Dynamic Recrystallization (DRX) - Springer Link

Journal of ELECTRONIC MATERIALS, Vol. 38, No. 9, 2009

Regular Issue Paper

DOI: 10.1007/s11664-009-0882-4 2009 TMS

Dynamic Recrystallization (DRX) as the Mechanism for Sn Whisker Development. Part II: Experimental Study P.T. VIANCO1,2 and J.A. REJENT1 1.—Sandia National Laboratories, Albuquerque, NM, USA. 2.—e-mail: [email protected]

In Part I of this study, a dynamic recrystallization (DRX) model was proposed to describe the development of metal whiskers. A diffusion-assisted, dislocation-based mechanism would support the DRX steps of grain initiation (refinement) and grain growth. This, Part II, describes experiments investigating the time-dependent deformation (creep) of Sn under temperature conditions (0C, 25C, 50C, 75C, and 100C) and stresses (1 MPa, 2 MPa, 5 MPa, and 10 MPa) that are commensurate with Sn whisker development, in order to parameterize the DRX process. The samples, which had columnar grains oriented perpendicular to the stress axis similar to their morphology in Sn coatings but of larger size, were tested in the as-fabricated condition as well as after 24 h annealing treatments at 150C or 200C. The steady-state creep behavior fell into two categories: low (107 s1). The apparent activation energy (DH) at low strain rates was 8 ± 9 kJ/mol for the as-fabricated condition, indicating that an anomalously or ultrafast diffusion mass transport mechanism assisted deformation. Under the high strain rates, the DH was 65 ± 6 kJ/mol (as-fabricated). The rate kinetics were not altered significantly by the annealing treatments. The critical strain (ec) and Zener–Hollomon parameter (Z) confirmed that these stresses and temperatures were nearly capable of causing cyclic DRX in the Sn creep samples, but would certainly do so in Sn coatings with the smaller grain size. The effects of the annealing treatments, coupled with the DRX model, indicate the need to maximize the creep strain rate during stress relaxation so as to avoid conditions that would favor whisker growth. This study provides a quantitative methodology for predicting the likelihood of whisker growth based upon the coating stress, grain size, temperature, and the similarity assumption of creep strain. Key words: Sn whiskers, dynamic recrystallization, creep

INTRODUCTION In Part I of this study, a model was proposed that attributed metal whisker growth to dynamic recrystallization (DRX). Creep deformation, which resulted from the compressive stress, supports cyclic DRX and, in particular, the grain growth step. The characteristics of DRX fit very nicely with the qualitative behaviors of whisker growth. Part I examined prior studies on diffusion processes and (Received March 16, 2009; accepted June 17, 2009; published online July 7, 2009)

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time-dependent deformation in Sn and other metal and alloy systems to identify a mass transport mechanism. It was concluded that the mechanism was anomalously or ultrafast diffusion assisting dislocation motion. It could not be ruled out that anomalously fast diffusion is the mechanism by itself, since very little is known of the details of this phenomenon. The objective of the present, Part II, investigation was to obtain the creep strain and creep rate kinetics properties for Sn under temperature and compressive stresses commensurate with whisker development. Then, those creep properties were

Dynamic Recrystallization (DRX) as the Mechanism for Sn Whisker Development. Part II: Experimental Study

further investigated with respect to the proposed DRX model. The creep experiments were performed on bulk Sn samples in order to control the applied stress levels. The test samples had a grain structure similar to that of Sn coatings, that is, columnar grain boundaries across the test volume thickness and oriented perpendicular to the stress axis. The difference was that the grain size was considerably larger than those of coatings. The present experiments were not intended to cause DRX, explicitly, because the DRX would interfere with the rate kinetics analyses, given the relatively small strains. Although the strains and strain rates were capable of initiating cyclic DRX in Sn, the relatively large grain size would preempt any large-scale cyclic DRX from continuing under these test parameters. EXPERIMENTAL PROCEDURES Materials The test samples were fabricated from certified 99.99% pure Sn by casting into cylinders having nominal dimensions of 10 mm diameter and 20 mm length. Density measurements confirmed the absence of porosity in the samples. The ends of the cylinders were machined for parallelism. The center of each sample was drilled to create a hollow cylinder with a wall thickness of 2 mm. Although the cylinder wall thickness did not conform to the ASTM E9-89A specification, the strains were sufficiently small such that buckling did not take place.1 The grain structure in the cylinder walls is shown in Fig. 1. The grain boundaries were oriented largely perpendicular to the direction of the applied compressive stress so as to replicate the relationship between the columnar structure and the compressive stress direction observed in Sn coatings. Because a consistent correlation remains to be determined between Sn coating texture and whisker formation, it was not deemed critical to document the texture of the grains in the test samples at this juncture. The grain widths (dimension parallel to

