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Creep of nanocrystalline nickel: A direct comparison between uniaxial and nanoindentation creep
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Scripta Materialia 62 (2010) 175–178 www.elsevier.com/locate/scriptamat
Creep of nanocrystalline nickel: A direct comparison between uniaxial and nanoindentation creep C.L. Wang,a Y.H. Lai,b J.C. Huangb and T.G. Nieha,* a
Department of Materials Science and Engineering, The University of Tennessee, Knoxville, TN 37996, USA b Department of Materials and Optoelectronic Science, Center for Nanoscience and Nanotechnology, National Sun Yat-Sen University, Kaohsiung 804, Taiwan, ROC Received 17 August 2009; revised 25 September 2009; accepted 12 October 2009 Available online 20 October 2009
Uniaxial and nanoindentation creeps were conducted on nanocrystalline nickel with an as-deposited grain size of 14 nm at 398 K and the results were directly compared. The stress exponent under the two test conditions was found to be almost the same, indicating a similar creep mechanism. However, the strain rate measured by nanoindentation creep was about 100 times faster than that by uniaxial creep. The rate difference was discussed in terms of stress states and the appropriate selection of the Tabor factor. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Creep; Nanoindentation; Compression test
The nanoindentation method has been widely used for the study of mechanical properties and deformation behavior in a small volume of materials, particularly the study of rate-dependence processes such as nanoindentation creep [1–5]. Goodall and Clyne [6] recently pointed out that the data measured from nanoindentation creep did not correlate well with those reported in the literature obtained by conventional creep. There appears to be a large discrepancy between the two creep rates and, specifically, the nanoindentation creep rate (_ei ) is faster than the uniaxial creep rate (_eu ). To the best of our knowledge, only Poisl et al. [7] made efforts to correlate the indentation strain rate with the effective strain rate experienced by the material under the indenter; other groups have made comparisons with data essentially obtained from different studies. It is noted that Poisl et al. studied amorphous Se, which behaved as a Newtonian fluid (stress exponent n ¼ 1) at temperatures above Tg (the glass transition temperature). The linear relationship between strain rate and stress for Newtonian flow made the determination of the correlation coefficient b, where b ¼ ð_eu =_ei Þ, relatively easy, and b 0:09 in their study. For crystalline materials, however, the determination of b is not straightforward since the n value is usually not unity. For
* Corresponding author. Tel.: +1 865 974 5328; fax: +1 865 974 4115; e-mail: [email protected]
example, Lucas and Oliver [1] carried out nanoindentation creep experiments on indium and observed a stress exponent value of about 5, which is similar to that reported in uniaxial creep [8]. The magnitude of their two creep strain rates differed greatly, however, and no reason for this was given. In the present study, we conducted both uniaxial creep and nanoindentation creep on nanocrystalline (nc) Ni and directly compared the two sets of data. ncNi films with a grain size of 14 nm were prepared by direct current electrodeposition. However, the grain size increased to about 60 nm during heating and stabilization of the temperature prior to creep testing at 398 K (125 °C) [5]. All samples were prepared initially by mechanical grinding with SiC papers, with a final polishing done with a microcloth using a slurry of 1.0 lm alumina. Nanoindentation creep tests were done directly on the mirror-finish sample. Micropillars for uniaxial creep were fabricated from the polished Ni films by a focused ion beam (FIB) system following the prototypical methodology of Uchic et al. [9]. The pillars had an aspect ratio of 2:1 and a nominal diameter of 2 lm was taken at half of the height of the pillars. The pillars were made to reside at the center of a 45 lm crater to provide enough space for the indenter moving down during the test and to guarantee that only the tip contacted the pillar of interest. Both creep tests were carried out with the same Triboindenter (Hysitron, Inc., Minneapolis, MN)
1359-6462/$ - see front matter Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2009.10.021
176
C. L. Wang et al. / Scripta Materialia 62 (2010) 175–178
