A QUANTITATIVE STUDY OF VACANCY DEFECTS IN QUENCHED PLATINUM BY FIELD ION MICROSCOPY AND ELECTRICAL

RESISTIVITY-II. ANALYSIS AND INTERPRETATIONT

A. S. BERGERZ D. N. SEIDMAN,: and R. W. BALLUFFIf

In the previous paper (Part I) it w8s demonstrated by means of field ion microscopy that the measured retie of divctcency concentration to monovaccmcy concentration (c,,/c,,) in high purity platinum speci- mens quenched from 1700 f 1°C w8s 0.06 & 0.02. In the present paper (Part II), 8 detailed series of qu8ntit8tive celculations w8s first performed which showed that this retie w8s not affected by specimen storage and electropolishing procedures subsequent to the quench and hence represented cpv/clc at the “freeze-out” temper8ture (T*). It was then shown th8t all combinations of the divecancy binding energy (E,,“) and binding entropy (Ulna) which lead to 8 quenched-in value of ce./cIF equal to 0.06 f 0.02 for a quench temperature of 1700°C 8re governed by the linear relationship

(0.23 & 0.03) + 6.17. lo-* y ( )I eV,

where k is Boltzmann’s constant. This linear equation implies that the divacenoy binding free energy (G,,a) w8s 0.23 eV at T* = 443.4’C. This value of G,,a, calouleted from our experimental data, depends only on 8 knowledge of the monovecancy migration energy ctnd an approximate value of a frequency factor. The existing tracer diffusion data for pl8tinum were re-examined in terms of 8 monovacancy and div8oancy model in en sttempt to decompose Grob into E,,b and S1, b terms, and it was concluded that the availubEe data 8re not sufficiently accurate to make this decomposition significant.

UNE ETUDE QUANTITATIVE AU MICROSCOPE A EMISSION D’IONS ET PAR RESISTIVITE ELECTRIQUE DES DEFAUTS LACUNAIRES DANS LE PLATINE-II. ANALYSE ET INTER-

PRETATION Dans 1s premiere partie, les auteurs ont dbmonte& 8u moyen du microscope B &mission d’ions que le

r8pport entre 18 concentration des bilacunes et celle des monol8ounes (c,&,) qui a 6ti mesurb dans des bchantillons de pletine de haute puretb tremp& B partir de 1700 f 1°C est de 0,06 i 0,02. Dans cettc deuxieme pertie, une s&ie d&ail&e de calouls qusntitstifs 8 6ti d’8bord effect&e, oes calculs ont montre que ce rapport n’est ~8s affect6 per un stooksge de l’hhantillon et p&r des polissages Blectrolytiques f8is8nt suite & 18 tremp8, et que, p8r suite, oe r8pport correspond B la temp&ature “d’immobilisetion” (!I’*). Ensuite, les auteurs montrent que toutes lea combineisons de 1’8nergie de lieison (Ezwb) et de l’entropie de liaison des bilaounes (Slub) qui conduisent 8. une valeur ap&s trempe de cec/cla &gale $ 0,06 f 0,02 pour une temp&ature de trempe de 17OO’C obbissent ZL la relation linbaire

E,,” = (0,23 f 0,03) + 6,17 * lo-* ‘$ ( )I eV,

oh k est la constante de Boltzmann. Cette Equation li&;aire suppose que 1’6nergie libre de liaison des bilacunes (G,,b) est de 0,28 eV B T* = 443,4’C!. Cette valeur de Gtvb, calcul&e. B partir de nos r&ultats exp&imentaux, depend seulement d’une conneiss8noe de l’bnergie de migration des monolecunes, et. d’une veleur approchb d’un facteur de frhquence. Les r&n&&s d&j& existent obtenus psr diffusion de treceurs pour le platine ont Qti rbxami&s B l’aide d’un modhle de monolacunes et de bilacunes per- mettant d’essayer de d&composer (32-b en E tVb et S1,,b, et les 8uteurs concluent que les valeurs utilisebles ne sont pas assez prbcises pour que cette d&composition ait un sens.

EINE QUANTITATIVE UNTERSUCHUNG VON LEERSTELLENDEFEKTEN IN ABGESCHRECKTEM PLATIN MIT HILFE DER FELDIONENMIKROSKOPIE UND

ELEKTRISCHER WIDERSTANDSMESSUNGEN-II. ANALYSE UND INTERPRETATION

In der vorhergehenden Arbeit (Teil I) wurde mit Hilfe der Feldionenmikroskopie gezeigt, d8B in hochreinen, von 1700 f 1°C abgeschreckten Platinproben das Verhiiltnis von Doppelleerstellen-Konzen- tration zu Einfachleerstellen-Konzentration (c&,,) 0,06 f 0,02 ist. In der vorliegenden Arbeit (Teil II) wird 8nh8nd ausfiihrlicher quantitcrtiver Berechnungen nechgewiesen, da13 dieses gemessene Verhiiltnis nicht durch Probenaufbewahrung und Elektropolieren nach dem Abschreoken beeinfluDt wird und somit ca,,/cla bie der ‘ ‘Ausfrier”-Temperatur (!Z’*) darstellt. AuBerdem wird gezeigt, daO alle Kombinationen der Doppelleerstallen-Bindungsenergie (Pevb) und Bindungsentropie (SsOb), die zu dem eingeschreokten Wert c.&~~ = 0,06 += 0,02 fiir eine Abschrecktemperatur von 1700°C fiihren, durch die folgende lineare Beziehung beschrieben werden:

E,,” = (0,23 f 0,03) + 6,17. 10B- ‘$ [ ( )I eV.

