The diffusional creep of binary copper alloys

THE

DIFFUSIONAL B.

CREEP

BURTONt:

OF BINARY and

B.

D.

COPPER

ALLOYS*

BASTOWt

Low stress creep has been investigated in the single phase copper alloys Cu-Xi at 1163°K and a-phase Cu-Sn at 1009°K in the range where the stress directed lattice diffusion of vacancies is expected to control. The absence of a primary creep stage is confirmed and creep rate and stress are shown to be linearly related. Further, the creep rate is shown to vary as the reciprocal of the square of the grain size as predicted by the Nabarro-Herring equation. Additions of nickel and tin are shown to cause respective decreases and increases in creep rate corresponding to their influence upon diffusirities in these alloys. The choice of the appropriate diffusion coefficient to use in the diffusion creep equation is discussed and comparisons are made with diffusion coefficients calculated from creep rates of these alloys and also from a previous study on Cu-Zn alloys. Factors which can give rise to discrepancies between experiment and theory are discussed. FLUAGE

DE

DIFFUSION

DANS

LES

ALLIAGES

DE

CUIVRE

BINAIRES

Le fluage sous faible contrainte a 6tB Btudie dans les alliages de cuivre B une seule phase Cu-Ni & 1163°K et dans la phase aCu-Sn B 1009°K dans le domaine oh on suppose que ce ph&om&ne est contrBle par la diffusion des lacunes dens le r&au sous l’influence de la contrainte. L’absence d’un stade de fluage primaire est conf?rmke, et les auteurs montrent que la vitesse et la contrainte de fluage sont reli4es linbairement. En outre, ils montrent que la vitesse de fluage varie comme l’inverse du car& de la 11s montrent que des additions de nickel et taille des grains suivant l’bquation de Nabarro-Herring. d’6tain provoquent respectivement une diminution et une augmentation de la vitesse de fluage, ce qui correspond & leur influence respective sur les diffusivit& dans ces alliages. Le choix du coefficient de diffusion appropri6 qui doit 6tre utilisk dans 1’Qquation du fluage de diffusion est discut6, et des comparaisons sont effect&es avec les coefficients de diffusion calcul& B partir des vitesses de fluage de ces alliages et Bgalement h partir d’une 6tude anterieure effect&e sur les alliages Cu-Zn. Les facteurs pouvant dormer des &arts entre la theorie et l’exp&ience sont discut&. DIFFUSIONSKRIECHEN

BINARER

KUPFERLEGIERUNGEN

Kriechen bei kleinen Spannungen wurde in einphasigen Cu-Ni-Legierungen bei 1163’K und in der a-Cu-Sn-Legierung bei 1009°K untersucht; in diesem Bereich wird der Kriechvorgang durch spannungsinduzierte Leerstellendiffusion in der Matrix kontrolliert. Es wird best&t&t, da0 keine prim&e Kriechstufe auftritt und daD ein linearer Zusammenhang zwischen Kriechgeschwindigkeit und Spannung besteht. AuDerdem ist die Kriechgeschwindigkeit wie von der Nabarro-Herring-Gleichung vorausgesagt umgekehrt proportional zum Quadrat der KorngriX3e. Nickel- und Zinnbeilegierungen verursachen je nach ihrem Eitiull auf die Diffusivitiiten in diesen Legierungen eine Ab- bzw. Zunahme der Kriechgeschwindigkeit. Es wird die Wahl des geeigneten Diffusionskoeffizienten fiir die Diffusions-Kriechgleichung diskutiert; diese Diffusionskoeffizienten werden verglichen mit entsprechenden aus den Kriechgeschwindigkeiten dieser Legierungen berechneten Werten und mit Ergebnissen friiherer Untersuchungen an Cu-Zn-Legierungen. Es werden Faktoren diskutiert, die zu einer Diskrepanz zwischen Theorie und Experiment fiihren k8nnen. INTRODUCTION

constant which depends on the particular grain geometry but usually takes a value of ~10. For finer grained material or at lower temperatures grain boundary diffusion can predominate and under these conditions creep rates have been shown@-s) to agree with the theoretical creep equation of Cable,(‘)

