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Internal friction in silver solid solutions
INTERNAL FRICl’ION IN SILVER SOLID SOLUTIONS* T. J. TURNER and G. P. W-ILLMS,
JR.t
Zener relaxation studies have heen made on three silver solid solutions with a torsion pendulum and an elastic-after-effect apparatus. over the range of temperatures studied an Arrhenius equation describes the results. An Arrhenius plot yields for 32% Cd in Ag tr = lo- 16.6se~p(~~$oo/RT); for 44% Au in Agt, = 10-‘6*5” exp (42,3W/RT). For each concentration the relaxation exp (44,2OO/RT); and for 68% Au in Ag t,. = lo- 16.28 time t,. is compared with t the mean time of stay obtained from diffusion measurements of other investigators. In each case the relaxation time is clearly associated with the extrapolated mean time of stay of the faster moving component. Over the temperature range studied the constant relating these times is aPPrnXin’IatelY unity. Grain boundary relaxation was studied in the system 42% Au in Ag. The relaxation process can be described by tr = 10-14,0 exp (49,3OO/RT). AMORTISSEMENT
INTERIEUR
DANS DES SOLUTIONS
SOLIDES
D’ARGENT
Des Etudes de la relaxation de Zener ont I% effect&s sur trois solutions solides d’argent, B l’aide d’un pendule de torsion et d’un appareil mesurant l’effet Blastique retard& Pour la zone de temtiratures ktudiks, une dquation d’Arrhenius d&it les rCsultats. On obtient ainsi l’equation du temps de relaxation pour cliff& rentes concentrations en cadmium et en or: pour 32% CdT, = 10-16,88exp (35.OOO/RT) pour 44% Au en Agr, = l&-W66 exp (44.200/RT) et pour 68% Au en Ag 7r = 10-16,es exp (42,30O/RT). Pour chaque concentration, le temps de relaxation 7r est compare avec le temps moyen de sdjour d’un atome dans un site pa.rticulier du r&au;.ce dernier temps est d6duit des mesures de diffusion effect&es par d’autres chercheurs. Dam chaque cas, le temps de relaxation est t&s nettement en relation avec le temps moyen de skjour des atomes de l’&ment le plus mobile. Pour les temp6ratures &udi&s, le rapport entre ces temps est approximativement &al ii l’unit& La relaxation aux joints de grains a 6ti Btudi& dans le systtme Ag-Au contenant 42% d’or. Ia relaxation peut &re d&rite par la relation 71 = lo-“,0 exp (49,30O/RT). INNERE
REIBUNG
FESTER
LdSUNGEN
IN SILBER
An drei festen tisungen in Silber wurden mit Hilfe eines Torsionspendels und eines Apparates zur Messung der elastischen Nachwirkung Untersuchengen fiber das Zenersche Relaxationsphtiomen angestellt. uber den untersuchten Temperaturbereich lassen sich die Ergebnisse durch eine Arrhenius-Gleichung wicdergeben. Eine derartige Darstellung ergibt fiir 32% Cd in Ag 7r = 10-15@ exp (35,0OO/RT), fiir 44% Au in Ag TT= 10-16.66 exp (44,2OO/RT) und fiir 68% Au in Ag ICY= 10-15@ exp (42,3OO/RT). Fiir jede Konzentration wird die Relaxationszeit 7r verglichen mit 7, der mittleren Aufenthaltszeit, wie sie sich aus den Diffusionsmessungen anderer Forscher ergibt. In jedem Fall hlngt die Relaxationszeit klar mit der mittleren Aufenthaltszeit der schneller beweglichen Komponente zusammen. Im untersuchten Temperaturbereich ist die Konstante, die diese beiden Zeiten verknilpft, nahezu eins. Korngrenzen-Relaxation wurde untersucht im System 42% Au in Ag. Der RelaxationsprozeB liil3t sich beschreiben durch 7r = 10-14,0 exp (49,3OO/RT).
JNTRODUCTION
Anelastic
one to measure a relaxation time tr, characteristic of the metal, which is the time required for an appropriate redistribution of atoms as the result of an applied stress. It can be described Anelastic
studies enable
by an Arrhenius
type equation.
has enabled
z, = zrO exp. (HJRT).
