.~ ~'-

'

10 2 .~m ,~ .~ '~ 10

.o Table I. Silver diffusion in pure lead and lead-silver solid solutions T t C Ag D Ag (K) (s) (10 -6 at/at) (m2- s-I) 500 5460 92 6.5 x I0-': 500* 5460 6.2 x 10-': 453 4080 92 1.2 x 10-'2 453* 4080 1.4 X 10 - 1 2 361 270240 361" 270240 *Diffusion in pure lead

10

2.0 x 10 -14 2.4 x 10-~4

"2 I~ (/3

@ 1

I 25

X

I 50 |JLI ml

Fig. 3. Penetration profile of "°Ag in Pb--Ag (92 x 10-6at/at) during the precipitation of the excess of silver: T = 349 K; t = 18.54 103 s.

MENAI et al.: SILVER PRECIPITATION FROM DILUTE Pb--Ag

4053

Table 2. Silver diffusion during the precipitation of the super saturated solid solutions (Pb--Ag) T CAs t k D* (K) (10-6at/at) (s) (s a) (m2.s i) 327 92 88860 1.635x 10_4 1.6 x l0 ]4 327 92 236960 1.635× 10 _4 2.0 × 10 -14

1

349 381 381 338 363

Z 0.1

t, o,• , 0

I i 4 T IM E ( h. )

I 8

Fig. 4. Determination of parameters k and n by using the relation (8) in Pb-Ag (92 x 10-6 at/at): Tq = 558 K; TA = 393 K.

precipitation occurring during the diffusion, the experimental profile does not fit the solution (3), whereas it fits very well the equation (7) as reported in Fig. 3 for the specific activity measurements, The diffusion coefficients D* can be calculated from the slopes of the curves log A/Ao = f ( x ) if the rate constants k are known. The parameters k have been measured by a conduction heat flux differential calorimeter by observing, after quenching, the precipitation of P b - A g alloys over the same range of temperature than that used in the present diffusion experiments. When only one activated process is present, the rate change of C, the concentration of free atoms, during an annealing at temperature TA is given by dC(t, TA) dt = -kf(C) = -kCL

C(t, T^) ~ exp - kt.

(9)

Co being the precipitated silver concentration at the end of ageing. Since Q(t, TA), the quantity of heat still stored at time t, and Q0, the total quantity of released heat are respectively proportional to C(t, TA) and Co, the thermograms can be standardized [17] to reveal the variable

F(t, TA) .

Q(t, TA) C(t, TA) . . . Q0 Co

When atoms precipitation occurs with an order reaction n = 1, the quantity log F as a function of time presents a linear variation in agreement with equation

18540 7200 21600 86400 49740

4.9 × 19 × 19 x 9.5 × 33.3 ×

10 _4 10 -4 10 -4 10 _4 10 _4

5.1 × 1.910 × 2.6 x 4 x 1.4 x

10 -14 l 0 13 10 13 10 t4 10-13

(8). Figure 4 shows this linear variation, down to F = 0 . 0 5 , for an alloy P b - A g (92 x 10-6at/at) quenched from Tq = 558 K and aged at TA = 323 K. Table 2 reports the different D* diffusion coefficients together with k values deduced from the calorimetric measurements. Figure 5 shows the variation of D* as a function of temperature. As it can be seen, the D* coefficients are slightly higher than the extrapolated D coefficients relative to silver diffusion in pure lead, lead-silver solid solutions, and previously precipitated super saturated solid solutions; they can be described by the equation 49.4 kJ/mol D*(m 2- s -~) = 0.015 x 10-~4 exp ka T 5. DISCUSSION The diffusion coefficients D* are similar in the two alloys. The D* values are not far from D values measured in pure lead and in P b - A g solid solutions; they correspond to the coefficients of the defect involved in the higher temperature stage of precipitation of lead-silver alloys. It supports the view that most of the Ag atoms at Tq are dissolved as an interstitial defect Xi, that is responsible for silver

(8)

