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Philosophical Magazine Letters Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tphl20
Deformation mechanism transitions in nanoscale fcc metals a
a
Robert J. Asaro , Petr Krysl & Bimal Kad
a
a
Department of Structural Engineering , University of California , San Diego, 9500 Gilman Drive, La Jolla, California 92093-0085, USA Published online: 04 Jun 2010.
To cite this article: Robert J. Asaro , Petr Krysl & Bimal Kad (2003) Deformation mechanism transitions in nanoscale fcc metals , Philosophical Magazine Letters, 83:12, 733-743, DOI: 10.1080/09500830310001614540 To link to this article: http://dx.doi.org/10.1080/09500830310001614540
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/ page/terms-and-conditions
PHILOSOPHICAL MAGAZINE LETTERS, December 2003 VOL. 83, NO. 12, 733–743
Deformation mechanism transitions in nanoscale fcc metals Robert J. Asaro, Petr Krysly and Bimal Kad Department of Structural Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0085, USA [Received in final form 22 July 2003 and accepted 25 July 2003 ]
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Abstract We consider possible mechanisms that lead to transitions in the mechanisms of deformation in fcc metals and alloys. In particular, we propose that, when grain sizes are below a critical size (i.e. below 100 nm), deformation can occur via the emission of stacking faults from grain boundaries into the intragranular space. A model is developed that accounts for observed experimental data and which, in turn, shows how stacking-fault energy together with shear modulus determines achievable strength. A mechanism is proposed based on this model for transitions at both high and quasistatic strain rates, including grain-boundary sliding.
} 1. Introduction When the grain size in metals and alloys changes from micrometres down to nanometres there are accompanying transitions in the mechanisms of inelastic deformation. There is direct experimental evidence for these transitions as well as suspicions that derive from what is known about the mechanisms of plastic deformation in crystalline metals and alloys. For example, in fcc metals with grain sizes in the micron and large size range, plastic deformation occurs via the generation and motion of intragranular slip, that is dislocation motion. This process is evidently shut off at grain sizes below the micron size range. To understand this, it suffices to note that the crystallographic shear stresses required to move the dislocation segments that exist within the well-characterized networks which evolve during plastic flow are of the order of Gb=‘ where G is the elastic shear modulus, b the Burgers vector magnitude of the slip dislocation and ‘ the dislocation segment length. However, if dislocations are to be confined to the intra granular space (which might be taken as the definition of a grain), ‘ must be less than the grain diameter d, say ‘ ¼ d=2 or d/3, and thus (2 3)G (b/d). This suggests a required shear stress larger than is observed or actually required as we show below. (Microscopic observations, together with well-known dislocation models for the operation of dislocation segments, have clearly shown that ‘ must be smaller than d.) The two mechanisms that appear to operate on nanoscale grain diameters involve, firstly, grain-boundary emission (GBE) (and subsequent absorption) of dislocations and, secondly, grain-boundary sliding (GBS). yAuthor for correspondence. Email: [email protected]. Philosophical Magazine Letters ISSN 0950–0839 print/ISSN 1362–3036 online # 2003 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/09500830310001614540
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The purpose of this letter is to prove a simple and yet compelling mechanistic picture of why these transitions occur and to show how lattice and chemical properties of the metals influence the transitions and help to set the levels of achievable strength. In particular the role of stacking-fault energy is shown to be of first order, as well as that of elastic modulus; the effect of the latter is, of course, expected. The stacking-fault energy in fact serves to set limits to the strength levels that may be realized, as well as to explain the distinct differences between metals with simple fcc structures. We shall develop arguments based principally around the data available for nominally pure Cu and Ni and provide some correlations for other metals such as Pd and Ag, at least as far as the existing data allows.
} 2. Dislocation models for intragranular deformation From what has been noted above, it should already be expected that observations via transmission electron microscope would show that grain matrices in nanoscale metals are essentially free of dislocation. Dislocations do reside in the boundaries, often in addition to those present in the boundaries and these account for the boundary misorientations. Such boundaries are often referred to as nonequilibrium boundaries. An example is shown in figure 1 taken from the work of Huang et al. (2001) and their physical basis has been described in the elegant molecular dynamic simulations of Van Swygenhoven et al. (1999) and Van Swygenhoven and Derlet (2001). As dislocations within the boundary of this type are able to move, under sufficient stress, into the interior of the grains, they can contribute to increments of strain. Figure 2 illustrates a possible mechanism for this. It is important to note that the segment of dislocation that is to move across the grain in question must be part of a loop, whose structure is unknown for now, but this in itself suggests that more must be understood about the nature of these grain boundary dislocations. As figure 2 depicts, the dislocation moves across the grain a distance dx and creates two segments, each of length dx. For now, but soon to be revised, we take this dislocation segment to be a perfect lattice dislocation with a Burgers vector b ¼ (a/2) [110]; it thus has an energy (per unit length) of E ¼ 12 Gb2 . As noted in the figure, this leads to a prediction for the resolved shear stress to accomplish this: b ¼G : d
ð1Þ
It should be noted that interesting d1 scaling which results from this is irrespective of the details but comes about solely via the fact that the area over which work can be done (to supply the energy needed to create additional defect) scales with d. In other words, any embellishments to the precise description of the dislocation segments produced will not change the d1 scaling. The problem with the formula as it stands is that it predicts critical resolved shear stresses that are too large, as described later. Some typical numbers for Cu are listed in the figure. Then without further consideration we seek to find how these large stresses can be mitigated. A key to this is to recall that the grain sizes of interest, say d < 50 nm, are of size scales approaching the spacings between fcc partial dislocations. Accordingly, we now examine the scenario whereby a partial, that is an extended dislocation, is emitted and traverses the nanograin.
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Deformation mechanism transitions
Figure 1.
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Transmission electron micrographs of a non-equilibrium grain boundary.