(1)
Where zrOis a frequency factor, H,. is the activation energy and RT has the usual significance. * held was 7
Paper 9 presented at the Conference on Internal Friction on 10 and 11 July, 1961, at Cornell University. This work supported in part by the U. S. Atomic Energy Commission. Department of Physics, Wake Forest College, WinstonSalem, N. C., U.S.A. ACTA METALLURGICA,
VOL. 10, APRIL
studies
of interstitial
solid solutions
yield
activation energies H, equal to those obtained in diffusion studies. WerF has shown that z, = 213 t for an interstitial solution where z is the mean time of stay of an atom at a particular lattice site. This investigators
to study
atomic
mobilities
at relatively low temperatures. In the case of substitutional solid solutions which give rise to Zener relaxation the situation is more complicated. Using the pair orientation theory proposed by Zenerc2) to explain stress-induced ordering in substitutional solid solutions, LeClaire(s) obtained expressions relating z, and z. For a stress applied along the (111) direction he deduced ‘tr = 11/8t, and
1962 t3051
306
ACTA METALLURGICA,
for the (001) and (101) directions he obtained rr = 11/12 t. In the case of o brass considered by LeClaire, t is the mean time of stay of the zinc atom, the faster diffusor of the two components. Unfortunately, insufficient experimental data was available to check these results. Nowick(4) has suggested that the relaxation process is controlled by the slower moving component. His reasoning was based on the large probability that a fast moving atom vacating a lattice site would be replaced by an identical fast moving species. Thus no contribution would be made to the redistribution of atoms necessary for the relaxation process. He performed extensive anelastic measurements on a series of silver-rich, silver-zinc solid solutions. The results indicated a slight deviation from an Arrhenius law, but he quoted a mean heat of activation for each concentration studied. Later diffusion measurements by Lazarus and Tomizuka(5) on 30 per cent Zn in Ag yielded activation energies several kcal/mole greater than the value quoted by Nowick for the same concentration. Consequently, neither the mean time of stay of the silver nor of the zinc atoms could be related to tl since each had a different temperature dependence. Hino ef ~1.‘~)performed a comprehensive study of internal friction and diffusion in ,31 per cent alpha brass in an attempt to relate rr and r. The internal friction results showed no deviation from an Arrhenius law, but the activation energy for the relaxation process was less, by several kcal/mole, than that for the diffusion of either constituent into the solid solution. However, the value obtained for the activation energy was found to be closest to that for chemical diffusion of zinc into alpha brass. The authors pointed out that this agreement may be fortuitous. These results indicate a need for further anelastic and diffusion experiments on other systems. In particular the Ag-Au system appears ideal since Zener relaxation has been observed”) and there is no concentration limitation because of solubility considerations. In addition extensive diffusion studies have been made by Slifkin et u/.‘~’ and thermodynamic data is readily available. The formal theory of the Zener relaxation, sometimes referred to as stress-induced ordering, has recently been reviewed by Nowick and Seraphimu’). Upon the application of a static elastic stress there appears in addition to the elastic strain, a time de-
VOL. 10, 1962
pendent anelastic strain E”. Upon removal of the applied stress the anelastic strain relaxes according to the equation E” = ei’ exp (-
t/r,).
(2)
This is termed the elastic-after-effect. This same relaxation time can be determined from internal friction studies, i.e. a measure of the phase difference between stress and strain in a dynamic experiment. This phase difference is called the internal friction and is given by
@ is the angle by which the strain lags the stress, 6 is the logarithmic decrement, o the circular frequency and A, is the relaxation strength, a measure of the anelastic properties. Equation (6) of Nowick and Seraphim(e) reduces to equation (3) above for A,< 1. In the systems studied A, s. 0.01 validating the use of equation (3) in the determination of rr. The relaxation time t,. will be compared with the extrapolated mean-time-of-stay of an atom at a given lattice site. This mean-time-of-stay t can be obtained from the diffusion coefficient D. From random walk considerations for a f.c.c. lattice.
D=+g
(4)
where a is the lattice parameter and the correlation factor f of Bardeen and Herring has been included. Assuming that the diffusion process can be represented by an Arrhenius type equation one has
*=- fa” 120
1
~fa” exp (H/RT). 120,
(5)
Relating t and rf by a geometrical factor C, one can see from equation (1) and (6) provided that H= H, that c
=
12Do~o fa"
(6)
Uncertainties in the correlation factor f as well as experimental errors in Do and z,~ make a determination of C quite difficult. However, in addition to calculating C, a useful comparison might be made by showing the temperature of r and r7 on the same plot. This will be done for one concentration of cadmium in silver and two concentrations of gold in silver.
TURNER
AND
EXPERIMENTAL
WILLIAMS: PAPER 9 OF INTERNAL METHOD
(a) Specimen preparation and analysis The Ag-Au alloy specimens were 0.030 in. diameter wires 6-8 in. long. These were rolled and drawn from sections taken from $ in. or 8 in. diameter single crystals destined for other experiments and grown by the Bridgemen technique from 99.999 per cent pure gold and silver. A g-in. length of the drawn wire was sealed in a quartz capillary at a pressure of 6 x 10mgmm Hg and passed through a furnace at a rate of 1 cm per hour at a temperature approximately 25°C above the melting point of the alloy. One half-inch was cut from each end of the large-grained, or single crystal wire resulting. To facilitate handling and mounting in the apparatus the ends of the remaining 8 in. specimens were silver soldered into 1 in. long copper chucks made from $ in. dia. rod. The concentration of the Ag-Au alloys was calculated assuming a linear relationship between density and concentration. The density was determined prior to rolling and at the conclusion of all measurements on a given wire specimen by weighing it in air and in distilled water using a micro-balance. Agreement in concentrations calculated before and after the relaxation time measurements was accepted as sufficient evidence that there was no evaporation loss of either component during the experiment. The alloys of silver with cadmium were analysed before and after the experiment by chemical methods. (b) Torsion pendulum To measure relaxation times in the range 0.1-1.0 set the specimen was mounted as the torsion member of an inverted torsion pendulum of the tyoe descibed by Wert, and surrounded by a tube type furnace in a vacuum chamber capable of an ultimate vacuum of approximately 3 x 1O-5 mm Hg. The natural frequency of vibration was controlled by shifting and changing weights on a steel inertia arm initially excited by solenoids placed near by. After several hours of vacuum annealing in situ below the recrystallization temperature, the decrement measurements were begun. The amplitude decay was observed visually using a 3 m optical lever, with strains less than 5 x 1O-6. The temperature was controlled by a platinum resistance element as a Wheatstone Bridge arm wound non-inductively on the center section of the
