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Self-diffusion of gold in gold-nickel alloys
SELF-DIFFUSION
OF GOLD
IN GOLD-NICKEL
A. D. XURTZ, 13. L. A~R3A~~,
and MURRIS
ALLOYS* COHEN?
The self-diffusion rate of gold in gold-nickel alloys was measured as a function of composition and temperature in the region of complete solid solubility, using an autoradiographic method to trace the diffusion of Au’Qs. The diffusion data reveal no anomalies that correlate with the miscibility gap. However, the DA,*, D,,A,* and QA~*values show maximum deviations from the linear averages of the pure metal values in the composition range corresponding to the minimum in the solidus curve. AUTODIFFUSION
DE L’OR DANS LES ALLIAGES OR-NICKEL
La vitesse d’autodiffusion de l’or dans les alliages or-nickel dans le domaine de solubilite totale a et& mesuree par autoradiographie avec A+* en fonction de Idcomposition et de la temperature. Les constantes de diffusion ne montrent aucune anomalie en relation avec la lacune de miscibilite. Par contre, &art de Dnu*, D,,,.k,* et QA~* par rapport B la variation lineaire entre les valeurs ~orr~spondant aux deux metaux purs, presente un maximum pour le minimum du solidus. DIE SELBSTDIFFUSION
VON GOLD IN GOLD-NICKEL
LEGIERUNGEN
Mit Hilfe einer autoradiografischen Methode, die es gestattet, die Spur von Au’98 bei der Diffusion zu verfolgen, wurde das Mass der Selbstdiffusion von Gold in Gold-Nickel Legierungen in AbhLngigkeit von der Zusammensetzung und der Temperatur im Bereich vijlliger Mischkristallbildung gemessen. Die Ergebnisse tiber die Diffusion zeigen keine Anomalien, die mit der Mischungslticke in Widerspruch stehen. Im Zusammensetzungsbereich, der dem Minimum in der Soliduskurve entspricht, zeigen jedoch stehen. Im Zusammensetzungsbereich, der dem Minimum in der Solidus-kurce entspricht, zeigen jedoch die DA,,*-, D,,A~*- und QA”*-Werte maximale Abweichungen von den linearen Mitteln der reinen Metalle.
1. INTRODUCTION
Relatively few measurements have been made of self (radioactive) diffusion coefficients in an alloy system as a function of composition. Although such data are worth while in themselves, the present work on goldnickel alloys was ‘uridertaken with broader objectives in mind. This system exhibits a miscibility gap below 84O’C; above this temperature, a complete series of face-centered-cubic solid solutions exists. There are suitable radioactive isotopes for the determination of the self-diffusion coefficients for both gold and nickel, and the thermodynamic activities have been measured by Seigle.’ Thus this system appears quite app~priate for an experimental test of the equation proposed by Darken2 and Le Claire,3 which relates the self-diffusivities DA,,* and DNi*, the interdiffusion coefficient D,, and the thermodynamic driving force for interdiffusion. The first step in this sequence of experiments involved the determination of DAM* as a function of composition and temperature, and these data are reported here. 2. EXPERIMENTAL
METHOD
Specimen Preparation Alloys were prepared from fine gold (99.96) and vacuum-cast nickel (99.986) by induction melting in alumina crucibles under an argon atmosphere, and arranging for directional solidification from the bottom.
