Kirkendall effect and diffusion in the gold-platinum system—II

KIRKENDALL THE

EFFECT

CONCENTRATION

AND DIFFUSION PENETRATION

IN THE

CURVES

AND

GOLD-PLATINUM THE

DIFFUSION

SYSTEM-II COEFFICIENTS*

A. BOLKt During the interdiffusion of gold and platinum a phase boundary appears, which can be seen from the concentration penetration curves; these curves are discussed in this part of the paper. The overall diffusion coefficient has been calculated from the curves according to the graphical method of Matano. From the observed Kirkendall displ~ement and the data concerning the concentration gradient and the concentration in the marker interface, the partial diffusion coefficients could be calculated. From the ratio between the partial diffusion coefficients a quantity has been derived which enabIes us to express the magnitude of an observed Kirkendall effect in a numerical value between 1 and 0. EFFET KIRKENDALL ET DIFFUSION DANS LE SYSTEME OR-PLATINE-II DETERMINATION DES COURBES PENETRATION-CONCENTRATION ET DU COEFFICIENT DE DIFFUSION Lors de l’interdiffusion de l’or et du platine, il apparait un saut de concentration qui peut Qtre observe sur les courbes concentration-penetration; ces courbes sont examin& dans une partie de cet article. Le coefficient global de diffusion a Bte calcule It partir de ces courbes p&&ration-concentration et suivant la methode graphiqm de Matano. A partir de l’effet Kirkendall observe et des don&es relatives au gradient de concentration et la concentration It l’interfaee initial, les coefficients de diffusion partiels peuvent &tre calculb. En partant du rapport entre les coefficients partiels de diffusion, l’auteur a pu determiner une valeur qui donne l’ordre de grandeur de l’effet Kirkenddl observe; cettc valeur est comprise entre I et 0. KIRKENDALL-EFFEKT DIE KURVEN

UND DIFFUSION IM SYSTEM KONZENTRATION/EINDRINGTIEFE DIFFUSIONSKOEFFIZIENTEN

GOLD-PLATIN-II UND DIE

Wahrend der Ineinanderdiffusion von Gold und Platin tritt sine Phasengrenze auf, die in den Kurven Konzentration/Eindringtiefe sichtbar wird; diese Kurven werdon irn ersten Teil der Arbeit diskutiert. Mit der graph&hen Methode von Matano wird aus den Kurven der Gesamt-Diffusionskoeffizient berechnet. Aus der beobachtcten Kirkendall-Verschiebnng und den Ergebnissen iiber den Konzentrationsgradienten und die Konzentration in der Markierungsflache lieDan sich die partiellen Diffusionskoeffizienten berechnen. Aus dem Verhiiltnis der partiellen Diffusio~koeffizienten wird eine GroDe hergeleitet, die die St&rke eines ~o~hteten K~kendall-Effektes durch eine Zahl zwischen 1 und 0 auszudriicken gestattet.

INTRODUCTION

calculated from the equations according to Darken(a).

In the first part of this paper(r) attention has been paid to the Kirkendall effect and the accompanying phenomena. In this part the concentration penetration curves will be discussed. The overall diffusion coefficient has been calculated from these curves according to the graphical method of Matano(2) by means of the equation jj,=

1

ax

2t

aN~~

--.-.

s NAu i

x ~~A~

VOL.

9, JULY

1961

-

%t)

dflAU __ dX

(2)

(3)

From the diffusion coefficients at different temperatures the frequency factors and activation energies have been calculated.

(1)

* This paper contains a part of the author’s thesis, Delft, 1959. Received May 12, 1960; revised December 14, 1960. t Laboratory for Physical Chemistry, Teohnical University, Delft, Holland. Now at Philips’ Research Laboratories, Eindhoven, Holland. ~ETALLURGICA,

tBAu

5 = NAu&t + N@A,,.

NAu,

A. EXPERIMENTAL

the concentration N,, being expressed in atomic fraction. The partial diffusion coefficients DA, and D,, for the concentration in the marker interface can be

ACTA

v =

PART

The concentration penetration curves have been determined according to the method described by the author.@) The diffusion couple, embedded in ureaformaldehyde, has been fixed in a specimen holder P, which can be screwed into the holder H (Fig. 1) or into a von Laue back reflection camera. Holder f?r can be screwed into the chuck of a lathe, so that it is possible to turn off very thin layers from the diffusion couple. In the gold-rich part of the couple these layers are

643

ACTA

644

METALLURGICA,

VOL.

