Electrical resistivity of an incrementally-fatigued lead alloy

Electrical resistivity of an incrementally-fatigued lead alloy

Scripta METALLURGICA et M A T E R I A L I A Vol. 25, pp. 1 7 1 3 - 1 7 1 7 , 1991 Printed in t h e U . S . A . Pergamon Press plc All rights reserv...

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Scripta METALLURGICA et M A T E R I A L I A

Vol.

25, pp. 1 7 1 3 - 1 7 1 7 , 1991 Printed in t h e U . S . A .

Pergamon Press plc All rights reserved

ELECTRICAL RESISTIVITY OF AN I N ~ Y - F A T I G U E D LEAD ALLOY

Larry Lawson and M. Meshii Department of Materials Science, Northwestern University, Evanston, Illinois 60208 (Received April 26, 1 9 9 1 ) (Revised M a y 7, 1 9 9 1 )

Introduction From time to time the effect of plastic deformation on the resistivity of metals has been studied. Not long ago, Basinski and Basinski[1] pointed out the ambiguity of ascribing the cause of the resistivity increase seen in fatigue to a specific source by showing that, in the case of copper single crystals fatigued at 4.2 K, both a dislocation and a vacancy hypothesis gave the same resistivity result. Other researchers have also studied the contribution to resistivity produced by fatigue[2-5] and attempted to separate point defect from dislocation effects. In most cases, a previously unstrained metal was subjected to a large amount of plastic strain at a very low temperature. As a consequence, a dislocation structure evolved, amid immobile point defects, which was not that seen in ordinary fatigue; such conditions could give rise to ambiguities. But, evidence exists to show that the two sources of fatigue resistivity are not necessarily indistinguishable. VanBueren used a magnetoresistivity technique to estimate the dislocation-produced resistivity apart from point defect-produced resistivity in a monotonic tension test performed on copper[6]. He obtained a comparatively low value for the fraction of resistivity due to dislocations. He also found that mag,~toresistivity did not anneal out in stage V as he had expected were dislocations its cause. Polak found evidence corroborating the results of VanBueren in that all resistivity added by room-temperature fatigue did not anneal, in copper, I/, the stage V range[7]. l~e observed saturation in fatigue induced resistivity raises questions about what, if anything, is generated when a metal having a fully-developed dislocation structure is fatigued? It is probable that an equilibrium is reached between point defects and dislocations formed and those lost through interactions with other, moving, dislocations. But, there is an absence of data for the resistivity production and annealing behavior of small incremental amounts of fatigue under conditions where the dislocation structure is relatively stable. Such data is needed in order give experimental support to this commonly-held idea of equilibrium. The purpose of this experlmexlt was to fill this gap by exploring the annealing behavior of a pre-fatigued f.c.c metal under conditions of marginal additional fatigue. The need for data on excess vacancy production during the fatigue of solder afforded an opportunity to do this work. Material end Techni uqu~ The material used was a c~,mercial solder alloy, 96.SPb-3.5Sn. The alloy was first cast into small ingots, homogenized by armealing for i00 hours at 175°C, then sawed into bars, swaged end drawn into round wire having a di~eter of 0.862 ram. The overall reduction was about 12:1. The wire was then jacketed in a thermally-matched silicone polymer, General Electric GE372. After curing the insulation, the wires were ant~aled in air at 180°C for 24 hours and aged for I0 days to stabilize the microstructure. The wire was then assembled into the sample shown in Figure I. An outer casing made of zinc in order to match the thermal expansion coefficient of the insulated wire allows the wire to be compressed repeatedly without buckling. Based on cyclic hardening and crack growth data [8], samples were pre-fatigued for 50 cycles at a strain range of IX it, order to generate a reasonably stable dislocation structure without cracking. The initial resistence was adjusted at this point by adjusting the final strain. The testing apparatus is described in Figure Z. Strain was applied manually using the differential screw end a dial gauge calibrated in 2.54~4 m. increments. A strain rate of 2.25e-4/s was used for all tests. Cooling and subsequent annealing were performed at zero load by means of a disconnecting pin and a

