Suppressing fracture in microcompression

The effect of size on fracture has long been known in particle technology as the grinding limit, where further breakdown of particles ceases,^[ Reference Kendall ^{56 ]} and has been at the heart of most established theories of fracture where critical flaw sizes of ratios between surface area created and volume elastically relaxed are considered.^[ Reference Griffith ^{57 ]} It is the scaling laws in these concepts and the small volume, excluding pre-existing cracks, which allow the plastic deformation of even the most brittle materials by microcompression.

It was shown that where no pre-existing cracks are present, it is their nucleation—as a result of dislocation motion, intersection and lock formation—which must be avoided to achieve significant plasticity.^[ Reference Östlund, Howie, Ghisleni, Korte, Leifer, Clegg and Michler ^{2 ]} This is illustrated in Fig. 5 and of course depends on both the crystal orientation allowing single or multiple slip, as well as the relative stresses at which dislocations move or cracks extend, once nucleated. Within Fig. 3 the approach shown in Fig. 5 is applied in terms of the temperature-dependent flow stress and size of silicon micropillars and shows very good correlation of the experimentally observed and predicted ductile and brittle regimes. Other mechanisms and geometries of cracking have also been discussed in the literature^[ Reference Östlund, Rzepiejewska-Malyska, Leifer, Hale, Tang, Ballarini, Gerberich and Michler ^{1 ,} Reference Howie, Korte and Clegg ^{23 ]} and a size effect on the deformation behavior is also observed in the amorphous bulk metallic glasses, although its physical origin and magnitude is still under debate.^[ Reference Greer and De Hosson ^{58 ]}

Figure 5. Axial splitting and suppression of fast fracture in semiconductors and oxide ceramics. Splitting model/geometry from Ref. Reference Östlund, Howie, Ghisleni, Korte, Leifer, Clegg and Michler2 Reprinted with permission by Taylor & Francis Ltd. (www.tandfonline.com) from Ref. Reference Östlund, Howie, Ghisleni, Korte, Leifer, Clegg and Michler2, John Wiley and Sons from Ref. Reference Östlund, Rzepiejewska-Malyska, Leifer, Hale, Tang, Ballarini, Gerberich and Michler1 and Springer from Ref. 97, (Al₂O₃: unpublished).

Characterization of individual slip systems in hard and anisotropic materials

In hexagonal, ionic, and/or ordered structures, but especially in anisotropic crystals of which a substantial fraction has larger unit cells, it is important to be able to distinguish the properties of individual slip systems. Indentation is largely representative of the resistance governing three-dimensional (3D) accommodation of the tip, i.e., often the hardest slip system that is required to operate to fulfill the von Mises criterion. In contrast, failure—or in fact machinability as more desirable quality in brittle materials—is commonly governed by the weakest link, that is the softest slip system. A prominent example of this is given by the MAX phases,^[ Reference Barsoum and Radovic ^{59 ]} atomically layered ternary carbides with a critical resolved shear stress on the basal planes that is much lower than the commonly factor of the order of six assumed to link critical or flow shear stresses and hardness.

In microcompression, a near uniaxial stress state can be achieved and specific orientations for the single-crystal constituting the sample may be chosen and characterized individually (see Fig. 6); either by producing a well aligned single crystal and preparing several identical pillars by employing focused ion beam (FIB) milling or lithography, or alternatively by using EBSD to map the local crystal orientation followed by the site-specific FIB preparation. If the slip systems which operate in a given crystal structure are not known a priori, then indentation may nevertheless be exceedingly useful in a purposeful preparation of the more involved microcompression experiments. Correlated EBSD-nanoindentation maps may be used to identify statistically those slip planes on which deformation occurs if slip traces are formed on the surface. TEM, either conventional or high resolution, is easily performed after compression by the site-specific FIB milling and where necessary further thinning of a transparent membrane from the micropillar (see Figs. 3, 4, 6, and 10, for examples).