9,

1961

FIG. 1. Specimen holder.

about

0.005 cm thick;

in the neighbourhood

phase

boundary

are

they

about

of the

0.0005 cm.

The

cm and 0.00025 cm in gold and platinum, respectively. Thus it is possible to determine the concentration

in a

as the difference

layer with a thickness of about 0.00013 cm, assuming

between the total thickness of specimen plus holder P

that a reflection with an intensity lower than a ninth

thickness

of a layer

is measured

before and after turning off. This thickness is measured

of that of the reflection

with a very accurate micrometer.

seen on the photograph.

specimen

a back reflection

Cu-K,-radiation

After cleaning the

photograph

with nickel-filter.

between

this quantity

and the concen-

tration has been given by Darling et aZ.c5) (Fig. 2). follows

from

coefficient

the

value

of

that the intensity

the

X-ray

It

absorption

of Cu-Kg-radiation

to a ninth of its initial value after penetrating

B. RESULTS

falls

0.00027

The obtained concentrations

have been plottedversus

the readings from the micrometer drawn.

This, however,

introduces

the two horizontal

axis.

straight lines through

60

80

of

Because it is easier to draw

a number

I

40

a very great sub-

especially in the neighbourhood

4,05

20

and the curves were

jective inaccuracy,

d in 8,

0

be

is taken with

From the Debye-

Scherrer rings the lattice parameter can be calculated. The relation

from the surface, cannot

100

-weight%Au FIG. 2. Lattice parameter in the gold-platinum system (according to Dwling(6)).

of points,

we have

BOLK:

looked for a method N,,

into another

to transform

quantity

gives a linear function

KIRKENDALL

even

the concentration

in such a way,

of the penetration

that

x.

it

Earlier

1 + erfu.

(4)

Nr

where N,(z < 0) and N&x > 0) are the initial concentrations Plotting

of the two halves of the diffusion

u as a function

couple.

of x enabled them to draw

straight lines through the points with a great accuracy and to

“normalize”

the

curves by retransforming equation

concentration

penetration

u into N again by means of

(4).

For concentration the u-x-curves

independent

diffusion coefficients

pass through the origin, so giving the

relation u = x/1/(4Dt), which can be derived directly from the second law of Fick. diffusion coefficient Equation centration

From this relation the

can be calculated.

(4) can also be used to calculate dependent

diffusion

coefficient

a con-

from

the

obtained straight lines. * In this case the straight lines do not pass through the origin. If

a phase

boundary

(i.e.

appears during the diffusion, limited

miscibility

concentration

B,

are constants.

coefficients are concentration calculated

the gold-rich

consisting

penetration

is zero.

mathematical

the concentration

Transforming

N,,

If the diffusion they can be

has been applied

to

in the normal N,, (4).

-

Matano

according

gives a non-linear

=

NAu,

definitely

to Matano.

concentrations

are

the curves the

could be determined

integration

phase boundary

value of NAu

curves of some couples

After “normalizing”

interface

graphical

x

x diagram according to equation

The “normalized”

given in Fig. 4.

by

For the

the equilibrium

values

given by the phase diagram have been used. It appears that the u-x

curves from couples

sisting of pure gold and the gold-rich A/Ll/A

con-

alloy (series III,

in Part I(l)) pass through the origin, so that we

can conclude

that the diffusion

tration independent.

coefficient

is concen-

It can be calculated

by means

of the relation u = x/2/(4Dt). This result deviates from that obtained for the same concentration

region

of couples

consisting

of pure

gold and pure platinum by the method discussed above. A possible explanation

of this divergence

consisting

(series I, A/PG/A,

is suggested

of pure gold and pure and series II, A/PO/A,

in

The curves do not pass through the origin, the diffusion

from the concen-

could

not be

are concentration

dependent.