Figure 2.
Emission of a perfect dislocation.
2.1. The mechanics of extended dislocations Consider the primary slip system to be b ¼ ða=2Þ½101 ; its two partials are thereby bð1Þ ¼ ða=6Þ½21 1 and bð2Þ ¼ ða=6Þ½112 . Recall that for the perfect lattice dislocation the energy per unit length is, within the limits of linear elasticity, R E ¼ Kij ; bi bj ln : ð2Þ r0 To complete the geometry, let the slip plane normal be m ¼ (1/31/2) [111], and the slip direction be s ¼ ð1=21=2 Þ½101 ; this leaves the unit vector in the slip plane
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orthogonal to s as z ¼ ð1=61=2 Þ½1 21 . K is the energy factor matrix defined by Barnett and Asaro (1972) and, for an elastically isotropic material, Kmm ¼ Kzz ¼ G/4(1 ) and Kss ¼ G/4p. Now consider the extended dislocation, extended through the distance d and with the stacking-fault energy G. The energy of the extended dislocation is R R R ð1Þ ð2Þ ð2Þ ð1Þ ð2Þ b ln b b ln b b ln ð3Þ E ¼ Kij bð1Þ þ K þ 2K þ Gð ro Þ: ij i ij i j j j i r0 r0 r0 Minimizing E by choice of d yields 2E12 ; ð4Þ G ð1Þ ð2Þ where E12 Kij bj bj . Now assume the partial (1) to be fixed and let partial (2) move, that is extend relative to (1); if they are reversed, the signs of the driving stresses would change. We then consider the total energy change when the dislocation extends, including the work done by the applied stresses ms and mz: ð2Þ E ¼ 2E12 ln ð5Þ þ ðms bð2Þ s þ mz bz Þð r0 Þ þ ð r0 Þ: r0
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eq ¼
If E is minimized relative to d, the equilibrium spacing becomes ¼
2E12 ; G
ð6Þ
ð2Þ where G G þ ðms bð2Þ s þ mz bz Þ. It should be apparent that deviations from the Schmid rule of a critical resolved shear stress are inherent in any such analysis as is under development, but at this juncture we shall consider only the influence of the Schmid stress ms. We leave the exploration of such deviations to a later study and keep in mind the findings of Asaro and Rice (1977) of the importance of such deviations from what amounts to a normality rule for flow vis-a`-vis the promotion of localized plastic flow. Specifically, Asaro and Rice (1977) found that such deviations would promote the onset of intense localized plastic flow, either in the form of intense shear bands or ‘kinking-type’ bands. Equations (3)–(5) allow us to calculate the shear stress required to drive a partial dislocation across a grain of dimension d; that is d is now set to d ¼ d. We do this by assuming that the terms involving mz are of the same order in magnitude as those involving ms, since in the average grain this is as likely to be true as not. Note how the mechanics on this scale lead to a much less accurate picture for the notion of the Schmid rule. Consider figure 3, which illustrates the extension of a partial dislocation from the boundary into the intragranular region; we shall imagine that it traverses the entire grain. As it does, it produces two segments whose energy per unit length in a fcc crystal is, in fact, 13ð1=2Gb2 Þ. Thus this leads as before to a contribution to the required shear stress of 13Gðb=dÞ. We next define d/deq and with this, if we set d ¼ d, equation (6) yields, when combined with the stress required to create the additional two segments, the result
1b 1 G þ : G 3d Gb
ð7Þ
Note that for Cu this leads to required stresses considerably lower than does the result given earlier in equation (1). For example, if d ¼ 25.6 nm, b/d 1/100 and thus the required shear stress suggested by equation (1) would be G/100 versus the G/300
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Deformation mechanism transitions
Figure 3.
Table 1.
Emission of a partial dislocation.
Numerical predictions from equations (7) and (8) Stress (MPa)
Cu Ni Ag Pd
d ¼ 50 nm
d ¼ 30 nm
d ¼ 20 nm
d ¼ 10 nm
deq (nm)
G/Gb
211 960 125 719
2481 1027 158 763
291 1115 198 820
421 1381 321 988
1.6 0.5 3.4 0.54
1/250 1/100 1/447 1/75
from the first term in equation (7). As G/Gb 1/250 for Cu, the second term in equation (7) yields a maximum contribution of G/250 which, in combination with the contribution of G/300, yields a total result approximately 33% less than found from equation (1). Of course, as the grain size decreases even further, the differences between the predictions increase. The other obvious feature of the result in equation (7) is the effect of stacking-fault energy which appears as a primary influence on the required stress. We note for interest that, for a fcc crystal, equation (4) yields, for deq, eq ¼
1 Gb : 12p G
ð8Þ
It will be useful to note what this yields, inter alia, regarding equilibrium partial dislocation extensions without stress in what follows. Some numerical predictions from (7) and (8) are listed in table 1. The trends shown in table 1 demonstrate the role of stacking-fault energy in setting limits to achievable strength metals with a grain nanometric size. For example, it is helpful to compare not just the magnitudes
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Figure 4.