Over the range of studied precipitation temperatures, the solid solubility of silver remains lower than 10 -5 at/at [8]. Neglecting this quantity, one can assume C(oo, T A ) ~ 6 0 and one obtains as integrated form of equation (8) for n = 1

92 92 92 184 184

10-11 ~ ~_" - -E

10Ja

la

10.13

\

tef.(8)--~

~

ref (8)--

10.14

10"I" 20

,

.I 25

"'-..~.. "'~, ,.,~ . ~ "t.

i

104/T (K-1)

I 30

Fig. 5. Diffusion coefficients of 'l°Ag as a function of the temperature: D in pure lead (straight line Ref. [8] and A); D in lead-silver solid solutions (V); D in a previously precipitated super saturated solid solution ((3, 1520x 10-6at/at); D* during the precipitation of silver ( e , 184 x 10-6at/at, II, 92 × 10-6 at/at).

4054

MENAI et al.: SILVER PRECIPITATION FROM DILUTE Pb-Ag

transport during the precipitation. Thus, referring to equation (1), D* may be identify to D i. Consistent with a lower activation energy QD. compared with QD,D* coefficients are slightly higher than D values obtained over the same range of temperature. The Qo* value, which is in agreement with the activation energy of precipitation E a previously measured by Mena~ [17] and Nozato [16], represents the migration enthalpy of the defect Xi and there is now a conclusive evidence that the growth rate of silver precipitates in P b - A g is slightly higher than that calculated from the tracer diffusivity of silver in pure lead, in contrast to the precipitation behaviour of gold in Pb-Au. The diffusivity of gold D]o measured during the precipitation of P b - A u is smaller than that observed in pure lead [21-23]. The decrease of D*u relative to DAu, as CAu increased, has been explained by the presence of (mui-Aui) dimers, the probability to form this defect (or higher order clusters) increasing with Au concentration. In what follows we will show that the presence of (Ag-Pb) mixed dumbbells explain all the results obtained in pure lead and the most dilute lead-silver alloys under equilibrium conditions or during the precipitation of silver while one must take also into account the formation of (Ag-V) pairs during the precipitation of silver at temperatures higher than TA ~ 200 K when silver concentration is higher than (130-150)10-6at/at.