They have been calculated of Matano.

from

by means of

The value dx/dN,,

equation

(4)

by

could be

differentiation

with

respect to x, giving

penetration

Matano interface where the

coefficients

the method

dX

em

-_

of pure gold and pure

(u2h/rr

-’

‘I

h

dNA,

considering u=hx+k

the meaning of and

erfu=-

s

yxp

(- a2) da

where h is the slope of the straight line and lc a constant.

by

means

of

an

equation

The differences between the values calculated

from

for the p-phase and written

equation (5) and those obtained from graphical determination of the slope of the NA,, - x curves vary

-

from -20

as NAu

have

u from a point on

curve is obtained by re-transforming

the straight line into the corresponding

sections.

used for our calculations. identical to the equation

we

curves by

could be drawn much easier, so that a better NAU -

so that

So this equation

procedure, penetration

obtaining curves like Fig. 3. The straight-line sections

platinum

It appears from this calculation

concentration.

concentration

NAU into u by means of equation (4). The

that the diffusion coefficient increases with increasing gold

the

obtained values of u have been plotted versus x, thus

found

B, could be calculated

tration in the provisional

“normalize”

Part I(l)) the u--17:curves consist of two straight-line

region of the concentration

curve of a couple

a purely

For the couples

(a-phase)

independent,

for the b-phase

to

in the discussion.

from these equations.(‘)

The equation

platinum.

phase:

(B-phase)

I and

of a

in the solid state, two equations

1 B,

jump)

as a consequence

must be used, one for each homogeneous

where

As

645

curves in this way.

transforming

N - N,

2---=

II

impossible

penetration “normalized”

da Silva and MehV6) used an equation of the form

N, -

EFFECT

B,[l

-

erf u]

relation between u and x.

So it is

* This method has been proposed by L. D. HALL, J. Chew Phys. 21, 87 (1953).

per cent to +20

per cent as a consequence

of the inaccuracy of the last method. The value of the integral in equation (1) has been measured graphically from curves.

the

“normalized”

concentration

penetration

646

ACTA

METALLURGICA,

I-

VOL.

9, 1961

X. IO4 in cm

ocouple .couple . qcouple ncoupla

13

12 46 47

FIQ. 3. The u-x diagram.

I

005

- 200

-100

0

100

200

300

400

-

xiny

(4

NAu1to

09

t 098

-xinp (b) FIG. 4. Some concentration penetration curves.

BOLK:

KIRKENDALL

TABLE 1. The diffusion coefficient according to Matano in the diffusion couples of series III ._ _

1055 1020 970 926

4.59 2.4O 1.48 5.43

x x x x

W’C) -

10-10 1O-‘o 10-10 lo-”

1055 1020 970 926

The values of the diffusion coefficients calculated for the couples belonging to series III are given in Table 1. As an example of the dependence on concentration the values of the calculated diffusion coefficients in the couples 7I and 7” (the two separate diffusion zones from sandwich-couple 7) of series I and in the couples 18 and 19 of series II are given in Fig. 5 as a function of the gold concentration. The diffusion coefficients in the platinum-rich phase of the couples of the series I and II have been taken constant in view of the inaccuracy of this region; they are given in Table 2. By means of the relations

D = D, exp (-Q/RT)

II

647

TABLE 2. The diffusion coefficient according to Matano in the platinum-rich phase of the couples of series I and II

ij (cme/sec)

WC)

EFFECT

E (cma/sec) series I (A/PG/A) 3.4 2.5 1.6 9.2

x x x x

D”(cmB/sec) series II (A/PO/A) 1.6 8.6 2.9 1.3

lo-” lo-” 10-l’ 10-12

x x x x

lo-” lo-‘* IO-‘2 10-12

activation energy & can be calculated;

the value log

D,, can be read off from the log z-axis.

The relation

between D and T for the couples of series III is given by 5 = 2.09 x 10e2 exp (-46,50O/RT) with a standard deviation ~2; = 0.080. For the series I and II the values of i and 5, are given as a function of the gold concentration NAU in Table 3.

(6)

the activation energy & and the constant 5, have been deduced. In Fig. 6 some values of log 5 in the couples of the three series are given as a function of the reciprocal temperature, together with the results of Jo&(*). From the slope of the log 5 -

l/T curve the

1

E.lO’O

c’-fxec

I 305

3F‘-

?ZB-

I

Q9 -NAu

Q8

FIR 6. The diffusion coefficient according to Matano as a function of the concentration at 102O’C.

t5

90

9.5__c

IO.0 y7 IO4

FIU. 6. Log 0” as a function of the reciprocal temperature.