Macroscopic shear zone formation.
of the predicted stress levels but also their ratio to the shear modulus. For Cu at a grain size of, say, 20 nm, /G 7.2 103 whereas, for Ni, /G 1.31 102. This together with the higher modulus for Ni, leads to the nearly four-fold increase in predicted stress for Ni versus Cu! The estimates for uniaxial tensile strength that follow from the predicted shear strength values listed in table 1 would be estimated as y 2, and this would produce estimates that are typically higher than reported except those measured at high strain rates. For example, Dalla Torre et al. (2002) showed a significant dependence on strain rate for nanostructured Ni where the initial flow stresses are of the order of 1500 MPa at strain rates of the order of 10 s1, but of the order of 2400 MPa at strain rates of the order 103 s1. These values are indeed consistent with the estimates shown in table 1. At strain rates of the order of 104–103 s1, the flow stresses were only of the order of 800–1200 MPa, and this needs to be explained as discussed below. One quite reasonable approach is to recognize that GBS provides an alternative deformation mechanism, and we explore this transition next. 2.2. The onset of grain-boundary sliding Conrad and Narayang (2000) have recently provided a correlation between a simple phenomenological model of GBS in nanostructured metals and available data; we shall use that model to attempt to provide a rationale for the transition between dislocation-induced deformation and GBS. This will turn out, not surprisingly, to lead to a picture that is strain rate dependent. Figure 4 illustrates the scheme, consistent with their model. They construct what they refer to as the macroscopic shearing rate as 6bvD v F _ ¼ sin exp ; ð9Þ kT kT d where vD is a typical lattice vibrational frequency (equal to attend 103 s1), v is an atomic volume (taken as b3 by them) and F is an activation energy for localized
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Figure 5.
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Predicted shear stress levels.
lattice–GB diffusion. We ignore any other contributions to the driving stress other than what is required by equation (9) to drive the GBS process. Equation (9) may be solved for , or in fact /G, to compare with the estimates from equation (7). This we have done and display the results in figure 5. Before reviewing these results it is worth noting that the scaling of shear rate with d1 simply reflects the scaling with the ratio of the grain-boundary area to volume. Models such as these have, in fact, been used to rationalize what appear to be decreases in strength with decreases in grain size below critical grain sizes. Our present model allows some preliminary commentary on this possibility as well. We have co-plotted in figure 5 the segment of the curve generated from equation (7) covering the grain-size range 10 nm 103 , no such transition should be realized. Figure 5 also contains one similar plot for Ni using again the data for equation (9) supplied by Conrad and Narayan (2000); the plot of the results from table 1 is likewise plotted. Two trends are evident, namely dislocation-induced flow dominates at higher strain rates and the grain sizes at which transitions occur are lower. For example, at a strain rate of 3.44 104 s1, the transition from dislocation-induced deformation to GBS occurs at a grain size of approximately 15 nm. Of course, at this strain rate the critical grain size at the expected transition would be smaller than the 32 nm described above and would be in the same range of 15–20 nm as well. For Ni these results indicated that within the grain sizes of, say 10 to 50 nm, transitions are expected at strain rates in the range 102–103 s1. This is essentially consistent with the data presented by Dalla Torre et al. (2002), shown in figure 6 below.
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Figure 6.
Tensile data for nanocrystalline Ni.
2.3. Other fcc metals The results shown in table 1 indicate the trends within a group of fcc metals regarding the influence of stacking-fault energy, as embodied in the quantity G/Gb. It would appear that Ag would probably not develop the same sort of very high homologous shear strength as, say, Ni or Pd for that matter. Pd has a particularly high stacking-fault energy (we used the value of 180 mJ m2) and thus has a high resistance to the injection of partial dislocation as envisaged herein. Cu is intermediate in terms of its potential to become ‘strong’ by nanostructuring. Precisely these trends are found in the data of Nieman et al. (1992), who show data for Ag, Cu and Pd all processed by inert-gas condensation. The data are subject to the usual provisos due to the tendency to introduce defects via this process; yet the ranking of these three metals is precisely in accord with the ordering derived from table 1. In no case did their materials achieve strength levels of the order listed in table 1 or figure 5. The strain rates of their tensile tests were between 105 and 6 104 s1 and were in the range, therefore, where GBS could influence the deformation.
} 3. Discussion Van Swygenhoven et al. (1999) performed a series of molecular dynamic simulations on the deformation of Cu and Ni and noted the importance of both the stacking-fault energy and the grain size in determining mechanisms; the present model is consistent with these results in several key aspects. Firstly, their simulations showed that, at a sufficiently small grain size, intragrain dislocation-induced slip occurs via the appearance of stacking faults within the grains, emanating from the boundaries. In the present model, we analyse the mechanics of partial dislocation emission and find that it results in a far more viable picture for slip. Van Swygenhoven et al. (1999) also found stacking faults i.e. partial dislocation) appearing at smaller grain sizes in Cu than Ni, and the current model would lead to this conclusion based on the trade-off between GBS and GBE as illustrated in figure 5. In our case, however, we note that the precise grain sizes at which this occurs depends on applied strain rate. The grain sizes at which transitions occurred in their studies were in the range of 8 nm for Cu and 15 nm for Ni, but this is surely influenced by the
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extreme strain rates induced in a molecular dynamics simulation. Still another feature of the phenomenology noted by Van Swygenhoven et al. (1999) was that the partial dislocation activity observed in their simulations did not appear to conform to something akin to a Schmid rule of a critical resolved shear stress, but this is completely understandable from the fundamental nature of the model presented herein. Equation (6) embodies the reasons for this, as can be seen by just noting the definition of G ; as Asaro and Rice (1977) explained in the context of partial dislocation motion and processes such as cross-slip, the influence on non-Schmid stresses is inherent in the process of dislocation motion. (This leads to inter alia deviations from the rule of normality in plastic flow, which in turn influence the stability of uniform plastic flow.) In fact, in the analysis by Asaro and Rice (1977), the approximate adherence to the Schmid rule, for fcc metals, rests to a significant extent on the amplifying effects of dislocation pile-ups on the primary slip plane, and such pile-up structures are not possible in nanostructured grains. The role of GBS needs to be better quantified. Furthermore, more precise models are necessary in