The ingots were machine into $-inch diameter rods, sealed in evacuated Vycor tubes, and annealed for about one week at 925 to 1000°C. After this treatment, the grain size was approximately 4 mm, which was sufficiently coarse to avoid undue grain-boundary diffusion. The faces of #-inch thick disc specimens were surface-ground on fine abrasive papers and a film of Aurg8 about 10012 thick was deposited by evaporation on the exposed faces of various alloy compositions. This amount of gold was too small to have a material effect on the chemical composition of the base alloy. A disc-specimen of identical composition was then welded to the radioactive face. Welding was accomplished by hot pressing in a stainless steel cylinder under an argon atmosphere. Satisfactory welds were obtained in one hour at 850°C. The diffusion anneals were of the order of 6 to 14 days, and the temperatures were controlled to =tl”C. Autoradiographic
Method
The penetration of the radioactive tracer was determined by an autoradiographic method.4 The diffusion samples were sectioned at an angle of about 3 degrees to the original interface, the selected angle depending on the extent of penetration. Eastman No-screen X-ray film was placed next to the sectioned surface and exposed for times ranging from five minutes to one hour. Microdensitometer traces were obtained for each autoradiogram; by seiecting a suitable amplification factor on the microphotometer, it was possible to distinguish differences in density between points 0.04 cm apart in the radiogram, corresponding to a
* Received December 16, 19.54; in revised form April 2, 1955. t Department of Metallurgy, Massachusetts Institute of Technology, Cambridge, Massachusetts. , ACTA METALLURGICA, VOL. 3, SEPTEMBER 1955 442
KURT&
AVERBACH,
AND
distance of 0.002 cm in the sample for a sectioning angle of 3 degrees. The blackening was converted to relative intensity, and the self-diffusion coefficient was determined from the intensity V~WZGS distance curve. The isotope Aulg8 decays to Hgis* by emitting a @particle (0.96 mev). There is a subsequent emission of a y-ray (0.41 mev), but only the p-ray was recorded in the autoradiogram since the emulsion was insensitive to y-rays of this wavelength. In determining the self-diffusion coefficient by autoradiography it is advisable to make a correction for the intensity originating beneath the surface. At a surface making an angle, CY,with t.he original interface, it is shown in the appendix that the observed intensity is given as a function of the distance, 1, along the interface (see Fig. 1) by the equation:* I(I) exp($ -__=IO
tanff+$Erf) [I -erfhJf, 2
FIG. 1. Geometry
of sectioned
(1)
couples.
where I= distance along the original interface, I(I) = intensity corresponding to a given value of I, 10-a constant including the initial intensity at the interface, D= diffusion coefficient, I= cliffusion time, p= mean linear absorption coefficient,
COHEN:
443
SELF-DIFFUSIOX
FIG. 2. Autoradiographic
intensity
curve.
values of hZ tan+ the correction term becomes neglibible and the slope becomes constant. A typical plot of log[1(Z)/Io)] ZJS(I tancr)” is shown in Fig. 2 for an alloy containing 80 atomic per cent nickel. At small values of the abcissa, the curve deviates from a straight line and exhibits a steeper slope, as indicated by the expression above. At larger values of the abcissa, the slope is constant and thus equal to -1/4L?t. Diffusion coefficients were obtained by utilizing the slope of the straight line portion of such plots. Because the isotope ALP decays with a half-life of only 2.7 days, a strict time schedule had to be maintained in order to obtain suitable autoradiograms. This necessitated the use of initial activities of the order of lo4 millicuries per gram. 3. RESULTS
The self-diffusion coefficients of gold, DA,+, in various gold-nickel alloys at a series of temperatures are shown in Fig. 3. It is evident that the diffusion coefficients are smooth functions of composition with some tendency toward a maximum at 20 atomic per cent nickel. No anomaly appears at 70-80 atomic
and P tan% Z&J= --+~crl t 4Dt
z tancu+$L% >
The quantity, 1 tancu, is the distance from the original interface to various positions at the surface of the oblique section. It is also shown in the appendix that for reasonably large values of P the slope of a plot of ln[1(I)/lrJ w (I tancu)2 is given by
The absolute value of the slope is thus larger than 1/4X.% and dependent on 1, but at sufficiently large ~* The correction term previously publishecl*contains an error.
FIG. 3. Self-diffusion
of gold in gold-nickel
alloys.
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ACTA
METALLURGICA,
VOL.
3, 19.55
TABLEII. Entropies of self-diffusion of gold in gold-nickel alloys
At. % Ni
(IO&ecf
DO&P
2::
3.56
3.97 4.07
1.6
2:
4.17 %
4:g
t::
::
4.40
3.64 3.73
:::
100
;:::
3.56
1;::
The correlation of the diffusivities with the melting temperature is inherent in many of the recent interpretations of self-diffusion data.6*7*8 Following Zeners the self-difIusivity of substitutional atoms in face-centeredcubic metals may be written as: D=a2v exp(AS/R)
exp(-AE/.?W),
(2)
where fiG. 4. Effect of temperature on self-diffusion of goId in gold-nickel alloys.
per cent nickel, the composition at the top of the miscibility gap. Figure 4 shows that the diffusion data are consistent with the customary relationship
and the values for If&Au* and QnU* are given in Table I. There are minima in &,A~* and QA~* in the neighborhood of 20-50 atomic per cent nickel, as plotted in Fig. 5. 4. DISCUSSION
OF RESULTS
Although the self-diffusion coefficients, DA”*, (Fig. 3) appear to have maximum values near 20 atomic per cent nickel, the maximum deviation from linear average of the pure metal values lies at about 50 atomic per cent. This is approximately where the maximum lowering of the melting-point occurs, relative to the average (linear) melting-point of the pure metals. In a rough way, the self-diffusion coefficients seem to correlate with the melting-point variations as a function of composition.