D8bei ist k die Boltzmann-Konstante. Diese lineare Gleichung flirt zu dem Ergebnis, dsti die Bindung- senthalpie der Doppelleerstelle (Gp,‘) bei T * = 443,4’% 0.22 eV ist. Dieser 8us unseren experimentellen D&m bereohnete Wert von G,p hiingst nur von der Kenntnis der Wanderungsenergie der Einfachleer- stelle und einem Niiherungswert des Frequenzf8ktors 8b. Die vorliegenden Tracer-Diffusionsdaten fiir PI&in werden in Hinsicht auf ein Einfachleerstellen- und Doppelleerstellen-Model1 erneut untersucht mit dem Ziel, G,,b in E,,” und Szeb zu zerlegen; es zeigte sich, da13 die verfiigbaren Daten nicht geniigend gen8u sind, urn eine bedeutungsvolle Zerlegung zu erlrtuben.

t Received June 27, 1972. This work was supported by the U.S. Atomic Energy Commission. Addition81 support was received from the Advanced Research Projects Agency through the use of the technical f8cilities of the Materials Science Center at Cornell University.

$ Department of Materials Science and Engineering, Cornell University, Ithaca, New York 14850. $ John Simon Guggenheim Memorial Foundation Fellow 1972-73.

ACTA METALLURGICA, VOL. 21, FEBRUARY 1973 137

13s ACTA METALLURGICA, VOL. 21, 1973

1. INTRODUCTION

In Part 1’1) an experimental value was reported for the ratio of the divacancy to monovacancy con- centration (czV/cl,,) for platinum specimens quenched from 1700 + 1°C. The main purpose of the present paper is to demonstrate that this ratio is a direct measure of the divacancy binding free energy (GzUb) at a characteristic “freeze-out” temperature (T*) as described in Part I and that the experimental value of csV/ciu (i.e. 0.06 -& 0.02) can be used to determine a linear relationship between the divacancy binding energy (E,,“) and binding entropy (Ss,“). The cal- culation of this relationship only requires a knowledge of the migration energy of a monovacancy (E,,m) and an approximate value of a frequency factor (v).

The use of the value of cZV/clV to determine the quantity GZvb required proof that the measured cZ,,/cn, was not changed from its quenched-in value at T* as a result, of the specimen storage and electro- polishing procedures which followed the quench from 1700 f 1°C. Hence, a detailed series of calculations was first made (Section 2) which monitored the ratio c.Jc~~ during the total specimen thermal history. The main conclusion of these calculations was that the value of the ratio cs,,/ci,, for the specimens electro- polished by procedure B (Section 2.3.2 of Part I) was not changed by this treatment or any other steps involved in their thermal history, and therefore could be used to determine GZvb at T*.

Finally, in an attempt to decompose GZvb into EaVb and SZvb terms, the available tracer diffusion data for platinum were re-examined critically (Section 4.2) and it was concluded that the existing data are not sufficiently accurate to allow a significant decompos- ition.

2. THE RATIO cza/cn, AS A FUNCTION OF THE TOTAL THERMAL HISTORY

OF A SPECIMEN

The total thermal history of a specimen consisted of the following three stages:

1. A quench from 1700 & 1°C (T,,) to -5°C. 2. The storage of the specimens in bulk form at -20°C for a maximum of 30 days. 3. The preparation of a sharply pointed FIM tip

from a bulk specimen by either electropolishing pro- cedure A or B (see Sections 2.3.1 and 2.3.2 in Part I) followed by further storage at either ~20 or -5O’C.

The above three stages are illustrated schematically in Fig. 1.

2.1 Redistribution of a closed assembly of monovacancies and divacancies during the quench

The first. stage consisting of a quench from 1700 f 1 to -5’C, was schematized as a quench to 20°C

SCHEMATIC THERMAL HISTORY OF A SPECIMEN FOR BOTH ELECTRO-

zj POLISHING PROCEDURES

e

rELECTROPOLlSHlNG PROCEDURE A \ 325 TO 369-C FOR 6 TO 25.3 6m

JToREAq??_ - _ _ - - _

1 ELECTROPOLISHING PROCEDURE B ! r-20°C FOR - l-3 hr 1 \STORE AT - 2o’c 0R -50’~

OIJWCH+-ST~~E_l~o~~LK-+-ELECTROPOLISH 8 STORE----_-i

TIME

FIG. 1. A schematic diagram illustrating the three stages in the total thermal history of a specimen for both elec- tropolishing procedures A and B (see Sections 2.3.1 and

2.3.2 of Part Ill’).