equipped with a heating stage that could heat the specimen to 473 K. All creep tests were carried out at 398 K (125 °C). Prior to each creep test and after stabilizing at a prescribed temperature for at least 15 min, the indenter tip was brought into contact with the sample surface so that the tip could be heated and reach a steady-state temperature distribution. The continuous contact between sample and tip resulted in a negligible temperature variation (<0.1 K) and a low drift rate of 0.02–0.3 nm s–1. In addition, thermal drift was automatically corrected by the Triboindenter software during testing. The effect of thermal drift during nanoindentation has been discussed previously [5]. Uniaxial creep tests were conducted under fixed loads ranging from 3.6 to 5 mN using a customized flat punch, which was machined by the FIB technique out of a standard diamond Berkovich tip mounted on a machinable ceramic rod. The resulting punch has an equilateral triangle cross-section with a 8 lm inscribed circle. Nanoindentation creep tests were conducted using a standard Berkovich tip. The loading scheme was as follows: the specimen was uploaded to the preset maximum load with a fixed loading time of 1 s followed by a holding period of 60 s to examine the creep behavior. After the holding, the load was immediately reduced to zero. The data obtained from uniaxial creep were directly compared with that from nanoindentation creep. The morphology of the pillars before and after creep deformation was examined using scanning electron microscopy (SEM). Typical SEM micrographs of a pillar before and after creep at a fixed load of 5 mN are shown in Figure 1. It is noted from Figure 1(a) that the sidewall of the pillar is not parallel to its axis, but has a tapered angle of about 3° with respect to the test axis. This is an inherent byproduct of the current FIB technique. This taper effect was taken into account when analyzing the stress and strain in the pillars. One apparent feature of the deformed pillars is the uniform deformation, as shown in
Figure 1(b). In general, all crept pillar samples were uniformly deformed, in contrast to the inhomogeneous, localized slips observed in single crystalline materials [9]. The load and displacement recorded during uniaxial creep were converted into true stress and true strain by assuming a constant taper angle of 3° during creep. The uniaxial strain rate e_ u under uniaxial creep was calculated using the equation:
Figure 1. SEM micrographs of the nc-Ni pillar crept at a fixed load of 5 mN: (a) before and (b) after creep.
Figure 2. Strain–time curves for nc-Ni uniaxially crept at 398 K and different loads, as indicated in the graph.
de dl ¼ ð1Þ dt ldt where e is the true strain, t is time and l is the instantaneous length of the pillar. On the other hand, the nanoindentation strain rate e_ i was calculated through the following equation:
e_ u ¼
1 dh ð2Þ h dt where h is the indenter displacement monitored during creep. Practically, this nanoindentation strain rate e_ i can be computed by taking the slope of the displacement–time curve, i.e. the instantaneous displacement _ and dividing it by the displacement at any particrate h, ular point in time. A dimensional analysis indicates that the uniaxial creep rate is linearly scaled with the indentation creep rate, i.e. e_ i ¼
e_ u ¼ b_ei
ð3Þ
where b is the correlation coefficient as mentioned above. In fact, Poisl et al. [7], in their study of amorphous Se, demonstrated that the two creep rates were indeed linearly dependent and b 0:09. The strain–time plots for pillars crept under different loads are shown in Figure 2. It can be readily observed that the strain initially increases sharply with time, followed by a decreasing strain rate, and then exhibits a steady-state type behavior, i.e. the strain increases linearly with time. In the uniaxial creep tests we used a constant load, and the stress level at the end of each test was less than 1% lower than the stress at the beginning. Thus, it is essentially a constant stress creep condition. The uniaxial steady-state creep rate vs. stress relationship for nc-Ni plotted in a double-logarithmic graph is given in Figure 3. The stress exponent value is about 5.13. To make a direct comparison, data obtained from nanoindentation creep of nc-Ni are also plotted in Figure 3.
C. L. Wang et al. / Scripta Materialia 62 (2010) 175–178
Figure 3. Strain rate vs. stress or (Hardness/a) for nc-Ni from uniaxial creep and nanoindentation creep, respectively.