FRICTION CONFERENCE
307
furnace core. This control unit was of the off-on type and controlled power to three separate windings of the furnace. The current to each of these was separately manually regulated to provide uniform temperature along the specimen length. The temperature was determined from the average of the readings of three standardized chromel-alumel thermocouples placed at the top, center and bottom of the specimen. The criteria for uniform temperature over the entire length of the specimen was that the top and bottom theromocouples should read to within $ “C of the reading of the middle thermocouple before the decrement could be read at that temperature. Three independent decrement measurements were made at each temperature setting from which a curve of internal frictionvs. temperature was obtained. In spite of a broadening produced by a distribution of relaxation times making the peak approximately 20-30 per cent broader than predicted by theory, the temperature dependence of the maximum occurring in this plot was assumed to obey equation (3) above for Zener relaxation. Hence the relaxation time z, is the reciprocal of the circular frequency w. These torsion pendulum curves were analyzed by two different methods to determine the peak shift in temperature. In one method, the background was subtracted and the peak position located on each curve. In the other method, the identical shape of the leading wing of each curve made it possible to determine the shift in temperature of this leading edge. The two methods gave the same result to within * 1Q”c. (c) Elastic-after-effect
apparatus
For relaxation times between 100 and 5000 set the specimen was placed in the elastic-after-effect apparatus and again given several hours of annealing in situ below the re-crystallization point. The furnace temperature was controlled exactly as described for the torsion pendulum, and the temperature was also determined as previously described. The controller kept a constant temperature to within f 0S”C for runs of 1 hr or less and to within i 1°C for runs of several hours, up to one day, provided the room temperature was maintained constant to f 3°C. The elastic-after-effect runs were made observing change in strain under conditions of constant stress, a stress of zero actually being used. The specimen was twisted approximately 2” by energizing solenoids outside
ACI-A METALLURGlCA,
308
of the vacuum tube, thus exet’ting a torque on a small magnet fastened to a light extension rod attached to the chuck on the lower end of the specimen. An optical lever arrangement made it possible to observe angular motion of the specimen to less than 0.02”. When the specimen had been held in the twisted position using a constant solenoid current for a period corresponding to 3 or 4 times the estimated relaxation time tt, the current was quickly reduced to zero, and the strain relaxation observed and plotted as a function of logarithm of the time, according to equation (2). Following Nowick( the point of inflexion of this curve is used to determine the relaxation time. The points corresponding to these relaxation times and temperatures are then plotted on the same In z, vs. l/T plot made for the torsion pendulum data. Thus the range of the measurement of tl is extended over approximately 6 cycles of 10 and the slope, yielding Q, and the intercept,’ yielding zrO may be obtained with greater precision than is characteristic of values obtained from one method alone.
VOL.
10, 1962
ical torsion pendulum data for Ag-Cd. Typical curves for Ag-Au have been published earlier.(‘) Figure 2 is an Arrhenius plot of the data obtained for 32 at. % Cd in AgO The lower series of points
32 At 96 Cd in Ag
Mean Time --.-Diffusion -
24
32 At % Cd in Ag - Frequency 0.6 cps
16
8
4
I
/
/
/
I 1.800
FIG. 1. Internal
I
I 2.000 1000 T
I
I 2.200
I
I 2.400
friction as a function of temperature solid solution 32% Cd in Ag.
Diffusion
I
1
2.0
I
2.2 y
Stay
Cd into
of
A9 into A9
Aq.Cd Cd
Schoen) ,
2.4
1
2.6
(OK-I)
FIG. 2. Relaxation time as a function of inverse temperature for the solid solution 32% Cd in Ag. Mean times of stay are shown extrapolated from the diffusion data of SchoerW.
was obtained from torsion pendulum data and the higher set from elastic-after-effect data. For comparison the mean time of stay of each component is plotted. These times were obtained from equation (5) using the diffusion data of Schoen(l’). The highest concentration measured by Schoen was 28 per cent; therefore, an extrapolation of 4 per cent was required for comparison. Manning (12)has published correlation factors also obtained from Schoen’s data. Although a different temperature range is involved, reasonable estimates of the correlation factor can be obtained assuming the temperature dependence is not significant for these concentrations. These values along with a least squares analysis of the experimental data are shown in Table 1. The value of C was obtained from equation (6).
20
10 0 -
-
(after
EXPERIMENTAL RESULTS AND DISCUSSION
Typical elastic-after-effect curves are similar to those previously published.(134) Fig. 1 illustrates typ-
-
of of
for the
Figures 3 and 4 give the data obtained with 44 and 68% Au in Ag, respectively. As in Fig. 2, the mean time of stay of each component is plotted for comparison.
These
diffusion
data
including
cor-
TURNER
WILLIAMS:
AND
PAPER 9 OF 1NTERNAL TABLE 1 -___.__
‘---
%.
)
Hr
1 xld_T;
1
35,100 1 34,700 p.,o
)
FRlCTlON
309
CONFERENCE
-_
fCd 1
HA,
/
DoAg
1
fAB
1
c
-
[
0.90
1 OiF
__-_ ,,,,. _____-
j
relation factors were obtained from the work of Slifkin et al.@. Table 2 summarizes the data relating to Figs. 3 and 4.
0.70
1 0.14
ture may be higher than that observed in the Ag-Zn system .03) Quenching experiments will be attempted I04
lOp------
1
I03
103-
102
IO2-
? s
c f
c
C
7; i IO-l5 55 e 44,200/,Z3
IO’
IO’ -
I
100
) Loi
II I
10-l I.5
1’
Meon
-.-.
-
---
1.6
Time
of of
into
Ag
Au
Diffusion
of Au into
Ag
Au
(after
’
1.7
1 1.9
I.8 !y
IO0
Stay
Diffusion
Ag
Shfkm)
l 2.0
Mean -‘--
Time
of
Stay
Diffusion
of
Ag
into
Ag
Au
Diffusion
of Au
into
Ag
Au
(after
Slifkin)
10-I
I 2.1
(OK-l)
FIG. 3. Relaxation time as a function of inverse temperature for the solid solution 44% Au in Ag. Mean times of stay are shown extrapolated from the diffusion data of Slifkin et u1.@‘.