a=lattice parameter, v = frequency of vibration, AS= entropy of activation for self-diffusion, AH=enthalpy of activation for self-diffusion, R= gas constant, and T = diffusion temperature. Thus, the experimental values (Table I) of D*,A~* correspond to the Ca2p exp(AS/R)] and QA”* is equivalent to AH. Using the Debye frequency for Y, the entropies of di~usion were calculated and listed in Table II. The v values given in Table II were computed by Averbach* from experimental data on the excess relative entropy of mixing.l The values for the diffusion entropy calculated from &Au* are all positive. Zener” has interpreted self-diffusion data in terms of the fraction, X, of the free energy of activation required for the jumping of an atom from one site to an adjacent
TABLE I. Frequency factors and activation energies for self-diffusion of gold in gold-nickel alloys. At. y0 Ni 28 2: :: 100 t
Do.au*(cm, set-1)
0.26 0.05 0.063i 0.091 O&51-)
Qnu*(cal/mol) 45,300 40,200 42,700’F
;::
Determined by Reynolds6 in a related investigation.
FIG. 5. Self-diffusion of gold in goId-nickel alloys.
KURTZ,
AVERBACH,
vacant site and the fraction, (l-X), formation of the vacant site. The estimated to be:
AND
required for the entropy is then
aS=M(AH,‘r,),
(3)
where ~=~(~~E~)~~(T/T~), E is Young’s modulus, E. is Young’s modulus at OK, and T, is the melting temperature. The coefficient p arises from an attempt to estimate the temperature coefficient of the free energy associated with the straining of the lattice during the elementary diffusion jump. The value of p for gold is listed6 as 0.31 and a value of @=0.42 was obtained from data on Young’s modulus for nickel.lO A linear average of these values was used for the intermediate alloys. The X-quantities calculated from Eq. (3) are listed in Table III. Zener has concluded on empirical grounds that the best value for X in face-centered-cubic metals is 0.55. TABLE
III. Fraction
of activation energy for jump of gold atom in self-diffusion in gold-nickel alloys.
_-.-XC’>
At. % Ni
~--_.I_-
0
KW ~..
A(‘)
::
0.70 0.38 0.40
0.052 0.037 0.039
0.88 0.58 0.57
:z 1:
0.56 0.42 0.54 0.67
0.037 0.035 0.046 0.045
0.88 0.64 0.80 0.82
WCA similar approach was used by Bu~ngton7 except that the temperature-dependence of the lattice spacing as well as of the elastic constants is also considered. The diffusion equation becomes :
KEoao3X ( p- 3~2) D= a2v exp
(4) R
where a = lattice parameter,
SELF-DIFFUSION
445
Eq. (3), can then be evaluated from the measured frequency factor &Au. The quantities K and X derived from Eq. (4) are tabulated in Table III; Buffington selected the values K=O.OSl and X=0.64 as the most probable for face-centered-cubic metals. The assumptiol~s entering into the formulation of Eqs. (3) and (4) are applicable to the case of selfdiffusion in a pure metal, but discrepancies may be expected when these equations are used for the case of self-diffusion in an alloy. In addition, the parameters listed in Table V involve estimates of the values of p, ‘p, CX,and E. for solid solutions, and thus only qualitative agreement can be anticipated. Nevertheless, the scatter in the resulting X and K quantities for gold-nickel alloys is no worse than that obtained for similar calculations based on the published self-diffusivities in pure metals. These treatments assume that the displacements of the barrier atoms involved in the elementary jump of a given atom into a neighboring vacancy are the same for each atom, and that the elastic energy at the saddle point can be described in terms of a macroscopic elastic modulus. This simple situation does not prevail when more than one kind of atom participates in the diffusion act. In addition, entropy changes arising from a modification of the vibrational spectrum when the atom is at the saddle point are also neglected. When the diffusing atom is jammed between the barrier atoms there is a negative contribution to the entropy arising from the reduced amplitude of vibration of the jumping atom and the barrier atoms. This effect is partially compensated by the increased amplitudes in the vacancy created by the diffusing atom. However, with atoms of greatly different sizes, as in the goldnickel syst,em,g these effects may not cancel and the poorest compensation would be expected near the 50 atomic per cent composition. While the rough outlines of these interpretations may be useful here as a basis for discussion, considerable refinement will be required before they can be applied with confidence to the process of self-diffusion in alloys.