It is noted that, in this figure, the applied stresses under nanoindentation are computed through the classical Tabor relationship: H ¼ ar, a ¼ 3 [10], although there are indications that a may be less than 3 at high temperatures [11]. It is evident from Figure 3 that the nanoindentation creep follows a power-law with a stress exponent, ni , of about 5.78, which is almost the same as the value of nu (=5.13) obtained from the uniaxial creep. This result is similar to that reported in polycrystalline indium, in which both ni and nu are essentially identical (5) [1]. The fact that the stress exponent is almost identical under the two creep conditions was expected since the underpinned creep mechanism in both cases is the same. Although not shown here, the temperature dependence (i.e. the creep activation energy) for the two creeps is also the same. In fact, both correspond to the activation energy for grain-boundary diffusion in Ni [5], suggesting that both cases are dislocation climb creep, with the rate-controlling step being the grain-boundary diffusion. The only difference between the two creeps appears to be the magnitude of the creep rate. Strain rates at different stress levels for the two creeps are listed in Table 1. At a fixed stress, the nanoindentation creep rate is noted to be much faster than the uniaxial creep rate. Specifically, the ratio of the two creep rates (_eu =_ei ) is about 0.01 for the current nc-Ni. In comparison, the ratio was only about 0.09 for amorphous Se [7]. In other words, b (0.01) for nc-Ni appears to be about 1/10th of that for amorphous Se (b 0:09). The difference can be rationalized as follows. A uniaxial mechanical test monitors the response of a material under a relatively uniform stress state in a glo-
Table 1. Stress or (Hardness/a), strain rate and strain rate ratio under uniaxial and nanoindentation creep at 398 K. Stress or (Hardness/a) (GPa)
Strain rate (s1) Uniaxial creep, e_ u
Nanoindentation creep,_ei
1.388 1.53654 1.68637 1.84433
2.88 105 4.75 105 7.96 105 1.22 104
2.5 103 4.93 103 8.30 103 1.32 102
Strain rate ratio e_ u =_ei 11.5 103 9.63 103 9.59 103 9.24 103
177
bal way. By contrast, nanoindentation probes the material response from a local volume. In addition, the stress state under an indentation, which depends on the intrinsic properties of the material, is complex and is actually much more severe than that under the uniaxial condition. To the first approximation, it can be viewed as a biaxial stress state. It has been demonstrated [12] that, at the same equivalent stress, the deformation rate under the biaxial condition is faster than that under the uniaxial condition. Therefore, a faster nanoindentation creep rate as compared to the uniaxial creep rate in Se is anticipated, at least qualitatively. In the current case of ncNi, however, the gap between the two creep rates is much larger. It is noted that, to evaluate the nanoindentation creep rate, a ¼ 3 is commonly applied in the Tabor relation: H ¼ ar. At high temperatures, however, a would be expected to be lower because of a reduced strain hardening and enhanced thermal recovery. In the description of nanoindentation creep, the power-law equation can be written as [13]: n H e_ i ¼ Arn ¼ A ð4Þ a In this equation, the strain rate is inversely proportional to the Tabor factor a to the nth power. In the case of amorphous Se, which flows in a Newtonian fashion (n ¼ 1), a variation of a can only affect the actual creep rate in a linear and insignificant manner. By contrast, nc-Ni has a stress exponent of about 5, and the selection of a can shift the actual creep rate in a much more dramatic way. There are only limited parallel hardness–strength data as a function of temperature in the literature [11,14]. In the case of a Zr–2.5Nb alloy [11], in particular, it was found that the Tabor factor decreases as the temperature increases, and a 2:09 at 0.27Tm, where Tm is the melting temperature of the alloy. This homologous temperature is in fact similar to the creep temperature of 0.23Tm used in the current study of nc-Ni. Selecting a Tabor factor of 2 instead of 3 can change the nanoindentation creep rate by a factor of over 8 (leading to b 0:08), which reasonably accounts for the observed difference in the b (=_eu =_ei ) value between Se and nc-Ni. The nanoindentation creep rate vs. hardness compensated by various Tabor factors is presented in Figure 3. Three important points deserve further discussion. The first is the creep mechanism, in particular during nanoindentation. As we have already pointed out in our previous paper [5], several independent mechanisms – diffusional creep, grain-boundary sliding and powerlaw creep – operate concurrently during nanoindentation. However, diffusional creep or grain-boundary sliding was ruled out as the dominant mechanism, even they both might occur. Two main factors support this conclusion. First, the test temperature was 398 K (125 °C), which is only about 0.23Tm. Diffusional creep and grain-boundary sliding mechanisms dominate typically at around 0.7Tm and above. For nc-Ni this temperature may be lower, but it is unlikely to be as low as 0.23Tm. Second, assuming the three mechanisms operate