(see Fig. 1) for the analysis. In the analysis for the defect redistribution we treated the assembly of monovacancies and divacancies as a closed system, hence it was necessary to specify a value of the total vacancy concentration (ct) for the calculations. Thus, we first present a brief physical argument to demonstrate that the relevant value of ct is the

measured quenched-in value (ct*) and that the thermal equilibrium value at 17OO’C need not be known.?

The best value of 2 reported in Part I is a lower limit to the actual thermal equilibrium value at 1700°C because of the loss of vacancies to sinks which must occur in any real quench t2) from a temperature which was so close to the melting point (1769°C). The largest percentage of the total loss occurs at elevated temperatures and after this initial loss ct is then approximately a constant for the remainder of the quench (see Seidman and Balluffi(3*4) for a treatment of the diffusional aspects of this problem). The temper- ature at which the losses become negligible is higher than the critical temperature (T*) at which a closed assembly of monovacancies and divacancies can no longer maintain itself in quasi-equilibrium. The reason for this fact is that the diffusion distance between dislocations is appreciably greater than the diffusion distance between two monovacancies, so that the assembly is able to maintain itself in local equilibrium long after the vacancy losses to dislocations have be- come negligible. Thus, the calculated ratio c2V/clV at the end of a quench for a closed assembly is characteristic of an open assembly, if the value of ct employed in the calculation is the one measured after the quench.

t It is also to be noted that this approach avoids the prob- lem of assigning exact values to El,*, S,,f, S,,b and E,,b in the expression for ct [me equations (2)-(5) of Part I] at the initia- tion of the analysis.

The analysis starts with the approximate equa- most of the numbers reported in the literature. Hence

tionsc5) we calculated the quenched-in ratio c~~/c~~ for E,,” =

ac,, 1.33, 1.38 and 1.43 eV at constant values of y. The

dt = -2@,,2 + 2%~ (1) range considered for v was from lOI t.o lOlQ see-l and it. is noted that this IOO-foid variation for v is appreciablg greater than the possible range of this

(2) parameter expected in practice. The results of these calculations are displayed in Table 1. The numbers

where the K,‘s are given byc5)

K, = 84~ exp (-E,,“/kT) TABLE 1. The ratio of divacancies to monovacancies frozen-in

(3) at low temperatures after a quench from Z’* for a closed system of monovacancies and divwxmcies with

KZ = 14~ exp (S~~/k) exp [-(%,m + &,“)/W, (4) 2 = 2.64 - lo-* at.fr. for various values of the parameters V, E,,“’ and E,,* and t is time. The quantities Elqvm and Ezvrn are the monovacancy and divacancy migration energies, respectively, and v is a frequency factor which in- cludes an entropy of migration factor. The free energy GZWb is given by

GZeb = Ezwb - TX,; (5)

where EZvb and StQb are defined by equations (5) and (6) of Part I. The total vacancy concentration is given

bY Ct = Clv + 2c,,, (6)

%,6 (eV

0.05 0.10 0.20 0.30 0.40

(a)

(c,,/c,,.) for Y = 1Oz3 set-’ and SaVb = 0

Elr.R (0V) 1.33 1.38 1.43

0.004 0.0039 0.0038 0.0095 0.009 0.0085 0.044 0.04 0.036 0.158 0.139 0.124 0.43i 0.38 0.338

lb) h&*af

for Elub = 1.38 eV and S,,b = 0

and since ct is constant for a closed assembly, it -@Lb follows that (eV) 10’s

y ‘;=g”’ 10’”

a% _2dc,, 0.05 0.0036 0.0039 0.0042 -= dt’

0.10 0.0078 0.009 0.010 at 0.20 0.031 0.04 0.052

0.30 0.099 0.139 0.194

Fujiwara@) was the first to solve the coupled nonlinear 0.40 0.261 0.38 0.552

differential equations (1) and (2) under idealized quenching conditions employing computer techniques. We have also solved these equations employing a com-

listed in Table l(a) demonstrate that for Elv* equal to

puter, but have used the following expression for 1.33-1.43 eV (v = lOI se&), the ratio cZa/clo is

the therma history during quenching which was rather insensitive to the value of Elem when compared

derived from the experimentally determined quen- to its strong dependence on Ez,,b. Table l(b) shows

ching curve shown in Fig. 1 of Part I: the sensiti~ty of the ratio cZojclo to Y for El,“’ = 1.38 eV. It is seen that the final value of c~,,/c~~ is not a

T(t) = 1700 exp (--[(13sf2 + 5((1Oa+ + (IO+ strong function of the value of v employed, when