order to make quantitative predictions of the transitional regimes between GBE and GBS, and to provide a full constitutive framework for describing rate-, temperature- and time-dependent deformation in nanostructured metals and alloys. The very thoughtful model and calibrated data of Conrad and Narayan (2000) is a good starting point, but much more experimental observation is required of the phenomenology of GBS. For example, it would be of value to explore the response of nanostructed metals so calibrated by their model of the application of stress levels with intensities of, say, 90% or 80% of those that induce strain rates of, say, 104 or 105 s1, that is to explore the creep regime in more details to test the range of applicability of formulas such as equation (9). This is necessary to confirm the more general applicability of models such as expressed in equation (9) for describing the phenomenology of GBS. For example, while it is evident that the calibration of equation (9) has been done for strain rates in the quasistatic range of, say, 103–105 s1, it is not clear that any such formula holds for higher strain rates above, say 101–103 s1. An obvious dilemma with the numerology shown in figure 5 is that, although the trends in all aspects regarding the effects of stacking-fault energy and modulus are consistent with experiment, the magnitudes of the strain rates through which the apparent transitions in deformation mechanism occur are too low. For example, the data of Dalla Torre et al. (2002) demonstrated that the flow stresses in nanostructured Ni decrease with strain rates falling through the range 102–101 s1, and not just through the range 102–104 s1, shown in figure 5. An example of their data is shown in figure 6, where it should be noted that these particular data were obtained from tensile tests performed on what they called their 20 mm samples. These specimens were ‘dogbone’-like with cross-sections of dimensions 2.5 mm 0.2 mm; they thus were of a shape that would have promoted a strain state between uniaxial and plane strain. Dalla Torre et al. (2002) did in fact note differences between the response of this type of specimen and their more equiaxed (and smaller) specimens with cross-sections of dimension 0.2 mm 0.25 mm. None the less, the data indicate that there are reductions in flow stress between the strain rates 103 and 101 s1. Their data also show continued reductions in flow stress in the more quasistatic strain-rate range between 102 and 105 s1, although the reduction at the higher rates is larger in magnitude. There is then a strain-rate sensitivity that is yet to be accounted for by the specific models presented herein. A possible source for this may be found in the
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Figure 7.
Vickers hardness for nanocrystalline Ag, Cu and Pd.
mechanics of stacking-fault emission, or even perfect dislocation emission, and involving possible mechanisms of relaxation of the residual dislocation structures that are envisaged to form in figures 2 and 3. To understand what this would mean, vis-a`-vis the model presented here, we note that Dalla Torre et al. (2002) showed data that listed initial flow stresses of nanostructured Ni with grain sizes in the 20 nm size range as low as 1000–1400 MPa in response to strain rates in the range 105– 102 s1. In the strain-rate range this would mean that the contributions of the first term in equation (7) would need to vanish. If the residual segments depicted to form in figures 2 or 3 were to vanish, say, via annihilation, through reactions, with other structure in the boundary, this may occur. This type of process may indeed involve very-short-range diffusion or atomic shuffling and thus be rate dependent. This is, however, not possible to quantify without a far more detailed understanding of the nature of the boundaries that exist in nanostructured metals. This is another topic that needs to be researched further. Experimental data for Ag, Cu and Pd from Nieman et al. (1992) are shown in figure 7. In this case, hardness measurements were made rather than tensile testing. The ranking with respect to strength versus stacking-fault energy is clear as explained above. Moreover, since the shear moduli of Pd and Cu are very similar and that of Cu is only modestly less (34 GPa for Ag versus 41 GPa for Cu), the ranking most clearly reflects that of stacking-fault energy per se. The stacking-fault energy of Pd is approximately 180 mJ m2, which places it above Ag and Cu, as seen. The perspective offered by the model analysis presented here suggests that there are at least two levels of transition in deformation mechanism in nanostructured metals, in particular the nanostructured fcc metals examined here. These transitions appear to occur in the strain-rate regimes at high and again at quasistatic strain rates. In the strain-rate range between 10 and 103 s1 the transition may occur via a process involving relaxation within the boundaries, which mitigates the excess energy that would be formed when a partial dislocation is emitted into intragranular spaces. Within the context of the simple model presented here, this would effectively reduce or eliminate the first term in equation (7). This would, in fact, explain at least in terms of magnitude, the reduction in flow stress level shown in figure 6. A more quantitative perspective on this may come from molecular dynamics simulations,
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coupled to experimental observations that confirm that other mechanisms such as GBS do not occur at these high strain rates. At quasistatic strain rates, say, in the range 101–105 s1 or below, GBS—at least according to the calibrations of equation (9) offered by Conrad and Narayan (2000)—can provide a lower-stress mechanism for deformation. This too needs further confirmation via experiment and more detailed modelling. We further note that the model presented here for deformation by stacking-fault emission has implications for deformation phenomenology such as texture development and the ‘yield criteria’. It has already been noted that the Schmid law would be replaced by a multiple-shear-stress criteria for the activity of faulting on a given slip plane; this will require experimental investigation as well. In some ways the mechanics of deformation in nanostructured metals and alloys may be easier to understand and quantify than in coarser-grained ductile crystalline materials. A reason for this is that the complexity of three-dimensional arrays of dislocation networks is absent. The phenomenology of GBS, however, is complex and especially when a complete quantitative framework is desired—which, of course, it is. This is the subject of ongoing research.
ACKNOWLEDGEMENTS R.J.A. and P.K. would like to thank the National Science Foundation NIRT initiative, specifically under grant 0210173. REFERENCES ASARO, R. J., and RICE, J. R., 1977, J. Mech. Phys. Solids, 25, 309. BARNETT, D. M., and ASARO, R. J., 1972, J. Mech. Phys. Solids, 20, 353. CONRAD, H., and NARAYAN, J., 2000, Scripta mater., 42, 1025. DALLA TORRE, F., VAN SWYGENHOVEN, H., and VICTORIA, M., 2002, Acta mater., 50, 3957. HUANG, J. Y., ZHU, Y. T., JIANG, H., and LOWE, T. C., 2001, Acta mater., 49, 1497. KUMAR, K. S., SURESH, S., CHISHOLM, M. F., HORTON, J. A., and WANG, P., 2003, Acta mater., 51, 387. NIEMAN, G. W., WEERTMAN, J. R., and SIEGEL, R. W., 1992, Nanostruct. Mater., 1, 185. VAN SWYGENHOVEN, H., and DERLET, P. M., 2001, Phys. Rev. B, 64, 224 105–1. VAN SWYGENHOVEN, H., SPACZER, M., and CARO, A., 1999, Acta mater., 47, 3117.