ACTA

648

METALLURGICA,

TABLE 3. The calculated -

---)

activation

VOL.

)

G,, (cm2/sec)

I

0.105 0.11’ 0.10” 0.108 0.100 0.104 0.099 0.093 0.093 0.092 4 x 10-e

0.98 0.96

0.94 0.92 0.90 0.88 0.86 0.84 0.82 0.80 0.02-0.08

Series II (A/PO/A)

/

&(kcal/mol)

/

u”, (cm2/sec)

I

50.1 50.1 50.2 50.4 + 5.2 50.4 50.5 1.3

1020 970 1055 926

;::9” 6.9’ 1.13

1.16 0.54 1.51 0.19

1055 1020 970 926

6.1’ 3.3* 1.8& 0.76

1 70 1:07 0.24 0.19

36 31 77 40

1055 1020 970 926

9.0’ 5.62 2.7= 1.36

El

65 77 61 94

III

The standard deviations in the values of 0” are 0.080 and 0.040 for the series I and II, respectively. The partial diffusion coefficients DAu and D,, were calculated according to the method of Darken [equations (2) and (3)]. The marker velocity w could be calculated from the marker displacement A, by means of AM

The values of DAu and DPt are given in Table 4 together with the ratio n = D,,/D,,. The concentrations in the marker interfaces are put together in Table 5. The standard deviations in the calculated diffusion coefficients are : Series I Series II Series III Series I + II + III

=

0.050,,

=

0.07D8,

=

O.OlD,,

aD~u ‘D~u

ODau

a,

DA, Dpt

DA,, X 10’ (cm2/sec)

II

aA, v=dt=2t.

Dpt x lOlo (cmz/sec)

Pt

54.6 56.0 55.4 55.4 55.2 * 2.6 55.2 55.1 55.2 55.1 55.0 62.6 + 2.0

DA” and Dpt

T(“‘J

I

1 & (kcal/mol)

0.62 1.00 0.73 0.67 0.60 0.58 0.53 0.52 0.47 0.43 0.37

49.8

TABLE 4. The partial diffusion coefficients Series

1961

energies and factors for the series I and II

Series I (A/PG/A) NAU

9,

n=-

naV

45 45 46 60

~

0.45 0.14

49 * 4

I

46 f

74 * 7

TABLE 5. The gold concentrations Series

-

11

I(A/PG/A) II(A/PO/A) III(A/Ll/A)

in the marker interfaces NAU 0.952 * 0.004 0.949 + 0.002 0.967 * 0.002

equation (6) and are tabulated in Table 6. Log DAu and log Dpt have been plotted versus the reciprocal temperature in Figs. 7 and 8 together with the results of recent selfdiffusion experiments in gold and platinum. The values of Dpt of the three series have been used together to calculate QPt and DOPt,while they are not of such accuracy to justify a single calculation for every series. Activation energies can also be calculated from the magnitude of the Matano-area XMt* and the phase boundary displacement Apb. Analogous to the marker * The Matano-area

has been defmed by

= 0.140,,.

The activation energies Qa, and QPt and the frequency factors DOAuand Dopt were calculated from

where NM~ = the concentration in the Matano interface (x = 0). N1 and N., are the initial concentrations of the two halves of the diffusion couple.

BOLK:

KIRKENDALL

TABLE 6. The activation energies and frequency factors calculated from the partial diffusion coefficients together with those from recent self-diffusion experiments

way.

II

Equation

(2) can be written as

AM = D0 (cma/sec)

Author Au

Pt /

&(kcal/mol)

The

45.3 39.4 42.9 41.65

Gates and Kurtz~**) Okkerae(18’ Mead and Birchenallfiz4’ Makin”6) This investigation: A/PC/A; NA, = 0.952 A/PO/A; Nau = 0.949 A/Ll/A; N&,, = 0.967

0.265 0.031 0.14 0.087 0.21 0.32 0.32

45.0 & 3.0 47.0 * 4.0 45.6 & 0.7

Kidson and Ross”~’ This investigation:

0.33

68.1

0.09

NAu = o.g56

displacement

/

displacement

reciprocal

temperature

QMMt, Qpb Kzwtand Kpb

in Fig.

9.

a unit

can be expressed

are tabulated

MAu=

and the

on l/z/t

‘i,,,dt s0

equation

For M,,

(24

-

by the first law of Fick (7)

-DA+.

dNA~ dt

= -

tDAu82 s0

values

of

in Table

7

.