which can be involved for silver transport is the pair (Ag-Pb) because the probability to form (Ag-Ag) pairs is too small in these very dilute alloys. The presence of this defect may explain all the results obtained in pure lead and lead-silver solid solutions at equilibrium [3] as well as those observed during precipitation of silver at temperatures higher than T i> 290 K. As an example lead atoms associated with silver being in a dimer configuration will jump more easily in a neighbouring vacancy leading to a slight increase of the lead diffusion coefficients in the presence of silver. Furthermore, many atoms being involved in the jump process for a long range diffusion, the value of the isotope effect parameter will be very small as it has been observed by Miller et aL [25]. Referring to Dederich's and Takamura's calculations [26, 27], a stable mixed dumbbell can be created by interaction of an interstitial with a solute when the solute is undersized. Dederich had estimated the binding energies of these complexes as a function of the atomic size difference betweeen a solute and a solvent atom without relaxation of the lattice. More recently Takamura's calculation, performed by using the molecular dynamics technique, confirms the stability of the mixed dumbbell for undersized solutes and leads to a binding energy of the pair (Ag-Pb) higher than 1.5 eV [27]. Such 5.1. Nature o f the defects involved during silver stability afford them the opportunity to be present transport and to migrate even over the range of temperatures We have already mentioned that Q~ value is too investigated for silver radiotracer diffusion in lead large to represent the migration energy of (Agi) and lead-silver solid solutions. defects in the form of pure interstitials in pure lead. The D* coefficients measured in Pb--Ag The activation energies, QD. and Ea, although smaller (184 x 10 6 at/at) are similar to those obtained in than QD, remain large; thus the migration of silver in P b - A g (92 x 10 -6 at/at) and we can expect the same the form of pure (Ag i) interstitials during the precipi- defect to be responsible for the migration of silver. By tation must also be ruled out and the presence of using previous results obtained by calorimetry we can more complex defects ~ such as (Ag-V), (Ag-Ag) or have a better understanding of what happens in this (Ag-Pb) pairs has to be considered, alloy where the precipitation enthalpy for P b - A g In a previous work where silver precipitation in (184 x 10 6 at/at) is lower than the dissolution dilute (Pb-Ag) alloys were studied between enthalpy of silver in lead. Such a behaviour may be 288-363 K by calorimetric measurements [17], we explained by taking into account the precipitation of have noted the presence of only one stage of precipi- a fraction of silver atoms at lower temperatures by tation which is slightly dependent on silver concen- using a defect X~ more mobile than (Pb-Ag). This tration while the enthalpy AHp was found a function stage of precipitation at lower temperatures has been of this concentration: for the same value of quench- observed by Nozato et al. in P b - A g (380 x 10 -6 at/at) ing temperature (Tq = 558 K), AHp was respectively [16]. Moreover Yavari and Turnbull [28] have equal to 33.67kJ/mol in P b - A g (184 x 10-rat/at) reported that the critical yield stress of P b - A g and 48.99 kJ/mol in Pb--Ag (92 x 10 -6 at/at). For the alloys, quenched from 583 K (or 553 K) and annealed most dilute alloy, AHp is in fair agreement with for 15 min at 223 K before the yield stress measurethe dissolution enthalpy AHd measured for silver in ment at 77 K, remained independent of the silver lead (AHd = 49.22 kJ/mol [14], AH d = 50.49 kJ/moi concentration up to CAg = 150 x 10 -6 at/at and then [15]). This could indicate that in this alloy, rose steeply as CAg was further increased. Thus at silver begins to precipitate after the quench by very low temperatures, clustering or precipitation of using, during the annealing performed at low silver would only occur in alloys where temperatures TA, the defect present at equilibrium at CAg > 150 × 10 - 6 at/at, indicating that in P b - A g high temperatures Tq. As no defect positron trapping (184 × 10 -6 at/at) one would only measure, between was observed in quenched Pb--Ag (92 × 10 -6 at/at) 288 and 363 K, the thermic effect linked to the silver alloys [18], the only one ordinary complexe Xi fraction still free over this range of temperature, i.e.

MENAI et al.: SILVER PRECIPITATION FROM DILUTE Pb-Ag 150 x 10-6at/at. This assumption is in agreement with the following experimental features: • the quantity of heat released for one mole of silver in Pb-Ag (184 x 10-6at/at) corresponds to about 70% of the AHd value which would indicate that only 130 × 10 -6 at/at of silver atoms was still unprecipitated at the beginning of the calorimetric measurements at Ta. • Ham's analysis of the isothermal annealing curves shows that 70% of the total silver concentration is unprecipitated at the beginning of the annealing treatment [29]. The defect X~ which induces the first stage of silver precipitation at low temperatures when CAg> 150 X 10-6at/at must take form during the quench. Studying silver precipitation in more concentrated alloys (fAg= 1300 x 10-6at/at and CAg ----920 X l0 -6 at/at), Nozato and Yamaji [30] by resistivity measurements and Menai et al. [18] by positron annihilation technique have find again this stage where the defect X~ anneals out above 220 K. Menai"s positron lifetime study identifies this defect as the (Ag~-V) pair in 920 x l0 -6 at/at Pb-Ag alloys [31].