Philosophical Magazine Letters Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tphl20
Deformation mechanism transitions in nanoscale fcc metals a
a
Robert J. Asaro , Petr Krysl & Bimal Kad
a
a
Department of Structural Engineering , University of California , San Diego, 9500 Gilman Drive, La Jolla, California 92093-0085, USA Published online: 04 Jun 2010.
To cite this article: Robert J. Asaro , Petr Krysl & Bimal Kad (2003) Deformation mechanism transitions in nanoscale fcc metals , Philosophical Magazine Letters, 83:12, 733-743, DOI: 10.1080/09500830310001614540 To link to this article: http://dx.doi.org/10.1080/09500830310001614540
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/ page/terms-and-conditions
PHILOSOPHICAL MAGAZINE LETTERS, December 2003 VOL. 83, NO. 12, 733–743
Deformation mechanism transitions in nanoscale fcc metals Robert J. Asaro, Petr Krysly and Bimal Kad Department of Structural Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0085, USA [Received in final form 22 July 2003 and accepted 25 July 2003 ]
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Abstract We consider possible mechanisms that lead to transitions in the mechanisms of deformation in fcc metals and alloys. In particular, we propose that, when grain sizes are below a critical size (i.e. below 100 nm), deformation can occur via the emission of stacking faults from grain boundaries into the intragranular space. A model is developed that accounts for observed experimental data and which, in turn, shows how stacking-fault energy together with shear modulus determines achievable strength. A mechanism is proposed based on this model for transitions at both high and quasistatic strain rates, including grain-boundary sliding.
} 1. Introduction When the grain size in metals and alloys changes from micrometres down to nanometres there are accompanying transitions in the mechanisms of inelastic deformation. There is direct experimental evidence for these transitions as well as suspicions that derive from what is known about the mechanisms of plastic deformation in crystalline metals and alloys. For example, in fcc metals with grain sizes in the micron and large size range, plastic deformation occurs via the generation and motion of intragranular slip, that is dislocation motion. This process is evidently shut off at grain sizes below the micron size range. To understand this, it suffices to note that the crystallographic shear stresses required to move the dislocation segments that exist within the well-characterized networks which evolve during plastic flow are of the order of Gb=‘ where G is the elastic shear modulus, b the Burgers vector magnitude of the slip dislocation and ‘ the dislocation segment length. However, if dislocations are to be confined to the intra granular space (which might be taken as the definition of a grain), ‘ must be less than the grain diameter d, say ‘ ¼ d=2 or d/3, and thus (2 3)G (b/d). This suggests a required shear stress larger than is observed or actually required as we show below. (Microscopic observations, together with well-known dislocation models for the operation of dislocation segments, have clearly shown that ‘ must be smaller than d.) The two mechanisms that appear to operate on nanoscale grain diameters involve, firstly, grain-boundary emission (GBE) (and subsequent absorption) of dislocations and, secondly, grain-boundary sliding (GBS). yAuthor for correspondence. Email: [email protected]. Philosophical Magazine Letters ISSN 0950–0839 print/ISSN 1362–3036 online # 2003 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/09500830310001614540
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The purpose of this letter is to prove a simple and yet compelling mechanistic picture of why these transitions occur and to show how lattice and chemical properties of the metals influence the transitions and help to set the levels of achievable strength. In particular the role of stacking-fault energy is shown to be of first order, as well as that of elastic modulus; the effect of the latter is, of course, expected. The stacking-fault energy in fact serves to set limits to the strength levels that may be realized, as well as to explain the distinct differences between metals with simple fcc structures. We shall develop arguments based principally around the data available for nominally pure Cu and Ni and provide some correlations for other metals such as Pd and Ag, at least as far as the existing data allows.
} 2. Dislocation models for intragranular deformation From what has been noted above, it should already be expected that observations via transmission electron microscope would show that grain matrices in nanoscale metals are essentially free of dislocation. Dislocations do reside in the boundaries, often in addition to those present in the boundaries and these account for the boundary misorientations. Such boundaries are often referred to as nonequilibrium boundaries. An example is shown in figure 1 taken from the work of Huang et al. (2001) and their physical basis has been described in the elegant molecular dynamic simulations of Van Swygenhoven et al. (1999) and Van Swygenhoven and Derlet (2001). As dislocations within the boundary of this type are able to move, under sufficient stress, into the interior of the grains, they can contribute to increments of strain. Figure 2 illustrates a possible mechanism for this. It is important to note that the segment of dislocation that is to move across the grain in question must be part of a loop, whose structure is unknown for now, but this in itself suggests that more must be understood about the nature of these grain boundary dislocations. As figure 2 depicts, the dislocation moves across the grain a distance dx and creates two segments, each of length dx. For now, but soon to be revised, we take this dislocation segment to be a perfect lattice dislocation with a Burgers vector b ¼ (a/2) [110]; it thus has an energy (per unit length) of E ¼ 12 Gb2 . As noted in the figure, this leads to a prediction for the resolved shear stress to accomplish this: b ¼G : d
ð1Þ
It should be noted that interesting d1 scaling which results from this is irrespective of the details but comes about solely via the fact that the area over which work can be done (to supply the energy needed to create additional defect) scales with d. In other words, any embellishments to the precise description of the dislocation segments produced will not change the d1 scaling. The problem with the formula as it stands is that it predicts critical resolved shear stresses that are too large, as described later. Some typical numbers for Cu are listed in the figure. Then without further consideration we seek to find how these large stresses can be mitigated. A key to this is to recall that the grain sizes of interest, say d < 50 nm, are of size scales approaching the spacings between fcc partial dislocations. Accordingly, we now examine the scenario whereby a partial, that is an extended dislocation, is emitted and traverses the nanograin.