-t (1W811)oC, (8) compared to the sensitivity of this ratio to Etvb. Thus, it is concluded that it is B satisfactory pro-

where 5 = (t + 2 . 10m3) sec. cedure for the purposes of Section 2 to employ one set The variable parameters in this model which affect of “best” values (E,,m = 1.38 eV and v = 1013 sea-1)

the quenched-in ratio czD/cl_ are E,,“, Y, SZvb and for the calculation of c.&~ as a function of Ezvb. Ezum, because !l’, and ct are fixed by the experimental To emphasize the strong effect of Ezub on the ratio conditions. Since our objective was to determine c.&~~ at constant v and SZvbt we have plotted this E,,b from the value of cZv/cl,,, we calculated the sensi- ratio in Fig. 2 as a function of T(‘C) for Elvm = 1.38 tivity of this ratio to the remaining variable param- eV, SZnb = 0 and v = lOI see-l. Figure 2 illustrates eters for SZvb = 0 and showed that for al1 reasonable the following three important characteristics of a values of Elsm and v the value of czv/clv was essentially quench : determined by EZvb. A value of Elvm equal to 1.38 eV 1. At elevated temperatures, during a rapid quench, is favored as a best value by both Jackson(7) and the assembly is able to maintain itself in a quasi- Schumacher et al.,@) and the range 1.33-1.43 eV covers equilibrium state and the ratio cZa/clv obeys the

BERGER et al.: VACANCY DEFECTS IN QUEh’CHED PLATINUX-II 139

140 ACTA METALLURGICA, VOL. 21, 1973

~-0139

,633 1700 1500 1300 1100 900 700 500 300 loo 0

TEMPERATURE 1°C) DURING QUENCH

FIQ. 2. The ratio of divaoancy to monovaoancy oon- centration (cl,&,,) vs. temperature during a quench from 1700°C for Es,” = 0.40, 0.30, 0.20, 0.10 and 0.05 eV, SSSb = 0 and Y = 10ls BBC-~. The curves were calculated using equations (l)-(7) for a closed system of mono- vacancies and divacancies for an experimental quenching rate which was given by equation (8). The total concen- tration employed was the value of 2.64. IO-4 at.fr.

reported in Part I.

equilibrium relationship

c2vlclv = NV e=p (+f) exp (g). (9) 2. Upon cooling to lower temperatures during a

quench the value of cZ&Iv falls below the quasi- equilibrium ratio and eventually at T* the defects become “frozen-in” and the ratio cZ,,/cIv then becomes a constant independent of T.

3. The ratio cz,,/cIv is an extremely sensitive function of E2,,“. Note that cZv/clv varies by a factor of ~10~ as Eeyb is varied by a factor of 8 from 0.05 to 0.4 eV at constant SZvb.

2.2 Redistribution of a closed assembly of monovacuncies and &vacancies in a bulk specimen at constant temperature

The second step in the thermal history consisted of the specimen storage in bulk form at ~20°C (see Fig. 1) for a maximum period of 30 days. Prior results of isochronal recovery experiments on bulk specimens quenched from 1700% indicated that there were no appreciable losses of vacencies to sinks until ~180°C (see Jackson’s(‘) Fig. 7 and Schumacher et d’s(s)

Fig. l), hence our storage procedure could not have

resulted in any change in ct 7 But it is still possible

that there was some redistribution in the relative monovacancy and divacancy populations, hence we calculated the ratio cZa/cIV as a function of t at a fixed recovery T for a constant ct.

The differential equations governing the redis- tribution of monovacancies and divacancies

are given by equations (1) and (2) with the h’,‘s described by equations (3) and (4). The quantity c2v can be eliminated from equation (1) employing equation (6) to obtain

--&$j = (clu+g$2-a2 (10) where

a2 = (K22 + 8h’lh’,C,) 16K, ’

Next, making the substitution

K2 u = Clv + 4K’

1

(11)

(12)

and casting the differential equation in integral form yields

(13)

where the quantity u* at t = 0 is determined from the value of cl,, at T* (see Fig. 2). The integration of equation (13) yields

a 1 + (u* - a) exp (-4aK,t)

Cl,@) =

(u* + a) 1 h’2 -- [ ’

(u* - a) 1 4K1 - (U* + a) exp ( -4aKlt) (14)

for the time dependence of cIv at constant T and ct. Equation (14) was evaluated at T = 20°C for ct = 2.64 * 1OA at.fr., El,,” = 1.38 eV, v = 1013 set-l and for E2vb = 0.40, 0.30, 0.20, 0.10 and 0.05 eV.

The calculations showed that there was absolutely no redistribution of the monovacancy-divacancy assembly for a holding time of 30 days at ~20°C. This result held for all five values of E2,,” employed. Hence, it was concluded that the specimen storage in bulk form et room temperature did not cause a change in the velue of c2&,, frozen-in at T*.