At elevated temperatures and low stresses many metals behave as Newtonian fluids, that is the rate of deformation is linearly proportional to the applied stress. For relatively coarse grained material or at temperatures which are a sufficiently high fraction of the melting temperature, experimentally measured creep rate0 are in good agreement with the theoretical predictions of Nabarroc2) and Herringf31 who derived the rate of creep due to the stress directed lattice diffusion of vacancies to be d, = BaQDld2kT

E,,

(1)

* Received March 27, 1972; revised May 18, 1972.

t Department of Physical Metallurgy and Science of Materials, University of Birmingham, P.O. Box 363, Birmingha+m~~;gl~d. : C.E.G.B., Berkeley Nuclear Laboratories BeTkeley, Gloucestershire, England. METALLURGICA,

VOL.

21, JANUARY

1973

(2)

where w is the grain boundary width, D,, the grain boundary self-diffusion coefficient and B’ - 150/~. Although the numerous studies of diffusion creep in pure metals are consistent and in good agreement with theoretical predictions, the relatively scarce data on alloys is much more varied. On alloying, eqtiations (1) and (2) predict the creep rate for material of similar grain sizes to depend only on the diffusion coefficient. However in several two phase alloys (Mg-MgO,@) Au-A~,O,,~~) CU-AI,O,,‘~O) Mg-ZrH,(‘l) where the presence of particles should have a negligible effect upon D, creep rates were many orders of magnitude less than predicted and in some cases a threshold

where a is the applied stress, Q the atomic volume, D the lattice self-diffusion coefficient, a! the grain size and kT has the usual meaning. B is a numerical

ACTA

= B’aQmDGs/d3kT

13

14

ACTA

~~ETALLURG~CA,

stress was noted@*lc) below which no creep occurred. On the other hand in some other two phase alloys (Mg--crMn,c12)Cu-Ge0,‘r3)) creep rates agreed well with theory. It appears that the influence of certain second phase particles is to prevent boundaries sating as perfect sources and sinks for vacancies, thus invalidating one of the basic &~umptions in deriving equations (1) and (2). However the presence of particles does not appear to be the only condition for the inhibition of boundaries as vacancy sources and sinks. For instance Hondros and Lake(r*) noted that diffusion creep rates in dilute C&O, alloys were reduced by an amount greater than could be explained by changes in ~ffusi~ty and interpreted the results in terms of oxygen adsorption at grain boundaries. However in the dilute iron alloys, Fe-P, Fe-N(rs) and Fe-Si,(15*18)rates were in agreement with reported diffusion coefficients indicating that boundaries were acting as efficient sources and sinks in these oases, A further study by Bernstein’5) on the diffusion creep of Zr and Zr-1.4 y0 Sn also indicates apparently classic behaviour with creep rate depen~ng only upon D and a study of two Cu-Zn alloys by Burton and Greenwoodd7k shows a similar trend. At present there seems to be no systematic study of diffusion creep in single phase binary alloys over a wide composition range and the present work was undertaken to provide such data using Cu-Ni and a-phase Cu-Sn alloys. EXPERIMENTAL

METHODS

Since creep tests were to be performed at low stress levels it was convenient to use creep specimens in the form of helical springs to obtain extra sensit.ivity. In oases where i: and e are not linearly related analysis of creep data for such specimens is complex(i*) but where a linear relationship does exist the analysis is relat,iveIy simple since &-:-a behaviour is analogous to linear elastic behaviour. Moreover the viscosity 9 (which is the proportionality constant between shear stress and shear strain rate; or q = a/36 in the tensile case) can be obtained directly from measurements of relaxation time wit,hout precise knowledge of specimen geometry (see Appendix A). The preparation and analysis of the alloys used in this study has already been described in detail by Bastow and Kirkwood and will not be repeated here. The 7 mm alloy bars were reduced to specimen dimensions by cold swaging with intermediate vacuum anneals. The helix specimens were made by &ul.ing the wires around a machined former and giving a short anneal. The choice of dimensions for these specimens

VOL.