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C. L. Wang et al. / Scripta Materialia 62 (2010) 175–178
independently, the total creep rate e_ total can be estimated approximately by e_ total ¼ vd e_ d þ vgbs e_ gbs þ vp e_ p
ð5Þ
where e_ d ; e_ gbs and e_ p represent the creep rates caused by diffusional creep, grain-boundary sliding and power-law creep, respectively, vd , vgbs and vp represent the corresponding weighted volume fraction of region affected by each creep, and vd + vgbs + vp = 1. The exact values of vd , vgbs and vp are determined by the applied stress, the temperature and the properties of Ni, but they are probably within one order of magnitude. On the other hand, e_ d , e_ gbs and e_ p depend upon the applied stress raised to a power of 1, 2 and 5, respectively. Obviously, e_ p has the strongest stress dependence. This, together with the fact that the test temperature is only 0.23Tm, means that nc-Ni favors the power-law creep, even when the grain size is small. The second important point is the pillar sample damage by FIB process. One may expect the creep rate difference to be a result of sample surface contamination by Ga ions. However, it should be noted that the penetration depth for 30 keV Ga ions (i.e. the energy used for the current processing of Ni pillar) in Ni is only 10– 20 nm [15]. Compared with the diameter of the pillar (2000 nm), it is relatively small. Specifically, taking the penetration depth of 20 nm, the amount of the cross-sectional area affected by the Ga ions is only about 4% of the total cross-sectional area of the pillar. In this case, the unaffected region dominates the overall behavior of the pillar. The last question is possible surface oxidation during indentation. Ni does not exhibit noticeable oxidation until T > 400–500 °C, which is much higher than the current test temperature of 125 °C [16]. Thus, oxidation is expected to be negligible. In fact, the nc-Ni samples still look shiny after testing. In summary, nanoindentation creep and uniaxial creep are expected to have a similar stress exponent because the underpinning deformation mechanism is the same. However, the creep rate under nanoindentation is typically much faster than that under uniaxial creep. The faster creep rate was caused by the facts that the stress state under nanoindentation is more complex and severe than the uniaxial condition, and the conver-
sion factor used from hardness to stress (i.e. the Tabor factor) is less than the typical value of 3. In fact, in the current case of nc-Ni, after replacing with a proper Tabor factor, the difference between the two creeps can be brought into reasonable agreement with that previously observed in amorphous Se. This work was supported by the United States Department of Energy, Office of Basic Energy Sciences, under Contract DE-FG02-06ER46338 with the University of Tennessee. Instrumentation for the nanoindentation work was jointly funded by the Tennessee Agricultural Experiment Station and UT College of Engineering. Partial sponsorship from the National Science Council of Taiwan under the Project NSC 95-2221E-110-013-MY3 is gratefully acknowledged. [1] B. Lucas, W. Oliver, Metall. Mater. Trans. A 30 (1999) 601. [2] H. Li, A.H.W. Ngan, J. Mater. Res. 19 (2004) 513. [3] J. Alkorta, J.M. MartI´nez-Esnaola, J. Gil Sevillano, Acta Mater. 56 (2008) 884. [4] C.L. Wang, T. Mukai, T.G. Nieh, J. Mater. Res. 24 (2009) 1615. [5] C.L. Wang, M. Zhang, T.G. Nieh, J. Phys. D: Appl. Phys. 42 (2009) 115405. [6] R. Goodall, T.W. Clyne, Acta Mater. 54 (2006) 5489. [7] W.H. Poisl, W.C. Oliver, B.D. Fabes, J. Mater. Res. 10 (1995) 2024. [8] J. Weertman, Trans. AIME 218 (1960) 207. [9] M.D. Uchic, D.M. Dimiduk, J.N. Florando, W.D. Nix, Science 305 (2004) 986. [10] D.S. Tabor, The Hardness of Metals, Clarendon Press, Oxford, 1951. [11] T.R.G. Kutty, K. Ravi, C. Ganguly, J. Nucl. Mater. 265 (1999) 91. [12] D.R. Lesuer, J. Wadsworth, T.G. Nieh, Ceram. Int. 22 (1996) 381. [13] A.F. Bower, N.A. Fleck, A. Needleman, N. Ogbonna, Proc. R. Soc. Lond. A 441 (1993) 97. [14] J. Moteff, R. Bhargava, W. McCullough, Metall. Mater. Trans. A 6 (1975) 1101. [15] J.F. Ziegler, J.P. Biersach, U. Littmark, The Stopping and Range of Ions in Solids, Stopping and Range of Ions in Matter, Pergamon, New York, 1985. [16] G.C. Wood, I.G. Wright, J.M. Ferguson, Corros. Sci. 5 (1965) 645.