(8)

gradient. to be dependent

(8) becomes: dNAll

(

ds

.

(9)

16

we get in the same way: Mpt = 2tDpt

Substitution

of equations

(-1 dNA~

ax

(10)

t’

(9) and (10) into equation

(2a) gives Ai+f = -MA,

-

Mpt

or in absolute quantities

in the following

IAMI = IMA,I -

lo@pt

I I I Gatos and Kurtz(l9s4) 2OkkerseCl956) sMead and BirchenallCl9s7~ 4Makin c.s.C1957)

62

of time and passing regard to the marker

unit

of area, with

MA,=-2tD,,

together with the other activation energies and factors. The values of n = DA,/D,, are not highly accurate. More reliable values can be obtained

knoll

J&t)

During the time t there will have been diffused

versus the

The

per

interface

&I, = Kpbdt exp C-Q I$RT). are plotted

-

through

can be expressed as:

and log A,,/dt

iAU,

%A” =

S ‘Wit = KIWtdt exp ( -Q_d2RT)

Log fT,,/~t

WD,,

Assuming the concentration

(see Part I) the Matano-area

phase boundary

massflow

54 * 7

I-

649

EFFECT

I

(11)

IMP,I.

I

I

.‘\,

I Kidson and RossC 19573 2thls investigation NAu-0,95 0 * :

- 8,25

-II,5 \

2

I

- 850

.

-12,5,

- 9,251

’ 25.0

I 7,75

I 8.00

I 0.25

*

I 8.50 l/+104

FIQ. 7. Log DA,, as a function of the reciprocal temperature, together with data of recent selfdiffusion experiments.

Pm. 8. Log fit as a function of the reciprocal temperature, together with data of recent selfdiffusion experiments.

-95

660

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METALLURGICA,

-

‘/T IO4

VOL.

9,

1961

- 6,0

-

Fro. 9. Log ,S’,,/z/t and log Arb/dt as a

According to equation (11) it is possible to express the marker displacement in terms of the total mass diffused through the marker interface. Assuming n to be constant within the investigated temperature range, equation (11) can be written as

This expression enables us to calculate values for n from the observed values of AM and MA,. The results are given in Table 8 together with the values calculated from the diffusion coefficients. It appears that the ratio calculated from equation (12) is more accurate than the value calculated directly. C. DISCUSSION (a)

The concentration penetration curves

It appears from the concentration penetration curves that a phase boundary, i.e. concentration jump,

‘h

IO4

functionof the reciprocaltemperature. comes into existence in accordance with the phase diagram. Jedelefg) did not observe a phase boundary, in our opinion as a consequence of his experimental technique. He determined the concentration penetration curve by sectioning the gold-platinum diffusion couple and measuring the gold content in the layers by chemical analysis. The thickness of the layers was about 25-55 ,u with a total width of the diffusion zone of about 350 ,LL It is most likely that a part of the gold-rich and the platinum-rich phase were both present in one layer. The overall concentration in this layer was situated within the miscibility gap, which induced Jedele to conclude that the diffusion had not been influenced by the two-phase region. Consequently this author draws the concentration penetration curve as a continuous line. Thomas and Birchenallo”) pointedout the fundamental discrepancy between this continuous concentration penetration curve and the phase diagram. We have determined the gold concentration by

TABLE 7. Activation energies and factors calculated from several temperature dependent quantities Series I (A/PG/A) &

(kcal/mol)

50.4 * (NAu = 45.5 i_ 50.6 + 51 *

&M (kcal/mol) &art (kcal/mol)

Q_ppb (kcal/mol) lJ,

Series II (A/PO/A)

5.2 0.9) 2.3 4.0 7

0.100

(cm*/sec)

TABLE 8. The ratio n = DA”/&

--I(A/PG/A) II(A/PO/A) III(A/LI/A)

Nau

0.952 0.949 0.967

1 -

+ = * + *

2.6 9.9) 1.2 2.1 2.0

Series III (A/Ll/A) 46.5 4_ 5.345.6 & 1.6 45.1 + 5.5 0.0209

0.60 (NA” = 0.9) 0.52” 0.14 0.15

(%\ B = o.g) 0:079 0.083

KM (cm/set”*) K,i (cm/secN2) Kpb (cm/sec1’2)