5.2. Mechanism o f migration o f the pair (Ag-Pb) Three elementary jumps have been distinguished for the mixed dumbbells in a f.c.c, structure by Dederich et al. [26]: (a) By a jump of the solute from a position to an equivalent one, the mixed dumbbell can reorientate after dissociation and occupy six configurations corresponding to an octahedral cage centred at the octahedral site position [32]. By this process the mixed dumbbell remains confined to a cage centred at the octahedral position. This cage motion does not lead to long range diffusion, (b) After dissociation of the mixed dumbbell, by a jump of the interstitial solvent atom to a neighbouring host position, a normal (Pb-Pb) dumbbell can be formed. This dumbbell may later either dissociate or loop around the solute and reform a mixed dumbbell (Ag-Pb). In that case, the new defect (Ag-Pb) compared to the original one has been turned by 180°. (c) By a rotation on its site the pair may rotate without any dissociation by 90 ° with an activation energy Error. These three possibilities do not lead to long range motion but one of the last two together with the faster cage motion will lead to long range diffusion. By taking into account either the rotation of the pair on its site or the dissociation of the defect followed by the migration of the solute to a neigbouring octahedral site and the formation of a new dumbbell, Maury [33] arrived at the same conclusion: a long range migration requires mechanisms such as rotation without jumping or jump without rotation followed by the cage motion. It should be pointed out that Takamura's simulations have shown that the energies for the two possibilities are the same since

4055

the solute migrates along the same direction [27]. Thus the activation energy Ea=Qn. must be attributed either to the dissociation or to the rotation of the mixed dumbbell: (i) According to Dederich's calculation, the dissociation energy of a mixed dumbbell is about 50 times greater than the migration energy Epmbt~)of the self interstitial when the size factor is around 18%. The low temperature isochronal annealing of lead after electron irradiation at 1.5 K shows an annealing steep at about 4 K [34]. It is worth noting that in copper, where this stage annealing occurs in the range 15-55K, one admits, from theoretical calculation, a migration energy of the split-interstitial to be about 0.05eV. Thus it seems possible to assume a smaller value in lead (about 0.01 eV [35]). By using this value for Epmb(l), o n e obtains 0.5eV for the dissociation energy of (Ag-Pb) in fair agreement with the experimental values E a and Qo*. As the dissociation energy Edisscan be written Ediss = E ~ ) + EAg b Pb the solute-solvent binding energy EbAg-Pb would be 0.50 eV. This value seems to be not large enough to make the dumbbell stable up to the higher temperatures used in the diffusion experiments. Mixed dumbbells found in irradiated Al(Mn), Al(Mg)and Cu(Be) [36] alloys have a very large solute-solvent binding energy (for instance, 0.9 eV in Cu(Be) [26] and 0.89 eV in AI(Be) [37]) but they are only stable up to stage III (200 K) [36]. Consequently, in agreement with Takamura's estimation of the binding energy, it turns out that the Ag-Pb dumbbells migrate without any dissociation. The rotation should take place and leads, together with the faster cage motion, to the long range migration. (ii) According to Barbu [38, 39], the diffusion coefficient Dp of a mixed pair can be written as in irradiated alloys

Dp = O i =

D* = ~ 2 wr(5w3+ w~) + 5w3w2 3 wia 5w3w2W (5W3 "]-W2)(Wi "1-Wr)

where a is the crystal parameter, and where the different jump frequencies are respectively: wr = jump frequency pertaining to the rotation of the mixed pair; w~= cage-jump frequency between the solute confined to the center of the cage (at the octahedral position) and the neighbouring solvent making the cage; w3 = dissociative frequency of the type " a " pairs (i.e. the pairs where the solvent and the solute are in the first neighbor configuration); w2 = dissociative frequency of the type mixed pairs; w~=associative frequency of the mixed pair. For undersized impurities, the activation energy for the cage-jumps is always lower than that for dissociation or looping [26, 38]. Thus one can assume wi >>wr. As the binding energy of the pair (Ag-Pb) is

MENAI et al.: SILVER PRECIPITATION FROM DILUTE Pb-Ag

4056

very large, one can also expect w2 = 0 and express D O as I 2

AS

D* = Dp _--~a va exp ~

E

exp

TM

k8 T

AH~Ag_Pb) = 46.25 + 12.5 --~58.75 kJ. mol-I

where va is the "attempt" frequency, AS the activation entropy linked to the rotation of the pair and E m the migration energy of the pair. As the long range motion of the pair is due to the cage motion (mechanism "a" practically unactivated) and the rotation of the pair (mechanism "b"), the migration energy of the pair is quite equal to its rotation energy Ermt. By using the experimental values of D* and taking Ea = E TM = Eromt _-- 0 . 5 0 eV, one obtains AS 1012 s_l va exp ~ = 7.2 x . The value of the frequency factor, lower than the Debye's frequency vo, is probably linked to the modes of vibration associated with interstitials [40]. Assuming v~ to be equal to VD/8 [26], one can estimate AS to be ~ 3.4kB in agreement with the values generally obtained for the dumbbell (AS/k a = 1 _+ 2 [39]).