Scripta Materialia 62 (2010) 175–178 www.elsevier.com/locate/scriptamat
Creep of nanocrystalline nickel: A direct comparison between uniaxial and nanoindentation creep C.L. Wang,a Y.H. Lai,b J.C. Huangb and T.G. Nieha,* a
Department of Materials Science and Engineering, The University of Tennessee, Knoxville, TN 37996, USA b Department of Materials and Optoelectronic Science, Center for Nanoscience and Nanotechnology, National Sun Yat-Sen University, Kaohsiung 804, Taiwan, ROC Received 17 August 2009; revised 25 September 2009; accepted 12 October 2009 Available online 20 October 2009
Uniaxial and nanoindentation creeps were conducted on nanocrystalline nickel with an as-deposited grain size of 14 nm at 398 K and the results were directly compared. The stress exponent under the two test conditions was found to be almost the same, indicating a similar creep mechanism. However, the strain rate measured by nanoindentation creep was about 100 times faster than that by uniaxial creep. The rate difference was discussed in terms of stress states and the appropriate selection of the Tabor factor. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Creep; Nanoindentation; Compression test
The nanoindentation method has been widely used for the study of mechanical properties and deformation behavior in a small volume of materials, particularly the study of rate-dependence processes such as nanoindentation creep [1–5]. Goodall and Clyne [6] recently pointed out that the data measured from nanoindentation creep did not correlate well with those reported in the literature obtained by conventional creep. There appears to be a large discrepancy between the two creep rates and, specifically, the nanoindentation creep rate (_ei ) is faster than the uniaxial creep rate (_eu ). To the best of our knowledge, only Poisl et al. [7] made efforts to correlate the indentation strain rate with the effective strain rate experienced by the material under the indenter; other groups have made comparisons with data essentially obtained from different studies. It is noted that Poisl et al. studied amorphous Se, which behaved as a Newtonian fluid (stress exponent n ¼ 1) at temperatures above Tg (the glass transition temperature). The linear relationship between strain rate and stress for Newtonian flow made the determination of the correlation coefficient b, where b ¼ ð_eu =_ei Þ, relatively easy, and b 0:09 in their study. For crystalline materials, however, the determination of b is not straightforward since the n value is usually not unity. For
* Corresponding author. Tel.: +1 865 974 5328; fax: +1 865 974 4115; e-mail: [email protected]
example, Lucas and Oliver [1] carried out nanoindentation creep experiments on indium and observed a stress exponent value of about 5, which is similar to that reported in uniaxial creep [8]. The magnitude of their two creep strain rates differed greatly, however, and no reason for this was given. In the present study, we conducted both uniaxial creep and nanoindentation creep on nanocrystalline (nc) Ni and directly compared the two sets of data. ncNi films with a grain size of 14 nm were prepared by direct current electrodeposition. However, the grain size increased to about 60 nm during heating and stabilization of the temperature prior to creep testing at 398 K (125 °C) [5]. All samples were prepared initially by mechanical grinding with SiC papers, with a final polishing done with a microcloth using a slurry of 1.0 lm alumina. Nanoindentation creep tests were done directly on the mirror-finish sample. Micropillars for uniaxial creep were fabricated from the polished Ni films by a focused ion beam (FIB) system following the prototypical methodology of Uchic et al. [9]. The pillars had an aspect ratio of 2:1 and a nominal diameter of 2 lm was taken at half of the height of the pillars. The pillars were made to reside at the center of a 45 lm crater to provide enough space for the indenter moving down during the test and to guarantee that only the tip contacted the pillar of interest. Both creep tests were carried out with the same Triboindenter (Hysitron, Inc., Minneapolis, MN)
1359-6462/$ - see front matter Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2009.10.021
176
C. L. Wang et al. / Scripta Materialia 62 (2010) 175–178