1
p=--
dE -; Eo ( dT >
COHEN:
Eo=Young’s
modulus extrapolated
1 da ; (Y=---- a0 ( dT )
5. SUMMARY
to 0°K
ao= lattice parameter extrapolated
to O”K,
K= constant. Suitable values of the parameters (a and Q! are not always available even for pure metals, and in the case of gold-nickel alloys it was necessary to use a linear average of the best data on the pure metals. The constant K for each com~sition can be computed from the measured activation energy (QnU*= KE~ao3) and the quantity X, which has the same significance as in
The self-diffusion coefficient for gold has been determined as a function of composition and temperature in the gold-nickel system. The diffusion data reveal no anomalies that correlate with the miscibility gap. However, the DAM*, Do,*“* and QA,~*values show maximum deviations from the linear averages of the pure metal values in the composition range corresponding to the minimum in the solidus curve. The entropies of diffusion are positive and are qualitatively consistent with recent interpretations. 6. ACK~O~EDGMENTS
The authors are grateful to the Wright Air Development Center for sponsoring this investigation (under
446
ACTA
METALLURGICA,
Contract AF 33(038)-23281, Scope I) and to G. Pishenin, E. Keplin, and V. Czerniszow for their assistance with the experimental phases of the program. They would also like to acknowledge the aid received on several occasions from H. C. Gates, V. Griffiths, and J. E. Reynolds. The cooperation of the Brookhaven National Laboratory in providing the gold isotope is also greatly appreciated. The research represents a portion of a thesis submitted by A. D. Kurtz in partial fulfillment of the requirements for the degree S.M. in Metallurgy at the Massachusetts Institute of Technology, August, 1952.
VOL.
I(1) --_= IO
exp(j.61 tancu)
m
(4nDt) *
s ztana
The integration I(1) -=
m s
I(y) exp[-dy-1
Itana
tadI@,
I(l) -_=
IO I(y)=---(4uDt)texP where IO is the constant, D is the diffusion and t is the diffusion time. One obtains
(A-2) coefficient,
exp(d
tancu+p2Dt) [l-erf(zo)].
(A-4)
2
exp( - P tan2q’4Dt) (47rzo)4
Neglecting
3 ,_$+_. . .). o2 (2zl12)2
(A-5)
all but first powers of ZO, 1n55= IO
-(lt-+ln(4Tzo). 4Dt
(~-6)
For values of (1 tancu/2pDt)2<<1, 1i(1) _
(1 tana) F-$ln(
l+z)+consunt
(A-7)
IO and the slope of a plot of ln[I(l)/lo] becomes the expression given in the text.
(A-1)
where 1 is a distance along the original interface, y is the diffusion distance, and y-l tancu is the absorption length for radiation originating at y. The solution of the diffusion equation appropriate to the present boundary condition is
(A-3)
For large values of ZO,
APPENDIX
I(l)=
exp( - y2/4Dt-py)dy.
gives Eq. (1) in the text:
IO
IO
When the quantity l/p is not negligibly small compared to the diffusion distance, y, it is necessary to take into account the intensity arising from regions in the sample beneath the autoradiographic surface. Assuming an average linear absorption coefficient, /*, the contribution from each infinitesimal layer to the intensity observed at the surface must be computed. Introducing the parameters shown in Fig. 1, if the intensity at a given point in the sample is I(y), the contribution of all such regions to the surface intensity, I(l), is:
3, 1955
vs (1 tancr)2
REFERENCES 1. L. L. Seigle, M. Cohen, and B. L. Averbach, J. Metals 4, 1320 (1952). 2. L. S. Darken, Trans. A.I.M.E. 175, 184 (1948). 3. A. D. LeClaire, Progress in Metal Physics (Interscience Publishers, New York, 1949) Vol. I, Chap. 7. 4. H. Gatos and A. D. Kurtz, J. Metals 6, 616 (1954). 5. J. E. Reynolds, Dlffusion in Gold-Nickel Alloys. Sc.D. thesis, Department of Metallurgy, Massachusetts Institute of Technology. 6. C. Zener, Imperfection in Nearly Perfect Crystals (John Wiley, New York, 1952) Chap. 11. 7. F. S. Buffington and Morris Cohen, Acta Met. 2, 660 (1954). 8. A. D. Le Claire, Acta Met. 1, 438 (1953). 9. B. L. Averbach, P. A. Flinn, and M. Cohen, Acta Met. 2,92, 10. $%%ter,
Z. Metallk.
39, 1 (1948).