1713 0036-9748/91 $ 3 . 0 0 + .00 Copyright (c) 1 9 9 1 P e r g a m o n Press

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counterbalance which allowed free movement. Cooling to 77K was accomplished by immersing the sample in liquid nitrogen. Annealing was performed by lowering the cryostat and heating the sample with a heating mantle. The temperature was allowed to ramp to a preset value and maintained there by a controller. The instantaneous temperature of the sample was digitally recorded as a function of time and the entire thermal history was subsequently analyzed and fitted to a first order kinetic approximation. The range of annealing temperatures chosen was in Stage IV and the lower part of Stage V. These temperatures are high enough to avoid the large changes in activation energy seen at lower temperatures, eg. see Meshii end Kauffman [9]. Resistance was measured in liquid nitrogen using a ten-ampere current. The effect of Joule heating was calculated using the methods of Carslaw and Jaeger[lO]. It was found that the thermal time constant was short in comparison to the measurement timer and that the small offset introduced would cancel when the difference of two readings was taken. Tensile tests on the fatigued materlal at room temperat~e and 77K were compared using s scaled-up s-,,ple having a cross section of nearly 1 square centimeter. These data indicated that little change took place in the dislocation structure formed at room temperature when given the small amount of additional fatigue at 77K. This conclusion is based on these observatlons: First, the overall form of the tensile curves at both temperatures is quite similar; in both cases, the amount of elastic strain is small. Second, the change in the yield and flow stresses~ a factor of 1.8p takes place instantaneously upon cooling. No additional plastic strain appears to he needed. Consequently, it is unlikely that this increase results from e change in dislocation density or any significant change in dislocation structure. Third, the strain Im£dening exponent changes very little. At room temperature it is 0.22. At 77K it is 0.24. Commonly, if the dlslocation structure evolves at a low temperature, the strain hardening exponent is significantly higher than if the dislocation structure develops at a high temperature. This suggests that~ at least for the few cycles involved, the room temperature structure persists at liquid nitrogen temperature. Results and Discussion Amlealing data, derived from three samples are shown in Table 1.

TABLE l Resistance Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Test Treatment

Temp. Current

# .

.

°C .

2 2 4 4 4 5 5 5 5 6 6 6 6

.

.

.

.

.

.

.

.

.

.

.

.

none none none I0 cycles ammal 30°C none I0 cycles anneal -lO°C anneal 23°C none 10 cycles anneal -10°C anneal 23°C

.

.

.

.

.

.

24 -194 -194 -194 -194 -194 -194 -194 -194 -194 -194 -194 -194

Potential

Thermal ~

mV

mV

A .

.

.

.

.

.

.

.

.

.

0.0900 10.008 I0.000 I0.000 i0.000 10.000 I0.000 I0.000 I0.000 10.000 10.000 I0.000 I0.000

.

.

.

.

.

.

.

.

.

.

0.400 12.170 11.942 12.010 11.947 11.942 11.973 11.969 11.952 11.947 12.038 11.959 11.949

.

.

.

.

.

.

.

.

.

Resistance

mOhms .

.

0 -0.440 -0.443 -0.443 -0.443 -0.443 -0.443 -0.443 -0.443 -0.443 -0.443 -0.443 -0.443

.

.

.

.

.

.

.

.

.

.

.

.

.

.

4.37 1.260 1.2385 1.2453 1.2390 1.2385 1.2416 1.2412 1.2395 1.2390 1.2481 1.2400 1.2392

~he three test samples received a total of five anneals. Additionally, another sample given similar pre-fatigue was used to obtain the room-temperature resistivity by the Van der Pauw method[ll]. The thermal emf is that present when zero current is flowing and is subtracted from the measured potential when resistance is detemlned. It is seen that, within the accuracy of measurement, all the fatlgue-lnduced resistance anneals out. ~he first order contribution to annealing kinetics is the largest, as seen by Helgelend [4] emong others. This is expressed in equations (I) and (2) which were fitted to the data as follows. A numerical least-squares fit was performed for each whole number value of 8ctlvation energy between 30 and 60 KJ/M using the digitally recorded temperature vs. time records. For each fit, the root mean square of the fractional error, expressed as seconds of error/second of annealing time at 20"C,was calculated. The fitted values are those which ml, lmlzed this error.