21,

1973

was dictated by sensitivity and also by the requirement that the maximum shear stress on the top coil due to the self weight was sufficiently low such that dislocation movement did not contribute significantly to creep. The dimensions chosen were 64 turns of wire of 2.6 mm diameter wound to a radius of curvat.ure of 13 mm. This gave a range of shear stresses at the wire surface between 0.12 and 0.72 MNlm2. Creep tests were performed in a vacuum of better than 10m5torr and the total specimen deflection was measured by a vernier microscope focussed on a marker wire in the transparent load chamber. The majority of the stress on the specimen was supplied by the self-weight and since the shear stress per coil varied between the top and bottom of the specimen the stress dependence of creep could be assessed simply by measuring the coil spacings at intervals throughout the test. After each test the specimens were weighed and the spring constant (elastic deflection/unit load) measured. The grain size of each specimen was measured using standard metallog~phic techniques. RESULTS

Creep tests were performed at 1163°K for the Cu-Ni and 1009*K for the Cu-Sn specimens. Extension-time plots were essentially linear with negligible evidence for any primary creep occurring. In prolonged tests or at higher temperatures when si~i~oant grain growth occurred deflection rates did decrease but this decrease could be quant.itatively accounted for by grain growth. Creep rates varied linearly with applied stress and this is demonstrated in Fig. 1 where the results for a Cu-Ni specimen are plotted as the fraction of the total creep strain which occurred in a particular coil versus the surface shear stress on that coil. A precise linear variation is not,ed

5.30

-

0.25

-

0.20

-

0

q/c, 0

is-

O.lO-

0.55

-

SHEAR

STRESS,

MN/m2

The vari&ion of creep strain (plotted aa the fraction of total strain) with applied stress for & Cu-Ni alloy spring at three different tots1 extensions. Frc.

1.

BURTOK

AXD

BASTOW:

DIFFUSIOKAL

CREEP

OF BIKART

COPPER

ALLOTS

15

TABLE 1 Solute (at. So)

Weight per coil (g)

External load (g)

1Xi 4.2X-i 7.6X 10.3X 13.2Ki 17.0X 41.2X lOONi*

3.3 3.3 3.3 3.3 ?Z

2.1 2.2 2.2 2.3 2.0 2.1 2.1 3.8 2.3 2.3 2.5 2.6 2.4

3:3 1.6

0.62Sn

?$ 3:4 3.4 3.4

2.4Sn 3.95Sn 5.65Sn 7.3Sn

Spring constant (mm/g) 4 4 4 4 3.1 3.1 2.4 1 3 3.4 3.8 5 6

s x x x x x x x x x x x x

Grain size (Ccm)

10-a 10-S IO-3 10-s 10-S 10-Z 1O-3 10-Z 10-S 10-S 10-S 10-S 1O-3

82 48 88 53 65 43 i2 150 63 66 13i 95 128

Deflection rate (mm/set) 5.2 4 4.3 2.2 1.4 2.2 1.7 3.4 7.3 x.3 4.‘) 2.8 4.6

x s x x x x \, h s M :< s s

10-d 10-Z 10-a 10-a 10-S IO-3 10-a 10-j 10-4 10-a IO-” IO-3 1O-3

* Wire diameter 2 mm, coil radius 10 mm, 11 coils. All other specimens wire diameter 2.6 mm, coil radius 13 mm, 63 coils.

at the three times (corresponding to total specimen extensions of 5, 10 and 15 mm) at which the test, was interrupted. In cases where primary creep is of importance its contribution predominates in the early stages of creep and thus Fig. 1 demonstrates its relative unimportance here since a linear variation is obeyed which is independent of the time at which the test was stopped. Experimental data for all the specimens are tabulated in Table 1 and viscosities calculated from this data are plotted in Fig. 2 as the ratio of the grain size compensated viscosity of the alloy to that of the pure metal. The grain size compensated viscosity r,l/d” ( = a/3 d dn) is simply derived from equations (1) or (2), n taking the value of 2 or 3 depending whether lattice or grain boundary diffusion predominates. In the present case n = 2 gives the more systematic

variation demonstrating that lattice diffusion creep is occurring. Furthermore Fig. 2 shows clearly the influence of solute additions on creep rat,e; the addition of nickel which decreases diffusivities in Cu-Ni alloys is shown to similarly reduce creep rates (i.e. increase viscosities) and the reverse trend is shown for the addition of tin which increases diffusivities. In order to compare creep rates with reported values of diffusion coefficients it is convenient to calculate an effective coefficient for creep D,, given by D, = t,d2kT/BaQ or in terms of viscosities D, = d2kT/3BC2q, where g,,, and 1;1,are the experimentally measured values. D, values have been calculated and are plotted in Figs. 3 and 4 for comparison with diffusion coefficients obtained from conventional measurements. COMPARISON DIFFUSION