OF GOLD
IN GOLD-NICKEL
A. D. XURTZ, 13. L. A~R3A~~,
and MURRIS
ALLOYS* COHEN?
The self-diffusion rate of gold in gold-nickel alloys was measured as a function of composition and temperature in the region of complete solid solubility, using an autoradiographic method to trace the diffusion of Au’Qs. The diffusion data reveal no anomalies that correlate with the miscibility gap. However, the DA,*, D,,A,* and QA~*values show maximum deviations from the linear averages of the pure metal values in the composition range corresponding to the minimum in the solidus curve. AUTODIFFUSION
DE L’OR DANS LES ALLIAGES OR-NICKEL
La vitesse d’autodiffusion de l’or dans les alliages or-nickel dans le domaine de solubilite totale a et& mesuree par autoradiographie avec A+* en fonction de Idcomposition et de la temperature. Les constantes de diffusion ne montrent aucune anomalie en relation avec la lacune de miscibilite. Par contre, &art de Dnu*, D,,,.k,* et QA~* par rapport B la variation lineaire entre les valeurs ~orr~spondant aux deux metaux purs, presente un maximum pour le minimum du solidus. DIE SELBSTDIFFUSION
VON GOLD IN GOLD-NICKEL
LEGIERUNGEN
Mit Hilfe einer autoradiografischen Methode, die es gestattet, die Spur von Au’98 bei der Diffusion zu verfolgen, wurde das Mass der Selbstdiffusion von Gold in Gold-Nickel Legierungen in AbhLngigkeit von der Zusammensetzung und der Temperatur im Bereich vijlliger Mischkristallbildung gemessen. Die Ergebnisse tiber die Diffusion zeigen keine Anomalien, die mit der Mischungslticke in Widerspruch stehen. Im Zusammensetzungsbereich, der dem Minimum in der Soliduskurve entspricht, zeigen jedoch stehen. Im Zusammensetzungsbereich, der dem Minimum in der Solidus-kurce entspricht, zeigen jedoch die DA,,*-, D,,A~*- und QA”*-Werte maximale Abweichungen von den linearen Mitteln der reinen Metalle.
1. INTRODUCTION
Relatively few measurements have been made of self (radioactive) diffusion coefficients in an alloy system as a function of composition. Although such data are worth while in themselves, the present work on goldnickel alloys was ‘uridertaken with broader objectives in mind. This system exhibits a miscibility gap below 84O’C; above this temperature, a complete series of face-centered-cubic solid solutions exists. There are suitable radioactive isotopes for the determination of the self-diffusion coefficients for both gold and nickel, and the thermodynamic activities have been measured by Seigle.’ Thus this system appears quite app~priate for an experimental test of the equation proposed by Darken2 and Le Claire,3 which relates the self-diffusivities DA,,* and DNi*, the interdiffusion coefficient D,, and the thermodynamic driving force for interdiffusion. The first step in this sequence of experiments involved the determination of DAM* as a function of composition and temperature, and these data are reported here. 2. EXPERIMENTAL
METHOD
Specimen Preparation Alloys were prepared from fine gold (99.96) and vacuum-cast nickel (99.986) by induction melting in alumina crucibles under an argon atmosphere, and arranging for directional solidification from the bottom.