Figure 6. Characterization of individual slip systems in highly anisotropic crystals, here MgO. By choice of crystal orientation, like in macroscopic single-crystal studies, slip on specific sets of systems can be activated, analyzed by TEM and quantitative measurements of the critical resolved shear stresses achieved. A comparison of the soft ${\rm \{ 110\}} {\textstyle\,{{\rm 1} \over {\rm 2}}\,}\langle \overline {\rm 1} {\rm 10}\rangle $ and hard ${\rm \{ 001\}} {\textstyle\,{{\rm 1} \over {\rm 2}}\,}\langle \overline {\rm 1} {\rm 10}\rangle $ slip systems in MgO^[ Reference Korte and Clegg ^{70 ]} highlights the reduced relative importance of plasticity size effects on yielding on slip systems with high intrinsic strength governed by the lattice resistance rather than discrete obstacles and source length.

Size effects in hard crystal: dislocation motion and nucleation

The stresses measured are affected by size effects on plasticity.^[ Reference Kraft, Gruber, Mönig and Weygand ^{19 ,} Reference Uchic, Shade and Dimiduk ^{53 ]} Here, an advantage lies in a high bulk intrinsic strength governed by the lattice resistance due to the intrinsically much smaller length scale of double kink formation. Similarly, materials already strengthened by exploiting a length scale significantly below the pillar size, e.g., in dispersion and precipitation strengthened alloys,^[ Reference Uchic, Dimiduk, Florando and Nix ^{17 ,} Reference Baptiste, Andreas, Carl and Eduard ^{60 ]} exhibit a smaller or negligible size effect within the range of sizes studied by microcompression. This is shown for a range of materials in Fig. 7, including a fit using one of the simplest expressions for size effects due to source size as given by Parthasarathy et al.^[ Reference Parthasarathy, Rao, Dimiduk, Uchic and Trinkle ^{61 ]} The higher intrinsic strength therefore reduces the factor by which small-scale characterization overestimates the bulk material strength, producing more directly applicable data. This is also shown in a direct comparison of the critical resolved shear stress of MgO on the soft and hard slip systems with macroscopic data in Fig. 6, where a much more pronounced deviation is found on the soft system for identical processing and testing conditions.

Figure 7. Comparison of the effect of size on critical resolved shear stress across material classes, including FCC Al (unpublished), BCC Mo,^[ Reference Schneider, Frick, Arzt, Clegg and Korte ^{98 ]} the soft ${\rm \{ 110\}} {\textstyle\,{{\rm 1} \over {\rm 2}}\,}\langle \overline {\rm 1} {\rm 10}\rangle, $ and hard ${\rm \{ 001\}} {\textstyle\,{{\rm 1} \over {\rm 2}}\,}\langle \overline {\rm 1} {\rm 10}\rangle $ slip systems in MgO,^[ Reference Korte and Clegg ^{70 ]} the oxide ceramics MgAl₂O₄ spinel, and Al₂O₃ alumina (unpublished) as well as the semiconductors InAs (unpublished) and Si.^[ Reference Korte, Barnard, Stearn and Clegg ^{31 ]}

The hard materials are also often less prone to FIB damage in the form of formation of dislocations as seen frequently in the soft metals.^[ Reference Kiener, Motz, Rester, Jenko and Dehm ^{52 ]} Only a blessing at first glance, this does mean that nucleation must be considered to play a substantial role in determining the stresses required for deformation. In metals, where the stress required to move a dislocation, even a short segment, is usually much lower than the nucleation stress, the regimes of dislocation motion and nucleation are often readily distinguished.^[ Reference Johanns, Sedlmayr, Sudharshan Phani, Mönig, Kraft, George and Pharr ^{62 ]} In very hard materials, the critical stresses for dislocation nucleation (i.e., a fraction of the shear modulus), and the stress to overcome the lattice resistance may be close enough to not allow a distinction based on stress level alone. This is due to the experimental scatter and uncertainties of the stress calculation, such as the choice of representative cross-sectional area in tapered micropillars. As a result of the similar activation volumes of double kink formation and dislocation nucleation,^[ Reference Zhu, Li, Samanta, Leach and Gall ^{63 ]} a distinction based on variable rate testing also appears difficult. It should be noted, however, that in many cases stable deformation can be achieved even in brittle crystals. Under such conditions, continuous formation of new dislocations is required and where surface steps have been formed as a result of slip crossing the pillar, the nucleation stresses will fall significantly^[ Reference Li and Xu ^{64 ]} to or below the stresses required for nucleation motion.