WITH REPORTED COEFFICIENTS

The majority of reported diffusion coefficients are obtained from studies of diffusion in a concentration gradient. Diffusion creep occurs by stress-directed vacancy diffusion and this complicates comparisons of the coeflicients measured by these two techniques. For diffusion down a concentration gradient in a binary system the flux of atoms may be described by the interdiffusion coefficient D. This is related to the intrinsic diffusion coefficients of the separate species by the Darken equation:

b = X,D, + X,D,

ATOM

PERCENT

SOLUTE

FIG. 2. The variation of the grain size compensated viscosity (q/d*) for copper-nickel and copper-tin alloys expressed in terms of the viscosity of pure copper.

(3)

where Xi, X2 are the atom fractions of components 1 and 2. While this coefficient generally reflects the change in atomic mobility on alloying (see Figs. 3 and 4), its use to predict diffusion creep rates is strictly incorrect. For the Cu-Ni alloys the values of a plotted at 1173”K(2c) are typical of several reported in the

16

ACTA

METALLURGICA,

VOL.

21,

1973

Herring(a) has indicated that the correct diffusion coefficient to use for a multicomponent alloy undergoing diffusional creep is : i, j = 1 . . . n

DC= = [Cij (D-l)ji* XiXj]-l

(4)

where (D-l)jiH is the reciprocal of the diffusion coefficient matrix in the multicomponent flux equations : Ji = - g

EDijH ‘T (pi -

mu,)

(5)

where iV is the number of unit cells/unit, volume, ,uuiis the chemical potential of component i and ,uL,is the vacancy chemical potential. For a binary system in the case when the DiJE matrix is symmetrical and when the cross coefficients are not negligible equation (4) becomes : Da= c D22

t ‘”

I

I

20

40

I

60

I

ATo& Ni

FIG. 3. The comparison of interdiffusion coefficients and the theoretical predictions of equations (7) and (8) with values of diffusion coefficients derived from viscosity data for copper-nickel alloys.

literature although one recent measurement at 1179°K(21) which shows a more rapid variation in nickel rich regions is also plotted. For the Cu-Sn alloys reported measurements are more limited but the two available values obtained at 1006”K’22) and 1009°K’23) are shown. Also, from the approximate Darken relation for tracer coefficients the relevant data in Figs. 3 and 4 are consistent with the correlation between D, and b. -...-E Ref.21 A a Ref.22

cu

I

2

I 4 6 AT ‘lo Sn 1

I

DII=Dz~= - D,,=D,,=

H

H Xl2

-

242

X,X,

-I-

D,,

(6) X22

If the DijH matrix is diagonal the expression further reduces to :

‘Ni

80

H

I

8

FIQ. 4. The comparison of interdiffusion coefficients in the copper-tin system with coefficients calculated from viscosity data.

D== c

DnHD22= D,,HX,2

i-

D,HX,2

(7)

In earlier studies of diffusional creep in dilute iron alloys(15*fa)where X, < Xl, DcR -h DllH and DllH was identified with the relevant tracer diffusion coefficient D *. In the present case however the condition X,“‘< X, does not hold and it is necessary to use equation (7) to calculate DcH values. Assuming, with earlier workers, that DllH = Di* values of DcH have been calculated for the Cu-Ni alloys using values of Die at 1163°K extrapolated from the radiotracer data of Monma et uZ.(~) The results are shown in Fig. 3. Although the overall change in D, and Doa is similar this figure indicates that the precise shapes of the curves are not in such good agreement. For the Cu-Sn alloys only Dsn* has been measured (25) and comparisons of D, and D,” are not possible. Tracer diffusion coefficients have however been measured for Cu-Zn alloys (summarised in Ref. 26) and it is thus possible to evaluate equation (7) and to compare the calculated values of DcH with those obtained from previously published diffusion creep data.“‘) The results are shown in Fig. 5. Again extrapolation was necessary to obtain Di* at the test temperature. In this case both the range of values and the shape of the experimental and theoretical lines are in reasonable agreement although absolute values are not so close. Again rather surprisingly, values of .,?j (taken from Refs. 27, 28) appeared to fit experimental points more closely. More recently Weertman(29*30) has noted that