FIG. 4. Relaxation time as a function of inverse temperature for the solid solution 68% Au in Ag. Mean times of stay are shown extrapolated from the diffusion data of Slifkin et uL@).
the elastic-after-effect apparatus to note the relaxation strength as one approaches the critical temperature. Grain boundary activation energies were initially identified with those obtained from diffusion. However, in this investigation grain boundary relaxation studies on 42 Au in Ag yield a relaxation time r, 1CP4*0 exp (4,93O/RT). This value of 49.3 kcal/mole fits the Marx-Wert(14) plot and is considerably greater than the diffusion with
Li and Nowick(13) have discussed the critical temperature for stress-induced ordering T, given by the equation To
&f=.._. T-
T,
(7)
Where d, is the relaxation strength shown in equation (3), To is a constant and T the temperature. Rough measurements of the relaxation strength in the silver-gold system indicated that the critical tempera-
ACTA METALLURGICA,
310
values HAB= 43.5 kcal/mole and H_,, = 45.5 kcal/ mole obtained by Slifkin et al.@). Finally, it should be noted that doubt arose shortly after publication’71 with regard to the accuracy of the activation energy reported for a 58.5% Au in Ag alloy because of improperly corrected thermocouple readings in the raw data. Although the discrepancy was small, a check run was indicated. Unfortunately, this specimen had been cold worked at the conclusion of the original measurements, and hence, required an anneal. During this anneal silver was inadvertently evaporated from the surface, resulting in a the~ally etched surface, rendering a bona fide repetition of the earlier run impossible. Nevertheless, the measurements were made. The results yielded the activation energy and r,, values compatible with those of an 80% Au in Ag alloy. The inference is not only that silver had indeed evaporated, but that the observed relaxation was characteristic of the gold rich layer near the surface, as would be expected. However, preliminary studies indicate that this surface layer in which the major portion of the relaxation occurs is surprisingly thin.
VOL. 10, 1962
have been shown for all systems. Values quoted in Tables 1 and 2 for the other concentrations were obtained by interpolation or extrapolation. A comparison between tr and r based on Figs. 2-4 appears to be better than that indicated by the values of C given in the tables because of uncertainties in z~, and which were in calculating
CONCLUSIONS I
Although uncertainty exists regarding what is actually being measured in a Zener relaxation experiment, even the origin of the effect being uncertain, though in the Ag-Au system, it has been attributed previously(‘) to a difference in compressibility resulting in a net size difference-it appears on the basis of the data presented that one is measuring the diffusion of the faster moving component. Figs, 3-5 show what must be regarded as surprising agreement between tl and r the mean time of stay of the fast diffusor, even though the difference in activation energies is at the limit of the combined experimental error, Therefore, for these systems over the range of temperatures studied, one can use relaxation data to obtain the mean time of stay of the faster moving component. These relaxation times can then be used in equation (5) to obtain the diffusion coefficient. A direct comparison between diffusion coefficients obtained in this manner and those measured by Shfkin et &.(*) is shown in Fig. 5. It is unfortunate that the 68% Au in Ag specimen happened to be the only concentration duplicated for both anelastic and diffusion experiments. Otherwise such a plot would
BOO
I
I
1.000 1.200
I
I
I.400 1.600 1.800 2.t 10
!!F°K-‘) FIG. 5. Diffusion coefficient of silver diffusing into the solid
solution 68% Au in Ag as a function of inverse temperature. Diffusion coefficients are calculated from internal friction data, assuming the relaxation time to be equal to the mean time of stay.
Tomizuka eb &.(16)have recently studied self-diffusion of 15% Zn in Ag, reporting an activation energy for the faster moving component, zinc, agreeing with the value obtained by Nowickc4) for the same concentration. These data, from four systems, lead one to believe that anelastic methods can be used to extend diffusion measurements in substitutional solid solutions to relatively low temperatures.
We would like to record our appreciation to Professor L. Slifkin for helpful discussions, and for supplying Ag-Au crystals from which some of the wire specimens were drawn.
TURNER
AND
WILLIAMS: PAPER 9 OF INTERNAL FRICTION CONFERENCE
REFERENCES
1. 2. 3. 4. 5. 6. 7. 8.
C. A. WERT, Phys. Rev. 79, 601 (1950). C. ZENER, Phys. Rev. 71, 34 (1947). A. D. LECLAIRE, Phil. Mag. 42, 673 (1951). A. S. NOWICK, Phys. Rev. 88, 925 (1952). D. LAZARUSand C. TOMIZUKA,Phys. Rev. 103,1155 (1956). J. HINO, C. TOMIZUKAand C. WERT, Aclu Met. 5,41 (1957). T. J. TURNERand G. P. WILLIAMS,Acta Met. 8, 891 (1960). L. SLIFKIN, W. MALLARD, R. BASSand A. GARDNER, to be published.
311
9. A. S. NOWICK and D. P. SERAPHIM,Actu Met. 9, 40 (1961). 10. J. BARDEENand C. HERRING,Imperfections in Nearly Perfect Crystals (Ed. by W. SHOCKLEY)p. 261. Wiley, New York (1952). 11. A. SCHOEN,Thesis, Univ. of Illinois (1958). 12. J. R. MANNING, Phys. Rev. 116, 69 (1959). 13. C. Y. LI and A. S. NOWICK, Actu Met. 9, 49 (1961). 14. C. WERT and J. MARX, Acta Met. 1, 113 (1953). 15. C. TOMIZUKA,R. ROBERTSand R. DICKERSON,Bull. Amer. Phys. Sot. 6, 172 (1961); and private communication.