BERGER et ~2.: VACANCY DEFECTS IS QUESCHED PLATINUM-II 141

2.3 Redistribution of an open assembly of monovacancies and d~v~n~~~s during the electrol.'olisfaing iprocedure B

!I%e third stage in the thermal history (see Fig. 1) of one group of specimens consisted of the preparation of a sharply pointed tip (~100 .& radius) by electro- polishing at *2O”C and subsequent storage at either ~26 or --5O’C (see Section 2.3.2 in Part I). As a result a surface was introduced which was in the immediate vicinity of the monova~ncy~iv~eancy population which was subsequently examined by the FIM pulse dissection technique. Thus, monovacan- eies and divacancies may have diffused out of the sharply pointed tip to its surface during the electro- polishing and subsequent storage. In order to analyze this possibility quantitatively, a model was constructed based on the following assumptions :t

1. The FIM tip was approximated by a sphere of radius R.

2. The surface of the sphere was the only sink present for both monovacancies and divacancies and it maintained the defect concentrations in local equilibrium during the holding period.

3. The initial distribution of monovacancies and divacancies was uniform throughout the tip.

4. The initial value of ct was taken as the experi- mental value of 2.64 . 10m4 atfr.

5. The concentration gradient had no angular dependence.

The coupled nonlinear partial differential equations governing this situation are

where the K,‘s are defined by equations (3) and (4) and the Dnv’s by

or+, = a2rlv exp f -.&,“lkT), (17)

D2, = G vpz; exp (- E,,“/kT) W)

where a is the lattice parameter, and vn, and yayz, are the frequency factors for monova~ncy and divacancy migration respectively which include entropy of migration terms. Equations (15) and (16) were solved numerically on a comput,er at fixed values of R and T to obtain ci,,(r, t) and c&r, t). These latter quantities

t It is readily shown (see next section) that any losses which occurred during the relrttively short eleotro~li~i~ period were negligible and that we need consider only the storage period after the tip was formed.

were then integrated over the volume of the sphere to obtain their mean values as a function of holding time at a particular T. The following values of the variable parameters were employed; E,,” = 1.38 eV, vie = r2, = lOi se&, Ezf? = 0.40, 0.30, 0.20, 0.10 and 0.05 eV and 5’,2 = 0. These values are identical to those used previously in Sections 2.1 and 2.2. The only additional param- eter to be specified was Es,“. A value of 1.11 eV for Ezpna has been suggested bp both Jacksonti) and Schumacher et aI., although effective energies as low as ~1.0 eV have been measured in recovery experiments for specimens quenched from near the melting point (Politk@) and Schumacher el al.(*)). In view of this all calculations were performed for E,,m equal to both 1.11 and 1.0 eV, even though it is highly probable that 1 .O eV does not correspond to the migration energy of the divacancy.

Some of the calculated results are given in Fig. 3 which shows plots of the normalized ratio c,(t)/cz9(0) vs. time at 20°C for the two values of Es,,” for a specimen with a 100 A tip radius. The initial values of er, and czv were obtained from Fig. 2. The following results were obtained at 20%:

1. For B,,m = 1.11 eV the initial value of cau decreased by only ~4.5 per cent (see Fig. 3) and q, remained completely unchanged after a total holding time of 600 hr.

2. For E,” = 1.0 eTr the initial value of csu decreased by ~51.5 per cent (see Fig. 3) and ciu remained unchanged after a total holding time of 600 hr.

3. The percentage decrease in c2v for both values of Egum was independent of EZvb (see Fig. 3).

At T = --50% the calculations showed that

I.0

TIME

FIG. 3. divacency concentration [es,(t)] constant temperature

spherical field ion microscope tip of 100 A radius. The curves were calculated from equations

boundary conditions and assumptions described in Section 2.3. The two curves were cdculated

Es,,“’ = 1.11 eV snd ES-m = 1.0 eV for an initial total concentration of vacancy defects of 2.64 * lo-‘ at.fr.

142 ACTA METALLURGICA. VOL. 21, 1973

both clV and cpv were completely unchanged after a total holding time of 600 hr for E2,,m equal to either 1.11 and 1.0 eV. In terms of the experimental results (Table 3 in Part I) this implied that for specimens 7 and 8 the measured value of c&~~ was characteristic of that at T*.

For specimen 6 (Table 3 in Part I) the possibility arises that if E2,,“’ is as small as 1.0 eV, then there would have been a zone denuded of divacancies next to the original electropolished surface. But it is noted that ~50-100 A of material was removed from this specimen during the low temperature field evaporation process which was used to bring the tip to its final end form. Hence, the denuded material was probably removed prior to the pulse dissection examination for vacancies. Thus, it is believed that even for the improbable case of Ezvm = 1.0 eV, the material examined for vacancies had a value of csu which was close to that present originally at T*. We note that the experimental result that there are no significant differences between the divacancy con- centrations measured in specimens 6-8 (Table 3 in Part I) is consistent with this conclusion.

2.4 Redistribution of an open assembly of mono- vacancies and divacancies during the electropolishing procedure A

The situation during electropolishing procedure A was more complicated, since appreciable defect redistribution may have occurred during the rel- atively high temperature electropolish.

We first show that rapid redistribution without losses occurred during electropolishing in regions well away from the surface and that a quasi-equilib- rium monovacancy-divacancy population typical of the electropolishing temperature must have been established in those regions. Equation (13) was used to calculate the time required to reach the quasi. equilibrium values of cl,, and csv at 325°C for ct = 2.64 * lo-* at.fr. The times required were ~6-7, ~2,

BERGER et al.: VACANCY DEFECTS 1,1‘ QUENCHED PLXTISUhI-II

the defect diffusion distance over t according

‘El0 = 2[~~*~(~~ w’2, and

?i& = Z&?,,(t) dp-.