BURTON

AND

BASTOW:

DIFFUSIONAL

r

FIG. 5. The comparison of interdiffusion ooefficients and the theoretical prediotions of equations (7) and (8) with values of diffusion ooefscients calculated from previously published data on”‘) the viscosities of copper-zinc alloys.

equation (7) must be modified for use with experimentally measured diffusion coeficients to : D‘wL=l E

w2 XlD2 + x201

(8)

The derivation of this equation is outlined in Appendix B and the approximations which must be made and which were not discussed by Weertman are also noted. Values of Dew have been calculated and are plotted in Figs. 3 and 5 where it is indicated that 0,” N Dez in these cases and no si~ificantly improved fit is obtained, DISCUSSION

has been shown that while there is general overah agreement between the range of values of diffusion coefficients calculated from creep rates and those calculated from equations (7) and (8); the absolute magnitudes and the precise variations with composition are not closely com~rable. Several possible reasons exist for these discrepancies and will be discussed. Firstly it is possible that grain boundary diffusion contributes to creep in certain cases. Since the activation energy for grain boundary diffusion is less than that for lattice diffusion and grain boundary diffusion creep depends more strongly upon grain size, then grain boundary ~ffu~on will be more irn~~ant for fine grained material at lower values of the homologous temperature TIT,. Equations (1) and (2) predict the ratio of the contributions from grain boundary and lattice diffusion to be : 8,,/i, = 50 D&d D,. Now the ratio of the pm-exponential factors D,aB/DoL N 0.1 and the ratio of the activation energies QaBjQI, N 0.6. Writing the empirical relationship QL fz~ 150 T, It

2

CREEP

OF BINARY

COPPER

ALLOYS

17

J/mole and w N 0.5 nm enables the ratio B,,/dL to be evaluated. Using the solidus lines from Hansen and Anderkoc3n for Cu-Sn and Cu-Zn and that of Bastow and Kirkwood for Cu-Ni it can be demonstrated that grain boundary diffusion should give an insignificant contribution at these test temperatures for material with the grain sizes shown in Table 1. Secondly it is possible that concentration gradients set up by unequal diffusion of the fast,er moving species ma.y affect the creep process. Such an effect, has been noted in sintering@2) when segregation of the faster moving species led to precipitation of a second phase in the region of the sin&zing neck. However in the present case only the Cu-7.3 at.% Sn alloy was close to the solvus limit (7.7 at. %) and no precipitation was noted; nor could any composition differences be detected using electron probe microanalysis. This is perhaps not surprising in view of the small expected magnitude of the effect@) especially for the Cu-Sn alloy since the tracer diffusivities of copper and tin in pure copper are close. A further possibility is that in certain composition ranges the effect of solutes may be to reduce the efficiency of gram boundaries as vacancy sources and sinks. This may be the case with the Cu-Ni alloys where the diffusion coefficient measured from creep in the composition range 3-70 at.% Ni is less than that predicted from radiotracer data. Since the operation of boundaries as vacancy sources and sinks is generally considered to occur by the movement, of line defects in the boundary and interaction energies between line defects and solute atoms is greatest for elastically hard solutes ; this might explain why suppressed rates are obtained on adding nickel but not zinc to copper. However in view of the approximations involved in deriving the various expressions and errors involved in experimental measurements this suggestion must be regarded as somewhat speculative. CONCLUSIONS