An example of the surface affecting scatter and strength of silicon micropillars is shown as part of Fig. 3, where testing of pristine 2 µm pillars at room temperature led to appreciable scatter, presumably due to the need to nucleate dislocations from a fairly pristine surface damaged mostly in terms of amorphization during FIB milling, but with no line defects visible by TEM in the undeformed volume.^[ Reference Korte, Barnard, Stearn and Clegg ^{31 ]} Consistently, testing at the same temperature after exposure to some high-temperature surface oxidation resulted in a much smaller scatter and average strength values toward the lower bound of those measured in the pristine pillars.^[ Reference Korte, Barnard, Stearn and Clegg ^{31 ]}

In general, the effect of size on strength is less pronounced in hard materials, but also more difficult to quantify. This is due to the missing overlap in sample size tested in brittle materials, where samples of the order of microns can be deformed plastically, but no bulk samples can be prevented from fracturing at low temperatures. A representative bulk yield stress then needs to be estimated from macroscopic data collected at high temperatures and extrapolated toward lower temperatures. This approach includes significant errors in the application of any type of equation describing thermal activation and the resulting Peierls barrier or lattice resistance at temperatures above 0 K, such as^[ Reference Bhakhri, Wang, Ur-rehman, Ciurea, Giuliani and Vandeperre ^{65 ,} Reference Clegg, Vandeperre and Pitchford ^{66 ]}

(1) $$\tau (T,\rho _{\rm m}, \dot \gamma ) = \tau _{\rm P} + \displaystyle{{kT} \over V}\ln \left( {\displaystyle{{\dot \gamma} \over {\rho _{\rm m} b^2 \nu _{\rm A}}}} \right),$$

where τ _P is the Peierls stress (i.e., the lattice resistance at 0 K), k the Boltzmann constant, T the absolute temperature, V the activation volume, $\dot \gamma $ the shear strain rate, ρ _m the mobile dislocation density, b the Burgers vector, and ν _A an attempt frequency of the order of 10¹¹ s⁻¹.^[ Reference Bhakhri, Wang, Ur-rehman, Ciurea, Giuliani and Vandeperre ^{65 ,} Reference Frost and Ashby ^{67 ]} There are several parameters, which are difficult to determine accurately in a small-scale experiment, even if the microsample has been machined from the same bulk material, as the dislocation density may have been changed by FIB milling^[ Reference Shim, Bei, Miller, Pharr and George ^{68 ]} and be affected by dislocation exhaustion during deformation.^[ Reference Kiener and Minor ^{69 ]} In addition, the shear strain rate may vary locally for a tapered pillar. A more detailed discussion of this difficulty may be found in Ref. Reference Korte and Clegg70. A direct comparison of different slip systems in the same material, while maintaining a constant level of FIB damage and similar effects of taper, confirms the general trend of an increased size effect at the reduced bulk strength (Fig. 6).

A last difficulty in achieving a direct comparison between bulk and microscale deformation may occur where dislocations move as dissociated partials with an increasing dissociation distance as the stress on the slip plane is raised.^[ Reference Wessel and Alexander ^{71 ]} Where the sample size is reduced, the leading partial may cross the entire volume without nucleation of the trailing partial, as shown, for example, in GaAs.^[ Reference Michler, Wasmer, Meier, Ostlund and Leifer ^{26 ]} The implications for the stresses measured must then be assessed carefully in each case. Possible strategies to avoid this problem include careful choice of the crystal orientation such that the Schmid factor becomes larger for the trailing partial and selection of as large a pillar size as possible.