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Deformation mechanism transitions
Figure 1.
735
Transmission electron micrographs of a non-equilibrium grain boundary.
Figure 2.
Emission of a perfect dislocation.
2.1. The mechanics of extended dislocations Consider the primary slip system to be b ¼ ða=2Þ½101 ; its two partials are thereby bð1Þ ¼ ða=6Þ½21 1 and bð2Þ ¼ ða=6Þ½112 . Recall that for the perfect lattice dislocation the energy per unit length is, within the limits of linear elasticity, R E ¼ Kij ; bi bj ln : ð2Þ r0 To complete the geometry, let the slip plane normal be m ¼ (1/31/2) [111], and the slip direction be s ¼ ð1=21=2 Þ½101 ; this leaves the unit vector in the slip plane
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orthogonal to s as z ¼ ð1=61=2 Þ½1 21 . K is the energy factor matrix defined by Barnett and Asaro (1972) and, for an elastically isotropic material, Kmm ¼ Kzz ¼ G/4(1 ) and Kss ¼ G/4p. Now consider the extended dislocation, extended through the distance d and with the stacking-fault energy G. The energy of the extended dislocation is R R R ð1Þ ð2Þ ð2Þ ð1Þ ð2Þ b ln b b ln b b ln ð3Þ E ¼ Kij bð1Þ þ K þ 2K þ Gð ro Þ: ij i ij i j j j i r0 r0 r0 Minimizing E by choice of d yields 2E12 ; ð4Þ G ð1Þ ð2Þ where E12 Kij bj bj . Now assume the partial (1) to be fixed and let partial (2) move, that is extend relative to (1); if they are reversed, the signs of the driving stresses would change. We then consider the total energy change when the dislocation extends, including the work done by the applied stresses ms and mz: ð2Þ E ¼ 2E12 ln ð5Þ þ ðms bð2Þ s þ mz bz Þð r0 Þ þ ð r0 Þ: r0
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eq ¼
If E is minimized relative to d, the equilibrium spacing becomes ¼
2E12 ; G
ð6Þ
ð2Þ where G G þ ðms bð2Þ s þ mz bz Þ. It should be apparent that deviations from the Schmid rule of a critical resolved shear stress are inherent in any such analysis as is under development, but at this juncture we shall consider only the influence of the Schmid stress ms. We leave the exploration of such deviations to a later study and keep in mind the findings of Asaro and Rice (1977) of the importance of such deviations from what amounts to a normality rule for flow vis-a`-vis the promotion of localized plastic flow. Specifically, Asaro and Rice (1977) found that such deviations would promote the onset of intense localized plastic flow, either in the form of intense shear bands or ‘kinking-type’ bands. Equations (3)–(5) allow us to calculate the shear stress required to drive a partial dislocation across a grain of dimension d; that is d is now set to d ¼ d. We do this by assuming that the terms involving mz are of the same order in magnitude as those involving ms, since in the average grain this is as likely to be true as not. Note how the mechanics on this scale lead to a much less accurate picture for the notion of the Schmid rule. Consider figure 3, which illustrates the extension of a partial dislocation from the boundary into the intragranular region; we shall imagine that it traverses the entire grain. As it does, it produces two segments whose energy per unit length in a fcc crystal is, in fact, 13ð1=2Gb2 Þ. Thus this leads as before to a contribution to the required shear stress of 13Gðb=dÞ. We next define d/deq and with this, if we set d ¼ d, equation (6) yields, when combined with the stress required to create the additional two segments, the result
1b 1 G þ : G 3d Gb
ð7Þ
Note that for Cu this leads to required stresses considerably lower than does the result given earlier in equation (1). For example, if d ¼ 25.6 nm, b/d 1/100 and thus the required shear stress suggested by equation (1) would be G/100 versus the G/300
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Deformation mechanism transitions
Figure 3.
Table 1.
Emission of a partial dislocation.
Numerical predictions from equations (7) and (8) Stress (MPa)
Cu Ni Ag Pd
d ¼ 50 nm
d ¼ 30 nm
d ¼ 20 nm
d ¼ 10 nm
deq (nm)
G/Gb
211 960 125 719
2481 1027 158 763
291 1115 198 820
421 1381 321 988
1.6 0.5 3.4 0.54
1/250 1/100 1/447 1/75
from the first term in equation (7). As G/Gb 1/250 for Cu, the second term in equation (7) yields a maximum contribution of G/250 which, in combination with the contribution of G/300, yields a total result approximately 33% less than found from equation (1). Of course, as the grain size decreases even further, the differences between the predictions increase. The other obvious feature of the result in equation (7) is the effect of stacking-fault energy which appears as a primary influence on the required stress. We note for interest that, for a fcc crystal, equation (4) yields, for deq, eq ¼
1 Gb : 12p G
ð8Þ
It will be useful to note what this yields, inter alia, regarding equilibrium partial dislocation extensions without stress in what follows. Some numerical predictions from (7) and (8) are listed in table 1. The trends shown in table 1 demonstrate the role of stacking-fault energy in setting limits to achievable strength metals with a grain nanometric size. For example, it is helpful to compare not just the magnitudes
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Figure 4.