5.3. Formation enthalpies of the defects (Ag-Pb)and Ag(s) At the beginning of silver precipitation in the most dilute alloys P b - A g (92 x l0 -6 at/at), the quite non mobile substitutional silver atoms, with a concentration C ~ , and the pairs (Ag-Pb), with a concentration C~q = C~g_Pb) are present. If the pair (Ag-Pb) is the defect responsible for the diffusion of silver in lead as well as in lead-silver alloys during silver precipitation, the diffusion coefficient D* of the pair (Ag-Pb), measured during the precipitation, is linked to the effective diffusion coefficient D by the relation (1) c~q

D =~

c~q

eq C(Ag-Pb)

D*.

D i = C~q + ~seqD i = C,~g_Pb) "Jr-Cseq

Thus it is possible to calculate, from the experimental values D and D*, the atomic fraction of silver present as mixed dumbbells as a function of temperature

C(¢~g-Pb) ~-- 12.2 exp - 12.5 kJ • m o l - t / k a T. C~g

(10)

It should be pointed out that in the diffusion experiments carried out during silver precipitation, the ratioC~qs_pb)/C~g is fixed by the quench temperature (573 K), whatever the diffusion temperature may be. (i) Expression (10) leads to a ratio C~,8_pb)/C~g equal to 0.88 at 573 K. This value is in agreement with that estimated from isotopic effect measurements by Miller [25] (0.8 at 573 K). By using the relation Cis = A e x p - AHd/k a T

with the mean value A H a = 46.25 kJ. tool-~ [14, 15], the true formation enthalpy of the dumbbell (Ag-Pb) is obtained by

(ii) From the variation of C~_ab)/C~g as a function of temperature the ratio oc = C'~g_pb)/C~qgts) can be calculated as eq eq CA~Igpb C(Ag_Pb)/CAg ~= = c~,q~s) 1 - C(Ag-Pb)/CA ~q ~q8 AH' 21.5 kJ. mo1= A ' e x p - ks T = 312.7 exp ka T f f As AH'=AHtA~Pb)--AHAgt~), one deduces for the formation enthalpy AH~,gc~ ) the value A H f ~ ) = 14.5 k J . mol - l .

(iii) It should be pointed out that by using the f value of AHtAg_Pb) = 58.75 kJ" mol -~ and equation (10), one can provide the quantity of heat which will be released during the precipitation after a quench carried out at Tq q=

C~Pb)c AH~Ag-Pb) • As

As an example, for T q = 5 5 8 K , one calculates for q the value 48.86 kJ .tool -l in fair agreement with the experimental value obtained from calorimetric measurements for P b - A g (92 x l0 -6 at/at): AHp = 48.90 kJ. m o l - ' . In the same way taking into account that only 70% of the total silver concentration are concerned with the studied precipitation stage in P b - A g (184 x 10 -6 at/at), one obtains for q the values 32.85 kJ. mol -Z in agreement with the experimental AHp value equal to 33.67kJ . m o l - ' .