Thermal activation of plasticity

In case of hard materials, the study of thermal activation is of particular interest for three reasons: firstly, high-temperature mechanical properties are often of interest in those to be applied at high temperatures. Secondly, the underlying deformation mechanisms and their rate-limiting steps may be understood in greater detail by studying the associated activation volumes and therefore rate and temperature dependence [cf. Eq. (1)] and, thirdly, correlation with macroscopic experiments is usually confined to reference data at high temperatures, in which case the interpolated gap should be minimized.

An example of high-temperature microcompression testing is shown in Fig. 8. Having first been demonstrated in air^[ Reference Korte and Clegg ^{35 ]} based on the oxide ceramic MgAl₂O₄ and a few years later in vacuum^[ Reference Korte, Stearn, Wheeler and Clegg ^{36 ]} initially using a superalloy, it is now routinely available to several hundred degrees and being developed based on similar efforts in nanoindentation to 1000 °C in order to achieve temperatures relevant to operation in turbines.^[ Reference Beake, Harris, Fox-Rabinovich, Rauh, Davies, Armstrong and Vishnyakov ^{72 ]} Especially in the highly anisotropic superalloys, the advantage of uniaxial testing is again apparent in that macroscopic yield stresses are reproduced in uniaxial compression (cf. pillar and macroscopic yield stress in Fig. 8), while in nanoindentation the triaxial stress-state causes significant deviation from uniaxial test results even if the 8% flow stress is taken as a reference, consistent with the change in superalloy properties based on orientation.

Figure 8. High-temperature microcompression experiments. First demonstrated in air on spinel^[ Reference Korte and Clegg ^{35 ]} to 400 °C (left) and in vacuum to 630 °C on the superalloy CMSX-4 tested^[ Reference Korte, Stearn, Wheeler and Clegg ^{36 ]} (right). Extensive plasticity at 200 °C in spinel, about 1500 °C below significant plasticity is commonly achieved in conventional uniaxial testing without hydrostatic pressure, highlights again the effect of size on fracture, while the data obtained on the superalloy reveals the importance of uniaxial testing and choice of representative strain in comparison with macroscopic data and high-temperature nanoindentation affected by the 3D stress-state in the highly anisotropic crystal. Image of spinel micropillar (left) adapted from Ref. Reference Korte and Clegg35 and plotted data taken from [I],^[ Reference Korte, Stearn, Wheeler and Clegg ^{36 ]} [II],^[ Reference Gibson, Schröders, Zehnder and Korte-Kerzel ^{99 ]} [III],^[ Reference Reed ^{11 ]} [IV],^[ Reference Wee and Suzuki ^{100 ]} [V].^[ Reference Conforto, Molénat and Caillard ^{101 ]} Images reprinted with permission by Elsevier from Ref. Reference Korte and Clegg35 and Cambridge University Press from Ref. Reference Korte, Stearn, Wheeler and Clegg36.

Exemplified above in the case of silicon, high-temperature tests can give valuable information regarding transitions in deformation mechanism, thermal activation and properties in a certain temperature range. However, studies of rate sensitivity at a given temperature as well as creep/relaxation testing at the extremes of slow rates/long times are often of interest, either where not enough measurements can be taken in a specific temperature regime to extract a gradient or where creep mechanisms are of interest.^[ Reference Maier, Merle, Göken and Durst ^{42 ,} Reference Wehrs, Deckarm, Wheeler, Maeder, Birringer, Mischler and Michler ^{73 –} Reference Maier, Durst, Mueller, Backes, Höppel and Göken ^{76 ]} Strain rate jump tests and creep/relaxation hold segments may therefore be employed in microcompression in a similar way to macroscopic testing (Fig. 9). Points to consider when performing such experiments usually include the change in cross-sectional area in ex situ experiments, compatibility of tip and sample material at elevated temperatures^[ Reference Wheeler and Michler ^{77 ]} and long contact times as well as the effect of temperature on gallium, both positive and negative, by encouraging diffusion from the sample surface into the bulk or formation of low-melting phases.