Calculations based upon equation (21)

to

(21)

(22)

indicate

c T, =1700’C; ~~~2.64 x10-4At Fr =CONSTANT IO-’ m E,, = 1.3&V; Y = 10 13 -I set

negligible denudation for the slower diffusing mono- vacancy. However, equation (22) predicted denuda- tion widths for the fas-ter diffusing divacancies as large as several hundred angstroms and we therefore pre- diet that appreciable divacancy losses may have occurred.

The experimental results appear to be in qualita- tive agreement with the above predictions. Only one divaeancy was detected among the 321,298 sites examined in the specimens prepared by procedure A (see Table 2 in Part I). This should be contrasted with the 8 divacancies detected among the 593,794 sites examined {see Table 3 in Part, I) in the speci- mens prepared by procedure B.

It is concluded generally that the ratio cz,/cl, measured for the specimens polished by procedure A (Nos. 1-5 in Table 2 of Part I) was not characteristic of the value frozen-in at T* during a quench, but that its value was appreciably decreased as a result of pre- ferential diffusional losses of divacancies to the surface during the high temperature polishing treatment (see Fig. 1). Furthermore, we conclude that our combined calculations and experimental results are completely consistent with a divacancy which is more mobile than the monovacancy as discussed in Part I.

3. THE DIVACANCY BINDING ENERGY

The results of the calculations presented in Section 2 demonstrate that the quenched-in value of cZ,,]clu determined for specimens 6-8 (Table 3 in Part I) can be used to determine EZvb via the method dis- cussed in Section 2.1. All the calculations performed up to this point were made on the basis of an assumed value of S,, b = 0, but it is clear that the equilibrium value of c~,,/c~~ at T,, as well as the quenched-in value at room temperature, is reaily determined by Gsnb. Hence, a search was made for all possible combinations of EZvb and S,,& which gave it quenched-in value of

c2u/c1v = 0.06 for the parameters 2 - 2.64 - lo* at.fr., E,,,m = 1.38 eV and Y = 1013 se+. Some typical results are shown in Fig. 4, for Ezvb = 0.40, 0.23, 0.11 and 0.05 eV. These calculations showed that the following linear relationship exists between EZvb and SZvb,

J&b = (0.23 rf 0.03) _t 6.17 * 1O-2 eV, (23)

1700 1500 1300 1100 900 700 500 300 too 0

TEMPERATURE (‘Cl DURING OUENCH

Fra. 4. The ratio of divacancy to monovacancy concon- tration (c,,/c,,.) vs. temperature during a quench from 1700°C for different combinations of Ezub and S,,” which lead to the same value of c,,/c,,, at, room temperature. The curves were calculated for a closed system of mono- vacancies and divacancies with a total concentration of 2.64. lo-*at.fr., Elom = 1.38eV, 1’ = lOIS see-’ and an

experimental quenching rate given by equation (8).

which is shown plotted in Fig. 5 for a wide range of possible values of EzDb and SZvb.

4. DISCUSSION

4.1 The relationship between the &vacancy binding energy (E2,b) and binding entropy (SZvb)

The linear relationship between E2u6 and Sllvb obtained in Section 3 implies that for all combinations, of these two quantities which lead to a fixed c2JcIu for a quench from a single temperature GZv” is a constant which is characteristic of a single T*. This can be seen by noting that under the above conditions equation (5) reads

Gzeb(T*) = ~2~b(T~) - T~S2*~(T*) = const. (24)

Hence, a simple calculation showed that T* = 443’C and that G2,b(4430C) = 0.23 eV for our experiments. We note that the further decomposition of G2: into EzUb and S2,b terms requires a knowledge of a second value of G2,P for a different T*. This could be ac. complished experimentally by quenching from the same T, at a faster quenching rate than the one employed or alternatively quenching from a different T,. The main experimental problem with the latter procedure would involve obtaining a statistically signi6cant value of c2v/clv if T, were decreased ap- preciably below 1700°C.

There have been no experimental determinations, to date, of S 2$ for any metal and the theoretical situation is in a rather uncertain state. Schottky

144 ACTA XETALLXJRGICA, VOL. 21, 1913

0.40

i

0.35

S?&k

Fxo, 5. The div#%noy binding energy {El*“) vs. the quan- t~t~~~~b~~ when, Stpb is the divmey binding entropy and k is Boltzm&~‘a constant. The line&z relationship between I@,,* and (S,,b{k) was obtained from Fig. 4 by finding the locus of all CO~~~E~~~~~ of Es** and 51Wb w&h

lead to B value of opV/eia = 0.06 at room temperature.