5.4. Comments on the diffusion in lead-silver solid solutions On the basis of the present results it may be interesting to come back to the silver equilibrium diffusion measurements performed in lead-silver solid solutions. It is generally reported in the literature that a decrease of the solute diffusivity, increasing the solute concentration, occurs as the result of the attraction between interstitial and substitutional solutes while, in contrast, no effect of solute concentration on solute diffusivity will occur when an attraction between a solute a n d a solvent atom lead to the formation of a solute-solvent dimer [2, 3, 8]. In practice the choice between the two possibilities is often difficult owing to the low solubility of fast diffusers which restricts experiments to a very dilute concentration range [2, 8]. All the studies of silver diffusion in P b - A g solid solutions present only a slight variation of the diffusion coefficients with respect to silver concentration. At first glance, these results look to be in disagreement with those observed during silver precipitation where we observe a clear enhancement of

MENAI et al.: SILVER PRECIPITATION FROM DILUTE Pb-Ag silver diffusion. On further examination, this difference of behaviour is only due to the small concentration of mixed dumbbells present in equilibrium with Ag(s) in the solid solutions in comparison with that present at the beginning of silver precipitation, As previously discussed, the concentration of mixed dumbbells is in effect fixed, in the second case, by the temperature Tq of the pre-annealing performed before the diffusion. As an example, at TA= 361 K where the solubility of silver in lead is equal to 15 x 10 -6 at/at [8], there are in the Pb-Ag (10 × 10 -6 at/at) solid solution 1.8 x 10 -6 at/at of (Ag-Pb) dimers (18%) while this concentration is equal to .81 x 10 -6at/at (88%) at the beginning of silver precipitation in the Pb-Ag (92 × 10 -6 at/at) alloy at the same temperature. Such an assertion is supported by the fact that (i) present measurements in saturated solid solutions (cf. Table 1) show a slight increase of silver diffusivity at high temperature only; (ii) the higher the temperature, the greater the enhancement of silver diffusion in the presence of silver in the set of experiments carried out by Cohen and Warburton [14]. Furthermore, on the basis of this argument, it is also possible to explain the discrepancy between Cohen's value concerning the activation energy of precipitation E a (which is equal to the diffusion activation energy of silver (Qo) in pure lead [15] and those measured by Menai et al. [17] and Nozato et al. [16]. In the later two sets of experiments, the samples were quenched from the same temperature Tq = 558 K. This temperature was high enough to create the dumbbells which will migrate later during the ageing at TA; thus it has been possible to measure over the range of temperature 327 K ~< TA ~<363 K the migration energy of the dumbbells. On the contrary, the temperature of quench, which is not indicated by Cohen and Turnbull [15], must be rather low: in that case, the dumbbell concentration created at Tq is not large enough. However this defect will be mainly formed during the annealing at TA leading to a precipitation activation energy equal to the sum of the formation and the migration energy of the defect (i.e. a value close to the diffusion activation energy of silver Qo in pure lead). 6. CONCLUSION The results obtained in this study may be summarized as follows: (i) The diffusivity (D*) of silver during silver precipitation in dilute Pb-Ag alloys is slightly higher than the diffusion coefficients (D) measured in pure lead or lead-silver solid solutions. The diffusion activation energy Q~. = 49.4 kJ/mol is slightly smaller than the diffusion activation energy ( Q o = 60kJ/mol) of silver in lead but is in fair agreement with the activation energy for precipitation of silver previously measured by the authors (Ea = 50 kJ/mol), AM 42/12 J

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(ii) For CAg smaller than (130-150) × 10-6at/at, the presence of (Ag-Pb) mixed dumbbells explain well all the silver diffusion results obtained in pure lead and lead-silver alloys either under equilibrium conditions (diffusion in solid solutions), or during silver precipitation. The activation energies E a and Qo. have been attributed to the rotation of the mixed pair (Pb-Ag) which must migrate without any dissociation. (iii) For CAg greater than (130-150)× 10-6at/at, the formation of another defect during the quench must be taken into account. The percentage of this defect strongly depends on the rate and the temperature of the quench. From previous positron lifetime measurements, we can assume that (Ag-V) pair, more mobile than (Pb-Ag) pair, corresponds to this second defect. (iv) From the experimental measurements of D, D* and C~g, the formation enthalpy of (Ag-Pb) f pb) = defect has been calculated: AH(Ag 58.75 kJ.mo1-1.

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