1. At low stresses and elevated temperatures Cu-Ni and Cu-Sn alloys behave like Newtonian fluids. 2. Viscosity varies as the square of the grain size as predicted by the Nabarro-Herring theory. 3. Additions of nickel and tin cause respective decreases and increases in creep rate corresponding to to their influence upon diffusivities in these alloys. 4. Diffusion coefficients calculated from creep rates are in reasonable agreement with reported values of interdiffusion coefficient. 5. The more strictly correct value of diffusion coefhcient to use in the diffusion creep equation is obtained from the expression due to Herring as

ACTA

18

ItfETALLURGICA,

described by Weertman. The range of values of this coefficient calculated from tracer data for the Cu-Ni alloys and for a previously reported Cu-Zn alloys are in general agreement with those calculated from creep. However the absolute magnitude and in the case of Cu-Ni the precise form of the variation are not in such good agreement. 6. The interpretation by earlier workers of Herring’s equation for the diffusion coefficient to be used in describing diffusion creep is inaccurate for concentrated alloys but the use of the correct equation does not give a significantly improved fit. ACKNOWLEDGEMENTS

The authors are grateful to the University of Birmingham for a Research Fellowship (B. B.) and the Science Research Council for a Research Associateship (B. D. B.) and to Professor R. E. Smallman for provision of research facilities. REFERENCES 1. H. JOKES, Mat. Sci. Engr 4, 106 (1969). 2. F. R. N. NABARRO, Report of a Conference on Strength of Solids, p. 76, Physical Society (1948). 3. C. HERRING, J. appl. Phys. 21, 437 (1950). 4. R. B. JONES, Nature 207, 70 (1965). 5. I. M. BERNSTEIX, Trans. Am. Inst. Min. Engrs 299, 1518 (1967). 6. B. BURTON and G. W. GREENWOOD, Metal. Sci. J. 4, 215 (1970). 7. R. L. COBLE, J. appt. Phy8. 54, 1679 (1963). 8. W. VICKERS and P. GREENFIELD, J. nucl. Mater. 27. 73 (1968). and E. S. CHEN, Oxide Dispersion 9. F. K. SALTER Strengthening, edited by G. S. ANSELL, p. 495. Gordon and Breach (1968). 10. B. BURTON. Metal. Sci. J. 5. 11 (1971). 11. J. E. HARR&, R. B. JONES, b. W. GREENWOOD and M. J. WARD, J. Au&. Inst. Metals 14, 154 (1968). 12. R. B. JONES, Quantitative Relation between Properties and Microstructure, edited by D. G. BRANDOX and A. ROSEN, p. 343. Israel University Press (1969). 13. R. RAJ, Ph.D. Thesis, Harvard University (1971). 14. E. D. HONDROS and C. R. LAKE, J. Mater. Sci. 5, 374

(1970).

15. E. D. HONDROS, Phys. Status Solidi 21, 375 (1967). 16. H. JONES and G. M. LEAK, Acta Met. 14. 21 (1966). 17. B. BKTRTOS and G. W. GREENWOOD. Acta a%iet. 18.I 1237 (1970). 18. F. D. BOARDMAN, F. P. ELLEN and J. A. WILLIA~~SOX, J. Strain Anatysia 1, 140 (1960). 19. B. D. BASTOW and D. H. KIRKWOOD, J. Inst. Metals 99, 277 (1971). 20. D. M. MAHER, M.S. Thesis, University of California (1962). 21. G. BRUNEL, G. CIZERON and P. LACOMBE, C. T. Lebd. Seam. Acad. Sci. Paris 269(C), 895 (1969). 22. L. C. C. DA SILVA and R. F. MEHL, Trans. Am. Inst. Min.

Engr8 191, 155 (1951).

and D. H. KIRKWOOD, J. Inst. Metals 100, 23. B. D. BASTOW 24 (1972). 24. K. MONMA, H. SUTO and H. OIKAWA, J. Jap. Inst. Metals 23, 192 (1964). 25. Quoted bv H. J. STEPPER and H. WEVER, J. Phus. Ghem. Solids 26,” 1103 (1967). 26. Y. ADDA and J. PHILIBERT, La Diffusion dans les Solidea, Vol. II. Presses Universitaires de France (1966). 27. G. T. HORNE and R. F. MEHL, TTCZTU.Am. Inst. Min. Engrs 203, 88 (19%). 28. R. RESNICK and R. W. BALLUFFI, TTan8. Am. Inst. Min. Engrs 203, 1004 (1955).