et uZ.tl” have calculated values? of S,,& for copper, silver snd gold of -1.8k, -l&k and -2.2k, respct- ively. Burton hss employed a method originally developed by Land and Goodmanus) to calculate X,,b for B &vacancy in solid argon and obtained a value of 0.34k. In addition, Burton@@ has criticized the Einstein ~pp~xim~tion employed by Schottky eb ul. in their calculations of S,,ib for the noble metals. In view of this controversial theoretical situation we have chosen not to decompose our measured value of C,,# on the basis of a. t~e~et~~Z value for SzDb.

defects available for platinum are the xesults of two high ~rn~r&tn~ tracer diffusion experiments. The first of these experiments was performed by Kidson and l%os@) using the sectioning technique(i6) and they

obt&ed the following expression for the tracer diffusion e~~~ient (L),)

D, = 0.33 exp (-2.96 & 0.06 ~~~k~~ cm2 set-f (25)

in the temperature range 132%1606% (see Pig. 6). The second experiment was performed by Cattaneo

7 Note that the convention used by Schottky et al.“‘) and Burton

BERGER el al.: VACASCT DEFECTS IX QUESCHED PLATIXUhI-II 143

TEMPERATURE (“Cl 750 1700 1650 1600 1550 1500 1450 1400 1350 1300 1250 lIIIlIlll~l~l’~‘ll I I

r

t x991 T, 0.7467:

6) - CALCULATED FIT

I I I I I I I I I I I I I I l I II 11 11 11 I l l ’ 11 1 ” 3 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5

x IO4 (“K-l)

FIG. 6. The tracer diffusion coefficient (Dr) vs. (l/T%) for all data available for platinum. The solid curves were calculated for e monovacancy and divacancy defect model which included a temperature dependence of the active- tion energies and entropies, and was described by equations (27)-(34). The values of the different parameters

employed to calculate DT for the various sets of E,,* end S,,* are listed in Table 2.

where

% = ~2ylvOfiV exp (Sl,m/N exp (S,,r/N, (28)

&I = &vf + Elo"' (2%

a = ida, + aF) (30)

Q21 = Eluf - Elvm + E2vm - E2vB3 (31) id

(374

The f,,, are the correlation factors for tracer diffusion (;r;, = 0.781(22) and f2, = 0.475@)), ~5’,,,~ the entropies of migration, v,,,O the frequency factors and +(a, + aF) the temperature dependence of the migration and formation energies and entropies. The expressions used for the temperature dependence of the energies

and entropies are identical to the ones employed by Schumacher et al. and are given by expressions of the form

and

E(T) = E(T,) + aE(T - To), (33)

S(T) = S( To) + ak In (T/T,) (34)

where To is a reference temperature which was set equal to 300°K. The quantity a was assumed to be identical for both monovacancies and &vacancies.

The values for the different sets of the five variables [equations (28) and (32)] used to calculate D, are listed in Table 2. The fourth column contains the best values given by Schumacher et al., and the re- maining columns are for other values of EZvb and S2: which are consistent with our equation (23). It is noted that changing EZvb and SZvb only affects Q21 and D, in equation (27), and, as seen below, the

146 ACTA METALLURGICA, VOL. 21, 1973

TABLE 2. Values of the five basic parameters of the monovacanoy and divacancy defect model for different combinations of Eteb and S,,b

am 0.0-t 0.051 0.11; 0.15t 0.23t 0.4% ** -3.7211 -2.925k -2.M -1.3k O.Oh 2.7%

D,,(cmZ see-l) 0.14 0.14 0.14 0.14 0.14 0.14 QI@V) 2.87 2.87 2.87 2.87 2.87 2.87

0.14 0.14 0.14 0.1: 0.14 0.14 1.22 1.17 1.11 1.07 0.99 0.82

118.46 53.44 35 10.53 2.88 0.187

t The values of Ep,” and S,, a in this column are consistent with equation (23). $ The values in this column are from Table 2 of Schumacher et c&~~!

equation (23) within the estimated experimental error of 0.03 eV.] [We note that the Etvb andS2,.B values are consistent with

There appear to be two arithmetic errors in Schumacher et aZ.‘s Table 2. The first one is that the values which they list for vI,o cm* see-l.

exp (Sl,m/k) and S,,f/k are not consistent with D,, = 0.14 The second one is that the value of D,, calculated from the listed quantities which comprise this expression

[equation (32)] does not correspond to 35. Nevertheless, we have taken D,, = 0.14 cm3 see-l and D,, = 35.