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the cylinder axis) varied, but were 350 lm as determined by the linear intercept method. The samples were tested in the as-fabricated condition or after one of two 24 h isothermal annealing treatments at 125C or 150C. The treatments did not alter the grain size or overall structure of the samples. Test Parameters Creep tests were performed on a servohydraulic frame under constant-load control. The test temperatures were 0C, 25C, 50C, 75C, and 100C and the stresses were 1 MPa, 2 MPa, 5 MPa, and 10 MPa. Duplicate samples were tested per condition. The test procedure and data analysis details are described in Ref. 2. Tests were halted under either one of the following two criteria: (a) the test time exceeded 150,000 s or (b) the strain exceeded 0.10 to avoid buckling. The sinh law (Eq. 1) was the primary expression used to describe the secondary or steady-state creep rate de=dtmin ¼ A sinhn ðarÞexpðDH=RT Þ:

(1)

In Eq. 1, A is a constant (s1), n is the sinh exponent, a is the stress coefficient (MPa1); r is the applied stress (MPa), DH is the apparent activation energy, R is the universal gas constant (8.314 kJ/mol K), and T is temperature (K). The parameters A, n, and DH were determined from a multivariable, linear regression analysis performed on the logarithm of Eq. 1. The dependent variable was ln(de/dtmin) while ln[sinh(ar)] and 1/T were the independent variables. The error terms on the parameter coefficients were based on a 95% confidence interval. This confidence interval is more stringent than the traditional 63% value to assure significance of the observed trends. The optimized value of the parameter a had an error of ±0.001 as determined by maximizing the square of the correlation coefficient (R2). Reference will also be made to the power-law equation for steady-state creep rate as a function of stress and temperature shown below: de=dtmin ¼ A0 r p expðDH=RT Þ:

(2)

This expression was used to obtain additional information on the creep behavior because the stress exponent, p, has been associated with likely dislocation mechanisms in metals and alloys. The microstructures of selected tested samples were documented by optical microscopy (see, e.g., Fig. 1) and scanning electron microscopy (SEM). RESULTS AND DISCUSSION Fig. 1. Optical micrograph showing the nominal grain structure of the hollow cylinder test sample. This sample was in the as-fabricated condition. The arrows indicate the direction of the applied compressive stress.

Creep Strain Thijssen determined that the initiation of DRX in Sn requires a minimum strain of 0.21 during

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stress–strain tests having 105 s1 to 104 s1 strain rates.3 The test temperatures were 140C and 190C, which are high for whisker growth. Similar strains were required to initiate DRX during the stress–strain tests of other materials.4,5 The literature was examined for DRX that was initiated by creep in order to compare with tests in the current study. Gifkins examined the relationship between creep and recrystallization strain in pure Pb at 20C (Th = 0.49) and stresses of 1.8 MPa to 10 MPa.6 Recrystallization was marked by bursts in the creep curves after reaching strains of 0.02 to 0.10. Similar results were observed for Pb by other authors cited by Richardson et al.7 In that latter work, the creep–DRX relationship was also investigated for very pure Ni at 965C (homologous temperature, Th, equal to 0.71) and stress of 12 MPa to 20 MPa. The creep strains at which DRX was observed, were in the range of 0.02 to 0.07, which are similar to those of the lower-melting-temperature Pb in Ref. 6. The grain sizes were approximately 180 lm for both materials. Therefore, DRX during creep required strains smaller by an order of magnitude than those needed under the faster strain rates of stress–strain tests. Figure 2 shows the inelastic strain values for samples in the as-fabricated condition after 10,000 s (2.78 h) of testing as a function of nominal applied stress. This time duration was selected because it included the most data across stress–temperature conditions. The dotted arrows indicate that the strain exceeded the 0.10 limit at the next stress prior to reaching 10,000 s. A survey of the creep curves representing the lowest temperatures and

Fig. 2. Graph of inelastic creep strain at 10,000 s as a function of nominal applied stress for each of the test temperatures. The specimens were in the as-fabricated condition. The dotted arrow and ‘‘>0.10’’ indicates that, at the higher stresses, the sample exceeded a strain of 0.10 in less than 10,000 s, at which point the particular tests were halted.