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Q

In equation (i) and following, rho is the resistivity, proportional to the sample resistance, k 2 is an annealing constant, Q is the activation energy, k is Boltzmann's constant end T is the absolute temperature. The annealing constant is related to the decay time constant, tau, the time required for the concentration to decay to 1/e times its original value, by equation (2). From the fit an activation energy of 42K.I/mol and a decay time constant of 520 seconds adjusted to 25°C were obtained. Figure 3 illustrates this result. Figure 3 shows a line fitted to the data points of Table I. The data points are plotted with error boxes. The horizontal dimensions represent uncertainties due to the accuracy of time and temperature measurement, equivalent to ±3K, while those in the vertical direction represent uncertainties in the resistance measurement. ~here remains a considerable amount of scatter. This scatter is not unexpected and has been discussed by Pelffer in terms of variations in the initial dislocation structure[12]. The sharpness of the minlmumin rms. error obtained for the activation energy m a y b e expressed in terms of the increase in magnitude of error per KJ/mol deviation from the error-minlmlzing value, 42. That amount is O.051mollKJ. The rms. fractional error itself is 0.563 st 42KJ/mol. Additional measurements were made in order to compare the rate of resistivity production with a vacancy formation hypothesis. The modulus at 77K was estimated from tests at 25°C and in liquid nitrogen using the ratio of the slopes and a previous value obtained for this alloy byVaynman[13]. From this and from associated peak load data, the total plastic strain per cycle in the tests at 77K was estimated to be I.az. The room temperature resistivity of the fatigued alloy was found to be 23 ±i microOhm-om at 24°C. From Table I, the average change in resistivity is found to be 1.84E-3 microOhm-om/~straln. Reale has calculated a value of 5.82 microOhm-cm/Zsite fraction[14] for the resistivity due to vacancies in lead. Taking the ratio of these gives 3.2E-6 site fraction/~strain. This is quite close to the estimate of 4E-6 site fraction/Zstrain often quoted for f.c.c, metals[15] and supports the hypothesis that the primary effect of a marginal increment of fatigue strain to an already established room-temperature dislocation structure is the generation of vacancies. The value obtained for the activation energy of resistivity annealing, 42KJ/mol, is close to that for the migration of vacancies, 41.4 KJ/mol, as determined by Schroeder and Schilling[15] for lead. This further enhances the possibility that the increased resistivity is due to vacancies. Annealing stages II through V were studied by schroeder and Schilling. (Stage I was not discovered at the time of their article.) Vacancies become mobile in stage III while vacancy-dlslocatlon complexes anneal in stage V. Since room temperature is located in stage V for lead, it stands by definition that stable dislocation structures formed in room temperature fatigue do not anneal out in stage V. The heat of vacancy formation is much higher than that for migration. Rice-Evans give this as 52 KJ/mol[17]. Consequently, thermally generated vacancies contributed less than 1/700 of the resitance produced by strain. Although the dislocation density of the samples was not measured, the density of sinks may he inferred from the decay time constant using the Lomer and Cottrell approximation for the number of jumps a point defect makes in its llfetlme[17]. This is given by equation (3).