Macroscopic shear zone formation.
of the predicted stress levels but also their ratio to the shear modulus. For Cu at a grain size of, say, 20 nm, /G 7.2 103 whereas, for Ni, /G 1.31 102. This together with the higher modulus for Ni, leads to the nearly four-fold increase in predicted stress for Ni versus Cu! The estimates for uniaxial tensile strength that follow from the predicted shear strength values listed in table 1 would be estimated as y 2, and this would produce estimates that are typically higher than reported except those measured at high strain rates. For example, Dalla Torre et al. (2002) showed a significant dependence on strain rate for nanostructured Ni where the initial flow stresses are of the order of 1500 MPa at strain rates of the order of 10 s1, but of the order of 2400 MPa at strain rates of the order 103 s1. These values are indeed consistent with the estimates shown in table 1. At strain rates of the order of 104–103 s1, the flow stresses were only of the order of 800–1200 MPa, and this needs to be explained as discussed below. One quite reasonable approach is to recognize that GBS provides an alternative deformation mechanism, and we explore this transition next. 2.2. The onset of grain-boundary sliding Conrad and Narayang (2000) have recently provided a correlation between a simple phenomenological model of GBS in nanostructured metals and available data; we shall use that model to attempt to provide a rationale for the transition between dislocation-induced deformation and GBS. This will turn out, not surprisingly, to lead to a picture that is strain rate dependent. Figure 4 illustrates the scheme, consistent with their model. They construct what they refer to as the macroscopic shearing rate as 6bvD v F _ ¼ sin exp ; ð9Þ kT kT d where vD is a typical lattice vibrational frequency (equal to attend 103 s1), v is an atomic volume (taken as b3 by them) and F is an activation energy for localized
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Deformation mechanism transitions
Figure 5.
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Predicted shear stress levels.
lattice–GB diffusion. We ignore any other contributions to the driving stress other than what is required by equation (9) to drive the GBS process. Equation (9) may be solved for , or in fact /G, to compare with the estimates from equation (7). This we have done and display the results in figure 5. Before reviewing these results it is worth noting that the scaling of shear rate with d1 simply reflects the scaling with the ratio of the grain-boundary area to volume. Models such as these have, in fact, been used to rationalize what appear to be decreases in strength with decreases in grain size below critical grain sizes. Our present model allows some preliminary commentary on this possibility as well. We have co-plotted in figure 5 the segment of the curve generated from equation (7) covering the grain-size range 10 nm
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Figure 6.
Tensile data for nanocrystalline Ni.
2.3. Other fcc metals The results shown in table 1 indicate the trends within a group of fcc metals regarding the influence of stacking-fault energy, as embodied in the quantity G/Gb. It would appear that Ag would probably not develop the same sort of very high homologous shear strength as, say, Ni or Pd for that matter. Pd has a particularly high stacking-fault energy (we used the value of 180 mJ m2) and thus has a high resistance to the injection of partial dislocation as envisaged herein. Cu is intermediate in terms of its potential to become ‘strong’ by nanostructuring. Precisely these trends are found in the data of Nieman et al. (1992), who show data for Ag, Cu and Pd all processed by inert-gas condensation. The data are subject to the usual provisos due to the tendency to introduce defects via this process; yet the ranking of these three metals is precisely in accord with the ordering derived from table 1. In no case did their materials achieve strength levels of the order listed in table 1 or figure 5. The strain rates of their tensile tests were between 105 and 6 104 s1 and were in the range, therefore, where GBS could influence the deformation.
} 3. Discussion Van Swygenhoven et al. (1999) performed a series of molecular dynamic simulations on the deformation of Cu and Ni and noted the importance of both the stacking-fault energy and the grain size in determining mechanisms; the present model is consistent with these results in several key aspects. Firstly, their simulations showed that, at a sufficiently small grain size, intragrain dislocation-induced slip occurs via the appearance of stacking faults within the grains, emanating from the boundaries. In the present model, we analyse the mechanics of partial dislocation emission and find that it results in a far more viable picture for slip. Van Swygenhoven et al. (1999) also found stacking faults i.e. partial dislocation) appearing at smaller grain sizes in Cu than Ni, and the current model would lead to this conclusion based on the trade-off between GBS and GBE as illustrated in figure 5. In our case, however, we note that the precise grain sizes at which this occurs depends on applied strain rate. The grain sizes at which transitions occurred in their studies were in the range of 8 nm for Cu and 15 nm for Ni, but this is surely influenced by the
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extreme strain rates induced in a molecular dynamics simulation. Still another feature of the phenomenology noted by Van Swygenhoven et al. (1999) was that the partial dislocation activity observed in their simulations did not appear to conform to something akin to a Schmid rule of a critical resolved shear stress, but this is completely understandable from the fundamental nature of the model presented herein. Equation (6) embodies the reasons for this, as can be seen by just noting the definition of G ; as Asaro and Rice (1977) explained in the context of partial dislocation motion and processes such as cross-slip, the influence on non-Schmid stresses is inherent in the process of dislocation motion. (This leads to inter alia deviations from the rule of normality in plastic flow, which in turn influence the stability of uniform plastic flow.) In fact, in the analysis by Asaro and Rice (1977), the approximate adherence to the Schmid rule, for fcc metals, rests to a significant extent on the amplifying effects of dislocation pile-ups on the primary slip plane, and such pile-up structures are not possible in nanostructured grains. The role of GBS needs to be better quantified. Furthermore, more precise models are necessary in order to make quantitative predictions of the transitional regimes between GBE and GBS, and to provide a full constitutive framework for describing rate-, temperature- and time-dependent deformation in nanostructured metals and alloys. The very thoughtful model and calibrated data of Conrad and Narayan (2000) is a good starting point, but much more experimental observation is required of the phenomenology of GBS. For example, it would be of value to explore the response of nanostructed metals so calibrated by their model of the application of stress levels with intensities of, say, 90% or 80% of those that induce strain rates of, say, 104 or 105 s1, that is to explore the creep regime in more details to test the range of applicability of formulas such as equation (9). This is necessary to confirm the more general applicability of models such as expressed in equation (9) for describing the phenomenology of GBS. For example, while it is evident that the calibration of equation (9) has been done for strain rates in the quasistatic range of, say, 103–105 s1, it is not clear that any such formula holds for higher strain rates above, say 101–103 s1. An obvious dilemma with the numerology shown in figure 5 is that, although the trends in all aspects regarding the effects of stacking-fault energy and modulus are consistent with experiment, the magnitudes of the strain rates through which the apparent transitions in deformation mechanism occur are too low. For example, the data of Dalla Torre et al. (2002) demonstrated that the flow stresses in nanostructured Ni decrease with strain rates falling through the range 102–101 s1, and not just through the range 102–104 s1, shown in figure 5. An example of their data is shown in figure 6, where it should be noted that these particular data were obtained from tensile tests performed on what they called their 20 mm samples. These specimens were ‘dogbone’-like with cross-sections of dimensions 2.5 mm 0.2 mm; they thus were of a shape that would have promoted a strain state between uniaxial and plane strain. Dalla Torre et al. (2002) did in fact note differences between the response of this type of specimen and their more equiaxed (and smaller) specimens with cross-sections of dimension 0.2 mm 0.25 mm. None the less, the data indicate that there are reductions in flow stress between the strain rates 103 and 101 s1. Their data also show continued reductions in flow stress in the more quasistatic strain-rate range between 102 and 105 s1, although the reduction at the higher rates is larger in magnitude. There is then a strain-rate sensitivity that is yet to be accounted for by the specific models presented herein. A possible source for this may be found in the
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Figure 7.