Figure 9. Studies of rate sensitivity in nickel, including rate jump testing, creep, and relaxation hold segments in microcompression inside the SEM.^[ Reference Wheeler, Armstrong, Heinz and Schwaiger ^{38 ]} Reprinted with permission by Elsevier from Ref. Reference Wheeler, Armstrong, Heinz and Schwaiger38.

Extracting general principles of plasticity

Where fundamental deformation mechanisms and the effect of local atomic arrangement and bonding character are of interest, experiments are often correlated with ab initio calculations in order to elucidate the underlying atomic environment.^[ Reference Gouriet, Carrez, Cordier, Guitton, Joulain, Thilly and Tromas ^{78 ,} Reference Emmerlich, Music, Braun, Fayek, Munnik and Schneider ^{79 ]} However, in crystals with large unit cells and especially where repeating sub-units are also large, the use of ab initio approaches is limited by the size of the required cell. Other modeling approaches, such as atomistic modeling requiring suitable interatomic potentials^[ Reference Hammerschmidt, Drautz and Pettifor ^{80 ]} and novel methods for extraction of dislocation core structure^[ Reference Walker, Gale, Slater and Wright ^{81 ]} and lattice resistance^[ Reference Cordier, Amodeo and Carrez ^{9 ]} are advancing continuously. Their application to complex crystals will be of particular interest where not only computation time for ab initio calculations is an issue, but also where there is a direct application of commonly accepted concepts. This applies for example to the use of the generalized stacking fault energy.^[ Reference Vítek ^{82 ]} Usually computed for a few selected planes, these may not be straightforward to interpret, as they do not give dislocation core structures directly and also do not immediately highlight complicated deformation mechanisms relying on coordinated movement on adjacent planes, e.g., synchroshear,^[ Reference Chisholm, Kumar and Hazzledine ^{14 ]} or mechanisms based on shuffling of atoms, as proposed for the giant metadislocation cores of quasicrystal approximants.^[ Reference Heggen and Feuerbacher ^{83 ]}

However, the addition of modeling to the basic approach of nanomechanical testing and simple microstructural characterization is nevertheless powerful, particularly where either the modeling or more in-depth TEM-based methods are individually too time-consuming in execution or subsequent analysis to perform for a vast matrix of crystal orientations, lattice planes, etc. While microcompression can enable the quantitative characterization of activated slip systems in a crystal of unknown slip geometries in combination with TEM, this approach does not directly give defect structures. These are analyzed by time-intensive characterization methods such as two-beam imaging or LACBED analysis of Burgers vectors^[ Reference Morniroli ^{84 ]} and HR-TEM of defects to obtain their atomic structure. Guided by initial experimental results—such as operative slip planes from microcompression with EBSD and/or conventional TEM plane trace analysis—ab initio calculations can be focused based on this experimental data. The calculations can, for example, identify slip planes for generalized stacking fault calculations, which in turn guides high-resolution work in identifying potential Burgers vectors and stacking faults, e.g., the choice of membrane orientation. In this way, the combination of both approaches also offers a strategy to find correlations between crystal structure and bonding conditions on the one hand and critical stresses, active slip systems, and defect structures on the other, even in complex crystals where current understanding is very limited. An example of such an approach is summarized in Fig. 10, where the study of plasticity in the hard coating material and superconductor Mo₂BC has benefited from the closely linked analysis by both ab initio, nanomechanical and electron microscopy methods.^[ Reference Korte-Kerzel, Klöffel and Meyer ^{85 ]}

Figure 10. Combination of ab initio calculations and microcompression with SEM/EBSD and TEM. Gamma surface calculated by B. Meyer and T. Klöffel.^[ Reference Korte-Kerzel, Klöffel and Meyer ^{85 ]}