Note added in proof Subsequent data indicate that HR is actually greater than HAB at the lower concentrations, although relaxation time is still clearly identified with the extrapolated mean time of stay of the faster moving component.
the
JR.t
Zener relaxation studies have heen made on three silver solid solutions with a torsion pendulum and an elastic-after-effect apparatus. over the range of temperatures studied an Arrhenius equation describes the results. An Arrhenius plot yields for 32% Cd in Ag tr = lo- 16.6se~p(~~$oo/RT); for 44% Au in Agt, = 10-‘6*5” exp (42,3W/RT). For each concentration the relaxation exp (44,2OO/RT); and for 68% Au in Ag t,. = lo- 16.28 time t,. is compared with t the mean time of stay obtained from diffusion measurements of other investigators. In each case the relaxation time is clearly associated with the extrapolated mean time of stay of the faster moving component. Over the temperature range studied the constant relating these times is aPPrnXin’IatelY unity. Grain boundary relaxation was studied in the system 42% Au in Ag. The relaxation process can be described by tr = 10-14,0 exp (49,3OO/RT). AMORTISSEMENT
INTERIEUR
DANS DES SOLUTIONS
SOLIDES
D’ARGENT
Des Etudes de la relaxation de Zener ont I% effect&s sur trois solutions solides d’argent, B l’aide d’un pendule de torsion et d’un appareil mesurant l’effet Blastique retard& Pour la zone de temtiratures ktudiks, une dquation d’Arrhenius d&it les rCsultats. On obtient ainsi l’equation du temps de relaxation pour cliff& rentes concentrations en cadmium et en or: pour 32% CdT, = 10-16,88exp (35.OOO/RT) pour 44% Au en Agr, = l&-W66 exp (44.200/RT) et pour 68% Au en Ag 7r = 10-16,es exp (42,30O/RT). Pour chaque concentration, le temps de relaxation 7r est compare avec le temps moyen de sdjour d’un atome dans un site pa.rticulier du r&au;.ce dernier temps est d6duit des mesures de diffusion effect&es par d’autres chercheurs. Dam chaque cas, le temps de relaxation est t&s nettement en relation avec le temps moyen de skjour des atomes de l’&ment le plus mobile. Pour les temp6ratures &udi&s, le rapport entre ces temps est approximativement &al ii l’unit& La relaxation aux joints de grains a 6ti Btudi& dans le systtme Ag-Au contenant 42% d’or. Ia relaxation peut &re d&rite par la relation 71 = lo-“,0 exp (49,30O/RT). INNERE
REIBUNG
FESTER
LdSUNGEN
IN SILBER
An drei festen tisungen in Silber wurden mit Hilfe eines Torsionspendels und eines Apparates zur Messung der elastischen Nachwirkung Untersuchengen fiber das Zenersche Relaxationsphtiomen angestellt. uber den untersuchten Temperaturbereich lassen sich die Ergebnisse durch eine Arrhenius-Gleichung wicdergeben. Eine derartige Darstellung ergibt fiir 32% Cd in Ag 7r = 10-15@ exp (35,0OO/RT), fiir 44% Au in Ag TT= 10-16.66 exp (44,2OO/RT) und fiir 68% Au in Ag ICY= 10-15@ exp (42,3OO/RT). Fiir jede Konzentration wird die Relaxationszeit 7r verglichen mit 7, der mittleren Aufenthaltszeit, wie sie sich aus den Diffusionsmessungen anderer Forscher ergibt. In jedem Fall hlngt die Relaxationszeit klar mit der mittleren Aufenthaltszeit der schneller beweglichen Komponente zusammen. Im untersuchten Temperaturbereich ist die Konstante, die diese beiden Zeiten verknilpft, nahezu eins. Korngrenzen-Relaxation wurde untersucht im System 42% Au in Ag. Der RelaxationsprozeB liil3t sich beschreiben durch 7r = 10-14,0 exp (49,3OO/RT).
JNTRODUCTION
Anelastic
one to measure a relaxation time tr, characteristic of the metal, which is the time required for an appropriate redistribution of atoms as the result of an applied stress. It can be described Anelastic
studies enable
by an Arrhenius
type equation.
has enabled
z, = zrO exp. (HJRT).