Series

55.2 (N&, 51.0 54.2 54.0

I

’

0.202 0.016 -

calculated from equation (12) (I/n)

0.9783 * 0.0011 0.9741 & 0.0010 0.9860 & 0.0009

n

46 + 3.4 39 & 1.5 71 & 5

n

from Table 4 49 * 4 46 & 11 74 & 7

BOLK:

KIRKENDALL

means of X-rays in very thin layers of about 1 ,u. This makes it possible to localize the phase boundary very accurately, especially when very thin layers have been turned off. A second advantage of this technique is the fact that the concentration difference is very low over the thickness of a layer. The Matano interface indicating the plane through which an equal number of gold and platinum atoms have been displaced, has moved with regard to the undiffused ends of the diffusion couple as a consequence of changes in lattice parameter. In fixing the place of the Matano interface by the graphical method

r

N2

JN,

xdN=O

EFFECT

651

II

the diffusion is faster in rolled polycrystalline platinum than in not-rolled platinum. In Part I we have met with a similar difference with regard to the marker displacement. We have interpreted that difference as a consequence of an extra diffusion transport along the crystal boundaries in polycrystalline platinum. An indication for the correctness of this interpretation lies in the fact mentioned above that the diffusion coefficients for the diffusion in the platinum-rich phase (see Tables 2 and 3 and Fig. 6c-d) for the couples of series I (gold/rolled platinum) are higher than those for the couples of series II (gold/not-rolled platinum). It is interesting to compare our results with those of Jost(@ and Jedele(@, which have been corrected by Matanool). In Fig. 10 the various diffusion coefficients are plotted as a function of the gold concentration for a temperature of 900°C. In spite of the incorrect technique and interpretation of Jedele there exists a rather good agreement in the gold-rich part of the diagram. The partial diffusion coefficient of gold D,, in a gold-platinum alloy with about 95 at.% gold is somewhat lower than the selfdiffusion coefficient of gold as obtained by Gatos and K~rtz(~~), Okkerseos), Mead and Birchenall (l4) and Makin et oZ.(is) (see Fig. 7 and Table 6). The activation energies are somewhat higher. The partial diffusion coefficient of platinum DPt, in a gold-rich alloy with about 95 at.% gold, is about 100 times higher than the selfdiffusion coefficient of pure platinum as obtained by Kidson and Ros@).

this physical significance has been lost. It is necessary to consider the difference between the two positions of the Matano interface, determined by the physical and graphical method. By means of the lattice parameters of gold and platinum and assuming that the system follows Vegard’s law (see Fig. 2), it is possible to calculate this difference for a certain Matano area. For the couples 12 and 13 this area is 3.025 x 10e3 cm3/cm2. By replacing this volume of platinum by an equal number of gold atoms, the volume increases to 3.143 X 10e3 cm3/cm2. The difference, 1.2 x 10”’ cm3/cm2, corresponds to a displacement of the Matano interface over 1.2 (u. This displacement can be neglected compared to the accuracy of the penetration measurement (about 3 ,u). So it is not necessary to replace the penetration in centimeters by that in number of atomic distances as has been done earlier by &.,z,I~~~ da Silva and Mehlc6). “I”” (b) The diffusion coeficients according to

0 Matano-Jedele m JostCl933)

Matano and Darken

Considering the diffusion coefficients as given in Tables 1 and 3, we conclude to a defective agreement between the results of the three series of couples. The diffusion coefficients are concentration independent for the diffusion couples of series III, concentration dependent, however, for the couples of the series I and II. This cannot be a consequence of the existence of a phase boundary, since the phase-boundary displacement is proportional to the square root of the diffusion time (see Part I, discussion(i)). We believe that the discrepancy is due to the behaviour of the gold-rich alloys, which contained many little pores, formed during the homogenizing of the slowly cooled alloys. Comparison of the values of the diffusion coefficients for couples of the series I and II indicates that

I 0

0,25

Au - PtCtw

I 950 -

>

I

I

0,75

I,00 NAu

FIG. 10. The diffusion coefikient according to Matano as a function of the concentration at 900°C compared with data from the literature.