equipped with a heating stage that could heat the specimen to 473 K. All creep tests were carried out at 398 K (125 °C). Prior to each creep test and after stabilizing at a prescribed temperature for at least 15 min, the indenter tip was brought into contact with the sample surface so that the tip could be heated and reach a steady-state temperature distribution. The continuous contact between sample and tip resulted in a negligible temperature variation (<0.1 K) and a low drift rate of 0.02–0.3 nm s–1. In addition, thermal drift was automatically corrected by the Triboindenter software during testing. The effect of thermal drift during nanoindentation has been discussed previously [5]. Uniaxial creep tests were conducted under fixed loads ranging from 3.6 to 5 mN using a customized flat punch, which was machined by the FIB technique out of a standard diamond Berkovich tip mounted on a machinable ceramic rod. The resulting punch has an equilateral triangle cross-section with a 8 lm inscribed circle. Nanoindentation creep tests were conducted using a standard Berkovich tip. The loading scheme was as follows: the specimen was uploaded to the preset maximum load with a fixed loading time of 1 s followed by a holding period of 60 s to examine the creep behavior. After the holding, the load was immediately reduced to zero. The data obtained from uniaxial creep were directly compared with that from nanoindentation creep. The morphology of the pillars before and after creep deformation was examined using scanning electron microscopy (SEM). Typical SEM micrographs of a pillar before and after creep at a fixed load of 5 mN are shown in Figure 1. It is noted from Figure 1(a) that the sidewall of the pillar is not parallel to its axis, but has a tapered angle of about 3° with respect to the test axis. This is an inherent byproduct of the current FIB technique. This taper effect was taken into account when analyzing the stress and strain in the pillars. One apparent feature of the deformed pillars is the uniform deformation, as shown in
Figure 1(b). In general, all crept pillar samples were uniformly deformed, in contrast to the inhomogeneous, localized slips observed in single crystalline materials [9]. The load and displacement recorded during uniaxial creep were converted into true stress and true strain by assuming a constant taper angle of 3° during creep. The uniaxial strain rate e_ u under uniaxial creep was calculated using the equation:
Figure 1. SEM micrographs of the nc-Ni pillar crept at a fixed load of 5 mN: (a) before and (b) after creep.
Figure 2. Strain–time curves for nc-Ni uniaxially crept at 398 K and different loads, as indicated in the graph.
de dl ¼ ð1Þ dt ldt where e is the true strain, t is time and l is the instantaneous length of the pillar. On the other hand, the nanoindentation strain rate e_ i was calculated through the following equation:
e_ u ¼
1 dh ð2Þ h dt where h is the indenter displacement monitored during creep. Practically, this nanoindentation strain rate e_ i can be computed by taking the slope of the displacement–time curve, i.e. the instantaneous displacement _ and dividing it by the displacement at any particrate h, ular point in time. A dimensional analysis indicates that the uniaxial creep rate is linearly scaled with the indentation creep rate, i.e. e_ i ¼
e_ u ¼ b_ei
ð3Þ
where b is the correlation coefficient as mentioned above. In fact, Poisl et al. [7], in their study of amorphous Se, demonstrated that the two creep rates were indeed linearly dependent and b 0:09. The strain–time plots for pillars crept under different loads are shown in Figure 2. It can be readily observed that the strain initially increases sharply with time, followed by a decreasing strain rate, and then exhibits a steady-state type behavior, i.e. the strain increases linearly with time. In the uniaxial creep tests we used a constant load, and the stress level at the end of each test was less than 1% lower than the stress at the beginning. Thus, it is essentially a constant stress creep condition. The uniaxial steady-state creep rate vs. stress relationship for nc-Ni plotted in a double-logarithmic graph is given in Figure 3. The stress exponent value is about 5.13. To make a direct comparison, data obtained from nanoindentation creep of nc-Ni are also plotted in Figure 3.
C. L. Wang et al. / Scripta Materialia 62 (2010) 175–178
Figure 3. Strain rate vs. stress or (Hardness/a) for nc-Ni from uniaxial creep and nanoindentation creep, respectively.