(1)
Where zrOis a frequency factor, H,. is the activation energy and RT has the usual significance. * held was 7
Paper 9 presented at the Conference on Internal Friction on 10 and 11 July, 1961, at Cornell University. This work supported in part by the U. S. Atomic Energy Commission. Department of Physics, Wake Forest College, WinstonSalem, N. C., U.S.A. ACTA METALLURGICA,
VOL. 10, APRIL
studies
of interstitial
solid solutions
yield
activation energies H, equal to those obtained in diffusion studies. WerF has shown that z, = 213 t for an interstitial solution where z is the mean time of stay of an atom at a particular lattice site. This investigators
to study
atomic
mobilities
at relatively low temperatures. In the case of substitutional solid solutions which give rise to Zener relaxation the situation is more complicated. Using the pair orientation theory proposed by Zenerc2) to explain stress-induced ordering in substitutional solid solutions, LeClaire(s) obtained expressions relating z, and z. For a stress applied along the (111) direction he deduced ‘tr = 11/8t, and
1962 t3051
306
ACTA METALLURGICA,
for the (001) and (101) directions he obtained rr = 11/12 t. In the case of o brass considered by LeClaire, t is the mean time of stay of the zinc atom, the faster diffusor of the two components. Unfortunately, insufficient experimental data was available to check these results. Nowick(4) has suggested that the relaxation process is controlled by the slower moving component. His reasoning was based on the large probability that a fast moving atom vacating a lattice site would be replaced by an identical fast moving species. Thus no contribution would be made to the redistribution of atoms necessary for the relaxation process. He performed extensive anelastic measurements on a series of silver-rich, silver-zinc solid solutions. The results indicated a slight deviation from an Arrhenius law, but he quoted a mean heat of activation for each concentration studied. Later diffusion measurements by Lazarus and Tomizuka(5) on 30 per cent Zn in Ag yielded activation energies several kcal/mole greater than the value quoted by Nowick for the same concentration. Consequently, neither the mean time of stay of the silver nor of the zinc atoms could be related to tl since each had a different temperature dependence. Hino ef ~1.‘~)performed a comprehensive study of internal friction and diffusion in ,31 per cent alpha brass in an attempt to relate rr and r. The internal friction results showed no deviation from an Arrhenius law, but the activation energy for the relaxation process was less, by several kcal/mole, than that for the diffusion of either constituent into the solid solution. However, the value obtained for the activation energy was found to be closest to that for chemical diffusion of zinc into alpha brass. The authors pointed out that this agreement may be fortuitous. These results indicate a need for further anelastic and diffusion experiments on other systems. In particular the Ag-Au system appears ideal since Zener relaxation has been observed”) and there is no concentration limitation because of solubility considerations. In addition extensive diffusion studies have been made by Slifkin et u/.‘~’ and thermodynamic data is readily available. The formal theory of the Zener relaxation, sometimes referred to as stress-induced ordering, has recently been reviewed by Nowick and Seraphimu’). Upon the application of a static elastic stress there appears in addition to the elastic strain, a time de-
VOL. 10, 1962
pendent anelastic strain E”. Upon removal of the applied stress the anelastic strain relaxes according to the equation E” = ei’ exp (-
t/r,).
(2)
This is termed the elastic-after-effect. This same relaxation time can be determined from internal friction studies, i.e. a measure of the phase difference between stress and strain in a dynamic experiment. This phase difference is called the internal friction and is given by
@ is the angle by which the strain lags the stress, 6 is the logarithmic decrement, o the circular frequency and A, is the relaxation strength, a measure of the anelastic properties. Equation (6) of Nowick and Seraphim(e) reduces to equation (3) above for A,< 1. In the systems studied A, s. 0.01 validating the use of equation (3) in the determination of rr. The relaxation time t,. will be compared with the extrapolated mean-time-of-stay of an atom at a given lattice site. This mean-time-of-stay t can be obtained from the diffusion coefficient D. From random walk considerations for a f.c.c. lattice.
D=+g
(4)
where a is the lattice parameter and the correlation factor f of Bardeen and Herring has been included. Assuming that the diffusion process can be represented by an Arrhenius type equation one has
*=- fa” 120
1
~fa” exp (H/RT). 120,
(5)
Relating t and rf by a geometrical factor C, one can see from equation (1) and (6) provided that H= H, that c
=
12Do~o fa"
(6)
Uncertainties in the correlation factor f as well as experimental errors in Do and z,~ make a determination of C quite difficult. However, in addition to calculating C, a useful comparison might be made by showing the temperature of r and r7 on the same plot. This will be done for one concentration of cadmium in silver and two concentrations of gold in silver.
TURNER
AND
EXPERIMENTAL
WILLIAMS: PAPER 9 OF INTERNAL METHOD
(a) Specimen preparation and analysis The Ag-Au alloy specimens were 0.030 in. diameter wires 6-8 in. long. These were rolled and drawn from sections taken from $ in. or 8 in. diameter single crystals destined for other experiments and grown by the Bridgemen technique from 99.999 per cent pure gold and silver. A g-in. length of the drawn wire was sealed in a quartz capillary at a pressure of 6 x 10mgmm Hg and passed through a furnace at a rate of 1 cm per hour at a temperature approximately 25°C above the melting point of the alloy. One half-inch was cut from each end of the large-grained, or single crystal wire resulting. To facilitate handling and mounting in the apparatus the ends of the remaining 8 in. specimens were silver soldered into 1 in. long copper chucks made from $ in. dia. rod. The concentration of the Ag-Au alloys was calculated assuming a linear relationship between density and concentration. The density was determined prior to rolling and at the conclusion of all measurements on a given wire specimen by weighing it in air and in distilled water using a micro-balance. Agreement in concentrations calculated before and after the relaxation time measurements was accepted as sufficient evidence that there was no evaporation loss of either component during the experiment. The alloys of silver with cadmium were analysed before and after the experiment by chemical methods. (b) Torsion pendulum To measure relaxation times in the range 0.1-1.0 set the specimen was mounted as the torsion member of an inverted torsion pendulum of the tyoe descibed by Wert, and surrounded by a tube type furnace in a vacuum chamber capable of an ultimate vacuum of approximately 3 x 1O-5 mm Hg. The natural frequency of vibration was controlled by shifting and changing weights on a steel inertia arm initially excited by solenoids placed near by. After several hours of vacuum annealing in situ below the recrystallization temperature, the decrement measurements were begun. The amplitude decay was observed visually using a 3 m optical lever, with strains less than 5 x 1O-6. The temperature was controlled by a platinum resistance element as a Wheatstone Bridge arm wound non-inductively on the center section of the