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stresses confirmed that little additional deformation had occurred between 10,000 s and 150,000 s. The inelastic strains in Fig. 2 appear to fall into one of two regimes demarked by 0.01. Most of the strain values were less than 0.01. The remaining strains significantly exceeded 0.01, some reaching values as high as 0.09. When the data in Fig. 2 were compared with the literature results cited above, the majority of stain, being less than 0.01, would not likely have initiated DRX. The strain–time curves were examined for fluctuations that would indicate the occurrence of cyclic DRX. The total creep strain curve in Fig. 3a represents a case of inelastic strain less than 0.01 (as-fabricated, 25C, 2 MPa). In general, fluctuations that exceed the measurement error were observed; yet, they were very small when compared with the amplitudes of 0.02 observed during DRX creep strain bursts in Pb.6,7 The total creep strain curve in Fig. 3b represents the high inelastic strain regime (as-fabricated, 100C, 5 MPa). There were

Fig. 3. Total creep strain versus time plot for one of two samples tested in the as-fabricated condition at test parameters of (a) 25C, 2 MPa and (b) 100C, 5 MPa.


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no significant fluctuations in that curve to indicate the DRX process. Similar observations pertained to creep curves from the annealed samples. Therefore, if cyclic DRX had occurred in any of the test specimens, it did so on a negligible scale. Metallographic cross-sections were made of samples that were tested under different stress, temperature, and annealing parameters. A small amount of grain boundary motion was observed, together with some additional twinning. The SEM images of the sample surfaces identified a few, isolated regions of grain refinement. However, these observations confirmed the absence of any largescale cyclic DRX in the microstructures. The results of other literature studies were used to determine the ‘‘proximity’’ of the current creep tests to initiating cyclic DRX. Toshihide et al. observed DRX strain bursts in polycrystalline Sn during compression creep tests at 120C and stresses of 3.5 MPa to 5.5 MPa.8 The grain size was 1480 lm. The higher temperature initiated DRX, implying that the current maximum temperature of 100C and stresses up to 5 MPa represented a lower limit below which DRX is not likely to take place. Darby and Ashby created a plot of log(s/G) versus log(D/b) using DRX data normalized from a number of metals (Cu, Ni, and Fe) and nonmetallic materials.9 The following parameters, which represented the present creep samples, were mapped in that plot: (a) D (grain size): 200 lm to 500 lm (present study) (b) G (shear modulus): E/2(1 + m), where E (MPa) = 76,087 109T, T(K),10 (c) m (Poisson’s ration): 0.33 (assumed) (d) b (Burgers vector): 0.32 nm to 0.66 nm, depending on the slip system11 The parameter space created by the ranges of D, b, and T (0C to 100C) values placed the present creep sample just outside the window predicted to initiate DRX. In fact, the higher temperature of 120C used by Toshihide would have decreased G sufficiently to push the current tests into the DRX envelope. Therefore, the cited literature confirmed that the present creep tests were likely just outside the window for large-scale DRX in the Sn hollow cylinders. The same creep strain analysis was performed on samples exposed to the 24 h annealing treatment at either 150C or 200C prior to testing. The curves are shown in Fig. 4a and b, respectively. The same two strain regimes were observed. At temperatures of 50C, 75C, and 100C, the stress levels at which the strain ‘‘broke-away’’ to very high values were lower, implying that the annealing treatments reduced the creep resistance of Sn at these temperatures. A different trend was observed at 0C and 25C. The creep strains exhibited relatively small increases at 5 MPa and only modest increases at 10 MPa between the as-fabricated and 150C annealed conditions. The strain jump still occurred

Fig. 4. Graph of inelastic strain at 10,000 s as a function of applied stress for each of the test temperatures and the two annealing conditions: (a) 150C, 24 h and (b) 200C, 24 h.