It is noted that, in this figure, the applied stresses under nanoindentation are computed through the classical Tabor relationship: H ¼ ar, a ¼ 3 [10], although there are indications that a may be less than 3 at high temperatures [11]. It is evident from Figure 3 that the nanoindentation creep follows a power-law with a stress exponent, ni , of about 5.78, which is almost the same as the value of nu (=5.13) obtained from the uniaxial creep. This result is similar to that reported in polycrystalline indium, in which both ni and nu are essentially identical (5) [1]. The fact that the stress exponent is almost identical under the two creep conditions was expected since the underpinned creep mechanism in both cases is the same. Although not shown here, the temperature dependence (i.e. the creep activation energy) for the two creeps is also the same. In fact, both correspond to the activation energy for grain-boundary diffusion in Ni [5], suggesting that both cases are dislocation climb creep, with the rate-controlling step being the grain-boundary diffusion. The only difference between the two creeps appears to be the magnitude of the creep rate. Strain rates at different stress levels for the two creeps are listed in Table 1. At a fixed stress, the nanoindentation creep rate is noted to be much faster than the uniaxial creep rate. Specifically, the ratio of the two creep rates (_eu =_ei ) is about 0.01 for the current nc-Ni. In comparison, the ratio was only about 0.09 for amorphous Se [7]. In other words, b (0.01) for nc-Ni appears to be about 1/10th of that for amorphous Se (b 0:09). The difference can be rationalized as follows. A uniaxial mechanical test monitors the response of a material under a relatively uniform stress state in a glo-
Table 1. Stress or (Hardness/a), strain rate and strain rate ratio under uniaxial and nanoindentation creep at 398 K. Stress or (Hardness/a) (GPa)
Strain rate (s1) Uniaxial creep, e_ u
Nanoindentation creep,_ei
1.388 1.53654 1.68637 1.84433
2.88 105 4.75 105 7.96 105 1.22 104
2.5 103 4.93 103 8.30 103 1.32 102
Strain rate ratio e_ u =_ei 11.5 103 9.63 103 9.59 103 9.24 103
177
bal way. By contrast, nanoindentation probes the material response from a local volume. In addition, the stress state under an indentation, which depends on the intrinsic properties of the material, is complex and is actually much more severe than that under the uniaxial condition. To the first approximation, it can be viewed as a biaxial stress state. It has been demonstrated [12] that, at the same equivalent stress, the deformation rate under the biaxial condition is faster than that under the uniaxial condition. Therefore, a faster nanoindentation creep rate as compared to the uniaxial creep rate in Se is anticipated, at least qualitatively. In the current case of ncNi, however, the gap between the two creep rates is much larger. It is noted that, to evaluate the nanoindentation creep rate, a ¼ 3 is commonly applied in the Tabor relation: H ¼ ar. At high temperatures, however, a would be expected to be lower because of a reduced strain hardening and enhanced thermal recovery. In the description of nanoindentation creep, the power-law equation can be written as [13]: n H e_ i ¼ Arn ¼ A ð4Þ a In this equation, the strain rate is inversely proportional to the Tabor factor a to the nth power. In the case of amorphous Se, which flows in a Newtonian fashion (n ¼ 1), a variation of a can only affect the actual creep rate in a linear and insignificant manner. By contrast, nc-Ni has a stress exponent of about 5, and the selection of a can shift the actual creep rate in a much more dramatic way. There are only limited parallel hardness–strength data as a function of temperature in the literature [11,14]. In the case of a Zr–2.5Nb alloy [11], in particular, it was found that the Tabor factor decreases as the temperature increases, and a 2:09 at 0.27Tm, where Tm is the melting temperature of the alloy. This homologous temperature is in fact similar to the creep temperature of 0.23Tm used in the current study of nc-Ni. Selecting a Tabor factor of 2 instead of 3 can change the nanoindentation creep rate by a factor of over 8 (leading to b 0:08), which reasonably accounts for the observed difference in the b (=_eu =_ei ) value between Se and nc-Ni. The nanoindentation creep rate vs. hardness compensated by various Tabor factors is presented in Figure 3. Three important points deserve further discussion. The first is the creep mechanism, in particular during nanoindentation. As we have already pointed out in our previous paper [5], several independent mechanisms – diffusional creep, grain-boundary sliding and powerlaw creep – operate concurrently during nanoindentation. However, diffusional creep or grain-boundary sliding was ruled out as the dominant mechanism, even they both might occur. Two main factors support this conclusion. First, the test temperature was 398 K (125 °C), which is only about 0.23Tm. Diffusional creep and grain-boundary sliding mechanisms dominate typically at around 0.7Tm and above. For nc-Ni this temperature may be lower, but it is unlikely to be as low as 0.23Tm. Second, assuming the three mechanisms operate