FRICTION CONFERENCE
307
furnace core. This control unit was of the off-on type and controlled power to three separate windings of the furnace. The current to each of these was separately manually regulated to provide uniform temperature along the specimen length. The temperature was determined from the average of the readings of three standardized chromel-alumel thermocouples placed at the top, center and bottom of the specimen. The criteria for uniform temperature over the entire length of the specimen was that the top and bottom theromocouples should read to within $ “C of the reading of the middle thermocouple before the decrement could be read at that temperature. Three independent decrement measurements were made at each temperature setting from which a curve of internal frictionvs. temperature was obtained. In spite of a broadening produced by a distribution of relaxation times making the peak approximately 20-30 per cent broader than predicted by theory, the temperature dependence of the maximum occurring in this plot was assumed to obey equation (3) above for Zener relaxation. Hence the relaxation time z, is the reciprocal of the circular frequency w. These torsion pendulum curves were analyzed by two different methods to determine the peak shift in temperature. In one method, the background was subtracted and the peak position located on each curve. In the other method, the identical shape of the leading wing of each curve made it possible to determine the shift in temperature of this leading edge. The two methods gave the same result to within * 1Q”c. (c) Elastic-after-effect
apparatus
For relaxation times between 100 and 5000 set the specimen was placed in the elastic-after-effect apparatus and again given several hours of annealing in situ below the re-crystallization point. The furnace temperature was controlled exactly as described for the torsion pendulum, and the temperature was also determined as previously described. The controller kept a constant temperature to within f 0S”C for runs of 1 hr or less and to within i 1°C for runs of several hours, up to one day, provided the room temperature was maintained constant to f 3°C. The elastic-after-effect runs were made observing change in strain under conditions of constant stress, a stress of zero actually being used. The specimen was twisted approximately 2” by energizing solenoids outside
ACI-A METALLURGlCA,
308
of the vacuum tube, thus exet’ting a torque on a small magnet fastened to a light extension rod attached to the chuck on the lower end of the specimen. An optical lever arrangement made it possible to observe angular motion of the specimen to less than 0.02”. When the specimen had been held in the twisted position using a constant solenoid current for a period corresponding to 3 or 4 times the estimated relaxation time tt, the current was quickly reduced to zero, and the strain relaxation observed and plotted as a function of logarithm of the time, according to equation (2). Following Nowick( the point of inflexion of this curve is used to determine the relaxation time. The points corresponding to these relaxation times and temperatures are then plotted on the same In z, vs. l/T plot made for the torsion pendulum data. Thus the range of the measurement of tl is extended over approximately 6 cycles of 10 and the slope, yielding Q, and the intercept,’ yielding zrO may be obtained with greater precision than is characteristic of values obtained from one method alone.
VOL.
10, 1962
ical torsion pendulum data for Ag-Cd. Typical curves for Ag-Au have been published earlier.(‘) Figure 2 is an Arrhenius plot of the data obtained for 32 at. % Cd in AgO The lower series of points
32 At 96 Cd in Ag
Mean Time --.-Diffusion -
24
32 At % Cd in Ag - Frequency 0.6 cps
16
8
4
I
/
/
/
I 1.800
FIG. 1. Internal
I
I 2.000 1000 T
I
I 2.200
I
I 2.400
friction as a function of temperature solid solution 32% Cd in Ag.
Diffusion
I
1
2.0
I
2.2 y
Stay
Cd into
of
A9 into A9
Aq.Cd Cd
Schoen) ,
2.4
1
2.6
(OK-I)
FIG. 2. Relaxation time as a function of inverse temperature for the solid solution 32% Cd in Ag. Mean times of stay are shown extrapolated from the diffusion data of SchoerW.
was obtained from torsion pendulum data and the higher set from elastic-after-effect data. For comparison the mean time of stay of each component is plotted. These times were obtained from equation (5) using the diffusion data of Schoen(l’). The highest concentration measured by Schoen was 28 per cent; therefore, an extrapolation of 4 per cent was required for comparison. Manning (12)has published correlation factors also obtained from Schoen’s data. Although a different temperature range is involved, reasonable estimates of the correlation factor can be obtained assuming the temperature dependence is not significant for these concentrations. These values along with a least squares analysis of the experimental data are shown in Table 1. The value of C was obtained from equation (6).
20
10 0 -
-
(after
EXPERIMENTAL RESULTS AND DISCUSSION
Typical elastic-after-effect curves are similar to those previously published.(134) Fig. 1 illustrates typ-
-
of of
for the
Figures 3 and 4 give the data obtained with 44 and 68% Au in Ag, respectively. As in Fig. 2, the mean time of stay of each component is plotted for comparison.
These
diffusion
data
including
cor-
TURNER
WILLIAMS:
AND
PAPER 9 OF 1NTERNAL TABLE 1 -___.__
‘---
%.
)
Hr
1 xld_T;
1
35,100 1 34,700 p.,o
)
FRlCTlON
309
CONFERENCE
-_
fCd 1
HA,
/
DoAg
1
fAB
1
c
-
[
0.90
1 OiF
__-_ ,,,,. _____-
j
relation factors were obtained from the work of Slifkin et al.@. Table 2 summarizes the data relating to Figs. 3 and 4.
0.70
1 0.14
ture may be higher than that observed in the Ag-Zn system .03) Quenching experiments will be attempted I04
lOp------
1
I03
103-
102
IO2-
? s
c f
c
C
7; i IO-l5 55 e 44,200/,Z3
IO’
IO’ -
I
100
) Loi
II I
10-l I.5
1’
Meon
-.-.
-
---
1.6
Time
of of
into
Ag
Au
Diffusion
of Au into
Ag
Au
(after
’
1.7
1 1.9
I.8 !y
IO0
Stay
Diffusion
Ag
Shfkm)
l 2.0
Mean -‘--
Time
of
Stay
Diffusion
of
Ag
into
Ag
Au
Diffusion
of Au
into
Ag
Au
(after
Slifkin)
10-I
I 2.1
(OK-l)
FIG. 3. Relaxation time as a function of inverse temperature for the solid solution 44% Au in Ag. Mean times of stay are shown extrapolated from the diffusion data of Slifkin et u1.@‘.