The ingots were machine into $-inch diameter rods, sealed in evacuated Vycor tubes, and annealed for about one week at 925 to 1000°C. After this treatment, the grain size was approximately 4 mm, which was sufficiently coarse to avoid undue grain-boundary diffusion. The faces of #-inch thick disc specimens were surface-ground on fine abrasive papers and a film of Aurg8 about 10012 thick was deposited by evaporation on the exposed faces of various alloy compositions. This amount of gold was too small to have a material effect on the chemical composition of the base alloy. A disc-specimen of identical composition was then welded to the radioactive face. Welding was accomplished by hot pressing in a stainless steel cylinder under an argon atmosphere. Satisfactory welds were obtained in one hour at 850°C. The diffusion anneals were of the order of 6 to 14 days, and the temperatures were controlled to =tl”C. Autoradiographic
Method
The penetration of the radioactive tracer was determined by an autoradiographic method.4 The diffusion samples were sectioned at an angle of about 3 degrees to the original interface, the selected angle depending on the extent of penetration. Eastman No-screen X-ray film was placed next to the sectioned surface and exposed for times ranging from five minutes to one hour. Microdensitometer traces were obtained for each autoradiogram; by seiecting a suitable amplification factor on the microphotometer, it was possible to distinguish differences in density between points 0.04 cm apart in the radiogram, corresponding to a
* Received December 16, 19.54; in revised form April 2, 1955. t Department of Metallurgy, Massachusetts Institute of Technology, Cambridge, Massachusetts. , ACTA METALLURGICA, VOL. 3, SEPTEMBER 1955 442
KURT&
AVERBACH,
AND
distance of 0.002 cm in the sample for a sectioning angle of 3 degrees. The blackening was converted to relative intensity, and the self-diffusion coefficient was determined from the intensity V~WZGS distance curve. The isotope Aulg8 decays to Hgis* by emitting a @particle (0.96 mev). There is a subsequent emission of a y-ray (0.41 mev), but only the p-ray was recorded in the autoradiogram since the emulsion was insensitive to y-rays of this wavelength. In determining the self-diffusion coefficient by autoradiography it is advisable to make a correction for the intensity originating beneath the surface. At a surface making an angle, CY,with t.he original interface, it is shown in the appendix that the observed intensity is given as a function of the distance, 1, along the interface (see Fig. 1) by the equation:* I(I) exp($ -__=IO
tanff+$Erf) [I -erfhJf, 2
FIG. 1. Geometry
of sectioned
(1)
couples.
where I= distance along the original interface, I(I) = intensity corresponding to a given value of I, 10-a constant including the initial intensity at the interface, D= diffusion coefficient, I= cliffusion time, p= mean linear absorption coefficient,
COHEN:
443
SELF-DIFFUSIOX
FIG. 2. Autoradiographic
intensity
curve.
values of hZ tan+ the correction term becomes neglibible and the slope becomes constant. A typical plot of log[1(Z)/Io)] ZJS(I tancr)” is shown in Fig. 2 for an alloy containing 80 atomic per cent nickel. At small values of the abcissa, the curve deviates from a straight line and exhibits a steeper slope, as indicated by the expression above. At larger values of the abcissa, the slope is constant and thus equal to -1/4L?t. Diffusion coefficients were obtained by utilizing the slope of the straight line portion of such plots. Because the isotope ALP decays with a half-life of only 2.7 days, a strict time schedule had to be maintained in order to obtain suitable autoradiograms. This necessitated the use of initial activities of the order of lo4 millicuries per gram. 3. RESULTS
The self-diffusion coefficients of gold, DA,+, in various gold-nickel alloys at a series of temperatures are shown in Fig. 3. It is evident that the diffusion coefficients are smooth functions of composition with some tendency toward a maximum at 20 atomic per cent nickel. No anomaly appears at 70-80 atomic
and P tan% Z&J= --+~crl t 4Dt
z tancu+$L% >
The quantity, 1 tancu, is the distance from the original interface to various positions at the surface of the oblique section. It is also shown in the appendix that for reasonably large values of P the slope of a plot of ln[1(I)/lrJ w (I tancu)2 is given by
The absolute value of the slope is thus larger than 1/4X.% and dependent on 1, but at sufficiently large ~* The correction term previously publishecl*contains an error.
FIG. 3. Self-diffusion
of gold in gold-nickel
alloys.
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METALLURGICA,
VOL.
3, 19.55
TABLEII. Entropies of self-diffusion of gold in gold-nickel alloys
At. % Ni
(IO&ecf
DO&P
2::
3.56
3.97 4.07
1.6
2:
4.17 %
4:g
t::
::
4.40
3.64 3.73
:::
100
;:::
3.56
1;::
The correlation of the diffusivities with the melting temperature is inherent in many of the recent interpretations of self-diffusion data.6*7*8 Following Zeners the self-difIusivity of substitutional atoms in face-centeredcubic metals may be written as: D=a2v exp(AS/R)
exp(-AE/.?W),
(2)
where fiG. 4. Effect of temperature on self-diffusion of goId in gold-nickel alloys.
per cent nickel, the composition at the top of the miscibility gap. Figure 4 shows that the diffusion data are consistent with the customary relationship
and the values for If&Au* and QnU* are given in Table I. There are minima in &,A~* and QA~* in the neighborhood of 20-50 atomic per cent nickel, as plotted in Fig. 5. 4. DISCUSSION
OF RESULTS
Although the self-diffusion coefficients, DA”*, (Fig. 3) appear to have maximum values near 20 atomic per cent nickel, the maximum deviation from linear average of the pure metal values lies at about 50 atomic per cent. This is approximately where the maximum lowering of the melting-point occurs, relative to the average (linear) melting-point of the pure metals. In a rough way, the self-diffusion coefficients seem to correlate with the melting-point variations as a function of composition.