between 5 MPa and 10 MPa. Between the 150C and 200C annealing treatments, there was a modest strain increase at 5 MPa; yet, the large strain jump remained between 5 MPa and 10 MPa. Interestingly, the strain values at 10 MPa after the 200C annealing treatment were less than those measured after the 150C treatment. To summarize, the annealing treatments reduced the creep resistance of the Sn samples for the 50C, 75C, and 100C test temperatures, as determined by the stress at which the strain jumped above 0.01. The creep strains at 0C and 25C test temperatures were generally less sensitive to annealing treatments, having only small to moderate increases at 5 MPa. At 10 MPa, those creep strains obtained at 0C and 25C exhibited a nonmonotonic trend between the three sample conditions. As noted earlier, the annealing treatments did not noticeably affect grain structure. Therefore, the effects of the

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annealing treatment as observed by comparing Figs. 2 and 4 were due to changes to primarily the defect structures within the specimens, perhaps by a small degree of static recovery. The creep strain behaviors in Figs. 2 and 4 had the characteristics of a threshold stress phenomenon. In Fig. 2, an apparent threshold stress would be hypothesized at 2 MPa to 5 MPa for 100C and 5 MPa to 10 MPa for the lower temperatures (0C to 75C). However, these stress values are too high for threshold stresses based upon internal friction or dislocation pinning mechanisms.12,13 Threshold stress values are 106 to 105 times the shear modulus for polycrystalline materials, which would be 0.02 MPa to 0.2 MPa for Sn. Instead, the threshold-stress-like phenomenon in Figs. 2 and 4 more likely represent a mechanism change to generalized deformation behavior. The range of stress (5 MPa to 10 MPa) in which the jump in creep strain took place (Figs. 2 and 4) encompasses 8 MPa, which has been suggested as a minimum compressive stress to grow Sn whiskers.14 According to the DRX model, however, lower strains could also result in cyclic DRX because strain rate, temperature, and grain size are also critical factors. Therefore, it is not possible at this time to draw that strong correlation between the 5 MPa and 10 MPa behavior in Figs. 2 or 4 and the 8 MPa benchmark noted in the literature.

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Fig. 5. Plot of steady-state strain rate as a function of stress for individual test samples in the as-fabricated condition. The dashed line demarks the 107 s1 strain rate benchmark.

Steady-State Creep Rate Kinetics The steady-state creep rate was measured from the strain–time curves. Multivariable linear regression analyses determined that the optimum value of a was 1.000 ± 0.001 for all test conditions. Initially, all of the strain rate data were included in the regression analysis. Then, the analysis considered the two effects of (a) minimum strain rate regime and (b) temperature regime. The minimum strain rate is plotted in Fig. 5 as a function of true stress for as-fabricated samples. There were two regimes, demarked by 107 s1. The same benchmark applied to samples tested after exposure to either annealing treatment. The temperature effect was addressed by performing the multivariable analyses on data according to the following temperature regimes: (a) (b) (c) (d) (e)

low temperatures (0C to 50C) low–medium temperatures (0C to 75C) entire temperature range (0C to 100C) medium–high temperatures (50C to 100C) high temperatures (75C to 100C)

The regression analyses examined the apparent activation energy, DH, which is plotted in Fig. 6 as a function of temperature regime and annealing treatment when all strain rate data were used in these regression analyses. The one-half error bars (95% confidence interval) were placed on the as-fabricated and 200C (24 h) annealing data; the

Fig. 6. Apparent activation energy, DH, as a function of temperature regime and annealing treatment for all strain rates. The half-error bars are 95% confidence intervals and were added only to the as-fabricated and 200C annealed data.

error bars were similar for test samples annealed at 150C (24 h). There was no statistically significant effect of annealing treatment. The DH values were in the range of 33 kJ/mol to 35 kJ/mol for the entire temperature range of 0C to 100C and all three sample conditions. The apparent activation energies, which are less than 0.4 to 0.5 of the bulk diffusion value of Sn (100 kJ/mol to 110 kJ/mol) indicate that an anomalously or ultrafast mass transport mechanism supported creep deformation. Interestingly, it was such unexpectedly faster rate kinetics that similarly caused Harper and Dorn as well as other investigators to abandon the Nabarro– Herring stress-directed (bulk) diffusion mechanism,