TEMPERATURE (“Cl

CT 5 I ” ” ” ” ” ’ “1 1 ’ t 1 t 1 t 3 t 1 I k 4.8% 490 4.92 4.94 4.96 4.98 5.00 502 5.04 5.06 5.06 5 to 5.12 5.14 513s

(tfTf x IO4 f”K-‘f

Fxa. 7. The tracer diffusion coefficient (Dp) vs. (I/T”K) in the temperature range 1865-1769°C (the melting point). The calculated curves for DT are for a monovaaancy and divacancy model described by equations (27)~(34) employing various sets of E,,b and SIvb values which are consistent with equation (23). This temperature range is an extremely important one with respect to the monovacancy-divecancy defect model. since it is the regime

which is the most sensitive to the values of E,,‘J and S,,‘J.

quantity

% e=p t - &dkT) (35)

will only become important at values of T close to the melting point. In Fig. 6 the D, curve calculated for the Schumacher et al. values and the Dp curve calculated for Esmb = 0.23 eV, S,,tl = 0, IZzeb = 0 and S,b = -3.72k are superimposed on the original tracer diffusion data. At ~125O”C, or less, the sbsol- ute value of D,, exp (-&,,/kT) is 10.01 for all the combinations of Ezvb and SZvb listed in Table 2, hence it is only useful to examine the values of D, for T 2 1250°C in the attempt to decompose Gzsb. The Arrhenius plot for D, in the temperature regime 1665-1769’C is shown in Fig. 7 as it is in this regime that one would expect to see the largest effects due to the existence of divacancies. At T, the increase in

U, in going from the values EZob = 0.4 eV and Szvb = 2.75k to E: Zpib = 0 and SzVb = -3.72k is only , which sre available in this important high temperature regime have an absolute accuracy of only f13.6 per cent [equation (ZS)]. In view of this situation we must conclude that the existing data for D, are not suffiei- ently accurate to allow a significant decomposition of Gsea into Eztp and SaVb terms.

ACKNOWLEDGEMENT

We would like to thank Dr. Kyoon H. Lie for writing the computer programs for the calculations involved in sections 2.1-2.3.

BERGER el al.: VACANCY DEFECTS IS QUENCHED PLATISU?rI-II 11;

1.

2.

3.

4.

5.

REFERENCES 12. A. S. BEROER, D. N. SEI~MAN and R. 1%‘. BALIXFFI. 13* Acta Met. flil, 123 (1973). R. W. BALLUFFI, K. H. Lx, D. N. SEIDMAN and R. W. SIEGEL, in Vacanck a& Interetitials in Met&, edited by

it* - *

A. SEEBER, D. SCFIUM~CHIZR, W. SCHILLING and J. DIEHL, pp. 125-167. North-Holland (1970). D. N. SEIDMAN and R. W. BALLUFFI, Phys. Status Solidi 17, 531 (1966). 16. D. N. SEIDXAX and R. W. BALLUFFI, in Interactions 17. Between Lattice Defects, edited by R. R. HASIGUTI, pp. 9 1 l-960. Gordon and Breach (1968). 18. For example, see, A. C. DAMASK and G. 5. DIENES, Point Defects in Metate, pp. 110-119. Gordon and 19. Breach (1962). ^.

6. H. FVJIXVARA, Technical Report, Contract NONR 2”. 1834(12) 46-22-55-363, Department of Physics, University of Illinois, Urbana, Illinois.

J. J. BVRTOS, J. phys. Chem. S&de 35, 589 (1972). P. L. LAXD and B. GOODMAF, J. phys. Chem. Solids 28, 113 (19671. J. J.~B~~R~ox, Phy8. Rev. B5,2948 (1972). C. IT\‘. KIDSOX and R. Ross, in Proceed&as of the Infer- n&ionaE Conference an Radioisotopes in Sciekijk Research, edited by R. C. EXTERMANX, Vol. 1, pp. 185-193. Per- Barnon Press (1957). C. T. TOMIZUKA, Neth. Exp. Phys. 6, 364 (1959). F. CATTANEO, E. GERMAGXOLI and F. GRASSO, Phil. Xag. 7, 1373 (1962). C. G. WAXG, D. N. SEIDXAX and R. 1V. BALLUFFI, Phys. Rec. 169,653 (1968). A. SEEGER and H. NEERER, Phy8. Sfafm i!?olidi 29, 231 (1968). N. PETERSOX, in Solid State Phusk. edited bv F. SEITZ and D. TURHBULL, Vol. 22, pp. 409-512. ‘Xcademic Press (1968).

7.

8.

J. J. JACKSON, in Lattice Defecta in Quenched Metals, 21. editedbyR.M.J. COTTERILL, M. DOYAMA,J.J.JACKSON

A. SEEGER and H. MEHRER, in T’acnncies and Interstitials in Metals, edited by A. SEEGER, D. SCHUMACHER, 1%‘. SCHILLIKG and J. DIEHL, pp. l-58. Sorth-Holland (lY?O).

and MCI, MESHII, pp. 467-479. Aeademio Press (1965). D. SGXXVEZACHER, A. SEECZR and 0. H~RL~x, .Phys. Statwr So&ii 25, 359 (1968). 22.

9. J. POL~X, Phys. Status Solidi 21, 681 (1967). 10. A. S. BJZRQER, Ph.D. The&, Cornell University (1971). 23. G. SCHOTTGY, Phys. Lett. 12, 95 (1964). 11. G. SCHOTTKY, A. SEEGER and G. SCHMID, Whys. Status

Solidi 4, 439 (1964).

K. CO~~PAAN and Y. HAVEX, Trans. Faraday Sot. 52, 786 (1956).