FIG. 4. Relaxation time as a function of inverse temperature for the solid solution 68% Au in Ag. Mean times of stay are shown extrapolated from the diffusion data of Slifkin et uL@).
the elastic-after-effect apparatus to note the relaxation strength as one approaches the critical temperature. Grain boundary activation energies were initially identified with those obtained from diffusion. However, in this investigation grain boundary relaxation studies on 42 Au in Ag yield a relaxation time r, 1CP4*0 exp (4,93O/RT). This value of 49.3 kcal/mole fits the Marx-Wert(14) plot and is considerably greater than the diffusion with
Li and Nowick(13) have discussed the critical temperature for stress-induced ordering T, given by the equation To
&f=.._. T-
T,
(7)
Where d, is the relaxation strength shown in equation (3), To is a constant and T the temperature. Rough measurements of the relaxation strength in the silver-gold system indicated that the critical tempera-
ACTA METALLURGICA,
310
values HAB= 43.5 kcal/mole and H_,, = 45.5 kcal/ mole obtained by Slifkin et al.@). Finally, it should be noted that doubt arose shortly after publication’71 with regard to the accuracy of the activation energy reported for a 58.5% Au in Ag alloy because of improperly corrected thermocouple readings in the raw data. Although the discrepancy was small, a check run was indicated. Unfortunately, this specimen had been cold worked at the conclusion of the original measurements, and hence, required an anneal. During this anneal silver was inadvertently evaporated from the surface, resulting in a the~ally etched surface, rendering a bona fide repetition of the earlier run impossible. Nevertheless, the measurements were made. The results yielded the activation energy and r,, values compatible with those of an 80% Au in Ag alloy. The inference is not only that silver had indeed evaporated, but that the observed relaxation was characteristic of the gold rich layer near the surface, as would be expected. However, preliminary studies indicate that this surface layer in which the major portion of the relaxation occurs is surprisingly thin.
VOL. 10, 1962
have been shown for all systems. Values quoted in Tables 1 and 2 for the other concentrations were obtained by interpolation or extrapolation. A comparison between tr and r based on Figs. 2-4 appears to be better than that indicated by the values of C given in the tables because of uncertainties in z~, and which were in calculating
CONCLUSIONS I
Although uncertainty exists regarding what is actually being measured in a Zener relaxation experiment, even the origin of the effect being uncertain, though in the Ag-Au system, it has been attributed previously(‘) to a difference in compressibility resulting in a net size difference-it appears on the basis of the data presented that one is measuring the diffusion of the faster moving component. Figs, 3-5 show what must be regarded as surprising agreement between tl and r the mean time of stay of the fast diffusor, even though the difference in activation energies is at the limit of the combined experimental error, Therefore, for these systems over the range of temperatures studied, one can use relaxation data to obtain the mean time of stay of the faster moving component. These relaxation times can then be used in equation (5) to obtain the diffusion coefficient. A direct comparison between diffusion coefficients obtained in this manner and those measured by Shfkin et &.(*) is shown in Fig. 5. It is unfortunate that the 68% Au in Ag specimen happened to be the only concentration duplicated for both anelastic and diffusion experiments. Otherwise such a plot would
BOO
I
I
1.000 1.200
I
I
I.400 1.600 1.800 2.t 10
!!F°K-‘) FIG. 5. Diffusion coefficient of silver diffusing into the solid
solution 68% Au in Ag as a function of inverse temperature. Diffusion coefficients are calculated from internal friction data, assuming the relaxation time to be equal to the mean time of stay.
Tomizuka eb &.(16)have recently studied self-diffusion of 15% Zn in Ag, reporting an activation energy for the faster moving component, zinc, agreeing with the value obtained by Nowickc4) for the same concentration. These data, from four systems, lead one to believe that anelastic methods can be used to extend diffusion measurements in substitutional solid solutions to relatively low temperatures.
We would like to record our appreciation to Professor L. Slifkin for helpful discussions, and for supplying Ag-Au crystals from which some of the wire specimens were drawn.
TURNER
AND
WILLIAMS: PAPER 9 OF INTERNAL FRICTION CONFERENCE
REFERENCES
1. 2. 3. 4. 5. 6. 7. 8.
C. A. WERT, Phys. Rev. 79, 601 (1950). C. ZENER, Phys. Rev. 71, 34 (1947). A. D. LECLAIRE, Phil. Mag. 42, 673 (1951). A. S. NOWICK, Phys. Rev. 88, 925 (1952). D. LAZARUSand C. TOMIZUKA,Phys. Rev. 103,1155 (1956). J. HINO, C. TOMIZUKAand C. WERT, Aclu Met. 5,41 (1957). T. J. TURNERand G. P. WILLIAMS,Acta Met. 8, 891 (1960). L. SLIFKIN, W. MALLARD, R. BASSand A. GARDNER, to be published.
311
9. A. S. NOWICK and D. P. SERAPHIM,Actu Met. 9, 40 (1961). 10. J. BARDEENand C. HERRING,Imperfections in Nearly Perfect Crystals (Ed. by W. SHOCKLEY)p. 261. Wiley, New York (1952). 11. A. SCHOEN,Thesis, Univ. of Illinois (1958). 12. J. R. MANNING, Phys. Rev. 116, 69 (1959). 13. C. Y. LI and A. S. NOWICK, Actu Met. 9, 49 (1961). 14. C. WERT and J. MARX, Acta Met. 1, 113 (1953). 15. C. TOMIZUKA,R. ROBERTSand R. DICKERSON,Bull. Amer. Phys. Sot. 6, 172 (1961); and private communication.
Note added in proof Subsequent data indicate that HR is actually greater than HAB at the lower concentrations, although relaxation time is still clearly identified with the extrapolated mean time of stay of the faster moving component.
the