a=lattice parameter, v = frequency of vibration, AS= entropy of activation for self-diffusion, AH=enthalpy of activation for self-diffusion, R= gas constant, and T = diffusion temperature. Thus, the experimental values (Table I) of D*,A~* correspond to the Ca2p exp(AS/R)] and QA”* is equivalent to AH. Using the Debye frequency for Y, the entropies of di~usion were calculated and listed in Table II. The v values given in Table II were computed by Averbach* from experimental data on the excess relative entropy of mixing.l The values for the diffusion entropy calculated from &Au* are all positive. Zener” has interpreted self-diffusion data in terms of the fraction, X, of the free energy of activation required for the jumping of an atom from one site to an adjacent
TABLE I. Frequency factors and activation energies for self-diffusion of gold in gold-nickel alloys. At. y0 Ni 28 2: :: 100 t
Do.au*(cm, set-1)
0.26 0.05 0.063i 0.091 O&51-)
Qnu*(cal/mol) 45,300 40,200 42,700’F
;::
Determined by Reynolds6 in a related investigation.
FIG. 5. Self-diffusion of gold in goId-nickel alloys.
KURTZ,
AVERBACH,
vacant site and the fraction, (l-X), formation of the vacant site. The estimated to be:
AND
required for the entropy is then
aS=M(AH,‘r,),
(3)
where ~=~(~~E~)~~(T/T~), E is Young’s modulus, E. is Young’s modulus at OK, and T, is the melting temperature. The coefficient p arises from an attempt to estimate the temperature coefficient of the free energy associated with the straining of the lattice during the elementary diffusion jump. The value of p for gold is listed6 as 0.31 and a value of @=0.42 was obtained from data on Young’s modulus for nickel.lO A linear average of these values was used for the intermediate alloys. The X-quantities calculated from Eq. (3) are listed in Table III. Zener has concluded on empirical grounds that the best value for X in face-centered-cubic metals is 0.55. TABLE
III. Fraction
of activation energy for jump of gold atom in self-diffusion in gold-nickel alloys.
_-.-XC’>
At. % Ni
~--_.I_-
0
KW ~..
A(‘)
::
0.70 0.38 0.40
0.052 0.037 0.039
0.88 0.58 0.57
:z 1:
0.56 0.42 0.54 0.67
0.037 0.035 0.046 0.045
0.88 0.64 0.80 0.82
WCA similar approach was used by Bu~ngton7 except that the temperature-dependence of the lattice spacing as well as of the elastic constants is also considered. The diffusion equation becomes :
KEoao3X ( p- 3~2) D= a2v exp
(4) R
where a = lattice parameter,
SELF-DIFFUSION
445
Eq. (3), can then be evaluated from the measured frequency factor &Au. The quantities K and X derived from Eq. (4) are tabulated in Table III; Buffington selected the values K=O.OSl and X=0.64 as the most probable for face-centered-cubic metals. The assumptiol~s entering into the formulation of Eqs. (3) and (4) are applicable to the case of selfdiffusion in a pure metal, but discrepancies may be expected when these equations are used for the case of self-diffusion in an alloy. In addition, the parameters listed in Table V involve estimates of the values of p, ‘p, CX,and E. for solid solutions, and thus only qualitative agreement can be anticipated. Nevertheless, the scatter in the resulting X and K quantities for gold-nickel alloys is no worse than that obtained for similar calculations based on the published self-diffusivities in pure metals. These treatments assume that the displacements of the barrier atoms involved in the elementary jump of a given atom into a neighboring vacancy are the same for each atom, and that the elastic energy at the saddle point can be described in terms of a macroscopic elastic modulus. This simple situation does not prevail when more than one kind of atom participates in the diffusion act. In addition, entropy changes arising from a modification of the vibrational spectrum when the atom is at the saddle point are also neglected. When the diffusing atom is jammed between the barrier atoms there is a negative contribution to the entropy arising from the reduced amplitude of vibration of the jumping atom and the barrier atoms. This effect is partially compensated by the increased amplitudes in the vacancy created by the diffusing atom. However, with atoms of greatly different sizes, as in the goldnickel syst,em,g these effects may not cancel and the poorest compensation would be expected near the 50 atomic per cent composition. While the rough outlines of these interpretations may be useful here as a basis for discussion, considerable refinement will be required before they can be applied with confidence to the process of self-diffusion in alloys.