and later the grain boundary Coble mechanism, as responsible for creep at low stresses and high temperatures.15–17 The mean value of 34 kJ/mol observed for the as-fabricated condition was compared to DH values published in other creep studies of Sn under like test conditions.18–22 Similar DH values were observed for the same conditions in single-crystal and polycrystalline Sn. A value of 34 kJ/mol was measured by Pawlicki using impression creep tests of Sn.23 The particular attribute of these experiments was that the author annealed the polycrystalline sample so that the creep test was effectively performed on single grains. In a separate study, dislocation slip was observed in individual grains in impression creep experiments.24 Several authors proposed that these low DH values, which certainly represent anomalously fast mass transport, indicate dislocation motion, but one that was likely assisted by this anomalously fast diffusion. The possible processes are dislocation climb or dislocation motion that is assisted by dislocation core diffusion, with the latter providing the anomalously fast contribution. The 95% confidence interval error bars in Fig. 6 had a relatively consistent magnitude except for the 75C to 100C temperature regime. The large error bars could not be attributed to the reduced number of data points used in the regression analysis. In fact, the square of the correlation coefficient, R2, for this regime and for the 50C to 100C regime with smaller error bars were both 0.96. Mathematically, the independent variable of temperature (or, more precisely, 1/T) caused this behavior; very little scatter was attributed to the stress (ar) variable. The increase of error bars is even more significant because only two temperatures were used in the 75C to 100C analysis, which tends to improve the regression fit. Physically, it appeared that an unknown characteristic of the Sn microstructure had introduced an enhanced sensitivity of the creep deformation to temperature in the range of 75C to 100C. The sinh exponent, n, was evaluated from the regression analysis of data obtained from all strain rates and all three sample conditions; the results are shown in Fig. 7. There was no statistically significant dependence of n on the sample condition. The value of n increased as a function of temperature regime. Unfortunately, the sinh exponent does not have a well-developed correlation to possible creep mechanisms. Dislocation movement theories have predicted values of the stress exponent, p, in Eq. 2 to be 3 to 5 for pure metals.25,26 Therefore, the multivariable regression analysis was performed using Eq. 2. Good fits were confirmed by high R2 values. The values of p were in the range of 3.8 to 4.3, which indicated the presence of dislocation motion theorized for pure metals. There was no significant dependence upon annealing treatment. The powerlaw expression produced the same DH value.

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Fig. 7. Sinh exponent as a function of temperature regime and annealing treatment. All strain rate data were used in the regression analysis. The half-error bars are 95% confidence intervals and were added only to the as-fabricated and 200C annealed data.

Next, the rate kinetics analysis was broken down according to steady-state strain rate and temperature regimes. The strain rate benchmark was 107 s1 for all conditions. Figure 8a shows the DH values (sinh equation) for strain rates less than 107 s1. The one-half error bars were placed on the as-fabricated and 200C (24 h) annealed data. Although a contributing factor to the larger error bars of the 50C to 100C and 75C to 100C temperature regimes was the reduced number of data points, the R2 values (Fig. 8b) suggested that there were other contributing phenomena. For example, the R2 values do not show the sharp drop-off that would be expected by the immediate increase of error bar at those temperature regimes. Also, R2 increased significantly with annealing condition; yet, the error bars remained large. Therefore, the large scatter for the 50C to 100C and 75C to 100C regimes reaffirmed that the creep deformation process had undergone a change at these higher temperatures. The following discussion addresses specifically the three temperature regimes: 0C to 50C, 0C to 75C, and 0C to 100C. The DH values in Fig. 8a were very low, indicating that an anomalously fast mass transport mechanism supported the creep deformation. There was no statistically significant difference in DH between annealing treatments. R2 values were relatively low for the as-fabricated condition, but then increased with the annealing treatments (Fig. 8b). This trend indicates that the annealing treatments provided a more consistent deformation behavior as a function of temperature at these low strain rates. R2 also increased with decreasing temperature regime, in spite of fewer data points, thereby confirming the lessening effect

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Fig. 9. Graph of the apparent activation energy DH as a function of temperature regime and annealing treatment for strain rates greater than 107 s1. The half-error bars are 95% confidence intervals and were added only to the as-fabricated and 200C annealed data.