Vickers hardness for nanocrystalline Ag, Cu and Pd.
mechanics of stacking-fault emission, or even perfect dislocation emission, and involving possible mechanisms of relaxation of the residual dislocation structures that are envisaged to form in figures 2 and 3. To understand what this would mean, vis-a`-vis the model presented here, we note that Dalla Torre et al. (2002) showed data that listed initial flow stresses of nanostructured Ni with grain sizes in the 20 nm size range as low as 1000–1400 MPa in response to strain rates in the range 105– 102 s1. In the strain-rate range this would mean that the contributions of the first term in equation (7) would need to vanish. If the residual segments depicted to form in figures 2 or 3 were to vanish, say, via annihilation, through reactions, with other structure in the boundary, this may occur. This type of process may indeed involve very-short-range diffusion or atomic shuffling and thus be rate dependent. This is, however, not possible to quantify without a far more detailed understanding of the nature of the boundaries that exist in nanostructured metals. This is another topic that needs to be researched further. Experimental data for Ag, Cu and Pd from Nieman et al. (1992) are shown in figure 7. In this case, hardness measurements were made rather than tensile testing. The ranking with respect to strength versus stacking-fault energy is clear as explained above. Moreover, since the shear moduli of Pd and Cu are very similar and that of Cu is only modestly less (34 GPa for Ag versus 41 GPa for Cu), the ranking most clearly reflects that of stacking-fault energy per se. The stacking-fault energy of Pd is approximately 180 mJ m2, which places it above Ag and Cu, as seen. The perspective offered by the model analysis presented here suggests that there are at least two levels of transition in deformation mechanism in nanostructured metals, in particular the nanostructured fcc metals examined here. These transitions appear to occur in the strain-rate regimes at high and again at quasistatic strain rates. In the strain-rate range between 10 and 103 s1 the transition may occur via a process involving relaxation within the boundaries, which mitigates the excess energy that would be formed when a partial dislocation is emitted into intragranular spaces. Within the context of the simple model presented here, this would effectively reduce or eliminate the first term in equation (7). This would, in fact, explain at least in terms of magnitude, the reduction in flow stress level shown in figure 6. A more quantitative perspective on this may come from molecular dynamics simulations,
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coupled to experimental observations that confirm that other mechanisms such as GBS do not occur at these high strain rates. At quasistatic strain rates, say, in the range 101–105 s1 or below, GBS—at least according to the calibrations of equation (9) offered by Conrad and Narayan (2000)—can provide a lower-stress mechanism for deformation. This too needs further confirmation via experiment and more detailed modelling. We further note that the model presented here for deformation by stacking-fault emission has implications for deformation phenomenology such as texture development and the ‘yield criteria’. It has already been noted that the Schmid law would be replaced by a multiple-shear-stress criteria for the activity of faulting on a given slip plane; this will require experimental investigation as well. In some ways the mechanics of deformation in nanostructured metals and alloys may be easier to understand and quantify than in coarser-grained ductile crystalline materials. A reason for this is that the complexity of three-dimensional arrays of dislocation networks is absent. The phenomenology of GBS, however, is complex and especially when a complete quantitative framework is desired—which, of course, it is. This is the subject of ongoing research.
ACKNOWLEDGEMENTS R.J.A. and P.K. would like to thank the National Science Foundation NIRT initiative, specifically under grant 0210173. REFERENCES ASARO, R. J., and RICE, J. R., 1977, J. Mech. Phys. Solids, 25, 309. BARNETT, D. M., and ASARO, R. J., 1972, J. Mech. Phys. Solids, 20, 353. CONRAD, H., and NARAYAN, J., 2000, Scripta mater., 42, 1025. DALLA TORRE, F., VAN SWYGENHOVEN, H., and VICTORIA, M., 2002, Acta mater., 50, 3957. HUANG, J. Y., ZHU, Y. T., JIANG, H., and LOWE, T. C., 2001, Acta mater., 49, 1497. KUMAR, K. S., SURESH, S., CHISHOLM, M. F., HORTON, J. A., and WANG, P., 2003, Acta mater., 51, 387. NIEMAN, G. W., WEERTMAN, J. R., and SIEGEL, R. W., 1992, Nanostruct. Mater., 1, 185. VAN SWYGENHOVEN, H., and DERLET, P. M., 2001, Phys. Rev. B, 64, 224 105–1. VAN SWYGENHOVEN, H., SPACZER, M., and CARO, A., 1999, Acta mater., 47, 3117.
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