1
p=--
dE -; Eo ( dT >
COHEN:
Eo=Young’s
modulus extrapolated
1 da ; (Y=---- a0 ( dT )
5. SUMMARY
to 0°K
ao= lattice parameter extrapolated
to O”K,
K= constant. Suitable values of the parameters (a and Q! are not always available even for pure metals, and in the case of gold-nickel alloys it was necessary to use a linear average of the best data on the pure metals. The constant K for each com~sition can be computed from the measured activation energy (QnU*= KE~ao3) and the quantity X, which has the same significance as in
The self-diffusion coefficient for gold has been determined as a function of composition and temperature in the gold-nickel system. The diffusion data reveal no anomalies that correlate with the miscibility gap. However, the DAM*, Do,*“* and QA,~*values show maximum deviations from the linear averages of the pure metal values in the composition range corresponding to the minimum in the solidus curve. The entropies of diffusion are positive and are qualitatively consistent with recent interpretations. 6. ACK~O~EDGMENTS
The authors are grateful to the Wright Air Development Center for sponsoring this investigation (under
446
ACTA
METALLURGICA,
Contract AF 33(038)-23281, Scope I) and to G. Pishenin, E. Keplin, and V. Czerniszow for their assistance with the experimental phases of the program. They would also like to acknowledge the aid received on several occasions from H. C. Gates, V. Griffiths, and J. E. Reynolds. The cooperation of the Brookhaven National Laboratory in providing the gold isotope is also greatly appreciated. The research represents a portion of a thesis submitted by A. D. Kurtz in partial fulfillment of the requirements for the degree S.M. in Metallurgy at the Massachusetts Institute of Technology, August, 1952.
VOL.
I(1) --_= IO
exp(j.61 tancu)
m
(4nDt) *
s ztana
The integration I(1) -=
m s
I(y) exp[-dy-1
Itana
tadI@,
I(l) -_=
IO I(y)=---(4uDt)texP where IO is the constant, D is the diffusion and t is the diffusion time. One obtains
(A-2) coefficient,
exp(d
tancu+p2Dt) [l-erf(zo)].
(A-4)
2
exp( - P tan2q’4Dt) (47rzo)4
Neglecting
3 ,_$+_. . .). o2 (2zl12)2
(A-5)
all but first powers of ZO, 1n55= IO
-(lt-+ln(4Tzo). 4Dt
(~-6)
For values of (1 tancu/2pDt)2<<1, 1i(1) _
(1 tana) F-$ln(
l+z)+consunt
(A-7)
IO and the slope of a plot of ln[I(l)/lo] becomes the expression given in the text.
(A-1)
where 1 is a distance along the original interface, y is the diffusion distance, and y-l tancu is the absorption length for radiation originating at y. The solution of the diffusion equation appropriate to the present boundary condition is
(A-3)
For large values of ZO,
APPENDIX
I(l)=
exp( - y2/4Dt-py)dy.
gives Eq. (1) in the text:
IO
IO
When the quantity l/p is not negligibly small compared to the diffusion distance, y, it is necessary to take into account the intensity arising from regions in the sample beneath the autoradiographic surface. Assuming an average linear absorption coefficient, /*, the contribution from each infinitesimal layer to the intensity observed at the surface must be computed. Introducing the parameters shown in Fig. 1, if the intensity at a given point in the sample is I(y), the contribution of all such regions to the surface intensity, I(l), is:
3, 1955
vs (1 tancr)2
REFERENCES 1. L. L. Seigle, M. Cohen, and B. L. Averbach, J. Metals 4, 1320 (1952). 2. L. S. Darken, Trans. A.I.M.E. 175, 184 (1948). 3. A. D. LeClaire, Progress in Metal Physics (Interscience Publishers, New York, 1949) Vol. I, Chap. 7. 4. H. Gatos and A. D. Kurtz, J. Metals 6, 616 (1954). 5. J. E. Reynolds, Dlffusion in Gold-Nickel Alloys. Sc.D. thesis, Department of Metallurgy, Massachusetts Institute of Technology. 6. C. Zener, Imperfection in Nearly Perfect Crystals (John Wiley, New York, 1952) Chap. 11. 7. F. S. Buffington and Morris Cohen, Acta Met. 2, 660 (1954). 8. A. D. Le Claire, Acta Met. 1, 438 (1953). 9. B. L. Averbach, P. A. Flinn, and M. Cohen, Acta Met. 2,92, 10. $%%ter,
Z. Metallk.
39, 1 (1948).