Fig. 8. (a) Apparent activation energy, DH, as a function of temperature regime and annealing treatment for strain rates of less than 107 s1. The half-error bars are 95% confidence intervals and were added only to the as-fabricated and 200C annealed data. (b) The square of the correlation coefficient, R2.

of the high-temperature ‘‘change’’ mentioned above. The data and analysis pertaining to Fig. 8a were examined in the context of likely microstructural processes. The grain boundaries were oriented predominantly perpendicular to the applied stress; thus, grain boundary diffusion (Coble) creep played a very limited role in the direction of the stress. In addition, the DH values were too low for grain boundary diffusion, and also for a Nabarro–Herring bulk diffusion creep mechanism. Rather, it appeared that the anomalously fast mass transport mechanism had assisted dislocation motion within the grains. This mechanism would have a role in the climb of dislocations or even their glide, although the latter mechanism has not been discussed in the

literature. By either account, dislocation motion would remain within the Sn grains, as shown schematically in Fig. 5 of Part I.* The values of the sinh exponent were also determined for strain rates of less than 107 s1. The values were 0.39 to 0.44 for the as-fabricated condition; 0.82 to 0.87 for the 150C, 24 h annealing condition; and 0.91 to 1.0 for the 200C, 24 h annealing condition. The values also increased slightly as a function of the temperature regime within each sample condition. The corresponding power-law stress exponent, p, was: 1.2 ± 0.6 as-fabricated, 1.9 ± 0.9 for 150C/24 h annealing, and 2.5 ± 0.7 for 200C/24 h annealing. The value of p for the as-fabricated condition (1.2 ± 0.6) suggests that Sn behaved very nearly like a viscous material. However, because the DH value was very low, the creep was not diffusion controlled per the Nabarro–Herring (bulk diffusion) mechanism as noted above.15,16 The annealing treatments increased the mean values of p towards, although not reaching, the theoretical values of 3 to 5. The rate kinetics were investigated next for strain rates greater than 107 s1. The DH values are plotted in Fig. 9. In all cases, the R2 values were greater than 0.90. Moreover, the value was 0.95 for the highest temperature regime of 75C to 100C, which still had the largest error bars. In general, the values of DH in Fig. 9 are greater than those in Fig. 8a, indicating a change in the *Although the authors’ model utilizes dislocation motion as the underlying mass transport medium, we recognize that anomalously fast diffusion could be a variant of bulk diffusion or an indirect consequence of dislocations, e.g., very fast diffusion along their cores.


deformation mechanism. At the lowest temperatures, 0C to 50C, the values of DH were statistically different between the as-fabricated samples (70 ± 8 kJ/mol) versus the two 24 h annealing treatments (150C, 43 ± 13 kJ/mol and 200C, 47 ± 19 kJ/mol). Thus, it appears that the annealing treatment caused the creep deformation to have a significant contribution by the anomalously fast mechanism. If it is assumed that dislocation densities were reduced by (a) the annealing treatment eliminating dislocations and (b) the low test temperatures slowing the generation of dislocations, then a characteristic of the anomalously fast mechanism is that it prefers low dislocation densities, which is a condition that is particularly applicable at low strain rates. The values of DH were in the range of 55 kJ/mol to 70 kJ/mol for the entire temperature range, 0C to 100C. These values are approximately one-half the DH value for bulk diffusion, which implies that the mass-transport-assisted dislocation motion was based upon a more ‘‘traditional’’ short-circuit dislocation climb process. Corroborating DH data were obtained by Nagasaka, who measured the tensile stress–strain curves of Sn at a strain rate of 1.7 9 105 s1, which is similar to the present data.27 Nagasaka calculated a slightly lower DH of 48 kJ/mol, which was based upon the temperature dependence of the yield point over the range of 73C to 207C. The mean DH values in Fig. 9 exhibited large error bars for the 75C to 100C regime, in spite of R2 values that exceeded 0.975. The large variability implied that the deformation mechanism had become highly sensitive to test temperature, as was similarly observed at the low strain rates (Fig. 8a). The mean DH values in Fig. 9 also exhibited a strong sensitivity to annealing treatment for the 75C to 100C regime. However, the trend was opposite to that shown in Fig. 8a for low strain rates. Clearly, the annealing treatments had different effects that depended on the strain rate; however, the details remain elusive at this time. The sinh and power-law stress exponents are plotted together in Fig. 10 as a function of strain rate (de/dt) regime. The half error bars (95% confidence interval) were placed only on the as-fabricated and 200C, 24 h annealed sample data; the values were similar for the 150C annealed samples. The consistency of the sinh exponents across all strain rate regimes exemplifies the avoidance of the power-law breakdown afforded by Eq. 1. Power-law breakdown was evident as p increased from 1 to 3 at the low strain rates to values of 6.5 to 8.5 at the high strain rates. Therefore, although the increase of DH with strain rate discussed above (Fig. 8a versus Fig. 9) would suggest a loss of thermally activated mobility, the significant increase of p implies that more mobile dislocations were generated by the applied stress that, in turn, produced the faster strain rates.