VOL.

21,

1973

29. J. WEERTMAX, Trans. Am. Sot. MetaEs 61, 681 (1968). 30. J. WEERTNAN, Trans. Am. Inst. ilfin. Engrs 218, 208

(19601. ,----I.

ASDERKO, Constitution of Binary McGraw-Hill (1958). G. C. KUCZYNSKI, G. MATSUMUR~ and’ B. D. CULLITT, Acta Met. 8. 210 (1960). P. G. SHE~&OX;, Diffusion in Solids. McGraw-Hill (1963). R. E. HOWARD and A. B. LIDIARD, Rep. Prog. Phys. 28, 161 (1964). J. E. LANE and J. S. KIRKALDT, Can. J. Phys. 42, 1643 (1964). J. P. SABATIER and A. VIGSES, Xem. Scient. Revue M&all. 64, 225 (1967). A. W. LAWSOX, .I. Chem. Phys. 22, 1948 (1954). A. G. Guy and V. LEROY, The Electron Xicroprobc, o. _ . 543. J. Wilev (1966). RI. A. -D~YAN;)A and R. E. GRACE, Trans. Bm. Inst. Hin. Engrs 299, 1287 (1965). M. A. DAYANDA, P. F. KIRSCH and R. E. GRACE, Trans.

31. M.

HANSEX

and

K.

Allous, 2nd edition.

32. 33. 34. 35. 36.

Xi: 39. 40.

Am. Inst. Min. Engrs 242, 885 (1968). Adv. Mater. Res. 4, 55 (1970)

41. J. S. KIRWLDT,

APPENDIX

A

When the rate of shear of a material 9 is linearly proportional to the applied shear stress a, the material is said to be a Newtonian fluid and the proportionality constant between a, and 9 is the viscosity q. Such behaviour is analogous to linear elastic behaviour with q equivalent to the shear modulus G i.e. G = a,/6 where 6 is the angle of shear caused by the shear stress as and q = a,/(dv/dr) where (dv/dr) is the velocity gradient. Now if a relaxation mechanism operates such that the initially applied shear stress is relaxed in time 7, then G = a,/r(dO/dt) and the rate of shear dO/dt is identical to the velocity gradient dvldr. It thus follows that 9 = Gr. Thus in order t,o measure the viscosity of a helix it is necessary to know the elastic deflection on loading, the subsequent rate of plastic deformation and the shear modulus. It is of interest to note that precise knowledge of stress distribution throughout the specimen is not required so long as the material deforms in a linear elastic and linear viscous manner in all parts. If the spring constant of a specimen is S and the creep test is performed under a load M than the initial elastic deflection is SM. If the rat’e of creep deflection is 8 this elastic deflection will be relaxed in time 7 = SMI8 and thus 17= SMGI8. (It should be noted that S is usually measured at room temperature and thus 7 will be lower than if the more strictly correct test t#emperaturevalue was used. However the correct value for 11 will be obtained if the room temperature value of G is used since this will be the same fraction too large as S was too small.) In the case where the weight of the specimen is no longer insignificant when compared with the load it is necessary to compute the extra contribution to M. The elastic deflection of a light helical spring is given by: S = CRsMn/Gad, where C is a numerical

BURTOX

AXD

BASTOW:

DIFF-USIONAL

constant, R the coil radius, a the wire diameter, 2cpthe applied load and 12 the number of turns. Thus the deflection of the ith coil of a heavy spring of weight/coil rn under an applied load M is cVi= K{M + (n - i)m.} where K = CR3/Gab and i = 1 represents the top coil. The total elastic deflection is thus 6 = K (nM + m, Qa> where T;,n = 1 + 2 + . . . + 2~. Now the deflection of the spring due to an external load of value (nm + M) is 6’ = Kn(nm + M). Thus

CREEP

OF

BINARY

COPPER.