178
C. L. Wang et al. / Scripta Materialia 62 (2010) 175–178
independently, the total creep rate e_ total can be estimated approximately by e_ total ¼ vd e_ d þ vgbs e_ gbs þ vp e_ p
ð5Þ
where e_ d ; e_ gbs and e_ p represent the creep rates caused by diffusional creep, grain-boundary sliding and power-law creep, respectively, vd , vgbs and vp represent the corresponding weighted volume fraction of region affected by each creep, and vd + vgbs + vp = 1. The exact values of vd , vgbs and vp are determined by the applied stress, the temperature and the properties of Ni, but they are probably within one order of magnitude. On the other hand, e_ d , e_ gbs and e_ p depend upon the applied stress raised to a power of 1, 2 and 5, respectively. Obviously, e_ p has the strongest stress dependence. This, together with the fact that the test temperature is only 0.23Tm, means that nc-Ni favors the power-law creep, even when the grain size is small. The second important point is the pillar sample damage by FIB process. One may expect the creep rate difference to be a result of sample surface contamination by Ga ions. However, it should be noted that the penetration depth for 30 keV Ga ions (i.e. the energy used for the current processing of Ni pillar) in Ni is only 10– 20 nm [15]. Compared with the diameter of the pillar (2000 nm), it is relatively small. Specifically, taking the penetration depth of 20 nm, the amount of the cross-sectional area affected by the Ga ions is only about 4% of the total cross-sectional area of the pillar. In this case, the unaffected region dominates the overall behavior of the pillar. The last question is possible surface oxidation during indentation. Ni does not exhibit noticeable oxidation until T > 400–500 °C, which is much higher than the current test temperature of 125 °C [16]. Thus, oxidation is expected to be negligible. In fact, the nc-Ni samples still look shiny after testing. In summary, nanoindentation creep and uniaxial creep are expected to have a similar stress exponent because the underpinning deformation mechanism is the same. However, the creep rate under nanoindentation is typically much faster than that under uniaxial creep. The faster creep rate was caused by the facts that the stress state under nanoindentation is more complex and severe than the uniaxial condition, and the conver-
sion factor used from hardness to stress (i.e. the Tabor factor) is less than the typical value of 3. In fact, in the current case of nc-Ni, after replacing with a proper Tabor factor, the difference between the two creeps can be brought into reasonable agreement with that previously observed in amorphous Se. This work was supported by the United States Department of Energy, Office of Basic Energy Sciences, under Contract DE-FG02-06ER46338 with the University of Tennessee. Instrumentation for the nanoindentation work was jointly funded by the Tennessee Agricultural Experiment Station and UT College of Engineering. Partial sponsorship from the National Science Council of Taiwan under the Project NSC 95-2221E-110-013-MY3 is gratefully acknowledged. [1] B. Lucas, W. Oliver, Metall. Mater. Trans. A 30 (1999) 601. [2] H. Li, A.H.W. Ngan, J. Mater. Res. 19 (2004) 513. [3] J. Alkorta, J.M. MartI´nez-Esnaola, J. Gil Sevillano, Acta Mater. 56 (2008) 884. [4] C.L. Wang, T. Mukai, T.G. Nieh, J. Mater. Res. 24 (2009) 1615. [5] C.L. Wang, M. Zhang, T.G. Nieh, J. Phys. D: Appl. Phys. 42 (2009) 115405. [6] R. Goodall, T.W. Clyne, Acta Mater. 54 (2006) 5489. [7] W.H. Poisl, W.C. Oliver, B.D. Fabes, J. Mater. Res. 10 (1995) 2024. [8] J. Weertman, Trans. AIME 218 (1960) 207. [9] M.D. Uchic, D.M. Dimiduk, J.N. Florando, W.D. Nix, Science 305 (2004) 986. [10] D.S. Tabor, The Hardness of Metals, Clarendon Press, Oxford, 1951. [11] T.R.G. Kutty, K. Ravi, C. Ganguly, J. Nucl. Mater. 265 (1999) 91. [12] D.R. Lesuer, J. Wadsworth, T.G. Nieh, Ceram. Int. 22 (1996) 381. [13] A.F. Bower, N.A. Fleck, A. Needleman, N. Ogbonna, Proc. R. Soc. Lond. A 441 (1993) 97. [14] J. Moteff, R. Bhargava, W. McCullough, Metall. Mater. Trans. A 6 (1975) 1101. [15] J.F. Ziegler, J.P. Biersach, U. Littmark, The Stopping and Range of Ions in Solids, Stopping and Range of Ions in Matter, Pergamon, New York, 1985. [16] G.C. Wood, I.G. Wright, J.M. Ferguson, Corros. Sci. 5 (1965) 645.