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Fig. 10. Sinh and power-law exponents as a function of strain rate regime and each of the annealing condition. The half error bars (95% confidence interval) were added only to the as-fabricated and 200C, 24 h annealing data.

In summary, the steady-state creep rate data were analyzed to determine the rate kinetics that would support the DRX mechanism for whisker growth. The analysis was categorized according to strain rate as well as by temperature regime. When all strain rates were considered, the DH values were 35 ± 10 kJ/mol for 0C to 100C across all annealing treatments. A single microstructural process across all strain rates would have an anomalously fast mass transport (diffusion) mechanism that either assisted dislocation motion within the Sn grains or was associated with the dislocation structures. An explicit dislocation motion process was indicated by a power-law stress exponent in the range of 3 to 5. The second approach separated the creep data into two regimes according to strain rate. When strain rates were less than 107 s1, the mean DH values were in the range of 8 kJ/mol to 16 kJ/mol for the three annealing treatments over the range of 0C to 100C. The 95% confidence intervals were 9 kJ/mol to 11 kJ/mol. These low values confirmed an anomalously fast mass transport mechanism that either assisted the motion of dislocations (climb, or even possibly glide) or resulted from the dislocation structure. The power-law stress exponent was in the range of 1 to 2.5. At strain rates greater than 107 s1, the creep mechanism changed, as indicated by DH values in the range of 50 kJ/mol to 70 kJ/mol (0C to 100C regime) as well as by power-law exponents of 6.5 to 8.5. In this case, a more traditional, short-circuit diffusion process took place during deformation. Overall, the annealing treatments affected the kinetics parameters slightly, but still within the 95% confidence interval that defined statistical significance.

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Zener–Hollomon and Critical Strain Parameters As described in Part I, the Zener–Hollomon parameter (Z), or temperature-compensated strain rate, de/dt exp(DH/RT), provides a metric that determines the likelihood of cyclic DRX, which has the grain growth step that is proposed to be responsible for whisker growth. A second parameter, the critical strain, ec, determines whether DRX will occur at all, irrespective of it being continuous or cyclic in nature. The critical strain is calculated by Eq. 3: n ec ¼ ADm o Z ;

(3)

where A, m, and n are constants as described in Part I. Calculations are presented below for ec and Z. The analysis began by using the single DH value of 35 kJ/mol that covers all strain rates. Then, it was repeated for the more representative condition of two strain rate regimes, that is, DH values of 8 kJ/mol and 65 kJ/mol for de/dt < 107 s1 and de/dt > 107 s1, respectively. In both cases, the DH values represented the entire temperature range of 0C to 100C. These analyses were limited to the asfabricated sample data; relatively small quantitative differences were expected for the annealed sample data. The Z parameter is plotted as a function of nominal stress in Fig. 11, representing all strain rates. The Z values were tens of orders of magnitude less than the 108–1020 s1 values typically observed in DRX experiments using stress–strain tests (strain rates of 105–10 s1) or those observed in other creep experiments.4,7,28–32 The cited works assumed DH values for bulk diffusion, which are typically 200 kJ/mol to 300 kJ/mol for pure Cu, pure iron, and stainless steels. Thijssen’s stress–strain curves showed cyclic DRX at 140C and 3 9 105 s1.3 Using the present DH value of 35 kJ/mol, the limiting Z value was calculated to be 0.76 based upon those experiments.** Therefore, it would be predicted from Fig. 11 that the present creep test conditions (which replicate Sn whisker growth conditions), would generate cyclic DRX under this DH value for all but the 10 MPa nominal stress. At 10 MPa, the higher strain rates would curtail cyclic DRX. The likelihood for DRX to occur, whether continuous or cyclic, is determined by the calculation of ec using Eq. 3. Unfortunately, the parameters A, m, and n were not available for Sn. Therefore, ec was estimated using the data of Thijssen.3 The grain sizes in Ref. 3 were comparable to those in the present hollow cylinder creep specimens. The logarithm was taken of Eq. 3. The term, lnA + mlnDo **The initial yield stress of that cited experiment was 2.5 MPa to 3.0 MPa, which is within the same range of stresses used in the current creep experiments.