ALLOYS

19

where D,, = (I),,’ - D,,‘). In a binary system it is equations of this form which define the two intrinsic diffusion coefficients and from equations (B.3) and (B.4) :

Compa~ng this with the relation Lij = Dii~~~~T it is clear that the direct coefficient DiiH is only comparable to the binary intrinsic coefficient Dii when the cross-coefficients Lij are zero and when the solution is 0.5(nn-\y + f) 6 2% + -jT= ideal. In this case DiiH = XiDii. n2 Herring’s) pointed out that assumption of negligible wherefis the ratio of M and m. Now 6’ = S(nm + M) cross-coefficients is most justifiable for interstitial sothats = ([05(n + 1) +f]/(n +f))S(n~ + .&f)and solid solutions (e.g. Fe-W5)) where the two compothis value is used instead of SM as in the ease of light nents diffuse via different mechanisms. For substitusprings. tional alloys calculations using simple atomic models APPENDIX B and experiments in ternary systems indicate that In order to obtain the correct, interpretation of while these coefficients are small compared to the equation (4) it is necessary to distinguish between direct coefficients, they are not’ negligible.(35*36) two alternative formulations of the diffusion equations It should also be noted that while Herring shows that the effect of pressure on the diffusion-flux-is included Ji = - xjLij atpj - ~~)/ax (B.1)in the chemical po~ntial terms, in the derivation of (B-2) equation (B.5) it is nesessary to use the thermody= - ~1~~~~8 axjjas namic relation between ,ui and Xi. In doing so the where equation (B-1) applies for any region where pressure term in the relation d,ui = (CJ,q/aP)dP + lattice sites are conserved (33). Equation (B.2) is used, (a,uu,/i3Xi) dX, has been neglected. It is difficult to in a suitably reduced form, for the evaluation of justify this, except in the cause of simplicit,y, since it is experimental diffusion profiles. The matrix of the pressure directed vacancy flux which causes difphenomenologicai coefficients L,,, is symmetric by fusion creep but the lack of reliable data, even for pure virtueofthe Onsagerreciprocalrelations.(~) Corning metals, on the effect on diffusion of stresses acting both equations (5) and (B.l) it is seen that Lii = D~j3~~~T perpendicular and parallel to the diffusion direction@‘) that is the coefficients in the original Herring formula makes this approximation necessary. are not directly identifiable with experimentally For the case where tracer diffusion coefficients, measured coefficients and the assumption that they Di*, are used in the calculation of DcR the relation are, which has been made by earlier workers, is only Dii = .Di*(l + aln y,/a In Xi) may besubstitutedinto valid in the following limiting cases. equation (B.3) to give Di* = (L,,/X, - L~j~X~)~T~~. In isothermal, isobaric diffusion (i.e. under the The relation between Dii and Di* is only an approxiconditions in which diffusion ~oe~cien~ are generally mation and is discussed more fully in Ref. 34 but the measured) it is assumed that. pV _N 0. This has been errors involved are negligible in the present application. discussed by Shewmon (3~ for experiments where there Hence, when Lij = 0, we have D,*Xi = Dilw and is a net vacancy flow producing a Kirkendall effect this may be substituted in equation (7) to give: and he concludes that any error introduced because of this assumption is not detected experimentally. D1*D2* By applying the Gibbs-Duhem relation to equation (B.l), for a binary system it may be shown that (34): which is equivalent to equation (8) when Di = Di*, i.e. for an ideal or dilute solution. In ternary systems intrinsic coefficients have been measured in very few systems w-~O) and in general it where yd is the activity coefficient of i. appears that cross-coefficients in the flux equations Applying the law of mass conservation to equation Ji = - NzTz: Dij dXj/‘dXmay be ignored in comparison with direct coefficients. However, even when this (B-2) is not the case it should be remembered that, as in

fn = +fn

(

20

ACTA

XETALLURGICA,

binary s37stems, the intrinsic coefficients measured in ternary diffusion experiments fDij) are not identical to those defined in equation (5). The relationship between these coefficients has been given by ICirkaldy(41) Hence equation (4) becomes a very complex relation in ternary or higher order systems when the crosscoefficients are not negligible and it seems desirable to

VOL. 21, 1913

assume that the Dpi matrix is diagonal even though this may only result in a crude approximation. Inithis case, for an n-component allaS:

I&D,*

D, = r_i(Di*T]ijxj)

i = 1 . *. % i f j

where Il is the product operator.