ACTA

652

METALLURGICA,

(c) Comparison of the Kirkendall effect in the gold-platinum system with that in other systems In the discussion of Part I(l) we have compared the Kirkendall effect in the gold-platinum system with that in other systems by plotting values of log A,ll/t versus the reciprocal temperature. So we can see from Fig. 10, Part I, that the Kirkendall effect in the copperzinc system is about 30,000 times larger than in the gold-platinum system at the same temperature and in the same diffusion time. It is incorrect to conclude from this fact that the Kirkendall effect in the copperzinc system is very pronounced and in the goldplatinum very small. It is much more correct to compare an observed Kirkendall effect with the total amount diffused. By means of the factor (1 - l/n) from equation (13) we can express the magnitude of an observed Kirkenda11 effect in a numerical value. For n = 1 (equal partial diffusion coefficients) the factor is 0, the smallest Kirkendall effect we ever can observe. For n = co (one of the partial diffusion coefficients is 0) the factor will be 1, the maximum possible Kirkendall effect. All other values lie between 0 and 1. The values obtained from our investigation are given in Table 9, together with some other values for other systems calculated from the observed diffusion coefficients. According to this way of comparison with other systems the Kirkendall effect in the goldTABLE 9. Comparison of the Kirkendall effect in several systems

_ System Cu-x-brass Cu-Zn Al-Mg Ag-Au U-Zr Au-Pt(A/PG/A) Au-Pt(A/PO/A) Au-Pt(A/Ll/A)

1 -

l/n

0.565 0.979 0.445 0.615 0.917 0.978 0.974 , 0.986

Author Darken’8) Heumann and Kottmanno7) Heumann and Kottmanno’) Heumann and Kottmannci7~ Adda and Philibert”*) This investigation This investigation This investigation

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platinum system proves to be the largest effect observed till now, in contradistinction to the way of comparison stated in Part I. ACKNOWLEDGMENTS

The author is much indebted to Professor Dr. W. G. Burgers for his guidance and continuous interest during the investigations carried out in his laboratory, and to Dr. T. J. Tiedema for valuable discussions about the subject of this paper. This work is part of the research programme of the Research-group “Metalen F.O.M.-T.N.O.” of the “Stichting voor Fundamenteel Onderzoek der Materie” (Foundation for Fundamental Research of MatterF.O.M.) and was also made possible by financial support from the “Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek” (Netherlands Organization for Pure Research-Z.W.O.). REFERENCES 1. A. BOLR, A&Met. 9, 632 (1961). 2. C. MATANO, J. Phws. Jazxwz 8. 109 (1933). 3. L. S. DARKEN, T&s. &WY-..Inst.‘ Mi&. (Metall.) Engrs 175, 184.(1948). 4. A. BOLK, ActnMet. 6, 69 (1958). 5. A. S. DARLINO, R. A. MINTERN and J. C. CRASTON, J. Inst. Met. 81, 125 (1952). 6. L. C. CORREA DA SILVA~~~R. F.MEHL, J.MetalsN.Y.3, 155 (1951). 7. W. SEITH,Diffusion in Metallen. Springer, Berlin (1955). 8. W. JOST,2. Phys. Chem. B21, 158 (1933). A. JEDELE, 2. Elektrochem. 39, 691 (1933). 1:: D. E. THOMAS and C. E. BIRCHENALL,J. Metals N. Y. 4, 867, (1952). 11. C. MATANO, PTOC.Phys. Math. Sot. Ja,pan 15, 405, (1933). 12. H. C. GATOS and A. D. KURTZ. J. Metals N.Y. 8. ~1 616.I (1954). 13. B. OKKERSE, Phys. Rev. 103, 1246, (1956). 14. H. W.MEAD ~~~C.E.BIRCHENALL,J.MetolsN.Y.9,874, (1957). 15. $. M: MAKIN, A. H. ROWE and A. D. LE CLAIRE,Proc. Phys. Sot., Lond. B70, 545, (1957). 16. G. V. KIDSON and R. Ross, reportedat Int. Conf. on Radioisotopes in Sci. Res., Paris 1957. UNESCO-Rapp. No. 216. 17. TH. HEUMANN and A. KOTTMANN, 2. Metallk. 4.4, 139, (1953). La Diflusion. dans les Mdtaux 18. Y. ADDA and J.PHILIRERT, p. 85. Philips Technical Library, Emdhoven (1957).