Fig. 11. Plot of the Zener–Hollomon parameter (Z) as a function of nominal stress for each of the test temperatures. The DH value was 35 kJ/mol, which was that obtained when all of the strain rate data were used from over the entire 0C to 100C test temperature range. The samples were in the as-fabricated condition.

was considered a constant, K. A linear regression analysis was performed on the data from Ref. 3, with the dependent variable, lnec, and the independent variable, nlnZ. The strain rate was 3 9 105 s1 from Ref. 3. The resulting values of K and n were 2.425 and 0.371, respectively. The R2 value was 0.952. These K and n values were then used to calculated ec for the Z parameters for the current creep tests. The plot in Fig. 12 shows the calculated critical strain, ec (solid symbols), and the experimentally measured mean, maximum strain, emax (open symbols) as a function of nominal stress for each test temperature. The maximum strains were achieved at the completion of the test (not after 10,000 s as shown in Fig. 2). Also, the emax values would be applicable to Sn coatings as well as to the current creep specimens. Maximum strains that exceeded ec were likely to initiate DRX. At 1 MPa, the emax values are well below ec (logarithmic scale) for all temperatures except 100C; thus, DRX would be unlikely to take place in all but that latter case. At 10 MPa, DRX was unlikely at all temperatures. At 2 MPa, the ec and emax values are very close but still, on a temperature-by-temperature basis, emax was less than ec. The likelihood for DRX would be slightly higher for 5 MPa and at 75C and 100C where emax exceeded ec. Therefore, given the approximations used in Eq. 3 and DH equal to 35 kJ/mol, the data in Fig. 12 indicate that the strains resulting from compressive stresses and temperatures were certainly within range of, and in a few cases exceeding, the ec value needed to generate DRX. The preceding discussion indicates that, for those cases in which emax exceeded ec, the Z parameter would predict cyclic DRX.


Fig. 12. Calculated critical strain, ec, and experimental maximum strain, emax, as functions of nominal applied stress for each test temperature. The single DH value of 35 kJ/mol was used, representing all strain rates. The other parameters were calculated from literature data.3 The samples were in the as-fabricated condition.

Fig. 13. Calculated critical strain, ec, and experimental maximum strain, emax, as functions of nominal applied stress for each test temperature. The value of DH was 8 kJ/mol for strain rates less than 107 s1 and 65 kJ/mol for strain rates greater than 107 s1. The other parameters were calculated from literature data.3 The samples were in the as-fabricated condition.

The ec parameters were recalculated using the two DH values representing the two strain rate categories: 8 kJ/mol and 65 kJ/mol for de/dt < 107 s1 and de/dt > 107 s1, respectively. The critical strains, ec, along with the maximum strains, emax, are plotted in Fig. 13. The dashed lines identify the two data sets. Although not shown for clarity, the same stipulation pertains here: emax values above ec indicate the likelihood of DRX and emax values below emax imply that DRX is unlikely to

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Fig. 14. Plot of the Zener–Hollomon parameter (Z) as a function of true stress for each of the test temperatures. The value of DH used in the Z parameter was 8 kJ/mol for strain rates less than 107 s1 and 65 kJ/mol for strain rates greater than 107 s1. The samples were in the as-fabricated condition.

take place. The two strain rate regimes result in a step in the critical strain values at 5 MPa. At the lower stresses of 1 MPa and 2 MPa, the maximum strains exceeded the critical strains, indicating that DRX would be predicted to take place. Although the magnitude of emax was limited at these low stresses, the slower strain rate and reduced DH provided the conditions to initiate DRX. On the other hand, the faster strain rates at 10 MPa would not initiate DRX, in spite of higher emax values. In summary, the data in Fig. 13 indicate that DRX can take place at the low stress and temperature conditions under which whiskers are observed of Sn coatings, owing to the anomalously fast mass transport mechanism supporting the creep deformation. Optical microscopy and scanning electron microscope (SEM) analyses detected isolated regions of grain refinement in samples tested at the lower stresses (