where tan is the decay time at temperature, T and nu is the base jmnp frequency, approximately the DeBye frequency. Evaluating this expression gives n, the number of jumps, as 2e8. The average density of atoms per square meter in lead is le19. Consequently there are about 5elO sinks per square meter. Not all sinks are dislocations and stress fields around dislocations alter the mobility of vacancies near them. Nevertheless, a dislocation density on the order of IElOm/sq.m. seems reasonable for a fatigued f.c.c. metal. Summary and Conclusion The annealing behavior of resistivity induced by incremental fatigue strain at 77K in a face centered cubic lead alloy prefatlgued at room temperature seems consistent with an hypothesis that the incremental resisEance is largely the result of vacancies. The resistivity in these experiments totally annealed out in the lower part of stage V with an activation energy essentially equal to that of vacancy migration. This would not be expected if resistivity resulted from an increase in the density of dislocations. Also, the decay time constant was reasonable for a process in which excess vacancies are consttmed at dislocation sinks. Taken together, these observations suggest that the dislocation structure formed by cold work and

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fatigue at room temperature was not significantly altered or increased byadditional fatigue at 77K. Rather, fatigue produced point defects, predominately vacancies, which were consumed by sinks probably dislocations. Acknowled~eunt The author thanks IBM Corporation for support. References I. Z.S. Hesinsky and S.J. Heslnsky, Acta. Metall., 37, 3275 (1989). 2. E.W. Johnson and H.H. Johnson, Trans. Metall. S,c. A.I.M.E., 233, 1333 (1965). 3. H. Rosanberg, Vacancies and Other Point Defects in Metals and Alloys, Institute of Metals Report #23, Institute of Metals, London, 1958. 4. O. Helgeland, Trans. A.I.M.E., 239, 2001 (1967). 5. J.B. Vander Sande, PhD. Dissertation, Northwestern University, Evanston, IL., 1971. 6. H. Van Busren, Philips Rea. Rap. 12, 190 (1957). 7. J. Polak, Czech. J. Phys., B19, 315 (1969). 8. L. Lawson, M.E. Fine and D. Jeannotte, Met. Trans. in press. 9. M. Msehli and J.W. Kauffman, Acta Metal1., 8, 815 (1960). I0. H. Carslaw and J. Jaeger, Conduction of Heat in Solids, Clarendon Press, Oxford, 1986. II. L. Van der Pauw, Phillps Rea. Rap. 13, I (1958). 12. H. Peiffer, J. Appl. Phys. 43, 298 (1963). 13. S. Vaynman, Ph.D. Dissertation, Northwestern University, Evanston, IL., 1987. 14. C. Reale, Phys. Lett. 2, 268 (1962). and G. Dienes, Point Defects in Metals, Gordon and Breach, New York, NY., 1963. 15. A. ~ - k 16. H. Schroeder and W. Schilling, Radiation Effects, 30, 243 (1973), 17. P. Rice-Evans, I. Chaglar and F. Khangl, Philos. Meg. A, 38, 54 (1978). 18. M. Lomer and A. Cottre11, Philos. Mag. 29, 738 (1958).

currentlead silico~

:oppertab f potentiallead zinc housing

polyimidefilm

/

FIG. I. Fatigue-resistivity sample. ~he copper tabs, used to grip the sample, are covered with polyimide film to electrically insulate them from the fatigue machine The zinc housing is perforated and has a shallow acme thread cut on its interior to allow for expansion of the silicone during compression.

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dlfferenttal screw

°'°

ryostat

I

[

!

FIG. 2. Fatigue apparatus. To avoid the jitter of servo-hydraulic control, the sample is deformed mammlly using the handle shown attached to the d/fferential screw. ~ sample is shown ~,m~rsed in liquid nitrogen in the cryostat. ~hermocouples attached to the sample monitor its temperature during annealing. I J m. u. J,

,. "-~

~

r~

-o ~0.1

0i,

500

1000

1500

2000

Effective t i m e at 20 ° C (S)

FIG. 3. Fraction of fatigue induced resistivity r~m~i~_ing after annealing for a 20°C time equivalent based on flrst-order kinetics and a constant act/ration activation energy of 4ZEJ/mol.Each point represents a separate test. ~ e scatter is believed to be caused by variations in the initial dislocation structure rather ~,n a departure from first-order kinetics. in the Inlt/al dislocation structure